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STUDIES IN LOGIC AND THE FOUNDATIONS OF MATHEMATICS VOLUME 137 S. ABRAMSKY I S. ARTEMOV I R.A. SHORE I A.S. TROELSTRA E...

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STUDIES IN LOGIC AND THE FOUNDATIONS OF MATHEMATICS

VOLUME 137 S. ABRAMSKY I S. ARTEMOV I R.A. SHORE I A.S. TROELSTRA EDITORS

HANDBOOK OF PROOF THEORY

SAMUEL R. BUSS (Editor)

ELSEVIER AMSTERDAM' LAUSANNE' NEW YORK, OXFORD' SHANNON' SINGAPORE' TOKYU

HANDBOOK HANDBOOK OF OF PROOF PROOF THEORY THEORY

STUDIES IN IN LOGIC LOGIC STUDIES AND AND THE FOUNDATIONS OF MATHEMATICS MATHEMATICS THE FOUNDATIONS OF VOLUME 137 137 VOLUME

H o n o r a r y Editor: Editor: Honorary P. SUPPES P. SUPPES

Editors" Editors: S. A BRAMSKY, London S. ABRAMSKY, London S. A R T E M O V , Moscow Moscow S. ARTEMOV, R.A. R . A . SHORE, S H O R E , Ithaca Ithaca

A.S. A . S . TROELSTRA, T R O E L S T R A , Amsterdam Amsterdam

ELSEVIER ELSEVIER AMSTERDAM.9LAUSANNE LAUSANNE·9NEW NEW YORK YORK·~ OXFORD OXFORD·~ SHANNON SHANNON·9SINGAPORE SINGAPORE·~ TOKYO TOKYO AMSTERDAM

ELSEVIER SCIENCE B.V. Sara Burgerhartstraat 25 P.O. P.O. Box 211, 1000 1000 AE Amsterdam The Netherlands

Library of Congress Cataloging-in-Publication Data Library of Congress Cataloging-in-Publication Data Handbook of proof theory p.

cm.

// edited by Samuel R. Buss. Buss.

-- (Studies in logic and the foundations of mathematics; mathematics ; v. 137) 137)

Includes bibliographic references and indexes. ISBN 0-444-89840-9 (alk. paper)

I. 1. Proof theory.

QA9.54.H35 QA9.54.H35

I. Buss, Samuel R.

II. Series.

1998 1998

51l .3--dc21 511.3--dc21

98-18922 CIP

ISBN: 0-444-89840-9 © V. All rights reserved. 9 1998 1998 Elsevier Science B. B.V. No part of this publication may be reproduced, stored in a retrieval system or transmitted in any form or by any means, electronic, mechanical, photocopying, recording or otherwise, without without the the prior prior written written permission permission of of the the publisher, publisher, Elsevier Elsevier Science Science B.V., Copyright & & Permissions Department, P.O. Box 521, 1000 1000 AM Amsterdam, The Netherlands. Netherlands. Special regulations for readers in the U.S.A. U.S.A. This publication has been registered with the Copyright Clearance Center Inc. (Ccq, (CCC), 222 Rosewood Drive, Danvers, MA 01923. 01923. Information can be obtained from the CCC about conditions under which photocopies of parts of this publication may be made in the U.S.A. All other copyright questions, including photocopying outside of the U.S.A., should be referred to the copyright owner, Elsevier Science B.V., unless otherwise specified. No responsibility is assumed by the publisher for any injury and/or damage to persons or property as a matter of products liability, negligence or otherwise, or from any use or operation of any methods, products, instructions or ideas contained in the material herein. 9 O The The paper used in this publication meets the requirements of ANSIINISO ANSI/NISO Z39.48-1992 (Permanence of Paper). Paper).

Printed in The Netherlands

P reface Preface

Proof Proof theory theory is is the the study study of of proofs proofs as as formal formal objects objects and and is is concerned concerned with with aa broad range range of of related related topics. topics. ItIt isis one one of of the the central central topics topics of of mathematical mathematical logic logic and and broad has has applications applications in in many many areas areas of of mathematics, mathematics, philosophy, philosophy, and and computer computer science. science. Historically, was developed by mathematicians Historically, proof proof theory theory was developed by mathematicians and and philosophers philosophers as as aa formalization formalization for for mathematical mathematical reasoning; reasoning; however, however, proof proof theory theory has has gradually gradually become become increasingly increasingly important important for for computer computer science, science, and and nowadays nowadays proof proof theory theory and and theoretical computer science science are as being closely connected. theoretical computer are recognized recognized as being very very closely connected. This volume contains contains articles broad spectrum spectrum of of proof with an an articles covering covering aa broad proof theory, theory, with This volume emphasis on its mathematical mathematical aspects. aspects. The The articles articles should should not be interesting emphasis on its not only only be interesting to specialists theory, but but should should also be accessible to aa diverse diverse audience, to specialists in in proof proof theory, also be accessible to audience, including mathematicians, computer scientists and including logicians, logicians, mathematicians, computer scientists and philosophers. philosophers. We We have attempted to central topics proof theory; have attempted to include include many many of of the the central topics in in proof theory; but but have have opted to to have have self-contained self-contained expository expository articles, articles, rather rather than than to to have have encyclopedic encyclopedic opted coverage. Thus, Thus, aa number number of of important important topics topics have have been been largely largely omitted, omitted, but but with with coverage. the the advantage advantage that that the the included included material material is is covered covered in in more more detail detail and and at at greater greater depth. depth. The chapters chapters are are arranged arranged so so that that the two introductory introductory articles articles come come first; first; these these The the two are the core are then then followed followed by by articles articles from from the core classical classical areas areas of of proof proof theory; theory; finally finally the the handbook handbook ends ends with with articles articles that that deal deal with with topics topics closely closely related related to to computer computer science. science. This handbook handbook was was initiated initiated at at the the suggestion suggestion of of the the publisher, publisher, as as aa partial partial This successor successor to to the the very very successful successful Handbook of Mathematical Logic, edited edited by by J. J. Barwise. Barwise. Only Only one one quarter quarter of of the the 1977 1977 Handbook Handbook of Mathematical Logic Logic was was devoted devoted to to proof proof theory, theory, and and since since then then there there has has been been considerable considerable progress progress in in this this area; area; as as aa result, there there is is remarkably remarkably little little overlap overlap between between the the contents contents of of the the Handbook of result, Mathematical Logic Logic and and the the present present volume. volume. Sam Buss Buss Sam La Jolla, Jolla, California California La November 1997 1997 November

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List List of of Contributors Contributors

J. Avigad, Carnegie Carnegie Mellon University University ((Ch. Ch. V) S.R. Ch. I and S.R. Buss, Buss, University of California, California, San Diego Diego ((Ch. and II) R.L. Constable, Cornell Ch. X) R.L. Constable, Cornell University University ((Ch. M. University of Sheffield ((Ch. Ch. III) M. Fairtlough, Fairtlough, University S. Feferman, Stanford University University ((Ch. S. Feferman, Ch. V) G. Jager, Ch. IX) G. J~ger, Universitiit Universitiit Bern ((Ch. G. Japaridze, Ch. VII) G. Japaridze, University University of Pennsylvania ((Ch. D. de Ch. VII) D. de Jongh, Jongh, University University of Amsterdam ((Ch. Ch. VIII) P. Pudhik, Pudls Academy of Sciences Sciences of the Czech Czech Republic Republic ((Ch. P. W. Pohlers, Westfiilische Universitiit ((Ch. Ch. IV) W. Pohlers, Westf~lische WilhelmsWilhelms-Universit~t St~rk, Universitiit UniversitSt Freiburg Freiburg ((Ch. R.F. Stark, Ch. IX) University of Amsterdam ((Ch. A.S. Troelstra, University Ch. VI) S.S. Ch. III) S.S. Wainer, Wainer, University University of Leeds Leeds ((Ch.

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Table Table of of Contents Contents

Preface Preface .. . .. . .. . .. . ... ... . .. . . . . . . . . . . . . . . . . . . . . . . . List List of of Contributors Contributors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Chapter I. An Introduction Introduction to to Proof Proof Theory Theory Chapter I. An Samuel S a m u e l R. R. Buss B u s s .. .. .. .. . . . . . . . . . . . .. .. .. .. . . . . . . . Chapter II. First-Order First-Order Proof Chapter II. Proof Theory Theory of of Arithmetic Arithmetic Samuel S a m u e l R. R. Buss B u s s .. .. .. .. . . . . . . . . . . . .. .. .. .. .. .. .. . . . . . Chapter III. III. Hierarchies Hierarchies of of Provably Provably Recursive Recursive Functions Functions Chapter

vv ll vii V

. . . . . . . . .

11

. . . . . . . . .

79 79

Matt M a t t Fairtlough Fairtlough and and Stanley Stanley S. S. Wainer W a i n e r .. .. .. . . . . . . . ..

. . . . . . . .

149 149

Chapter Chapter IV. IV. Subsystems Subsystems of of Set Set Theory Theory and and Second Second Order Order Number Number Theory Theory Wolfram W o l f r a m Pohlers Pohlers .. .. .. .. . . . . . . . . . . . .. .. .. .. .. .. .. . . . . . . . . . .. .. .. .. .. .. . . 209 209 Chapter "Dialectica") Interpretation Chapter V. V. Godel's G5del's Functional Functional (("Dialectica") Interpretation 337 JJeremy e r e m y Avigad A vigad and and Solomon S o l o m o n Feferman F e # r m a n .. .. .. . . . . . . . . . .. .. .. .. . . . . . . . .. . . 337 Chapter Realizability Chapter VI. VI. Realizability A nnne n e S. S. Troelstra Troelstra .. .. .. .. . . . . . . . . . . . .. .. .. .. .. .. .. . . . . . . . . . .. .. .. .. .. .. . . 407 407 A Chapter VII. VII. The The Logic Logic of of Provability Provability Chapter Giorgi Giorgi Japaridze Japaridze and and Dick D i c k de de Jongh Jongh

. . . . . . . . . . . . . . . . . .

475 475

. . . . . . . . . . . . . . . . .

547 547

Chapter Chapter VIII. VIII. The The Lengths Lengths of of Proofs Proofs

Pavel Pavel Pudltik P u d l d k .. .. .. .. .. . . . . . .. . . .. .. ..

Chapter Chapter IX. IX. A A Proof-Theoretic Proof-Theoretic Framework Framework for for Logic Logic Programming Programming Gerhard Gerhard Jager J @ e r and and Robert Robert F. F. Stark StSrk .. .. .. .. . . . . . . . .. .. .. .. ..

. . . . . .

639 639

. . . . .

683 683

Chapter Chapter X. X. Types Types in in Logic, Logic, Mathematics Mathematics and and Programming Programming Robert Robert L. L. Constable Constable .. .. .. .. . . . . . . . . . . . .. .. .. .. .. .. . . . . . . . . . . .

Name Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Name Index Subject Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Subject Index

787 787 797 797

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CHAPTER C H A P T E R II

An Introduction Introduction to to Proof P r o o f Theory Theory An Samuel R. R. Buss Buss Samuel California, San Diego Departments of Mathematics and Computer Science, University of California, Jolla, California 92093-0112, 92093-0112, USA La Jolla,

Contents Contents

1. Proof P r o o f theory theory of of propositional propositional logic logic . . . . . . . . . . . . . . . . . . . . . . . . . 1. 1.1. Frege Frege proof p r o o f systems systems .. .. .. .. .. . . . . . . . . . . . . . . . . . . . . . . . . . 1.1. 1.2. The T h e propositional propositional sequent sequent calculus calculus . . . . . . . . . . . . . . . . . . . . . . 1.2. 1.3. Propositional P r o p o s i t i o n a l resolution resolution refutations refutations .. . . . . . . . . . . . . . . . . . . . . . 1.3. 2. P r o o f theory theory of of first-order first-order logic logic .. . . . . . . . . . . . . . . . . . . . . . . . . . . 2. Proof 2.1. Syntax S y n t a x and and semantics semantics .. .. .. .. .. . . . . . . . . . . . . . . . . . . . . . . . . 2.1. 2.2. Hilbert-style Hilbert-style proof proof systems systems .. .. . . . . . . . . . . . . . . . . . . . . . . . . 2.2. 2.3. The T h e first-order first-order sequent sequent calculus calculus . . . . . . . . . . . . . . . . . . . . . . . . 2.3. 2.4. Cut Cut elimination elimination .. .. .. .. .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4. 2.5. Herbrand's H e r b r a n d ' s theorem, theorem, interpolation interpolation and and definability definability theorems theorems . . . . . . . . 2.5. 2.6. 2.6. First-order First-order logic logic and and resolution resolution refutations refutations . . . . . . . . . . . . . . . . . . 3. 3. Proof P r o o f theory theory for for other other logics logics .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1. 3.1. Intuitionistic Intuitionistic logic logic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2. Linear Linear logic logic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2. References References .. .. .. .. .. .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

HANDBOOK O F PPROOF R O O F THEORY THEORY HANDBOOK OF Edited S. R. Edited by by S. R. Buss Buss © 1998 1998 Elsevier Elsevier Science Science B.V. B.V. All All rights rights reserved reserved

33 55 10 10 18 18 26 26 26 26 29 29 31 31 36 36 48 48 59 59 64 64 64 64 70 70 74 74

22

s. Buss Buss S.

Proof Proof Theory Theory is is the the area area of of mathematics mathematics which which studies studies the the concepts concepts of of mathematmathemat ical proof proof and and mathematical mathematical provability. provability. Since Since the the notion notion of of "proof" "proof" plays plays aa central central ical role role in in mathematics mathematics as as the the means means by by which which the the truth truth or or falsity falsity of of mathematical mathematical propositions is is established; established; Proof Proof Theory Theory is, is, in in principle principle at at least, least, the the study study of of propositions the foundations foundations of of all all of of mathematics. mathematics. Of Of course, course, the the use use of of Proof Proof Theory Theory as as aa the foundation for for mathematics mathematics is is of of necessity necessity somewhat somewhat circular, circular, since since Proof Proof Theory Theory is is foundation itself aa subfield subfield of of mathematics. mathematics. itself There are are two two distinct distinct viewpoints viewpoints of of what what aa mathematical mathematical proof proof is. is. The The first first view view There is is that that proofs proofs are are social social conventions conventions by by which which mathematicians mathematicians convince convince one one another another of say, aa proof expressed in of the the truth truth of of theorems. theorems. That That is is to to say, proof is is expressed in natural natural language language plus possibly possibly symbols symbols and and figures, figures, and and is is sufficient sufficient to to convince convince an an expert expert of of the the plus correctness of of aa theorem. theorem. Examples Examples of of social social proofs proofs include include the the kinds kinds of of proofs proofs that that correctness are presented presented in in conversations conversations or or published published in in articles. articles. Of Of course, course, itit is is impossible impossible to to are precisely precisely define define what what constitutes constitutes aa valid valid proof proof in in this this social social sense; sense; and, and, the the standards standards for valid proofs may vary vary with with the the audience audience and and over over time. The second second view view of of proofs proofs for valid proofs may time. The is more narrow narrow in this view, string of of symbols symbols which which is more in scope: scope: in in this view, aa proof proof consists consists of of aa string satisfy some precisely stated rules and and which which prove prove a a theorem, theorem, which itself must must satisfy some precisely stated set set of of rules which itself also expressed as of symbols. this view, view, mathematics also be be expressed as aa string string of symbols. According According to to this mathematics can can be regarded 'game ' played played with with strings strings of be regarded as as a a 'game' of symbols symbols according according to to some some precisely precisely defined of the the latter latter kind kind are defined rules. rules. Proofs Proofs of are called called "formal" "formal" proofs proofs to to distinguish distinguish them them from from "social" "social" proofs. proofs. In practice, practice, social social proofs proofs and formal proofs very closely closely related. related. Firstly, In and formal proofs are are very Firstly, proof can can serve as aa social social proof proof (although (although it tedious and and aa formal formal proof serve as it may may be be very very tedious unintuitive) provided provided it is trusted. unintuitive) it is is formalized formalized in in a a proof proof system system whose whose validity validity is trusted. Secondly, the Secondly, the standards standards for for social social proofs proofs are are sufficiently sufficiently high high that, that, in in order order for for a a proof socially accepted, proof to to be be socially accepted, it it should should be be possible possible (in (in principle!) principle!) to to generate generate aa formal formal proof corresponding to Indeed, this proof corresponding to the the social social proof. proof. Indeed, this offers offers an an explanation explanation for for the the fact fact that there are generally the implicit that there are generally accepted accepted standards standards for for social social proofs; proofs; namely, namely, the implicit requirement requirement that that proofs proofs can can be be expressed, expressed, in in principle, principle, in in aa formal formal proof proof system system enforces enforces and and determines determines the the generally generally accepted accepted standards standards for for social social proofs. proofs. Proof Theory Proof Theory is is concerned concerned almost almost exclusively exclusively with with the the study study of of formal formal proofs: proofs: this this is is justified, justified, in in part, part, by by the the close close connection connection between between social social and and formal formal proofs, proofs, and necessitated by only formal and it it is is necessitated by the the fact fact that that only formal proofs proofs are are subject subject to to mathematical mathematical analysis. analysis. The The principal principal tasks tasks of of Proof Proof Theory Theory can can be be summarized summarized as as follows. follows. First, First, to to formulate formulate systems systems of of logic logic and and sets sets of of axioms axioms which which are are appropriate appropriate for for formalizing formalizing mathematical mathematical proofs proofs and and to to characterize characterize what what results results of of mathematics mathematics follow follow from from certain certain axioms; axioms; or, or, in in other other words, words, to to investigate investigate the the proof-theoretic proof-theoretic strength strength of of particular particular formal formal systems. systems. Second, Second, to to study study the the structure structure of of formal formal proofs; proofs; for for instance, instance, to to find find normal normal forms forms for for proofs proofs and and to to establish establish syntactic syntactic facts facts about about proofs. proofs. This This is is the the study study of of proofs proofs as as objects objects of of independent independent interest. interest. Third, Third, to to study study what what kind kind of of additional additional information information can can be be extracted extracted from from proofs proofs beyond beyond the the truth truth of of the the theorem theorem being being proved. proved. In In certain certain cases, cases, proofs proofs may may contain contain computational computational or or constructive constructive information. information. Fourth, Fourth, to to study study how how best best to to construct construct formal formal proofs; proofs; e.g., e.g., what what kinds kinds of of proofs proofs can can be be efficiently efficiently generated generated by by computers? computers?

Introduction to to Proof Theory

3

The The study study of of Proof Proof Theory Theory is is traditionally traditionally motivated motivated by by the the problem problem of of formaliz formalizing ing mathematical mathematical proofs; proofs; the the original original formulation formulation of of first-order first-order logic logic by by Frege Frege [1879] [1879] was the the first first successful successful step step in in this this direction. direction. Increasingly, Increasingly, there there have have been been attempts attempts was to extend extend Mathematical Mathematical Logic Logic to to be be applicable applicable to to other other domains; for example, example, to domains; for intuitionistic logic logic deals deals with with the the formalization formalization of of constructive constructive proofs, proofs, and and logic logic intuitionistic programming is is aa widely widely used used tool tool for for artificial artificial intelligence. intelligence. In In these these and and other other programming domains, domains, Proof Proof Theory Theory is is of of central central importance importance because because of of the the possibility possibility of of computer computer generation generation and and manipulation manipulation of of formal formal proofs. proofs. This This handbook handbook covers covers the the central central areas areas of of Proof Proof Theory, Theory, especially especially the the math mathematical ematical aspects aspects of of Proof Proof Theory, Theory, but but largely largely omits omits the the philosophical philosophical aspects aspects of of proof theory. This This first first chapter is intended to be be an an overview overview and and introduction introduction to to proof theory. chapter is intended to mathematical proof theory. theory. It proof theory mathematical proof It concentrates concentrates on on the the proof theory of of classical logic, logic, especially propositional propositional logic logic and and first-order first-order logic. logic. This This is is for for two two reasons: reasons: firstly, firstly, especially classical is by by far far the the most most widely widely used used framework framework for for mathematical mathematical classical first-order first-order logic logic is reasoning, reasoning, and and secondly, secondly, many many results results and and techniques techniques of of classical classical first-order first-order logic logic frequently frequently carryover carryover with with relatively relatively minor minor modifications modifications to to other other logics. logics. This This introductory introductory chapter chapter will will deal deal primarily primarily with with the the sequent sequent calculus, calculus, and and resolution, resolution, and and to to lesser lesser extent, extent, the the Hilbert-style Hilbert-style proof proof systems systems and and the the natural natural rst examine propositional logic, deduction proof deduction proof system. system. We We fi first examine proof proof systems systems for for propositional logic, then then proof proof systems systems for for first-order first-order logic. logic. Next Next we we consider consider some some applications applications of of cut cut elimination, elimination, which which is is arguably arguably the the central central theorem theorem of of proof proof theory. theory. Finally, Finally, we we review review the logic and the proof proof theory theory of of some some non-classical non-classical logics, logics, including including intuitionistic intuitionistic logic and linear linear logic. logic. 1 logic 1.. Proof P r o o f ttheory h e o r y of o f propositional propositional logic

Classical logic, also Classical propositional propositional logic, also called called sentential sentential logic, logic, deals deals with with sentences sentences and and propositions as distinct True /False values. propositions as abstract abstract units units which which take take on on distinct True/False values. The The basic basic syntactic syntactic units units of of propositional propositional logic logic are are variables which which represent represent atomic atomic propo propoTrue or sitions sitions which which may may have have value value either either True or False. False. Propositional Propositional variables variables are are also called ) : aa kk-ary -ary Boolean function combined combined with with Boolean Boolean functions functions ((also called connectives connectives): T, F} where T, F}k is is aa mapping mapping from from {{T, F} k to to {{T,F} where we we use use T T and and F F to to represent represent True and and False. The The most most frequently frequently used used examples examples of of Boolean Boolean functions functions are are the the connectives connectives T T and respectively; the and ..L _L which which are are the the O-ary 0-ary functions functions with with values values T T and and F, F , respectively; the binary binary connectives V, ::) connectives 1\., A, V, D,, t-+ ++ and and ED @ for for "and" "and",, "or" "or",, "if-then" "if-then",, "if-and-only-if" "if-and-only-if" and and "parity" negation. Note "parity";; and and the the unary unary connective connective -, -~ for for negation. Note that that V is is the the inclusive-or inclusive-or and and ED @ is is the the exclusive-or. exclusive-or. We We shall shall henceforth henceforth let let the the set set of of propositional propositional variables variables be be V V= - {PI (pl,, P2 p2,, P3 p3,...}; , . . .} ; however, hold also however, our our theorems theorems below below hold also for for uncountable uncountable sets sets of of propositional propositional variables. variables. The The set set of of formulas formulas is is inductively inductively defined defined by by stating stating that that every every propositional propositional variable variable is ,A) , (AI\.B ) , (AV B ) , (A ::) ), is aa formula, formula, and and that that if if A and and B are are formulas, formulas, then then ((~A), (AAB), (AVB), DB B), etc., A truth assignment consists / False values etc., are are formulas. formulas. A consists of of an an assignment assignment of of True True/False values to truth assignment T, F} to the the propositional propositional variables, variables, i.e., i.e., aa truth assignment is is aa mapping mapping T T :: V V� --+ {(T, F}..

44

s. S. Buss

A truth assignment domain the A truth assignment can can be be extended extended to to have have domain the set set of of all all formulas formulas in in the the obvious according to 1; we or(A) for obvious way, way, according to Table Table 1; we write write ~(A) for the the truth truth value value of of the the formula formula A A induced by induced by the the truth truth assignment assignment T. T. Table Table 11 Values Values of of aa truth truth assignment assignment or

A B B (..., (-~A) (A 1\ A B) B) A) (A A

T T T T F F F F

T T F F T T F F

F F F F T T T T

T T F F F F F F

(A V V B) B) (A

T T T T T T F F

(A D B) B) (A:)

T T F F T T T T

(A ++ ~ B) B) (A

T T F F F F T T

(A @ B) (AEBB)

F F T T T T F F

A A formula formula A A involving involving only only variables variables among among PI P l ,, .. ... ., P,Pk k defines defines aa k-ary k-cry Boolean Boolean A , by A (xl , . . . , xk ) equal function function f fA, by letting letting f fA(Xl,...,Xk) equal the the truth truth value value or(A) ~(A) where where T( T(pi) = Xi Pi ) = for for all all ii.. A A language is is a a set set of of connectives connectives which which may may be be used used in in the the formation formation of of L L-formulas. A language language L L is is complete if if and and only only if if every every Boolean Boolean function function can can be be -formulas. A defined defined by by an an L-formula. L-formula. Propositional Propositional logic logic can can be be formulated formulated with with any any complete complete ((usually usually finite time being, finite)) language language L L - - for for the the time being, we we shall shall use use the the language language -' 9,, 1\, A, V and and :J D.. A A propositional propositional formula formula A A is is said said to to be be aa tautology or or to to be be (classically) valid if if A A is is assigned assigned the the value value T T by by every every truth truth assignment. assignment. We We write write 1= ~A A to to denote denote that that A A is is a a tautology. tautology. The The formula formula A A is is satisfiable if if there there is is some some truth truth assignment assignment that that gives gives it it value value T T.. If If r F is is a a set set of of propositional propositional formulas, formulas, then then r F is is satisfiable if if there there is is some some truth truth assignment assignment that that simultaneously simultaneously satisfies satisfies all all members members of of r F.. We We say say A, or or r F 1= ~ A, A, if if every every truth truth assignment assignment which which satisfies satisfies r F also also rF tautologically implies A, satisfies satisfies A A.. One One of of the the central central problems problems of of propositional propositional logic logic is is to to find find useful useful methods methods for for satisfiable, this this A is recognizing tautologies; since recognizing tautologies; since A A is is aa tautology tautology if if and and only only if if -, ~A is not not satisfiable, is essentially the finding methods is essentially the same same as as the the problem problem of of finding methods for for recognizing recognizing satisfiable satisfiable formulas. decidable, since formulas. Of Of course, course, the the set set of of tautologies tautologies is is decidable, since to to verify verify that that a a formula formula A A with distinct propositional with n n distinct propositional variables variables is is aa tautology, tautology, one one need need merely merely check check that that the distinct truth these variables the 22nn distinct truth assignments assignments to to these variables all all give give A A the the value value T T.. This This brute-force brute-force 'method 'method of of truth-tables truth-tables'' is is not not entirely entirely satisfactory; satisfactory; firstly, firstly, because because it it can can involve involve an an exorbitant exorbitant amount amount of of computation, computation, and and secondly, secondly, because because it it provides provides no no intuition intuition as as to to why the the formula formula is, is, or or is is not, not, a a tautology. tautology. For For these these reasons, reasons, it it is is often often advantageous advantageous to to prove that that A A is is aa tautology tautology instead instead of of using using the the method method of of truth-tables. truth-tables. The The next next three three sections sections discuss discuss three three commonly commonly used used propositional propositional proof proof systems. systems. The The so-called so-called Frege Frege proof proof systems systems are are perhaps perhaps the the most widely used ponens. The most widely used and and are are based based on on modus modus ponens. The sequent sequent calculus calculus systems systems provide both the possibility of provide an an elegant elegant proof proof system system which which combines combines both the possibility of elegant elegant proofs proofs and and the the advantage advantage of of an an extremely extremely useful useful normal normal form form for for proofs. proofs. The The resolution resolution refutation refutation proof proof systems systems are are designed designed to to allow allow for for efficient efficient computerized computerized search search for for proofs. Later, we we will proofs. Later, will extend extend these these three three systems systems to to first-order first-order logic. logic.

Introduction Introduction to to Proof Proof Theory Theory

55

1.1. 1.1. Frege Frege proof p r o o f systems systems

The commonly used propositional proof The mostly mostly commonly used propositional proof systems systems are are based based on on the the use use of of modus ponens as as the the sole of inference. inference. Modus Modus ponens ponens is is the inference rule, rule, which which modus ponens sole rule rule of the inference allows, allows, for for arbitrary arbitrary A A and and B B,, the the formula formula B B to to be be inferred inferred from from the the two two hypotheses hypotheses A pictorially represented A ::> 3 B B and and A; A; this this is is pictorially represented as as A A A AD ::>BB B B In addition to In addition to this this rule rule of of inference, inference, we we need need logical logical axioms axioms that that allow allow the the inference inference of choices for of 'self-evident 'self-evident'' tautologies tautologies from from no no hypotheses. hypotheses. There There are are many many possible possible choices for sets of of axioms: axioms: obviously, obviously, we we wish wish to to have have aa sufficiently sufficiently strong strong set set of of axioms axioms so so that that sets every tautology tautology can can be derived from from the the axioms axioms by by use use of of modus modus ponens. In addition, addition, every be derived ponens. In we wish wish to to specify specify the the axioms axioms by by a a finite finite set set of of schemes. schemes. we 1.1.1. Definition. A 1.1.1. Definition. A substitution substitution a a is is aa mapping mapping from from the the set set of of propositional propositional variables formulas. If variables to to the the set set of of propositional propositional formulas. If A A is is a a propositional propositional formula, formula, then then the the result result of of applying applying a a to to A A is is denoted denoted Aa Aa and and is is equal equal to to the the formula formula obtained obtained by by simultaneously replacing each simultaneously replacing each variable variable appearing appearing in in A A by by its its image image under under a a..

An An example example of of aa set set of of axiom axiom schemes schemes over over the the language language -' ~,, 1\, A, V and::> and D is is given given in in the next next definition. definition. We We adopt adopt conventions conventions for for omitting omitting parentheses parentheses from from descriptions descriptions the of of formulas formulas by by specifying specifying that that unary unary operators operators have have the the highest highest precedence, precedence, the the connectives connectives 1\ A and and V have have second second highest highest precedence, precedence, and and that that ::> D and and +-* ~ have have lowest same precedence are to be associated lowest precedence. precedence. All All connectives connectives of of the the same precedence are to be associated from from ,B ::> is aa shorthand the formula right right to to left; left; for for example, example, A A ::> D-~B DC C is shorthand representation representation for for the formula (A D ((-,B) ((-~B) ::> D C)) C)).. (A::> 1.1.2. 1.1.2. Definition. Definition.

Consider Consider the the following following set set of of axiom axiom schemes: schemes:

PI ) PI P2 ::> Pl ::> :) ((P2 :)Pl) (PI ::> ((Pl (P2 ::> PI ::> PI ::> P3) P2 ) ::> P3)) ::> ~P2) :) ((Pl :)(P2 ~P3)) :)(Pl ~P3) V PI Pl ::> D pPIl V pP2 2 PI V P2 P2 P2 ::> D Pl Vp2 ((p~ PI ::> PI V P2 ::> P3) P3) ::> P2 ::> P3) ::> DP3) D ((P2Dp3) D ((p~VP2DP3)

(PI ::> ((Pl PI ::> P2 ) ::> --PI -'P2 ) ::> :)P2) :)(Pl :)~P2) :)~Pl ((~Pl) PI :)Pl -'-'PI ) ::> 1\pP22 ::> PI plA D PPI l PI P2 ::> Pl 1\ Ap2 D P2 P2 PI p~ ::> Dp~.Dp~Ap2 P2 ::> PI 1\ P2

The system :F its axioms The propositional propositional proof proof system jc is is defined defined to to have have as as its axioms every every substitution substitution instance of the above formulas and to have modus ponens as its instance of the above formulas and to have modus ponens as its only only rule. rule. An An :F -proof of -axiom ~'-proof of a a formula formula A A is is a a sequence sequence of of formulas, formulas, each each of of which which is is either either an an F ~-axiom or or is is inferred inferred by by modus modus ponens ponens from from two two earlier earlier formulas formulas in in the the proof, proof, such such that that the the formula in in the the proof proof is is A. fifinal nal formula We just IWe write write � ~ A, A, or or just ~- A, A, to to say say that that A A has has an an F-proof. ~'-proof. We We write write r F � ~ A A,, or has aa proof or just just r F fk- A, A, to to say say that that A A has proof in in which which each each formula formula either either is is deduced deduced according according the the axioms axioms or or inference inference rule rule of of :F ~" or or is is in in r F.. In In this this case, case, we we say say that that A A is is proved proved from from the the extra-logical extra-logical hypotheses hypotheses r F;; note note that that r F may may contain contain formulas formulas which which are not not tautologies. tautologies. are

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1.1.3. Soundness Soundness and and completeness completeness of of :F. ~'. 1.1.3.

easy to to prove prove that that every every ItIt isis easy

~'-provable formula is is aa tautology, tautology, by by noting noting that that all all axioms axioms of of :F ~" are are valid valid and and that that :F -provable formula

modus ponens ponens preserves preserves the the property property of of being being valid. valid. Similarly, Similarly, whenever whenever rF f:1' ~ A A,, modus then then rF tautologically tautologically implies implies A A.. In In other other words, words, :F ~ is is {implicationally} (implicationally) sound; sound; which which means means that that all all provable provable formulas formulas are are valid valid (or, (or, are are consequences consequences of of the the extra-logical extra-logical hypotheses r). F). hypotheses Of Of course, course, any any useful useful proof proof system system ought ought to to be be sound, sound, since since the the purpose purpose of of creating creating proofs is is to to establish establish the the validity validity of of aa sentence. sentence. Remarkably, Remarkably, the the system system :F ~" is is also also proofs complete complete in in that that it it can can prove prove any any valid valid formula. formula. Thus Thus the the semantic semantic notion notion of of validity validity and the the syntactic syntactic notion notion of of provability provability coincide, coincide, and and aa formula formula is is valid valid if if and and only only if if and is provable provable in in :F. ~'. itit is

Theorem. T h e o r e m . The propositional proof system :F ~ is is complete and is implicationally complete; namely, If A A is a tautology, then f:1' ~ A A .. (1) If If r FF ~ A, A, then r F f:1' ~ A A.. (2) If

The The philosophical philosophical significance significance of of the the completeness completeness theorem theorem is is that that aa finite finite set set of of (schematic) axioms axioms and and rules rules of of inference inference are are sufficient sufficient to to establish establish the the validity validity of of (schematic) any tautology. tautology. In In hindsight, hindsight, it it is is not not surprising surprising that that this this holds, holds, since since the the method method of of any truth-tables already provides truth-tables already provides an an algorithmic algorithmic way way of of recognizing recognizing tautologies. tautologies. Indeed, Indeed, the the proof proof of of the the completeness completeness theorem theorem given given below, below, can can be be viewed viewed as as showing showing that that the the method method of of truth truth tables tables can can be be formalized formalized within within the the system system :F. ~'. We that part We first first observe observe that part (2) (2) of of the the completeness completeness theorem theorem can can be be (1) by compactness theorem theorem reduced reduced to to part part (1) by a a two two step step process. process. Firstly, Firstly, note note that that the the compactness propositional logic that if if Fr ~F A A then finite subset subset F0 ra of r which for logic states states that then there there is is a a finite of F which for propositional also tautologically tautologically implies implies A. A. A for also A topological topological proof proof of of the the compactness compactness theorem theorem for below. Thus r may, without loss of generality, propositional sketched in in 1.1.5 propositional logic logic is is sketched 1.1.5 below. Thus F may, without loss of generality, be assumed assumed to to be finite set set of of formulas, formulas, say note that Bb . . . , B d. Secondly, be be aa finite say r F= - {{B1,..., Bk}. Secondly, note that :::> A A is is aa tautology. tautology. So, So, by by part (1), the the latter latter r ~F A A implies that B1 B1 D :::> B2 B2 D :::> ... D F implies that part (1), formula has has an an JC-proof, :F-proof, and and by by kk additional additional modus modus ponens ponens inferences, inferences, Fr kf- A. A . (To (To formula f- instead instead of of ~f:1'.).) simplify notation, notation, we we write write ksimplify It remains remains to to prove prove part part (1). (1). We begin by by establishing establishing aa series series of of special special cases, cases, It We begin (a)-(k) , of of the the completeness completeness theorem, theorem, in in order order to to "bootstrap" "bootstrap" the the propositional propositional (a)-(k), :F . We use symbols symbols r¢J , r'Ij;, X X for for arbitrary arbitrary formulas formulas and and II II to to represent represent any any system ~'. system We use set set of of formulas. formulas. 1.1.4. 1.1.4. Proof. Proof.

• • •

(a) (a) ef- r¢J ~:::> r¢J . three axioms Combine the axioms ((r¢J D ( ¢J D ( ¢J D :::> ¢Jr ) D Proof: Combine :::> ¢Jr ) D :::> (r the three ( ¢J D :::> r¢J D Proof: :::> r¢J) D :::> (r :::> r¢J) , :::> (r modus ponens. ponens. and r¢J D :::> (r (¢J D :::> r¢J) D :::> r¢J with with two two uses uses of of modus :::> (r (¢J D :::> r¢J) and r¢J D (b) (b) Deduction Deduction Theorem: Theorem: F, r, r¢J ~f- r'Ij; ifif and and only only ifif Fr kf- r¢J D :::> r'Ij;. The reverse reverse implication implication isis trivial. trivial. To To prove prove the the forward forward implication, implication, suppose suppose Proof: The Proof: Ck isis an and that that each each an ~'-proof :F-proof of of r'Ij; from from F, r, r¢J. This This means means that that Ck Ck isis r'Ij; and C1, C2,..., C2 , , Ck C1, • • .

Introduction to Proof Theory

7

6'/is is in in F, r , is is an an axiom, axiom, or or isis inferred inferred by by modus modus ponens. ponens. ItIt is is straightforward straightforward to to Ci is r¢J , is prove, by by induction induction on on i,i , that that Fr ~f- r¢J J Ci for for each each Ci. prove, D Ci Ci .

( c) r¢J J f- --,'ljJ -,¢J . (c) ~ r'ljJ ~r J ~ -r By the the deduction deduction theorem, theorem, itit suffices suffices to to prove that ¢Jr J -,'ljJ Ff- --r -,¢J . To prove Proof: By prove that D 'ljJ, r --r To prove (¢J J -,'ljJ) and and (r (¢J J (¢J J -,'ljJ) J and three three this, use use the the two two axioms axioms -,'ljJ this, --r J D (r D 9r D r'ljJ) J D (r D --r D -,¢J 9r and

uses of of modus modus ponens. ponens. uses

(d) r¢J , 9r (d) F- r -'¢Jf-'ljJ. From the axiom ¢Jr J (-,'ljJJ have ¢Jr Ff- -,'ljJ Thus, by (c) we get Proof: From the axiom D (-~r D ¢Jr ) , we we have -~r J D ¢Jr . Thus, by (c) we get r¢J f--,¢J F- 9r J D ----r and by by (b), (b) , r162 ¢J , -,¢J Ff- -~-~r 'ljJ Finally Finally modus modus ponens ponens with with the the axiom axiom -,-,'ljJ , and gives the the desired desired result. result. ----r'ljJJ D r'ljJ gives -,-,

.

-,-,

(e) -,¢J (e) 9 r 1f-6 2¢J1J 6 2'ljJ1 6and 2 'ljJ f- ¢J J'ljJ . This former former follows follows from from (d) (d) and and the the deduction deduction theorem, theorem, and and the the latter latter follows follows Proof: This from the axiom (¢J J from the axiom 'ljJ rJ D (r D 'ljJ) r . (f) -,(¢J J (f) ¢Jr , -,'ljJ ~r f~ ~(r ~ 'ljJ) r . Proof: It suffices to to prove f- -,'ljJ J --(r by (c) (c) and and the the deduction deduction It suffices prove ¢Jr ~ --r D D 'ljJ) r . Thus, Thus, by -,(¢J J f- r'ljJ. The The latter is immediate theorem, itit suffices to prove prove r¢J, ¢Jr J theorem, suffices to D 'ljJ r Flatter assertion assertion is immediate from from

modus modus ponens. ponens.

(g) (g) ¢Jr 1, 'ljJ 6 2f- ¢Jr 11\6 2'ljJ . uses of of modus modus ponens the axiom Proof: Two Two uses ponens with with the axiom ¢Jr J D 'ljJ rJ D (¢J (r A r . 1\ 'ljJ) (h) -,¢J f- -,(r -,(¢J A 1\ 'ljJ) and -,'ljJ (h) --r Fr and --r f--,(¢J F- 9(r A 1\ r'ljJ) . Proof: For (¢J 1\ For the the first first part, part, it it suffices suffices to to show show IF- -,¢J 9r J D --(r A 'ljJ) r , and and thus, thus, by by (c), (c), it it suffices suffices to to show show f~ (¢J (r 1\ A 'ljJ) r J D ¢Jr , which which is is an an axiom. axiom. The The proof proof that that -,'ljJ --r fF- -,( 9(r¢J 1\ A 'ljJ) r is is -,

similar. similar.

'ljJ f- ¢J V 'ljJ . f- ¢J V 'ljJ and (i) ¢JCF-r162 (i) andCF-r162 Proof: ¢Jr J D (¢J (r V V 'ljJ) r and and 'ljJ rJ D (¢J (r V V 'ljJ) r are are axioms. axioms. (j) -'¢J -,(¢J V ~r , -,'ljJ ~r fe -(r v 'ljJ) r . Proof: It this, by It suffices suffices to to prove prove -,¢J --r fF- -,'ljJ 9r J D -,(¢J 9(r V V 'ljJ) r , and and this, by (c), (c), follows follows from from -,¢J 9r f~ (¢J (r V V 'ljJ) r J D 'ljJ r . For For this, this, we we combine combine (i) (i) -,¢J --r fF- ¢Jr J D 'ljJ r , by by (e) (e),, (ii) f-F- 'ljJJ'ljJ r D r , by by (a) (a),, (iii) f-F- (¢J (¢J VV 'ljJ) (r J D 'ljJ) r J D ('ljJ (r J D 'ljJ) r J D (((r r J D 'ljJ) r , an an axiom, axiom,

with with two two uses uses of of modus modus ponens. ponens.

(k) (k) ¢Jr fF- -,-,¢J -,9r . Proof: By , -,-,¢J f-F- -,-,¢J By (d) (d),, ¢Jr , -,¢J 9r fF- -'-' ----r¢J , and and obviously, obviously, ¢Jr162 9-,r . So So ¢Jr f-F- -,-,¢J 9-,r follows follows from from the the next next lemma. lemma. 1.1.4.1. 1.1.4.1. Lemma. Lemma.

F, ¢Jr f-F-'ljJ r and rF,, -,¢J 9r f-~ 'ljJ, r then rF f-F-'ljJ r. IIff r,

88

S. Buss

P r o o f . By By (b) (b) and and (c), (c), the the two two hypotheses hypotheses imply imply that that r F f~- .'IjJ 9 r :::> D . D Proof. 9-~r D . D (.'IjJ ( 9 r :::> D •• -~9r D •• 9 9 r'IjJ and and •• 9 9 r'IjJ :::> D 'IjJ r •• give F Fr [:] 0 give r f- 'IjJ. 1.1.4.2. Lemma. L e m m a . Let the formula formula A A involve only the propositional propositional variables among 1.1.4.2. i . Then, either p l ,, .. ... ., p, Pnn. · For 11 ::; < i ::; ~ n n,, suppose that Bi is either either P Pii or 'P 9pi. PI B 1 , . . . , B~ F- A

or or

B 1 , . . . , B~ t- 9A.

P r o o f . Define Define rT to to be be aa truth truth assignment assignment that that makes makes each each Bi true. true. By By the the soundness soundness Proof. theorem, A A (respectively, (respectively, .A) 9A),, can can be be proved proved from from the the hypotheses hypotheses B B1,... only theorem, ll . . . ,,Bn Bn only if 7'(A) y(A) = - T (respectively (respectively 7'(A) ~(A) = - F). F). Lemma Lemma 1.1.4.2 1.1.4.2 asserts asserts that that the the converse converse holds holds if too. too. The lemma lemma is is proved proved by by induction induction on on the the complexity complexity of of A. A. In In the the base base case, case, The i ' Now A i : this A is is just just P pi" this case case is is trivial trivial to to prove prove since since Bi is is either either P Pii or or 'P 9pi. Now suppose suppose A is is aa formula formula Al A1 V A A2. If o-(A) a(A) = - T T,, then then we we must must have have r(Ai) T(Ai) -T for for some some A = T 2 . If e {{1,2}; the induction induction hypothesis hypothesis implies implies that that B B 1ll, . . ..,,BBn n fF- Ai and and thus, thus, by by (i) (i) I, 2 } ; the i E above, B B 1ll, .. .... , , Bn B~ fF- A. A. On On the the other other hand, hand, if if r(A) T(A) = - - F, F, then then r(A T(A1) = r(A T ( A 2 )2 ) = - - F, F, above, I) = so the the induction induction hypothesis hypothesis implies implies that that B B ~I ,, . . .., , Bn B~ fF- .Ai 9Ai for for both both i = - 11 and and i = = 2. 2. so From this, this, (j) (j) implies implies that that B B 1I ,, .. .. .., ,BBn ~ f~ .A. 9A. The The cases cases where where A A has has outermost outermost From connective 1\, A, :::> D or or • 9 are are proved proved similarly. similarly. D [::].. connective We are are now now ready ready to to complete complete the the proof proof of of the the Completeness Completeness Theorem Theorem 1.1.3. 1.1.3. We Suppose A Lemma 1.1.4.2 Suppose A is is a a tautology. tautology. We We claim claim that that Lemma 1.1.4.2 can can be be strengthened strengthened to to have have B1,...,B~ b A

is either either Pi or 'P but now now 0 ::; is permitted. permitted. where, before each where, as as before each Bi is 9pi,i , but _ kk ::; < n is Pi or = n n,, nn- We this by We prove prove this by induction induction on on kk = 1 , .1,. .... , 1 ,, 1,0. O. For For k = = n n,, this this is is just just Lemma 1.1.4.2. For the the induction induction step, step, note note that follows from from Lemma 1.1.4.2. For that B B 1ll, .. .. .., ,BBkk kf- A A follows B ll . . . , Bk, Pk+ l Ff- A and BB1ll, .. ...., B, B~k, B1,...,Bk,pk+l A and , 9p~+1 F- A A by Lemma 1.1.4.1. 1.1.4.1. When - O, by Lemma When k = 0, 'Pk+ 1 fwe have that that bf- A, which proves proves the the Completeness Completeness Theorem. Theorem. we have A, which Q.E.D. Theorem 1.1.3 1.1.3 Q.E.D. Theorem 1.1.5. It still still remains remains to to prove the compactness theorem for propositional logic. logic. 1.1.5. It prove the compactness theorem for propositional This theorem theorem states: This states:

Compactness Theorem. r be a set set of propositional formulas. formulas. C ompactness T h e o r e m . Let F of propositional (1) Fr is is satisfiable satisfiable if and only only ifif every finite subset subset of r is is satisfiable. satisfiable. (1) if and every finite of F (2) Fr ~FAA ifif and and only only if if there there is is aa finite subset F0 roof r such such that that F0 r0 ~FA. (2) finite subset of F A.

Since Fr ~FA A is is equivalent equivalent to to Fr U U {gA} being unsatisfiable, unsatisfiable, (2) (2) is is implied implied by by (1). (1) . It It is is Since {.A} being fairly easy easy to to prove prove the the compactness compactness theorem theorem directly, directly, and and most most introductory introductory books books fairly mathematical logic logic present present such proof. Here, Here, we we shall shall instead, give aa proof proof based based in such aa proof. instead, give in mathematical on the the Tychonoff Tychonoff theorem; theorem; obviously obviously this this connection connection to to topology topology is is the the reason reason for for the the on name name 'compactness 'compactness theorem.' theorem.'

Introduction to to Proof Proof Theory Introduction

99

Proof. Let Let V V be be the the set set of of propositional propositional variables variables used used in in r; the sets sets r and V V Proof. F; the F and need not not necessarily necessarily be be countable. countable. Let Let 22yv denote need denote the the set set of of truth truth assignments assignments on on V V and endow endow 22yv with with the the product product topology topology by by viewing viewing itit as as the the product product of of IV] IVI copies copies and the two two element element space space with with the the discrete discrete topology. topology. That That isis to to say, say, the the subbasis subbasis of of the B p,i -= (T: {T : 7"(p) T(p) -= i} i} for for pp EE Y V and and ii EE (T, {T, F}. F } . Note Note elements of of 22yv are are the the sets sets Bp,i elements that these these subbasis subbasis elements elements are are both both open open and and closed. closed. Recall Recall that that the the Tychonoff Tychonoff that theorem states states that that an an arbitrary arbitrary product product of of compact compact spaces spaces is is compact; compact; in in particular, particular, theorem is compact. compact. (See ( See Munkres Munkres [1975] [1975] for for background background material material on on topology.) topology. ) 22yv is For r¢ EE r, define De Dq, == (T {T EE 22yv :: TT ~1= r¢}. Since Since r¢ only only involves involves finitely finitely many many For F, define variables, each each Dq, is both both open open and and closed. closed. Now Now Fr is is satisfiable satisfiable ifif and and only only ifif McerDr nq,EfDq, variables, De is v , the is non-empty. non-empty. By By the the compactness compactness of of 22V, the latter latter condition condition is is equivalent equivalent to to the the is sets MCeroDr nq,EfoDq, being being non-empty non-empty for all finite finite ro Cr. This, in in turn turn is is equivalent equivalent to to sets for all F0 C F. This, 0 of Fr being being satisfiable. satisfiable. [] each finite subset subset ro each finite F0 of The compactness compactness theorem logic is more difficult; difficult; aa purely The theorem for for first-order first-order logic is more purely modelmodel see, e.g., [1977]). We include aa theoretic proof proof can can be given with with ultrafilters theoretic be given ultrafilters ((see, e.g., Eklof Eklof [1977]). We include proof-theoretic proof proof of of the the compactness compactness theorem theorem for logic for for countable countable proof-theoretic for first-order first-order logic languages in in section section 2.3.7 languages 2.3.7 below. below.

1.1.6. There to give 1.1.6. Remarks. Remarks. There are are of of course course aa large large number number of of possible possible ways ways to give sound and complete proof proof systems sound and complete systems for for propositional propositional logic. logic. The The particular particular proof proof system system F ~ used used above above is is adapted adapted from from Kleene Kleene [1952]. [1952]. A A more more detailed detailed proof proof of of the the completeness completeness theorem theorem for for F ~" and and for for related related systems systems can can be be found found in in the the textbook textbook of of Mendelson Mendelson [1987] [1987].. The The system system F 9r is is an an example example of of aa class class of of proof proof systems systems called called Frege Frege proof proof systems: systems: aa Frege Frege proof proof system system is is any any proof proof system system in in which which all all axioms axioms and and rules ally sound rules are are schematic schematic and and which which is is implication implicationally sound and and implicationally implicationally complete. complete. Most Most of of the the commonly commonly used used proof proof systems systems similar similar to to F ~" are are based based on on modus modus ponens ponens as non-Frege) systems as the the only only rule rule of of inference; inference; however, however, some some ((non-Frege) systems also also incorporate incorporate aa version deduction theorem version of of the the deduction theorem as as aa rule rule of of inference. inference. In In these these systems, systems, if if B B has has been been inferred inferred from from A, A, then then the the formula formula A A ::> DB B may may also also be be inferred. inferred. An An example example of of such such aa system system is is the the propositional propositional fragment fragment of of the the natural natural deduction deduction proof proof system system described described in in section section 2.4.8 2.4.8 below. below. Other Other rules rules of of inference inference that that are are commonly commonly allowed allowed in in propositional propositional proof proof systems systems include include the the substitution substitution rule rule which which allows allows any any instance instance of of ¢ r to to be be inferred inferred from from ¢, r and and the the extension extension rule rule which which permits permits the the introduction introduction of of abbreviations abbreviations for for long long formulas. formulas. These These two two systems systems appear to to be be more more powerful powerful than than Frege Frege systems systems in in that that they they seem seem to to allow allow substantially substantially shorter shorter proofs proofs of of certain certain tautologies. tautologies. However, However, whether whether they they actually actually are are significantly significantly more more powerful powerful than than Frege Frege systems systems is is an an open open problem. problem. This This issues issues are are discussed discussed more more fully fully by by Pudllik Pudls in in Chapter Chapter VIII. VIII. There There are are several several currently currently active active areas areas of of research research in in the the proof proof theory theory of of propo propositional sitional logic. logic. Of Of course, course, the the central central open open problem problem is is the the P P versus versus N N P question question of of whether whether there there exists exists aa polynomial polynomial time time method method of of recognizing recognizing tautologies. tautologies. Research Research on on the the proof proof theory theory of of propositional propositional logic logic can can be, be, roughly roughly speaking, speaking, separated separated into into three three problem problem areas. areas. Firstly, Firstly, the the problem problem of of "proof-search" "proof-search" is is the the question question of of

s S.. Buss Buss

10 10

what are are the the best best algorithmic algorithmic methods methods for for searching searching for for propositional propositional proofs. proofs. The The what proof-search problem problem is is important important for for artificial artificial intelligence, intelligence, for for automated automated theorem theorem proof-search proving programming. The propositional proof proving and and for for logic logic programming. The most most common common propositional proof systems systems used algorithms are resolution system used for for proof-search proof-search algorithms are variations variations of of the the resolution system discussed discussed in .3 below. second, related in 11.3 below. A A second, related research research area area is is the the question question of of proof proof lengths. lengths. In In this area, area, the the central central questions questions concern concern the the minimum minimum lengths lengths of of proofs proofs needed needed for for this tautologies in particular tautologies in particular proof proof systems. systems. This This topic topic is is treated treated in in more more depth depth in in Chapter Chapter VIn VIII in in this this volume. volume. A A third third research research area area concerns concerns the the investigation investigation of of fragments fragments of of the the propositional propositional proof proof system system F. ~'. For For example, example, propositional propositional intuitionist intuitionist logic logic is is the the logic logic which which is is axiomatized axiomatized by by the the system system F ~" without without the the axiom axiom scheme scheme -,-,A -~--AJ D A A.. Another Another important example example is is linear linear logic. logic. Brief Brief discussions discussions of of these these two two logics logics can can be be found found important in in section section 3. 3.

1.2. 1.2. The T h e propositional p r o p o s i t i o n a l sequent sequent calculus calculus

The introduced by The sequent sequent calculus, calculus, first first introduced by Gentzen Gentzen [1935) [1935] as as an an extension extension of of his his earlier earlier natural deduction proof natural deduction proof systems, systems, is is arguably arguably the the most most elegant elegant and and flexible flexible system system for for writing In this the propositional writing proofs. proofs. In this section, section, the propositional sequent sequent calculus calculus for for classical classical logic logic is 2.3 below. is developed; developed; the the extension extension to to first-order first-order logic logic is is treated treated in in 2.3 below. In In the the Hilbert-style Hilbert-style systems, systems, each each line line in in aa proof proof is is aa formula; formula; however, however, in in sequent sequent calculus calculus proofs, proofs, each each line line in in aa proof proof is is aa sequent: sequent: a a sequent sequent is is written written in in the the form form 1.2.1. 1.2.1. Sequents Sequents and and Cedents. Cedents.

A1, . . . , A~----~ B1, . . . , B t

where where the the symbol symbol � ~ is is aa new new symbol symbol called called the the sequent sequent arrow arrow (not (not to to be be confused confused with the the implication symbolJ) with implication symbol D ) and and where where each each Ai Ai and and Bj B j is is aa formula. formula. The The intuitive intuitive ' S implies the disjunction meaning the sequent sequent is that the the conjunction the Ai meaning of of the is that conjunction of of the Ai's implies the disjunction of of the Bj s. Thus, Thus, aa sequent the B j ''s. sequent is is equivalent equivalent in in meaning meaning to to the the formula formula

1\

kk

t

i=1

j=l

l

Ai J V 1\ AA, VBBj.. i =1 j=1

The The symbols symbols A and and V represent represent conjunctions conjunctions and and disjunctions, disjunctions, respectively, respectively, of of multiple convention that multiple formulas. formulas. We We adopt adopt the the convention that an an empty empty conjunction conjunction (say, (say, when when - 00 above) above) has has value value "True" "True",, and and that that an an empty empty disjunction disjunction (say, (say, when when £~ = = 00 k = above) above) has has value value "False" "False".. Thus Thus the the sequent sequent � --~ A A has has the the same same meaning meaning as as the the formula is false. formula A A,, and and the the empty e m p t y sequent sequent � ~ is false. A A sequent sequent is is defined defined to to be be valid valid or or aa tautology only if corresponding formula tautology if if and and only if its its corresponding formula is. is. The called the The sequence sequence of of formulas formulas A A1,...,Ak is called the antecedent a n t e c e d e n t of of the the sequent sequent I ' . . . ' Ak is displayed ...,B called its both referred displayed above; above; Bl, B1,..., Btl is is called its succedent s u c c e d e n t . . They They are are both referred to to as as cedents. cedents.

V

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Introduction to Proof Theory

1.2.2. 1.2.2. Inferences Inferences and and proofs. proofs. We We now now define define the the propositional propositional sequent sequent calculus calculus proof system system PK. PK. A A sequent sequent calculus calculus proof proof consists consists of of aa rooted rooted tree tree (or (or sometimes sometimes aa proof directed acyclic acyclic graph) graph) in in which which the the nodes are sequents. sequents. The The root root of of the the tree, written directed nodes are tree, written at the the bottom, bottom, is is called called the the endsequent and and is is the the sequent sequent proved proved by by the the proof. proof. The The at leaves, tree, are leaves, at at the the top top of of the the tree, are called called initial sequents or or axioms. Usually, Usually, the the only only initial sequents allowed allowed are are the the logical logical axioms axioms of of the the form form A A---F A,, where where we we further further initial sequents ---+ A require require that that A A be be atomic. atomic. Other than than the the initial initial sequents, sequents, each each sequent sequent in in aa PK-proof PK-proof must must be be inferred inferred by by Other one below. A one of of the the rules rules of of inference inference given given below. A rule rule of of inference inference is is denoted denoted by by a a figure figure -~ or Ssl) S S2 s~ indicating indicating that that the the sequent sequent 8 S may may be be inferred inferred from from 81 $1 or or from from the the � or S S $2.. The is called called the conclusion, 8, pair pair 81 $1 and and 8 S, is the lower sequent of of the the inference; inference; each each 2 The conclusion, hypotheses is the inference. rules of hypotheses is an an upper sequent of of the inference. The The valid valid rules of inference inference for for PK PK denote arbitrary arbitrary are essentially schematic, are as as follows; follows; they they are are essentially schematic, in in that that A A and and B B denote formulas etc. denote denote arbitrary arbitrary cedents. formulas and and r F,, � A,, etc. cedents.

Weak Structural Rules Weak Structural Rules rF, A � A,, B B,, II---+ II---FA Exchange:left ' F, A, II---+ II--FA r, B, B, A, � A, A, A, r---+ Contraction:left A, F"-F � A Contraction:left A, F----FA A, r---+ � r---+ � F-"} A Weakening:left Weakening:left A, r---+ F ---}� A A,

r---+� , A, B, A xchange:right F---+A, A, B, A Exchange:right E r---+ � , B, B, A, F---FA, A, A A

� , A, F----FA, A, A A Contraction:right r---+ Contraction:right F---FA, A r---+ �, A Weakening:right

r---+ � F---} A F--~ � A,, A A r---+

The structural rules rules. The The weak weak structural rules are are also also referred referred to to as as just just weak inference inference rules. The rest rest of rules are called strong inference rules. The rules consist consist of of the the rules are called inference rules. The structural rules of the the weak weak structural rules and structural rules and the the cut cut rule. rule. The The Cut Cut Rule Rule

rF---F ---+ �, A A, rF----F ---+ � A,A A, A r ---+ � F----F A

l T h e Propositional P r o p o s i t i o n a l Rules Rules 1 The -1d:left eft -

r---+ F ---+� A,, A A ...,-~A, A , r---+ � F---+A

-~:right ...,.Tight

A:left A:left

A, � A, B, B, r---+ F---}A AA A B, F----~A B, r---+ � A

r---+ �, B F---~A, B A:right r---+ F - + A�, , A A A:right F---+A, AA AB B r---+ �, A

B, r---+ � V:lefl A, A, r---+ F---+ � A B, F----FA V:left A � A VB B, r---+ F---FA F---FA, A B, F---FA r---+ �, A B, r---+ � :J:left D :left A :J B, r---+ � A D B, P--+A

A, A, r---+ F---} � A r---+ � , ...,-~A A F---+A,

V :right V:right

r---+ F---} � A,, A, A, B B r---+ �, A F---+A, A VB B

D :right :J:right

A, A, r---+ F~ � A,, B B r---+ F - + A�, , A A :J DB B

11 We have have stated the 1\ A:left :left and the V:right V:right rules rules differently differently than the traditional form. form. The traditional definitions definitions use the following following two V:right V:right rules rules of inference inference

s . Buss Buss S.

12 12

The above above completes completes the the definition definition of of PK. PK. We We write write PK PK ~I- r --+ A b.. to to denote denote that that The F--+ the sequent sequent r--+ b.. has has aa PK-proof. PK-proof. When When A A is is aa formula, formula, we we write write PK PK FI- AA to to the F--~ A I- --+ A. mean that that PK PK Fmean ~ A. The cut cut rule rule plays plays aa special special role role in in the the sequent sequent calculus, calculus, since, since, as as isis shown shown in in The section 1.2.8, 1 .2.8, the the system PK is is complete complete even without the the cut cut rule; rule; however, however, the the use use section system PK even without of the the cut cut rule rule can can significantly significantly shorten shorten proofs. proofs. A A proof proof is is said said to to be be cut-free cut-free if if does does of not contain contain any any cut cut inferences. inferences. not 1.2.3. A Ancestors, All of of the the 1.2.3. n c e s t o r s , ddescendents e s c e n d e n t s aand n d tthe h e ssubformula u b f o r m u l a pproperty. roperty. All principal formula formula which which inferences of of PK, PK, with with the the exception exception of of the the cut cut rule, rule, have inferences have aa principal is, is, by by definition, definition, the the formula formula occurring occurring in in the the lower lower sequent sequent of of the the inference inference which which is is not in in the the cedents cedents Fr or or A b.. (or (or H II or or A). A). The The exchange two principal principal not exchange inferences inferences have have two formulas. Every inference, except except weakenings, weakenings, has has one one or or more more auxiliary auxiliary formulas formulas formulas. Every inference, which which are are the the formulas formulas A A and and B, B , occurring occurring in in the the upper upper sequent(s) sequent (s ) of of the the inference. inference. The formulas F, A, A are are called formulas of The formulas which which occur occur in in the the cedents cedents r, b.. , II II or or A called side formulas of the the inference. The two of aa cut inference are the cut cut formulas. inference. The two auxiliary auxiliary formulas formulas of cut inference are called called the formulas. We define the the notions notions of of descendents in We now now define descendents and and ancestors ancestors of of formulas formulas occurring occurring in sequent calculus First we we define define immediate immediate descendents descendents as as follows: is aa sequent calculus proof. proof. First follows: If If C C is formula in in an say C i-th subformula subformula of aa side side formula an upper upper sequent sequent of of an an inference, inference, say C is is the the i-th of only immediate immediate descendent the corresponding aa cedent F, II, H, b.. A or or A, C's's only descendent is is the corresponding A , then then C cedent r, occurrence of the same formula formula in in the the same same position in the same cedent occurrence of the same position in the same cedent in in the the lower sequent of the the inference. inference. If If C C is is an an auxiliary auxiliary formula inference except lower sequent of formula of of any any inference except an an exchange or cut cut inference, inference, then then the the principal the inference inference is is the the immediate immediate exchange or principal formula formula of of the descendent of of C inference, the of the A descendent C.. For For an an exchange exchange inference, the immediate immediate descendent descendent of the A or upper sequent or B B in in the the upper sequent is is the the A A or or B B,, respectively, respectively, in in the the lower lower sequent. sequent. The The cut cut formulas formulas of of a a cut cut inference inference do do not not have have immediate immediate descendents. descendents. We We say say that that C C is is an an immediate ancestor of of D D if if and and only only if if D D is is an an immediate immediate descendent descendent of of C C.. Note Note that that the the only only formulas formulas iinn a a proof proof that that do do not not have have immediate immediate ancestors ancestors are are the the formulas the principal formulas in in initial initial sequents sequents and and the principal formulas formulas of of weakening weakening inferences. inferences. The The ancestor relation relation is is defined defined to to be be the the reflexive, reflexive, transitive transitive closure closure of of the the immediate relation; thus, immediate ancestor ancestor relation; thus, C C is is an an ancestor ancestor of of D D if if and and only only if if there there is is a a chain ancestor of chain of of zero zero or or more more immediate immediate ancestors ancestors from from D D to to C. C. A A direct ancestor of D D is is an an ancestor such that that C the same same formula ancestor C C of of D D such C is is the formula as as D. D. The The concepts concepts of of descendent descendent and and direct descendent are are defined defined similarly similarly as as the the converses converses of of the the ancestor ancestor and and direct direct ancestor ancestor relations. relations. A A simple, simple, but but important, important, observation observation is is that that if if C C is is an an ancestor ancestor of of D, D, then then C C is is aa subformula subformula of of D. D. This This immediately immediately gives gives the the following following subformula subformula property: property: F----} A,A r=t F---} Il, A,A AVB

and and

r--+ A F----} Il, A,A r=t VA F---} Il, A, B BV

and two dual rules rules of inference inference for A A:left. :left. Our method has the advantage of reducing reducing the number of of rules rules of of inference, inference, and and also also simplifying simplifying somewhat somewhat the the upper upper bounds bounds on on cut-free cut-free proof proof length length we we obtain obtain below. below.

Introduction to Proof Theory

13 13

1.2.4. (The 1.2.4. Proposition. Proposition. (The Subformula Subformula Property) Property) If If P P is is aa cut-free cut-free PK-proof, PK-proof, then then every every formula formula occurring occurring in in P P is is aa subformula subformula of of aa formula formula in in the the endsequent endsequent of of P P. .

There 1.2.5. 1.2.5. Lengths L e n g t h s of of proofs. proofs. There are are aa number number of of ways ways to to measure measure the the length length of of a a sequent sequent calculus calculus proof proof P; P; most most notably, notably, one one can can measure measure either either the the number number of of symbols or or the the number number of of sequents sequents occurring occurring in in P P . . Furthermore, Furthermore, one one can can require require symbols P to to be be tree-like tree-like or or to to be be dag-like; in the case of of dag-like dag-like proofs proofs no no sequent sequent needs needs to to dag-like; in the case P abbreviates 'directed be 'Dag' abbreviates be derived, derived, or or counted, counted, twice. twice. (('bag' 'directed acyclic acyclic graph graph',' , another another '.) name name for for such such proofs proofs is is 'sequence-like 'sequence-like'.) For this chapter, For this chapter, we we adopt adopt the the following following conventions conventions for for measuring measuring lengths lengths of of sequent sequent calculus calculus proofs: proofs: proofs proofs are are always always presumed presumed to to be be tree-like, tree-like, unless unless we we explicitly and we explicitly state state otherwise, otherwise, and we let let IIPII IIPII denote denote the the number number of of strong s t r o n g inferences inferences in in aa tree-like tree-like proof proof P P . . The The value value IIIPII IPI] is is polynomially polynomially related related to to the the number number of of sequents, then, then, of sequents sequents in in P P . . If If P P has has n sequents, of course, course, IIPII IIPII < < n n.. On On the the other other hand, hand, it is is not not hard to prove prove that that for for any any tree-like tree-like proof proof P P of of aa sequent sequent r F--~ A, there there is is --t �, it hard to 2 sequents and aa ((still still tree-like proof of --t�' tree-like)) proof of an an endsequent endsequent r' F'---~ A' with with at at most most IIIPI1 IPII 2 sequents and PII instead with with r' F' � C_r F and and �' A' � c_ �. A. The The reason reason we we use use IIIIPII instead of of merely merely counting counting the the bounds on actual actual number number of of sequents sequents in in P P , , is is that that using using lIP IIPIIII often often makes makes bounds on proof proof size size signifi cantly more significantly more elegant elegant to to state state and and prove. prove. PI I dag to Occasionally, Occasionally, we we use use 1IIIPIIdag to denote denote the the number number of of strong strong inferences inferences in in aa dag-like dag-like proof proof P P .. The 1.2.6. 1.2.6. Soundness S o u n d n e s s Theorem. Theorem. The propositional propositional sequent sequent calculus calculus PK P K is is sound. sound. That That is is to to say, say, any any PK-provable PK-provable sequent sequent or or formula formula is is aa tautology. tautology.

The The soundness soundness theorem theorem is is proved proved by by observing observing that that the the rules rules of of inference inference of of PK PK preserve of sequents preserve the the property property of sequents being being tautologies. tautologies. The holds. If The implicational implicational form form of of the the soundness soundness theorem theorem also also holds. If 6 | is is aa set set of of sequents, let let an be calculus proof proof in which sequents sequents, an 6-proof | be any any sequent sequent calculus in which sequents from from 6 | are are permitted initial sequents in addition ) . The permitted as as initial sequents ((in addition to to the the logical logical axioms axioms). The implicational implicational then soundness soundness theorem theorem states states that that if if aa sequent sequent r--t F---~ � A has has an an 6-proof, | then r--t F--~ � A is is made made true true by by every every truth truth assignment assignment which which satisfies satisfies 6. | 1.2.7. The 1.2.7. The T h e inversion inversion theorem. theorem. The inversion inversion theorem theorem is is aa kind kind of of inverse inverse to to the the implicational implicational soundness soundness theorem, theorem, since since it it says says that, that, for for any any inference inference except except weakening valid, then weakening inferences, inferences, if if the the conclusion conclusion of of the the inference inference is is valid, then so so are are all all of of its its hypotheses. hypotheses.

Theorem. propositional inference, T h e o r e m . Let Let I I be be aa propositional inference, aa cut cut inference, inference, an an exchange exchange inference inference or lower sequent or aa contraction contraction inference. inference. If If II ' s's lower sequent is is valid, valid, then then so so are are are are all all of of I I ' s's upper upper sequents. sequents. Likewise, Likewise, if if I I ' s's lower lower sequent sequent is is true true under under aa truth truth assignment assignment r, T, then then so are are all of of II ' 's s upper sequents.

The The inversion inversion theorem theorem is is easily easily proved proved by by checking checking the the eight eight propositional propositional inference inference rules; rules; it it is is obvious obvious for for exchange exchange and and contraction contraction inferences. inferences.

1144

S . Buss S.

Note Note that that the the inversion inversion theorem theorem can can fail fail for for weakening weakening inferences. inferences. Most Most authors authors ned them define the 1\ A :left and and V V :right rules rules of of inference inference differently differently than than we we defi defined them defi ne the for PK, PK, and and the the inversion inversion theorem theorem can can fail fail for for these these alternative alternative formulations formulations ((see the for see the footnote 1). footnote on on page page 111). 1.2.8. The 1.2.8. The T h e completeness c o m p l e t e n e s s theorem. theorem. The completeness completeness theorem theorem for for PK P K states states that tautology) can that every every valid valid sequent sequent ((tautology) can be be proved proved in in the the propositional propositional sequent sequent calculus. This, together theorem, shows calculus. This, together with with the the soundness soundness theorem, shows that that the the PK-provable PK-provable sequents precisely the sequents are are precisely the valid valid sequents. sequents.

Theorem. K-proof in T heorem. I Iff r F -+.6~ A is is aa tautology, tautology, then then it it has has aa P PK-proof in which which no no cuts cuts appear. appear.

In order to .2.8 we stronger lemma 1.2.9. 1.2.9. In order to prove prove Theorem Theorem 11.2.8 we prove prove the the following following stronger lemma which which includes includes bounds bounds on on the the size size of of the the PK-proof. PK-proof. Lemma. sequent in which there are m L e m m a . Let r-+.6F--~ A be a valid sequent m occurrences occurrences of of logical logical -proof P connectives. connectives. Then Then there there is is aa tree-like, tree-like, cut cut free free PK PK-proof P of of r F -+.6- + A containing containing fewer m strong fewer than than 22'n strong inferences. inferences. Proof. induction on m. In P r o o f . The The proof proof is is by by induction on m. In the the base base case, case, m m = - 00,, the the sequent sequent r -+.6logical connectives connectives and F---+ A contains contains no no logical and thus thus every every formula formula in in the the sequent sequent is is a a propositional Since the propositional variable. variable. Since the sequent sequent is is valid, valid, there there must must be be some some variable, variable, p p,, which in .6-. -+.6which occurs occurs both both in in r F and and in A. Thus Thus r F---+ A can can be be proved proved with with zero zero strong strong -+ p. p. inferences inferences from from the the initial initial sequent sequent p p----} The The induction induction step, step, m m > > 00,, is is handled handled by by cases cases according according to to which which connectives connectives are are used used as as outermost outermost connectives connectives of of formulas formulas in in the the cedents cedents r F and and .6-. A. First First suppose suppose there in r. there is is a a formula formula of of the the form form (-,A) (-~A) in F. Letting Letting r' F' be be the the cedent cedent obtained obtained from from r F -+.6occurrences of by by removing removing occurrences of -,A, -~A, we we can can infer infer r F---+ A by: by: r'-+ A F' ---F .6-, A,A -,A, -+ .6-~A,r' F'---~ A r-+ F---~ .6A where double line where the the double line indicates indicates a a series series of of weak weak inferences. inferences. By By the the inversion inversion theorem, theorem, r' -+.6-, F'---~ A, A A is is valid, valid, and and hence, hence, since since it it has has at at most most m m -- 11 logical logical connectives, connectives, the the induction induction hypothesis hypothesis implies implies that that it it has has a a cut-free cut-free proof proof with with fewer fewer than than 2m2m-11 strong strong inferences. -+.6inferences. This This gives gives r F--~ A a a cut-free cut-free proof proof with with fewer fewer than than 2m2m-11 + + 11 � <_ 22m m strong strong inferences. inferences. The The case case where where a a formula formula (-,A) (~A) occurs occurs in in .6A is is handled handled similarly. similarly. Second, Second, consider consider the the case case where where a a formula formula of of the the form form A A 1\ AB B appears appears in in r. F. Then, Then, -+.6letting letting r' F' be be the the cedent cedent r F minus minus the the formula formula A A 1\ AB B,, we we can can infer infer r F---> A by: by: A, B, r'-+ .6A,B,F'---> A A -+ .6A A1\BB, , r' F'-"F A r-+ .6P---F A By induction hypothesis, hypothesis, A, B, r'-+.6By the the inversion inversion theorem theorem and and the the induction A, B, F' ~ A has has a a cut-free cut-free -1 strong strong inferences. -+.6proof with proof with with fewer than 2m proof fewer than 2m-1 inferences. Thus Thus r F----F A has has a a cut-free cut-free proof with Third, suppose fewer fewer than than 2m 2 m strong strong inferences. inferences. Third, suppose there there is is a a formula formula of of the the A A 1\ AB B

Introduction to to Proof Proof Theory Theory Introduction

15 15

appearing appearing in in the the succedent succedent A. �. Letting Letting A' �' be be the the the the succedent succedent A � minus minus the the formula formula AA /\ B, B , we we can can infer infer A

f-tA',A �', A f-tA �', F--> F---> ' , BB f-tA',A �', A A /\ B B F---> Ff-t� -->A By the the inversion inversion theorem, theorem, both both of of upper upper sequents sequents above above are are valid. valid. Furthermore, Furthermore, they they By each have have fewer fewer than than m m logical logical connectives, connectives, so so by by the the induction induction hypothesis, hypothesis, they they each have m-1 strong have cut-free cut-free proofs proofs with with fewer fewer than than 22m-1 strong inferences. inferences. This This gives gives the the sequent sequent f -tA � aa cut-free cut-free proof proof with with fewer fewer than than 22 m m strong strong inferences. inferences. F--+ The remaining remaining cases cases are are when when aa formula formula in in the the sequent f -tA � has has outermost outermost The sequent F---> connective ~. These inversion theorem connective V or orJ. These are are handled handled with with the the inversion theorem and and the the induction induction hypothesis similarly to to the the above above cases. cases. [] 0 hypothesis similarly

1.2.10. The bounds bounds on on the the proof proof size size in in Lemma Lemma 1.2.9 can can be be improved improved somewhat somewhat 1.2.10. The make by counting only the occurrences occurrences of distinct subformulas in f � . To by counting only the of distinct subformulas in F -t ~ A. To make this precise, precise, we need to to define define the the concepts positively and and negatively negatively occurring occurring this we need concepts of of positively B of A , and and aa subformulas. formula A subformulas. Given Given a a formula A,, an an occurrence occurrence of of a a subformula subformula B of A, occurrence of a in A , we is negatively bound by by a if occurrence of a a logical logical connective connective c~ in A, we say say that that B B is negatively bound ~ if either (1) a is aa negation negation sign, sign, .-~, ." and and B B is is in in its its scope, or (2) ~a is implication sign, either (1) ~ is scope, or is an an implication sign, J, and ~, and B B is is aa subformula subformula of of its its first first argument. argument. Then, Then, B B is is said said to to occur occur negatively (respectively, bound by odd (respectively, (respectively, positively) in in A A if if B B is is negatively negatively bound by an an odd (respectively, even) even) number f-t number of of connectives connectives in in A A.. A A subformula subformula occurring occurring in in aa sequent sequent F ~ � A is is said said to to negatively in be be positively positively occurring occurring if if it it occurs occurs positively positively in in � A or or negatively in f F;; otherwise, otherwise, it it occurs negatively occurs negatively in in the the sequent. sequent.

Lemma. L e m m a . Let f F -t ~ � A be a valid sequent. Let m m'' equal the number of of distinct il subformulas subformulas occurring occurring positively positively in in the the sequent sequent and and m m" equal equal the the number number of of distinct distinct i Then there subformulas subformulas occurring occurring negatively negatively in in the the sequent. sequent. Let Let m m = =- m m '~+ +m m".l . Then there is is aa K -proof P tree-like, tree-like, cut cut free free P PK-proof P containing containing fewer fewer than than 2 m strong strong inferences. inferences. Proof. .2.9 built P r o o f . (Sketch) (Sketch) Recall Recall that that the the proof proof of of Lemma Lemma 11.2.9 built aa prooffrom proof from the the bottom bottomup, up, by by choosing choosing aa formula formula in in the the endsequent endsequent to to eliminate eliminate (i.e., (i.e., to to be be inferred) inferred) and and thereby thereby reducing reducing the the total total number number of of logical logical connectives connectives and and then then appealing appealing to to the the induction induction hypothesis. hypothesis. The The construction construction for for the the proof proof of of the the present present lemma lemma is is exactly exactly the the same, same, except except that that now now care care must must be be taken taken to to reduce reduce the the total total number number of of distinct distinct positively positively or or negatively negatively occurring occurring subformulas, subformulas, instead instead of of just just reducing reducing the the total total number number of of connectives. connectives. This This is is easily easily accomplished accomplished by by always always choosing choosing aa formula formula from from the the endsequent endsequent which which contains contains aa maximal maximal number number of of connectives connectives and and which which is is therefore therefore not not aa proper proper subformula subformula of of any any other other subformula subformula in in the the endsequent. endsequent. 0 [] 1.2.11. 1.2.11. The The cut cut elimination elimination theorem theorem states states that that if if aa sequent sequent has has aa PK-proof, PK-proof, then then it it has has aa cut-free cut-free proof. proof. This This is is an an immediate immediate consequence consequence of of the the soundness soundness and and completeness completeness theorems, theorems, since since any any PK-provable PK-provable sequent sequent must must be be valid, valid, by by the the soundness soundness theorem, theorem, and and hence hence has has aa cut-free cut-free proof, proof, by by the the completeness completeness theorem. theorem.

16 16

s . Buss Buss S.

This is is aa rather rather slick slick method method of of proving proving the the cut cut elimination elimination theorem, theorem, but but unforunfor This tunately, tunately, does does not not shed shed any any light light on on how how aa given given PK-proof PK-proof can can be be constructively constructively transformed into into aa cut-free cut-free proof. proof. In In section section 2.3.7 2.3.7 below, below, we we shall shall give give aa step-by-step step-by-step transformed procedure procedure for for converting converting first-order first-order sequent sequent calculus calculus proofs proofs into into cut-free cut-free proofs; proofs; the the same methods methods work work also also for for propositional propositional sequent sequent calculus calculus proofs. proofs. We shall not, not, same We shall however, describe this constructive constructive proof proof transformation transformation procedure procedure here; here; instead, we however, describe this instead, we will state, without will only only state, without proof, proof, the the following following upper upper bound bound on on the the increase increase in in proof proof (A proof proof length which which can can occur occur when when aa proof proof is is transformed transformed into into aa cut-free cut-free proof. proof. (A length can given using can be be given using the the methods methods of of Sections Sections 2.4.2 2.4.2 and and 2.4.3.) 2.4.3.) Cut-Elimination Theorem. Suppose P P be a (possibly dag-like) PK-proof PK-proof of of C ut-Elimination T h e o r e m . Suppose � . Then Then F r ~--t A� has a cut-free, cut-free, tree-like PK-proo] PK -proof with less than than or equal F r ~--t A. l idag strong to 2211P IIPIIdag strong inferences. inferences. 1.2.12. Let G <5 be be aa set and, as as above, define 1.2.12. FFree-cut r e e - c u t eelimination. l i m i n a t i o n . Let set of of sequents sequents and, above, define an to be sequent calculus proof which may contain sequents from from G as an G-proof <5-proof to be aa sequent calculus proof which may contain sequents <5 as initial sequents, sequents, in addition to to the logical sequents. sequents. If If II is is aa cut initial in addition the logical cut inference inference occurring occurring in P , , then we say say II' 'ss cut are directly descended from from G <5 if if they they cut formulas formulas are in an an G-proof <5-proof P then we have as a a formula formula in an initial have at at least least one one direct direct ancestor ancestor which which occurs occurs as in an initial sequent sequent which which said to to be be free ifif neither neither of of II ' 'ss auxiliary auxiliary formulas directly is in in <5 A cut cut II is is | . A is said formulas is is directly descended from proof is is free-cut free-cut free if if and only if if itit contains cuts. (See ( See descended from |<5 . A A proof and only contains no no free free cuts. cut elimination elimination theorem theorem Definition 2.4.4.1 for definition of of free Definition 2.4.4.1 for aa better better definition free cuts. cuts.)) The The cut can be generalized show that the free-cut free-cut free fragment of of the sequent calculus can be generalized to to show that the free fragment the sequent calculus is is implicationally complete: implicationally complete:

Free-cut F r e e - c u t Elimination E l i m i n a t i o n Theorem. T h e o r e m . Let S S be a sequent and <5 G a set of of sequents. If If <5 -proof of S G F ~ <5 G,, then there is a free-cut free-cut free <5 G-proof S..

We We shall shall not not prove prove this this theorem theorem here; here; instead, instead, we we prove prove the the generalization generalization of of this this for for first-order first-order logic logic in in 2.4.4 2.4.4 below. below. This This theorem theorem is is essentially essentially due due to to Takeuti Takeuti [1987] [1987] based based on on the the cut cut elimination elimination method method of of Gentzen Gentzen [1935] [1935].. 1.2.13. 1.2.13. Some S o m e rremarks. e m a r k s . We We have have developed developed the the sequent sequent calculus calculus only only for for classical classical propositional logic; however, calculus is propositional logic; however, one one of of the the advantages advantages of of the the sequent sequent calculus is its its flexibility flexibility in in being being adapted adapted for for non-classical non-classical logics. logics. For For instance, instance, propositional propositional intuitionistic intuitionistic logic logic can can be be formalized formalized by by aa sequent sequent calculus calculus P PJ J which which is is defined defined exactly K except exactly like like P PK except that that succedents succedents in in the the lower lower sequents sequents of of strong inferences inferences minimal logic are are restricted restricted to to contain contain at at most most one one formula. formula. As As another another example, example, minimal logic is formalized formalized like like P P J, J, except except with with the the restriction restriction that that every every succedent succedent contain contain is exactly logic, relevant logic, modal exactly one one formula. formula. Linear Linear logic, relevant logic, modal logics logics and and others others can can also also be be formulated formulated elegantly elegantly with with the the sequent sequent calculus. calculus. 1.2.14. 1.2.14. The T h e Tait Tait calculus. calculus. Tait Tait [1968] [1968] gave gave aa proof proof system system similar similar in in spirit spirit to to the the sequent sequent calculus. calculus. Tait's Tait's proof proof system system incorporates incorporates aa number number of of simplifications simplifications with with regard regard to to the the sequent sequent calculus; calculus; namely, namely, it it uses uses sets sets of of formulas formulas in in place place of of sequents, sequents,

17

Introduction to to Proof Theory Introduction

it allows allows only only propositional propositional formulas formulas to to be be negated, negated, and and there there are are no no weak weak structural structural it rules at at all. all. rules Since Since the the Tait Tait calculus calculus isis often often used used for for analyzing analyzing fragments fragments of of arithmetic, arithmetic, eses pecially in in the the framework framework of of infinitary infinitary logic, logic, we we briefly briefly describe describe itit here. here. Formulas Formulas pecially p, from from negated negated atomic atomic formulas formulas -~p, -'p, and and with with are built built up up from from atomic atomic formulas formulas p, are 1\ and and V. A A negated negated atomic atomic formula formula is is usually usually denoted denoted ~; p; and and the connectives connectives A the the negation negation operator operator is is extended extended to to all all formulas formulas by by defining defining ~p to to be be just just p, p, and and the 1\ B 1\ BB and inductively defining defining A V BB and and A A B to to be be A A and A V B, B , respectively. respectively. inductively Frequently, the the Tait Tait calculus calculus is is used used for for infinitary infinitary logic. logic. In In this this case, case, formulas formulas Frequently, r . The The and A F. are defined defined so so that that whenever whenever Fr is is aa set set of of formulas, formulas, then then so so are are V Fr and are intended meaning meaning of of these these formulas formulas is is the the disjunction, disjunction, or or conjunction, conjunction, respectively, respectively, intended of all all the the formulas formulas in in F. r. of Each Each line line in in aa Tait Tait calculus calcuIus proof proof is is aa set set Fr of of formulas formulas with with the the intended intended meaning meaning of of Fr being being the the disjunction disjunction of of the the formulas formulas in in F. r. A A Tait Tait calculus calculus proof proof can can be be tree-like tree-like or dag-like. The The initial or logical logical axioms, proof are sets of of the form ru p, p or dag-like. initial sets, sets, or axioms, of of a a proof are sets the form FU{p, ~}.. In the the infinitary infinitary setting, setting, there there are three rules rules of namely, In are three of inference; inference; namely,

A

A

A

A

V

1\

{ }

F U {Aj }

{ViEIAi} rU F U {{Aj: Aj'j j E e II} } rF u U {l\ { A jj~EIAAj} j} rF u ru U {A} {A} F U {A} {A} ru

I,

where where jj EE I,

(there are are III ]II many many hypotheses), hypotheses), and and (there

the cut cut rule. rule. the rF In In the the finitary finitary setting, setting, the the same same rules rules of of inference inference may may also also be be used. used. It It is is evident evident that calculus is isomorphic to that the the Tait Tait calculus is practically practically isomorphic to the the sequent sequent calculus. calculus. This This is is because -+ b. because aa sequent sequent r F---+ A may may be be transformed transformed into into the the equivalent equivalent set set of of formulas formulas containing containing the the formulas formulas from from b. A and and the the negations negations of of the the formulas formulas from from r F.. The The exchange exchange and and contraction contraction rules rules are are superfluous superfluous once once one one works works with with sets sets of of formulas, formulas, the the weakening weakening rule rule of of the the sequent sequent calculus calculus is is replaced replaced by by allowing allowing axioms axioms to to contain contain extra extra side side formulas formulas (this, (this, in in essence, essence, means means that that weakenings weakenings are are pushed pushed up up to to the the initial initial sequents sequents of of the the proof) proof).. The The strong strong rules rules of of inference inference for for the the sequent sequent calculus calculus translate, translate, by by this this means, means, to to the the rules rules of of the the Tait Tait calculus. calculus. Recall Recall that that we we adopted adopted the the convention convention that that the the length length of of aa sequent sequent calculus calculus proof proof isis equal equal to to the the number number of of strong strong inferences inferences in in the the proof. proof. When When we we work work with with tree-like tree-like proofs, proofs, this this corresponds corresponds exactly exactly to to the the number number of of inferences inferences in in the the corresponding corresponding Tait-style Tait-style proof. proof. The The cut cut elimination elimination theorem theorem for for the the (finitary) (finitary) sequent sequent calculus calculus immediately immediately implies implies the the cut cut elimination elimination theorem theorem for for the the Tait Tait calculus calculus for for finitary finitary logic; logic; this this isis commonly commonly called called the the normalization normalization theorem theorem for for Tait-style Tait-style systems. systems. For For general general infinitary infinitary logics, logics, the the cut cut elimination/normalization elimination/normalization theorems theorems may may not not hold; hold; however, however, Lopez-Escobar Lopez-Escobar [1965] [1965] has has shown shown that that the the cut cut elimination elimination theorem theorem does does hold hold for for infinitary ably infinite infinitary logic logic with with formulas formulas of of count countably infinite length. length. Also, Also, Chapters Chapters III III and and IV IV

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S. Buss Buss S.

of this this volume volume discuss discuss cut cut elimination elimination in in some some infinitary infinitary logics logics corresponding corresponding to to of theories of of arithmetic. arithmetic. theories

1.3. Propositional Propositional resolution resolution refutations refutations 1.3. The Hilbert-style Hilbert-style and and sequent sequent calculus calculus proof proof systems systems described described earlier earlier are are quite quite The powerful; however, however, they they have have the the disadvantage disadvantage that that it it has has so so far far proved proved to to be be powerful; very difficult difficult to to implement implement computerized computerized procedures procedures to to search search for for propositional propositional very Hilbert-style or or sequent sequent calculus calculus proofs. proofs. Typically, Typically, aa computerized computerized procedure procedure for for Hilbert-style proof search search will will start start with with aa formula formula A for for which which aa proof proof is is desired, desired, and and will will proof then construct construct possible possible proofs proofs of of A A by by working working backwards backwards from from the the conclusion conclusion A A then towards initial axioms. When When cut-free cut-free proofs proofs are are being being constructed constructed this this is is fairly fairly towards initial axioms. straightforward, but but cut-free cut-free proofs proofs may may be be much much longer longer than than necessary necessary and and may may straightforward, even be be too long to to be be feasibly feasibly constructed constructed by by aa computer. computer. General, General, non-cut-free, non-cut-free, even too long proofs may be quite quite short; however, the the difficulty difficulty with with proof proof search search arises arises from from the the proofs may be short; however, need to to determine determine what what formulas formulas make make suitable suitable cut cut formulas. formulas. For For example, example, when when need trying construct aa proof proof of of r F--~ A that that ends ends with with aa cut cut inference; inference; one one has has to to --+ .0. trying to to construct consider all all formulas formulas C C and and try try to to construct construct proofs proofs of of the the sequents sequents r--+ F ~ .0., A, C C and and consider C, r--+ F--~ .0.. A. In In practice, it has has been been impossible impossible to to choose choose cut cut formulas formulas C C effectively effectively C, practice, it enough to effectively effectively generate generate general general proofs. proofs. A A similar difficulty arises arises in in trying trying to to enough to similar difficulty construct construct Hilbert-style Hilbert-style proofs proofs which which must must end end with with aa modus modus ponens ponens inference. inference. Thus Thus to to have have aa propositional propositional proof proof system system which which would would be be amenable amenable to to comput computerized erized proof proof search, search, it it is is desirable desirable to to have have aa proof proof system system in in which which (1) (1) proof proof search search is efficient and does not require require too many 'arbitrary 'arbitrary'' choices, and (2) proof lengths is efficient and does not too many choices, and proof lengths are not Of course, the latter latter requirement intended to are not excessively excessively long. long. Of course, the requirement is is intended to reflect reflect the the amount of computer memory memory and and time; thus proofs many millions amount of available available computer time; thus proofs of of many millions of of steps might might well acceptable. Indeed, Indeed, for for computerized proof search, search, having having an an steps well be be acceptable. computerized proof easy-to-find proof which is is millions of steps long may may well easy-to-find proof which millions of steps long well be be preferable preferable to to having having aa hard-to-find hard-to-find proof proof which which has has only only hundreds hundreds of of steps. steps. The principal which meets the above above requirements The principal propositional propositional proof proof system system which meets the requirements is is based on on resolution. resolution. As As we we shall shall see, see, the the expressive power and and implicational implicational power power of based expressive power of resolution logic; in particular, resolution resolution resolution is is weaker weaker than than that that of of the the full full propositional propositional logic; in particular, is, in in essence, essence, restricted to formulas formulas in in conjunctive conjunctive normal However, resolution is, restricted to normal form. form. However, resolution has the the advantage advantage of of being being amenable to efficient search. has amenable to efficient proof proof search. Propositional and and first-order first-order resolution resolution were introduced in in the the influential influential work work Propositional were introduced of [ 1965b] and and Davis Davis and and Putnam Putnam [1960]. [ 1960] . Propositional Propositional resolution resolution proof proof of Robinson Robinson [1965b] systems are are discussed discussed immediately immediately below. below. A A large large part part of of the the importance importance of of proposiproposi systems tional resolution resolution lies lies in in the the fact that itit leads leads to to efficient efficient proof proof methods methods in in first-order first-order tional fact that logic: logic: first-order first-order resolution resolution is is discussed discussed in in section section 2.6 2.6 below. below.

1.3.1. Definition. A A literal literal is is defined defined to to be be either either aa propositional propositional variable or 1.3.1. Definition. variable Pi or the negation negation of of aa propositional propositional variable "'Pi ' The The literal literal "'Pi is also also denoted denoted ~Pi;; and and ifif the variable -~Pi. ~Pi is x is the the literal literal ~Pi, , then then 5x denotes denotes the the unnegated unnegated literal literal Pi. Pi . The The literal literal 5x is is called called the the x is

Introduction to Proof Theory

19

negated variable; complement complement of of x. x. A A positive literal literal is is one one which which is is an an un unnegated variable; a a negative literal literal is is one one which which is is a a negated negated variable. variable. A A clause C C is is a a finite finite set set of of literals. literals. The The intended intended meaning meaning of of C C is is the the disjunction disjunction of its its members; members; thus, thus, for for a a a a truth truth assignment, assignment, a( a(C) equals True if if and and only only if if C) equals of a(x) = True for a(x) for some some x x E E C. C. Note Note that that the the empty empty clause, clause, 00,, always always has has value value False. Since both xx and true, it Since clauses clauses that that contain contain both and x 5 are are always always true, it is is often often assumed assumed w.l.o.g. w.l.o.g. that no no clause clause contains contains both both x x and and x. 5. A A clause clause is is defined defined to to be be positive (respectively, (respectively, that negative) if only positive if it it contains contains only positive (resp., (resp., only only negative) negative) literals. literals. The The empty empty clause clause is the the only only clause clause which which is is both positive and and negative. negative. A A clause clause which which is is neither neither is both positive positive positive nor nor negative negative is is said said to to be be mixed. A non-empty set r F of of clauses is used used to to represent the conjunction conjunction of of its its members. members. A non-empty set clauses is represent the Thus a(r) a(F) is is True if if and and only only if if a(C) is True for for all all C C E E r. F. Obviously, Obviously, the the meaning meaning Thus a(C) is of of r F is is the the same same as as the the conjunctive conjunctive normal normal form form formula formula consisting consisting of of the the conjunction conjunction of of the the disjunctions disjunctions of of the the clauses clauses in in r. F. A A set set of of clauses clauses is is said said to to be be satisfiable if if there there is is at at least least one one truth truth assignment assignment that that makes makes it it true. true. able: this Resolution Resolution proofs proofs are are used used to to prove prove that that aa set set r F of of clauses clauses is is unsatisfi unsatisfiable: this is ned below) is done done by by using using the the resolution resolution rule rule (defi (defined below) to to derive derive the the empty empty clause clause from r. F. Since Since the the empty empty clause clause is is unsatisfiable unsatisfiable this this will will be be sufficient sufficient to to show show that that r F from is is unsatisfiable. unsatisfiable. 1.3.2. 1.3.2. Definition. Definition. Suppose Suppose that that C C and and D D are are clauses clauses and and that that x x E EC C and and x 5 E ED D are literals. The are literals. The resolution rule applied applied to to C C and and D D is is the the inference inference D C C D (C (C \\ {x})U {x}) U (D (D \\ {x}) {5})

The of C The conclusion conclusion (C ( C \\ {x}) {x}) U U (D (D \\ {x}) {5}) is is called called the the resolvent resolventof C and and D D (with (with respect respect to x x).2 to ) .2 Since Since the the resolvent resolvent of of C C and and D D is is satisfied satisfied by by any any truth truth assignment assignment that that satisfies satisfies both C resolution rule both C and and D D,, the the resolution rule is is sound in in the the following following sense: sense: if if r F is is satisfiable satisfiable and satisfiable. Since Since the and B B is is the the resolvent resolvent of of two two clauses clauses in in r, F, then then r FU U {B} (B} is is satisfiable. the empty empty clause clause is is not not satisfiable, satisfiable, this this yields yields the the following following definition definition of of the the resolution resolution proof system. system. proof Definition. Definition. A A resolution refutation refutation of of r F is is aa sequence sequence C}, C1, C C 22,, . . . ,, Ck Ck of of clauses clauses such such that that each each Ci Ci is is either either in in r F or or is is inferred inferred from from earlier earlier member member of of the the sequence sequence by by the the resolution resolution rule, rule, and and such such that that Ck C~ is is the the empty empty clause. clause. . • •

1.3.3. 1.3.3. Resolution Resolution is is defined defined to to be be aa refutation refutation procedure procedure which which refutes refutes the the satis satisfiability procedure for fiability of of aa set set of of clauses, clauses, but but it it also also functions functions as as aa proof proof procedure for proving proving the propositional formulas; the validity validity of of propositional formulas; namely, namely, to to prove prove aa formula formula A A,, one one forms forms a a A of tautology if A is set set r FA of clauses clauses such such that that A A is is a a tautology if and and only only if if r FA is unsatisfiable. unsatisfiable. Then Then 2Note optional) convention 2Note that xx is uniquely uniquely determined by C and D if we we adopt the ((optional) convention that clauses never never contain contain any literal and its complement. complement.

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Buss S. Buss

aa resolution resolution proof proof of of A A is, is, by by definition, definition, aa resolution resolution refutation refutation of of FA. rA . There There are are two two A from from A. A . The The first first method method is is to to express express the the negation negation principal means means of of forming forming r principal FA of A A in in conjunctive conjunctive normal normal form form and and let let FA rA be be the the set set of of clauses clauses which which express express that that of conjunctive normal normal form form formula. formula. The The principal principal drawback drawback of of this this definition definition of of FA r A is is conjunctive that that the the conjunctive conjunctive normal normal form form of of -~A -,A may may be be exponentially exponentially longer longer than than A, A , and and r A may may have have exponentially exponentially many many clauses. clauses. FA extension, introduced introduced by by Tsejtin Tsejtin [1968], [1968] , The second second method method is is the the method method of of extension, The which involves involves introducing introducing new new propositional propositional variables variables in in addition addition to to the the variables variables which occurring in in the the formula formula A. A . The The new new propositional propositional variables variables are are called called extension extension occurring variables and and allow allow each distinct subformula subformula B B in in A A to to be be assigned assigned aa literal literal XxsB variables each distinct by the the following following definition: definition: (1) (1) for for propositional propositional variables variables appearing appearing in in A, A , xp xp is is p; p; by (3) for any other other subformula subformula B, B , Xs is ~--6; is (2) for negated for any (2) for negated subformulas subformulas -~B, -,B , X-~S XB ; (3) XB is X�B is A to to contain contain the the following following clauses: clauses: new propositional propositional variable. variable. We then define define r aa new We then FA (1) (1) the the singleton singleton clause clause {~-X}, (2) for for each each subformula subformula of of the the form form SB A /\ C C in in A, A , the the {x A } , (2) clauses clauses and {XB {XB XB}, {XBAC, , x--6, XC , XBA {XBAC , Xc {XBAC, Xc}} and {~--6, XBAC}; AC , XB}, C };

C in A , the the clauses and for each each subformula subformula of of the the form form B BVC and (3) (3) for in A, clauses {XBvc,-2"-6},

{XBvC, X--6} and

{XB, Xc, XBvC}.

there are are additional additional logical A then of clauses be If there If logical connectives connectives in in A then similar similar sets sets of clauses can can be C (respectively, (respectively, B used. It It is is easy easy to check that that the the set set of of three three clauses clauses for for B used. to check B /\ AC B VC C)) iiss satisfied by exactly exactly those those truth truth assignments assignments that satisfied by that assign assign XBA XB^C (respectively, XB XBvC) VC ) C (respectively, the truth value value as as XB (resp., xB truth assignment assignment satisfi es the same same truth xB /\ A xe xc). Thus, a a truth satisfies XB V x c ) . Thus, Xc (resp., r A only and, conversely, FA only if if it it falsifies falsifies A A and, conversely, if if aa truth truth assignment assignment falsifies falsifies A A then then there there is is aa unique unique assignment assignment of of truth truth values values to to the the extension extension variables variables of of A A which which lets lets it it satisfy satisfy r FA A.. The The primary primary advantage advantage of of forming forming r F AA by by the the method method of of extension extension A is A . The is is that that r FA is linear-size linear-size in in the the size size of of A. The disadvantage, disadvantage, of of course, course, is is that that the the introduction of introduction of additional additional variables variables changes changes the the meaning meaning and and structure structure of of A A.. 1.3.4. 1.3.4. Completeness C o m p l e t e n e s s Theorem T h e o r e m for for Propositional P r o p o s i t i o n a l Resolution. Resolution. unsatisfiable unsatisfiable set of of clauses, then there is a resolution refutation refutation of of r F..

If If r F is an

Proof. usual, Proof. We We shall shall briefly briefly sketch sketch two two proofs proofs of of this this theorem. theorem. The The first, first, and and usual, proof is based on the Davis-Putnam procedure. First note, that by the Compactness proof is based on the Davis-Putnam procedure. First note, that by the Compactness Theorem Theorem we we may may assume assume w.l.o.g. w.l.o.g, that that r F is is finite. finite. Therefore, Therefore, we we may may use use induction induction on on the the number number of of distinct distinct variables variables appearing appearing in in r F.. The The base base case, case, where where no no variables variables appear appear in in r F is is trivial, trivial, since since r F must must contain contain just just the the empty empty clause. clause. For For the the induction induction step, step, let let p p be be aa fixed fixed variable variable in in r F and and define define r' F' to to be be the the set set of of clauses clauses containing containing the the following following clauses: clauses: a. a. For For all all clauses clauses C C and and D D in in r F such such that that p p E EC C and and p ~ E E D, D, the the resolvent resolvent of of C C and and D D w.r.t. w.r.t, p p is is in in r' F',, and and b. b. Every Every clause clause C C in in r F which which contains contains neither neither p p nor nor p ~ is is in in r'. F'.

Introduction to Proof Proof Theory Theory

21 21

Assuming, without without loss loss of of generality, generality, that that no no clause clause in in f F contained contained both both P p and and p ~,, it it Assuming, is clear clear that that the the variable variable P p does does not not occur occur in in f' U.. Now, Now, it it is is not not hard hard to to show show that that f' F' is is satisfi satisfiable if and and only only if if f F is, is, from from whence whence the the theorem theorem follows follows by by the the induction induction is able if hypothesis. hypothesis. The second second proof proof reduces reduces resolution resolution to to the the free-cut free-cut free free sequent sequent calculus. calculus. For For this, this, The if C C is is aa clause, clause, let let t1e Ac be be the the cedent cedent containing containing the the variables variables which which occur occur positively positively if in C C and and ITe He be be the the variables variables which which occur occur negatively negatively in in C. C. Then Then the the sequent sequent in He--~ Ac is is aa sequent sequent with with no no non-logical non-logical symbols symbols which which is is identical identical in in meaning meaning ITe ---t t1e to ---t Pb P3 to C C.. For For example, example, if if C C = - {Pl (pl,, P2 p-2,, P3} p3},, then then the the associated associated sequent sequent is is P2 p2--~pl, P3.' Clearly, if if C C and and D D are are clauses clauses with with aa resolvent resolvent E E,, then then the the sequent sequent ITE liE ---t ~ / kt1E E is is Clearly, obtained obtained from from lIe 1-Iv---t --~ t1e Ac and and lID liD ---t ~ / kt1D D with with aa single single cut cut on on the the resolution resolution variable. variable. Now suppose suppose f F is is unsatisfiable. unsatisfiable. By By the the completeness completeness theorem theorem for for the the free-cut free-cut free free Now sequent calculus, calculus, there there is is aa free-cut free-cut free free proof proof of of the the empty sequent from from the the sequents sequents sequent empty sequent Hc--~ Ac with with C C E Ef F.. Since Since the the proof proof is is free-cut free-cut free free and and there there are are no no non-logical non-logical ITe ---t t1e symbols appearing appearing in in any any initial initial sequents, sequents, every every cut cut formula formula in in the the proof proof must must be be symbols atomic. Therefore Therefore no no non-logical non-logical symbol symbol appear appear anywhere anywhere in in the the proof proof and, and, by by atomic. identifying the the sequents sequents in in the the free-cut free-cut free free proof proof with with clauses clauses and and replacing replacing each each cut cut identifying inference with with the the corresponding corresponding resolution resolution inference, inference, aa resolution resolution refutation refutation of of the the inference empty clause clause is is obtained. [:] empty obtained. 0 1.3.5. R e s t r i c t e d forms forms of of rresolution. esolution. One of of the the principal advantages of of 1.3.5. Restricted One principal advantages resolution resolution is is that that it it is is easier easier for for computers computers to to search search for for resolution resolution refutations refutations than than to search search for for arbitrary arbitrary Hilbert-style Hilbert-style or or sequent sequent calculus calculus proofs. proofs. The The reason reason for for this this to is is that that resolution resolution proofs proofs are are less less powerful powerful and and more more restricted restricted than than Hilbert-style Hilbert-style and and sequent particular, there sequent calculus calculus proofs proofs and, and, in in particular, there are are fewer fewer options options on on how how to to form form resolution proofs. This situation that that aa less-powerful proof resolution proofs. This explains explains the the paradoxical paradoxical situation less-powerful proof system can be preferable Thus it it makes makes sense system can be preferable to to aa more more powerful powerful system. system. Thus sense to to consider consider further restrictions on on resolution resolution which which may reduce the the proof more. further restrictions may reduce proof search search space space even even more. Of course there there is is aa tradeoff tradeoff involved involved in in using using more restricted forms forms of of resolution, resolution, Of course more restricted since one may find restricted proofs proofs are easier to to search search for, since one may find that that although although restricted are easier for, they they are are aa lot lot less of proof proof search is more less plentiful. plentiful. Often, Often, however, however, the the ease ease of search is more important important than than the existence of proofs; in sometimes even even preferable preferable to proof the existence of short short proofs; in fact, fact, itit is is sometimes to use use aa proof system which provided its proofs are easy to system which is is not not complete, complete, provided its proofs are easy to find. find. Although we do this until until section the second main advantage advantage of of second main Although we do not not discuss discuss this section 2.6, the resolution propositional refutations refutations can 'lifted ' to to first-order first-order refutations resolution is is that that propositional can be be 'lifted' refutations of of first-order formulas. formulas. It It is is important important that the restricted restricted forms forms of of resolution resolution discussed discussed first-order that the next next also also apply apply to to first-order first-order resolution resolution refutations. refutations. One implicit in first proof One example example of of aa restricted restricted form form of of resolution resolution is is implicit in the the first proof of of the the Completeness Theorem 1.3.4 based on 1.3.4 based on the the Davis-Putnam Davis-Putnam procedure; procedure; namely, namely, for for any any Completeness Theorem it can can be be required required that that aa resolution resolution refutation refutation ordering of of the the variables variables PPl, ordering l , ..... ,. P, Pm m , , it has Pl , then then resolutions resolutions with with respect respect to to p2, P2 , etc., etc., has first first resolutions resolutions with with respect respect to to Pl, This particular particular strategy strategy is not concluding to Pro. is not Pm . This concluding witti witli resolutions resolutions with with respect respect to particularly useful useful since since itit does does not not reduce reduce the the search search space space sufficiently. sufficiently. We consider particularly We consider next several strategies strategies that have been somewhat more next several that have been somewhat more useful. useful.

22 22

Buss S. Buss

1.3.5.1. ubsumption. A A clause clause CC isis said said to to subsume subsume aa clause clause D D ifif and and only only ifif 1.3.5.1. SSubsumption. C C_ D . The The subsumption subsumption principle principle states states that that ifif two two clauses clauses CC and and D D have have been been C � D. derived subsumes D, D , then then D D should should be be discarded discarded and and not not used used further further derived such such that that CC subsumes in in the the refutation. refutation. The The subsumption subsumption principle principle is is supported supported by by the the following following theorem: theorem:

Theorem. If Fr is unsatisfiable unsatisfiable and and if if C C cc D, D , then then F' r' = (F (r \\ {D}) {D}) U {C} is also T h e o r e m . If U {C} unsatisfiable. Furthermore, Furthermore, F' r' has has a resolution resolution refutation refutation which is no longer than than the unsatisfiable. shortest refutation refutation of r. shortest of F. 1.3.5.2. Robinson [1965a] [1965a] introintro 1.3.5.2. PPositive o s i t i v e rresolution e s o l u t i o n aand n d hhyperresolution. yperresolution. Robinson duced positive positive resolution resolution and and hyperresolution. hyperresolution. A A positive positive resolution resolution inference inference is is one one duced in positive clause completeness of in which which one one of of the the hypotheses hypotheses is is aa positive clause.9 The The completeness of positive positive resolution is is shown shown by: by: resolution

Theorem. (Robinson [1965a]) [1965a]) If r has a refutation refutation contain T h e o r e m . (Robinson If r F is unsatisfiable, then F containing only positive positive resolution resolution inferences. inferences. Proof. It will will suffice suffice to to show show that that itit is is impossible impossible for unsatisfiable Fr to to P r o o f . (Sketch) (Sketch) It for an an unsatisfiable be closed under under positive inferences and contain the the empty Let be closed positive resolution resolution inferences and not not contain empty clause. clause. Let A be be the the set set of clauses in in F; r; A A must must be be non-empty since Fr is unsatisfiable. A of positive positive clauses non-empty since is unsatisfiable. Pick aa truth truth assignment T that es all in A and assigns assigns the the minimum Pick assignment T that satisfi satisfies all clauses clauses in A and minimum r \\ A A which which is is falsified falsified by possible number of of "true" "true" values. values. Pick Pick aa clause clause L possible number L in in F by rT and the minimum of negative negative literals; literals; and and let let ~ one of negative and has has the minimum number number of p be be one of the the negative L . Note Note that L exists exists since since Fr is is unsatisfiable unsatisfiable and that T(p) T(p) -= True. Pick Pick literals in in L. literals that L and that clause JJ EE A A that that contains the rest its members by TT;; aa clause contains pp and and has has the rest of of its members assigned assigned false false by such such aa clause clause J J exists exists by by the the choice choice of of T T.. Considering Considering the the resolvent resolvent of of J J and and L L,, we we obtain obtain aa contradiction. contradiction. D []

Positive resolution is Positive resolution is nice nice in in that that it it restricts restricts the the kinds kinds of of resolution resolution refutations refutations that that need need to to be be attempted; attempted; however, however, it it is is particularly particularly important important as as the the basis basis for for hyperres hyperresbasic idea idea behind olution. The The basic behind hyperresolution hyperresolution is is that that multiple multiple positive positive resolution resolution inferences conclusion. To inferences can can be be combined combined into into a a single single inference inference with with aa positive positive conclusion. To justify that ifif R is aa positive justify hyperresolution, hyperresolution, note note that R is positive resolution resolution refutation refutation then then the the inferences inferences in in R R can can be be uniquely uniquely partitioned partitioned into into subproofs subproofs of of the the form form B A A1l BII A B A22 B22 A3 B3 Aa Ba B4 B4 A Ann

A An+l n+!

B Bnn

where where each each of of the the clauses clauses AI, A 1 ,.. .. ..,, A An+l are positive positive (and (and hence hence the the clauses clauses B B l1,,.. .. ..,, B Bnn n+ ! are are are not not positive) positive).. These These n + + 11 positive positive resolution resolution inferences inferences are are combined combined into into the the single single hyperresolution inference inference

Introduction to Proof Theory A1 A2 A3

""

23 23

An B1

An+l

((This This construction construction is is the the definition definition of of hyperresolution hyperresolution inferences. inferences.)) It immediately from It follows follows immediately from the the above above theorem theorem that that hyperresolution hyperresolution is is complete. complete. The importance importance of of hyperresolution hyperresolution lies lies in in the the fact fact that that one one can can search search for for refutations refutations The containing only only positive positive resolutions resolutions and and that that as as clauses clauses are are derived, derived, only only the the positive positive containing clauses clauses need need to to be be saved saved for for possible possible future future use use as as hypotheses. hypotheses. Negative resolution resolution is is defined defined similarly similarly to to positive positive resolution resolution and and is is likewise likewise complete. complete. 1.3.5.3. Semantic S e m a n t i c resolution. resolution. Semantic resolution, resolution, independently independently introduced introduced 1.3.5.3. Semantic by Slagle Slagle [1967] [1967] and and Luckham Luckham [1970], [1970], can can be be viewed viewed as as aa generalization generalization of of positive positive by resolution. For For semantic semantic resolution, resolution, one one uses uses aa fixed fixed truth truth assignment assignment ((interpretaresolution. interpreta tion permissible resolution resolution inferences. tion)) T T to to restrict restrict the the permissible inferences. A A resolution resolution inference inference is is said said to be be T-supported T-supported if if one one of of its hypotheses is given value value False by by T T.. Note Note that that at at most most to its hypotheses is given one one hypothesis hypothesis can can have have value value False, since since the the hypotheses hypotheses contain contain complementary complementary occurrences of of the the resolvent resolvent variable. variable. occurrences A A resolution resolution refutation refutation is is said said to to be be T-supported T-supported if if each each of of its its resolution resolution inferences inferences are truth assignment are T-supported. ~--supported. If If T TF is the the truth assignment which which assigns assigns every every variable variable the the value value F is False, then ally the then a a TF-supported Tf-supported resolution resolution refutation refutation is is definition definitionally the same same as as a a positive resolution refutation. positive resolution refutation. Conversely, Conversely, if if r F is is aa set set of of clauses clauses and and if if T T is is any any truth truth assignment, assignment, then then one one can can form form aa set set r' F' by by complementing complementing every every variable variable in in r F which which has has T-value T-value True: clearly, clearly, aa T-supported T-supported resolution resolution refutation refutation of of r F is is isomorphic isomorphic to to aa positive positive resolution resolution refutation refutation of of r' U.. Thus, Thus, Theorem Theorem 1.3.5.2 is is equivalent equivalent to to the the following Completeness Completeness Theorem Theorem for for semantic semantic resolution: resolution: following

Theorem. -supported T h e o r e m . For For any TT and r F,, r F is unsatisfiable unsatisfiable if if and only if if r F has a T T-supported resolution resolution refutation. refutation.

It It is is possible possible to to define define semantic-hyperresolution semantic-hyperresolution in in terms terms of of semantic semantic resolution, resolution, just just as hyperresolution hyperresolution was was defined defined in in terms terms of of positive positive resolution. resolution. as 1.3.5.4. Wos, [1965] intro 1.3.5.4. Set-of-support S e t - o f - s u p p o r t resolution. resolution. Wos, Robinson Robinson and and Carson Carson [1965] introduced resolution as principle for guiding aa search duced set-oj-support set-of-support resolution as another another principle for guiding search for for resolution resolution refutations. refutations. Formally, Formally, set set of of support support is is defined defined as as follows: follows: if if r F is is a a set set of of clauses clauses and and satisfiable, then is aa set support for if II II c C r F and and r F \\ II II is is satisfiable, then II II is set of of support for r F;; a a refutation refutation R R of of r F if possibly indirectly is is said said to to be be supported supported by II H if if every every inference inference in in R R is is derived derived ((possibly indirectly)) from An alternative, from at at least least one one clause clause in in II. II. ((An alternative, almost almost equivalent, equivalent, definition definition would would be be to to require require that that no no two two members members of of r F \\ II II are are resolved resolved together. together.)) The The intuitive intuitive idea one should idea behind behind set set of of support support resolution resolution is is that that when when trying trying to to refute refute r F,, one should concentrate concentrate on on trying trying to to derive derive a a contradiction contradiction from from the the part part II II of of r F which which is is not not known known to to be be consistent. consistent. For For example, example, r F \\ II II might might be be aa database database of of facts facts which which is is presumed consistent, and trying to presumed to to be be consistent, and II II a a clause clause which which we we are are trying to refute. refute.

24 24

S. Buss

Theorem. has a T h e o r e m . If If r F is unsatisfiable unsatisfiable and II II is a set of support for .for r F,, then r F has refutation refutation supported by II II..

This This theorem theorem is is immediate immediate from from Theorem Theorem 1.3.5.3. 1.3.5.3. Let Let T T be be any any truth truth assignment assignment which supported by which satisfies satisfies r F \\ II II,, then then a a T-supported T-supported refutation refutation is is also also supported by II II.. The The main main advantage advantage of of set set of of support support resolution resolution over over semantic semantic resolution resolution is is that that it does not it does not require require knowing knowing or or using using aa satisfying satisfying assignment assignment for for r F \\ II. II. 1.3.5.5. A 1.3.5.5. Unit Unit and input input resolution. resolution. A unit unit clause clause is is defined defined to to be be a a clause clause containing aa single literal; aa unit containing single literal; unit resolution inference is is an an inference inference in in at at least least one one

of of the the hypotheses hypotheses is is a a unit unit clause. clause. As As a a general general rule, rule, it it is is desirable desirable to to perform perform unit unit resolutions resolutions whenever whenever possible. possible. If If r F contains contains aa unit unit clause clause {x} {x},, then then by by combining combining unit unit resolutions resolutions with with the the subsumption subsumption principle, principle, one one can can remove remove from from r F every every clause clause which which contains contains x x and and also also every every occurrence occurrence of of x9 from from the the rest rest of of the the clauses clauses in in r F.. (The (The situation situation is is aa little little more more difficult difficult when when working working in in first-order first-order logic, logic, however.) however.) This This completely completely eliminates eliminates the the literal literal x x and and reduces reduces the the number number of of and and sizes sizes of of clauses clauses to to consider. consider. A unit unit resolution resolution refutation refutation is is a a refutation refutation which which contains contains only only unit unit resolutions. resolutions. A Unfortunately, Unfortunately, unit unit resolution resolution is is not not complete: complete: for for example, example, an an unsatisfiable unsatisfiable set set r F with no no unit unit clauses clauses cannot cannot have have aa unit unit resolution resolution refutation. refutation. with An An input input resolution resolution refutation refutation of of r F is is defined defined to to be be a a refutation refutation of of r F in in which which every every Obviously, aa minimal resolution resolution inference inference has has at at least least one one of of its its hypotheses hypotheses in in r F.. Obviously, minimal length length input input refutation refutation will will be be tree-like. tree-like. Input Input resolution resolution is is also also not not complete; complete; in in fact, fact, it it can can refute refute exactly exactly the the same same sets sets as as unit unit resolution: resolution: Theorem. T h e o r e m . (Chang (Chang [1970]) [1970]) A A set set of of clauses clauses has has aa unit unit refutation refutation if if and and only only if if it it has has a input input refutation. refutation. 1.3.5.6. 1.3.5.6. Linear L i n e a r resolution. r e s o l u t i o n . Linear Linear resolution resolution is is aa generalization generalization of of input input resolu resolution being complete: tion which which has has the the advantage advantage of of being complete: aa linear resolution refutation refutation of of r F is is either in aa refutation refutation Al A1,, A A 22,,.. .. ..,, A A,-1, A,n = = 0 0 such such that that each each Ai Ai is is either in r F or or is is obtained obtained n -b A by by resolution resolution from from Ai-l Ai-1 and and Aj Aj for for some some jj < < ii -- I1.. Thus Thus a a linear linear refutation refutation has has

the the same same linear linear structure structure as as an an input input resolution, resolution, but but is is allowed allowed to to reuse reuse intermediate intermediate clauses clauses which which are are not not in in r F.. Theorem. T h e o r e m . (Loveland (Loveland [1970] [1970] and and Luckham Luckham [1970]) [1970]) If If r r is unsatisfiable, then r r has a linear resolution refutation. refutation.

Linear rst proof Linear and and input input resolution resolution both both lend lend themselves themselves well well to to depth-fi depth-first proof search search strategies. Linear still complete conjunction with strategies. Linear resolution resolution is is still complete when when used used in in conjunction with set-of set-ofsupport resolution. support resolution. Further F u r t h e r reading. r e a d i n g . We We have have only only covered covered some some of of the the basic basic strategies strategies for for proposition proposition resolution resolution proof proof search. search. The The original original paper paper of of Robinson Robinson [1965b] [1965b] still still provides provides an an excellent this and excellent introduction introduction to to resolution; resolution; this and many many other other foundational foundational papers papers on on

Introduction Introduction to to Proof Proof Theory Theory

25 25

this this topic topic have have been been reprinted reprinted in in Siekmann Siekmann and and Wrightson Wrightson [1983]. [1983]. In In addition, addition, the the textbooks [1992] give textbooks by by Chang Chang and and Lee Lee [1973], [1973], Loveland Loveland [1978], and and Wos Wos et et al. al. [1992] give aa more description of resolution than more complete complete description of various various forms forms of of resolution than we we have have given given above. above. H o r n clauses. clauses. A A Horn Horn clause clause is is a a clause clause which which contains contains at at most most one one positive positive literal. literal. Horn Qn } Thus ql , . . . , q-~} Thus aa Horn Horn clause clause must must be be of of the the form form {p, (p, q7,..., or of of the the form form {( ~Ql, ,. ." . ," ~-g~} qn } or with n n � _> O0.. If If aa Horn clause is is rewritten rewritten as as sequents sequents of of atomic atomic variables, variables, it it will will with Horn clause have have at at most most one one variable variable in in the the antecedent; antecedent; typically, typically, Horn Horn clauses clauses are are written written in in reverse-sequent reverse-sequent format format so, so, for for example, example, the the two two Horn Horn clauses clauses above above would would be be written written as as implications implications p~

ql,...,qn

and and r ql, . . . , qn"

In In this this reverse-sequent reverse-sequent notation, notation, the the antecedent antecedent is is written written after after the the � r , and and the the commas conjunctions ((A's). /\ 's) . Horn commas are are interpreted interpreted as as conjunctions Horn clauses clauses are are of of particular particular interest interest both because handle many situations and both because they they are are expressive expressive enough enough to to handle many situations and because because deciding the the satisfiability satisfiability of of sets sets of of Horn clauses is is more more feasible feasible than than deciding deciding the the deciding Horn clauses satisfiability arbitrary sets clauses. For reasons, many logic programming satisfiability of of arbitrary sets of of clauses. For these these reasons, many logic programming environments PROLOG are environments such such as as PROLOG are based based partly partly on on Horn Horn clause clause logic. logic. In In propositional propositional logic, logic, it it is is an an easy easy matter matter to to decide decide the the satisfiability satisfiability of of aa set set of Horn clauses; of Horn clauses; the the most most straightforward straightforward method method is is to to restrict restrict oneself oneself to to positive positive unit in which one of unit resolution. resolution. A A positive positive unit unit inference inference is is aa resolution resolution inference inference in which one of the hypotheses the hypotheses is is aa unit unit clause clause containing containing aa positive positive literal literal only. only. A A positive positive unit unit refutation refutation is is a a refutation refutation containing containing only only positive positive unit unit resolution resolution inferences. inferences. Theorem. positive unit T heorem. A A set set of of Horn Horn clauses clauses is is unsatisfiable unsatisfiable if if and and only only if if it it has has aa positive unit resolution resolution refutation refutation.. Proof. unsatisfiable set clauses. rF must P r o o f . Let Let r F be be an an unsatisfiable set of of Horn Horn clauses. must contain contain at at least least one one positive positive unit unit clause clause {p} {p},, since since otherwise otherwise the the truth truth assignment assignment that that assigned assigned False False to to all variables variables would would satisfy satisfy r F.. By By resolving resolving {p} {p} against against all all clauses clauses containing containing p ~,, and and all then p, one then discarding discarding all all clauses clauses which which contain contain p p or or ~, one obtains obtains aa smaller smaller unsatisfiable unsatisfiable set set of of Horn Horn clauses. clauses. Iterating Iterating this this yields yields the the desired desired positive positive unit unit refutation. refutation. 0 [:]

Positive resolutions are Positive unit unit resolutions are quite quite adequate adequate in in propositional propositional logic, logic, however, however, they they do do not not lift lift well well to to applications applications in in first-order first-order logic logic and and logic logic programming. programming. For For this, this, aa more more useful useful method method of of search search for for refutations refutations is is based based on on combining combining semantic semantic resolution, linear resolution, linear resolution resolution and and set-of-support set-of-support resolution: resolution: 1.3.5.7. Henschen 1.3.5.7. Theorem. Theorem. Henschen and and Wos Wos [1974]. [1974]. Suppose Suppose r F is is an an unsatisfiable unsatisfiable set set of of Horn Horn clauses clauses with with IT H � C_ r F aa set set of of support support for for r F,, and and suppose suppose that that every every clause clause in in r F \\ IT H contains contains aa positive positive literal. literal. Then Then r F has has aa refutation refutation which which is is simultaneously simultaneously aa negative negative resolution resolution refutation refutation and and aa linear linear refutation refutation and and which which is is supported supported by by IT II..

s. Buss S.

26

Note that that the the condition condition that that every every clause clause in in Fr \\ II contains contains aa positive positive literal literal means means Note True to to every satisfies Fr \\ H. II . Thus Thus that the the truth truth assignment assignment rr that that assigns that assigns True every variable variable satisfies aa negative negative resolution resolution refutation refutation isis the the same same as as aa T-supported r-supported refutation refutation and and hence hence is is II . supported by by H. supported The theorem theorem is is fairly fairly straightforward straightforward to to prove, prove, and and we we leave leave the the details details to to the the The II is is presumed presumed to to contain contain aa reader. However, However, note note that that since since every clause in in Fr \\ H reader. every clause positive literal, literal, itit is is impossible impossible to to get get rid rid of of all all positive positive literals literals only only by by resolving resolving positive C such such that that II . Therefore, Therefore, H II must contain aa negative clause C against clauses clauses in in Fr \\ H. against must contain negative clause there resolves against against clauses \ II C , always always resolves clauses in in Fr\II there is is aa linear linear derivation derivation that that begins begins with with C, yielding yielding negative negative clauses clauses only, only, and and ending ending with with the the empty empty clause. clause. The The resolution resolution refutations of of Theorem Theorem 1.3.5.7, 1.3.5.7, or or rather rather the the lifting lifting of of these these to to Horn Horn clauses clauses described described refutations in in section section 2.6.5, can can be be combined combined with with restrictions restrictions on on the the order order in in which which literals literals are are resolved to give what is is commonly commonly called called SLD-resolution. SLD-resolution. resolved to give what 2. Proof 2. P r o o f theory t h e o r y oof f ffirst-order i r s t - o r d e r logic logic

2.1. Syntax Syntax and 2.1. and semantics semantics

First-order logic logic is is aa substantial substantial extension extension 2.1.1. of first-order first-order logic. logic. First-order 2.1.1. Syntax S y n t a x of of propositional logic, and and allows allows reasoning reasoning about about individuals using functions functions and and of propositional logic, individuals using predicates that that act act on on individuals. individuals. The The symbols symbols allowed allowed in in first-order first-order formulas predicates formulas include the the propositional propositional connectives, quantifiers, variables, symbols, con include connectives, quantifiers, variables, function function symbols, constant symbols symbols and and relation relation symbols. symbols. We We take :J as the allowed stant take -~, 1\, A, V V and and D as the allowed propositional connectives. connectives. There There is symbols; we use propositional is an an infinite infinite set set of of variable variable symbols; we use x , yy,, zz,, . .., . and and a,b,c,.., as metasymbols for variables. The quantifiers quantifiers are are the the x, a, b, c, . . . as metasymbols for variables. The existential ers, (3x) existential quantifi quantifiers, (3x),, and and the the universal universal quantifiers, quantifiers, (V'x) (Vx),, which which mean mean "there "there exists x" and ". A exists x" and "for "for all all x x". A given given first-order first-order language language contains contains aa set set of of function function g, h, symbols symbols of of specified specified arities, arities, denoted denoted by by metasymbols metasymbols j f,, g, h , .. ... . and and aa set set of of relation relation Q, R, ..... . . . Function symbols symbols of of specified specified arities, arities, denoted denoted by by metasymbols metasymbols P, P, Q, Function symbols symbols of of arity arity zero zero are are called called constant constant symbols. Sometimes Sometimes the the first-order first-order language language contains contains aa distinguished distinguished two-place two-place relation relation symbol symbol = = for for equality. equality. built up The first-order logic logic are Firstly, terms The formulas formulas of of first-order are defined defined as as follows. follows. Firstly, terms are are built up from symbols, constant from function function symbols, constant symbols symbols and and variables. variables. Thus, Thus, any any variable variable x x is is aa tk are term. Second, term, and tk) is t I , . . . ,,tk) term, and if if tt lI ,, ..... . ,,tk are terms terms and and j f is is k-ary, k-cry, then then j( f(tl,... is aa term. Second, atomic formulas formulas are are defined defined to to be be of of the the form form P(t P(tl,..., for P P aa k-ary k-cry relation relation I , . . . , tk) for symbol. symbol. Finally, Finally, formulas formulas are are inductively inductively to to be be built built up up from from atomic atomic formulas formulas and and logical logical connectives; connectives; namely, namely, any any atomic atomic formula formula is is aa formula, formula, and and if if A A and and B B are are formulas, A ) , (A I\ B) , (AV formulas, then then so so are are (..., (~A), (AAB), (A V E) B),, (A (A :J D B) S),, (V'x)A (Vx)A and and (3x)A (3x)A.. To To avoid avoid writing writing too too many many parentheses, parentheses, we we adopt adopt the the conventions conventions on on omitting omitting parentheses parentheses in in propositional propositional logic logic with with the the additional additional convention convention that that quantifiers quantifiers bind bind more more tightly tightly than than binary binary propositional propositional connectives. connectives. In In addition, addition, binary binary predicate predicate symbols, symbols, such such as as = - and and < <,, are are frequently frequently written written in in infix infix notation. notation. Consider (x =1= Consider the the formula formula (x (x = = 0V V (V'x) (Yx)(x ~- ff (x))) (x))).. (We (We use use x x =1= -~ ff (x) to to abbreviate abbreviate (...,x (-~x = = ff (x)) (x)).).) This This formula formula uses uses the the variable variable x x in in two two different different ways: ways: on on one one hand, hand, it it ..."

.

Introduction to Proof Proof Theory Theory Introduction

27 27

asserts something something about about an an object object x x;; and and on on the the other other hand, hand, it it (re (re)uses the variable variable x asserts )uses the to state state aa general general property property about about all all objects. objects. Obviously Obviously it it would would be be less less confusing confusing to to to write instead instead (x = = 0V V (\ly)(y (Vy)(y f= ~= f(y))) f(y)));; however, however, there there is is nothing nothing wrong, wrong, formally formally write speaking, with with using using x in in two two ways ways in in the the same same formula. formula. To To keep keep track track of of this, this, we we speaking, need to to define define free free and bound occurrences occurrences of of variables. variables. An An occurrence occurrence of of aa variable variable x need and bound in aa formula formula A is is defined defined to to be be any any place place that that the the symbol symbol x occurs occurs in in A A,, except except in in in quantifer (- . . ) quantifer symbols symbols (Qx) (Qx).. (We (We write write (Qx) to to mean mean either either (\Ix) (Vx) or or (3x) (3x).).) If If (Qx) (Qx)(...) is aa subformula subformula of of A A,, then then the the scope scope of of this this occurrence occurrence of of (Qx) is is defi defined to be be the the is ned to . . . ) . An subformula denoted denoted ((...). An occurrence occurrence of of x in in A is is said said to to be be bound bound if if and and only only if if subformula it is is in in the the scope scope of of aa quantifier quantifier (Qx) (Qx);; otherwise otherwise the the occurrence occurrence of of x is is called called free. free. If If it is aa bound bound occurrence, occurrence, it it is is bound bound by the the rightmost rightmost quantifier quantifier (Qx) which which it it is is in in x is the the scope scope of. of. A A formula formula in in which which no no variables variables appear appear freely freely is is called called aa sentence. The intuitive intuitive idea idea of of free free occurrences occurrences of of variables variables in in A A is is that that A A says says something something The about about the the free free variables. variables. If If tt is is a a term, term, we we define define the the substitution of t for x in A A,, denoted A(t/x) A(t/x) to to be be the the formula formula obtained obtained from from A by by replacing replacing each each free free occurrence occurrence denoted of x in in A by by the the term term tt.. To To avoid avoid unwanted unwanted effects, effects, we we generally generally want want t to to be be of for x x,, which which means means that that no no free free variable variable in in t becomes becomes bound freely substitutable for bound in A as as a a result result of of this this substitution; substitution; formally formally defi defined, this means means that that no no free free ned, this in occurrence of of x in occurs in scope of of aa quantifier quantifier (Qy) with with y a a variable variable in A occurs in the the scope occurrence occurring in . . . , tk for Xl, occurring in tt.. The The simultaneous substitution of tl, tl,...,tk Xl,...,Xk A,, . . . , Xk in A denoted A(ttfXI A(tl/Xl,..., tk/Xk),, is is defined defined similarly similarly in in the the obvious obvious way. way. denoted , . . . , tk/Xk) To simplify simplify notation, notation, we we adopt adopt some conventions for for denoting denoting substitution. substitution. To some conventions Firstly, Firstly, if if we we write write A(x) and and A(t) in in the the same same context, context, this this indicates indicates that that A A= = A(x) is a a formula, formula, and and that that A(t) is is A(t/x) A(t/x).. Secondly, Secondly, if if we we write write A(s) and and A(t) in in the the is same context, context, this this is is to to mean mean that that A is is formula, formula, x is is some some variable, variable, and and that that A(s) is is same A(s/x) and A(t/x). . A(s/x) and A(t) is is A(t/x) The sequent calculus, discussed in in 2.3, different conventions The sequent calculus, discussed 2.3, has has different conventions on on variables variables than than the systems discussed in 2.2; has distinct classes of of the Hilbert-style Hilbert-style systems discussed in 2.2; most most notably, notably, itit has distinct classes symbols for for free variables and and bound bound variables. variables. See See section section 2.3.1 2.3.1 for the discussion discussion of symbols free variables for the of the usage of calculus. the usage of variables variables in in the the sequent sequent calculus.

In define the semantics, In this this section, section, we we define the semantics, or of first-order Since this is really really model model theory or 'meaning', 'meaning', of first-order formulas. formulas. Since this is theory instead instead of of proof proof theory, and and since semantics of of first-order first-order logic logic is is well-covered well-covered in in any any introductory introductory theory, since the the semantics textbook on mathematical logic, we we give give only only a a very very concise concise description description of of the the textbook on mathematical logic, notation and used in notation and conventions conventions used in this this chapter. chapter. In order order to to ascribe ascribe aa truth truth value to aa formula, formula, itit is is necessary necessary to to give give an an interpreinterpre In value to tation of of the the non-logical non-logical symbols symbols appearing appearing in in it; it; namely, namely, we we must must specify specify aa domain tation or meanings to each variable or universe of of objects, objects, and and we we must must assign assign meanings to each variable occurring occurring freely each function relation symbol appearing in freely and and to to each function symbol symbol and and relation symbol appearing in the the formula. formula. A A structure ~4 M (also (also called called an an interpretation), for for aa given given language language L, L , consists consists of of the the following: following: A non-empty non-empty universe M of of objects, objects, intended intended to to be be the the universe of objects objects over over (1) universe M universe of (1) A which which variables variables and and terms terms range; range; 2.1.2. of first-order logic. 2.1.2. Semantics S e m a n t i c s of f i r s t - o r d e r logic.

28

s. S. Buss

(2) For each each k-ary k-cry function function f f of of the the language, language, an an interpretation interpretation (2) For and and

fM f ~ : 9M M kk H ~ M M;;

(3) For each each k-ary k-cry relation relation symbol symbol P P of of the the language, language, an an interpretation interpretation pM P~ � C_ M M kk (3) For containing all hold. If containing all k-tuples k-tuples for for which which P P is is intended intended to to hold. If the the first-order first-order language language contains contains the the symbol symbol for for equality, equality, then then = - ~M must must be be the the true true equality equality predicate on predicate on M M..

We We shall shall next next define define the the true/false true/false value value of of a a sentence sentence in in a a structure. structure. This This is is possible possible since since the the structure structure specifies specifies the the meaning meaning of of all all the the function function symbols symbols and and relation relation symbols symbols in in the the formula, formula, and and the the quantifiers quantifiers and and propositional propositional connectives connectives take take their their M aa structure, usual usual meanings. meanings. For For A A a a sentence sentence and and Ad structure, we we write write M j~4 F ~ A ,4 to to denote denote A A being being true true in in the the structure structure M .h4.. In In this this case, case, we we say say that that M A/[ is is aa model model of of A A,, or or that that A A is is satisfied satisfied by by M Ad.. Often, Often, we we wish wish to to talk talk about about the the meaning meaning of of a a formula formula A A in occur. For this, we structure, but in which which free free variables variables occur. For this, we need need not not only only a a structure, but also also an an which is is a a mapping mapping 0' a from from the the set set of of variables variables (at (at least least the the ones ones object assignment, which free free in in A A)) to to the the universe universe M M. . The The object object assignment assignment 0' a gives gives meanings meanings to to the the freely freely being true occurring occurring variables, variables, and and it it is is straightforward straightforward to to define define the the property property of of A A being true in in a a given given structure structure M J~4 under under a a given given object object assignment assignment 0' a,, denoted denoted M Ad F ~ A[a] A[a].. To definition of [a] , we To give give the the formal formal definition of M .M F ~ A A[a], we first first need need to to define define the the interpre interpretation terms, i.e, ne the tation of of terms, i.e, we we need need to to formally formally defi define the manner manner in in which which arbitrary arbitrary terms terms ne tM [0'] by induction on represent objects in represent objects in the the universe universe M M.. To To this this end, end, we we defi define t~[a] by induction on the [0'] is the complexity complexity of of tt.. For For x x a a variable, variable, xM x~[a] is just just a(x) a(x).. For For tt a a term term of of the the form form define t~[a] tM [a] to equal fM (t{'f [0'], . . . , tt~[a]). ;' [a]) . If f(tl , . . . , tk) f(tl,..., tk),, we we define to equal f~(tl~[a],..., If tt is is a a closed closed term, term, i.e., contains no [a] is independent of denoted by just tM i.e., contains no variables, variables, then then tM t~[a] is independent of 0' a and and is is denoted by just t ~ .. If If 0' a is is an an object object assignment assignment and and m m E EM M,, then then a(m/x) a(m/x) is is the the object object assignment assignment which identical to which is is identical to 0' a except except that that it it maps maps x x to to m m.. We We are are now now ready ready to to define define the induction on the truth truth of of A A in in M AJ with with respect respect to to 0' a,, by by induction on the the complexity complexity of of A A.. For For A atomic formula tk) , then holds if only if A an an atomic formula P(tl' P ( t l , ..... ., , tk), then M ,h4 F ~ A[a] A[a] holds if and and only if the the k-tuple k-tuple (t{'f[a], . . . , t;' [a]) is is in F ...,-~A[a] A [a] holds [a] , where (tl~[a],..., tk~[a]) in pM P ~ . . Also, Also, M Ad ~ holds if if and and only only M j~4 jt ~ A A[a], where jt is A0 [a] with is the the negation negation of of F ~.. Likewise, Likewise, the the value value of of M .M F~ A | B B[a] with 0 | one one of of the the binary binary propositional propositional connectives connectives depends depends only only on on the the truth truth values values of of M M F ~ A[a] A[a] and in the obvious way and M A/[ F ~ B[a] B[a],, in the obvious way according according to to Table Table 11 above. above. If If A A is is (Qx)B (Qx)B with denoting either Y, then [a] holds with Q Q denoting either :3 3 or or V, then M M F ~ A A[a] holds if if and and only only if if the the property property M [a(m/x)] holds holds for .M F~ B B[a(m/x)] for some some (respectively, (respectively, for for all) all) m m E EM M.. We We say say that that a a formula formula A A is is valid valid in in M A4,, if if M A/[ F ~ A[a] A[a] holds holds for for all all appropriate appropriate object ned to object assignments. assignments. A A formula formula is is defi defined to be be valid valid if if and and only only if if it it is is valid valid in in all all structures. structures. If If r F is is a a set set of of formulas, formulas, we we say say r F is is valid valid in in M A/I,, written written M Ad F ~ r F,, if if and and only only if if every every formula formula in in r F is is valid valid in in M M. . When When M J~4 F ~ r F,, we we say say that that r F is is satisfied satisfied by M . The set r is satisfiable if and only if it is satisfied by some model. by A/[. The set F is satisfiable if and only if it is satisfied by some model. For A, For r F a a set set of of formulas formulas and and A A a a formula, formula, we we define define r F logically logically implies implies A A,, r F F~ A, to hold if which rF is to hold if A A is is valid valid in in every every structure structure in in which is valid. valid. If If r F is is the the empty empty set, set, we we A , which which means means of that A write write just just F ~ A, of course course that A is is valid. valid, 3a 3This definition definition of r F FI=A A has the slightly unexpected result that free free variables variables appearing in r F are treated as if they are universally universally quantified. quantified. For example, example, according to our definition, definition, we have

Introduction to Proof Theory

29 29

2.1.3. For 2.1.3. For languages languages which which contain contain the the equality equality symbol, symbol, = - , , the the interpretation interpretation of of = ~ in in any any structure structure M A/[ must must be be true true equality. equality. Of Of course, course, this this restriction restriction influences influences =M the the definition definition of of r F F ~ A A in in languages languages which which contain contain equality. equality. Let Let F' ~ indicate indicate logical logical

implication implication defined defined with with respect respect to to all all structures, structures, including including structures structures in in which which equality is is not not interpreted interpreted by by true true equality. equality. It It is is an an elementary, elementary, but but important, fact equality important, fact that F ~ can can be be defined in terms of P~. let EQ EQ be be the the set set of of equality that defined in terms of F'. Namely, Namely, let equality axioms axioms defined in section section 2.2.1 below; below; then A holds holds if if and and only if r, F, EQ EQ F' ~ A A holds. holds. defined in then rF FP A only if

A rst-order theory A fi first-order theory is is a a set set T T of of sentences sentences closed closed under under logical logical implication; implication; i.e., i.e., if if T T F PA A then then A A E ET T.. An An axiomatization axiomatization of of T T is is a a set set r F of of sentences sentences logically implied implied by such such that that T T is is precisely precisely the the set set of of sentences sentences logically by r. F. 2.1.4. 2.1.4. Definition. Definition.

2.2. 2.2. Hilbert-style H i l b e r t - s t y l e proof p r o o f systems systems 2.2.1. A A proof p r o o f system. system. We give give here here an an example example of of a a proof proof system system FF :FRO for 2.2.1. We O for fifirst-order rst-order logic, logic, which logic. In addition to which is is an an extension extension of of F $" to to first-order first-order logic. In addition to the the axioms axioms of of F $" and and the the rule rule modus modus ponens, ponens, there there are are two two axiom axiom schemes: schemes:

A(t) J D (3x)A(x) (3x)A(x) A(t)

and and

(Vx)A(x) J D A(t) A(t) (\fx)A(x)

where A A may may be be any any formula formula and and tt any any term. term. There There are are also also two two quantifier quantifier rules rules of of where inference: inference: C A(x) C J D A(x) A(x) and A(x) J DC C and C (3x)A(x) CJ D (\fx)A(x) (Vx)A(x) (3x)A(x) J DC C where, both inferences, where, in in both inferences, x x may may not not appear appear freely freely in in C C.. If If the the first-order first-order language language under under consideration consideration contains contains the the distinguished distinguished equality equality sign (( = = )), , then then the the equality equality axioms axioms must must also also be be included; included; namely, namely, sign ( W ) ((x x = = x) ~) (\fx) = (\fX) ( v ~ ) ((\f v ~i!) ) ( (Xl ~l = - Yl v, 1\ A . .. .. . 1\ ^ Xk xk = = Yk y~ J ~ / (f~ (x) ) = / (f( b )i!) ) = J P( . . . (\fX) (\fi!) (Xl = (v~)(v~7)(Xl = Yl y, 1\ A . . . 1\ A Xk xk = Yk y~ 1\ A P(x) P ( ~ ) D p(y-*)) i!) )

where their arity. where f f and and P P are are arbitrary arbitrary function function and and predicate predicate symbols symbols and and k k is is their arity. We We leave leave it it to to the the reader reader to to check check that that the the symmetry symmetry and and transitivity transitivity of of equality equality follow third equality follow from from these these axioms, axioms, since since in in the the third equality axiom, axiom, P P may may be be the the equality equality sign. We A having sign. We write write FFO $'FO f~- A A to to denote denote A having an an FFO-proof. JcFO-proof. A(x) F~ (Vx)A(x) (Vx)A(x),, and and thus thus A(x) F~ A(y) A(y).. An An alternative alternative definition definition of of logical logical implication implication is is often often used (but not this chapter!) which free used (but not in in this chapter!) in in which free variables variables occurring occurring in in r F are are treated treated as as syntactically syntactically as constants; under this definition definition A(x) F~ (Vx)A(x) (Vx)A(x) does does not hold hold in general. as The advantage our choice choice for the definition definition of F~,, is that it yields yields a simpler and more elegant proof-theory. since, instead instead of proof-theory. This This choice choice does does not not involve involve any any loss loss of of expressive expressive power power since, of free free variables one can can use use new new constant constant symbols, thus obtain variables in in r, F, one symbols, and and thus obtain the the same same effect effect as as the the alternative alternative definition definition of of F~.. In In any any event, event, the the two two definitions definitions of of F~ coincide coincide when when r F is is aa set set of of sentences.

s. S. Buss Buss

30 30 2.2.2. 2.2.2. The T h e Soundness S o u n d n e s s Theorem. Theorem.

(1) (1) If If FFO $'FO Ik- A A,, then then F ~A A.. set of -proof of (2) (2) Let Let r F be be aa set of sentences. sentences. If If there there is is an an FFO ~'FO-proof of A A using using sentences sentences from from rF as as additional additional axioms, axioms, then then r FF ~A A.. The Soundness Theorem The Soundness Theorem states states that that FFO $'FO proves proves only only valid valid formulas. formulas. Both Both parts parts of induction on of the the soundness soundness theorem theorem are are readily readily proved proved by by induction on the the number number of of lines lines in in aa proof. proof. 2.2.3. 2.2.3. The T h e Completeness C o m p l e t e n e s s Theorem. Theorem.

(1) (1) If If F ~A A,, then then FFO .T'FO I~A A.. -proof of (2) (2) Let Let r F be be aa set set of of formulas. formulas. If If r F F ~ A A,, then then there there is is an an FFO ~FO-proof of A A using using sentences sentences from from r F as as additional additional axioms. axioms. The The completeness completeness theorem theorem and and soundness soundness theorem theorem together together show show that that FFO JcFO is is an an adequate logic. The theorem was adequate system system for for formalizing formalizing first-order first-order logic. The completeness completeness theorem was proved originally by G5del [1930]; textbook proof proved originally by GSdel [1930]; the the now-standard now-standard textbook proof is is due due to to Henkin Henkin [1949] [1949].. We We shall shall not not give give a a proof proof for for the the completeness completeness of of FFO $'FO here; here; instead, instead, we we LK shall shall prove prove the the completeness completeness of of the the cut-free cut-free fragment fragment of of L K in in 2.3.7 2.3.7 below. below. It It is is straightforward straightforward to to see see that that FFO $'FO can can simulate simulate the the LK L K proof proof system, system, and and this this gives gives an indirect proof an indirect proof of of the the completeness completeness of of FFO 9vFO.. 2.2.4. 2.2.4. A A set set r F of of sentences sentences is is consistent consistent if if and and only only if if there there is is no no sentence sentence A A such such that both A A are and, equivalently, that both A and and ..., -~A are provable provable from from r F and, equivalently, if if and and only only if if there there is is some some formula formula A A such such that that there there is is no no proof proof of of A A from from r F.. The The soundness soundness and and completeness completeness theorems theorems immediately immediately imply imply that that r F is is consistent consistent if if and and only only if if r F is is satisfiable. satisfiable. Frege Frege [1879] [1879] gave gave the the first first full full formulation formulation of of first firstorder order logic. logic. Frege Frege used used aa pictorial pictorial representation representation of of propositional propositional connectives connectives and and "I- A" quantifiers; quantifiers; one one remnant remnant of of his his notation notation that that is is still still in in use use is is "kA",, which which was was Frege's Frege's notation notation for for "A "A is is asserted asserted to to be be true" true".. Frege Frege used used the the pictorial pictorial notation notation ~ A A A to proposition A notation I-to express express aa proposition A;; whereas whereas he he used used the the notation I A to to express express the the assertion or has ) . Negation, assertion that that the the proposition proposition A A is is true true ((or has been been proved proved). Negation, implication, implication, and and universal universal quantification quantification were were represented represented by by Frege Frege with with pictures pictures such such as as 2.2.5. 2.2.5. Historical Historical remarks. remarks.

--

--,A, i A,

L� L BA

and and

~ --&-

A A,,

which A, B A , respectively. which represent represent the the propositions propositions ..., ~A, B :::> D A A,, and and (V'x) (Vx)A, respectively. These These constructions constructions can can be be iterated iterated to to form form arbitrary arbitrary first-order first-order expression; expression; that that is is to to say, say, be replaced in the the pictures in pictures above, above, A A and and B B may may be replaced by by arbitrary arbitrary pictorial pictorial propositions. propositions. The The addition addition of of aa vertical vertical line line to to the the left left end end of of aa proposition proposition indicates indicates that that the the e.g., proved ) . Thus, proposition proposition has has been been established established to to be be true true ((e.g., proved). Thus, for for instance instance

Introduction to Proof Proof Theory Theory Introduction i i ~-v--A B

31 31

and and

I i , ,~r-- A I B denote the the proposition proposition B B :J D (3x)A (3x)A and and the the assertion assertion that that this this proposition proposition has has been been denote established, respectively. respectively. established, Frege also also developed developed extensions extensions to to his his first-order first-order logic logic to to include include functionals functionals and and Frege predicates; however, however, these these extensions extensions were were later later discovered discovered by by Russell Russell to to introduce introduce predicates; set-theoretic paradoxes. paradoxes. set-theoretic Peano [1889] introduced, independently independently of of Frege, Frege, aa fragment fragment of of first-order logic Peano [1889] introduced, first-order logic for reasoning reasoning about about integers. integers. Based Based on on the the work work of of Frege Frege and and Peano, Peano, Whitehead Whitehead for and Russell Russell [1910] [1910] gave gave aa consistent consistent framework framework for for formalizing formalizing mathematics mathematics based based and on first-order first-order logic, logic, using using universal universal and and existential existential quantifiers quantifiers denoted denoted (x) (x) and and (3x) (3x).. on The 'Hilbert-style 'Hilbert-style'' system system given given above above is is apparently apparently so-named so-named because because it it is is closely closely The related to to aa system system used used by by Hilbert and Ackermann Ackermann [1928] [1928],' which which is is based based on on earlier earlier related Hilbert and lectures of of Hilbert. Hilbert. The The later later work work of of Hilbert Hilbert and and Bernays Bernays [1934-39] [1934-39],' however, however, lectures used the the f-calculus e-calculus instead of a a 'Hilbert-style 'Hilbert-style'' system. system. The The f-calculus e-calculus contains contains no no used instead of quantifiers, but but in in their their place place uses uses symbols which are intended to denote quantifiers, symbols fex(A(x)) (A(x)) which are intended to denote x an object object x x such such that that A(x) A(x) holds, holds, if if there there is is such such an an object. Note that that other other free free object. Note an variables and and other other uses uses of of fe symbols symbols may may appear appear in in A(x) A(x).. The The e-symbol can be be variables f-symbol can used to to express express quantifiers quantifiers by by the the informal informal equivalences equivalences (3x)A(x) (3x)A(x) {:} 4:~ A(f A(ex(A(x))) used x (A(x))) and ('v'x)A(x) (Vx)A(x) {:} ~ A(f A(e~(~A(x))). and x (..,A(x))) . The The Hilbert-style Hilbert-style system system we we used used above above is is essentially essentially that that of of Kleene Kleene [1952] [1952].. 2.3. The T h e ffirst-order i r s t - o r d e r sequent s e q u e n t calculus calculus 2.3. In this sequent calculus, section 1.2, is In this section, section, the the propositional propositional sequent calculus, introduced introduced in in section 1.2, is enlarged to system for enlarged to aa proof proof system for first-order first-order logic. logic. 2.3.1. Free Free and a n d bbound o u n d variables. The first-order first-order sequent calculus has variables. The sequent calculus has two two 2.3.1. classes and bound bound variables. variables. There There are infinitely classes of of variables, variables, called called free free variables variables and are infinitely many variables each type; type; free are denoted denoted by the metavariables many variables of of each free variables variables are by the metavariables a, b, y, x, The essential b, cc,, .. ... ., , and and bound bound variables variables by by the the metavariables metavariables z, z, y, x, .... . . . . The essential idea idea is is that free not be be quantified, quantified, while while bound bound variables variables may may not occur freely freely that free variables variables may may not not occur in formulas. formulas. The The syntactic free and variables necessitates in syntactic distinction distinction between between free and bound bound variables necessitates aa change to the the definitions of terms terms and Firstly, terms terms are are now now defined defined change to definitions of and formulas. formulas. Firstly, as from free semiterms as being being built built up up from free variables variables and and function function symbols; symbols; whereas, whereas, the the semiterms are free and are defined defined as as being being built built up up from from free and bound bound variables variables and and function function symbols. symbols. Secondly, only only bound bound variables may ever be quantified. quantified. The The set set of of formulas Secondly, variables may ever be now formulas isis now redefined requirement that variables may redefined with with the the additional additional requirement that only only free free variables may occur occur freely freely in in formulas. formulas. Semi.formulas Semiformulas are are like like formulas, formulas, except except that that bound bound variables variables may may occur occur freely We henceforth as metavariables metavariables for for terms, terms, and and freely in in semiformulas. semiformulas. We henceforth use use r,r, s, tt,, ..... . as A, B, C C,, ... .. . as as metavariables metavariables for for formulas. formulas. Note that that in in general, general, aa subformula subformula of of aa formula formula will will be be aa semiformula semiformula instead instead of of Note aa formula. formula.

s. S. Buss Buss

32

One One advantage advantage of of these these conventions conventions on on free free and and bound bound variables, variables, is is that that it it avoids avoids some some of of the the difficulties difficulties involved involved in in defining defining A(t) A ( t ) , , since since it it is is always always the the case case that that the term term tt will be freely freely substitutable substitutable for for bb in in aa formula formula A( A(b). without these these the will be b) . Also, Also, without conventions, the the cut cut elimination elimination theorem theorem proved proved below below would would have have to to be be reformulated reformulated conventions, slightly. slightly. For For example, example, it it is is not not hard hard to to see see that that P(x, P ( x , yy) ) - -� - + ((::Iy) 3 y ) ((::Ix)P(y, 3 x ) P ( y , x x) ) would would this example not not have have a a cut-free cut-free proof proof ((this example is is from from Feferman Feferman [1968]). [1968]). 2.3.2. 2.3.2. Definition. Definition. The The first-order first-order sequent sequent calculus calculus LK L K is is defined defined as as an an extension extension of propositional system of the the propositional system PK. PK. LK L K contains contains all all the the rules rules of of inference inference of of PK P K plus plus the the following following additional additional rules rules of of inference: inference: The T h e Quantifier Q u a n t i f i e r Rules Rules

V:left V:left

r-+A A(t), r�� (Vx)A(x), , r�� F---+A (Vx)A(x)

r��, r--+A, A(b) A(b) V:right V:right r��, (Vx)A(x) F---+A, (Vx)A(x)

3:left ::I:left

A(b) A(b),, r�� F---+A ( 3 x ) A ( x ) , , r�� F---~A (::Ix)A(x)

3:right r��, F---+A, A(t) A(t) ::I:right r��, (::Ix)A(x) F---~A, (3x)A(x)

In In quantifier quantifier rules, rules, A A may may be be an an arbitrary arbitrary formula, formula, tt an an arbitrary arbitrary term, term, and and the the free free variable variable bb of of the the V:right V :right and and ::I3 :left :left inferences inferences is is called called the the eigenvariable eigenvariable of of the the inference inference and and must must not not appear appear in in r, F, � A.. The The propositional propositional rules rules and and the the quantifier quantifier rules rules are are collectively collectively called called logical logical rules. rules. Most of the syntactic defi nitions of PKK carry Most of the syntactic definitions of P carry over over to to LK L K . . For For example, example, the the direct ) descendents notions notions of of ((direct) descendents and and ancestors ancestors are are identically identically defined; defined; and and proof proof lengths lengths are are still still measured measured in in terms terms of of the the number number of of strong strong inferences inferences in in the the proof. proof. The course, considered The quantifier quantifier inferences inferences are, are, of of course, considered strong strong rules rules of of inferences. inferences. If If S S is is a a sequent sequent r F� ---} � A,, then then we we let let A A ss be be the the formula formula (/\ (A r) F) :J D (V (V �) A).. Taking Taking S S to to have have the the same same meaning meaning as as A A ss, , all all the the definitions definitions of of 'validity 'validity'' and and 'logical 'logical implication to sequents. implication'' of of section section 2.1.2 apply apply also also to sequents. Let Let the the free free variables variables of of A A ss be b, As (X) , of be b, so so A A ss = - A As(b-'). We let let VS VS denote denote the the universal universal closure, closure, (Vi) (V~)As(~), of the the s (b) . We formula formula A A ss. . 2.3.3. LK initial sequents �A with 2.3.3. L K is is defined defined to to allow allow only only initial sequents of of the the form form A A----} A with A atomic. However, initial sequents; A atomic. However, it it is is often often convenient convenient to to allow allow other other initial sequents; so so if if <5 | is is aa set set of of sequents, sequents, we we define define LKe; LKG to to be be the the proof proof system system defined defined like like LK L K , , but but allowing allowing initial initial sequents sequents to to be be from from <5 | too. too. An rst-order logic An important important example example is is the the theory theory LKe LKe for for fi first-order logic with with equality. equality. LKe with the addition of L K e is is LK L K with the addition of the the following following initial initial sequents sequents for for equality: equality:

�s - - - ~ S= = Ss s~1 8

� f( = it~, sk = = tk tk---~ f (s-*) = f(i) f (t-) = 8) = t , .. .. .. ,, Sk

Sl = = tl tl,, ·. .. .. ,, Sk sk = = tk t~,, P( P(s-)-'+P(t-) 8) � P(i) Sl

Introduction Introduction to to Proof Proof Theory Theory

33 33

We ---t A(a) Ll(a) is We say say that that a a set set 6 | is is closed closed under under substitution, substitution, if if whenever whenever r(a) F(a)--+ is term, then ---t Ll(t) in in 6 | and and tt is is a a term, then r(t) F(t)---} A(t) is is also also in in 6. | 2.3.4. When 2.3.4. When P P is is aa proof, proof, we we write write P(a) P(a) and and P(t) P(t) to to indicate indicate that that P(t) P ( t ) i is s the the result result of of replacing replacing every every free free occurrence occurrence of of aa in in formulas formulas of of P P with with tt.. Theorem. T h e o r e m . Let Let 6 ~ be be aa set set of of sequents sequents which which is is closed closed under under substitution. substitution. If If P(b) P(b) -proof, and is is an an LKs LK~-proof, and if if neither neither bb nor nor any any variable variable in in tt is is used used as as an an eigenvariable eigenvariable in -proof. in P(b) P(b),, then then P(t) P(t) is is aa valid valid LKs LK~-proof.

2.3.5. of aa proof is called 2.3.5. Definition. Definition. A A free free variable variable in in the the endsequent endsequent of proof is called aa parameter parameter variable variable of of the the proof. proof. A A proof proof P P is is said said to to be be in in free free variable variable normal normal form form provided provided that (1) (1) no no parameter parameter variable variable is is used used as as an an eigenvariable, eigenvariable, and and (2) every other other free free that (2) every variable in P is used once as and appears variable appearing appearing in P is used exactly exactly once as an an eigenvariable eigenvariable and appears in in P P only only in in sequents sequents above above the the inference inference for for which which it it is is used used as as an an eigenvariable. eigenvariable.

In In this this chapter, chapter, we we consider consider only only tree-like tree-like proofs proofs and and thus thus any any proof proof may may be be put put in in free free variable variable normal normal form form by by merely merely renaming renaming variables. variables. 2.3.6. 2.3.6. Soundness S o u n d n e s s Theorem. T h e o r e m . Let Let r F ---t ~ Ll A be be an an arbitrary arbitrary sequent. sequent. -proof, then then rF-----} ---t Ll (1) If If r F ---t ~ Ll A has has an an LK LK-proof, A is is valid. valid. (1) ---t Ll (2) Let (2) Let 6 | be be aa set set of of sequents. sequents. If If r F----} A has has an an LKs LK~ -proof, -proof, then then 6 | F ~r F ---t ~ Ll A.. The induction on The soundness soundness theorem theorem is is readily readily proved proved by by induction on the the number number of of inferences inferences in in a a proof. proof. 2.3.7. completeness of 2.3.7. Completeness C o m p l e t e n e s s of of cut-free c u t - f r e e LK L K . . We We next next prove prove the the completeness of the the cut-free and, more Since our cut-free fragment fragment of of LK L K and, more generally, generally, the the completeness completeness of of LKs L K ~ . . Since our countable) compactness proofs completeness also proofs of of completeness also give give aa proof proof of of the the ((countable) compactness theorem, theorem, we we the completeness include the compactness part (2) of include the compactness theorem theorem as as part of the completeness theorem. theorem. Cut-free C u t - f r e e Completeness C o m p l e t e n e s s Theorem. T h e o r e m . Let Let r F ---t ~ Ll A be be aa sequent sequent in in aa first-order first-order lan language guage L L which which does does not not contain contain equality. equality. -proof. (1) (1) If If r F ---t ~ Ll A is is valid, valid, then then it it has has aa cut-free cut-free LK LK-proof. ---t Ll be aa set (2) Let Let II be set of of sentences. sentences. If If II II logically logically implies implies r F----} A,, then then there there are are (2) ---t Ll G has aa cut-free -proof. C 1l ,, .. .. .., , G Ckk E E II II such such that that G C1,. Ck, A has cut-free LK L K-proof. k . rF--'~ l , . . 9. , G Corollary. be aa set Corollary. Let Let 6 | be set of of sequents. sequents. If If 6 | logically logically implies implies r F ---t ~ Ll A,, then then r F ---t ~ Ll A has -proof, has an an LKs LKG-proof.

Note -proof may Note that that in in the the corollary, corollary, the the LKs LK6-proof may not not be be cut-free. cut-free. An An important important special special case complete. Although, general, the case of of the the corollary corollary is is that that LKe LKe is is complete. Although, in in general, the cut-free cut-free fragment complete; we shall see fragment of of LKs L K s may may not not be be complete; we shall see later later that that it it is is always always possible possible to to get get LKs LK~ proofs proofs which which contain contain no no 'free 'free cuts cuts'.' .

34 34

s. Buss S.

The corollary corollary isis an an immediate immediate consequence consequence of of the the cut-free cut-free completeness completeness theorem. theorem. The � , then then part part (2) (2) of of the the theorem theorem implies implies that that there there are are This isis because, because, ifif |(5 ~1= Pr ~---+ A, This Sk 6E |(5 so so that that VSx,...,VSk, VSI , . . . , VSk , F---~A r ---+ � has has an an LK-proof. LK-proof. But But clearly clearly each each $SI1 ,, .. ... ., , S~ ---+ VSi has has an an LK6-proof, LKs -proof, so so with with kk further further cut cut inferences, inferences, F---> r ---+ A � also also has has an --'+VSi an LKs -proof. -proof. LK~ Proof. Part Part (2) (2) of of the the cut-free cut-free completeness completeness theorem theorem clearly clearly implies implies part part (1); (1); we we Proof. (2) only only for for the the case case where where H II isis countable countable (and (and hence hence the the language language LL is, is, shall prove prove (2) shall w.l.o.g., II logically logically implies implies Fr ~---+ A. � . The The general general idea idea of of the the w.l.o.g., countable). countable). Assume Assume II r ---+ � from the the bottom bottom up, up, working working argument is is to to try try to to build build an an LK-proof LK -proof of of P--~ argument A from r ---+ A � to to initial initial sequents. This is, is, of of course, course, analogous analogous to to the the backwards from backwards from F--+ sequents. This 1 .2.8; but but because because of of the the presence presence of of quantifiers, quantifiers, procedure used used to to prove prove Theorem Theorem 1.2.8; procedure the process process of of searching searching for proof of of F--~ r ---+ A � is is no no longer longer finite. finite. Thus, Thus, we we shall shall the for aa proof have to show show the the proof-search proof-search process process eventually eventually tterminates this will will be be done done by have to e r m i n a t e s- this by � is not valid. showing that if if the the proof-search does not not yield showing that proof-search does yield a a proof, proof, then then r F ---+ ~ A is not valid. the language is countable, countable, we L-formulas as as Since Since the language is we may may enumerate enumerate all all L-formulas AI A2 , Aa,... so that that every every L-formula L-formula occurs occurs infinitely infinitely often often in the enumeration. A3 , A1,, A2, so in the enumeration. Likewise, we enumerate all all L-terms L-terms as t I , t2, t2 , ta,.., h, . . . with repeated infinitely infinitely Likewise, we enumerate as tl, with each each term term repeated ---+ � The construction construction often. We We shall shall attempt to construct cut-free proof proof PP of often. attempt to construct aa cut-free of r F---+ A.. The in stages: initially, PP consists of just the sequent r ---+ A; � ; at of of P P will will proceed proceed in stages: initially, consists of just the sequent F---> at each be modified each stage, stage, P P will will be modified (we (we keep keep the the same same name name P P for for the the new new partially partially constructed constructed P P).) . A A sequent sequent in in P P is is said said to to be be active provided provided it it is is aa leaf leaf sequent sequent of of P P and and there there is is no no formula formula that that occurs occurs in in both both its its antecedent antecedent and and its its succedent. succedent. Note Note that leaf sequent that aa non-active non-active leaf sequent is is derivable derivable with with aa cut-free cut-free proof. proof. diagonal enumeration; enumeration; each We We enumerate enumerate all all pairs pairs (Ai, (Ai, tj) tj) by by aa diagonal each stage stage of of the the construction construction of of P P considers considers one one such such pair. pair. Initially, Initially, P P is is the the single single sequent sequent r P ---+ ~ � A.. At At each each stage stage we we do do the the following: following: • • •

Loop: Loop: Let Let (Ai, (Ai, tj) tj) be be the the next next pair pair in in the the enumeration. enumeration. Step Step (1): (1): If If Ai Ai is is in in II II,, then then replace replace every every sequent sequent r' F' ---+ ---} �' A' in in P P with with the the sequent sequent r', F', Ai Ai ---+ '-} �' A'.. Step (2): If If Ai Ai is is atomic, atomic, do do nothing nothing and and proceed proceed to to the the next next stage. stage. Otherwise, Otherwise, we we will will modify modify P P at at the the active active sequents sequents which which contain contain Ai Ai by by doing doing one one of of the the following: following:

Case B , then Case (2a): (2a): If If Ai Ai is is ..., -~B, then every every active active sequent sequent in in P P which which contains contains Ai Ai,, say say B , r" ---+ �' of of form form r', F', ..., -~B, F"---> A',, is is replaced replaced by by the the derivation derivation r', B , r" F', ..., -~B, F" ---+ ---> �', A', B B F', -~B, F"---+ A' r', ...,B , r" ---+ �' and ---+ �', B , �" and similarly, similarly, every every active active sequent sequent in in P P of of the the form form r' F'--+ A', ..., ~B, A" is is replaced replaced by by the the derivation derivation B, r' P' ---+ ~ �', A', ...,~B, A" B, B , �" F' ---+ ---} �', A', ...,~B, A" r' B , �"

Introduction to Proof Theory

35 35

Case (2b): (2b): IfIf A/ A i isis of Case of the the form form BB V V C, C , then then every every active active sequent sequent in in PP of of the the form F', fl, BB VV C, G, F"--+ f" -t A', /:1' , isis replaced replaced by by the the derivation derivation form B, B, F', fl, BB VV C, G, F" f" --+ -t A' /:1'

C, G, F', fl, BB VV C, G, F" f" --+ -t A' /:1'

F', + A' f', BB VV C, G, F" f" --t /:1'

And, every every active active sequent sequent in in PP of of the the form form F' f' ---} -t A', /:1', BB VV C, G, A" /:1" is is replaced replaced And, by the the derivation derivation by

fl -t A', /:1', BB Vv C, G, A", /:1", B, B, CG F'-+ fl -t A', /:1', BB VV C, G, A" /:1" F'--+ Cases (2c)-(2d): (2c)-(2d): The The cases cases where where Ai A i has has outermost connective D:J or or A1\ are are Cases outermost connective similar (dual) ( dual) to to case case (2b), ( 2b) , and and are are omitted omitted here. here. similar Case (2e): (2e): If If Ai A i is is of of the the form form (3x)B(x), (3x)B(x) , then then every every active sequent in in PP of of the the Case active sequent form F', fl, (3x)B(x), f" -t A' /:1' is is replaced replaced by form (3x)B(x),F"---} by the the derivation derivation B(~), (3~)B(~), , r"--+ zx' B(c), r', fl, (3x)B(x) f" -t /:1' r', r"--+ A, fl, (3z)B(x), (3x)B(x), f" -t /:1' where cc is is a a new new free free variable, not used in P yet. In In addition, any active active where variable, not used in P yet. addition, any sequent of of the the form form f' P'--+ A" is is replaced replaced by by the the derivation derivation -t A', /:1', (3x)B(x), (3x) B (x), /:1" sequent

fl -t A', /:1', (3x)B(x) /:1", B(tj) F'"-+ (3x)B(x),, A", B(tj) f' -t /:1', (3x)B(x) , /:1" P'--+ A', (3x)S(x), A" Note Note that that this, this, and and the the dual dual V V :left :left case, case, are are the the only only cases cases where where tj t3 is is used. used. These These two two cases cases are are also also the the only only two two cases cases where where it it is is really really necessary necessary to to keep keep the the formula formula A Aii in in the the new new active active sequent; sequent; we we have have however, however, always always since it kept kept A Aii since it makes makes our our arguments arguments aa little little simpler. simpler. Case Case (21): (2f): The The case case where where A Aii begins begins with with aa universal universal quantifier quantifier is is dual dual to to 2e) . case case ((2e). Step Step (3): (3): If If there there are are no no active active sequents sequents remaining remaining in in P P ,, exit exit from from the the loop; loop; otherwise, otherwise, continue continue with with the the next next loop loop iteration. iteration. End End loop. loop. If If the the algorithm algorithm constructing constructing P P ever ever halts, halts, then then P P gives gives aa cut-free cut-free proof proof of of -t /:1 Gl non-active) initial C 1, ,.. .. .. ,, Gk, Ck, f F---} A for for some some Gl C 1, ,.. .. .. ,, Gk Ck E 6 II I1.. This This is is since since each each ((non-active) initial sequent sequent has has aa formula formula A A which which appears appears in in both both its its antecedent antecedent and and succedent succedent and and -t AA isis derivable since, since, by by induction induction on on the the complexity complexity of of A A,, every every sequent sequent A A--+ derivable with with aa cut-free cut-free proof. proof. It It remains remains to to show show that that if if the the above above construction construction of of P P never never halts, halts, then then the the -t /:1 sequent sequent fF--+ A is is not not logically logically implied implied by by II I1.. So So suppose suppose the the above above construction construction of of P P never never halts halts and and consider consider the the result result of of applying applying the the entire entire infinite infinite construction construction process. except process. From From the the details details of of the the construction construction of of P P ,, P P will will be be an an infinite infinite tree tree ((except -t /:1 in in the the exceptional exceptional case case where where fF---+ A contains contains only only atomic atomic formulas formulas and and II II is is empty, empty,

36 36

Buss s.S. Buss

in which which case case P P isis aa single single sequent) sequent).. If If II II isis empty, empty, then then each each node node in in the the infinite infinite in tree P P will will be be aa sequent; sequent; however, however, in in the the general general case, case, each each node node in in the the infinite infinite tree tree will will be be aa generalized generalized sequent sequent of of the the form form r/, F ~,II II--~ A ~with with an an infinite infinite number number -+ �' tree of formulas formulas in in its its antecedent. antecedent. (At (At each each stage stage of of the the construction construction of of P P ,, the the sequents sequents of contain contain only only finitely finitely many many formulas, formulas, but but in in the the limit limit the the antecedents antecedents contain contain every every formula from from II II since since these these are are introduced introduced by by step step (1).) (1).) P P is is aa finitely finitely branching branching tree, tree, formula so by by Konig's Khnig's lemma, lemma, there there is is at at least least one one infinite infinite branch branch 7rr in in P P starting starting at at the the roots roots so and proceeding proceeding up up through through the the tree tree (except, (except, in in the the exceptional exceptional case, case, 7rr is is to to contain contain and just the the endsequent). endsequent). We We use use 7rr to to construct construct aa structure structure M Az[ and and object object assignment assignment (Ja,, just for which which rF--~ A is is not not true. true. The The universe universe of of M A/[ is is equal equal to to the the set set of of L-terms. L-terms. for -+ � The object object assignment assignment (Ja just just maps maps aa variable variable aa to to itself. itself. The The interpretation interpretation of of aa The function symbol symbol is is defined defined so so that that ffM(rl,... the term term ff ((rrb l , .. ... ., , rr~). Finally, M (r l, . . . ,, rrk) function k) isis the k ) . Finally, the interpretation interpretation of of aa predicate predicate symbol symbol P P is is defined defined by by letting letting pM P ~ ((rrl,l , .. .. ..,, rra) hold the k) hold and only only the the formula formula P P(rl,..., appears in in an an antecedent antecedent of of aa sequent sequent contained contained ifif and (rb . . . , rrk) k ) appears in the the branch branch 7r r .. in To finish finish the the proof proof of of the the theorem, theorem, it it suffices suffices to to show show that that every every formula formula A A To occurring occurring in in an an antecedent antecedent (respectively, (respectively, aa succedent) succedent) along along 7r r is is true true (respectively, (respectively, false) in in M A4 with with respect respect to to (J a;; since since this this implies implies that that r F---~ A is is not not valid. valid. This This claim claim -+ � false) is proved proved by by induction induction on on the the complexity complexity of of A A.. For For A A atomic, atomic, it it is is true true by by definition. definition. is Consider the the case case where where A A is is of of the the form form (3x)B(x) (3x)B(x). . If If A A appears appears in in an an antecedent, antecedent, Consider then so does does a a formula formula of of the the form form B(c) B(c);; by by the the induction hypothesis, B(c) B(c) is is true true then so induction hypothesis, is. If in in M Ad w.r.t. w.r.t, (J a,, so so hence hence A A is. If A A appears appears in in an an succedent, succedent, then, then, for for every every term term tt,, appears in in an succedent; hence every B(t) B(t) is is false in M w.r.t. (J B(t) eventually appears an succedent; hence every false in Az[ w.r.t, a,, B(t) eventually which implies implies A also false. that A A cannot appear in in both antecedent and which A is is also false. Note Note that cannot appear both an an antecedent and 7r , since since the the nodes nodes in in 7r initial sequents. aa succedent succedent along along ~, ~ were were never never active active initial sequents. The The A , are and we we omit omit their their rest of of cases, for different different outermost outermost connectives connectives of of A, rest cases, for are similar similar and proofs. proofs.

2.3.8. There There are are number number of of semantic semantic tableau tableau proof proof systems, systems, independently independently due due to to 2.3.8. [1956], Hintikka Hintikka [1955], [1955], Kanger Kanger [1957] [1957] and and Schutte [1965], which are very very similar similar Beth [1956], Beth Schiitte [1965], which are to the the first-order first-order cut-free cut-free sequent sequent calculus. calculus. The The proof proof above, above, of of the the completeness completeness of of to the cut-free cut-free sequent sequent calculus, calculus, isis based based on on the the proofs proofs of of the the completeness completeness of of semantic semantic the tableau systems systems given given by by these these four four authors. authors. The The original original proof, proof, due due to to Gentzen, Gentzen, was was tableau based on on the the completeness completeness of of LK LK with with cut cut and and on on aa process process of of eliminating eliminating cuts cuts from from based aa proof proof similar similar to to the the construction construction of of the the next next section. section. 2.4. CCut The cut-free cut-free completeness completeness theorem, theorem, as as established established above, above, 2.4. u t eelimination. l i m i n a t i o n . The has aa couple couple of of drawbacks. drawbacks. Firstly, Firstly, we we have have proved proved cut-free cut-free completeness completeness only only for for has pure LK LK and and itit isis desirable desirable toto also also establish establish aa version version of ofcut-free cut-free completeness completeness (called (called pure 'free-cut free' free ' completeness) completeness) for for LKe LKe and and for for more more general general systems systems LK6. LKs . Secondly, Secondly, 'free-cut the proof proofisis completely completely non-constructive non-constructive and and gives gives no no bounds bounds on on the the sizes sizes of ofcut-free cut-free the proofs. Of Of course, course, the the undecidability undecidability of of validity validity in in first-order first-order logic logic implies implies that that the the proofs. size size of of proofs proofs (cut-free (cut-free or or otherwise) otherwise) cannot cannot be be recursively recursively bounded bounded inin terms terms of of the the formula formula being being proved; proved; instead, instead, we we wish wish toto give give an an upper upper bound bound on on the the size size of of aa

Introduction Introduction to to Proof Proof Theory Theory

37 37

cut-free LK-proof terms of of the of a a general general LK-proof. cut-free LK -proof in in terms the size size of LK -proof. Accordingly, the cut cut elimination Accordingly, we we shall shall give give a a constructive constructive proof proof of of the elimination theorem theorem this proof proof will will give give an an effective effective procedure for converting converting a a general general LK-proof into aa this procedure for LK -proof into cut-free LK-proof. As part part of of our our analysis, analysis, we we compute compute an an upper upper bound on the the size size cut-free LK -proof. As bound on of of the the cut-free cut-free proof proof constructed constructed by by this this procedure. procedure. With With some some modification, modification, the the procedure free-cut free procedure can can also also be be used used to to construct construct free-cut free proofs proofs in in LK LKee or or LKs L K ~ ; ; this this will will be be discussed discussed in in section section 2.4.4. 2.4.4. It It is is worth worth noting noting that that the the cut cut elimination elimination theorem theorem proved proved next, next, together together with with the the Completeness Completeness Theorem Theorem 2.2.3 for for the the Hilbert-style Hilbert-style calculus, calculus, implies implies the the Cut-free Cut-free Completeness Theorem 2.3.7. This because LK simulate Hilbert-style Completeness Theorem This is is because L K can can easily easily simulate Hilbert-style proofs and and is is thus thus complete; and then, since any any valid valid sequent sequent has has an an LK LK-proof, it -proof, it proofs complete; and then, since LK -proof. also also has has aa cut-free cut-free LK-proof.

to equal equal the the height 2.4.1. 2.4.1. Definition. Definition. The The depth, depth, dp(A) dp(A),, of of aa formula formula A A is is defined defined to height of of the the tree tree representation representation of of the the formula; formula; that that is is to to say: say: •9 dp(A) dp(A) = - 0O,, for for A A atomic, atomic, }, dp(A 1\ A B) B) = = dp(A dp(A V V B) B) = = dp(A dp(A :> ~ B) B) = = 1+ + max max{dp(A), dp(B)}, {dp(A) , dp(B) •9 dp(A •9 dp( dp(-~A) = dp( d p ( ((3x)A 3 x ) A )) = dp( d p ( ((Vx)A V x ) A )) = 1 + + dp(A) dp(A).. -,A) = The The depth depth of of aa cut cut inference inference is is defined defined to to equal equal the the depth depth of of its its cut cut formula. formula. Definition. Definition. The The superexponentiation superexponentiation function function 2i 2~,, for for i, i, x x � _ 00,, is is defined defined inductively inductively 2f . Thus by by 20 2~ = - x x and and 2i+l 2i~+1= - 222T. Thus 2i 2T is is the the number number which which is is expressed expressed in in exponential exponential ' s with notation notation as as aa stack stack of of ii many many 22's with an an x x at at the the top. top.

2.4.2. -proof and 2.4.2. Cut-Elimination C u t - E l i m i n a t i o n Theorem. T h e o r e m . Let Let P P be be an an LK LK-proof and suppose suppose every every cut cut formula formula in in P P has has depth depth less less than than or or equal equal to to d d.. Then Then there there is is aa cut-free cut-free LK L K - -proof proof P P ** with with the the same same endsequent endsequent as as P P,, with with size size 91iPii < 2211PII IILP*II IP* II < d+2 '

Most this theorem original proof Gentzen [1935] Most proofs proofs of of this theorem are are based based on on the the original proof of of Gentzen which which involved involved making making local local changes changes to to a a proof proof to to reduce reduce the the depth depth of of cuts, cuts, the the number number of of cuts, cuts, or or the the so-called so-called rank rank of of a a cut. cut. We We present present here here a a somewhat somewhat differently differently structured the depth structured proof proof in in which which the depth or or number number of of cuts cuts is is reduced reduced by by making making global global changes changes to to aa proof. proof. We We feel feel that that our our approach approach has has the the advantage advantage of of making making the the overall overall cut cut elimination elimination process process clearer clearer and and more more intuitive. intuitive. The The main main step step in in proving proving the the cut cut elimination elimination theorem theorem will will be be to to establish establish the the following following lemma: lemma:

Let LK -proof with Let P P be be an an LK-proof with final final inference inference aa cut cut of of depth depth d d such such that that every every other other cut cut in in P P has has depth depth strictly strictly less less than than d d.. Then Then there there is is an an LK -proof P same endsequent endsequent as LK-proof P ** with with the the same as P P with with all all cuts cuts in in P P ** of of depth depth less less than than d d and and with with ItiP* < 1jjpjj2. 1 P 1 12 . I P * IitI < 2.4.2.1. 2.4.2.1. Lemma. Lemma.

S. Buss

38

P r o o f . The The proof proof P P ends ends with with aa cut cut inference inference Proof.

' 9.. :. 9. .. ..QQ ' ,

,

.

.R

..

9 ..R ,

o

,

A,A A, r F--+ A rF---~ -+ �, A A, -+ � F---~ A r -+ � where the the depth depth of of the the cut cut formula formula A A equals equals dd and and where where all all cuts cuts in in the the subproofs subproofs where and R have have depth depth strictly strictly less less than than d d.. The The lemma lemma is is proved proved by by cases, cases, based based on on Q and the outermost outermost logical logical connective connective of of the the cut cut formula formula A A.. We We can can assume assume w.l.o.g. w.l.o.g, that that the both Q and and R R contain contain at at least least one one strong strong inference; inference; since since otherwise, otherwise, we we must must have have both A in in r F or or in in � A,, or or have have aa formula formula which which occurs occurs in in both both r F and and � A,, and and in in the former A the former case, the the sequent sequent r F---~ A is is obtainable obtainable by by weak weak inferences inferences from from one one of of the the upper upper -+ � case, sequents and and the the cut cut can can therefore therefore be be eliminated, eliminated, and and in in the the second second case, case, r F--~ A can can -+ � sequents be inferred inferred with with no no cut cut inference inference at at all. all. The The proof proof P P is is also also assumed assumed to to be be in in free free be variable normal normal form. form. variable Case (a): Suppose Suppose A A is is aa formula formula of of the the form form ..., -~B. We shall shall form form new new proofs proofs Case B . We Q** and and R R** of the sequents sequents B, B, r F--~ A, B B,, which which can can then then be combined Q of the -+ A � and and rP--~ -+ �, be combined *, with aa cut cut inference inference of of depth depth d d -- 1 to to give give the the proof proof P P ** of of r F--~ A.. To To form form Q Q*, with -+ � first form form Q' by by replacing every sequent sequent II II -+ ~ A A in in Q with with the the sequent sequent H, B -+ ~ A A-,, replacing every II, B first where AA- is is obtained from A A by by removing removing all all direct direct ancestors ancestors of of the the cut cut formula formula A A.. where obtained from Off course, course, Q' is is not not a a valid valid proof; proof; for for example, -~ :right :right inference inference iinn Q of of the the form form O example, aa ..., B, II 1-I-+ ---~ A A B, H -+ ---+ A, A, ..., --B II B could become in Q' could become in B, B---~ B, II, II, B -+ AAII, + A A-II, B B--+ This can be be modified modified This is is not, not, strictly strictly speaking, speaking, aa valid valid inference; inference; but but of of course, course, itit can to be valid by by inserting some exchanges and a a contraction. contraction. In this fashion, it is to be valid inserting some exchanges and In this fashion, it is straightforward to proof Q Q** by so that that itit becomes becomes aa valid valid proof by removing removing some some straightforward to modify modify Q' so inserting some some weak it to to the reader to -, :left :left inferences inferences and and inserting weak inferences. inferences. We We leave leave it the reader to ..., check that can be validly formed way: it be noted that we we are are using check that Q* Q * can be validly formed in in this this way: it should should be noted that using *, the assumption that initial initial sequents sequents e -+ C e must have C e atomic. proof, R the assumption that C--+ must have atomic. The The proof, R*, of F--~ A, B B is from R. new cuts are of r -+ �, is formed formed in in aa similar similar fashion fashion from R. Obviously, Obviously, no no new cuts are introduced by this process process and, and, since since we we do do not count weak weak inferences, inferences, IIIQ*II I Q * I I :::; IQ I I introduced by this not count _< IIIQll :::; II IPll I PI I . 9 and hhas as II II PP** II II -< has only only cuts cuts of of depth depth << dd and thus Pp** has :::; II IRll; IRI I ; thus and II IR*II IR* I I _< and (b): Now Now suppose suppose the the cut cut formula formula A A is is of of the the form form B e . We We define define Q' Q' as as aa Case (b): Case BV V C. tree r -+ A, �, B, B, C, e , by by replacing replacing every every sequent sequent II---~ II -+ A A tree of of sequents, sequents, with with root root labeled labeled F--~ is A A minus minus all all occurrences occurrences of of direct direct in Q in Q with with the the sequent sequent II--~ II -+ A-, A - , B, B, C, e , where where AA - is ancestors inferences from ancestors of of the the cut cut formula. formula. By By removing removing some some formerly formerly Y V :right :right inferences from Q' Q' and and by by adding adding some some weak weak inferences, inferences, Q' Q' can can be be transformed transformed into into aa valid valid proof proof Q*. Q* . Now construct construct RB RB from R by by replacing replacing every occurrence in in R R of of B BV V Ce as as aa direct direct Now from R every occurrence ancestor way that B . One One way that RB RB can can fail fail to to ancestor of of the the cut cut formula formula with with just just the the formula formula B. be V :left :left inference inference be aa valid valid proof proof is is that that an an V

39 39

Proof Theory Theory Introduction to Proof B, II II---+ A C, II I1---+ A ---+ A C, ---+ A B, B vV C, C, II II---+ A B ---+ A may become become just just may

B, II II---+ A C, II H---} A ---+ A C, ---+ A B, B, II II---} A B, ---+ A

in RB RB.. This This is is no no longer longer aa valid valid inference, inference, but but it it can can be be fi fixed up by by discarding discarding the the xed up in inference and and its its upper upper right right hypothesis, hypothesis, including including discarding discarding the the entire entire subproof subproof of of inference the upper right hypothesis. hypothesis. The The only only other changes needed needed to to make make RB valid valid are are the upper right other changes the the addition addition of of weak weak inferences, inferences, and and in in this this way, way, aa valid valid proof proof RB of of B, ~ � A is is B, rP ---+ formed. A A similar similar process process forms forms aa valid valid proof Re of of C, C, r F---+ A.. The The proof proof P P ** can can ---+ � formed. proof Re now be be defi defined to be be now ned to

* ' 9.. :9 .. . .QQ * A, B, C C rF--} ---+ �, o . .

. Re '. .. :9 ...Re C, rF---+ /k C, ---+ � . . .

A,B rP---} ---+ �, B

... : ... RB B, F---+ A

-+A rF ----+ � * , RB The processes processes of of forming forming Q Q*, RB,, and and Re Re did did not not introduce any new new cuts cuts or or any any The introduce any new strong strong inferences. inferences. Thus Thus we we clearly clearly have have that that every every cut cut in in p P ** has has depth depth < < d, d, new I Q I I + I IR I I + IQ I I , IIIRll IRI I � IQI I + 2 1 IR I I + 2 . Since IPII = and Since I[[P[I and that that III[PP ** [I II _::; IIJQJJ+2JJRI[+2. = IIIQ]J+IJR[[ +11 and and IIlQl], > 11,, this suffices suffices to to prove prove the the lemma lemma for for this this case. case. this Cases (c}, (c),(d): The cases cases where where A A has has outermost outermost connective connective 1\ A or or :) D are are very very Cases (d): The similar to to the previous case, case, and are omitted. omitted. similar the previous and are Case (e): (e): Now Now suppose suppose A A is is of of the the form form (:3x)B(x) (3x)B(x).. First First consider consider how how the the formula formula Case (3x)B(x) can can be be introduced introduced in in the the subproof subproof Q Q.. Since Since it it is is not not atomic, atomic, it it cannot cannot be be (:3x)B(x) introduced in in an an initial initial sequent; sequent; thus, thus, it it can can only only be be introduced introduced by by weakenings and introduced weakenings and inferences. Suppose Suppose that there are :right inferences by 3:3 :right by :right inferences. that there are kk � > 00 many many :3 3:right inferences in in Q which have have their their principal formula. These can which principal formula formula aa direct direct ancestor ancestor of of the the cut cut formula. These can be enumerated as as be enumerated

Hi --~ Ai, B(ti)

a,, (:3x)B(x) IIi ---+ Ai, with _< k. we locate all the the 3:left have with 11 _::; i/ ::; k. Similarly, Similarly, we locate all :3 :left inferences inferences in in R R which which have principal formula aa direct formula and enumerate these principal formula direct ancestor ancestor of of the the cut cut formula and enumerate these as as B(a,),II{--+ A~: B ( ai) , II: ---+ A (3x)B(x), II~: ---+ --~ A a~: (:3x)B(x), II for 1 < for ::; ii<::;g .£ . For each each ii ___ ::; k, k , we we form form aa proof proof Ri R; of of the the sequent sequent B(ti), B(ti), F--+ r ---+ � For A by by replacing replacing everywhere in in R, R , replacing replacing every all every direct direct all g£ of of the the variables variables aj aj with with the the term term titi everywhere ancestor of the cut cut formula formula (3x)B(x) (:3x)B(x) in in R R with with B(t~), B(ti) , and and then then removing the ge 3:left :3 :left ancestor ofthe removing the inferences. fact that inferences. It It is is easy easy to to see see that that this this yields yields aa valid valid proof; proof; note note that that the the fact that PP is is with titi will aj ' s with in free variable normal normal form form ensures ensures that that replacing replacing the the aj's in free variable will not not impact impact the inferences in the eigenvariable eigenvariable conditions conditions for for inferences in R. R. Now, Now, form form Q' Q' from from Q Q by by replacing replacing each each sequent sequent II---} II ---+ A A iinn Q Q with with the the sequent sequent A. where AA- is is A A minus minus all all direct direct ancestors ancestors of of the the cut cut formula formula A. II, II, F---+ r ---+ A, �, A-, A - , where

40 40

S. Buss Buss S.

Clearly, -+ Ll, Clearly, Q' Q' ends ends with with the the sequent sequent r, F, rF----> A, Ll A;; however, however, it it is is not not aa valid valid proof. proof. To To fix it it up up to to be be aa valid valid proof, proof, we we need need to to do do the the following. following. First, First, an an initial initial sequent sequent fix in -+ Ll, in Q' Q' will will be be of of the the form form A, A, rF--~ A, A; A; this this can can be be validly validly derived derived by by using using the the initial initial sequent -+ AA followed sequent A A---+ followed by by weakenings weakenings and and exchanges. exchanges. Second, Second, for for 11 � < ii � < kk,, the the i-th :33 ::right inference enumerated enumerated above, above, will will be be of of the the form form right inference i-th IIi, IIi, rr --+ + Ll, A, Ai, Ai, B(ti) B(ti) IIi, Hi, rF -+ --} Ll, A, Ai Ai in Q Q.. This This can can be be replaced replaced by by the the following following inferences inferences in R; . ..R~ :

IIi, -+ Ll B(ti) -+ Ll, II~,rp--+ A, Ai, A~,B(ti) B(h) B(h),' rF--~ A IIi, Hi, r[' -+ --'~ Ll, A, Ai Ai Note that that this this has has replaced replaced the the :33 ::right inference of of Q Q with with aa cut cut of of depth depth dd - 11 and and Note right inference -

some weak weak inferences. inferences. some No No further further changes changes are are needed needed to to Q' Q' to to make make it it aa valid valid proof. proof. In In particular, particular, the eigenvariable eigenvariable conditions conditions still still hold hold since since no no free free variable variable in in r F -+ ~ Ll A is is used used as as an an the eigenvariable eigenvariable in in Q Q.. By By adding adding some some exchanges exchanges and and contractions contractions to to the the end end of of this this proof, the the desired desired proof proof P P ** of of r F -+ ~ Ll A is is obtained. obtained. It It is is clear clear that that every every cut cut in in P P ** proof, inspection of has has depth depth < < dd.. From From inspection of the the construction construction of of P P **, , the the size size of of P P ** can can be be bounded by by bounded IIIP*ll IP* II � IQI I · ( I IRII + 1) < I P W2.· _< IIlqll'(llRIl+l) < IIIPII Case completely dual Case ((f): The case case where where A A is is of of the the form form (Vx)B(x) (Vx)B(x) is is completely dual to to case case (e), (e), f): The and is is omitted omitted here. and here. Case Finally, consider consider the the case case where where A A is is atomic. atomic. Form R' from R by by Case (g): (g): Finally, Form R' from R II -+ A in R with the the sequent sequent II-, II - , rF ~-+ A, Ll, A where II replacing every every sequent sequent II---+ replacing A in R with A,, where His occurrences of direct ancestors ancestors of A. R' with the sequent is II II minus minus all all occurrences of direct of A. R' will will end end with the sequent Ll, A Ll and and will will be be valid valid as as aa proof, proof, except for its sequents. The initial -+ A, r, F, r F---+ except for its initial initial sequents. The initial -+ B R , with with B the cut formula A, A , become sequents sequents B B--~ B in in R, B not not aa direct direct ancestor ancestor of of the cut formula become r -+ A, Ll, B B in in R'; R' ; these these are are readily readily inferred inferred from from the the initial initial sequent sequent B---> B -+ BB with with B, B, F--~ A -+ A A in in R R only weak inferences. On On the the other other hand, hand, the the other other initial initial sequents sequents A--+ only weak inferences. of become F---} r -+ A, Ll, A A which which is is just just the the endsequent endsequent of of Q. Q . The The desired desired proof proof P* P * of become Ll is is thus thus formed formed from from R' R' by by adding adding some some weak inferences and and adding adding some some r -+ A F---> weak inferences copies of of the the subproof subproof Q Q to to the the leaves leaves of of R', R' , and and by by adding adding some some exchanges exchanges and and copies contractions contractions to to the the end end of of R'. R' . Since Q Q and and RR have have only only cuts cuts of of degree degree << dd (i.e., (i.e., have have no no cuts, cuts, since since dd == 0), 0), Since likewise has has only only cuts cuts of of degree degree << d. d . Also, Also, since since the the number number of of initial initial sequents sequents PP** likewise in in R' R' isis bounded bounded by by [IRII I IRII ++ 1, 1 , the the size size of of Pp** can can be be bounded bounded by by

I JPW2 ( I IR I I ++ 1) 1) << (lIQll I I P * III I � IIRll I I R I I ++ IIQ[I" I IQ I I · (lIRJI ( I I Q I I ++ 1)(lIRll 1 ) << IIPJJ liP* l ) (I IR I I ++ 1) That That completes completes the the proof proof of of Lemma Lemma 2.4.2.1. 2.4.2.1.

41 41

Introduction to Proof Theory

Lemma 2.4.2.1 2.4.2.1 shows shows how how to to replace replace aa single single cut cut by by lower lower depth depth cut cut inferences. inferences. Lemma By iterating iterating this this construction, it is is possible possible to to remove remove all all cuts cuts of of the the maximum maximum By construction, it depth d d in in a a proof. proof. This This is is stated stated as as Lemma Lemma 2.4.2.2: 2.4.2.2: the the Cut-Elimination Theorem depth Cut-Elimination Theorem is lemma. is an an immediate immediate consequence consequence of of this this lemma.

2.4.2.2. If LK -proof with d, there 2.4.2.2. Lemma. Lemma. If P P is is an an LK-proof with all all cuts cuts of of depth depth at at most most d, there is is same endsequent an LK-proof P** with with the the same endsequent which which has has all all cuts cuts of of depth depth strictly strictly less less an LK -proof P 2 11PII .9 P** I[[I < < 2 2211Vll than d and with size I[II P P r o o f . Lemma Lemma 2.4.2.2 2.4.2.2 will be proved proved by by induction on the the number number of of depth depth d d cuts cuts Proof. will be induction on in P P.. The The base base case, case, where where there there are are no no depth depth d d cuts cuts is is trivial, trivial, of of course, course, since since in 21 1P1I .. For IPI] < < 222~lPll For the the induction induction step, step, it it suffices suffices to to prove prove the the lemma lemma in in the the case case where where IIIPII P ends ends with with a a cut cut inference inference P

.Q .R '9.. : 9 ...Q '... : 9 ...R A, -t .6. r -t .6., A F---} A,A A, r F---} A r .6. F - ---t > A where the cut formula A where the cut formula A is is of of depth depth d d.. First suppose suppose that that one one of of the the subproofs, subproofs, say say R, R, does does not not have have any any strong strong in inFirst ferences; IRI I = ferences; i.e., i.e., II[R]I = O0.. Therefore, Therefore, R R must must either either contain contain the the axiom axiom A A -t ~ A A,, or or must have have direct direct ancestors ancestors of of the the cut cut formula formula A A introduced introduced only only by by weakenings. weakenings. In In must the be obtained obtained the former former case, case, A A must must appear appear in in .6. A,, and and the the desired desired proof proof P P ** can can be from adding some from Q Q by by adding some exchanges exchanges and and a a contraction contraction to to the the end end of of Q Q.. In In the the second second case, obtained from case, P P ** can can be be obtained from R R by by removing removing all all the the Weakening:left Weakening:left inferences inferences that possibly removing that introduce introduce direct direct ancestors ancestors of of the the cut cut formula formula A A (and (and possibly removing some some exchanges 's) . A exchanges and and contractions contractions involving involving these these A A's). A similar similar argument argument works works for for the the 211PII . case I]]QI] = O0.. In In both both cases, cases, I[IP*[I < I[IPII < 2221'Pl'. IP* II < case IPII < IQ I I = IP I I are Second, Second, suppose suppose that that IIIQII and IIIPII are both both nonzero. nonzero. B Byy the the induction induction hypoth hypothIQ I I and esis, esis, there there are are proofs proofs Q Q** and and R R** of of the the same same endsequents, endsequents, with with all all cuts cuts of of depth depth less less than than d d,, and and with with 1 1 RII < 2211QII 2.iQil and < 22 22iIRil and I][Q*][ IQ * I I < I][R*[] IR* I I < Applying Applying Lemma Lemma 2.4.2.1 2.4.2.1 to to the the proof proof 9

.

9

9

o

9

.. . '9 . ..RR * * * * '9.. :9 .. ..QQ .6., A A, rF---+ -t .6. rF--+ -t A,A A, A rF---~ -t .6. A .

.

,

,

.

.

gives -t .6. d, so gives a a proof proof P P ** of of r F---} A with with all all cuts cuts of of depth depth < < d, so that that 2

(

211QII + _/_ 2 2 211RI 211RllI - 11 *II + 1) 2 � < (IIQ (/[Q*I[ + I[IR*II + 1) ~ ( 222''Q'' I R* I I + i P * IIII < lliP* _

) 22

The since IIIQII, IQ I I , IIIRII IRII :2: 11.. The final final inequality inequality holds holds since Q.E.D. Q.E.D. Lemma Lemma 2.4.2.2 2.4.2.2 and and the the Cut Cut Elimination Elimination Theorem. Theorem.

IQII+IIRII+I 21IQII+IIRII+1 < 2 221 <

=

21 1P11 . 22211PII

4422

S. Buss Buss S.

2�:lI2

2.4.3 " A A general g e n e r a l bound b o u n d on on cut cut elimination. e l i m i n a t i o n . The The upper upper bound bound ~IIPII in the the Cut Cut 2.4.3. in ~2d+2 Elimination Theorem Theorem is is based based not not only only on on the the size size of of P P ,, but but also also on on the the maximum maximum Elimination depth of of the the cut cut formulas formulas in in P P.. However, However, there there is is aa general general method method that that allows allows the the depth bound to to be be expressed expressed in in terms terms of of just just I]lPII. This is is based based on on the the following: following: bound IP I I . This

Proposition. -proof of ---+ � P r o p o s i t i o n . Suppose Suppose P P is is an an LK LK-proof of the the sequent sequent rF--~ A.. Then Then there there is is aa * I I << 911Pal cut-free cut-free proof proof P P** of of the the same same sequent sequent with with size size I]]I P P'HI "211P]l"

2��1I '

P r o o f . The The proposition proposition is is obviously obviously true true if if P P has has no no cuts, cuts, so so suppose suppose it it contains contains Proof. at at least least one. one. We We need need some some definitions: definitions: Two Two semiformulas semiformulas are are said said to to be be term term variants variants if if they they have have identical identical logical logical structure structure and and thus thus one one can can be be transformed transformed into the the other other by by changing changing only only its its semiterms. semiterms. Obviously Obviously the the 'term 'term variant variant'' relation relation into is an an equivalence equivalence relation. relation. For For any any equivalence equivalence class class of of term term variants, variants, there there is is aa is 71 , . . . , 7k) formula formula skeleton skeleton R( R(T1,..., Tk) such such that that the the equivalence equivalence class class contains contains exactly exactly the the semiformulas semiformulas of of the the form form R(tl, R ( t l , .. .. .., , tk) tk) where where tl, t l , .. .. .., , ttkk are are semiterms. semiterms. Let be nite sequence · ·) aa Let Ccl, be an an infi infinite sequence of of new new free free variables. variables. For For R(· R(...) l , Cc2, 2 , Cc3,.., 3, formula formula skeleton skeleton for for aa term term variant variant class, class, we we pick pick aa new new predicate predicate symbol symbol PR PR.. Given Given aa member member R( R ( tt ll ,, .. ... ., , tk) tk) of of the the term term variant variant class, class, we we can can form form the the atomic atomic semiformula semiformula PR(t correspond to PR(tl,..., t~) and and its its bound bound variables variables correspond to the the bound bound variables variables which which are are l , ' . . , tk) freely occurring occurring in in R(i) R(t-).. freely A formula formula A A is is defined defined to to be be active active in in P P if if some some strong strong inference inference in in P P has has a a term term A variant variant of of A A as as aa principal principal formula. formula. Suppose Suppose A A is is non-atomic non-atomic and and not not active active in in P P;; let · ·) give A. Consider let R(· R(..-) give the the term term variant variant class class of of A. Consider the the following following transformation transformation tk) which of P P:: for for each each term term variant variant R(tl, R ( t l , ... .. ,. , tk) which appears appears as as aa subformula subformula of of a a formula formula of , . . . , tk) in in P P , , replace replace it it with with PR(tl PR(tl,..., tk).' It It can can be be checked checked that that this this transformation transformation yields yields aa valid proof, which which contains exactly the the same same kinds kinds of as PP. . valid proof, contains exactly of inferences inferences as By this transformation, proof P' pi is in which which By repeatedly repeatedly applying applying this transformation, a a valid valid proof is obtained obtained in every atomic or or is every subformula subformula either either is is atomic is a a term term variant variant of of a a principal principal formula formula of of aa strong inference in P'. P' . Since there are are at at most in pi IP I I many strong inference in Since there most II IPII many strong strong inferences inferences in P',, including at least cut inference, and since since initial initial sequents atomic including at least one one cut inference, and sequents contain contain only only atomic formulas, this this implies implies that that every formula in in P' pi has has depth depth less less than than ]lPl]. I I PI I . (To (To prove prove formulas, every formula the last last assertion, assertion, note note that that every every formula formula in in P' pi either either is is atomic atomic or or has has an an atomic atomic the Therefore, by by the the cut cut elimination elimination theorem, theorem, there there is is aa cut-free cut-free proof proof P" P" ancestor. ) Therefore, ancestor.) ~llPII IP"I with the same endsequent of size I I with the same endsequent of size liP"l] < < "211P]l" This proof proof P" P" has has the the desired desired size, size, but but itit may may no no longer longer be be aa proof proof of of F--~ r ---+ A, �, This However, we we may may PR(it , . . . ,tk). , tk) ' However, since itit may may contain contain subformulas subformulas of of the the form form PR(tl,... since (iteratively) replace replace all all such such subformulas subformulas in in P" P" with with the the formula formula RR(tl, tk) ' This This (iteratively) ( t l , ... .. ,. , t~). 0 yields r ---+ A. � . [::] yields the the desired desired proof proof of of F--~ . • •

2��1 I'

2.4.4. FFree-cut next investigate investigate the the possibility possibility of of eliminating eliminating 2.4.4. r e e - c u t eelimination. l i m i n a t i o n . We We next cuts in in LK~-proofs; LK(5-proofs; that that isis to to say, say, in in proofs proofs in in which which initial initial sequents sequents may may come come cuts from |(5 . The The set set |(5 isis presumed presumed to to be be aa fixed fixed set set of ofsequents sequents closed closed under under substitution. substitution. from An important important example example of of such such an an |(5 isis the the set set of of equality equality axioms axioms of of LK~; LKe ; however, however, An |(5 can can also also be be the the axioms axioms of of any any first-order first-order theory. theory.

Introduction to Proof Proof Theory

43 43

The -proofs; on The Cut Cut Elimination Elimination Theorem Theorem 2.4.2 applied applied only only to to LK LK-proofs; on the the other other hand, hand, the the proof proof of of Corollary Corollary 2.3.7 gave gave a a partial partial cut cut elimination elimination theorem theorem for for LKs LK~- proofs. significantly improved notion of proofs. These These results results can can be be significantly improved by by introducing introducing a a notion of 'free' 'free' cuts construction eliminating cuts and and giving giving a a construction eliminating free free cuts cuts from from LKs-proofs LKe-proofs by by a a method method similar similar to to the the proof proof of of Theorem Theorem 2.4.2. 2.4.2. 2.4.4.1. Definition. Definition. Let P P be be an an LKs LKe-proof. A formula formula occurring occurring in in P P is is 2.4.4.1. Let -proof. A anchored by an ) if anchored ((by an <5-sequent | if it it is is a a direct direct descendent descendent of of a a formula formula occurring occurring in in an an initial sequent initial sequent in in <5 | . A A cut cut inference inference in in P P is is anchored anchored if if either: either: ((i) i ) the not atomic the cut cut formula formula is is not atomic and and at at least least one one of of the the two two occurrences occurrences of of the the cut cut formula formula in in the the upper upper sequents sequents is is anchored, anchored, or or ((ii) ii) the the cut cut formula formula is is atomic atomic and and both both of of the the occurrences occurrences of of the the cut cut formula formula in in the the upper upper sequents sequents are are anchored. anchored. A is said A cut cut inference inference which which is is not not anchored anchored is said to to be be free. free. The The proof proof P P is is free-cut free-cut free free if if it it contains contains no no free free cuts. cuts.

2.4.4.2. An 2.4.4.2. An occurrence occurrence of of a a formula formula in in a a proof proof P P is is said said to to be be only only weakly weakly introduced introduced in P P if if it it does not have have a a direct in an an initial initial sequent sequent or or which which in does not direct ancestor ancestor which which appears appears in is a a principal principal formula formula of of a a strong strong inference. inference. It It is is often often convenient convenient to to assume assume that that a a is proof es the condition that proof P P satisfi satisfies the condition that no no cut cut formula formula is is only only weakly weakly introduced; introduced; that that is is to to say, say, that that every every cut cut inference inference satisfies satisfies the the condition condition that that neither neither occurrence occurrence of of its its cut cut formula formula is is only only weakly weakly introduced introduced in in P. P. -proof, then Note that that if if P P is is an an LKs LKe-proof, then P P can can be be assumed assumed w.l.o.g. w.l.o.g, to to satisfy satisfy this this Note extra extra condition condition without without an an increase increase in in the the size size of of the the proof proof or or in in the the depth depth of of cuts cuts in in the the proof. proof. However, However, conforming conforming to to this this convention convention might might increase increase the the number number and and depth depth of of free free cuts cuts in in the the proof proof P P. . To To see see this this suppose suppose that that P P contains contains the the cut cut inference introduced; for inference with with one one of of its its cut cut formulas formulas weakly weakly introduced; for instance, instance, suppose suppose that that an an inference inference rF----~ -+ A t.,, AA A -+ t. A,, r F---~ A rF---~ -+ t. A has has the the A A occurring occurring in in the the right right upper upper sequent sequent A A,, r F --+ + t. A only only weakly weakly introduced. introduced. -+ t., Now, Now, it it is is possible possible that that the the subproof subproof of of r F---+ A, A A causes causes a a formula formula B B in in r F or or t. A to to be be anchored, anchored, and and that that the the corresponding corresponding B B is is not not anchored anchored in in the the subproof subproof of of A A,, r F --+ - } t. A.. The The obvious obvious way way to to eliminate eliminate this this cut cut is is to to remove remove all all direct direct ancestors ancestors of -+ t. of the the A A in in the the right right upper upper sequent, sequent, thereby thereby getting getting a a proof proof of of r F---} A,, and and then then discard discard the the left left upper upper sequent sequent and and its its proof. proof. This, This, of of course, course, causes causes B B to to be be only only weakly weakly introduced introduced in in the the new new subproof subproof of of r F -+ ~ t. A.. In In other other words, words, the the occurrence occurrence of Then, if of B B in in the the sequent sequent r F -+ ~ t. A becomes becomes unanchored. unanchored. Then, if aa direct direct descendent descendent of of B B is elimination of could have is later later used used as as a a cut cut formula, formula, the the elimination of the the cut cut on on A A could have changed changed the the cut cut from from an an anchored anchored cut cut into into a a free free cut. cut. Our Our proof proof of of the the free-cut free-cut elimination elimination theorem theorem will will use use induction induction on on the the maximum maximum depth work, it depth of of free free cuts cuts appearing appearing in in a a proof. proof. For For this this induction induction to to work, it is is important important that cuts, introducing that anchored anchored cuts cuts do do not not change change into into free free cuts, introducing new new free free cuts cuts of of higher higher

44 44

S. Buss

depth. To avoid avoid this, we will assume that that no no cut cut formulas formulas in in the the proof are only only depth. To this, we will assume proof are weakly weakly introduced. introduced.

Let Let (5 | be be aa set set of of sequents sequents closed closed under under substitution. substitution. -proof of (1) ---7 � (1) If /]" LKs L K e fFr F---+ A,, then then there there is is aa free-cut free-cut free free LKs LK~-proof of r F ---7 ~ � A.. -proof satisfying (2) (2) Let Let P P be be an an LKs LKG-proof satisfying condition condition 2.4.4.2 2.4.4.2 and and suppose suppose every every anchored anchored less than cut formula cut formula in in P P has has depth depth less than or or equal equal to to d d.. Then Then there there is is aa free-cut free-cut free free same endsequent P, with LKs -proof P LK6-proof P** with with the the same endsequent as as P, with size size

2.4.5. 2.4.5. Free-cut Free-cut Elimination E l i m i n a t i o n Theorem. Theorem.

11P11 < 2911PII IIIP*ll IP* II < 2d+2 ' The Gentzen; Takeuti The Free-cut Free-cut Elimination Elimination Theorem Theorem is is essentially essentially due due to to Gentzen; Takeuti [1987J [1987] states states aa related related construction construction for for use use with with induction induction rules, rules, which which we we describe describe in in section section 2.4.6. 2.4.6. P r o o f . Obviously, Obviously, it it suffices suffices to to prove prove part part (2) (2) of of Theorem Theorem 2.4.5. 2.4.5. For For this this proof, proof, Proof. we shall give we shall give aa procedure procedure (Lemma (Lemma 2.4.5.1) 2.4.5.1) for for removing removing one one maximum maximum depth depth free free cut. is not not possible use the the procedure the proof cut. It It is possible to to use procedure from from the proof of of Lemma Lemma 2.4.2.1 2.4.2.1 without without modification, from modification, because because it it may may remove remove (5-sequents | from the the proof proof and and thereby thereby change change some some anchored anchored cuts cuts into into free free cuts cuts (and (and thereby thereby increase increase the the maximum maximum depth depth of of free cuts) cuts).. This This is is unacceptable unacceptable since since our our proof proof uses uses induction induction on on the the maximum maximum free situation can happen in (b) of depth depth of of free free cuts cuts in in P P. . This This undesirable undesirable situation can happen in case case (b) of the the proof of Lemma 2.4.2.1 connective of proof of Lemma 2.4.2.1 where where the the outermost outermost connective of the the cut cut formula formula A A is is V V,, since ---7 A since at at various various points, points, subproofs subproofs ending ending with with C, C, IT H---~ A or or with with B B,, IT II ---7 ---+A A are are just just discarded Subformulas in discarded in in R's R~ and and in in R'c R~,.. Subformulas in IT H and and A A may may become become unanchored unanchored by by this This can also in in the the similar cases (c) (c) and this process. process. This can happen happen also similar cases and (d) (d).. In In addition, addition, it it could right could also also happen happen in in cases cases (e), (e), (f) (f) and and (g) (g) if if the the cut cut formula formula occurring occurring in in the the right upper sequent only weakly since in these cases the subproof the upper sequent is is only weakly introduced, introduced, since in these cases the subproof on on the left since condition this never left is is completely completely discarded; discarded; however, however, since condition 2.4.4.2 2.4.4.2 holds, holds, this never occurs. occurs. Therefore, Therefore, we we need need the the following following analogue analogue of of Lemma Lemma 2.4.2.1: 2.4.2.1:

2.4.5.1. -proof with 2.4.5.1. Lemma. L e m m a . Let Let P P be be an an LKs LK6-proof with final final inference inference aa free free cut cut of of depth depth dd such such that that every every other other free free cut cut in in P P has has depth depth strictly strictly less less than than d d.. Then Then there there is is an -proof P same endsequent an LKs LK6-proof P** with with the the same endsequent as as P P with with all all free free cuts cuts in in P P** of of depth every formula depth less less than than d d and and with with IIII P P ** IIII < < 1I1P IPII1 1 22.. Furthermore, Furthermore, every formula occurring occurring still anchored in in the the endsequent endsequent of of P P which which was was anchored anchored by by an an (5 | -sequent is is still anchored in in the the proof proof P P **,, and and every every formula formula in in the the endsequent endsequent of of P P ** which which is is only only weakly weakly introduced introduced in in P P** was was already already only only weakly weakly introduced introduced in in P P.. Proof. P r o o f . We We indicate indicate how how to to modify modify the the proof proof of of Lemma Lemma 2.4.2.1 2.4.2.1.. Assume Assume that that the the proof proof P P ends ends with with aa free free cut cut inference inference . . . :9 . .. ..QQ . . :9. .. ..RR . rF---~ ---7 �, A A, ---7 � A,A A, r F---~ A rF---+ ---7 � A 9

.

.

.

.

.

,

.

Introduction to Proof Theory

45 45

If then Q contain at If the the cut cut formula formula A A is is nonatomic, nonatomic, then Q and and R R both both contain at least least one one strong strong inference inference with with principal principal formula formula equal equal to to A A.. As As before before the the proof proof splits splits into into cases cases depending depending on on the the outermost outermost connective connective of of A A.. When When A A has has outermost outermost connective connective -' -~,, :3 3 or or V V then then the the argument argument used used in in cases cases (a) (a),, (e) (e) and and (f) (f) of of Lemma 2.4.2.11 still still Lemma 2.4.2. works: works" we we leave leave it it to to the the reader reader to to check check that, that, in in these these cases, cases, any any formulas formulas in in the the sequent -+ t. sequent f F--~ A which which were were anchored anchored or or were were not not only only weakly weakly introduced introduced in in P P still still have have these these properties properties in in P P **. . construction is needed. In For For the the case case where where A A is is of of the the form form B B V VG C,, aa different different construction is needed. In -+ A, t., B, construction used this this case, case, form form a a proof proof Q Q** with with endsequent endsequent f F--~ B, G C by by the the construction used in case case (b) of the the proof proof of of Lemma Lemma 2.4.2.1 2.4.2.1.. Also Also form form R' R ~ from from R R be be replacing replacing every every in (b) of sequent II II-----} A in in R R with with the the sequent sequent H-, ~ A, A where H-- is is II II minus direct -+ A 11-, fF -+ t., A where 11 minus all all direct sequent principal formula ancestors ancestors of of the the cut cut formula formula B BV VG C.. An An V V :left :left inference inference in in R R with with principal formula B V VG C will no longer longer be be aa valid valid inference inference since, since, in in R' R ~ it it will will become become will no B

- , fP---+ - , fF---~ -+ t., A B, II II-, A,A C, II II-, A,A -+ t., A G, B, 11 II-,, f F---+ A,A -+ t., A This replacing it This can can be be transformed transformed into into a a valid valid proof proof by by replacing it with with .. : ..Q ..

9

.

.

. . Q

.

".

9 9 .

.. 9

- , fF--~ fF---+ -+ t., B,, CG A,B C, 11 l-I-, A,A ... :... G, -+ t., A , f -+ t., A, B II-, F---+ A , A , B B, II H-,, f F---+ A,A II B, -+ t., A - , fF---+ 11 -+ A,A t., A II-, Fixing adding some Fixing up up R' R' in in this this way, way, plus plus adding some weak weak inferences inferences yields yields a a valid valid proof proof R R** f, fF -+ t., t. yields the of of F, ~ A, A.. Appending Appending some some exchange exchange and and contraction contraction inferences inferences yields the desired 1) < desired proof proof p P **. . It It is is readily readily checked checked that that IliP* <- IIIRII + 1) < IIIPII I P * IIII :::; I R I I ·" ((llqll I PW2.. I IQ I I + The The final final case case to to consider consider is is the the case case where where A A is is atomic. atomic. Since Since the the cut cut is is not not anchored, will either anchored, the the cut cut formula formula will either not not be be anchored anchored in in Q Q or or not not be be anchored anchored in in R R.. IInn the the latter latter case, case, since since the the cut cut formula formula is is not not only only weakly weakly introduced, introduced, it it must must have have aa direct sequent A -+ A the argument direct ancestor ancestor in in an an initial initial sequent A---+ A of of R. R. Therefore, Therefore, the argument from from case (g) of the former the cut cut case (g) of the the proof proof of of Lemma Lemma 2.4.2.1 2.4.2.1 still still works. works. In In the former case, case, where where the formula not anchored in Q the dual formula is is not anchored in Q,, the dual argument argument works. works. That That completes completes the the proof proof of of Lemma Lemma 2.4.5.1. 2.4.5.1. The The Free-cut Free-cut Elimination Elimination Theorem Theorem follows follows immediately immediately from from this this lemma lemma in in the the same same way way that that the the Cut-Elimination Cut-Elimination Theorem Theorem followed followed from from Lemma Lemma 2.4.2.1. 2.4.2.1. Q.E.D. Q.E.D. Free-cut Free-cut Elimination Elimination Theorem. Theorem. 2.4.6. A 2.4.6. Free-cut F r e e - c u t elimination elimination with with induction induction rules. rules. A very very useful useful application application of of free-cut free-cut elimination elimination is is for for analyzing analyzing subtheories subtheories of of arithmetic arithmetic in in which which induction induction is is restricted restricted to to certain certain classes classes of of formulas. formulas. For For these these fragments fragments of of arithmetic, arithmetic, the the free-cut elimination theorem induction rules free-cut elimination theorem becomes becomes easier easier to to use use if if induction rules are are used used in in place place of of induction induction axioms. axioms. The The most most common common formulation formulation of of induction induction rules rules is is as as inferences form: inferences of of the the form:

46

s. S. Buss

A(b),, r F -t ~ D., A, A(b + 1) A(b) A(b + A(0),, F ~ D., A, A(t) A(t) r-t A(O) where where bb is is an an eigenvariable eigenvariable and and may may appear appear only only as as indicated, indicated, and and tt is is an an arbitrary arbitrary term. It term. It is is easily easily checked checked that, that, because because of of the the presence presence of of the the side side formulas formulas r F and and A,, the the induction induction rule rule for for A A is is equivalent equivalent to to the the induction induction axiom axiom for for A A:: D.

A(O) 1\ A ((Vx)(A(x) D A(x A(x + 1)) 1)) J D (V'x)A(x) (Vx)A(x).. A(O) V'x) (A(x) J

For indicates the For
of -proofs. of Theorem Theorem 2.4.5(2) 2.~.5(2) apply apply to to T T-proofs.

This This theorem theorem is is proved proved by by an an argument argument identical identical to to the the proof proof of of Theorem Theorem 2.4.5. The The theorem theorem also also applies applies without without modification modification to to theories theories axiomatized axiomatized with with PIND PIND in IND. It in place place of of IND. It is is also also straightforward straightforward to to modify modify the the free free cut cut elimination elimination to to work work with with sequent sequent calculus calculus formulations formulations of of inference inference rules rules other other than than induction. induction. For For example, the collection axioms for Peano arithmetic can example, the collection (replacement) (replacement) axioms for fragments fragments of of Peano arithmetic can be equivalently sequent calculus be equivalently formulated formulated as as sequent calculus rules. rules. The The obvious obvious straightforward straightforward

Introduction to Proof Theory

47 47

modification of of the the Free-cut Free-cut Elimination Elimination theorem theorem applies applies to to theories theories with with collection collection modification rules. rules. A particularly particularly useful useful corollary corollary of of the the Free-Cut Free-Cut Elimination Elimination Theorem Theorem is: is: A

Let Let (5 | be be aa set set of of sequents sequents and and 9 be be aa set set of of formulas formulas closed closed under term term substitution substitution and and under under subformulas, subformulas, such such that that every every sequent sequent in in (5 ~ contains contains under only formulas formulas from from (~.. Let Let T T be be the the theory theory (5 | + -IND ~-IND (or (or (5 | + + -PIND). (I)-PIND). Suppose Suppose only that r F --+ ~ � A is is aa consequence consequence of of T T and and that that every every formula formula in in r F--+ A is is in in (~.. Then Then --+ � that there -proof P there is is aa T T-proof P of of r F --+ ~ � A such such that that every every formula formula appearing appearing in in P P is is in in (~..

2.4.7. 2.4.7. Corollary. Corollary.

To prove prove this this corollary, corollary, just just note note that that it it ilOlds imlds for for every every free-cut free-cut free free T T-proof of -proof of To A;; this this is is because because every every formula formula in in the the proof proof must must be be an an ancestor ancestor of of either either aa rF--} --+ � cut formula formula or or aa formula formula in in the the endsequent. endsequent. cut 2.4.8. Natural N a t u r a l ddeduction. e d u c t i o n . Natural Natural deduction proof systems systems were were introduced by 2.4.8. deduction proof introduced by Gentzen the same same paper the first-order sequent calculus. Gentzen [1935] [1935] in in the paper which which introduced introduced the first-order sequent calculus. ' s motivation Gentzen's motivation in in defi defining natural deduction deduction was was (in (in his his words words)) "to "to set set up up Gentzen ning natural ,, The aa formula formula system system which which comes comes as as close close as as possible to actual actual reasoning. reasoning. ''44 The possible to importance of of natural natural deduction deduction was was established established by by the the classic classic result result ofPrawitz of Prawitz [1965] [1965] importance that aa version version of of the the cut cut elimination elimination holds holds also also for for natural natural deduction. that deduction. It is is fair fair to to say say that that the the process process of of constructing constructing natural natural deduction deduction proofs proofs is is indeed indeed It 'natural'' in in that that it it corresponds corresponds closely closely to to the the human human reasoning reasoning process. process. On On the the other other 'natural hand, a a fully fully constructed constructed natural natural deduction deduction proof proof can can be be very very confusing confusing to to read; read; in in hand, particular, because because of of the the non-local non-local nature nature of of natural natural deduction deduction proofs, proofs, it it is is difficult difficult particular, to quickly quickly ascertain ascertain which which formulas formulas depends depends on on which which hypotheses. hypotheses. to Natural deduction proof systems are are particularly elegant for for intuitionistic intuitionistic logic, logic, Natural deduction proof systems particularly elegant especially with respect to formulas-as-types interpretation interpretation (see (see especially with respect to the the Curry-Howard Curry-Howard formulas-as-types A good good modern modern treatment treatment of of applications applications of natural deduction deduction is is section 3.1.5). A of natural section given given by by Girard Girard [1989]. We give give aa short definition of of the natural deduction here; however, however, We short definition the natural deduction proof proof system system here; the reader should refer refer to to Prawitz full treatment and for the [1965] for for aa full treatment and for statements statements of of the the reader should Prawitz [1965] normalization theorems. theorems. The of terms terms and formulas and and the the conventions normalization The definitions definitions of and formulas conventions on and bound bound variables variables are same for as for on free free and are the the same for natural natural deduction deduction as for the the sequent sequent calculus, except except that negation, ..." 9, is new atomic calculus, that negation, is not not aa basic basic symbol; symbol; instead, instead, aa new atomic formula _1_ is used denote absurdity constant False), and ..., ~A abbreviates formula .1 is used to to denote absurdity (the (the constant False) , and A abbreviates A A ~::> _l_. .i . A A natural natural deduction deduction proof proof is is aa tree tree of of formulas; formulas; any any formula formula may may appear appear at at aa leaf, leaf, as as aa hypothesis. hypothesis. Various Various inferences inferences may may close close or or discharge discharge the the hypotheses; hypotheses; in in completed proof proof all all hypotheses hypotheses must must be be discharged discharged and and the proof is is aa proof proof of of the the aa completed the proof formula at the root node node at at the the bottom tree. It best to formula appearing appearing at the root bottom of of the the tree. It is is best to picture picture the the construction at any construction of of aa natural natural deduction deduction proof proof as as an an ongoing ongoing process; process; at any point point in in this this process some some hypotheses already be be discharged, discharged, whereas whereas others others remain remain open. open. process hypotheses may may already rules of of inference inference are given below; classified as introduction rules rules The valid rules are given below; they they are are classified as introduction The valid 4Gentzen [1969], [1969], p. p. 68

s. Buss Buss S.

48 48

Hypotheses discharged discharged by by an an inference inference are are shown shown in in square square or or elimination elimination rules. rules. Hypotheses brackets. brackets. AA AB B AA A BB A B A-intro B A-elim A A-elim A A-intro A AAABB A

V-intro v-intro

A A AVvBB A

B B

A A

B B AVvBB A

V-dim v-elim

[B] [B]

[A] [A]

AV vB B A

C C

C C

C C

[A]

::> -intro -intro

B B AD::>B B A

::>-elim D-elim

A A

V-intro V-intro

A(b) A(b) (Vx)A(x) (Vx)A(x)

V-elim V-elim

(Vx)A(x) (Vx)A(x) A(t) A(t)

3-intro 3-intro

A(t) A(t) (3x)A(x) (3x)A(x)

A AD::>B B B B

[A(b)] [A(b)] 3-elim 3-elim

(3x)A(x) (3x)A(x)

B B

B B

[A ::> D ..1] _L] [A ..1 _L -[-c _1_ ..1c ..1 A A A A In the and 3-elim 3-elim rules, free variable variable bb is an eigenvariable: eigenvariable: this In the V-intro V-intro and rules, the the free is an this means means it must must not not appear appear in non-discharged hypothesis the V-intro inference or or it in any any non-discharged hypothesis above above the V-intro inference in any non-discharged hypotheses hypotheses other other than than A(b) A(b) above the hypothesis hypothesis B of the the in any non-discharged above the B of ..1/ rule rule is is used used for for both intuitionistic and and classical classical logic; logic; the the 3-elim inference. inference. The The -1-I 3-elim both intuitionistic is included included for for classical logic. If rules for for ..1 omitted, then then minimal minimal •..1c rule rule is classical logic. If both both rules • are are omitted, logic logic is is obtained. obtained.

..1/ 2-i

2.5. 2.5. Herbrand's H e r b r a n d ' s theorem, t h e o r e m , interpolation i n t e r p o l a t i o n and a n d definability definability theorems theorems 2.5.1. 2.5.1. Herbrand's H e r b r a n d ' s theorem. t h e o r e m . One One of of the the fundamental fundamental theorems theorems of of mathematical mathematical logic logic is is the the theorem theorem of of Herbrand Herbrand [1930] [1930] which which allows allows aa certain certain type type of of reduction reduction of of first-order logic to first-order logic to propositional propositional logic. logic. In In its its basic basic form form it it states: states: Herbrand's H e r b r a n d ' s Theorem. T h e o r e m . Let Let T T be be aa theory theory axiomatized axiomatized by by purely purely universalformulas. universal formulas. Suppose (3Yl l . . . , Yk )B(x, Y) Suppose that that T T F ~ (Vi) (VZ)(3yl,..., yk)S(Z, ~ with with B(x, B(Z, Y) y-') aa quantifier-free quantifier-free formula. formula. Then Then there there is is aa finite finite sequence sequence of of terms terms ti ti,j,j = ---- ti ti,j(X), with 11 :::; < ii :::; < rr and and 11 :::; < jj :::; < kk ,j (X) , with

so so that that

T r (vx)

(�

)

B(x, ti ,l , . . . , ti,k) .

Proof. Proof. Since Since T T is is axiomatized axiomatized by by purely purely universal universal formulas, formulas, it it may, may, without without loss loss of er-free formulas of generality, generality, be be axiomatized axiomatized by by quantifi quantifier-free formulas (obtained (obtained by by removing removing the the universal universal quantifiers) quantifiers).. Let Let 'I ~7 be be the the set set of of sequents sequents � --~ A A with with A A aa (quantifier-free) (quantifier-free) axiom axiom of of T T.. Since Since T T F ~ (Vx) ( V ~ )(3Y)B(x, ( 3 ~ B ( ~ ,y) ~ , , there there is is aa LK'J;-proof LK~-proof of of the the sequent sequent

Introduction Introduction to to Proof Proof Theory Theory

49 49

there is (3g)B(g, if) ~ . . By By the the free-cut free-cut elimination elimination theorem, theorem, there is aa free-cut free-cut free free proof proof P P � (3if)B(ii, of of this this sequent. sequent. Since Since the the 'I-sequents T-sequents contain contain only only quantifier-free quantifier-free formulas, formulas, all all cut cut formulas formulas in in P P are quantifier-free. quantifier-free. Thus, Thus, any any non-quantifi non-quantifier-free formula in in P P must must be be of of the the form form are er-free formula . . . (3 j, · . . , YYk) k) with with 11 � (3yj)... ( 3 yYk)B(ii, ~ ) B ( g , tl t l ,,. .. . ., . , tj_ tj_~,! , Y yj,..., <_ j j < < k k.. We We claim claim that that P P can can be be (3Yj) modified modified to to be be a a valid valid proof proof of of a a sequent sequent of of the the form form --+ B(ii, tl,! , . . . , tl,k), . . . , B(ii, tr, ! , . . . , tr k). � , The general general idea idea is is to to remove remove all all 3 3 :right :right inferences inferences in in P P and and remove remove all all existential existential The quantifi ers, replacing bound variables terms. Since quantifiers, replacing the the bound variables by by appropriate appropriate terms. Since there there may may have been been contractions contractions on on existential existential formulas formulas that that are are no no longer longer identical identical after after terms terms have are substituted substituted for for variables variables it it will will also also be be necessary necessary to to remove remove contractions contractions and and add add are additional formulas to to the the sequents. sequents. To To do do this this more more formally, formally, we we know know that that any any additional formulas sequent in in P P is is of of the the form form r F---+ A, �' A ~ ((up to order order of of the the formulas formulas in in the the sequent sequent), sequent � �, up to ), where each each formula formula in in r F and and � A is is quantifier-free quantifier-free and and where where each each formula formula in in �' A' is is where not er-free but induction on not quantifi quantifier-free but is is purely purely existential. existential. We We can can then then prove prove by by induction on the the number of of lines lines in in the the free-cut free-cut free free proof proof of of r F� ~ �, A, �' A ~ that that there there is is an an rr ;::: > 0 and and a a number cedent cedent �" A" of of the the form form B ( d , t 1 , 1 , . . . , tl,k), . . . , B ( d , tr,1, . . . , tr,~)

� A, �, �" the rest the details the reader. such that rF---} such that A" is is provable. provable. We We leave leave the rest of of the details to to the reader. 0 [] We momentarily define We momentarily define an an instance i n s t a n c e of of a a universal universal formula formula (\lx)A(x) (V~)A(~) to to be be any any using cut quantifier-free quantifier-free formula formula A(i) A(t-).. It It is is not not hard hard to to see see using cut elimination, elimination, that that if if theory T then it aa quantifier-free quantifier-free formula formula C C is is a a consequence consequence of of a a universal universal theory T,, then it is is aa tautological finite set axioms of tautological consequence consequence of of some some finite set of of instances instances of of axioms of T T and and of of null theory, equality axioms. In where T equality axioms. In the the special special case case where T is is the the null theory, we we have have that that and C called aa C is is a a consequence consequence of of instances instances of of equality equality axioms axioms ((and C is is therefore therefore called C quasitautology q u a s i t a u t o l o g y ) .) . If, If, in in addition, addition, C C does does not not involve involve equality, equality, C C will will be be tautologically tautologically valid. valid. Thus, Thus, Herbrand Herbrand's' s theorem theorem reduces reduces provability provability in in first-order first-order logic logic to to generation generation of quasi) tautologies. of ((quasi)tautologies. 2.5.2. As 2.5.2. As stated stated above, above, Herbrand's Herbrand's theorem theorem has has limited limited applicability applicability since since it it applies applies only only to to II II2-consequences of universal universal theories: theories: fortunately, fortunately, however, however, there there 2 -consequences of are are several several ways ways to to extend extend Herbrand's Herbrand's theorem theorem to to more more general general situations. situations. In In 2.5.3 2.5.3 below, below, we we explain explain one one such such generalization; generalization; but but first, first, in in this this paragraph, paragraph, we we give give a a simpler method method of widening the applicability of theorem, based simpler of widening the applicability of Herbrand's Herbrand's theorem, based on on the the introduction introduction of of new new function function symbols, symbols, which which we we call call Herbrand H e r b r a n d and and Skolem S k o l e m functions, functions, that allow allow quantifier quantifier alternations alternations to to be be reduced. reduced. that For-notational For'notational convenience, convenience, we we will will consider consider only only formulas formulas in in prenex prenex form form in in this this paragraph; paragraph; however, however, the the definitions definitions and and proposition proposition below below can can be be readily readily generalized generalized to arbitrary arbitrary formulas. formulas. to

S. Buss Buss S.

50 50

Definition. Let Let (3x)A(x, (3x)A(x, 5) C) be be aa formula formula with with 5'c all all of of its its free free variables. variables. The The Skolem Skolem Definition. function for for (3x)A (3x)A isis represented represented by by aa function function symbol symbol f3xA hxA and and has has the the defining defining function axiom: axiom:

Sk-def(hxA) :: Sk-de](fSxA)

A( hxA (Y) , Y) ) . (A(x, Y) D (VY) (Vx) (A(x,y-') (V~7)(Vx) :::l A(f3,A(~,Y~).

Note that that Sk-def(f3~A) Sk-def(hxA) implies implies (Vy-') (VY) ((3x)A(x,g) A(hxA(Y) , Y) ) . ((3x)A(x, Y) ~f-+ A(f3~A(Y-),Y-)). Note Definition. Definition. Let Let A(c-) A(C) be be aa formula formula in in prenex prenex form. form. The The Skolemization, Skolemization, AAS(C) s (c-*),, of of A A is the the formula formula defined defined inductively inductively by: by: is

(1) If If A(c-) A(C) isis quantifier-free, quantifier-free, then then AS(c AS(C) is A(c-). A(C) . (1) -) is (2) If -) is If A(c-) A(C) isis (Vy)B(5', (Vy)B(c, y), y) , then then AS(c AS(C) is the the formula formula (Vy)BS(g, (Vy)BS(c, y). y) . (3) If If A(~ A(C) is is (3y)B(g,y), (3y)B(c, y) , then then AS(~ AS(C) is is BS(g, BS(c, fA(c-)), !A(C)) , where where fA fA isis the the Skolem Skolem A. function function for for A.

It is is aa simple, simple, but that A ASs 1=~ A. A. It but important important fact fact that The theory T T is the theory theory TS Ts = - {A Note that The Skolemization Skolemization of of aa theory is the {Ass :: AA E T}. T} . Note that TTSs is purely universal Incidentally, the all Sk - def axioms axioms of the Skolem is a a purely universal theory. theory. Incidentally, the set set of of all Sk-def of the Skolem functions is equivalent to aa set set of of universal universal formulas; however, they they are not included functions is equivalent to formulas; however, are not included in theory theory TS T s.. From From model-theoretic model-theoretic considerations, considerations, it it is is not not difficult difficult to to see see that that TS TS in contains contains and and is is conservative conservative over over T T.. We completely dual We next next define define the the concept concept of of 'Herbrandization' 'Herbrandization' which which is is completely dual to to the the notion of Skolemization: notion of Skolemization: Definition. Definition. Let Let (Vx)A(x, (Vx)A(x, C) ~ be be aa formula formula with with c 5' all all of of its its free free variables. variables. The The Herbrand Herbrand function function for for (Vx)A (Vx)A is is represented represented by by aa function function symbol symbol hv hvzA and has has the the xA and

defining defining axiom: axiom:

A ( hVxA (Y) , Y) ). (V A (x, Y) (V~7)(Vx) (~A(x, ~) :::l D ..., ~A(hwA(y-), Y-O). Y) (Vx) (..., Note hVxA (Y) , Y) Note that that this this implies implies (V (Vy-')((Vx)A(x, ~7) f-+ ~ A( A(hvxA(y-*), y-*)). The Herbrand Herbrand function function ) . The Y) ((Vx)A(x, Y) can can also also be be thought thought of of as as aa 'counterexample 'counterexample function function';' ; in in that that (Vx)A(x) (Vx)A(x) is is false false if if and and only only if if hhvxA provides aa value value xx which which is is aa counterexample counterexample to to the the truth truth of of (Vx}A (Vx)A.. VxA provides Definition. (C) , Definition. Let Let A(C) A(c~ bbee aa formula formula iinn prenex prenex form. form. The The Herbrandization, Herbrandization, A AH u(c-), of of A A is is the the formula formula defined defined inductively inductively by: by: (1) er-free, then (C) isis A(C) (1) If If A(C) A(~ is is quantifi quantifier-free, then AH A H(~ A(~.. (2) If c, y) If A(C) A(~ is is (3y)B(c, (3y)B(~, y) y),, then then AH A H(C) (~ is is the the formula formula (3y)BH (3y)S H((5', y).. (3) If If A(C) A(~ is is (Vy)B(c, (Vy)B(g,y), then AH AH(c BH(g, hA(c-)), where hA hA isis the the Herbrand Herbrand (3) y) , then (C)-) isis BH( c, hA (C)) , where

function function for for A A..

It It is is not not hard hard to to see see that that A A 1=~ AH A H.. Note Note that that AH A H isis purely purely existential. existential.

Introduction Introduction to to Proof Proof Theory Theory

51 51

Proposition. P r o p o s i t i o n . Let Let T T be be a a set set of o.f prenex prenex formulas formulas and and A A any any prenex prenex formula. formula. Then Then

the the following following are are equivalent: equivalent: (1) (1) T T ~FAA, , Ts F ~ A, A, (2) TS (3) T T F ~ A AH H,, Ts F ~ A AH H ,, (4) TS

This proposition is easily easily proved proved from from the the above above definitions and remarks. The This proposition is definitions and remarks. The importance of of the the proposition proposition lies lies in in the the fact fact that that TS T s is is a a universal universal theory theory and and importance that A AH H is is an an existential existential formula, formula, and and that that therefore therefore Herbrand Herbrand's's theorem theorem applies applies that to TS Ts F ~ A AH H.. Therefore, Therefore, Herbrand Herbrand's' s theorem theorem can can be be applied applied to to an an arbitrary arbitrary logical logical to implication implication T T F ~A A,, at at the the cost cost of of converting converting formulas formulas to to prenex prenex form form and and introducing introducing Herbrand Herbrand and and Skolem Skolem functions. functions.

2.5.3. 2.5.3. A A generalized g e n e r a l i z e d Herbrand's H e r b r a n d ' s theorem. t h e o r e m . Herbrand Herbrand actually actually proved proved aa more more general general version version of of Theorem Theorem 2.5.1 2.5.1 which which applied applied directly directly whenever whenever F ~A A,, for for A A aa general general formula, necessarily existential. His result formula, not not necessarily existential. His result avoids avoids the the use use of of Skolem Skolem/Herbrand /Herbrand functions functions but but is is somewhat somewhat more more difficult difficult to to state state and and comprehend. comprehend. The The theorem theorem we we ' s theorem state state next next is is aa generalization generalization of of Herbrand Herbrand's theorem which which is is quite quite similar similar in in spirit spirit and and power power to to the the theorem theorem as as stated stated originally originally by by Herbrand Herbrand [1930] [1930].. In section, we shall consider In this this section, we shall consider a a first-order first-order formula formula A A such such that that F ~A A.. Without Without loss loss of of generality, generality, we we shall shall suppose suppose that that the the propositional propositional connectives connectives in in A A are are restricted to that the the -, only in restricted to be be /\ A,, V V and and -' -~,, and and that -~ connective connective appears appears only in front front of of The only atomic atomic subformulas subformulas of of A A.. ((The only reason reason for for this this convention convention is is that that it it avoids avoids having ers appear having to to keep keep track track of of whether whether quantifi quantifiers appear positively positively and and negatively negatively in in A A.).) Definition. Let negations. An -expansion of Definition. Let A A satisfy satisfy the the above above convention convention on on negations. An V V-expansion of A A is applications of is any any formula formula that that can can be be obtained obtained from from A A by by a a finite finite number number of of applications of the the following following operation: operation: (a) -expansion A' (a) If If B B is is aa subformula subformula of of an an V V-expansion A ~ of of A A,, replacing replacing B B in in A' A ~ with with B BV VB B produces another produces another V-expansion V-expansion of of A A.. A -expansion of defined similarly, A strong strong V V-expansion of A A is is defined similarly, except except that that now now the the formula formula B B is is restricted restricted to to be be aa subformula subformula with with outermost outermost connective connective an an existential existential quantifier. quantifier. Definition. Let Definition. Let A A be be a a formula. formula. A A prenexification prenexification of of A A is is aa formula formula obtained obtained from bound variables from A A by by first first renaming renaming bound variables in in A A so so that that no no variable variable is is quantified quantified more operations to more than than once once in in A A and and then then using using prenex prenex operations to put put the the formula formula in in prenex prenex normal normal form. form.

Note Note that that there there will will generally generally be be more more than than one one prenexification prenexification of of A A since since prenex prenex operations operations may may be be applied applied in in different different orders orders resulting resulting in in aa different different order order of of the the quantifiers in in the the prenex prenex normal normal form form formula. formula. quantifiers

s. S. Buss Buss

52

Definition. be aa valid valid first-order prenex normal no Definition. Let Let A A be first-order formula formula in in prenex normal form, form, with with no variable quantified quantified twice A.. If If A A has _ 0 existential existential quantifiers, quantifiers, then A is is of of variable twice in in A has rr � then A the following with B the following form form with B quantifier-free: quantifier-free:

(VXl""" Xnl)(:::]yl)(VXnl+I

" " " Xn2)(~y2)

" " " (:~Yr)(VXnr+l

* " " Xnr+,)B(:~,

Y~

with with 0 � _< n nll � _~ n n22 � _ < .' ". . � _< nr nr+l. A witnessing witnessing substitution substitution for for A A is is a a sequence sequence of of +1 . A actually, semiterms terms terms ((actually, semiterms)) ttb l , .. .., . tr tr such such that that (1) each each ttii contains contains arbitrary arbitrary free free vari variables but only bound variables from ables but only bound variables from X Xbl , .. .. .., , xn• xn, and and (2) the the formula formula B(x, B(Z, tt lt ,, .. ...., , t.) t~) is i.e., aa tautological is a a quasitautology quasitautology ((i.e., tautological consequence consequence of of instances instances of of equality equality axioms axioms only ) . In sign, then (2) is only). In the the case case where where B B does does not not contain contain the the equality equality sign, then (2) is equivalent equivalent to being aa tautology. to B B being tautology. Let rst-order theory. over T Let T T be be a a fi first-order theory. A A sequence sequence of of terms terms is is said said to to witness witness A A over T if if replaced by the hold except condition (2) replaced the above above conditions conditions hold except with with condition by the the weaker weaker condition condition that that T T 1= ~ (Vx)B(x, (V~)S(x, i) t-).. Definition. A A Herbrand Herbrand proof proof of of a a first-order first-order formula formula A A consists consists of of a a prenexifi prenexificaDefinition. ca *. tion substitution (7a for tion A A** of of a a strong strong V-expansion V-expansion of of A A plus plus a a witnessing witnessing substitution for A A*. -proof of A A Herbrand Herbrand T T-proof of A A consists consists of of a a prenexification prenexification A A** ofa of a strong strong V-expansion V-expansion of substitution which A over of A A plus plus a a substitution which witnesses witnesses A over T T..

We We are are now now iinn aa position position to to state state the the general general form form of of Herbrand Herbrand's's theorem: theorem: Theorem. Theorem. A A first-order first-order formula formula A A is is valid valid if if and and only only if if A A has has aa Herbrand Herbrand proof. proof. and only has aa H erbrand More More generally, generally, if if T T is is aa universal universal theory, theory, then then T T 1= ~A A if if and only if if A A has Herbrand

T T -proof. -proof.

Proof. the frst first part part of the theorem the proof P r o o f . We We shall shall sketch sketch a a proof proof of of only only the of the theorem since since the proof of of the the second second part part is is almost almost identical. identical. Of Of course course it it is is immediate immediate from from the the definitions definitions that that if if A A has has a a Herbrand Herbrand proof, proof, then then A A is is valid. valid. So So suppose suppose A A is is valid, valid, and and therefore therefore shall modify has -proof P has a a cut-free cut-free LK LK-proof P.. We We shall modify P P in in stages stages so so as as to to extract extract a a Herbrand Herbrand proof proof of of P P. . The will involve The first first stage stage will involve restricting restricting the the formulas formulas which which can can be be combined combined by by aa contraction One problem that arises in this that inferences contraction inference. inference. One problem that arises in this regard regard is is that inferences implicit contractions with with two two hypothesis, hypothesis, such such as as V V :left :left and and 1\ A :right, :right, contain contain implicit contractions on on side side formulas. formulas. To To avoid avoid dealing dealing with with this this complication, complication, we we modify modify the the rules rules of of inference inference so occur; e.g., so that that no no implicit implicit contractions contractions occur; e.g., the the V V :left :left inference inference rule rule is is replaced replaced by by the the rule rule A, -t � B, -t �' A, f F--~ A B, f' F'---~ A' A -t �, �'' AV V B, B, f, P, f' P'--~ A,A

and and the the 1\ A :right :right and and :::> D :left :left are are modified modified analogously. analogously. It It is is easy easy to to see see that that changing changing the the rules rules of of inference inference in in this this way, way, makes makes no no difference difference to to what what strong strong inferences inferences are are needed needed in in aa proof: proof: instead, instead, it it merely merely makes makes contractions contractions explicit. explicit. In In particular, particular, the the cut elimination theorems cut elimination theorems still still hold hold with with the the new new inference inference rules. rules.

Introduction Introduction to Proof Proof Theory Theory

53

A resp., an A contraction contraction inference inference is is said said to to be be aa propositional propositional contraction contraction ((resp., an :3 3-contraction) contraction) provided provided that that the the principal principal formula formula of of the the contraction contraction is is quantifier quantifierfree resp., its ) . The free ((resp., its outermost outermost connective connective is is an an existential existential quantifier quantifier). The first first step step in in modifying modifying P P is is to to form form a a cut-free cut-free proof proof PI P1,, also also with with endsequent endsequent --+ --~ A A such such that that all all contraction contraction inferences inferences in in PI P1 are are propositional propositional or or 3-contractions. 3-contractions. The The construction construction of of PI P1 from from P P is is formally formally done done by by an an induction induction process process analogous analogous to to the the proof proof of of the the Cut-elimination ne the -depth of Cut-elimination Theorem Theorem 2.4.2. Namely, Namely, defi define the E E-depth of aa formula formula similarly similarly to to the definition definition of of depth in in 2.4.1, except except define define the the E-depth E-depth of of aa quantifier-free quantifier-free formula formula the

or or of of aa formula formula with with outermost outermost connective connective an an existential existential quantifier quantifier to to be be zero. zero. Then Then prove, by by double double induction induction on on the the maximum maximum E-depth E-depth d d of of contraction contraction formulas formulas prove, and and the the number number of of contractions contractions of of formulas formulas of of this this maximum maximum E-depth, E-depth, that that PI P1 can can be -depth be transformed transformed into into a a proof proof in in which which all all contractions contractions are are on on formulas formulas of of E E-depth zero. The The induction step consists consists of of removing removing aa topmost topmost contraction contraction inference inference of of zero. induction step the the maximum maximum E-depth E-depth d d.. For For example, example, suppose suppose that that the the following following inference inference is is aa topmost principal formula topmost contraction contraction with with principal formula of of E-depth E-depth d; d;

. . ..R rr ---++ �, ((W)B, Vx ) B, ((Vx)B Vx) B A, ((W)B r --+ �, Vx) B 9

*

. o R

Since Since PI P1 is is w.l.o.g. w.l.o.g, in in free free variable variable normal normal form form and and since since this this is is a a topmost topmost con contraction E-depth dd,, we traction of of E-depth we can can modify modify the the subproof subproof R R of of PI P1 by by removing removing at at :right inferences and/or changing most most two two V V:right inferences and/or changing some some Weakening:right Weakening:right inferences inferences to to get --+ �, get a a proof proof of of r F--~ A, B(a) U(a),, B(a') B(a'),, where where aa and and a' a' are are free free variables variables not not appearing appearing in R. Further in the the endsequent endsequent of of R. Further replacing replacing a' a ~ everywhere everywhere by by aa gives gives aa proof proof of of r --+ �, this get proof ending: F--~ A, B(a), B(a), B(a) B(a)": we we use use this get to to a a proof ending: ".

9 9 .

. 9

rr ---+ + �, B(a) , B(a) --+ �, A, B(a) S(a) rP--~ rr ---++ �, A, ((Vx)B Vx) B Thus -depth of similar procedure Thus we we have have reduced reduced the the E E-depth of the the contraction contraction inference. inference. A A similar procedure works works for for contractions contractions of of E-depth E-depth d d with with outermost outermost connective connective aa propositional propositional connective-we connective--we leave leave the the details details to to the the reader. reader. Note Note that that the the construction construction of of PI P1 depends on the fact that depends on the fact that propositional propositional inferences inferences and and V:right V:right inferences inferences can can be be pushed pushed downward downward in in the the proof. proof. It It is is not not generally generally possible possible to to push push 33 :right :right inferences inferences downward violating eigenvariable downward in in aa proof proof without without violating eigenvariable conditions. conditions. The The second second step step in in modifying modifying P P is is to to convert convert PI P1 into into aa cut-free cut-free proof proof P P22 of of some some strong strong V-expansion V-expansion A' A' of of A A such such that that every every contraction contraction in in P P22 is is propositional. propositional. This This is is done done by by the the simple simple expedient expedient of of replacing replacing every every 3-contraction 3-contraction in in Pt P1 with with an an V V :right :right inference, inference, and and then then making making the the induced induced changes changes to to all all descendents descendents of of the the principal principal formula formula of of the the inference. inference. More More precisely, precisely, starting starting with with a a lowermost lowermost 3-contraction in in PI P1,, say say 3-contraction rr---+ --+ �, A, ((3x)S, 3x) B, ((3x)S :3x) B ~ �, A, ((3x)B rF --+ :3x) B

54 54

S. S. Buss Buss

replace this this with replace with an an V V :left :left inference inference f ---+ b.. F'--+ A,, (3x)B, ( 3 x ) S , (3x)B (3x)S f ---+ b.. F-"+ A,, (3x)B (3x)S V V (3x)B (3x)S and then, then, in in order order to to get get aa syntactically syntactically correct correct proof, proof, replace, replace, as as necessary, necessary, subfor subforand mulas (3x)B' in P with (3x)B' we use the notation mulas ( 3 x ) B ' of of formulas formulas in P with (3x)B' V V (3x)B' ( 3 x ) S ' ((we use the notation B' B' since since terms in in B B may may be be different different in in the the descendents descendents).) . Iterating Iterating this this process process yields yields the the terms desired P2 of -expansion A' bound variables desired proof proof/)2 of aa strong strong V V-expansion A' of of A A.. By By renaming renaming bound variables in in P P22 we we can can assume assume w.l.o.g. w.l.o.g, that that no no variable variable is is quantified quantified twice twice in in any any single single sequent sequent in in P P2. 2• Thirdly, Thirdly, from from P P22 we we can can construct construct aa prenexification prenexification A A** of of A' A' together together with with aa witnessing witnessing substitution, substitution, thereby thereby obtaining obtaining aa Herbrand Herbrand proof proof of of A A.. To To do do this, this, we we iterate iterate the the following following procedure procedure for for pulling pulling quantifiers quantifiers to to the the front front of of the the proved proved which has already been formula. Find any quantifier inference formula. Find any lowest lowest quantifier inference in in P P22 which has not not already been handled: handled: this this quantifier quantifier inference inference corresponds corresponds to to aa unique unique quantifier, quantifier, (Qx) (Qx),, appearing appearing in the the endsequent endsequent of of the the proof proof ((and conversely, each each quantifier quantifier in in the the endsequent endsequent of of and conversely, in the proof corresponds quantifier inference, since all the proof corresponds to to aa unique unique quantifier inference, since all contraction contraction formulas formulas are (Qx) as are quantifier-free quantifier-free).) . Use Use prenex prenex operations operations to to pull pull (Qx) as far far to to the the front front of of the the endsequent but not endsequent formula formula as as possible possible ((but not past past the the quantifiers quantifiers that that have have already already been been Also, push moved moved to to the the front front of of the the endsequent endsequent formula formula).) . Also, push the the quantifier quantifier inference inference downward until it downward in in the the proof proof until it reaches reaches the the group group of of quantifier quantifier inferences inferences that that have have already been pushed pushed downward already been downward in in the the proof. proof. It It is is straightforward straightforward to to check check that that this this procedure procedure preserves preserves the the property property of of having having aa syntactically syntactically valid valid proof. proof. When When we we are are P3 of done done iterating iterating this this procedure, procedure, we we obtain obtain aa proof proof/)3 of aa prenexification prenexification ---+ --+ A A** of of A A.. It It remains remains to to define define aa witnessing witnessing substitution substitution for for A* A*,, which which is is now now easy: easy: for for each each existential quantifier (3 * , find find the existential quantifier (3y/) in A A*, the corresponding corresponding 33 :right :right inference inference Yi) in r--+ b.. A,, B(ti) B(t ) f---+ Lx,, (3 (3y,)B(y,) fr--+ Yi)B(Yi) ---+ b.. be from this inference. and let the term t and let the term tii be from this inference. That That this this is is aa witnessing witnessing substitution substitution for for A A** is is easily easily proved proved by by noting noting that that by by removing removing the the 33 :right :right inference inference from from P Pa, 3 , aa obtained where portion of proof proof of of A�(x A~(Z,, i) t-) is is obtained where A� A~ is is the the quantifier-free quantifier-free portion of A* A*.. 0 [::1

The obtain the The above above theorem theorem can can be be used used to to obtain the following following 'no-counterexample 'no-counterexample inter interpretation bounded arithmetic arithmetic ((see see pretation'' which which has has been been very very useful useful recently recently in in the the study study of of bounded Krajicek, section 3.3.2 of 1991] ' or Kraji~ek, Pudhik Pudls and and Takeuti Takeuti [[1991], or section of Chapter Chapter 11) II)..55 Corollary. Corollary. Let Let T T be be a a universal universal theory theory and and suppose suppose T T F ~ (3x) ( 3 x ) ((Vy)A(x, V y ) A ( x , y, y, C) ~ with with

A A a a quantifier-free quantifier-free formula. formula. Then Then there there is is aa k k > > 00 and and terms terms tl t l ((C) ~ ,, t2 t2(g, ( C, Yyl), l) , tt3(c', 2) , . . . , tk ( C, Yb 3 ( C, Yb Yl, YY2),..., tk(~, Yx, '. .. .. Yk-l) Yk-1) such such that that T F (VYl) [A(t1(C) , Yl , C) V v (VY2) [A(t2(c, yd, Y2, C)V T (V 2) , YY3, (Vy3)[A(ta(g, Yl,, Y Y2), c-*)V V ·. ·. .· V V (V (Vyk)[A(t~(g, yl,, ". . ". , Y yk-1), c-))]...]]] k - l) , yk, Yk, C)) Yk ) [A(tk(C, Yl ] . . .]]] Y3 ) [A(t3 (c, Yl 3 , C)

5This SThis corollary corollary is named named after after the more more sophisticated sophisticated no-counterexample no-counterexample interpretations interpretations of Kreisel Kreisel [1951,1952]. [1951,1952].

Introduction to to Proof Proof Theory Theory Introduction

55 55

To prove prove the the corollary, corollary, note note that that the the only only strong strong V-expansions V -expansions of of AA are are formulas formulas To and apply apply the the previous previous theorem. theorem. of the the form form VV(3x) (Vy)A(x, y,y, ~C) and of (3x)(Vy)A(x, 2.5.4. NNo is interesting interesting to to ask ask 2.5.4. o rrecursive e c u r s i v e bbounds o u n d s oon n nnumber u m b e r oof f tterms. erms. ItIt is whether itit isis possible possible to to bound bound the the value value of of rr in in Herbrand's Herbrand 's Theorem Theorem 2.5.1. 2.5.1. For For whether this, consider consider the the special special case case where where the the theory theory TT isis empty, empty, so so that that we we have have an an this, of ((3X x) where where BB isis quantifier-free. quantifier-free. There There are are two two ways ways xk)B(ii, ~) LLK K - -proof p r o o f PP of 3 Xll ,, .'. .. ,. , xk)B(g, which one one might might wish wish to to bound bound the the number number rr needed needed for for Herbrand's Herbrand's theorem: theorem: in in which as aa function function of of the the size size of of PP, , or or alternatively, alternatively, as as aa function function of of the the size size of of the the as For the the first first approach, approach, itit follows follows immediately immediately from from Theorem Theorem 2.4.3 2.4.3 formula formula (3~)B. (3x)B . For 2�lf»II ' For For the the second second approach approach, we we theorem, that that rr -<:::; "211Pl1" and 911PI[ and the the proof proof of of Herbrand's Herbrand ' s theorem, sketch aa proof proof below below that that rr can can not not be be recursively recursively bounded bounded as as aa function function of of the the shall sketch shall formula (3~)B. ( 3x)B . formula we To show show that that rr cannot cannot be be recursively recursively bounded bounded as as aa function function of of (3~)B, ( 3x)B , we To would give give aa decision decision procedure procedure shall prove that having having aa recursive recursive bound bound on on rr would shall prove that determining ifif aa given formula isis valid. valid. Since Since itit is is well well known known that that for determining for given existential existential formula validity of existential first-order formulas formulas isis undecidable, undecidable, this this implies implies that that rr cannot cannot validity of existential first-order be recursively bounded in in terms terms of of the the formula formula size. size. be recursively bounded What shall show that, given given aa formula formula BB as in Theorem 2.5.1 and given W h a t we we shall show is is that, as in Theorem 2.5.1 and given which make the formula formula > 0, 0 , itit isis decidable decidable whether whether there there are are terms terms t1 tr,k which an an rr > t l ,, !1,, .. .. .., , tr,k make the r

B(ii, titi,1,...,ti,~) ~lB(d, ,! , . . . , ti,k) V i=1 i=1

(1) (1)

aa tautology. that rr cannot tautology. 66 This This will will suffice suffice to to show show that cannot be be recursively recursively bounded. bounded. The The quantifier-free . . . , Dl ) quantifier-free formula formula B B is is expressible expressible as as aa Boolean Boolean combination combination C(D!, C(D1,...,DI) where where each each Di Di is is an an atomic atomic formula formula and and C(· C ( . .·. )·) is is aa propositional propositional formula. formula. If If the the formula Di(ii, titi,1,..., being !, , . . . , tit~,k) k) formula (1) (1) is is aa tautology, tautology, it it is is by by virtue virtue of of certain certain formulas formulas Di(g, being , identical. identical. That T h a t is is to to say say there there is is aa finite finite set set X X of of equalities equalities of of the the form form Di(g, ti,1,..., ti,k) = Di, (g, ti,,1,..., ti,,~)

such such that, that, any any set set of of terms terms t1 t l ,,!1, ,.. .. .., , tr tr,k which makes makes all all the the equalities equalities in in X X true true will will ,k which make make (1) (1) aa tautology. tautology. But But now now the the question question of of whether whether there there exist exist terms terms ttl,1, which satisfy satisfy such such ,k which 1 , ! , . 9. 9. 9, trtr,k aa finite finite set set X X of of equations equations is is easily easily seen seen to to be be aa first-order first-order unification unification problem, problem, as as described described in in section section 2.6.1 2.6.1 below. below. This This means means that that there there is is an an algorithm algorithm which which can can either either determine determine that that no no choice choice of of terms terms will will satisfy satisfy all all the the equations equations in in X X or or will will find aa most most general general unifier unifier which which specifies specifies all all possible possible ways ways to to satisfy satisfy the the equations equations find of of X X .. Since, Since, for for aa fixed fixed rr > > 00,, there there are are only only finitely finitely many many possible possible sets sets X X of of equalities, equalities, we we have have the the following following algorithm algorithm for for determining determining if if there there are are terms terms which which make make (1) (1) aa tautology: i.e., aa tautology: for for each each possible possible set set X X of of equalities, equalities, check check if if it it has has aa solution solution ((i.e., 66 This This was was first first proved proved by by Herbrand Herbrand [1930] [1930]by by the the same same argument argument that that we wesketch sketch here. here.

56 56

S. Buss Buss S.

most general general unifier) unifier),, and and if if so, so, check check if if the the equalities equalities are are sufficient sufficient to to make make (1) (1) aa most tautology. D [] tautology.

2.5.5. Interpolation Interpolation theorem theorem 2.5.5. Suppose we we are are given given two two formulas formulas A A and and B B such such that that A A ::> D B B is is valid. valid. An An Suppose interpolant for for A A and and B B is is aa formula formula C C such such that that A A ::> D C C and and C C ::> D B B are are both both interpolant

valid. It It is is aa surprising, surprising, and and fundamental, fundamental, fact fact that that it it is is always always possible possible to to find find an an valid. interpolant C C such such that that C C contains contains only only non-logical non-logical symbols symbols which which occur occur in in both both interpolant A and and B B.. A We shall shall assume assume for for this this section section that that first-order first-order logic logic has has been been augmented augmented to to We include the the logical logical symbols symbols T T and and .1 _l_. For this, this, the the sequent sequent calculus calculus has has two two new new include . For initial sequents ---+ ~ T T and and .1 2_---+. We write write L(A) L(A) to to denote denote the the set set of of non-logical non-logical ---+ . We initial sequents symbols occurring occurring in in A A plus plus all all free free variables variables occurring occurring in in A A,, i.e., i.e., the the constant, constant, symbols symbols, function function symbols, symbols, predicate predicate symbols symbols and and free free variables variables used used in in A A.. For For II II aa symbols, cedent, L(II) L(II) is is defined defined similarly. similarly. cedent,

Craig's Interpolation Interpolation Theorem. Theorem. Craig Craig [1957a] [1957a].. Craig'S (a) Let Let A A and and B B be be first-order first-order formulas formulas such such that that F ~ A D B B.. Then Then there there is is aa A ::> (a) formula C C such such that that L(C) L(C) � C_ L(A) L(A) n N LL(B) ( B ) and and such such that that F ~ A D C C and and formula A ::>

F~ CCD::>B .B .

(b) Suppose such (b) Suppose r F1, ~ L}.l, A1, L}. A22 is is aa valid valid sequent. sequent. Then Then there there is is aa formula formula C C such I > rF22 ---+ ) and such that rl , C and that L(C) L(C) is is aa subset of L(rl' L(r~, L}.l) A1) n n L(r L(F2, A2) and such that F1 --~ A1, C and that subset of , L}. ---+ L}.l 2 2 C, r F22 ---+ ~ L}. A22 are are both both valid. valid. C, Craig's interpolation can be straightforwardly from the cut elimination Craig's interpolation can be proved proved straightforwardly from the cut elimination theorem. We some of key points points of of the the proof, proof, but but leave full proof proof theorem. We shall shall outline outline some of the the key leave aa full to the reader. First it easy to see that that part part (a) (a) is is just just a a special special case case of of (b), (b), so so it it to the reader. First it is is easy to see suffices to to prove prove (b) (b), we first use to obtain suffices (b).. To To prove prove (b), we first use the the cut cut elimination elimination theorem theorem to obtain r2 ---+ A1, cut-free proof P of A 22 .. We then prove prove by by induction number of F1, rl > F2--~ We then induction on on the the number aa cut-free proof P L}.l , L}. of strong inferences in that there there is is aa formula formula C C with with only only the the desired desired non-logical non-logical of strong inferences in PP that Pl and and P2 P2 of of rF11 ---~ ---+ L}. 1 , CC and r2 ---+ fact, the symbols and there proofs P1 symbols and there are are proofs A1, and C, C, F2 ---~ L}. A2. In fact, the 2 . In proofs P P2 are are also cut-free and and have bounded by by the proofs P1l and and P2 also cut-free have lengths lengths linearly linearly bounded the length length ' s make of P an example example of of how the proof by induction induction goes, simplifying of P. . For For an how the proof by goes, let let's make the the simplifying assumption that no function function symbols our languages, and then assumption that there there are are no symbols in in our languages, and then assume assume that :right inference that the the final final strong strong inference inference of of PP is is an an 33 :right inference with with principal principal formula formula in L}.2 . That That is is to to say, say, suppose suppose PP ends ends with with the the inference inference in A2. . . . . . r~, r~--+ A,, A~, A(t) r l > F2 r2 ~---+ A1, L}. 1> L}.�, (3x)A(x) F1, A~, (3x)A(x) with L}. A22 is we are are assuming are no with is the the cedent cedent A~, L}.�, (3x)A(x). (3x)A(x) . Since Since we assuming there there are no function function symbols, tt is symbols, variable or constant symbol. is just just aa free free variable or aa constant symbol. The The induction induction hypothesis hypothesis states C(t) with with an an appropriate appropriate first-order first-order language language such such states that that there there is is an an interpolant interpolant C(t) 9 .

.

~

: ~

~

9

Introduction to Proof Theory

57 57

that F1 �2 , A(t) A(t) are r l ~----7 A1, are LK-provable. LK-provable. The The interpolant interpolant C* C* and C(t), C(t), F2 r2 ~----7 A2, �I , C(t) C(t) and that for the the endsequent endsequent of of PP isis defined defined as as follows: follows: ifif the the symbol symbol tt does does not not appear appear in in the the for r2 ~----7 A2, �2 ' then then C* C * isis (3y)C(y); (3y)C(y) ; otherwise, otherwise, ifif the the symbol symbol tt does does not not appear appear sequent F2 sequent in the the sequent sequent F1 rl ~----7 A1, �I ' then then C* C * isis (Vy)C(y); (Vy)C(y) ; and and ifif tt appears appears in in both both sequents, sequents, in C* isis just just C. C . ItIt can can be be checked checked that that in in all all three three cases, cases, the the sequents sequents F~ r I --~ ----7 A~, �I ' C* C * and and C* an interpolant interpolant for for the the C*,F2---~A2, (3x)A(x) are are LK-provable. LK -provable. Therefore, Therefore, C* C * isis an C * , r2 ----7 �2' (3x)A(x) P ; also, also, itit is is obvious obvious that that the the language language L(C) L( C) of of C C isis still still appropriate appropriate endsequent of of P; endsequent for the the endsequent. endsequent. for Secondly, suppose suppose PP ends ends with with the the inference inference Secondly, ~

o ~

o

~

r,, A(b) rI, F2 r2 --~ ----7 A~, �1, A'2, ��, (Vx)A(x) F~, with the the principal formula still still presumed presumed to to be be in in A2. �2 . The The induction hypothesis with principal formula induction hypothesis states that there there is an interpolant interpolant C with an an appropriate first-order language states that is an C with appropriate first-order language such such and C(b), C(b) , F2 r2 --~ ----7 � A(b) . Since, by the eigenvariable condition, condition, that rF1l ----7 that ---~ �I A1,, C(b) and A2, Since, by the eigenvariable 2 , A(b). bb does except as as indicated; indicated; we we get get immediately immediately LK-proofs the sequents does not not occur occur except LK -proofs of of the sequents r l ----7 �I, (Vy)C(y) and and (Vy)C(y), F2 r2 ~----7 A2, �2' (Vx)A(x). Therefore, (Vy)C(y) serves (Vx)A(x) . Therefore, F1 ~ A1, serves P. as for the the endsequent endsequent of of P. as an an interpolant interpolant for

There are are aa number number of of other other cases cases that that must must bbee considered, considered, depending depending on on the the There type the final in P type of of the final strong strong inference inference in P and and on on whether whether its its principal principal formula formula is is (an (an ancestor ancestor of) of) aa formula formula in in �I A1,' � A2, or r F2. These cases cases are are given given in in full full detail detail in in 2 ' rF1I or 2 . These textbooks textbooks such such as as Takeuti Takeuti [1987] [1987] or or Girard Girard [1987b]. [1987b]. It consider the It remains remains to to consider the case case where where there there are are function function symbols symbols in in the the language. language. The The usual usual method method of of handling handling this this case case is is to to just just reduce reduce it it to to the the case case where where there there are are no no function function symbols symbols by by removing removing function function symbols symbols in in favor favor of of predicate predicate symbols symbols which ne the which defi define the graphs graphs of of the the functions. functions. Alternatively, Alternatively, one one can can carry carry out out directly directly aa proof proof on on induction induction on on the the number number of of strong strong inferences inferences in in P P even even when when function function symbols symbols are are present. present. This This involves involves aa more more careful careful analysis analysis of of the the 'flow 'flow'' of of terms terms in in the the proofs, proofs, but but still still gives gives cut-free cut-free proofs proofs PI P1 and and P P22 with with size size linear linear in in the the size size of of P P.. We We leave leave the the details details of of the the function-symbol function-symbol case case to to the the reader. reader.

Other O t h e r interpolation i n t e r p o l a t i o n theorems. theorems. A A useful useful strengthening strengthening of of the the Craig Craig interpolation interpolation theorem theorem is is due due to to Lyndon Lyndon [1959]. [1959]. This This theorem theorem states states that that Craig's Craig's interpolation interpolation theorem theorem may may be be strengthened strengthened by by further further requiring requiring that that every every predicate predicate symbol symbol which which occurs occurs positively positively (resp., (resp., negatively) negatively) in in C C also also occurs occurs positively positively (resp., (resp., negatively) negatively) in in both both A A and and B B.. The The proof proof of of Lyndon's Lyndon's theorem theorem is is identical identical to to the the proof proof sketched sketched above, above, except except that that now now one one keeps keeps track track of of positive positive and and negative negative occurrences occurrences of of predicate predicate symbols. symbols. Craig Craig [1957b] [1957b] gives gives aa generalization generalization of of the the Craig Craig interpolation interpolation theorem theorem which which applies applies to to interpolants interpolants of of cedents. cedents. Lopez-Escobar Lopez-Escobar [1965] [1965] proved proved that that the the interpolation interpolation theorem theorem holds holds for for some some in infinitary proved finitary logics. logics. Barwise Barwise [1975,§III.6] [1975,w proved that that the the interpolation interpolation theorem theorem holds holds for for aa wider wider class class of of infinitary infinitary logics. logics. Lopez-Escobar's Lopez-Escobar's proof proof was was proof-theoretic, proof-theoretic,

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S. Buss s.

based on on aa sequent sequent calculus calculus formalization formalization of of infinitary infinitary logic. logic. Barwise's Barwise's proof proof was was based model-theoretic; Feferman Feferman [1968] [1968] gives gives aa proof-theoretic proof-theoretic treatment treatment of of these these general general model-theoretic; interpolation theorems, theorems, based based on on the the sequent sequent calculus. calculus. interpolation 2.5.6. Beth's B e t h ' s definability definability theorem theorem 2.5.6. Definition. Let Let P P and and pi P~ be be predicate predicate symbols symbols with with the the same same arity. arity. Let Let r(p) F(P) be be Definition. i , and an arbitrary arbitrary set set of of first-order first-order sentences sentences not not involving involving p P~, and let let r(pl) F(P ~) be be the the same same an set of of sentences sentences with with every every occurrence occurrence of of P P replaced replaced with with P' P~.. set The set r(P) is said to explicitly define the predicate P if there there is is aa formula formula A(~ The set r (P) is said to explicitly define the predicate P if A( C) such that that such r(P) I-~ (V' (V~)(A(~) ++ P( P(~)). r(p) X')). X') (A(X') f-t The set set r(p) F(P) is is said said to to implicitly define the the predicate predicate P P if if The

r(P) U u r(P')~ (v~)(P(~) +, PI( P'(~)). r(P) r(pl) 1= (V' X')). X') (P(X') f-t

The Definability Definability Theorem Theorem of of Beth Beth [1953] [1953] states states the the fundamental fundamental fact fact that that the the The notions of of explicit explicit and and implicit implicit definability definability coincide. coincide. One One way way to to understand understand the the notions importance of of this this is is to to consider consider implicit implicit definability definability of of P P as as equivalent equivalent to to being being importance able to to uniquely uniquely characterize characterize P P.. Thus, Thus, Beth's Beth's theorem theorem states, states, loosely loosely speaking, speaking, that that able if aa predicate predicate can can be be uniquely uniquely characterized, characterized, then then it it can can be be explicitly explicitly defined defined by by aa if formula not not involving involving P P.. formula One One common, common, elementary elementary mistake mistake is is to to confuse confuse implicit implicit definability definability by by aa set set of of with implicit sentences r(P) sentences F(P) with implicit definability definability in in aa particular particular model. model. For For example, example, consider consider the theory theory T T of of sentences sentences which which are are true true in in the the standard standard model model (N, (N, 0, 0, S, S, +) +) of of natural natural the numbers with with zero, zero, successor successor and and addition. addition. One One might might attempt attempt to to implicitly implicitly define define numbers multiplication in terms of zero and theory multiplication in terms of zero and addition addition letting letting F(M) r(M) be be the the theory

x)} TTUU {(Vx)(M(x,O) M(x,y)y) + x)) {(V'x) (M(x, 0) = 0), 0), (Vx)(Vy)(M(x,S(y)) (V'x) (V'y) (M(x, S(y)) = M(x, It is is true uniquely characterizes the multiplication multiplication function function M(x, in the It true that that this this uniquely characterizes the M (x, y) y) in the sense that there there is way to expand (N, 0, S, S, +) +) to of r(M); (N, 0, to aa model model of r(M) ; however, however, sense that is only only one one way to expand this implicit definition definition of of M M since since there there are which this is is not not an an implicit are nonstandard nonstandard models models of of T T which have more than one expansion to to a a model of F(M). r (M) . have more than one expansion model of

Beth's Definability TTheorem. r(p) implicitly implicitly defines P if only if it explicitly B e t h ' s Definability h e o r e m . F(P) defines P if and and only if it explicitly defines P defines P.. Proof. Beth's Beth's theorem theorem is is readily readily proved proved from from the the Craig Craig interpolation interpolation theorem theorem as as Proof. follows. First is explicitly definable, then then itit is is clearly clearly implicitly implicitly First note note that that if if PP is follows. explicitly definable, definable. For the assume that compactness, is implicitly implicitly definable. definable. By By compactness, definable. For the converse, converse, assume that PP is r(p) is is aa single single sentence. sentence. Then Then we we may may assume without loss loss of of generality generality that that F(P) we assume without we have that have that r(P) (~ ~ '(~. r(p) A 1= r(P') r(pl) ~:) PPI(C). 1\ PP(C)

By the Craig Interpolation Interpolation Theorem, Theorem, there there is interpolant A(~ A(C) for for r(P) r(P) ^1\ P(c-) By the Craig is aa interpolant P(C) and and 0 F(P') interpolant is desired formula r(pl) D :) P'(~. PI(C) . This This interpolant is the the desired formula explicitly explicitly defining defining P. P. D

Introduction to Proof Theory

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It is is also also possible possible to to prove prove the the Craig Craig Interpolation Interpolation Theorem Theorem from from the the Beth Beth Defi Definability nability It Theorem. In In addition, addition, both both theorems theorems are are equivalent equivalent to to the the model-theoretic model-theoretic Joint Joint Theorem. Consistency Theorem Theorem of of Robinson Robinson [1956] [1956].. Consistency 2.6. F i r s t - o r d e r logic logic and a n d resolution r e s o l u t i o n refutations refutations 2.6. First-order The importance importance of of the the resolution resolution proof proof method method for for propositional propositional logic logic (described (described The in section section 11.3) lies in in large large part part in in the the fact fact that that it it also also serves serves as as aa foundation foundation in .3) lies for theorem-proving theorem-proving in in first-order first-order logic. logic. Recall Recall that that by by introducing introducing Herbrand Herbrand and and for Skolem functions, functions, theorem-proving theorem-proving in in first-order first-order logic logic can can be be reduced reduced to to proving proving Skolem II2-formulas of of the the form form (\fx) (V~)(3y-~A(~, ~ with with A quantifier-free quantifier-free (see (see §2.5.2) w Also, Ih-formulas (:3Y)A(x, Y) . Also, by Herbrand's Herbrand's Theorem Theorem 2.5.1, 2.5.1, the the problem problem of of proving proving (\fx) (VZ)(3~)A(Z, y-~ is is reducible reducible by (:3Y)A(x, Y) to the the problem problem of of fi finding terms Tt, ?1, ?2,..., ?~ so so that that V Vii A A(Z, ~) is is tautologically tautologically to nding terms 1'2, . . . , Tk (x, i;) valid. I, . . . , T k , determining valid. Now, Now, given given the the terms terms T 71,..., ?k, determining tautological tautological validity validity is is 'merely' 'merely' problem in in propositional propositional logic; and hence is amenable amenable to to theorem theorem proving proving methods methods aa problem logic; and hence is such as as propositional propositional resolution. resolution. Thus Thus one one hopes hopes that that if if one one had had aa good good scheme scheme for for such choosing terms terms T r'l,..., ?k,, then then one could have have aa reasonable reasonable method method of of first-order first-order choosing one could I " ' " i'k theorem proving. proving. theorem This latter latter point point is is exactly exactly the the problem problem that that was was solved solved by by Robinson Robinson [1965b]; [1965b]; This namely, he he introduced introduced the the resolution resolution proof proof method method and and showed showed that that by by using using aa namely, unification to select select terms, terms, the the entire entire problem problem of of which which terms terms to to use use could could unifi cation algorithm algorithm to be be solved solved efficiently efficiently by by using using the the "most "most general" general" possible possible terms. terms. In In essence, essence, this this reduces the the problem problem of of first-order first-order theorem theorem proving proving to to propositional propositional theorem theorem proving. proving. reduces (Of (Of course, course, this this last last statement statement is is not not entirely entirely true true for for two two reasons: reasons: firstly, firstly, there there may may be number (not bounded) of are needed, needed, and be a a very very large large number (not recursively recursively bounded) of terms terms that that are and secondly, it is entirely of what what terms are sufficient, might secondly, it is entirely possible possible that that foreknowledge foreknowledge of terms are sufficient, might help guide the for a a propositional help guide the search search for propositional proof.) proof.)

2.6.1. Unification. We now describe describe the for finding finding 2.6.1. Unification. We shall shall now the unification unification algorithm algorithm for unifiers. We We shall term containing containing function most general general unifiers. shall let let tt denote denote a a term function symbols, symbols, most constant symbols symbols and and variables. substitution, a, is aa partial map from constant variables. A A substitution, a, is partial map from variables variables to write xxa a to a ( x ) , , and when x is is not of a a,, we we to denote denote a(x) and when not in in the the domain domain of to terms; terms; we we write let x a be be x. If aa is is aa substitution, ta denotes of simultaneously simultaneously let xa x . If substitution, then then to' denotes the the result result of replacing variable xx in in tt with x a . . We extend aa to atomic relations by replacing every every variable with xa We extend to atomic relations by , t k ) a denote denote RR((ttlla , t ka) . We concatenation aaT to denote denote letting R(tt letting R ( t l ,,. .. .. . ,tk)a a ,,. .. .. . ,tka). We use use concatenation T to the application followed by by an an application the substitution substitution which which is is equivalent equivalent to to an an application application of of aa followed of T. of T. Definition. Let Let A1,... AI, . . . ,Ak , Ak bbee atomic atomic formulas. formulas. A A unifierfor unifier for the the set set (A1,... {AI, " " , AAk} Definition. k} Aka where -= represents represents the the is aa substitution substitution aa such such that that AAla is la -= A2a A2a .=. . . -= A k a , , where property identical formula. property of of being being the the identical formula. A A substitution substitution is is said said to to be be aa variable variable renaming renaming substitution, substitution, ifif the the substitution substitution maps variables only maps variables only to to terms terms which which are are variables. variables. . . .

S. S. Buss Buss

60 60

A said to be aa mmost the same same A unifier unifier a a is is said to be o s t general g e n e r a l unifier u n i f i e r if, if, for for every every unifier unifier T r for for the set, there there is Note that set, is a a unifier unifier p p such such that that T T = - a a pp. . Note that up up to to renaming renaming of of variables, variables, a a most unique. most general general unifier unifier must must be be unique.

Unification Theorem. , . . . , Ad has Unification T h e o r e m . If I f {AI {A1,...,A~} has a a unifier u n i f i e r then t h e n it it has has a a most m o s t general general

unifier. uniter.

Proof. outlining an P r o o f . We We shall shall prove prove the the theorem theorem by by outlining an efficient efficient algorithm algorithm for for determin determining er exists so, finding finding aa most unifier. The ing whether whether a a unifi unifier exists and, and, if if so, most general general unifier. The algorithm algorithm is is described described as as an an iterative iterative procedure procedure which, which, at at stage stage ss has has a a set set Es Es of of equations equations and substitution as and a a substitution as.. The The equations equations in in Es Es are are of of the the form form a c ~== - f3 ~ where where a c~ and and f3 may terms. The may be be formulas formulas or or terms. The meaning meaning of of this this equation equation is is that that the the sought-for sought-for most most general unifier must general unifier must be be a a substitution substitution which which makes makes a c~ and and f3 ~ identical. identical. Initially, Initially, Eo E0 is is the and ao the set set of of k k- 11 equations equations Aj A j == - Aj A j ++I l and a0 is is the the identity. identity. Given Given Es Es and and as as,, the the algorithm algorithm does does any any one one of of the the following following operations operations to to choose choose Es Es+l and as as+l + ! and + ! :: (1) If If Es Es is is empty, empty, we we are are done done and and as as is is a a most most general general unifier. unifier. (1) (2) If Es Es contains contains an an equation equation of of the the form form (2) If

F(tl ti) F ( t l , ' .. .. .. ,, ti)

== " F(t~l, . . . , t~),

F(t�, . . . , t;) ,

then then as as+l = as as and and Es Es+l is obtained obtained from from Es Es by by removing removing this this equation equation and and +l = + ! is adding adding the the ii equations equations ti ti ==" t: t~.. Here Here F F is is permitted permitted to to be be a a function function symbol, symbol, a a constant constant symbol symbol or or a a predicate predicate symbol. symbol. (3) Suppose Suppose Es Es contains contains an an equation equation of of the the form form (3) x - t

or or

tt == - x, x,

with x, then with x x a a variable variable and and tt a a term. term. Firstly, Firstly, if if tt is is equal equal to to x, then this this equation equation ! is Es minus this equation and as ! = as . Secondly, if is discarded, so is discarded, so Es Es+l is Es minus this equation and as+l as. Secondly, if + + tt is is a a non-trivial non-trivial term term in in which which x x occurs, occurs, then then the the algorithm algorithm halts halts outputting outputting that that no no unifier unifier exists.7 exists. 7 Thirdly, Thirdly, if if x x does does not not occur occur in in tt,, then then let let [x/t] I x / t ] denote denote the the substitution substitution that that maps maps x x to to tt and and define define Es Es+l to be be the the set set of of equations equations + ! to s[x/t] ne as six~t] == - s'[x/t] s ' [ x / t ] such such that that ss == - s' s' is is in in E Es, and defi define as+l - as[x/t] as[x/t]. . . , and +! = We We leave leave it it to to the the reader reader to to prove prove that that this this algorithm algorithm always always halts halts with with a a most most general general unifier unifier if if a a unifier unifier exists. exists. The The fact fact that that the the algorithm algorithm does does halt halt can can be be proved proved by by noting noting that that each each iteration iteration of of the the algorithm algorithm either either reduces reduces the the number number of of distinct distinct variables variables in in the the equations, equations, or or reduces reduces the the sum sum of of the the lengths lengths of of the the terms occurring in terms occurring in the the equations. equations, 0 o The algorithm as efficient; however, The algorithm as given given above above is is relatively relatively efficient; however, the the sizes sizes of of the terms terms involved Indeed, there are the involved may may grow grow exponentially exponentially large. large. Indeed, there are unifi unifi- cation problems where exponential in cation problems where the the size size of of the the most most general general unifier unifier is is exponential in the the unification the size size of of the unification problem; problem; for for example, example, the the most most general general unifier unifier for for {J(Xb maps with { f ( x l , x 2 ,X2 . . . x, ~· )· ,· Xk), f f ( g(g(x ( x 2 ,2 ' X2) x 2 ) , ,. .". " , gg( ( xXk k ++l ,b X ~Xk + l+) )!))} } maps Xl Xl to to a a term term of of height height k k with 22kk atoms. atoms. 7This failure failure condition condition is known known as as the occurs occurs check. check.

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If one one is is willing willing to to use use a a dag dag (directed (directed acyclic acyclic graph) graph) representation representation for for terms, terms, If then this this exponential exponential growth growth rate rate does does not not occur. occur. Paterson Paterson and and Wegman Wegman [1978] [1978] have have then given on representing given an an efficient efficient linear-time linear-time unification unification algorithm algorithm based based on representing terms terms as as dags. dags.

We 2.6.2. 2.6.2. Resolution R e s o l u t i o n and a n d factoring f a c t o r i n g inferences. inferences. We now now describe describe the the resolution resolution inference inference used used by by Robinson Robinson [1965b] [1965b] for for first-order first-order logic. logic. The The starting starting point point is is Her Hertheorem: we brand's's theorem: we assume assume that that we we are are attempting attempting to to prove prove a a first-order first-order sentence, sentence, brand which without without loss loss of of generality generality is is of of the the form form (:3x)A(Y) (3Z)A(~7).. (This (This may may be be assumed assumed which without since, if Herbrand functions without loss loss of of generality generality since, if necessary, necessary, Herbrand functions may may be be introduced.) introduced.) We We assume, assume, in in addition, addition, that that A A is is in in disjunctive disjunctive normal normal form. form. Instead Instead of of proving proving this this sentence, we we shall shall instead instead attempt attempt to the sentence sentence (V'x) (VZ)(-~A(x)). Since A A is is (...,A (x)) . Since sentence, to refute refute the in disjunctive disjunctive normal normal form, form, we we may may view view ..., ~A as a a set set r FA of clauses clauses with with the the literals literals in A as A of in the the clauses clauses being being atomic atomic formulas formulas or or negated negated atomic atomic formulas. formulas. We We extend extend the the in definition definition of of substitutions substitutions to to act act on on clauses clauses in in the the obvious obvious way way so so that that {C1 ( C 1,, .. .. .., , Cda Ck}a is ...,C is defined defined to to be be {CIa, (Cla,..., Cka}. k a} . When we we refute refute ((V~)(-~A), we are are showing showing that that there there is is no no structure M which which When structure M V'x) (...,A ) , we satisfies satisfies ((VZ)(-~A). Consider aa clause clause C C in in r FA A.. The The clause clause C C is is a a set set of of atomic atomic V'x) ( ...,A) . Consider and and negated negated atomic atomic formulas, formulas, and and a a structure structure M M is is said said to to satisfy satisfy C provided provided it it satisfies immediate that satisfies ((vz)(Vcecr Thus, it it is is immediate that if if M M satisfies satisfies C C,, and and if if a a is is V'X) (V r/>EC ¢>(X)) . Thus, aa substitution, substitution, then then M M also also satisfies satisfies Ca Ca.. From From this, this, we we see see that that the the following following version of of resolution resolution is is sound sound in in that that it it preserves preserves the the property property of of being being satisfied satisfied by by version aa model model M M": If If B B and and C C are are clauses, clauses, if if ¢> r is is an an atomic atomic formula formula and and if if a a and and 7 T are are substitutions, let D be the the clause {¢>}) UU (CT (C7 \\ {"¢)}) substitutions, let D be clause (Ba \\ (r (r . It It is is easy easy to to verify verify that that if if B B and and C C are are satisfied satisfied in in M M , , then then so so is is D D.. Following use aa restricted Following Robinson Robinson [1965b] [1965b],, we we use restricted form form of of this this inference inference principle principle as as the sole rule rule of first-order resolution the sole of inference inference for for first-order resolution refutations: refutations: Definition. D e f i n i t i o n . Let Let B B and and C C be be clauses clauses and and suppose suppose P(sd P(gl),, P(S P ( g 22 )) ,, .. ... . P(Sk) P(gk) are are atomic atomic formulas P (i;.) , ...,p (i;) , . . . ...,P (i't) are formulas in in B B and and that that ..., ~P(~),-~P(~2),...-~P(Q) are negated negated atomic atomic formulas formulas in in C C.. Choose Choose a a variable variable renaming renaming substitution substitution 7 T so SO that that C7 CT has has no no variables variables in in common common P( Si) and P( �)7 have with with B B.. Also Also suppose suppose that that the the k k+ + £g formulas formulas P(gi) and P(~)T have a a most most general general unifier unifier a a.. Then Then the the clause clause D D defined defined by by

(Ba \\ {P(sda}) { P ( g l ) a } ) Uu (Cm (CTa \\ {""P(sdm}) {-~P(gl)Ta}) is is defined defined to to be be an an R-resolvent R-resolventof of B B and and C C.. The The reason reason for for using using the the renaming renaming substitution substitution 7 T is is that that the the variables variables iinn the the clauses clauses implicitly universally B B and and C C are are implicitly universally quantified; quantified; thus thus if if the the same same variable variable occurs occurs in in both both B B and and C C we we allow allow that that variable variable to to be be instantiated instantiated in in B B by by a a different different term term than than in in C C when when we we perform perform the the unification. unification. Applying Applying 7 T before before the the unification unification allows allows this this to to happen happen automatically. automatically. One One often often views views R-resolution R-resolution as as the the amalgamation amalgamation of of two two distinct distinct operations: operations: first, the the factoring operation operation finds finds a a most most general general unifier unifier of of a a subset subset of of clause, clause, and and first,

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second, the the unitary unitary resolution resolution operation operation which which resolves resolves two two clauses clauses with with respect respect to to second, ( a) choosing choosing subsets subsets of of the the clauses clauses single literal. literal. Thus, Thus, R-resolution R-resolution consists consists of of (a) aa single and factoring factoring them, them, and and then then (b) ( b) applying applying resolution resolution w.r.t, w.r.t. to to the the literal literal B and and CC and B obtained by by the the factoring. factoring. obtained Completeness of R R-resolution. A set set Fr of of first-order first-order clauses clauses isis unsatisfiable unsatisfiable ifif C o m p l e t e n e s s of -resolution. A

and only only ifif the the empty empty clause clause can can be be derived derived from from Fr by by R-resolution. R-resolution. and

This theorem theorem isis proved proved by by the the discussion discussion in in the the next next paragraph. paragraph. This Lifting ground to ffirst-order A ground ground literal literal isis 2.6.3. Lifting 2.6.3. g r o u n d resolution r e s o l u t i o n to i r s t - o r d e r rresolution. esolution. A defined to be be aa literal in which no variables variables occur; occur; aa ground ground clause is aa set set of of ground ground defined to literal in which no clause is literals. We assume, with with no no loss loss of of generality, that our our first-order first-order language language contains contains literals. We assume, generality, that constant symbol symbol and and that that therefore therefore ground ground literals literals exist. exist. Ground Ground literals literals may may aa constant independently be assigned truth values values8s and and therefore play the the same same role independently be assigned truth therefore play role that that literals literals played in propositional propositional logic. logic. A A ground ground resolution resolution refutation propositional-style played in refutation is is aa propositional-style refutation involving ground literals in in place place of propositional refutation involving ground clauses clauses only, only, with with ground ground literals of propositional resolution, aa set set of literals. By By the the Completeness Theorem 1.3.4 for literals. Completeness Theorem for propositional propositional resolution, of ground clauses clauses is is unsatisfiable unsatisfiable if and only only if if itit has has aa ground ground resolution resolution refutation. refutation. ground if and For For sets sets of of ground ground clauses, clauses, R-resolution R-resolution is is identical identical to to propositional-style propositional-style resolu resolution. Suppose, Suppose, however, tion. however, that that r F is is an an unsatisfiable unsatisfiable set set of of first-order first-order ((not necessarily not necessarily ground) clauses. ground) clauses. Since Since r F is is unsatisfiable unsatisfiable there there is, is, by by Herbrand's Herbrand's theorem, theorem, aa set set of of substitutions that each ground clauses clauses and substitutions O'l a l ,, .. ... ., , O'r ar so so that each rO'r Far is is aa set set of of ground and so so that that the the set ally unsatisfi able. Therefore set IT II = -- U [.Jii rO'i Fai of of clauses clauses is is proposition propositionally unsatisfiable. Therefore there there is is a a ground ground resolution resolution refutation refutation of of IT H.. To To justify justify the the completeness completeness of of R-resolution, R-resolution, we we shall shall show show that that any any ground ground resolution resolution refutation refutation of of IT H can can be be 'lifted' 'lifted' to to an an R-resolution R-resolution refutation refutation of of r F.. In In fact, fact, we we shall shall prove prove the the following: following: if if C1, C1, C2, (;'2,...,, Cn Cn = - 00 is is aa resolution resolution refutation refutation of of IT H,, D2, ,,Din Dm = then then there there are are clauses clauses D1, D1,D2,... - 0 which which form form an an R-resolution R-resolution refutation refutation of , 0'2, ' . . ,,am am SO of r F and and there there are are substitutions substitutions 0'1 al,a2,... so that that DiO'i D i a i= - Ci Ci.. We We define define Di Di and and O'i ai by by induction induction on on ii as as follows. follows. Firstly, Firstly, if if Ci Ci E E IT H,, then then it it must must be be equal equal to to DWi Dia~ for nition of for some some Di E Er F and and some some substitution substitution 0'; ai by by the the defi definition of IT H.. Secondly, Secondly, if if C; Ci is is inferred inferred from from Cj Cj and and Ck Ck,, with with j, j, kk < < ii by by resolution resolution w.r.t. w.r.t, the the literal literal P(fj P ( ~ , , then then define define Ej Ej to to be be the the subset subset of of Dj Dj which which is is mapped mapped to to P(fj P ( ~ by by O'j aj,, and and define define Ek Ek similarly. similarly. Now, Now, form form the the R-resolution R-resolution inference inference which which factors factors the the subsets subsets Ej and and Ek E~ of of Dj Dj and and Dk Dk and and forms forms the the resolvent. resolvent. This This resolvent resolvent is is Di and and it it is is straightforward straightforward to to show show that that the the desired desired O'i ai exists. exists. That That finishes finishes the the proof proof of of the the Completeness Completeness Theorem Theorem for for R-resolution. R-resolution. It It should should be be noted noted that that the the method method of of proof proof shows shows that that R-resolution R-resolution refutations refutations are are the the shortest shortest possible possible refutations, refutations, even even if if arbitrary arbitrary substitutions substitutions are are allowed allowed for for factoring factoring infer inferences. ences. Even Even more more importantly, importantly, the the method method by by which which ground ground resolution resolution refutations refutations were were 'lifted' 'lifted' to to R-resolution R-resolution refutations refutations preserves preserves many many of of the the search search strategies strategies that that • • •

• • •

8SWe We are = ) is are assuming assumingthat that the the equality equality sign sign ((=) is not not present. present.

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were discussed discussed in in section that these these search search strategies can be be used used were section 11.3.5. .3.5. This This means means that strategies can for for first-order first-order theorem theorem proving.9 proving. 9

The 2.6.4. 2.6.4. Paramodulation. Paramodulation. The above above discussion discussion of of R-resolution R-resolution assumed assumed that that equality was was not not present present in in the the language. language. In In the the case case where where equality equality is is in in the the equality language, language, one one must must either either add add additional additional initial initial clauses clauses as as axioms axioms that that express express the the equality axioms axioms or or one one must must add add additional additional inference inference rules. rules. For For the the first first approach, approach, equality one one could could add add clauses clauses which which express express the the equality equality axioms axioms from from section section 2.2.1 2.2.1;; for for instance the the third equality axiom axiom can can be be expressed expressed with with the the clause clause instance third equality {xl r Yl, ..., xk r y~,-~P(~), P ( ~ }, and the the other other equality equality axioms axioms can can similarly similarly be be expressed expressed as as clauses. More computa computaand clauses. More tional efficiency obtained with tional efficiency can can be be obtained with equality equality clauses clauses of of the the form form

{x # ~ y, y, A(x), A(x), A(y)} A(y) } {x

where (x) indicates where A A(x) indicates an an arbitrary arbitrary literal. literal. For For the the second second approach, approach, the the paramodulation inference inference is is used used instead instead of of equality equality clauses; this this inference is aa little define, but Suppose B clauses; inference is little complicated complicated to to define, but goes goes as as follows: follows: Suppose B and C C are are clauses clauses with with no no free free variables variables in in common common and and that that rr = - ss is is a a literal literal in in C; C; and let let tt be be a a term term appearing appearing somewhere somewhere in in B B and and let let a a be be aa most most general general unifier unifier of of rr and (or of ) ; let and tt (or of ss and and tt); let B' B ~ be be the the clause clause which which is is obtained obtained from from Ba Ba by by replacing replacing respectively) and occurrences occurrences of of ta ta with with sa sa (or (or with with ra ra,, respectively) and let let C' C' be be (C (C \\ {{rr = s})a. s))a. Under U C' Under these these circumstances, circumstances, paramodulation paramodulation allows allows B' B~U C ~ to to be be inferred inferred from from B be complete B and and C C.. Paramodulation Paramodulation was was introduced introduced and and shown shown to to be complete by by Robinson Robinson and completeness, paramodulation and Wos Wos [1969] [1969] and and Wos Wos and and Robinson Robinson [1973] [1973]:: for for completeness, paramodulation must must be be combined combined with with R-resolution, R-resolution, with with factoring factoring and and with with application application of of variable variable renaming substitutions. renaming substitutions. 2.6.5. 2.6.5. Horn H o r n clauses. clauses. An An important important special special case case of of first-order first-order resolution resolution is is when when the the clauses clauses are are restricted restricted to to be be Horn Horn clauses. clauses. The The propositional propositional refutation refutation search search strategies described section 11.3.5.6 .3.5.6 still still apply; apply; and, and, in strategies described in in section in particular, particular, an an unsatisfiable unsatisfiable set set r F of of Horn Horn clauses clauses always always has has a a linear linear refutation refutation supported supported by by a a negative negative clause clause addition, the portion of in in r F.. In In addition, the factoring factoring portion of R-resolution R-resolution is is not not necessary necessary in in refutations refutations of of r F.. A A typical typical use use of of Horn Horn clause clause refutations refutations is is as as follows: follows: a a set set � A of of Horn Horn clauses clauses is is assumed assumed as as a a 'database 'database'' of of knowledge, knowledge, such such that that every every clause clause in in � A contains contains a a positive positive the object literal. which is literal. A A query, query, which is an an atomic atomic formula formula P(S P(sl,l , ... ...,, Sk) sk),, is is chosen; chosen; the object is is to to determine determine if if there there is is an an instance instance of of P(S) P ( ~ which which is is aa logical logical consequence consequence of of � A.. In In other the object other words, words, the object is is to to determine determine if if (3X)P(S) ( 3 ~ ) P ( ~ is is a a consequence consequence of of � A where where x ~ is is the vector vector of solve this this problem, the clause the of variables variables in in P(S) P ( ~ . . To To solve problem, one one forms forms the clause {P(S) { P ( ~ )} 9Historically, 9Historically, it was was the desire to find strategies for first-order theorem proving and the ability ability to lift results results from from propositional theorem proving, proving, that motivated the research research into search search strategies for propositional propositional resolution.

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S. Buss s.

and lets lets r F be be the the set set � AU U {P( ( P ( ~S)}}; ; one one then then searches searches for for aa linear linear refutation refutation of of r F and which which is is supported supported by by P(S) P ( ~ . . If If successful, successful, such such aa linear linear refutation refutation R R also also yields yields aa substitution (J a,, such such that that � A I-F P( P(s-')a; and indeed, indeed, (J a is is the the most most general general substitution substitution substitution S)(J ; and such that that R R gives gives aa refutation refutation of of � AU U {P( { P ( S)(J} ~ a } .. From From this, this, what what one one actually actually has has is is aa such proof of of (Vi/) ( V ~ (P(S)(J) ( P ( ~ a ) where where 17 ~7is is the the vector vector of of free free variables variables in in the the terms terms P(S)(J. P(~a. Note Note proof that there there may may be be more more than than one one refutation, refutation, and and that that different different refutations refutations can can give give that different different substitutions substitutions (J a,, so so there there is is not not necessarily necessarily aa unique unique most most general general unifier. unifier. What we we have have described described is is essentially essentially aa pure pure form form of of PROLOG, PROLOG, which which is is aa logic logic What programming language language based based on on searching searching for for refutations refutations of of Horn Horn clauses, clauses, usually usually in in programming depth-first search. PROLOG PROLOG also also contains contains conventions conventions for for restricting restricting the the order order of of aa depth-fi rst search. the proof proof search search procedure. procedure. the For further f u r t h e r reading. reading. There There is is an an extensive extensive literature literature on on logic logic programming, programming, au auFor tomated reasoning reasoning and and automated automated theorem theorem proving proving which which we we cannot cannot survey survey here. here. tomated The paper paper of of Robinson Robinson [1965b] still provides provides aa good good introduction introduction to to the the foundations foundations The [1965b] still of logic logic programming; programming; the the textbooks textbooks of of Chang Chang and and Lee Lee [1973] [1973] and and Loveland Loveland [1978] [1978] of provide a a more more detailed treatment of of the the subject subject matter matter above, above, and and the the textbooks of provide detailed treatment textbooks of Kowalski [1979] [1979] and and Clocksin Clocksin and and Mellish Mellish [1981] [1981] provide provide good good detailed detailed introductions introductions Kowalski to logic logic programming programming and and PROLOG. Chapter IX, IX, by by G. G. Jager J~iger and and R. R. Stark, St~irk, discusses discusses to PROLOG. Chapter the proof-theory proof-theory and and semantics semantics of of extensions extensions of of logic logic programming programming to to non-Horn non-Horn the clauses. clauses. 3. P r o o f ttheory h e o r y for f o r oother t h e r logics logics 3. Proof In the the final final section section of of this this chapter, chapter, we we shall shall briefly briefly discuss discuss two two important important non nonIn classical logics, intuitionistic intuitionistic logic classical logics, logic and and linear linear logic. logic. 3.1. n t u i t i o n i s t i c logic 3.1. IIntuitionistic logic

subsystem of of classical which historically Intuitionistic logic logic is is aa subsystem classical logic logic which historically arose arose out out Intuitionistic of to formulate formulate aa more more constructive constructive foundation mathematics. of various various attempts attempts to foundation for for mathematics. For example, law of of the the excluded excluded middle, AV ~A,, does For example, in in intuitionistic intuitionistic logic, logic, the the law middle, A V -,A does not hold in general; furthermore, it to intuitionistically prove A not hold in general; furthermore, it is is not not possible possible to intuitionistically prove AV VB B unless already already at at least least one one of of A A or or B B is is already already intuitionistically intuitionistically provable. shall unless provable. We We shall discuss below primarily primarily mathematical of the istic logic, and shall shall discuss below mathematical aspects aspects of the intuition intuitionistic logic, and omit van Dalen omit philosophical philosophical or or foundational foundational issues: issues: the the books books of of Troelstra Troelstra and and van Dalen [1988] [1988] provide aa good good introduction introduction to to the the latter latter aspects aspects of of intuitionistic intuitionistic logic. logic. provide The J, is defined similarly similarly to classical sequent The intuitionistic intuitionistic sequent sequent calculus, calculus, LLJ, is defined to the the classical sequent calculus calculus LK, LK , except except with with the the following following modifications: modifications: (1) To simplify is not not aa (1) To simplify the the exposition, exposition, we we adopt adopt the the convention convention that that negation negation (-~) ( ) is propositional 1.. in in the the propositional symbol. symbol. In In its its place, place, we we include include the the absurdity absurdity symbol symbol _L 1.. is is aa nullary nullary propositional propositional symbol symbol which which is is intended intended to to always have language; _1_ language; always have False. The The two two ~-, rules rules of of LLK are replaced replaced with with the the single single _L 1.. :left ;left initial initial value False. value K are sequent, namely 1.. ----t . Henceforth, Henceforth, -~A -,A is is used used as as an an abbreviation abbreviation for for A AD :J _L. 1.. . sequent, namely _L---+. -,

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(2) In LJ, L J, the the V Y :right :right rule the definition definition of of LK L K in in section section 1.2.2 1.2.2 is is replaced replaced rule used used in in the (2) In by by the the two two rules rules

f ---7 A,A �, A fF ----7 A F---+ and + �, A,A and F---F A,A B A,B VA A f ---7 �, A VV B fF---F ---7 �, BV Otherwise, LJ defined like (3) LJ is is defined like LK L K ,, except except with with the the important important proviso proviso that that at at (3) Otherwise, most one in the most one formula formula may may appear appear in the antecedent antecedent of of any any sequent. sequent. In In particular, particular, this this means means that that rules rules which which have have the the principal principal formula formula to to the the right right of of the the sequent sequent arrow arrow may may not not have have any any side side formulas formulas to to the the right right of of the the sequent sequent arrow. arrow. 3.1.1. Cut C u t elimination. e l i m i n a t i o n . An An important important property property of of intuitionistic intuitionistic logic logic is is that that the the 3.1.1. cut elimination theorems cut elimination elimination and and free-cut free-cut elimination theorems still still apply: apply:

Theorem. Theorem. (1) ---7 A LJJ-provable. -provable. Then -proof of ---7 A (1) Let Let f F---+ A be be L Then there there is is aa cut-free cut-flee LJ L J-proof of f F'--+ A.. (2) Let Let <5 ~ be be aa set set of of sequents sequents closed closed under under substitution substitution such such that that each each sequent sequent in in <5 G (2)

has most one one formula LJJ augmented has at at most formula in in its its antecedent. antecedent. Let Let LJ6 L J6 be be L augmented with with initial initial ---7 A -proof of sequents sequents from from <5 | . If If LK6 L K s fF- f F---+ A,, then then there there is is aa free-cut free-cut free free LK6 LK~-proof of f ---7 A F---~ A..

The The (free) (free) cut cut elimination elimination theorem theorem for for LJ L J can can be be proved proved similarly similarly to to the the proof prooftheoretic theoretic proofs proofs used used above above for for classical classical logic. logic. 3.1.2. Intuitionistic 3.1.2. Constructivism C o n s t r u c t i v i s m and a n d intuitionism. intuitionism. Intuitionistic logic logic is is intended intended to to provide provide a a 'constructive' 'constructive' subset subset of of classical classical logic: logic: that that is is to to say, say, it it is is designed designed to reasoning. An to not not allow allow non-constructive non-constructive methods methods of of reasoning. An important important example example of of the aspect of logic is the constructive constructive aspect of intuitionistic intuitionistic logic is the the Brouwer-Heyting-Kolmogorov Brouwer-Heyting-Kolmogorov (BHK) (BHK) constructive constructive interpretation interpretation of of logic. logic. In In the the BHK BHK interpretation, interpretation, the the logical logical connectives connectives V V,, :3 3 and and J D have have non-classical, non-classical, constructive constructive meanings. meanings. In In particular, particular, in in order order to to have have aa proof proof of of (or (or knowledge knowledge of) of) aa statement statement (:3x)A(x) (3x)A(x),, it it is is necessary necessary to to (t) . Likewise, have have a a particular particular object object tt and and aa proof proof of of (or (or knowledge knowledge of) of) A A(t). Likewise, in in order order to BHK interpretation to have have aa proof proof of of A AV VB B the the BHK interpretation requires requires one one to to have have aa proof proof either either of A A or or of of B B,, along along with with a a indication indication of of which which one one it it is is a a proof proof of. of. In In addition, in addition, in of order order to to have have aa proof proof of of A AJ DB B,, one one must must have have aa method method of of converting converting any any proof proof of of A A into BHK interpretation similar to into aa proof proof B B.. The The BHK interpretation of of the the connectives connectives 1\ A and and '<:/ V are are similar to the the classical classical interpretation; interpretation; thus, thus, a a proof proof of of A A 1\ AB B or or of of ('<:/x)A(x) (Vx)A(x),, consists consists of of proofs proofs and B or of of constructing (t) for of of both both A A and B or of a a method method of constructing aa proof proof of of A A(t) for all all objects objects tt.. It It is is not not difficult difficult to to see see that that the the intuitionistic intuitionistic logic logic L L JJ is is sound sound under under the the BHK BHK -provable formula interpretation, interpretation, in in that that any any LJ L J-provable formula has has aa proof proof in in the the sense sense of of the the BHK BHK interpretation. interpretation. The The BHK BHK interpretation interpretation provides provides the the philosophical philosophical and and historical historical underpinnings underpinnings of logic; however, of intuitionistic intuitionistic logic; however, it it has has the the drawback drawback of of characterizing characterizing what what constitutes constitutes aa valid valid proof proof rather rather than than characterizing characterizing the the meanings meanings of of the the formulas formulas directly. directly. It It is possible to BHK interpretation is possible to extend extend the the BHK interpretation to to give give meanings meanings to to formulas formulas directly, directly,

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e.g., e.g., by by using using realizability; realizability; under under this this approach, approach, an an existential existential quantifier quantifier (3x) (3x) might might mean "there "there is is aa constructive constructive procedure which produces produces x x,, and and (possibly) (possibly) produces produces mean procedure which evidence evidence that that x x is is correct" correct".. This This approach approach has has the the disadvantage disadvantage of of requiring requiring one one to to pick pick aa notion notion of of 'constructive 'constructive procedure procedure'' in in an an ad-hoc ad-hoc fashion; fashion; nonetheless nonetheless it it can can be be very very fruitful fruitful in in applications applications such such aa intuitionistic intuitionistic theories theories of of arithmetic arithmetic where where there there are are natural natural notions notions of of 'constructive 'constructive procedure procedure'.' . For intuitionistic logic, logic, there model theory For pure pure intuitionistic there is is aa very very satisfactory satisfactory model theory based based on models provide intuitionistic formulas on Kripke Kripke models. models. Kripke Kripke models provide aa semantics semantics for for intuitionistic formulas which is is analogous analogous to to model model theory theory for for classical classical logic logic in in many many ways, ways, including including aa which model theoretic theorem, analogous model theoretic proof proof of of the the cut-free cut-free completeness completeness theorem, analogous to to the the proof proof of of Theorem Theorem 2.3.7 2.3.7 given given above. above. For accessible account BHK interpretation model semantics For an an accessible account of of the the BHK interpretation and and Kripke Kripke model semantics for Troelstra and for intuitionistic intuitionistic logic, logic, the the reader reader can can refer refer to to Troelstra and van van Dalen Dalen [1988,vol. [1988,vol. 1]; 1]; Chapter Chapter VI VI contains contains an an thorough thorough discussion discussion of of realizability. realizability. In In this this section section we we shall only the the following the statements shall give give only following theorem theorem of of Harrop, Harrop, which which generalizes generalizes the statements that A V B is intuitionistically valid if and only if either A or B is, and that A V B is intuitionistically valid if and only if either A or B is, and that that (3x)A(x) is intuitionistically intuitionistically valid valid if if and and only only if if there there is is aa term term tt such such that that A(t) A(t) is is (3x)A (x) is intuitionistically valid. valid. intuitionistically Definition. The 3.1.3. 3.1.3. Definition. The Harrop Harrop formulas formulas are are inductively inductively defined defined by: by: (1) Every Every atomic atomic formula formula is is aa Harrop (1) Harrop formula, formula, (2) If If B B is is aa Harrop Harrop formula formula and and A A is is an an arbitrary arbitrary formula, formula, then then A AJ ::) B B is is aa Harrop Harrop (2) formula. formula. (3) If (3) If B B is is aa Harrop Harrop formula, formula, then then (Vx)B (Vx)B is is aa Harrop Harrop formula. formula. (4) If (4) If A A and and B B are are Harrop Harrop formulas, formulas, then then so so is is A A 1\ AB B..

Intuitively, Intuitively, aa Harrop Harrop formula formula is is aa formula formula in in which which all all existential existential quantifiers quantifiers and and all all disjunctions disjunctions are are 'hidden 'hidden'' inside inside the the left-hand left-hand scope scope of of an an implication. implication. Theorem. formulas, and T h e o r e m . (Harrop (Harrop [1960]) [1960]) Let Let r F be be aa cedent cedent containing containing only only Harrop Harrop.formulas, and

let let A A and and B B be be arbitrary arbitrary formulas. formulas. (a) If LJJ f-F rF--~ -+ (3x)B(x) then there such that LJJ proves (a) /f L (3x)B(x),, then there exists exists aa term term tt such that L proves r F -+ ~ B(t) B(t) .. (b) If LJJ f-F rF---~ -+ A one of LJJ-provable. -provable. (b) If L AV VB B,, then then at at least least one of r F -+ ~ A A and and r F -+ ~ B B is is L

We We omit omit the the proof proof of of the the theorem, theorem, which which is is readily readily provable provable by by using using induction induction on on -proofs. the the length length of of cut-free cut-free LJ L J-proofs. 3.1.4. 3.1.4. Interpretation Interpretation into classical classical logic. logic. The The 'negative 'negative translation translation'' provides provides aa translation translation of logic; an corollary of of classical classical logic logic into into intuitionistic intuitionistic logic; an immediate immediate corollary of the the negative translation is simple, constructive negative translation is aa simple, constructive proof proof of of the the consistency consistency of of classical classical logic intuitionistic logic. logic. There logic from from the the consistency consistency of of intuitionistic There are are aa variety variety of of negative negative translations: independently discovered Kolmogorov, Godel translations: the the first first ones ones were were independently discovered by by Kolmogorov, GSdel and Gentzen. Gentzen. and

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Definition. Definition. Let Let A A be be aa formula. formula. The The negative negative translation, translation, A A --, , of of A A is is inductively inductively defined defined by: by: A is is atomic, atomic, then then AA- is is • -~-~A, (1) If A •A , (2) (.B) ( - , B ) - i is s .(B-) ~ ( B - ) .. (2) (3) (B (3) (B A A C) C)- is is (B-) (B-) A A (C-) (C-), , (4) (B (S ::) D C)C)- is is (B-) (S-) ::) D (C-) (C-),, (4) (5) (B V V C)C)- is is .(.(B-) ~(-~(B-) A A-~(C-)), (5) (B (C )) , (6) (VxB)(6) (VxB)- is is Vx(B-) Vx(S-),, .Vx(.(B - )) , (7) (3xB) (3xB)-- is is-~Vx(~(B-)), (7) .

-

Theorem. T h e o r e m . Let Let A A be be aa formula. formula. Then Then LK L K fFA A if if and and only only if if LJ L J fFA A --. .

then Clearly AA- is is classically equivalent to to A A,, so so if if A A-- is is intuitionistically intuitionistically valid, valid, then Clearly classically equivalent A classically valid. valid. For A is is classically For the the converse, converse, we we use use the the following following two two lemmas lemmas which which immediately imply the immediately imply the theorem. theorem. Lemma. A - --+ L e m m a . Let Let A A be be aa formula. formula. Then Then LJ L J proves proves •• -~-~A~ A A -- .. Proof. induction on will do Proof. The The proof proof is is by by induction on the the complexity complexity of of A A.. We We will do only only one difficult cases. one of of the the more more difficult cases. Suppose Suppose A A is is B B ::) D C. C. We We need need to to show show that that LJ L J proves proves -•• ~ ((B B -- ::) D C-) C - ) ---+ - } BB- - ::) D C C --, , for for which, which, it it suffices suffices to to prove prove •• (B::) C-) --+ C. By the induction hypothesis, •• C - --+ B-, -,-,(B- D C-) ~ C - . By the induction hypothesis, LJ L J proves proves -~-,C--} CC - ,, B-, so will suffice (B- - ::) -provable. To so it it will suffice to to show show that that B-, B - , -•• ~(B DC C --)),, -.C, C - ---+ + is is LJ L J-provable. To do do - , B-+ C- is -provable, which this suffices to this it it suffices to show show that that B B-, B - ::) D CC----~Cis LJ L J-provable, which is is easy easy to to prove. 0 [] prove. If r r- denotes Bi:k and F is is aa cedent cedent B1, B 1 , . . . , ,BBk ~ , , then then Fdenotes the the cedent cedent B1 B ~ -, ,. ... .. ,, B and .r ~Fdenotes , · . . , .Bi: denotes the the cedent cedent .B1 -~B~-,..., ~B~-.. • • •

Lemma. --+ b.. L e m m a . Suppose Suppose that that LK L K proves proves r F-----} A.. rF-,, .b.. -~A-- --+ ----} ..

Then proves the Then LJ L J proves the sequent sequent

Proof. Proof. The The proof proof of of the the lemma lemma is is by by induction induction on on the the number number of of inferences inferences in in an an ' ll do LK -proof, possibly possibly containing containing cuts. LK-proof, cuts. We We'll do only only one one case case and and leave leave the the rest rest to to the the reader. -proof ends reader. Suppose Suppose the the LK LK-proof ends with with the the inference inference B , Fr---+ + b.. A ,, C B, C r --+ b.. , B ::) F--+ A,B DC C

The induction hypothesis r- , .~A-,-~C----} b.. - , .C- --+ is The induction hypothesis is is that that B-, B-,F-, is provable provable in in LJ. LJ. By By - , rthe the previous previous lemma, lemma, the the induction induction hypothesis hypothesis implies implies that that B B-, F-,-,A---}Cis , .b.. - --+ C- is --+ is provable, provable, and and from from this this it it follows follows that that rF-,-~A-,-,(BD C-) C-)---} is also also LJ LJ, 'b.. - , .(B- ::) provable, provable, which which is is what what we we needed needed to to show. show. 0 D

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One One consequence consequence of of the the proof proof of of the the above above theorem theorem is is aa constructive constructive proof proof that that the consistency consistency of of intuitionistic intuitionistic logic logic is is equivalent equivalent to to the the consistency consistency of of classical classical the logic; thus thus intuitionistic intuitionistic logic logic does does not not provide provide aa better better foundations foundations for for mathematics mathematics logic; from the the point point of of view view of of sheer sheer consistency. consistency. This This is, is, however, however, of of little little interest interest to to from aa constructivist, constructivist, since since the the translation translation of of classical classical logic logic into into intuitionistic intuitionistic logic logic does does not not preserve preserve the the constructive constructive meaning meaning (under, (under, e.g., e.g., the the BHK BHK interpretation) interpretation) of of the the formula. formula. It is is possible possible to to extend extend the the negative negative translation translation to to theories theories of of arithmetic arithmetic and and to to It set theory; theory; see, see, e.g., e.g., Troelstra Troelstra and and van van Dalen Dalen [1988,vol. [1988,vol. 1]. 1]. set 3.1.5. Section 3.1.5. Formulas F o r m u l a s as as types. types. Section 3.1.2 3.1.2 discussed discussed aa connection connection between between intu intuitionistic itionistic logic logic and and constructivism. constructivism. An An important important strengthening strengthening of of this this connection connection is is the the Curry-Howard Curry-Howard formulas-as-types formulas-as-types isomorphism, isomorphism, which which provides provides aa direct direct correspon correspondence between between intuitionistic intuitionistic proofs proofs and and A-terms A-terms representing representing computable computable functions, functions, dence under under which which intuitionistic intuitionistic proofs proofs can can be be regarded regarded as as computable computable functions. functions. This This section section will will discuss discuss the the formulas-as-types formulas-as-types isomorphism isomorphism for for intuitionistic intuitionistic propositional propositional logic; logic; our our treatment treatment is is based based on on the the development development by by Howard Howard [1980]. [1980]. Girard Girard [1989] [1989] contains contains further further material, material, including including the the formulas-as-types formulas-as-types isomorphism isomorphism for for first-order first-order intuitionistic logic. intuitionistic logic. The T h e {{::) D }-fragment } - f r a g m e n t ... We We will will begin begin with with the the fragment fragment of of propositional propositional logic logic in in which which the the only only logical logical connective connective is is ::) D.. We We work work in in the the sequent sequent calculus, calculus, and and the the only only strong strong rules rules of of inference inference are are the the ::) D :left :left and and ::) D :right :right rules. rules. Firstly Firstly we we must must define define the set set of of types: types: the .

Definition. The types types are are defined Definition. The defined as as follows: follows: (a) Any propositional variable Pi propositional variable Pi is is aa type. type. ( a) Any

(b) a and is aa type. type. T ) is If a and T are are types, types, then then (a (b) If ( a --+ -+ T)

the types types are are exactly exactly the the same same as as formulas, formulas, since since If we we identify identify the the symbols symbols ~::) and and -+ If 4 , , the ::) is is the the only only logical logical connective allowed. We We shall shall henceforth henceforth make this identification identification connective allowed. make this of of types types and and formulas formulas without without comment. comment. has aa unique unique Secondly, we we must define the the terms terms of of the the A-calculus; A-calculus; each term tt has Secondly, must define each term to mean mean that that tt is is aa term term of of type type T. T. associated type. type. We write tF~ to associated We write Definition. For For each each type type T, T , there there isis an an infinite infinite set set of of variables, variables, x~, xl, x~, x2, x~,.., xa, . . of of Definition. T . Note Note that that ifif aa ~'I T, T , then then x~ xf and and x~ xi are are distinct distinct variables. variables. type T. type The The terms terms are are inductively inductively defined defined by: by: (a) term of of type type T. T. Any variable variable of of type type T isis aa term (a) Any .

U-7T and (b) If If sS~-~ and ttU~ are are terms terms of of the the indicated indicated types, types, then then (st) (st) isis aa term term of of type type T. T. (b) This term term may may also also be be denoted denoted (st) (sty~ or or even even (s~-~t~) (SU-7TtUY~.. This

term then then )~x~.t AXu. t isis aa term term of oftype type aa --+ -+ T. T . This This term term can can also also be be denoted denoted (c) (c) If If tt~T isis aa term ()~x~.t) ~ (AxU F )U-7T or (Ax~.t~) (AXU.tY~ -7T or

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Traditionally, Traditionally, a a type type a a -+ 7 T is is viewed viewed as as a a set set of of mappings mappings from from the the objects objects of type type a a into into the the objects objects of of type type 7 T.. For For intuitionistic intuitionistic logic, logic, it it is is often often useful useful to to of think of of a a type type a a as as being being the the set set of of proofs proofs of of the the formula formula o' a.. Note Note that that under under the the think BKH-interpretation this this is is compatible compatible with with the the traditional traditional view view of of types types as as a a set set of of BKH-interpretation mappings. mappings. The ne the The AX Ax connective connective serves serves to to bind bind occurrences occurrences of of X x;; thus thus one one can can defi define the notions notions of of bound bound and and free free variables variables in in terms terms in in the the obvious obvious way. way. A A term term is is closed closed provided provided it it has has no no free free variables. variables. The The computational computational content content of of aa term term is is defined defined by letting letting (st) (st) represent represent the the composition composition of of the the terms terms ss and and tt,, and and letting letting AXu Ax~.t. t by represent objects dd of represent the the mapping mapping that that takes takes objects of type type 7 T to to the the object object t( t (x x //dd)). . Clearly Clearly this gives gives a a constructive constructive computational meaning to to terms. this computational meaning terms. Theorem. B if T h e o r e m . Let Let B B be be aa formula. formula. LJ L J fF- B if and and only only if if there there is is aa closed closed term term of of type B B .. type

An An example example of of this this theorem theorem is is illustrated illustrated by by the the closed closed term term K K defined defined as as AX.Ay.X and yy are and 7T,, respectively, and therefore the term Ax.Ay.x where where x x and are of of type type a a and respectively, and therefore the term 7 -+ a ) , which K K has has type type a a -+ ~ ((T a), which is is a a valid valid formula. formula. A A second second important important example example is which is is defi ned by ((XZ) ( Yz)) where is the the closed closed term term S S which defined by AX.Ay.AZ. Ax.)~y.Az.((xz)(yz)) where the the types types of of x x,, and Zz are are a a -+ ~ ((T #),, a a -+ --+ 7 T and and a a,, respectively, respectively, and and therefore, therefore, S S has has type type yy and T -+ fJ) a -+ terms K --+ ((T --+ fJ)) #)) -+ --+ (( ((aa -+ ~ 7 r)) -+ -+ ((a --+ fJ)) #)) which which is is a a valid valid formula. formula 9 The The terms g ((aa -+ 7 -+ and S S are are examples examples of of combinators. combinators. and The The import import of of this this theorem theorem is is that that a a closed closed term term of of type type B B corresponds corresponds to to aa proof proof of B B.. We We shall shall briefly briefly sketch sketch the the proof for the the sequent sequent calculus; calculus; however, however, the the of proof of of this this for correspondence correspondence between between closed closed terms terms and and natural natural deduction deduction proofs proofs is is even even stronger, stronger, namely, namely, the the closed closed terms terms are are isomorphic isomorphic to to natural natural deduction deduction proofs proofs in in intuitionistic intuitionistic logic see Girard lemma: logic ((see Girard [1989]) [1989]).. To To prove prove the the theorem, theorem, we we prove prove the the following following lemma: Lemma. LJ --7 B Lemma. L J proves proves the the sequent sequent AI A1,' .. '. ". , An An---+ B if if and and only only if if there there is is aa term term tt B B of of the t " . . . ,, Xx�nA.n . the indicated indicated type type involving involving only only the the free free variables variables x xA1,. 9

9

Proof. induction on P r o o f . It It is is straightforward straightforward to to prove, prove, by by induction on the the complexity complexity of of tt B s ,, that that the LJJ-proof -proof exists. the desired desired L exists. For For the the other other direction, direction, use use induction induction on on the the length length of of the LJJ-proof -proof to exists. We will consider only the the case the L to prove prove that that the the term term tt B B exists. We will consider only case where where the LJJ-proof -proof is the last last inference inference of of the the L is

B, --7 C fF--+ --7 A A B, f F--> C --7 C A AJ D B, B, f F--+ C A (xrr)) and C (xf , xr ) where The induction induction hypotheses hypotheses give give terms terms rrA(x and SsC(xB,xr) where xrr actually actually The represents represents a a vector vector of of variables, variables, one one for for each each formula formula in in f F.. The The desired desired term term C (xt �B , xx rr)) is r ) .. D C ((xt � B rA (xr )) , xxr) ttC(xA~B, is defined defined to to be be ssC((xA~SrA(xr)), [] The .l}} -fragment. T h e {J, { D, _l_ - f r a g m e n t . The The above above treatment treatment of of the the formulas-as-types formulas-as-types isomorphism isomorphism allowed allowed only only the the connective connective J. D. The The formulas-as-types formulas-as-types paradigm paradigm can can be be extended extended to symbol .1 negation, by to allow allow also also the the symbol _L and and thereby thereby negation, by making making the the following following changes. changes.

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Firstly, enlarge the nition of types, by adding aa clause clause specifying specifying that Firstly, enlarge the defi definition of types, by adding that 00 is is aa type. type. Identifying Identifying 0@with with .1 2. makes makes types types identical identical to to formulas. formulas. The The type type 00 corresponds corresponds to to the BHK interpretation the empty empty set; set; this this is is consistent consistent with with the the BHK interpretation since since .1 2_ has has no no proof. proof. Secondly, definition of definition aa new Secondly, enlarge enlarge the the definition of terms terms by by adding adding to to the the definition new clause clause term f0 stating that, for stating that, for every every type type a a,, there there is is a a term f 0 --tu ~ of of type type 00 -+ a a.. Now the the Theorem Theorem and and Lemma Lemma of of section section 3.1.5 3.1.5 still still hold hold verbatim verbatim for for formulas formulas Now with definitions of terms. with connectives connectives {:J, { D, .1} 2_} and and with with the the new new definitions of types types and and terms. The :J , .1, T h e {{D, 2., I\}-fragment. A } - f r a g m e n t . To To further further expand expand the the formulas-as-types formulas-as-types paradigm paradigm to to intuitionistic intuitionistic logic logic with with connectives connectives :J D,, .1 2. and and 1\ A,, we we must must make make the the following following modifications modifications to to the the definitions definitions of of terms terms and and types. types. Firstly, Firstly, add add to to the the definition definition a x r) of of types, types, a a clause clause stating stating that that if if a a and and r T are are types, types, then then ((a T) is is a a type. type. By By identifying still have identifying x with with 1\ A,, we we still have the the types types identified identified with with the the formulas. formulas. Consistent Consistent with the the BHK-interpretation, BHK-interpretation, the the type type ((aa x r T)) may may be be viewed viewed as as the the cross-product cross-product with of of its its constituent constituent types. types. Secondly, Secondly, add add to to the the definition definition of of terms terms the the following following two two clauses: (i) ( i ) iif f sU s ~ and t r are are terms, terms, then (s,, t) is aa term term of of type type a a x r 7-,, and (ii) if clauses: and r then (s t) is and (ii) if SUXT and (11"2S)" The former the pairing s ~x~ is is a a term, term, then then (11"lS)U (rls) ~ and (r2s) ~ are are terms. terms. The former term term uses uses the pairing function; function; and and the the latter latter terms terms use use the the projection projection functions. functions. Again, the the Theorem Theorem and and Lemma Lemma above above still still holds holds with with these these new new definitions definitions of of Again, types types and and terms. terms. The :J , .1 1\, V}-fragment. T h e {{D, 2.,, A, V } - f r a g m e n t . To To incorporate incorporate also also the the connective connective V V into into the the formulas-as-types formulas-as-types isomorphism isomorphism we we expand expand the the definitions definitions of of types types and and terms terms as as follows. follows. Firstly, Firstly, add add to to the the definition definition of of types types that that if if a a and and r T are are types, types, then then ((aa + +r T)) is is a a type. type. To To identify identify formulas formulas with with types, types, the the symbol symbol + + is is identified identified with with V V.. The The type type a a + + r ~" is is viewed viewed as as the the disjoint disjoint union union of of a a and and r T,, consistent consistent with with the the BHK-interpretation. BHK-interpretation. Secondly, Secondly, add add to to the the definitions definitions of of terms terms the the following following two two clauses clauses -t (U+T) t) t (U+T) s) and •9 If If SU s ~ and and tT t r are are terms terms of of the the indicated indicated types, types, then then (Lr(~-~(~+r)s) and (L; (~-*(~+~)t) are are terms terms of of type type a a + + r T.. •9 For For all all types types a a,, r T and and J..L #,, there there is is a a constant constant symbol symbol d(ud (~-+')-~((~-+')-~((~+~)-+')) t/,)-t« T-t/,)-t« U+T)-t/,» of type. In of the the indicated indicated type. In other other words, words, if if su-t/, s ~-~" and and tT-t/' t ~-+" are are terms, terms, then then (dst) (dst) is is a a term term of of type type ((aa + + r T)) -+ --+ J..L #.. Once again, the holds with Once again, the Theorem Theorem and and Corollary Corollary of of section section 3.1.5 3.1.5 holds with these these enlarged enlarged definitions terms. definitions for for types types and and terms.

3.2. 3.2. Linear L i n e a r logic logic Linear 1987aJ as Linear logic logic was was first first introduced introduced by by Girard Girard [[1987a] as a a refinement refinement of of classical classical logic, logic, which which uses uses additional additional connectives connectives and and in in which which weakening weakening and and contractions contractions are are not not allowed. allowed. Linear Linear logic logic is is best best understood understood as as aa 'resource 'resource logic logic'' in in that that proofs proofs use each In this in linear in linear logic logic must must use each formula formula exactly exactly once. once. In this point point view, view, assumptions assumptions in in a a proof proof are are viewed viewed as as resources; resources; each each resource resource must must be be used used once once and and only only once once during the during the proof. proof. Thus, Thus, loosely loosely speaking, speaking, an an implication implication A A f~ B B can can be be proved proved in in

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linear if A A is is exactly exactly what what is is needed needed to to prove prove B B;; thus thus if if the the assumption assumption A A linear logic logic only only if is strong, it possible to linear logic is either either too too weak weak or or too too strong, it is is not not possible to give give aa linear logic proof proof of of 'if 'if ' As an A thtn then B B'.. As an example example of of this, this, A A f~ B B being being provable provable in in linear linear logic logic does does not not A generally generally imply imply that that A, A fF- B B is is provable provable in in linear linear logic; logic; this this is is because because A A fF- B B is is A, A asserting that that one one use use of of the the resource resource A A gives gives B B,, whereas whereas A, A fF- B B asserts asserts that that two two A, A asserting uses the resource gives B uses of of the resource A A gives B.. We shall shall introduce introduce only only the the propositional propositional fragment fragment of of linear linear logic, logic, since since already already We the propositional fragment. the main main features features of of linear linear logic logic are are present present in in its its propositional fragment. We We are are particularly interested interested in in giving intuitive understanding understanding of of linear linear logic; logic; accordingly, accordingly, particularly giving aa intuitive we shall frequently make make vague vague or or even even not-quite-true assertions, sometimes, sometimes, but but not not we shall frequently not-quite-true assertions, always, "loosely speaking" speaking" or always, preceded preceded by by phrases phrases such such as as "loosely or "intuitively" "intuitively",, etc. etc. 0 The initial Linear Linear logic logic will will be be formalized formalized as as a a sequent sequent caiculus. calculus. 1~~ The initial logical logical sequents sequents are contraction structural structural rules are just just A A --t -} A A;; the the weakening weakening and and contraction rules are are not not allowed allowed and and the structural rules rule. Since Since contraction the only only structural rules are are the the exchange exchange rule rule and and the the cut cut rule. contraction is is not not permitted, permitted, this this prompts prompts one one to to reformulate reformulate the the rules, rules, such such as as 1\ A :right :right and and V V :lejt, :left, which which contain contain implicit implicit contraction contraction of of side side formulas. formulas. To To begin begin with with conjunction, conjunction, linear logic logic has has two two different different different different conjunction conjunction symbols, symbols, denoted denoted ® | and and & &:: the the linear multiplicative multiplicative connective connective ® | does does not not allow allow contractions contractions on on side side formulas, formulas, whereas whereas the conjunction & does. The the additive additive conjunction & does. The rules rules of of inference inference for for the the two two varieties varieties of of conjunction conjunction are: are:

A, A, B, B, r-t� F---~A ®: | leit A A® | B, B~ r-t� F---}A and and

A, A, r-t� F---}A &:lejt &:left A&B A&B,, r-t� F---~A B, &:left B, r-t� F--'~A &:lejt A&B A&B,, r-t� F-"> A

@:right

FI---+A1, A F2---+A2, B F1, F2---~A1, A2, A | B

r-t�, F---}A, A A F---}A, B B . . ht r-t�, & .Ng &:right r-t� F--+A,, A&B A&B

The The obvious obvious question question now now arises arises of of what what the the difference difference is is in in the the meanings meanings of of the the two two different and & one should in terms different conjunctions conjunctions ® | and &.. For For this, this, one should think think in terms of of resource resource usage. Consider first both A usage. Consider first the the ® @ :lejt :left inference: inference: the the upper upper sequent sequent contains contains both A and and B B so both A so that that resources resources for for both A and and B B are are available. available. The The lower lower sequent sequent contains contains A A® | B B in stead; the intuition is for) A in their their stead; the obvious obvious intuition is that that the the resource resource ((for) A® | B B is is equivalent equivalent both A A and to having having resources resources for for both and B B.. Thus, Thus, loosely loosely speaking, speaking, we we translate translate ® | as as to meaning "both" meaning "both".. Now Now consider consider the the & & :lejt :left inference; inference; here here the the lower lower sequent sequent has has the the place of either A principal principal formula formula A&B A&B in in place of either A or or B B,, depending depending on on which which form form of of the the inference used. This This means that aa resource resource A&B inference is is used. means that A&B is is equivalent equivalent to to having having a a resource resource desired. So for for A A or or B B,, whichever whichever is is desired. So we we can can translate translate & & as as "whichever" "whichever".. The "both" and The translations translations "both" and "whichever" "whichever" also also make make sense sense for for the the other other inferences inferences for and, since since we for the the conjunctions. conjunctions. Consider Consider the the ® | :right :right inference inference and, we have have not not yet yet discussed the this case, case, the discussed the disjunctions, disjunctions, assume assume � A11 and and � A22 are are empty. empty. In In this the upper upper hypotheses hypotheses say say that that from from a a resource resource r F1l one one can can obtain obtain A A,, and and from from a a resource resource r F22 lOThe customary convention f- as 1~ convention for linear linear logic logic is to use "b" as the symbol symbol for the sequent -t " however. connection; we shall shall continue to use the symbol connection; symbol "" ~ however. "

"

S. Buss

72

conclusion then one can can obtain obtain B B.. The The conclusion then says says that that from from (the (the disjoint disjoint union union of) of) both both one these B. Now these resources, resources, one one obtains obtains both both A A and and B. Now consider consider a a & & :right :right inference inference

rF---~ --+ A rF---~ --+ B A B rF --+ ---+ A&B A&B For For the the lower lower sequent, sequent, consider consider an an agent agent who who has has the the responsibility responsibility of of providing providing (outputting) (outputting) a a resource resource A&B A&B using using exactly exactly the the resource resource r F.. To To provide provide the the resource resource A&B,, the the agent agent must must be be prepared prepared to to either either provide provide aa resource resource A A or or a a resource resource B B,, A&B whichever one one is is needed; needed; in in either either case, case, there there is is an an upper upper sequent sequent which which says says the the whichever agent agent can can accomplish accomplish this. this. Linear Linear logic logic also also has has two two different different forms forms of of disjunction: disjunction" the the multiplicative multiplicative dis disjunction junction � ~' and and the the additive additive disjunction disjunction ill @.. The The former former connective connective is is dual dual to to ® | and and the latter latter is is dual dual to to & &.. Accordingly Accordingly the the rule rule of of inference inference for for the the disjunctions disjunctions are: are: the

'left � ~ .:left and and

A, B, --+�l A, rl Ft---}A1 B, r F22 --+� "----}A2 2 A�B, --+�l � A ~ B , rl r l ,, r r2--'}At, A2 2 ' 2

1 ,4 A, F---~A A, r--+�

ill e :: leeIt J"

B, F---}A B, r--+� A r--+� A ill @ B, B,F---+A

r--+�, A, B :~.:right � Tight F'--} A, A, B r--+�, F---+ A, A�B A:~B

P-+A, A A . ht r--+�, ill rzg @::right F-----}A, A A ill @B B r--+�, F---+A, B B . ht r--+�, ill @:rzg :right F---} A, A A ill @B B r--+�,

By examining these By examining these inferences, inferences, we we can can see see that that the the multiplicative multiplicative disjunction, disjunction, � ~ ,, is is a a a a kind kind of of 'indeterminate 'indeterminate OR' OR' or or 'unspecifi 'unspecificc OR OR',' , in in that that aa resource resource for for A�B A~B consists consists of of a a either either a a resource resource either either for for A A or or for for B B,, but but without without aa specification specification of of which which one one it it is is a a resource resource for. for. On On the the other other hand, hand, the the additive additive conjunction, conjunction, ill @,, is is a a 'determinate OR' or 'specific OR 'determinate OR' or a a 'specific OR',' , so so that that a a resource resource for for A A ill (9 B B is is either either aa resource resource for for A A or or a a resource resource for for B B together together with with a a specification specification of of which which one one it it is is a a resource resource for. consider aa military military general for. To To give give a a picturesque picturesque example, example, consider general who who has has to to counter counter an an invasion will come the western invasion which which will come either either on on the western front front or or on on the the eastern eastern front. front. If If the the connective 'or' is connective 'or' is the the multiplicative multiplicative or, or, then then the the general general needs needs sufficient sufficient resources resources to to counter counter both both threats, threats, whereas whereas if if the the 'or 'or'' is is an an additive additive disjunction disjunction then then the the general general needs needs only only enough enough resources resources to to counter counter either either threat. threat. In In the the latter latter case, case, the the general general is told ahead occur and is told ahead of of time time where where the the invasion invasion will will occur and has has the the ability ability to to redeploy redeploy resources the actual resources so so as as to to counter counter the actual invasion; invasion; whereas whereas in in the the former former case, case, the the general general must must separately separately deploy deploy resources resources sufficient sufficient to to counter counter the the attack attack from from the the west west and and resources resources sufficient sufficient to to counter counter the the attack attack from from the the east. east. From From the the rules rules of of inference inference for for disjunction, disjunction, it it is is clear clear that that the the commas commas in in the the succedent succedent of of aa sequent sequent intended intended to to be be interpreted interpreted as as the the unspecific unspecific or, or, � ~ .. The The linear linear implication implication symbol, symbol, -0 --o,, is is defined defined by by letting letting A A -0 --o B B be be an an abbrevia abbreviabelow. tion tion for for A A •J.�B , where where A A 1. • is is defined defined below. nullary Linear logic logic also Linear also has has two two nullary nullary connectives, connectives, 11 and and T T,, for for truth truth and and two two nullary connectives, 00 and connectives, and .1 _k,, for for falsity. falsity. Associated Associated to to these these connectives connectives are are the the initial initial sequents --+ � sequents --+ ~ 11 and and 0, 0, r F--} A,, and and the the one one rule rule of of inference: inference:

rF----~ --+ � A 1, --+ � 1, r F---} A

73 73

Introduction to Proof Theory

The intuitive intuitive idea idea for for the the multiplicatives multiplicatives isis that that 11 has has the the null null resource resource and and that that The ..L . For For the the additive additive connectives, connectives, TT has has aa 'arbitrary' 'arbitrary' resource, resource, there isis no no resource resource for for _1_. there 'wild which means means that that anything anything isis aa resource resource for for T, T , whereas whereas any any resource resource for for 00 isis aa 'wild which card ' resource which may may be be used used for for any any purpose. purpose. card' resource which In A1-• which which is is involutive involutive in in that that In addition, addition, linear linear logic logic has has aa negation negation operation operation A (A• For each negation (A1-)1-• isis the the same same as as formula formula A. A . For each propositional propositional variable variable p, p, p• p1- isis the the negation of inductively defined of p. p. The The operation operation of of the the negation negation is is inductively defined by by letting letting (A (A |18> B) B)1-• be be AZ~B A1-�B1-z,, (A&B) (A&B)1-• be be A A1-• @ EB B, B , 111-• be be 3_, ..L , and and TT1-z be be 0. o. This This plus plus the the involution involution property two rules property defines defines the the operation operation A A1-• for for all all formulas formulas A. A . The The two rules of of inference inference associated with with negation negation are are associated

. A, r --+ � •1- :right :nght A, F - - } A

F---} r --+ A,A �, A1-•

and and

•

F~

A, A

Ai, F__.} A

Definition. The The multiplicative/additive multiplicative/additive fragment fragment of of propositional propositional linear linear logic logic is is Definition. denoted MALL and contains contains all all the the logical logical connectives, connectives, initial initial sequents sequents and and rules denoted M A L L and rules of of inference inference discussed discussed above. above.

The of weakening weakening and and contraction contraction rules, rules, makes the linear logic sequent The absence absence of makes the linear logic sequent calculus particularly particularly well well behaved; behaved; in in particular, particular, the the cut cut elimination theorem for for calculus elimination theorem MALL quite easy easy to to prove. prove. In fact, if if PP is MALL there is M A L L is is quite In fact, is a aM A L L proof, proof, then then there is aa shorter shorter MALL P** with with the the same no cut M A L L proof proof P same endsequent endsequent which which contains contains no cut inferences. inferences. addition, the the number number of of strong strong inferences inferences in bounded by by the number In In addition, in P P ** is is bounded the number connectives in in the the endsequent. endsequent. Bellin Bellin [1990] [1990] contains contains an an in-depth in-depth discussion discussion of of of of connectives MALL related systems. M A L L and and related systems. A 18> (BEBC) --+ and As an distributive laws As an exercise, exercise, consider consider the the distributive laws A| "+ (AI8>B)EB(AI8>B) (A|174 and (A|174 EB (A 18> B) --+ ~ A A|18> (BEBC) . The The reader reader should should check check that that these these are are valid valid under under (AI8>B) the intuitive intuitive interpretation interpretation of of the the connectives connectives given given above; above; and, and, as as expected, expected, they they the can Similar reasoning can be be proved proved in in MALL. M A L L . Similar reasoning shows shows that that & & is is distributive distributive over over � ~ .. On On the the other other hand, hand, EB (9 is is not not distributive distributive over over � ~ ; ; this this can can be be seen seen intuitively intuitively by by considering the meanings the connectives, the cut cut elimination considering the meanings of of the connectives, or or formally formally by by using using the elimination theorem. theorem. MALL (LL) has, M A L L is is not not the the full full propositional propositional linear linear logic. logic. Linear Linear logic logic (LL) has, in in addition, !! and addition, two two modalities modal• and ?? which which allow allow contraction contraction and and weakening weakening to to be be used used in situations. The in certain certain situations. The negation negation operator operator is is extended extended to to LL LL by by defining defining (!A)1(!A) • to to be ? (A 1-) . The four rules of inference for ! are: be ?(A• The four rules of inference for ! are: :weakeningo ,.v :wea kenzngo · 9

r--+� F---} A !A, � !A, r--+ F---+A

� !A, !A, !A, r--+ F---+A !A, ....,.-'-=---;--: !!:contraction : contraction --'-:-!A, !A, r--+� F---+A

A, A, r--+� F---}A '.!:weakening1 · :weakenzngl !A, r--+ F---+A !A, � !r--+?�, !F---~?A, A A !F---}?A, !A !A !r--+?�,

where respectively, ??A) �) represents where in in the the last last inference, inference, !r !F ((respectively, represents aa cedent cedent containing containing only only formulas respectively, the ). formulas beginning beginning with with the the !! symbol symbol ((respectively, the ?? symbol symbol). The The dual dual rules rules for for ?? are are obtained obtained from from these these using using the the negation negation operation. operation. One One should should intuitively intuitively think think of of aa resource resource for for !A !A as as consisting consisting of of zero zero or or more more resources resources

74 74

s. Buss Buss S.

for A. A . The The nature nature of of full full linear linear logic logic with with the the modalities modalities is is quite quite different different from from that that for of either either M MALL or propositional propositional classical classical logic; logic; in in particular, particular, Lincoln Lincoln et et al. al. [1992] [1992] of A L L or show that that LL LL is is undecidable. undecidable. show The above above development development of of the the intuitive intuitive meanings meanings of of the the connectives in linear linear logic logic The connectives in has not not been been as as rigorous as one one would would desire; desire; in in particular, particular, itit would would be be nice nice have has rigorous as have aa completeness theorem theorem for linear logic logic based based on on these these intuitive intuitive meanings. meanings. This This has has completeness for linear for some some fragments fragments of of linear linear logic logic by by Blass Blass [1992] [1992] and and Abramsky Abramsky and and been achieved for been achieved Jagadeesan [ 1994] , but but has has not not yet been attained for the the full full propositional propositional linear linear Jagadeesan [1994], yet been attained for logic. more information logic. In In addition addition to to the the references references above, above, more information on on linear linear logic logic may may [ 1992] although although his notation is different from the standard be found found in in Troelstra Troelstra [1992], be his notation is different from the standard ' notation we we have used. notation have used. Acknowledgements. We are are grateful grateful to to S. S. Cook, Cook, E. E. Gertz, Gertz, C. C. Guti~rrez, Gutierrez, A. A. JonasJonas A c k n o w l e d g e m e n t s . We son, C. C. Lautemann, Lautemann, A. A. Maciel, Maciel, R. R. Parikh Parikh C. C. Pollett, Pollett, R. R. St~irk, Stark, J.J. Tor~n Toran and and S. S. Wainer Wainer son, for suggesting corrections versions of of this chapter. Prepa Prepafor reading reading and and suggesting corrections to to preliminary preliminary versions this chapter. of this this article article was was partially partially supported supported by by NSF NSF grant ration ration of grant DMS-9503247 DMS-9503247 and and by by cooperative /ME-103 of NSF and the Czech Czech Republic cooperative research research grant grant INT-9600919 INT-9600919/ME-103 of the the NSF and the Republic Ministry of Education. Ministry of Education. References References s. ABRAMSKY A BRAMSKY AND AND R.. R. JAGADEESAN JAGADEESAN S. Games and and full completeness for logic, Journal of [1994] full completeness for multiplicative multiplicative linear linear logic, of Symbolic Logic, Logic, [1994] Games 59, pp. pp. 543-574. 59, 543-574. BARWISE J. BARWISE [[1975] 1975] Admissable AdmissableSets and Structures: An Approach to Definability Theory, Theory, Springer-Verlag, Springer-Verlag, Berlin. Berlin. Handbookof Mathematical Logic, Logic, North-Holland, North-Holland, Amsterdam. [[1977] 1977] Handbook

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[1971] Logical Logical Writings, D. [1971] D. Reidel, Dordrecht, Holland. ed. by W. Goldfarb. AND W. D D.. HILBERT HILBERT AND W. ACKERMANN ACKERMANN [1928] Grundziige Grundziige der theoretischen theoretischen Logik, Logik, Springer-Verlag, Springer-Verlag, Berlin. Berlin. [1928] AND P D D.. HILBERT HILBERT AND P.. BERNAYS BERNAYS [1934-39] Grundlagen Grundlagen der der Mathematik, Mathematik, II fj f_4II, II, Springer, Springer,Berlin. Berlin. [1934-39] J. HINTIKKA J. HINTIKKA [1955] [1955] Form Form and and content content in in quantification quantificationtheory, theory,two two papers papers on on symbolic symbolic logic, logic,Acta Acta Philo Philo8, pp. pp. 7-55. sophica sophica Fennica, Fennica, 8, 7-55. W. HOWARD W. A. A. HOWARD in: To H. B. Curry: Essays in Combi [1980] The The formulas-as-types formulas-as-types notion notion of of construction, construction, in: Combi[1980] natory Logic, J. R. Logic, Lambda Calculus and Formalism, J. J. P. P. Seldin Seldin and and J. R. Hindley, Hindley, eds., eds., Academic Academic Press, Press, New New York, York, pp. pp. 479-491. 479-491.

S. S. KANGER KANGER [1957] Provability in Logic, Logic, Almqvist Almqvist & & Wiksell, Wiksell, Stockholm. Stockholm. [1957] S. C. C. KLEENE S. KLEENE [1952] Introduction Introduction to Metamathematics, Metamathematics, Wolters-Noordhoff, Wolters-Noordhoff, Groningen Groningen and and North-Holland, North-Holland, [1952] Amsterdam. Amsterdam. R. KOWALSKI R. KOWALSKI [1979] Problem Solving, [1979] Logic Logicfor for Problem Solving,North-Holland, North-Holland, Amsterdam. Amsterdam. J. KRAJiCEK, TAKEUTI J. KRAJf(3EK, P P.. PUDLAK, PUDL~.K, AND AND G G.. TAKEUTI [1991] Bounded Bounded arithmetic arithmeticand and the the polynomial polynomial hierarchy, hierarchy,Annals Annals of of Pure Pure and and Applied Applied Logic, Logic, [1991] 52, pp. 52, pp. 143-153. 143-153. G. KREISEL G. KREISEL [1951] On On the the interpretation interpretationof of non-finitist non-finitistproofs-part proofs-part I, I, Journal Journal of of Symbolic Symbolic Logic, Logic, 16, 16, [1951] pp. pp. 241-267. 241-267. [1952] [1952] On On the the interpretation interpretationof of non-finitist non-finitistproofs, proofs, part part II. II. interpretation interpretationof of number number theory, theory, applications, applications, Journal of of Symbolic Symbolic Logic, Logic, 17, 17, pp. pp. 43-58. 43-58. P C. MITCHELL, AND N. N. SHANKAR P.. LINCOLN, LINCOLN, JJ.. C. MITCHELL, A A.. SCEDROV, SCEDROV, AND SHANKAR [1992] Decision Decision problems problems for for linear linear logic, logic, Annals Annals of of Pure and Applied Logic, Logic, 56, 56, pp. pp. 239-311. 239-311. [1992] E G. K. K. LOPEZ-EsCOBAR E.. G. LOPEZ-ESCOBAR [1965] An An interpolation interpolation theorem theorem for for denumerably denumerably long long formulas, formulas, Fundamenta Mathematicae, [1965] 57, pp. 57, pp. 253-272. 253-272. D W. LOVELAND D.. W. LOVELAND [1970] [1970] A A linear linear format format for for resolution, resolution, in: in: Symp. Syrup. on Automatic Automatic Demonstration, Demonstration, Lecture Lecture Notes Notes in in Mathematics Mathematics #125, #125, Springer-Verlag, Springer-Verlag, Berlin, Berlin, pp. pp. 147-162. 147-162. [1978] Automated Automated Theorem Proving: A Logical Logical Basis, North-Holland, North-Holland, Amsterdam. Amsterdam. [1978] D D.. LUCKHAM LUCKHAM Refinement theorems resolution theory, in: Symp. [1970] [1970] Refinement theorems in in resolution theory, in: Syrup. on Automatic Automatic Demonstration, Demonstration, Lecture Lecture Notes Notes in in Mathematics Mathematics #125, #125, Springer-Verlag, Springer-Verlag, Berlin, Berlin, pp. pp. 163-190. 163-190. R. R. C C.. LYNDON LYNDON predicate calculus, [1959] [1959] An An interpolation interpolation theorem theorem in in the the predicate calculus, Pacific Journal of of Mathematics, 9, 9, pp. pp. 129-142. 129-142. E. E. MENDELSON MENDELSON [1987] /Cole, Monterey. [1987] Introduction Introduction to Mathematical Logic, Logic, Wadsworth Wadsworth & & Brooks Brooks/Cole, Monterey. J. R. R. MUNKRES J. M UNKRES [1975] [1975] Topology: Topology: A First Course, Prentice-Hall, Prentice-Hall, Englewood Englewood Cliffs, Cliffs, New New Jersey. Jersey.

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M. S. S. PATERSON AND M. M. N M. PATERSON AND N.. WEGMAN WEGMAN [1978] Linear Linear unification, unification, J. Comput. System System Sci., 16, 16, pp. pp. 158-167. 158-167. [1978] G. G. PEANO PEANO [1889] Arithmetices Principia, noca methodo exposito, Thrin. Turin. English English translation translation in: in: van van [1889] Heijenoort Heijenoort [1967], [1967], pp. pp. 83-97. 83-97.

D. PRAWITZ D. PRAWITZ Theoretical Study, Almqvist [1965] Natural Deduction: A ProofProof-Theoretical Almqvist & & Wiksell, Wiksell, Stockholm. Stockholm. [1965] A. ROBINSON A. ROBINSON A result result on on consistency consistency and and it it application application to to the the theory theory of of definability, definability, Indagationes [1956] A Mathematicae, pp. 47-58. Mathematicae, 18, 18, pp. 47-58. AND L G G.. ROBINSON ROBINSON AND L.. WOS Wos [1969] Paramodulation Paramodulation and and theorem-proving theorem-proving in in first-order first-order theories theories with with equality, equality, in: in: Machine [1969] Intelligence pp. 135-150. Intelligence 4, PP. 135-150. J J..

A. ROBINSON A. ROBINSON [1965a] Automatic Automatic deduction deduction with with hyper-resolution, hyper-resolution, International International Journal of of Computer Math Math[1965a] ematics, 1, pp. 227-234. Reprinted in: in: Siekmann Siekmann and 1, pp. 227-234. Reprinted and Wrightson Wrightson [1983,vol [1983,vol l]. 1]. [1965b] A A machine-oriented machine-oriented logic logic based based on on the resolution principle, principle, J. Assoc. Comput. Mach., [1965b] the resolution 12, 23-41. Reprinted 12, pp. pp. 23-41. Reprinted in: in: Siekmann Siekmann and and Wrightson Wrightson [1983,vol [1983,vol l]. 1].

K K.. SCHUTTE SCHIJTTE [1965] Ein Ein System System des des verkniipfenden verkniipfenden Schliessens, Schliessens, Archiv for far Mathematische Mathematische Logik und Grund Grund[1965] lagenforschung, 2, pp. pp. 55-67. lagenforschung, 2, 55-67. J. SIEKMANN G. WRIGHTSON J. SIEKMANN AND AND G. WRIGHTSON [1983] 1&2, Springer-Verlag, [1983] Automation Automation of of Reasoning, Reasoning, vol. vol. l&2, Springer-Verlag, Berlin. Berlin. J. R. R. SLAGLE J. SLAGLE [1967] Automatic Automatic theorem theorem proving proving with with renamable renamable and and semantic semantic resolution, resolution, J. Assoc. Comput. [1967] Mach., 14, pp. 687-697. in: Siekmann 14, pp. 687-697. Reprinted Reprinted in: Siekmann and and Wrightson Wrightson [1983,vol [1983,vol l]. 1].

W. W. W. W. TAIT WAIT [1968] [1968] Normal Normal derivability derivability in in classical classical logic, logic, in: in: The Syntax and Semantics Semantics of of Infinitary Infinitary Languages, Lecture Notes in Mathematics Mathematics #72, #72, J. J. Barwise, Barwise, ed., ed., Springer-Verlag, Springer-Verlag, Berlin, Berlin, pp. pp. 204-236. 204-236. G. TAKEUTI G. TAKEUTI [1987] [1987] Proof Theory, North-Holland, North-Holland, Amsterdam, Amsterdam, 2nd 2nd ed. ed. S. TROELSTRA A. S. A. TROELSTRA [1992] Lectures on Linear Logic, Logic, Center Center for for the the Study Study of of Logic Logic and and Information, Information, Stanford. Stanford. [1992] D. VAN A. S. S. TROELSTRA A. TROELSTRA AND AND D. VAN DALEN DALEN [1988] [1988] Constructivism Constructivism in Mathematics: Mathematics: An An Introduction, Introduction, vol. vol. I&II, I&II, North-Holland, North-Holland, Amsterdam. Amsterdam.

G S. TSEJTlN G.. S. TSEJTIN derivation in [1968] [1968] On On the the complexity complexity of of derivation in propositional propositional logic, logic, Studies in Constructive Constructive Mathe Mathematics and Mathematical 2, pp. pp. 1115-125. 15-125. Reprinted in: Siekmann Mathematical Logic, Logic, 2, Reprinted in: Siekmann and and Wright Wrightson son [1983,vol [1983,vol 2]. 2]. A. N. N. WHITEHEAD B. RUSSELL A. WHITEHEAD AND AND B. RUSSELL 1, Cambridge [1910] [1910] Principia Mathematica, vol. vol. 1, Cambridge University University Press. Press. L. WOS, R. OVERBEEK, E. LUSK, AND J. J. BOYLE L. Wos, R. OVERBEEK, E. LUSK, AND BOYLE [1992] [1992] Automated Automated Reasoning: Introduction Introduction and Applications, McGraw-Hill, McGraw-Hill, New New York, York, 2nd 2nd ed. ed.

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L. WOS G. ROBINSON L. Wos AND AND G. ROBINSON [1973]J Maximal Maximal models models and and refutation refutation completeness: completeness: Semidecision Semidecision procedures procedures in in automatic automatic [1973 theorem proving, proving, in: Problems: Decision Problems and the Burnside Problem in theorem in: Word Problems: B. Cannonito, C. Lyndon, Group Theory, Theory, W. W. W. W. Boone, Boone, F. F. B. Cannonito, and and R. R. C. Lyndon, eds., eds., North-Holland, North-Holland, Amsterdam, Amsterdam, pp. pp. 609-639. 609-639. L. Wos, G. ROBINSON, ROBINSON, AND AND D D.. F. CARSON F. CARSON L. Wos, [1965] Efficiency and and completeness completeness of of the the set set of of support support stategy stategy in in theorem theorem proving, proving, J. Assoc. [1965 J Efficiency Comput. Mach., 12, 1983,vol I1]. J. 12, pp. pp. 201-215. 201-215. Reprinted Reprinted in: in: Siekmann Siekmann and and Wrightson Wrightson [[1983,vol

CHAPTER CHAPTER II II

First-Order Proof Theory Theory of of Arithmetic Arithmetic FirstOrder Proof Samuel R. R. Buss Buss Samuel Science, University of o] California, California, San Diego, Departments of Mathematics and Computer Science, La Jolla, Jolla, CA 92093-0112, 92093-0112, USA

Contents Contents

1. Fragments of of arithmetic a r i t h m e t i c .. .. .. .. .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1. Fragments Very weak weak fragments fragments of of arithmetic arithmetic . . . . . . . . . . . . . . . . . . . . . . 1.1. Very Strong fragments fragments of of arithmetic arithmetic . .. . . . . . . . . . . . . . . . . . . . . . . 1.2. Strong 1.3. Fragments Fragments of of bounded b o u n d e d arithmetic arithmetic .. . . . . . . . . . . . . . . . . . . . . . . Sequent calculus calculus formulations formulations of of arithmetic arithmetic . . . . . . . . . . . . . . . . . 1.4. Sequent GSdel incompleteness incompleteness . .. .. .. .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2. Giidel A r i t h m e t i z a t i o n of of metamathematics m e t a m a t h e m a t i c s .. . . . . . . . . . . . . . . . . . . . . 2.1. Arithmetization T h e Giidel GSdel incompleteness incompleteness theorems theorems .. . . . . . . . . . . . . . . . . . . . . 2.2. The On the the strengths strengths of of fragments fragments of arithmetic . . . . . . . . . . . . . . . . . . . . 3. On of arithmetic 3.1. Witnessing Witnessing theorems theorems .. .. .. .. .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2. Witnessing Witnessing theorem t h e o r e m for for S� S~ .. .. .. .. .. . . . . . . . . . . . . . . . . . . . . . . Witnessing theorems theorems and and conservation conservation results results for for T� T~ . . . . . . . . . . . . 3.3. Witnessing lEn n .. .. .. . . . . . . . . . . . . . . . . . . 3.4. Relationships Relationships between between BE BEnn and and/E Strong incompleteness incompleteness theorems theorems for for Iflo IA0 + + exp exp .. . . . . . . . . . . . . . . . . . . 4. Strong References References .. .. .. .. .. .. . . . . . . . . . . . . . .. .. .. .. .. .. .. .. .. .. . . . . . . . . . . . . . . . . .

HANDBOOK PROOF H A N D B O O K OF OF P R O O F THEORY THEORY Edited S. R. Edited by by S. R. Buss Buss © 1998 1998 Elsevier Elsevier Science Science B.V. B.V. All All rights rights reserved reserved

81 82 83 97 109 112 1113 13 1118 18 122 122 127 130 134 137 143 143

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This chapter chapter discusses discusses the the proof-theoretic proof-theoretic foundations foundations of of the the first-order first-order theory theory of of This the non-negative non-negative integers. integers. This first-order theory theory of of numbers, numbers, also also called 'first-order the This first-order called 'first-order arithmetic',' , consists consists of of the the first-order first-order sentences sentences which which are are true true about about the the integers. integers. arithmetic The study study of of first-order first-order arithmetic arithmetic is is important important for for several several reasons. reasons. Firstly, Firstly, in in the the The study of of the the foundation foundation of of mathematics, mathematics, arithmetic arithmetic and and set set theory theory are are two two of of the the study most first-order theories; indeed, the most important important first-order theories; indeed, the usual usual foundational foundational development development of of mathematical begins with mathematical structures structures begins with the the integers integers as as fundamental fundamental and and from from these these constructs constructs mathematical mathematical constructions constructions such such as as the the rationals rationals and and the the reals. reals. Sec Secondly, the the proof proof theory theory for for arithmetic is highly highly developed developed and and serves serves as as aa basis basis for for ondly, arithmetic is proof-theoretic many stronger proof-theoretic investigations investigations of of many stronger theories. theories. Thirdly, Thirdly, there there are are intimate intimate connections connections between between subtheories subtheories of of arithmetic arithmetic and and computational computational complexity; complexity; these these ' s discovery connections go connections go back back to to G6del Ghdel's discovery that that the the numeralwise numeralwise representable representable func functions the recursive tions of of arithmetic arithmetic theories theories are are exactly exactly the recursive functions functions and and are are recently recently of of great interest because because some theories of close connection great interest some weak weak theories of arithmetic arithmetic have have very very close connection of of feasible feasible computational computational classes. classes.

's second Because Because of of G6del Ghdel's second incompleteness incompleteness theorem theorem that that the the theory theory of of numbers numbers is good proof is not not recursive, recursive, there there is is no no good proof theory theory for for the the complete complete theory theory of of numbers; numbers; therefore, therefore, proof-theorists proof-theorists consider consider axiomatizable axiomatizable subtheories subtheories (called (called fragments) fragments) of of first-order arithmetic. arithmetic. These These fragments fragments range range in in strength strength from from the the very very weak weak theories theories first-order R and and Q Q up up to to the the very very strong strong theory theory of of Peano Peano arithmetic arithmetic ((PA). PA). R The The outline outline of of this this chapter chapter is is as as follows. follows. Firstly, Firstly, we we shall shall introduce introduce the the most most important arithmetic and their relative important fragments fragments of of arithmetic and discuss discuss their relative strengths strengths and and the the boot bootstrapping strapping process. process. Secondly, Secondly, we we give give an an overview overview of of the the incompleteness incompleteness theorems. theorems. Thirdly, Thirdly, section section 3 discusses discusses the the topics topics of of what what functions functions are are provably provably total total in in various various fragments arithmetic and strengths of fragments of of arithmetic and of of the the relative relative strengths of different different fragments fragments of of arith arithmetic. conclude with proof of J. Paris A. Wilkie Wilkie which metic. Finally, Finally, we we conclude with aa proof of aa theorem theorem of of J. Paris and and A. which 's incompleteness improves theorem by showing that improves G6del Ghdel's incompleteness theorem by showing that L�o IA0 + + exp exp cannot cannot prove prove the prerequisite for reading this this chapter knowledge of the consistency consistency of of Q Q.. The The main main prerequisite for reading chapter is is knowledge of the sequent Chapter II of this volume. the sequent calculus calculus and and cut-elimination, cut-elimination, as as contained contained in in Chapter of this volume. The The proof proof theory theory of of arithmetic arithmetic is is aa major major subfield subfield of of logic logic and and this this chapter chapter necessarily necessarily omits omits many many important important and and central central topics topics in in the the proof proof theory theory of of arithmetic; arithmetic; the the most most notable notable omission omission is is theories theories stronger stronger than than Peano Peano arithmetic. arithmetic. Our Our emphasis emphasis has has instead nitary proof instead been been on on weak weak fragments fragments of of arithmetic arithmetic and and on on fi finitary proof theory, theory, especially especially on on applications applications of of the the cut-elimination cut-elimination theorem. theorem. The The articles articles of of Fairtlough-Wainer, Fairtlough-Wainer, Pohlers, Pohlers, Troelstra Troelstra and and Avigad-Feferman Avigad-Feferman in in this this volume volume also also discuss discuss the the proof proof theory theory of of arithmetic. arithmetic. There There are are aa number number of of book book length length treatments treatments of of the the proof proof theory theory and and model model theory theory of of arithmetic. arithmetic. Takeuti Takeuti [1987], [1987], Girard Girard [1987] [1987] and and Schiitte Schiitte [1977] [1977] discuss discuss the the classical classical proof proof theory theory of of arithmetic, arithmetic, Buss Buss [1986] [1986] discusses discusses the the proof proof of of the the bounded bounded arithmetic, arithmetic, and and Kaye Kaye [1991] [1991] and and Hajek H~jek and and Pudlak Pudl~k [1993] [1993] treat treat the the model model theory theory of of arithmetic. arithmetic. The The last last reference reference gives gives an an in-depth in-depth and and modern modern treatment treatment both both of of classical classical fragments fragments of of Peano Peano arithmetic arithmetic and and of of bounded bounded arithmetic. arithmetic.

Proof Proof Theory Theoryof of Arithmetic Arithmetic

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Fragments o f arithmetic arithmetic 11.. Fragments of

This This section section introduces introduces the the most most commonly commonly used used axiomatizations axiomatizations for for fragments fragments of arithmetic. arithmetic. These These axiomatizations axiomatizations are are organized organized into into the the categories categories of of 'strong 'strong of fragments',' , 'weak 'weak fragments fragments'' and and 'very 'very weak weak fragments fragments'.' . The The line line between between strong strong fragments and and weak weak fragments fragments is is somewhat somewhat arbitrarily arbitrarily drawn drawn between between those those theories theories which which can prove prove the the arithmetized arithmetized version version of of the the cut-elimination cut-elimination theorem theorem and and those those which which can cannot; in in practice, practice, this this is is equivalent equivalent to to whether whether the the theory theory can can prove prove that that the the cannot; superexponential function function ii f-7 ~-~ 2} 21 is is total. total. The The very very weak weak theories theories are are theories theories which which superexponential do not not admit admit any any induction induction axioms. axioms. do Non-logical Non-logical symbols s y m b o l s for for arithmetic. a r i t h m e t i c . We We will will be be working working exclusively exclusively with with first firstorder theories theories of of arithmetic: arithmetic: these these have have all all the the usual usual fifirst-order symbols, including including order rst-order symbols, propositional connectives connectives and and quantifiers quantifiers and and the the equality equality symbol symbol ((= - )). . In In addition, addition, propositional they have have non-logical non-logical symbols symbols specific specific to to arithmetic. arithmetic. These These will will always always include include the the they constant constant symbol symbol 00,, the the unary unary successor successor function function S S,, the the binary binary functions functions symbols symbols + and and . 9for for addition addition and and multiplication, multiplication, and and the the binary binary predicate predicate symbol symbol � _< for for + 'less omitting parentheses 'less than than or or equal equal to to'.' . 11 Very Very often, often, terms terms are are abbreviated abbreviated by by omitting parentheses around around the the arguments arguments of of the the successor successor function, function, and and we we write write St instead instead of of S(t) S(t).. In In n addition, addition, for for n n 2: k 0 an an integer, integer, we we write write s S"tt to to denote denote the the term term with with n n applications applications of S to to tt.. of For For weak weak theories theories of of arithmetic, arithmetic, especially especially for for bounded bounded arithmetic, arithmetic, it it is is common common to include include further further non-logical symbols. These These include include aa unary unary function function l[sxJ to non-logical symbols. �1xJ for for division by two, two, a a unary unary function function IIxl which is is defined defined by by division by x l which I.,I = r~og~(n + 1)1, and 'smash')' ) which which we we define define by and Nelson's Nelson 's binary binary function function ## (pronounced (pronounced 'smash by

nl . m # n = 2 Iml·l Iml'lnl. m#n =

It iiss easy easy to to check check that that Inl Inl isis equal equal to to the the number number of of bits bits in in the the binary binary representation representation It

of of Tb. n.

An alternative alternative to to the the ## function function is is the the unary unary function function Wl, WI , which which is is defined defined by by An nllog2 nJ and and has has growth growth rate rate similar similar to to ##. . The The importance importance of of the the use use of of wl(n) wl (n) - n0og2nJ WI function function and and the the ## function function lies lies mainly mainly in in their their growth growth rate. rate. In In this this regard, regard, the wl the wl (n) (n) ~� nn#n and m m#n (max{m, n})). n})) . they are are essentially essentially equivalent equivalent since since Wl they # n and # n -= O(wl (max{m, Both of of these these functions functions are are generalizable generalizable to to faster faster growing growing functions functions by by defining defining Both 2 l x l #nlyl where where #2 #2 is is ##. . ItIt isis easy easy to to check check that that Wn-1(llog2 XJl and and x#n+ly X#n+lY -= 21*l#,lul wn(x ) -= xx~"-l(ll~ wn(x) and #~+1 #n+l are are equivalent equivalent in in the the sense sense that that any any term term involving involving the growth growth rates rates of of w~ the Wn and one of of the the function function symbols symbols can can be be bounded bounded by by aa term term involving involving the the other other function function one symbol. symbol. =

1 Many authors authors use use << instead instead of of <; :5; however, however, we we prefer prefer the the use use of of <:5 since since this this sometimes sometimes 1Many makes makes axioms axioms and and theorems theorems more more elegant elegant to to state. state.

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For strong strong theories theories of of arithmetic, it is is sometimes sometimes convenient convenient to to enlarge enlarge the the set set of of For arithmetic, it non-logical symbols symbols to to include include function function symbols symbols for for all all primitive primitive recursive recursive functions. functions. non-logical The usual usual way way to to do do this this is is to to inductively inductively define define the the primitive primitive recursive recursive functions functions The as as the the smallest smallest class class of of functions functions which which contains contains the the constant constant function function 0 and and the the successor successor function function S S,, is is closed closed under under aa general general form form of of composition, composition, and and is is closed closed under under primitive primitive recursion. recursion. The The closure closure under under primitive primitive recursion recursion means means that that if if g 1) -ary g and and h are are primitive primitive recursive recursive functions functions of of arities arities n n and and n n+ + 22,, then then the the (n (n + + 1)-ary function function f f defined defined by by

f/ ( (x, ~ , 0 O) ) =

g(x) g(~)

ff (x, (x, m)) (~, m m+ + 1) 1) = -- h(x, h(~, m, m, f f(~', m))

is called the is also also primitive primitive recursive. recursive. These These equations equations are are called the defining defining equations equations of of f f .. A bounded bounded quantifier quantifier is is of of the the form form (Vx � < t) t)(-..) or (3x � < t)(t)(...) where tt is is aa A · ·) where (- . . ) or term (x � term not not involving involving x x.. These These may may be be used used as as abbreviations abbreviations for for (Vx) (Vx)(x < tt :::l D ..--) . .) and (3x) (x � respectively; or, alternatively, the and (3x)(x <_ tt 1\ A ... .. ). ), , respectively; or, alternatively, the syntax syntax of of first-order first-order logic logic may be be expanded expanded to to incorporate quantifiers directly. In In the the latter latter case, case, may incorporate bounded bounded quantifi ers directly. the calculus is with additional additional inference .4. the sequent sequent calculus is enlarged enlarged with inference rules, rules, shown shown in in section section 11.4. A A usual usual quantifier quantifier is is called called an an unbounded unbounded quantifier; quantifier; when when Ixl Ix] is is in in the the language, language, aa t I) is called aa sharply bounded bounded quantifier quantifier of of the the form form (Qx � <_ IIt]) is called sharply bounded bounded quantifier. quantifier. bounded formulas. A theory theory is is said said to to be be bounded bounded ifit if it is is axiomatizable axiomatizable with with aa set set of of bounded formulas. A Since Since free free variables variables in in axioms axioms are are implicitly implicitly universally universally quantified, quantified, this this is is equivalent equivalent with aa set -sentences (which (which are defined in .2.1). to to being being axiomatized axiomatized with set of of Ill IIl-sentences are defined in section section 11.2.1). 1.1. 1.1. Very V e r y weak weak fragments f r a g m e n t s of of arithmetic arithmetic The The most most commonly commonly used used induction-free induction-free fragment fragment of of arithmetic arithmetic is is Robinson's Robinson's introduced by theory theory Q Q,, introduced by Tarski, Tarski, Mostowski Mostowski and and Robinson Robinson [1953] [1953].. The The theory theory Q Q has has non-logical symbols 00,, S axioms: non-logical symbols S,, + + and and · 9and and is is axiomatized axiomatized by by the the following following six six axioms: (Vx) 0) ( w ) ((~--,sSx ~ =F # 0) y) (Vx) ( w ) ((Vy) v y ) ((Sx s~ = = Sy s y :::l ~ x9 = : y)

(Vx) (x =F (v~)(~ # 0 :::l ~ (3y)(Sy (~y)(sy = = x)) ~)) (Vx) (x + (vz)(~ + 0= = x) ~) (Vx) (Vy)(x + (w)(vy)(~ + Sy sy = = S(x s(~ + + V) y))) (Vx) (x 0o = (v~)(~. = 0) o) .

(Vx) (x · Sy (v~)(~. sy = = x9 ·.yy + + x) ~)

Unlike Unlike most most of of the the theories theories of of arithmetic arithmetic we we consider, consider, the the language language of of Q does does not not contain contain the the inequality inequality symbol; symbol; however, however, we we can can conservatively conservatively extend extend Q Q to to include include _< by by giving giving it it the the defining defining axiom: axiom: � x� < Y y t+ ++ (3 ( ~ zz))((x ~ + + z= = V y).) .

Theory of Arithmetic Proof Theory

83 83

This conservative conservative extension extension of of Q Q is is denoted denoted Q� Q<.. This A yet yet weaker weaker theory theory is is the the theory theory R R,, also also introduced introduced by by Tarski, Tarski, Mostowski Mostowski A and Robinson Robinson [1953] [1953].. This This has has the the same same language language as as Q Q and and is is axiomatized axiomatized by by the the and following infi infinite set of of axioms, axioms, where where we we let let s � _ t abbreviate abbreviate (3z) (3z)(s + z= - t) t).. following nite set (s +

m o =InO Sin0 ~: S Sn0 s mo + m +n o nO = s Sin0 + S Sn0 = s Sm+~0 m o . 9S m.n o nO = s Sin0 S~0 = s Sm'~0

for all all 0 � _ m < < n, for for all all m, n � _ 0, for for all all m, n � _ 0, for mo V s mo � (Vx) (x � _ s Sin0 smo < x) for for all all m m � _ 0, and and (V'x) (vz)(z � < s s mmoo t-t + + .x = = 0 o V v x z = = SO so V v x z = - S s ~2o0 V v . .·. · · V v x z = = s s ~moo) ) (V'x)(x for all all m m � > o. O. for

We leave leave it it to to the the reader reader to to prove prove that that Q QF ~R R.. We 1.2. Strong S t r o n g fragments f r a g m e n t s of of arithmetic arithmetic 1.2. This section section presents presents the the definitions definitions and and the the basic basic capabilities capabilities of of some some strong strong This fragments of of arithmetic. arithmetic. These These fragments fragments are are defined defined by by using using induction induction axioms, axioms, fragments minimization axioms or or collection axioms; these these axioms axioms do do not not always always apply apply to to all all minimization axioms collection axioms; formulas, but but rather rather apply apply to to formulas formulas that that satisfy satisfy certain certain restrictions restrictions on on fifirst-order rst-order formulas, quantifier alternation. alternation. For For this this purpose, purpose, we we make make the the following following defi definitions: quantifier nitions: 1.2.1. Definition. Definition. A formula formula is is called called aa bounded bounded formula if if it it contains contains only only 1.2.1. A bounded quantifi quantifiers. formulas is For nn � >_ 0 0,, the bounded ers. The The set set of of bounded bounded formulas is denoted denoted A0. �o . For the classes ITn of of first-order formulas are defined by: classes En and IIn first-order formulas are inductively inductively defined by: �n and (1) ~0 �0o, , (1) = A �o == ri0 ITo = (2) �n+1 E~+I is is the set of of formulas formulas of (3s where A II~ and and si is (2) the set of the the form form (:Ji)A where AE E ITn is aa possibly possibly empty vector of empty vector of variables. variables. (3) II~+1 is the set of E~ and and ~ is aa possibly possibly the set of formulas formulas of of the the form form (3~)A (:Ji)A where where A EE �n i is (3) ITn + 1 is empty vector vector of empty of variables. variables. These classes classes �n En,, II~ These ITn form form the the arithmetic hierarchy.

1.2.2. Definition. Definition. The The induction axioms are axiom scheme; scheme; that 1.2.2. are specified specified as as an an axiom that

for all all formulas formulas A AC E ~.

S. Buss

84

for or replacement axioms for all all A E r . Likewise, Likewise, the the collection or axioms for for r are are denoted denoted -REPL O-REPL and and consist consist of of the the formulas formulas (Vx _ tt)(Sy)A(x,y) D (3z) (Sz)(Vx _ tt)(3y <_z)A(x, z)A(x, y), x� ) (3y � (\:Ix � ) (3y)A(x , y) J (\:I

. for all all A E ~. for The 1.2.3. 1.2.3. Definition. Definition. The above above axioms axioms form form the the basis basis for for aa hierarchy hierarchy of of strong strong fragments 0, S fragments of of arithmetic arithmetic over over the the language language containing containing the the non-logical non-logical symbols symbols 0, S,, � . The +, and _. The theory theory n:: /E,n iiss defined defined to to bbee the the theory theory axiomatized axiomatized by by the the eight eight + , . 9and axioms particular importance special case axioms of of Q� Q< plus plus the the I:n-IND E,-IND axioms. axioms. Of Of particular importance is is the the special case defined as o -IND axioms. axioms. The of I:n is of the the theory theory 11::. IA00 which which defined as Q� Q< plus plus the the 1::. A0-IND The theory theory L LE~ is -MIN axioms. defined the I:n defined to to be be the the theory theory 11::. IA00 plus plus the E~-MIN axioms. Similarly, Similarly, BI:n BEN is is the the theory theory consisting of 11::. IA00 plus plus the the I:n E,-REPL axioms. Other Other theories, theories, especially especially mn /II,,, Liin LH, -REPL axioms. consisting of and defined similarly. and BIIn BII,,, can can be be defined similarly. The theory theory of of Peano arithmetic, PA PA,, is is defined defined to to be theory Q plus inducbe the the theory plus induc The tion for for all all first-order first-order formulas. formulas. The The figure figure below below shows shows that that PA also also admits admits the the tion minimization minimization and and replacement replacement axioms axioms for for all all formulas. formulas. 1.2.4. 1.2.4. The The figure figure below below shows shows the the containments containments between between the the various various strong strong frag fragments that the the theory ments of of arithmetic, arithmetic, where where S S => =} T T indicates indicates that theory S logically logically implies implies the the n::n+ 1 => theory theory T T.. The The two two arrows arrows/E,+~ :=> BI:n+1 BEn+I and and BI:n+1 BE,+1 => = ~ /n::n E , do do not not reverse, reverse, i.e., i.e., the the containments containments are are proper. proper. These These facts facts are are due due to to Parsons Parsons [1970] [1970] and and Paris Paris and and The figure Kirby Kirby [1978] [1978].. ((The figure is is taken taken from from the the latter latter reference. reference.)) n::n+1 -~n+l .ij.

BI:n+ l � BIIn BE.+lV=~BII~ .ij.

n:: PE.n � ~ / 1 -m I. n � ~ L EL. I:n � r LIIn

Most .2.9. The Most of of these these containments containments are are proved proved in in section section 11.2.9. The fact fact that that BI:n+1 BE,+1 is is aa below. The subtheory subtheory of of/En+I is proved proved as as Theorem Theorem 1.2.9 1.2.9 below. The fact fact that that it it is is aa proper n::n +1 is subtheory n::n+1 is Theorem 3.4.2. subtheory of of/E,+I is proved proved as as Theorem 1.2.5. Some 1.2.5. I:� E + and and II� H + formulas. formulas. Some authors authors use use aa different different definition definition of of the the arithmetic .2.1 . These arithmetic hierarchy hierarchy than than definition definition 11.2.1. These alternative alternative classes, classes, which which we we denote denote I:� E + and and II� H + ,, are are inductively inductively defined defined by by (1) 0, (1) I:; + = = II; n o+ = = 1::. zx0, ending an block of (2) I:� (2) ++11 is E,+ is the the set set of of formulas formulas obtained obtained by by prep prepending an arbitrary arbitrary block of exis existential bounded universal universal quantifiers -formulas. tential quantifiers quantifiers and and bounded quantifiers to to II� II+-formulas. (3) II� ending an II++l is the the set set offormulas of formulas obtained obtained by by prep prepending an arbitrary arbitrary block block of of universal universal + l is quantifiers bounded existential -formulas. quantifiers and and bounded existential quantifiers quantifiers to to I:� E+-formulas.

Proof Theory Theory of of Arithmetic Arithmetic Proof

85 85

Thus + and + are Thus EE� and HlI� are defined defined analogously analogously to to En En and and Hn, lIn ' except except arbitrary arbitrary bounded bounded quantifiers may be be inserted inserted without without adding adding to to the the quantifier complexity. quantifiers may quantifier complexity. It ~ - R E P L proves E� -formula isis It isis straightforward straightforward to to prove prove that that EEn-REPL proves that that every every E+-formula equivalent to to aa E~-formula, En -formula, using using induction induction on on the the number number of of unbounded unbounded quantifiers quantifiers equivalent which which are are in in the the scope scope of of aa bounded bounded quantifier, quantifier, with with aa sub-induction sub-induction on on the the number number of of bounded quantifiers quantifiers which which have have the the outermost outermost unbounded unbounded quantifier quantifier in in their their scope. scope. bounded Therefore, /E rr;� is equivalent equivalent ttoo / Err;n and BE BE� is equivalent equivalent to to BEn. BEn . Therefore, + is ~ and + is

1.2.6. BBootstrapping It:.o , PPhase 1.2.6. o o t s t r a p p i n g IA0, h a s e 11 The many The axioms axioms of of Q Q are are very very simplistic simplistic and, and, by by themselves, themselves, do do not not imply imply many elementary facts facts about about addition addition and and multiplication, multiplication, such such as as commutativity commutativity and and asas elementary sociativity. sociativity. When When combined combined with with induction induction axioms, axioms, however, however, the the axioms axioms of of Q Q imply imply many many basic basic facts facts about about the the integers. integers. The The process process of of establishing establishing such such basic basic facts facts as as commutativity and associativity of of addition the transitivity of <�, , commutativity and associativity addition and and multiplication, multiplication, the transitivity of bootstrapping, named named after after the the expression the totality of subtraction, subtraction, etc. etc. is is called called bootstrapping, the totality of expression "to lift oneself oneself by by one's h a t is say, in in order order to use the of "to lift one's bootstraps". bootstraps" . TThat is to to say, to use the full full power power of aa set set of axioms, itit is is necessary necessary to to do do some some relatively relatively tedious tedious work work establishing establishing that that of axioms, the axioms axioms of Q are are sufficient base theory. theory. the of Q sufficient as as a a base This give a a sketch of the bootstrapping process process for for It:.o IA0;; to to keep keep This section section will will give sketch of the bootstrapping things brief, an outline outline will will be be given, given, with with most of the proofs left the things brief, only only an most of the proofs left to to the reader. Because Because It:.o theory of of all all the the strong fragments defined defined above, above, this reader. IA0 is is aa sub subtheory strong fragments this bootstrapping to all all of of them. them. bootstrapping applies applies equally equally well well to To begin the the bootstrapping bootstrapping process, process, show that the the following formulas are To begin show that following formulas are It:.o IA0 provable. provable. (a) Addition A d d i t i o n is is commutative: commutative: (\Ix, (Vx, y) y ) ((x x + + y y = = y y + + x) x ) . . In In order order to to prove prove this, this, first first (a) (0 + prove prove the the formulas formulas (a.l) (a.1) (\Ix) (Vx)(0 + x x = - x) x) and and (a.2) (a.2) (\Ix, (Vx, y) y ) ((Sx Sx + + y y = - SS(x (x + + y)) y)). . In In order induction on order to to prove prove (a.l), (a.1), use use induction on the the formula formula 00 + + x x = = x x,, and and to to prove prove (a.2) (a.2),, use induction on use induction on Sx Sx + + y y = - SS(x (x + + y) y) with with respect respect to to the the variable variable y y.. Note Note that that the the variable x x is is being being used used as as a a parameter parameter in in the the latter latter induction. From these these two, two, one one variable induction. From can use use induction on x + § y= - y+ + x x and and prove prove the the commutativity commutativity of of addition. addition. can induction on (b) (b) Addition Addition is is associative: associative: (\Ix, (Vx, y, y , zz) ) ( ((x (x + + y) y) + + zz = = x x + + (y (y + + z)) z)).. Use Use induction induction on on (x ( x ++y y) ) + z+= zx +=( yx++ z ) (y . + z) . (c) (c) Multiplication Multiplication is is commutative: commutative: (\Ix, (Yx, y)(x y ) ( x ·. y y = = y y .. x) x ) . . Analogously Analogously to to (a), (a), first first prove induction with prove (c.l) (c.1) O0 .·xx = = 00 and and (c.2) (c.2) (Sx) ( S x ) . y· y = - X x ·. yY + +y y by by induction with respect respect to to x x and and y, respectively. y, respectively. (d) (d) Distributive Distributive law: law: (\Ix, (Vz, y, y, z) z ) (((x (x + + y) y) .. zz = - X x ·. Zz + + yy . . z) z).. Use Use induction. induction. (e) (e) Multiplication Multiplication is is associative: associative: (\Ix, (Vx, y, y , zz)((x ) ( ( x , · y) y) .. zz = = X x . · (y ( y . . z)) z)).. Use Use induction. induction. (f) Cancellation Cancellation laws laws for .for addition: addition: (\Ix, (Vx, y, y, z) z ) ((x x + + zz = = Y y + + zz +-+ ~ x x = = y) y) and and (f) (\Ix, (Vx, y, y , zz)(x+z )(x+z � <_ y+z y + z +-+ ++ x x � <_ y) y).. Use Use induction induction w.r.t. w.r.t, zz for for the the forward forward implications. implications. (g) (g) Discreteness Discreteness of of � <_": (\Ix, (Yx, y)(x y)(x � < Sy Sy J D x x � <_ Y yV Vx x = -- Sy) S y ) . . This This can can be be proved proved from without any from Q Q without any induction: induction: if if x x � <_ Sy S y , , then then x x + + zz = - Sy S y for for some some zz;; either either zz = - 0 so so x x = - Sy S y , , or or there there is is aa u u such such that that u u = - Sz S z and and so so x x + + Su Su = - Sy S y , , in in which which case case xx + + u u -= - yy and and thus thus x x � _ y y..

s. S. Buss Buss

86 86

(h) Transitivity y, z) (h) Transitivity of of ::; <": (Vx, (Vx, y, z ) ((x x ::; < Y y /\ A Y y ::; < zz :J D x x ::; < y) y).. Follows Follows from from Q Q and and the the associativity associativity of of addition. addition. (i) Anti-idempotency laws: laws: (Vx, (Vx, y)(x y)(x + + y y = -- 00 :J D x x = = 00 /\ A yY = -- 0) 0) and and (i) Anti-idempotency (Vx, y)(x y ) ( x .. y y= - 0 0 :J Dx x = -- 00 V VY y= -- 0) 0).. These These both both follow follow from from Q Q without without any any induction. induction. (Vx, Use 0, then Use the the fact fact that that if if y y =1= ~= 0, then y y = - Sz S z for for some some zz.. (j) (x ::; (j) Reflexivity, Reflexivity, trichotomy trichotomy and and antisymmetry antisymmetry of of the the ::; < ordering: ordering: (Vx) (Vx)(x < x) x),, (Vx, y)(x y)(x ::; < y yV Vy y ::; _ x) x) and and (Vx, (Vx, y)(x y ) ( x ::; < y y /\ Ay y ::; <_ x x :J D x x = = y) y).. To To prove prove trichotomy, trichotomy, (Vx, use induction on depending on use induction on y y;; the the argument argument splits splits into into two two cases, cases, depending on whether whether x x ::; < y y or or y y + + S S zz = : x x for for some some zz.. To To prove prove antisymmetry, antisymmetry, reason reason as as follows: follows: if if x x + + u u = = y y and y y+ + vv = : x x,, then then x x+ + u u+ + vv = : x x,, so so by by the the cancellation cancellation law law for for addition, addition, u u+ + vv = - 00 and and potency uu = and by by anti-idem anti-idempotency : vv = = 00 and and thus thus x x = : y y.. (k) (k) Cancellation Cancellation laws laws for for multiplication: multiplication: (Vx, (Vx, yy,, z)(z z)(z =1= ~: 00 /\ Ax x . . zz = = y y .. zz :J D x x = = y) y) and (Vx, (Vx, y, y, z)(z z ) ( z =1= ~ 0 0 /\ Ax x .. zz ::; <__y y .. zz :J D x x ::; <_ y) y) .. These These can can be be proved proved using using (j) (j) to to have have and xx = or yy + Then use the distributive law, the = y y or or x x + + Sv Sv = = y y or + Sv Sv = = x x for for some some vv.. Then use the distributive law, the

anti-idem potency of anti-idempotency of multiplication, multiplication, and and the the cancellation cancellation laws laws for for addition. addition. (1) (1) Strict Strict inequality: inequality: ss < < tt is is an an abbreviation abbreviation for for SS(s) ( s ) ::; ~_ tt.. Thus, Thus, we we can can use use . . ) to bounded existential quantifiers quantifiers (Vx (Vx < < tK t)(.-.) to mean mean (Vx (Vx ::; < t) t ) ((x x < < tt :J D .. ... .)), , and and bounded existential similarly for bounded universal similarly for bounded universal quantifiers. quantifiers. Theorem. T h e o r e m . 1.6. IA f~- .6.o-MIN. Ao-MIN.

Proof. P r o o f . The The minimization minimization axiom axiom for for A A ((x) x ) is is easily easily seen seen to to be be equivalent equivalent to to complete complete induction induction on on ,A(x) -~A(x),, namely namely to to (Vy) (((Vz < A((y)) A((x) (Vy)(((Vz < y),A(z)) y)-~A(z)) :J D -, ~A y ) ) :J D (Vx) ( V x ), ~A x ) ..

This induction on bounded formula ( ,A(y)) , and This is is equivalent equivalent to to induction on the the bounded formula (Vy (Vy ::; < x) x)(-~A(y)), and therefore therefore is is provable provable in in 1.6.0 IA0.. 1.2.7. 1.2.7. Extending E x t e n d i n g the t h e language l a n g u a g e of o f arithmetic. arithmetic. We We now now introduce introduce two two useful useful means means of of conservatively conservatively extending extending the the language language of of arithmetic nitions of arithmetic with with defi definitions of new new predicate predicate symbols symbols and and function function symbols. symbols. It It will will be be of of particular particular importance importance that that we we can can use use the the new new function function and and predicate predicate symbols symbols in in induction induction formulas. formulas. Definition. Definition. A A predicate predicate symbol symbol R(x) R(~) is is .6. A0-defined if it it has has a a defining defining axiom axiom 0 -defined if R(x)

H

cp(x)

with with cp r a a .6.o-formula A0-formula with with all all free free variables variables as as indicated. indicated. The and The predicate predicate R R is is .6. A1-defined by a a theory theory T T if if there there are are �1 ]El formulas formulas cp(x) r and 1 - defined by 'If;(x) such (cp H r such that that R R has has the the defining defining axiom axiom above above and and T T f~ (Vx) (V~)(r ++ ''If;) 9r .

of Arithmetic Arithmetic Proof Theory of

87 87

Definition. Let Let TT be be aa theory theory of of arithmetic. arithmetic. A A function function symbol symbol ff (~) (x) is is E1 "'£. 1 -defined Definition. - defined by TT ifif itit has has aa defining defining axiom axiom by y

=

f (x)

+-+

4J(x, y),

where rcjJ is is aa 21 "'£. 1 formula formula with with all all free free variables variables indicated, indicated, such such that that TT proves proves where 2 (Vi) (3!y)cjJ(x, y) . (V~) (3!y)r y) ~ 9 The The ~l-definable "'£. 1 -definable functions functions of of aa theory theory are are sometimes sometimes referred referred to to as as the the provably provably recursive or or provably total functions functions of of the the theory. theory. To see that that this this aa reasonable reasonable recursive To see

definition for for "provably "provably recursive", recursive" , let let M M be be aa Turing Turing machine machine which which computes computes aa definition function yy -= M(x). M (x) . Also Also choose choose some some scheme scheme for for encoding encoding computations computations of of M M function and let let TM(X, TM(x, w, y) y) be be the the predicate predicate expressing expressing "w "w encodes encodes aa computation computation of of M M on on and input xx which which outputs outputs y." y ." From From the the bootstrapping bootstrapping below, below, itit can can be be seen seen that that the the input predicate can be Therefore, the function computed computed by be aa bounded bounded formula. formula. Therefore, the function by M M predicate TM can can ~l-defined Conversely, for can be be "'£. by the the (true) (true) formula formula (Vx)(3!y)(3w)(TM(x, (V'x) (3!y) (3w) (TM (X, w, y)). y)) . Conversely, for 1 -defined by any true true sentence ~-formula, the function (V'x) (3!y)cjJ(x, y) y) with with rcjJ aa "'£. function mapping mapping ~x to to any sentence (V~)(3!y)r 1 -formula, the yy can can be be computed machine that, given input input values values for searches for for computed by by aa Turing Turing machine that, given for Z, x, searches and for values of of the the existential existential quantified quantified variables variables in in rcjJ which value for for y and aa value for values which witness witness (3y)cjJ(x, y). y) . the truth of of (3y)r the truth of functions in lfio is possible possible to to give stronger In the case of functions in IA0,, it it is give a a stronger In the case of ~l-definability "'£. 1 -definability of equivalent condition; following theorem of Parikh Parikh [1971]: equivalent condition; this this is is based based on on the the following theorem of [1971]:

1.2.7.1 Theorem. Let A(~,y) A(x, y) be a bounded formula formula and TT be a 1.2.7.1.. Parikh's Parikh's T heorem. Let bounded theory. Suppose T (V'x) (3y)A(x, y) . Then there is a term bounded T f~-(V~)(3y)A(~,y). term t such that (3y :0:; T also proves (V'x) (V~)(3y _ t)A(x, t)A(~, y) y).. T The above above theorem theorem is is stated stated with with y a a single single variable, variable, but but it it also also holds for aa vector vector of of The holds for existentially quantified variables. proof of existentially quantified variables. A A proof-theoretic proof-theoretic proof of aa generalization generalization of of this this theorem theorem is is sketched sketched in in section section 1.4.3 1.4.3 below. below. It It is is easily easily seen seen that that 1fio IA0 is is a a bounded bounded theory, theory, since since the the defining defining axiom axiom for for :0:; < may may Ifio-provable) formula be replaced be replaced by by the the ((IA0-provable) formula

x :0:; y

+-+

(3z :O:; y) (x + z = y),

and and the the induction induction axioms axioms may may be be replaced replaced by by the the equivalent equivalent axioms axioms ) (A(x) � ((Vz)(A(0) A (V'x (Vx :0:; _< zz)(A(x) D A(Sx)) A(Sx)) � D A( A(z)). V'z) (A(O) 1\ z)) .

' s theorem Thus theorem. Its Thus applying applying Parikh Parikh's theorem gives gives the the following following theorem. Its proof proof is is straightfor straightforward ward and and we we leave leave it it to to the the reader. reader. . . " . This 2The 2The notation (3!y) (3!y) means means "there "there exists exists a unique unique y such such that ....". This is not part of the syntax rst-order logic; syntax of of fifirst-order logic; but but is is rather rather an an abbreviation abbreviation for for aa more more complicated complicated first-order first-order formula. formula.

88 88

S. Buss

1.2.7.2. 1.2.7.2. Theorem. Theorem. has a defining axiom

A A function function symbol ff (x) (Z) is � 5]l-defined by 1fio IAo if if and only if if it 1 -defined by

r y), yv=f(:) = f (x) ++ (x,

variables and it has a bounding and bounding term term t(x) t(Z) such that r is a fio A0 formula with all free variables (3!y ::; y) .3 indicated and and 1fio IAo proves (V'x) (VZ)(3!y _ t)(x, t)r y).a A -defined by A predicate predicate symbol symbol R R is is fi A1-defined by 1fio IA0 if if and and only only if if it it is is fio A0-defined by 1fio IA0 l -defined by (and furthermore, furthermore, 1fio IA0 can prove the equivalence of the two definitions).

The The next next theorem theorem states states the the crucial crucial fact fact about about � 5]:-definable functions that that they they 1 -definable functions may may be be used used freely freely without without increasing increasing the the quantifier quantifier complexity complexity of of formulas, formulas, even even when reexpressed in in the original language language of of arithmetic. arithmetic. when reexpressed the original 1.2.7.3. 1.2.7.3. Theorem. Theorem. (Gaifman 1982,Prop 2.3 ] , see 1986,Thm 2.2 ] .) (Gaifman and and Dimitracopoulos Dimitracopoulos [[1982,Prop 2.3], see also also Buss Buss [[1986,Thm 2.2].) (a) Let T T ;2 2 B�l BE1 be a theory of of arithmetic. Let T T be extended to a theory T+ T + in an enlarged language L ++ by adding fi A:-defined E1-defined function l -defined predicate symbols, � 1 -defined function symbols and their their defining equations. Then T+ T + is conservative over T T.. Also, if if i � > 11 -) formula in the enlarged language L ++ , then there and if if A A is a �i Ei- (respectively, IIi Hi-) is a formula formula A A -- in the language of T T such that A A is also in �i Ei (respectively, IIi) Hi) and such that that T+ T + fF- (A ++ ~ A-). A-). -

for T T ;2 ~ Q a bounded theory in the language 0, S, +, +,.' and (b) The same results holds for i = 0O,, i.i.e., e . , for the class fio Ao..

Proof. ) . The symbols Proof. We We first first sketch sketch the the proof proof for for part part (b (b). The proof proof shows shows that that the the new new symbols induction on can be eliminated eliminated from can be from aa formula formula A by by induction on the the number number of of occurrences occurrences of ned predicate of the the new new fio-defi A0-defined predicate symbols symbols and and � 5]:-defined function symbols symbols in in A A.. l -defined function Firstly, Firstly, any any atomic atomic formula formula involving involving aa fio-defined A0-defined R may may just just bbee replaced replaced by by the the defining defining equation equation for for R R.. Secondly, Secondly, eliminate eliminate � 5]:-defined function symbols symbols from from 1 -defined function (· · ·) terms in quantifier each bounded terms in quantifier bounds, bounds, by by replacing replacing each bounded quantifier quantifier (V'x (Vx ::; _ t) t)(...) by by (V'x (Vx ::; < t*)(x ::; _ t ::> D . . . )) where where t* is is obtained obtained from from t by by replacing replacing every every new new bounding term; similarly replacing � E:-defined function symbol symbol with with its its bounding term; and and by by similarly replacing 1 -defined function bounds bounds on on existential existential bounded bounded quantifiers. quantifiers. Thirdly, Thirdly, repeatedly repeatedly replace replace any any atomic atomic formula formula P P ((J f ((S)) ~ ) where where s does does not not involve involve any any new new function function symbols symbols by by either either of of the the formulas formulas ( z)) (3 (3zz ::; <_t ((S) ~ ))((A A lf((~s, z) A/\ P P(z)) • . .

or or

(Af( s, zz)) ::> (z )) , (Vz ::; _< t(S)) t(s-*))(Al(~, DP P(z)), (V'z

where where Af AI is is the the fio-formula A0-formula which which � El-defined and t is is the the bounding bounding term term of of ff.. l -defined ff,, and It It is is easy easy to to see see that that each each step step removes removes new new function function and and predicate predicate symbols symbols from and this ). from A and and preserves preserves equivalence equivalence to to A and this proves proves (b (b). 3The 3The notation notation (3!y (3!y :::; ___t)( t ) ( .. ... ·) ) means means "there "there is is a a unique unique y y such such that that " . . .', , and and this this y y is is :::; _< tt ""..

Proo] Theory of o] Arithmetic Proof

89

The proof proof of of part part ((a) is similar, similar, but but needs needs aa little little modification. modification. Most Most notably, notably, The a) is there there is is no no bounding bounding term term tt,, so so the the two two formulas formulas which which can can replace replace P(J(S)) P(f(s-)) use use an an unbounded quantification thus in Since A unbounded quantification of of zz and and are are thus in � E1l and and IT H:, respectively. Since Aff is is l , respectively. Ei-formula, it is is necessary necessary to to pick pick the correct one one of of the the two two formulas formulas for for replacing replacing aa �i -formula, it the correct �l and P ( f ((S)) ~ ) : : since since the the first first is is E: and the the second second is is 17: there is is always always an an appropriate appropriate P(J ITl there choice Also, in choice that that does does not not increase increase the the alternation alternation of of unbounded unbounded quantifiers. quantifiers. Also, in order order to to remove remove new new function function symbols symbols from from the the terms terms in in bounded bounded quantifiers, quantifiers, it it is is necessary to �l -replacement axioms. axioms. We the reader. necessary to use use the the E:-replacement We leave leave the the details details to to the reader. 0 [::l

As corollary to theorem, we As an an immediate immediate corollary to the the previous previous theorem, we get get the the following following important important bootstrapping bootstrapping fact. fact. Corollary. B�n for Corollary. Let Let T T be be 16.0 IAo,, �n IEn or or BEn for n n � >_ 11.. Then Then in in the the conservative conservative �l -defined function extension -defined function extension of of T T with with E1-defined function symbols symbols and and 6.1 A1-defined function symbols, symbols,

the induction, minimization the new new function function and and predicate predicate symbols symbols may may be be used used freely freely in in induction, minimization and replacement replacement axioms. axioms. and 1.2.8. 1.2.8. Bootstrapping Bootstrapping 16.0 IA0~, Phase Phase 2 2

To begin the To begin the second second phase phase of of the the bootstrapping bootstrapping for for 16.0 IA0,, several several elementary elementary functions relations are nable in functions and and relations are shown shown to to be be �l El-- and and 6.o-defi A0-definable in 16. IA0. 0, ((a) a) Restricted subtraction. The { O, xx -- y} y} can Restricted subtraction. The function function x x -'- y y which which equals equals max max(0, can be be � El-defined by 16.0 IA0 by by the the formula formula l -defined by

M(x, M(x, y, y, z) z) ¢:> r (y + + zz = - x) x) V V (x (x ::; <_Yy 1\ A zz = -- 0). The The existence existence of of zz follows follows immediately immediately from from the the trichotomy trichotomy of of ::; <;; thus thus 16.0 IA0 can can prove x, y) (jz ::; y, z) . Furthermore, prove (\i (Vx, y)(3z < x)M(x, x)M(x, y,z). Furthermore, 16.0 IA 0 can can prove prove the the uniqueness uniqueness of of zz satisfying satisfying M(x, M ( x , yy,, zz) ) using using the the cancellation cancellation law law for for addition. addition. This This then then is is a a � Ell definition of of the the restricted restricted subtraction function. definition subtraction function. ((b) b) Predecessor. -defined by Predecessor. The The predecessor predecessor function function is is easily easily �l El-defined by yy = - x x -'- l1.. ((c) c) Division. -defined by Division. The The division division function function (x, (x, y) y) H ~ Lx/yJ Lx/yJ can can be be �l El-defined by 6.0 A0 using the the formula formula using

M(x, M ( x , yy,, zz) ) ¢:> r (( yy ..zz<::; _ xxA1\x <x y<( zy(z + l )+ ) V1)) ( y =VO(yA z==0O1\) .z = 0) . Note division function total, we arbitrarily defined Note that that in in order order to to make make the the division function total, we have have arbitrarily defined x/O to equal equal O0.. The The existence existence of of zz is is proved proved using using induction induction on on the the formula formula x/O to (jz y, z) . The (3z ::; <_ x)M(x, x)M(x, y,z). The uniqueness uniqueness of of zz is is provable provable as as follows, follows, arguing arguing inside inside y, z) M(x, y, y, z') 16.0 IA0": suppose suppose M(x, M(x, y, z) and and M(x, z'),, w.l.o.g. w.l.o.g, zz ::; _< z' z';; thus, thus, using using restricted restricted subtraction (y + this, z' subtraction and and the the distributive distributive law, law, (z' (z' -'- z) z)(y + 11)) < < y y+ + 11;; and and from from this, z' = - zz follows easily. easily. follows Two particularly useful special cases division are divisor is Two particularly useful special cases of of division are when when the the divisor is two two or or four, four, i.e., i.e., L�x L 89J and and L�x ~4xJJ . ((d) d ) Remainder. Remainder. The remainder function The remainder function is is � El-definable in 16. IA 00 since since l -definable in xx mod relation, xly, ned by mod xx = mod yy = - x x -'- yy.. Lx/yJ Lx/yJ.. The The divides divides relation, x[y, is is defi defined by y y mod - O0..

S. Buss Buss S.

90 90

(e) Square Square root. root. The The square square root root function function xx ~+ I-t [v/~J l v'xJ isis ~-definable � 1 -definable IA0 I�o with with (e)

the formula formula the

M(x, ..:;. xx A/\ xx << (y M(x, y) (y ++ 1)(y l ) (y ++ 1). 1). y ) r{:} yy .. yY <_

(f) Primes. Primes. The The set set of primes is by the the formula formula (f) of primes is �o-definable A0-definable by (Vy (Vy _< ..:;. x)(ylx x) (ylx D:::> yy == xx VV yy == 1) 1) A /\ 11 << x. x.

IA0 prove many I�o can can prove many useful useful facts facts about about primes primes and and remainders. remainders. In In particular, particular, itit is prime prime and and xlab xlab then then xla xla or or xlb. xlb. The The sequence coding tools tools proves that that ifif xx is proves sequence coding developed below below will will enable enable I�o to prove prove the the unique unique factorization factorization theorem; theorem; however, however, developed IA0 to more bootstrapping is is needed needed before before we we can can even even express express the the unique unique factorization factorization more bootstrapping theorem in in first-order logic. theorem first-order logic. (g) Prime Prime powers. powers. The The predicate predicate "x "x isis prime prime and and yy is is aa power of x" x" isis A0�o (g) power of definable by by definable /\ zly xx is is prime prime and and (Vz (Vz _< ..:;. y)(1 y )( l << zz A xlz)).. zlY D:::> xlz

I�o can can prove simple properties properties about prime powers, powers, such such as as the that if if yy is IA0 prove simple about prime the fact fact that is aa is the the least least power power of of xx greater power of of the the prime then x. x yy is greater than than y. y . This This fact power prime xx then fact can can be proved by using with respect to yy.. be proved by using �o-minimization A0-minimization with respect to The sufficiently developed developed for for us us to give A0 The bootstrapping bootstrapping is is not not yet yet sufficiently to give �o definitions definitions of of powers of numbers; however, however, we we shall shall next next define define powers powers of of prime prime powers. powers. powers of composite composite numbers; (h ) Powers two, of four, and and of prime powers. already have (h) Powers of of two, of four, of prime powers. We We already have shown shown how how to define powers of of the two. For fours, we to define powers the prime prime two. For powers powers of of fours, we can can give give two two equivalent equivalent definitions: definitions: .

yy is is aa power of four four power of

1, is aa power power of mod 33 = r{:} yy is of two two and and y y mod - 1,

and and

lv'YJ ) 2 . is a a power power of of four four {:} r is a a power power of of two two and and y y= - (([v/-~J)2. yy is yy is The definitions can The equivalence equivalence of of these these definitions can be be proved proved using using �o-induction Ao-induction with with respect respect to to y y.. I�o IA0 can can also also prove prove that that when when y y< < y' y~ are are aa powers powers of of four, four, then then yyly and that that ly'~ and when when yy is is a a power power of of four, four, then then 4y 4y is is the the least least power power of of four four greater greater then then y y.. More More generally, generally, the the predicate predicate ""xx is is aa prime prime power power and and y y is is a a power power of of x" x" can can be be �o -defined by A0-defined by powers of and yy mod (3p (p is and yy are mod (x (3p ":;' _< x) x)(p is prime, prime, x x and are powers of p p,, and (x - 1) 1) = - 1) 1).. ((i) i ) The The LenBit L e n B i t function function is is defined defined so so that that LenBit(2i, LenBit(2 i, x) x) is is equal equal to to the the i-th i-th bit bit in in the the binary binary expansion expansion of of x x,, where where the the least least significant significant bit bit is is by by convention convention the the zeroth zeroth bit. This bit. This is is � ~l-definable by I�o IA0 since since LenBit(y, LenSit(y, x) x) = - lxjyJ Lx/y] mod mod 22.. Although Although it it is is l -definable by defined defined for for all all values values yy,, we we shall shall use use LenBit(y, LenBit(y, x) x) only only when when yy is is aa power power of of two. two.

The The next next theorem theorem states states that that I�o IA0 can can prove prove that that the the binary binary representation representation of of aa number number uniquely uniquely determines determines the the number. number. This This theorem theorem also also introduces introduces aa new new notation; notation; namely, namely, we we will will write write quantifiers quantifiers of of the the form form (V2i) (V2i) and and (32i) (32 ~) to to mean, mean, "for "for all note that all powers powers of of two" two" and and "there "there exists exists aa power power of of two." two." It It is is important important to to note that although although we we use use this this notation notation for for quantifying quantifying over over powers powers of of two, two, we we have have not not yet yet shown -define ii in shown how how to to �o A0-define in terms terms of of 2i 2i..

91 91

Proof Theory of of Arithmetic Arithmetic

Theorem. ILlo (, LenBit(2 LenBit(2i,i , y)) y))..

To induction with To prove prove the the theorem, theorem, ILlo IA0 uses uses strong strong induction with respect respect to to x x and and argues argues that t h a t if if x, then is the the greatest greatest power power of of two two less less tthan h a n x, then LenBit(2 LenBit(2 ii,, x) x) equals equals one, one, and and then, then, 22ii is when when LenBit(2 LenBit(2 ii,, y) y) also also equals equals one, one, applies applies the the induction induction hypothesis hypothesis to to x x -'- 22i/ and and i yy --'-2 2~.. 0 [] (j) the relation (j) We We next next show show how how to to Llo-define A0-define the relation x x = = 22Y as a a predicate predicate of of x x and and y y.. Y as As As a a preliminary preliminary step, step, we we consider consider numbers numbers of of the the form form P

mp -- ~--~22v i=0

{mp}p is for p p 2:: > 00 and and show show that t h a t these these numbers numbers are are Llo-definable. A0-definable. In In fact, fact, the the set set {mp}p is for defi nable by definable by the the formula formula LenBit(1, x) x) = = 00 /\ A LenBit(2, LenBit(2, x) x) = = 1/\ 1A LenBit(l, i i A(V2 i :::; _< x)(2 x)(2 < < 22 iJ D /\(,
As As an an immediate i m m e d i a t e corollary corollary we we get get a a Llo-formula A0-formula defining defining the the the the numbers numbers of of the the 2P ; namely, form x x = - 222P; namely, they they are are the the powers powers of of two two for for which which LenBit(x, LenBit(x, mp) mp) = - 11 holds holds form for for some some mp mp < < 2x 2x.. Now ning 22YY is notation as Now the the general general idea idea of of defi defining is to to express express y y in in binary binary notation as y y = = 22Pj • To distinct values Pj , and 2pl + + 2P 2p22 + + . .. .. . + + 2Pk 2pk with with distinct values pj, and thus thus define define x x = - 1-I~-122p~. To carry carry 2P1 u, x) this out, out, we we define define an an extraction extraction function function Ext( Ext(u, x) which which will will be be applied applied when when u u is is this 2P •. Formally of a a number n u m b e r of of the the form form 222p Formally we we define define of

n�=l

Ext(u, x) = Ext(u,x) = lx/uJ LxluJ mod m o d u. u. 2 then Ext( Note u, x) Note that t h a t when when u u = = 222pP ,, then Ext(u, x) returns returns the the number n u m b e r with with binary binary expansion expansion +1 2P-th bit ) -th bit will think think of equal equal to to the the 2P-th bit through t h r o u g h the the ((22Pp+I - l1)-th bit of of x x.. We We will of x x coding coding 2P, x) for the sequence sequence of of numbers numbers Ext(2 Ext(22P,x) for P p - = 0, 0, 1, 1,2, We also also define define Ext!(u, Ex](u,x)x) as as 2, .. ..... . We the u, x) Ext(uu22,, x) x);; this this is is of of course course the the number n u m b e r which which succeeds succeeds Ext( Ext(u, x) in in the the sequence sequence of of Ext( numbers numbers coded coded by by x x.. i We define We ii)) which We are are now now ready ready to to Llo-define A0-define x x = = 2 2 i .. We define it it with with a a formula formula if;(x, r which 2 states states there there are are numbers numbers a, b, b, c, c, dd :::; <_ xx 2 such such that t h a t the the following following hold: hold: >x (1) (1) a a is is of of the the form form mp mp and and a a > x.. 1 , Ext ( 2, cc)= ) = 00 and ) = 1. 1. (2) Ext(2, Ext(2, b) b)== 1, Ext(2, and Ext(2, Ext(2, dd)= (2) 2P such For all all u u of of the the form form 222~ such that that a > u u 22,, Ext! Ex~(u, b) = = 22. Ext( Ext(u, b),, (u, b) u, b) (3) For 2P such (4) For all all u u of of the the form form 222p such that that a a > > u u 22,, either either (4) For ((a) a) Ext!(u, c) = Ex~ (u, c) = Ext(u, Ext(u, c) c) and and Ext!(u, Ex~ (u, d) d) = = Ext(u, Ext(u, d) d),, or or u, a) b ) Ext! u, c) u, cc)) + u, b) (u, d) u, d) Ex~ ((u, c) = = Ext( Ext(u, + Ext( Ext(u, b) and and Ext! Ex~ (u, d) = = Ext( Ext(u, d) .. Ext( Ext(u, a).. ((b) 2P such (5) There is is a a u u < < a a of of the the form form 222p such that t h a t Ext( Ext(u, c) = - ii and and Ext( Ext(u, d) = - xx.. u, c) u, d) (5) There .

92 92

s. S. Buss

Obviously Obviously this this is is aa �o-formula; A0-formula; we we leave leave it it to to the the reader reader the the nontrivial nontrivial task task of of checking that I�o this definition checking that IA0 can can prove prove simple simple facts facts about about this definition of of 22ii ,, including including (1) (1) the the +j , fact that that if if ¢>(x, r i) and ¢>(x, r j) hold, then then ii = = jj ,, (2) (2) the the fact fact that that 22ii.. 22jj = - 22ii+j i) and j) both both hold, fact i (3) the fact fact that that if if x x is is a a power power of of two, two, then -- 22i for for some some ii < < x x,, and and (4) that if if (3) the then xx = (4) that r¢>(x, i) i) and and ¢>(y, r i ) tthen hen x x= = yy.. i) ((k) k) Length log2 (x + Length function. function. The The length length function function is is Ixl Ixl = = f[log2(x + 1)1 1)~ and and can can be be A0-defined in I�o IA0 as as the the value value ii such such that - 22ii is is the the least least power power of of two two greater greater �o-defi ned in that yy = than binary than x x.. Note Note that that 101 101 = = 00 and and for for x x > > 00,, Ixl Ixl is is the the number number of of bits bits in in the the binary representation representation of of x x.. The The reader reader should should check check that that I�o IA0 can can prove prove elementary elementary facts facts about about the the Ixl Ixl function, function, including including that that Ixl Ixl ::; _ x x and and that that 36 36 < < x x :J D Ixl Ixl22 < < x x.. (1) i, x ) function (1) The The Bit( Bit(i,x) function is is definable definable as as LenBit(2 LenBit(2i,i , x) x).. This This is is equivalent equivalent to to aa i �o-definition, i, x) is A0-definition, since since when when 22~ > x x,, Bit( B i t (i,i ,x) x ) -= oO.. Bit( Bit(i,x) is the the i-th i-th bit bit of of the the binary binary representation representation of of x x;; by by convention, convention, the the lowest lowest order order bit bit is is bit bit number number o0.. Sequence coding. coding. Sequences Sequences will will be be coded coded in in the the base base 44 representation representation used used ((m) m) Sequence by by Buss Buss [1986]; [1986]; many many prior prior works works have have used used similar similar encodings. encodings. A A number number x x is is viewed viewed as as aa bit bit string string in in which which pairs pairs of of bits bits code code one one of of the the three three symbols symbols comma, comma, "0" "0" or or "1".. The The i-th symbol from from the the end end is coded by by the the two two bits bits Bit(2i Bit(2i + + 11,, x) x) and and "1" i-th symbol is coded Bit(2i, Bit(2i, x) x).. This This is is best best illustrated illustrated by by an an example: example: consider consider the the sequence sequence (3, (3, 0, 0, 4) 4).. Firstly, comma is ended to Firstly, aa comma is prep prepended to the the the the sequence sequence and and the the entries entries are are written written in in base base two, string: ",11,,100" leading zeros two, preserving preserving the the commas, commas, as as the the string: ",11,,100";; leading zeros are are optionally optionally dropped dropped in in this this process. process. Secondly, Secondly, each each symbol symbol in in the the string string is is replaced replaced by by a a two two with "11" "0" with with "10" bit encoding bit encoding by by replacing replacing each each "I" "1" with "11",, each each "0" "10",, and and each each comma comma 101011 11010" in with with "01" "01".. This This yields yields "01111 "0111110101111010" in our our example. example. Thirdly, Thirdly, the the result result is is interpreted interpreted as as a a binary binary representation representation of of aa number; number; in in our our example example it it is is the the integer integer 32122. This then Godel number 32122. This then is is a a Ghdel number of of the the sequence sequence (3, (3, 0, 0, 4) 4).. This encoding sequences being relatively This scheme scheme for for encoding sequences has has the the advantage advantage of of being relatively efficient efficient from making it from an an information information theoretic theoretic point point of of view view and and of of making it easy easy to to manipulate manipulate sequences. sequences. It It does does have have the the minor minor drawbacks drawbacks that that not not every every number number is is the the Godel Ghdel number number of of aa sequence sequence and and that that Godel Ghdel numbers numbers of of sequences sequences are are non-unique non-unique since since it it is is allowable allowable that that elements elements of of the the sequence sequence be be coded coded with with excess excess leading leading zeros. zeros. Towards x) and Towards arithmetizing arithmetizing Godel Ghdel numbers, numbers, we we define define predicates predicates Comma(i, Comma(i,x) and Digit( Digit(i,i, x) x) as as

1, x) Comma(i, Comma(i, x) x) ¢} r Bit(2i Bit(2i + + 1, x) = = 00 /\ A Bit(2i, Bit(2i, x) x) = = 11 Digit(i, x) Digit(i,x)

= =

22.· (1 (1 -'1, x)) - Bit(2i Bit(2i + + 1, x ) ) ++ Bit(2i, Bit(2i, x). x).

Note Note Digit Digit equals equals zero zero or or one one for for encodings encodings of of "0" "0" and and "I" "1" and and equals equals 22 or or 33 otherwise. otherwise. 's Godel It It is is now now fairly fairly simple simple to to recognize recognize and and extract extract values values from from a a sequence sequence's Ghdel sEntry( i, j, number. number. We We define define I IsEntry(i, j, x) x) as as

(i = , x)) /\A Comma(j, (i - 00 V v Comma(i Comma(i -'- 11,x)) C o m m a ( j , xx) ) /\ A (Vk (Vk < < j)(k j)(k � > ii :J D Digit(j, Digit(j,x)x) ::; < 1) 1) which ) -st symbols which states states that that the the i-th i-th through through (j (j - l1)-st symbols coded coded by by x x code code an an entry entry in in

Proof Theory Theory of of Arithmetic Arithmetic Proof

93 93

the sequence. sequence. And And we we define define Entry(i, Entry( i, j,j, x) x) == yy by by the

Iyl <-::; Jj --'- ii A/\ (Vk (\fk << jj --'- ii)) ((Bit(k, lYl B i t ( k , y) y) == Digit(i Digit(i ++ k,k, x)). x)). When j, x) is true, j, x) of that in the When IsEntry(i, IsEntry(i, j, x) is true, then then Entry(i, Entry(i, j, x) equals equals the the value value of that entry entry in the sequence coded coded by by x. x . Checking Checking that that Entry Entry isis A0-definable �o -definable by by IA0 I�o is is left left to to the the sequence may be be replaced replaced by by aa sharply sharply bounded bounded reader; note note that that the the quantifier quantifier (Vk reader; (\fk << jj --'- i)i) may quantifier since, w.l.o.g., w.l.o.g., jj <_ ::; Ixl. Ixl . quantifier since, (n) Length-bounded Length-bounded counting and Numones. Numones. Although Although we we have defined Entry Entry (n) counting and have defined already, we we are are not not quite quite done done with with arithmetizing arithmetizing sequence sequence coding; coding; in in particular, particular, we we already, which equals equals the the i-th i-th entry entry would like like to to define define the the G5del G6del beta beta function, function, /3(i,x), f3(i, x) , which would of the the sequence sequence coded coded by by x. x . One One way to do do this this would would be be by by encoding encoding aa sequence sequence of way to al) as as the the sequence sequence (bn,..., (bn , . . . , bl) bl) where (an ' aann -- iI,, ... .. ,. , a~) where bi bi == (i, (i, ai). ai) . The The of numbers numbers (an, of drawback ai are are small, small, the the length length of of the the drawback of of this this approach approach isis that that when when the the values values ai (b) is is longer longer than than the the length length of of the the GSdel G6del G6del number number encoding encoding the the sequence GSdel sequence (b) number encoding the the sequence in fact, fact, it is longer longer by by aa logarithmic logarithmic factor factor and and number encoding sequence (ii) (~);; in it is thus the the function function (d) (ii) f-7 (b) cannot cannot be be A0-defined �o-defined by by I�o of the the function's function ' s thus ~-~ (b) IA0 by by virtue virtue of superlinear growth growth rate. rate. superlinear Upon reflection, reflection, one that the the basic basic difficulty defining the Upon one sees sees that difficulty in in defining the f3 ~ function function is difficulty of of commas commas encoded in a a G6del GSdel number is the the difficulty of counting counting the the number number of encoded in number This basically basically the ones in in of aa sequence. sequence. This the same same as as the the problem problem as as counting counting of of ones of x . Supposing Supposing xx has has binary binary representation representation the binary number x. the binary representation representation of of aa number xlxoh , we we would to be be able able to let a0 ao == Xo and ai ai -= ai (X then (XnXn_l''" would like like to to let x0 and ai-1 +xi and and then n Xn- 1 . . . XlX0)2, - l +Xi bl) . (i, ai) ai) and finally let be the the GSdel G6del number of the the sequence sequence ((bbnn,, .. .. .., , bl). let let bi bi = = (i, and finally let yy be number of the Now, I�o can prove prove that, exists, itit is is unique, unique, and and from from yy the the number number of of 1l's' s in Now, IA0 can that, if if y y exists, in the binary representation representation of of x x is is easily easily computed. computed. The The catch catch is is that, that, as as above, above, (b) (b) will will binary in in general general not not be be bounded bounded by by a a term term involving involving x x since since its its length length is is not not necessarily necessarily O(Ixl). . However, However, the the length length of G5del number number of of (b) (b-') is is O(lxI O(Ixl 22)) so so this this this this method method of the the G6del O(lxl) does does work work when when x x is is small; small; in in particular, particular, it it works works if if x x = = Iy lYll for for some some y y.. Thus, Thus, I�o IA0 -define the the function ned so can �l can El-define function LenNumones LenNumones defi defined so that that

LenNumones(y) LenNumones(y)

= -

the the number number of of ll's's in in the the binary binary representation representation of of Iyl lyl. .

ne aa NNumones To defi define u m o n e s function function that that works works for for all all numbers, numbers, we we use use a a trick trick that that To allows encoding of allows more more efficient efficient encoding of successive successive numbers. numbers. The The basic basic idea idea is is that that a a sequence encoded with sequence aI, al, a2, a2, a3, ca, . 9. 9. 9, ak ak of of numbers numbers can can be be encoded with fewer fewer bits bits if, if, when when writing writing the the number number ai+l ai+~,, one one only only writes writes the the bits bits of of ai ai+l which are are different different from from the the + l which corresponding bits corresponding bits in in ai ai.. This This works works particularly particularly well well when when we we have have ai ai ::; <_ ai+l ai+l ::; <_ ai+ ai+ 11 for ne the for all all ii;; in in this this case case we we formally formally defi define the succinct succinct encoding encoding as as follows: follows: for for ii > > 00,, defi ne ii** to ne 00** to define to bbee the the greatest greatest power power of of 22 which which divides divides ii;; and and defi define to equal equal o0.. Now Now ne a; to be be at if ai ai =1= :/: ai ai-- l1 or or to to be be 00 otherwise. otherwise 9(Example: (Example~. if if define define a� a~ = = ao a0 and and defi define a i' to a i* if - 24 24,, then then a' a' = - 88.).) Then Then the the sequence sequence ii d can can be be more more succinctly succinctly described described by by (t a'.. aa = It is is now now important important to to see see that that I�o IA0 can can extract extract the the sequence sequence (ii) (~) from from the the It sequence sequence ((t) ( s , at at least least in in aa certain certain limited limited sense. sense. In In particular, particular, we we have have that that if if xx = - (a�, ( a'k, . . .",, a� al,, a�) a~) and and if if IsEntry( IsEntry(i,i, j, j, x) x) and and if if this this entry entry is is the the entry entry for for a� a~,, then then the the ..

'

94 94

S. Buss

value i, j, describe this value at at can can be be E ~1-defined in terms terms of of i, j, x x.. To To describe this E ~l-definition, note that that I -defined in I -definition, note the k-th bit the k-th bit of of the the binary binary representation representation of of at at is is computed computed by by finding finding the the maximum maximum values jo, x) values io i0 < < jo j0 :::; _< ii such such that that IsEntry(io, IsEntry(io,jo, x) and and such such that that IEntry(io, [Entry(io, jo, jo, x) x)[1 > kk jo, x) and letting the k-th bit and letting the k-th k-th bit bit of of at ae equal equal the the k-th bit of of Entry(io, Entry(io, jo, x).. The (rt) is The whole whole point point of of using using (~) is to to give give aa sufficiently sufficiently succinct succinct encoding encoding of of (il) (~).. Of Of course, course, the the fact fact that that the the encoding encoding is is sufficiently sufficiently succinct succinct also also needs needs to to be be provable provable in G6del number 0* , 1*, in 1.6.0 IA0., It It is is easily easily checked checked that that the the Ghdel number of of the the sequence sequence ((0", 1 " , .. .. .., , x*) x*) uses bits; this uses exactly exactly 6x 6x - 2Numones(x) 2Numones(x) + + 22 many many bits; this is is proved proved by by first first showing showing that that there there are are 2x 2 x -- Numones(x) gumones(x) bits bits in in the the numbers numbers in in the the sequence, sequence, i.e., i.e., L ~i~0 1i*]I = = �=o W commas, and noting that 2x 2x - Numones(x) Numones(x), , and and second second noting noting that that there there are are x x+ + 11 commas, and noting that each each bit bit and and comma comma is is encoded encoded by by two two bits bits in in the the G6del Ghdel number. number. Furthermore, Furthermore, when when xx = = Iyl ]Yl for for some some yy,, 1.6.0 IA0 can can prove prove this this fact, fact, using using LenNumones LenNumones in in place place of of Numones. Numones. We We are are now now able able to to E ~l-define the function function Numones(x) Numones(x) equal equal to to the the number number of of l -define the x. This 1l's's in in the the binary binary representation representation of of x. This is is done done by by defining defining the the sequence sequence

uu

= =

k* ,, aa~), ) * , a�' _ I ) , . . . , ((0,0, a�)) ((k* ( (kk -- 11)*, (( �) , ((

such I , ao 0, and such that that k k= - Ix Ix[, a0 = = 0, and each each ai ai+l is equal equal to to ai ai + + Bit( Bit(i,i, x) x).. By By the the consider consider+ ! is ations ations in in the the previous previous paragraph, paragraph, 1.6. IA00 can can prove prove that that this this sequence sequence is is bounded bounded by by aa term term involving involving only only x x;; also, also, 1.6.0 IA0 can can compute compute the the values values of of 0, 0 , .. .. .., , kk from from 0* 0 ",, .. .. .., , k* k* and and therefore therefore can can compute compute the the values values of of ai ai as as aa function function of of ii and and u u.. (0) (o) Sequence Sequence coding. coding. Once Once we we have have the the Numones Numones function, function, it it is is an an easy easy matter matter ned so to to define define the the G6del Ghdel/3f3 function function by by counting counting commas. commas. The The/3f3 function function is is defi defined so that f3(m, x) that/3(m, x) = = am am provided provided x x is is the the G6del Ghdel number number of of aa sequence sequence ((al a l ,, ." . .", ak a~)) with with m m :::; _ k k.. It It is is also also useful useful to to define define the the length length function function Len(x) Len(x) which which equals equals kk when when xx is is as as above. above. These These are are defined defined easily easily in in terms terms of of the the Numones Numones function: function: the the value value f3( m, x) fl(m, x) equals equals Entry( Entry(i,i, j, j, x) x) where where there there are are m m -- 11 commas commas encoded encoded in in x x to to the the left left coded by of of bit bit ii;; and and Len(x) Len(x) equals equals the the number number commas commas coded by x x.. Once Once sequence sequence encoding encoding has has been been achieved, achieved, the the rest rest of of the the bootstrapping bootstrapping process process is is fairly fairly straightforward. straightforward. The The next next stage stage in in bootstrapping bootstrapping is is to to arithmetize arithmetize meta metamathematics, mathematics, and and this this is is postponed postponed until until section section 22 below. below. Stronger Stronger theories, theories, such such as as IE define all all primitive recursive functions: this is .2.10. /511, can define primitive recursive functions" this is discussed discussed in in section section 11.2.10. I , can 1.2.9. 1.2.9. Relationships R e l a t i o n s h i p s amongst a m o n g s t the t h e axioms a x i o m s of of PA PA

We are We are now now ready ready to to sketch sketch the the proofs proofs of of the the relationships relationships between between the the various various fragments fragments of of Peano Peano arithmetic arithmetic pictured pictured in in paragraph paragraph 1.2.4 1.2.4 above. above. Theorem. T h e o r e m . Let Let nn 2: > OO.. ((a) a) BrIn BII. F ~ BEn+ B~.+~. !' (b) (]9) IEn+ /~n-l-1! F~ BEn+ BEn+I. !. ((c) c) If A(x, 'Iii) Ee r.. En and tt is a. term, then th n B�n Br.. can prove that that ('
Proof Theory Theory of Arithmetic Arithmetic

95 95

Proof. Proof. To To prove prove (a), (a), suppose suppose A(x, y) is is aa formula formula in in �n ~n+l. We want want to to show show that that + 1 . We

<_u) u)(3y)A(x, D (3v) (3v)(Vx <_u) u)(3y _ v)A(x, y). (Vx � (3y � (Vx � (3y)A(x, y) :::>

is aa consequence consequence of of BIln BIIn.. This This is is proved proved by by the the following following trick: trick: take take all all leading leading is existential existential quantifiers quantifiers (3Z) (3~ from from the the beginning beginning of of A A and and replace replace these these quantifiers quantifiers and and the the existential existential quantifier quantifier (3y) (3y) by by aa single single existential existential quantifier quantifier (3w) (3w) which which is coding values is intended intended to to range range over over G6del Gbdel numbers numbers of of sequences sequences coding values for for all all of of the the ,8(1, w) w) = ,8(i + w) be variables variables yy and and z, ~', say say by by letting letting/~(1, - yy and and letting letting f~(i + 11,, w) be aa value value for for the i-th in z the collection the i-th variable variable in ~*.. Since Since yy = - ,8(1, fl(1, w) w) < < w w,, it it follows follows that that the collection axiom axiom for implies the collection axiom for this this new new formula formula implies the collection axiom for for A A.. Part Part (c) (c) is is proved proved by by induction induction on on n n.. Note Note that that (c) (c) is is obvious obvious when when n = - O0.. For For proved by using aa sequence (c) is is proved by noting noting that, that, by by using sequence to to code code multiple multiple values, values, we we n > 00,, (c) may assume assume without without loss loss of of generality generality that there is is only only one (unbounded) existential existential may that there one (unbounded) quantifier quantifier at at the the front front of of A A,, so so A A is is (3y)B with with B B E Il IIn_x. Then (Vx � <_t)A is is n - 1 . Then (Vx � (3y � induction hypothesis equivalent equivalent to to (3u) (3u)(Vx _ t) t)(3y _ u)B u)B;; and and finally finally by by using using the the induction hypothesis that (c) holds holds for that (c) for n n - 11,, we we have have that that (3y (3y � _ u)B is is equivalent equivalent to to aa Iln_ Ha_l-formula. 1 -formula. (Vx � From From this this (3u) (3u)(Vx <_t)A is is equivalent equivalent to to aa �n-formula. En-formula. We We prove prove (b) (b) by by induction induction on on n n.. Suppose Suppose A(x, y) is is aa �n 5]n+~-formula, possibly + l -formula, possibly containing containing other other free free variables. variables. We We need need to to show show that that .IT:n /En+l proves the the formula formula + l proves displayed displayed above, above, and and by by part part (a) (a) we we may may assume assume that that A A is is aa Iln-formula. Ha-formula. We We argue argue (3y)A(x, y) holds. holds. Let informally informally inside inside/En+~, assuming that that (Vx � <_u) u)(3y)A(x, Let r4>(a) be be the the .IT:n+ 1 , assuming formula formula -

(3v) (Vx � (3y � (3v)(Vx _ a) a)(3y < v)A(x, y) y)..

It and :::> It follows follows from from our our assumption assumption that that 4>(0) r and that that 4>(a) r D r4>(a + + 11)) for for all all aa < < u u.. The The induction induction hypothesis hypothesis that that .IT:n /En F ~ B�n B~n together together with with part part (c) (c) implies implies that that the the u) (3y <� v)A is to aa Iln formula formula (Vx (Vx � _ u)(3y is equivalent equivalent to H~_~-formula and thus thus 4> r is is equivalent equivalent - 1 -formula and -formula. Therefore, to to aa �n ~-formula. Therefore, by by induction induction on on 4>, r r4>( u) holds; holds; this this is is what what we we needed needed to to show. show. D [] With theorem, the relationships between With the the aid aid of of the the above above theorem, the other other relationships between fragments fragments of of Peano Peano arithmetic arithmetic are are relatively relatively easy easy to to prove. prove. To To prove prove that that .IT: /~nn implies implies Il IIn-IND, n-IND, let formula and let A(x) be be aa Iln Hn formula and argue argue informally informally inside inside .IT:n /En assuming assuming A(O) A(0) and and (Vx) (A(x) :::> (Vx)(A(x) ~ A(x + 1)) 1)).. Letting Letting aa be be arbitrary, arbitrary, and and letting letting B(x) be be the the formula formula ...,~A(aA (a � x) B (a) and B(x) :::> B (O) , x),, one one has has ..., -~S(a) and S(x) D B(x + 11).) . Thus, Thus, by by induction, induction, ..., ~S(0), and Since aa was and this this is is equivalent equivalent to to A(a) A(a).. Since was arbitrary, arbitrary, (Vx)A(x) follows. follows. A A similar similar mn implies argument argument shows shows that that/IIn implies �n-IND. En-IND. To -MIN axioms To show show that that the the �n En-MIN axioms are are consequences consequences of o f /.IT:n E n , , note note that that by by the the argument .2 . 6 above, argument given given at at the the end end of of section section 11.2.6 above, the the minimization minimization axiom axiom for for A(x) follows A (x) with follows from from induction induction on on the the formula formula (Vx � <_y)..., y)-~A(x) with respect respect to to the the variable variable yy.. ,A) is If the above � y)(... If A A E �n En, then then from from part part (c) (c) of of the above theorem, theorem, the the formula formula (Vx (Yx <_ y)(-~A) is ' equivalent minimization axiom equivalent to to aa Iln-formula, IIn-formula, so so the the minimization axiom for for A A is is aa consequence consequence of of m n= /1-In = / .IT:n En.· It the induction induction axiom It is is easy easy to to derive derive the axiom for for A A from from the the minimization minimization axiom axiom for for A A,, so so L�n LEa = = LIln LIIn = = / .IT:n En = =/m I / nn. ·

Buss S. Buss

96 96

Finally, the theorem theorem of of Clote Clote [1985] that the the strong strong En-replacement �n -replacement axioms axioms are are Finally, the [1 985] that consequences ooff / E�n can be be proved proved as as follows. follows. Assume Assume nn >� 11 and and A A EE E~ consequences n can �n and and consider the the strong strong replacement replacement axiom axiom consider (3w)(Vx ( :Jw)(\fx <_ A(x, fl(x :::; a)[(3y)A(x, a) [( :Jy)A(x, y) ~t-+ A(x, ,8(x ++ 1, 1 , w))]. w ))] .

for A. A . (Note (Note A A may may have variables other other than than x, x, y.) y .) Let Let NumA(u) NumA (u) be be aa E~�n for have free free variables formula which which expresses the property property that that there there exists exists aa w w for which A(x, A(x, ~(x+ formula expresses the for which 1 , w)) w )) ,8(x+ 1, holds for for at at least least uu many many values values of of xx <:::; a. a . Clearly Clearly NumA NumA (0) (O) holds holds and and gumA NumA(a + 2) holds (a + 2) fails. So, So, by by E~-maximization �n-maximization (which follows easily easily from from E~-minimization), �n-minimization), there is fails. (which follows there is maximum value value u0 Ua _< :::; aa ++ 11 for for which NumA (Uo) (Ua) holds. holds. A A value value w w that that works works for for aa maximum which NumA Ua satisfies satisfies the the strong strong replacement replacement axioms axioms for for A. A. this u0 this 1.2.10. D Definable 1.2.10. e f i n a b l e ffunctions u n c t i o n s oof f / �n E~.. When bootstrapping theories stronger IA0,, such a s /�n ~ n for > 00,, the such as for nn > the main main When bootstrapping theories stronger than than lila theorem of section 1.2.7 1.2.7 still applies, and and A Ill1-definable -definable predicates predicates and and �l -definable theorem of section still applies, El-definable functions may may be be introduced introduced into into the the language language of arithmetic and used freely in functions of arithmetic and used freely in Of particular particular induction axioms, axioms, without without increasing increasing the the strength strength of the theory. induction of the theory. Of importance recursive functions El-defined importance is is the the fact fact that that the the primitive primitive recursive functions can can be be � in l -defined in (any theory theory containing)/Ei. containing) �l ' (any

Definition. The The primitive recursive functions functions are functions on N and are inducinduc Definition. primitive recursive are functions on 1N and are tively as follows: tively defined defined as follows: (1) The The constant function with value 00 is primitive recursive. recursive. We We can view this (1) constant function with value is primitive can view this aa nullary nullary function. function. (2) The The unary unary successor successor function function S(x) = = x x+ + 11 is is primitive primitive recursive. recursive. (2) , , xn ) = projection function (3) For each each 11 :::; _ kk :::; < n n,, then then n-ary n-cry projection function 7l"k(Xl r'~(Xl,...,xn) = Xk xk is is (3) For primitive primitive recursive. recursive. n-ary primitive > " " hn are (4) If 9g is is an an n-cry primitive recursive recursive function function and and hi hl,...,hn are m-ary m-cry (4) If primitive m-ary function defined by primitive recursive recursive functions, functions, then then the the m-cry function 1 f defined by 1(31) (hl (31), . . . hn(31)) f(~) = gg(hi(~),.., hn(~)) is is primitive primitive recursive. recursive. (5) If -ary primitive (5) If n n � _> 11 and and 9 g is is an an (n ( n -- 1) 1)-cry primitive recursive recursive function function and and h h is is an an (n -ary primitive recursive function, then the n-ary function ned by: (n + + 1) 1)-cry primitive recursive function, then the n-cry function 1 f defi defined by: • • .

(31) 1(0, x) = gg(e) j'(o,e) l(m f (m + + 1, 1, 31) :F,) = h(m, h(m, I(m, f (m, x) :~),, x) :~) is is primitive primitive recursive. recursive. The The only only use use of of the the projection projection functions functions is is as as aa technical technical tool tool to to allow allow generalized generalized substitutions substitutions with with case case (4) (4) above. above. A A predicate predicate is is primitive primitive recursive recursive if if its its characteristic characteristic function function is is primitive primitive recur recursive. sive.

Proof Theory Theory of Arithmetic

97 97

Theorem. �l can �l -define the T h e o r e m . /El can E1-define the primitive primitive recursive recursive functions. functions. The converse converse to to this this theorem theorem holds holds as as well; well; namely, namely, /El can �l El-define exactly the the The �l can -define exactly primitive primitive recursive recursive functions. functions. This This converse converse is is proved proved later later as as Theorem Theorem 3.1.1. 3.1.1.

Proof. Proof. It It is is obvious obvious that that the the base base functions, functions, zero, zero, successor successor and and projection, projection, are are �l -definable iinn /�l It is �l -definable by El-definable E l . . It is easy easy to to check check that that set set of of functions functions El-definable b y /�l E l is is closed under composition. Finally suppose �l -definable in closed under composition. Finally suppose that that g g and and h h are are/El-definable i n /�l E l .. Then, the �l -defined Then, the function function f f defined defined from from g g and and h h by by primitive primitive recursion recursion can can be be El-defined with with the the following following formula formula expressing expressing f f (m, (m, i) 3) = = yy::

(3w)[Len(w) - m m+ + 11 /\ Ay y= -- (3(m ~(m + + 11,, w) w) /\ A (3(1, ~(1, w w)) = -- g(i) g(Z) /\ A (3w) [Len(w) =

))]. (3(i + (Vi < < m) m)(~(i + 2, w)== h(i, h(i,/~(i + 11,, w), w), i ~))]. (Vi ({3(i + 2, w)

This w, containing This formula formula expresses expresses the the condition condition that that there there is is aa sequence, sequence, coded coded by by w, containing (O, xx),..., ) , . . . , f(m, all all the the values values f f(O, f(m, i) Z),, such such that that each each value value in in the the sequence sequence is is correctly correctly computed computed from from the the preceding preceding value value and and such such that that the the final final value value is is y y.. The The theorem theorem of of .2.7 shows that the ning ff is equivalent to �l -formula. section section 11.2.7 shows that the above above formula formula defi defining is ((equivalent to)) aa El-formula. We leave leave it it to to the the reader reader to to check check that that /El can prove prove the the requisite requisite existence existence and and �l can We uniqueness definition of uniqueness conditions conditions for for this this definition of f f .. 0 [:3 As consequence, we As an an easy easy consequence, we have have Corollary. �l -definable by Corollary. Every Every primitive primitive recursive recursive predicate predicate is is A1-definable by �l IE1..

In the �l -definable. A In the theories theories � /Enn with with n n > > 11,, even even more more functions functions are are El-definable. A characterization of �l -definable in III of characterization of the the functions functions El-definable i n /� E nn is is given given in in Chapter Chapter III of this this volume. volume. Other Other proof-theoretic proof-theoretic characterizations characterizations of of these these functions functions can can be be found found in in Takeuti Takeuti [1987] [1987],, Buss Buss [1994] [1994] and and in in references references cited cited therein. therein. 1.3. 1.3. Fragments F r a g m e n t s of of bounded b o u n d e d arithmetic arithmetic

A A subtheory subtheory of of Peano Peano arithmetic arithmetic is is called called aa bounded bounded theory theory of of arithmetic, arithmetic, or or aa is axiomatized -formulas. The theory theory of of bounded bounded arithmetic, arithmetic, if if it it is axiomatized by by Ill IIl-formulas. The potential potential strength strength of of such such theories theories depends depends partly partly on on the the growth growth rates rates of of the the function function symbols symbols in in the the language, language, and and usually usually bounded bounded arithmetic arithmetic theories theories have have only only functions functions of of subexponential subexponential growth growth rate, rate, including including addition, addition, multiplication multiplication and and possibly possibly polyno polynomial mial growth growth rate rate functions functions such such as as WI Wl or or # # . . These These theories theories are are typically typically weaker weaker than than the the strong strong theories theories considered considered in in section section 1.2, 1.2, but but stronger stronger than than the the theories theories Q Q and and R R discussed discussed in in section section 1.1. 1.1. There There are are two two principal principal approaches approaches to to bounded bounded arithmetic. arithmetic. The The original original approach approach involved involved theories theories such such as as I�o IA0 and and I�o IA0 + + nl gtl;; more more recently, recently, bounded bounded theories theories such such as been extensively studied. One as S� S~ and and T� T~ have have been extensively studied. One of of the the main main motivations motivations for for studying is their studying bounded bounded arithmetics arithmetics is their close close connection connection to to low-level low-level computational computational complexity, complexity, especially especially regarding regarding questions questions relating relating expressibility expressibility and and provability provability in in

98 98

Buss S.S. Buss

bounded arithmetics arithmetics to to questions questions about about the the linear linear time time hierarchy hierarchy and and the the polynomial polynomial bounded time hierarchy. hierarchy. time 1.3. 1. I5o 1.6.0 aand nn 1.3.1. n d ~n

We have have already already defined defined IA0 1.6.0 and and described described its its bootstrapping bootstrapping process process in in fairly fairly We complete detail detail in in section section 1.2. 1 .2. One One corollary corollary of of the the bootstrapping bootstrapping process process isis that that the the complete of exponentiation exponentiation isis A0-definable .6.o-definable in in IA0; 1.6.0 ; that that isis to to say, say, there there isis aa bounded bounded graph of graph exp(x, y,y, z) z) which which expresses expresses the the condition condition xY xY -= zz and and such such that that IA0 1.6.0 can can formula exp(x, formula prove facts facts like like exp(x, exp(x, O, 0, 1), 1) , exp(x, exp(x, 1,l, x), x) , prove

exp(x, y,y, z) ^1\ exp(x, y', y' , z') z') => exp(x, yy ++ y', y' , z.z · z' ) and that that for for any any xx and and y, y , there there isis at at most most one one zz such such that that exp(x, exp(x, y,z). y, z) . The The and underlying idea idea of of the the A0-definition .6.o-definition of of exp(x, exp(x, y, z) isis to to defne define the the sequence sequence (xty/2'J)i (x Ly/2i J )i underlying where ii ranges ranges from from lYl down to to 0; 0 ; however, however, we we leave leave it the reader reader to supply where it to to the to supply Iyl down the details details behind behind this this A0-definition. .6.o-definition. The The fact fact that that exponentiation exponentiation is is A0-definable .6.o-definable the is essentially essentially due due to to Bennett Bennett [1962] [1962] and and was was first first (and ( and independently) independently) proved in the the is proved in setting by Gaifman and Dimitracopoulos Dimitracopoulos [1982]. setting of of 1.6.0 IA0 by Gaifman and [1982]. Once graph of exponentiation has Once the the graph of exponentiation has been been shown shown to to be be .6.o-definable, A0-definable, one one can can formulate the axioms axioms n Firstly, when working in in bounded bounded arithmetic, formulate the ~k. when working arithmetic, we we define define k • Firstly, to equal the greatest greatest yy such such that that 2Y 2Y <:5 x Then the the function Wl (x, (x, y) y) isis defined log log x x to equal the x.. Then function Wl defined to to equal equal x1og x l~ y .. Since Since IW1(X, Iwl (x, y) y)l1 = - 8(lxl O(Ixl.' lyl) lyl),, it it is is evident evident W Wll (X, (x, y) y) cannot cannot be be bounded bounded by aa polynomial polynomial of of x x and and y y.. Therefore, Therefore, by by Parikh Parikh's's Theorem Theorem 1.2.7.1, 1.2.7.1, the the function function W Wll by is -definable in shall see later, it desirable to is not not E1 ~El-definable in 1.6.0 IA0.• As As we we shall see later, it is is often often very very desirable to have have the the W Wll function function be be total; total; therefore therefore it it is is common common to to extend extend 1.6.0 IA0 to to aa stronger stronger theory theory containing containing the the axiom axiom n1 :

y)).. (\ix)(\iy) (3z) (z = Wl (X, y)) =

This 1. This stronger stronger theory theory is is called called 1.6.0 IA0 + + n ~1. The The function function W Wll has has what what is is called called polynomial polynomial growth growth rate, rate, i.e., i.e., for for any any term term t(ii) t(~) constructed constructed with with the the functions functions S S,, + + ,, . 9and and W wll there there is is aa polynomial polynomial Pt pt such such that that I) . There for ii) I :5___Ppt(lall,..., t ( lal l , . . . , Ian for all all ii, g, It( It(g)l lanl). There is is also also aa hierarchy hierarchy of of functions functions Wn wn,, n n :::: _> 11,, (log x,logy) . 9The (x, y) which which have have subexponential subexponential growth growth rates, rates, defined defined by by Wn+1 wn+l(x, y) = = 2wn 2~"0~176 The axioms axioms nn ~ are are Ih-axioms II2-axioms which which say say that that the the function function Wn w~ is is total. total. By By using using Parikh's Parikh's 2. 7.1, it Theorem Theorem 1. 1.2.7.1, it is is immediate immediate that that 1.6. IA00 + + nn ~ y: tz n ~ n++11.' Although Although the the Wn wn functions, functions, for for nn :::: >__22,, are are superpolynomial, superpolynomial, they they are are much much more more similar similar in in nature nature to to polynomial polynomial growth growth rate rate functions functions than than to to exponential exponential growth growth rate rate functions. functions. Using Using aa technique technique due due to to Solovay Solovay [1976], [1976], it it can can be be shown shown that, that, for for each each nn,, 1.6.0 IA0 can can define define an an inductive inductive cut cut on on which which the the Wn w~ function function is is provably provably total; total; for for an an explanation explanation of of this this construction, construction, see see Pudhik Pudl~k [1983], [1983], Nelson Nelson [1986] [1986] or or Chapter VIn of this volume. However, Paris and Dimitracopoulos [1982] Chapter VIII of this volume. However, Paris and Dimitracopoulos [1982] showed showed that that it it is is not not possible possible to to define define an an inductive inductive cut cut on on which which the the exponentiation exponentiation function function isis provably provably total. total. For For this this reason, reason, we we view view the the Wn w, functions functions as as being being more more akin akin to to

Proof Proof Theory Theory of of Arithmetic Arithmetic

99 99

feasible feasible polynomial polynomial growth growth rate rate functions functions than than to to the the infeasible infeasible exponential exponential function function (see (see Nelson Nelson [1986] [1986] for for aa strong strong expression expression of of this this viewpoint). viewpoint). 1.3.2. 1.3.2. �o-formulas A0-formulas and and the the linear-time linear-time hierarchy hierarchy There close connection connection between There is is aa very very close between �o-expressibility A0-expressibility and and computational computational complexity. Recall Recall that that the the linear linear time time hierarchy hierarchy consists consists of of those those predicates predicates which which complexity. can be be recognized recognized by by some some 'lUring Turing machine machine which which runs runs in in linear linear time time and and which which can makes aa bounded number of of alternations alternations between between existential existential and and universal universal states. states. makes bounded number Lipton [1978,sect. [1978,sect. 4], 4], building building on on work work of of Smullyan, Smullyan, Bennett Bennett and and Wrathall, Wrathall, proved proved Lipton which are that that the the �o A0 definable definable predicates predicates on on N 1N are are precisely precisely the the subsets subsets which are in in the the linear time hierarchy. linear time hierarchy. The Parikh [1971] The original original motivation motivation for for the the definition definition of of the the theory theory I�o IA0 by by Parikh [1971] was was to to give give aa proof proof theory theory that that would would be be appropriate appropriate to to linear linear bounded bounded automata, automata, i.e., bounded 'lUring i.e., to to predicates predicates computable computable by by linear linear space space bounded Turing machines. machines. It It is is still still an an open open problem problem whether whether the the linear linear time time hierarchy hierarchy equals equals linear linear space; space; although although it it is is commonly commonly conjectured conjectured that that they they are are not not equal. equal. It It is is known known that that the the linear linear time time hierarchy hierarchy contains contains log log space, space, and and also also contains contains the the predicates predicates which which can can be be computed computed by aa 'lUring Turing machine machine which which simultaneously simultaneously polynomial polynomial time time and and nn1-f 1-~ space space for for aa by Bennett [1962] NepomnjaSCii [1970]) constant constant €e > > 00 (see (see Bennett [1962] and and Nepomnja~ii [1970]).. 11.3.3. .3.3. The The theories theories S� S~ and and TJ T~ of of bounded b o u n d e d arithmetic arithmetic The The second second approach approach to to theories theories of of bounded bounded arithmetic arithmetic is is due due to to Buss Buss [1986] [1986] and and gives gives aa (conjectured) (conjectured) hierarchy hierarchy of of fragments fragments of of I�o IA0 + + 0 ~1, which are are very very closely closely 1 , which related related to to the the computational computational complexity complexity classes classes of of the the polynomial polynomial time time hierarchy. hierarchy. and TJ These fragments, These fragments, S� S~ and T~ and and others, others, use use the the language language 00,, S S,, + + ,,. ,. , # # , , lx Ixl, L 89J , i , L�x Ixl - lyl . and � _;; where where the the ## function function (pronounced 'smash')' ) is is defined defined so so that that x#y x#y = - 2 21xl.lyl. and (pronounced 'smash The # # function function was was first first introduced by Nelson Nelson [1986], [1986], and and it it is is evident evident that that the the # # The introduced by function function has has essentially essentially the the same same growth growth rate rate as as the the W wl-function. I -function. The The second second difference difference between between the the S� S~ and and the the T� T~ theories theories and and the the I�o IA0 + + 01 ~1 approach approach is is that that the the former former theories theories are are based based on on restricting restricting the the power power of of induction; induction; firstly firstly by by further further restricting restricting the the formulas formulas for for which which induction induction holds, holds, and and secondly secondly by by using using (apparently) (apparently) weaker weaker forms forms of of induction. induction. It It is is for for this this reason reason that that the the functions functions and L~xJ 1 are L�xj non-logical language, are included included in in the the non-logical language, since since they they are are needed needed to to Ixl and elegantly elegantly state state the the axioms axioms of of the the theories theories S� S~ and and TJ T~.. Before Before defining defining the the theories theories S� S~ and and T4 T~,, we we define define the the classes classes �� E~ and and rr� II~ of of formulas, formulas, which which are are defined defined by by counting counting alternations alternations of of bounded bounded quantifiers, quantifiers, ignoring ignoring sharply sharply bounded bounded quantifiers. quantifiers. (Bounded (Bounded and and sharply sharply bounded bounded quantifiers quantifiers are are defined defined in in section section 11 above.) above.) Definition. Definition. The The set set � A~g = = �g 2~ = = rrg H~ is is equal equal to to the the set set of of formulas formulas in in which which all all quantifiers bounded. For quantifiers are are sharply sharply bounded. For ii 2:: _ 11,, the the sets sets �� ~ and and rr� II~ are are inductively inductively defined defined by by the the following following conditions: conditions:

s. Buss S.

100 100

(a) (a) If If A A and and BB are are E~-formulas, E� -formulas, then then so so are are A AV V BB and and A A At\ B. B . If If A A isis aa II~ Il� formula formula and ~A E�-formula, then then A are E~-formulas. E�-formulas. and BB isis aa E~-formula, and A D:::> BB and ...,A are A isis aa II~_l-formula, IlL -formula, then then A A isis aa E~-formula. E� -formula. (b) If If A (b) A isis aa E~-formula E�-formula and and tt isis aa term, term, then then (Vx ("Ix <_ :::; Itl)A Itl)A isis aa E~-formula. E�-formula. (c) If If A (c) E�-formula. Note Note this this (d) (d) If If A A isis aa E~-formula E�-formula and and tt isis aa term, term, then then (3x (3x <:::; t)A t)A isis aa E~-formula. quantifier may may be be sharply sharply bounded. bounded. quantifier The four four inductive inductive conditions conditions defining defining II~ Il� are are dual dual to to (a)-(d) (a)-(d) with with the the roles roles of of The Il� and and Z~ Ef reversed. reversed. existential and and universal universal quantifiers quantifiers and and the the roles roles of of II~ existential

This is is aa good good place place to to justify justify the the presence, presence, in in bounded bounded arithmetic, arithmetic, of of the the ## This There are are essentially essentially three three reasons reasons why why itit function function or or the the equivalently equivalently growing growing wl. WI . There WI . Firstly, Firstly, itit gives natural bound bound to to the the Gbdel G6del number number is natural natural to to include include ## or or wl. is gives aa natural A(t) in in terms terms of of the the Gbdel G6del numbers numbers of of A A and and t;t ; namely, namely, the the number number of aa formula formula A(t) of A(t) is is bounded bounded by by the the product product of of the the numbers numbers of of symbols symbols in in A A and and of symbols symbols in in A(t) of in t.t . This This allows arithmetization of Secondly, itit arises in allows a a smooth smooth arithmetization of metamathematics. metamathematics. Secondly, arises naturally from consideration of versus sharply sharply bounded bounded quantifiers, quantifiers, since since naturally from consideration of bounded bounded versus it has exactly rate necessary necessary to to make following quantifier it has exactly the the growth growth rate make the the following quantifier exchange exchange property hold: property hold:

b)A(x, y) y) (("Iwx :::; < lal)(3y _< b)A(x, l al) (3y :::; 1, y)) y)) t\ 1, y) (A(x, (3 (x + (x + 1, SqBd(b, a)) (Vx :::; +4 (3w (3w :::; < SqBd(b, a))(Vx < lal) lal)(A(x, Z(x A (3 ~(x § 1, y) :::; <_b) b) ++ where where SqBd SqBd is is aa term term involving involving # # . . In In fact, fact, the the size size of of w w can can be be bounded bounded in in b, by terms terms of of aa and and b, by noting noting that that w w must must encode encode lal lal + + 11 many many numbers numbers of of at at most most c, and Ibl bits bits each; each; therefore, therefore, w w :::; < 2c·lal·lbl 2c'lal'lbl for for some some constant constant c, and SqBd SqBd can can easily easily Ibl be be expressed expressed using using # # . . The The quantifier quantifier exchange exchange property property allows allows sharply sharply bounded bounded quantifiers quantifiers to to be be pushed pushed inside inside non-sharply non-sharply bounded bounded quantifiers quantifiers (at (at least least when when the the (3 function function is is available). available). Thirdly, Thirdly, the the use use of of # # function function means means that that any any term term t(x) t(x) can x1c for can be be bounded bounded by by 221Ixl~ for some some constant constant cc,, and and conversely, conversely, any any 21x1c 2 Ixlc can can be be bounded bounded by by aa term term t(x) t(x) in in the the language language of of bounded bounded arithmetic. arithmetic. In In other other words, words, the the terms terms define define functions functions of of polynomial polynomial growth growth rate. rate. This This leads leads to to the the principal principal importance importance of of the the classes classes E� E~ and and Il� II~ of of formulas, formulas, which which is is that that they they express express precisely precisely the the corresponding corresponding classes classes of of the the polynomial polynomial time time hierarchy. hierarchy. This This fact fact is is discussed discussed in in more .3.6 below, more depth depth in in section section 11.3.6 below, but but in in brief, brief, aa set set of of natural natural numbers numbers is is definable definable by by aa E�-formula E~-formula (respectively, (respectively, aa m-formula) II~-formula) if if and and only only if if the the set set is is recognizable recognizable by by aa predicate predicate in in the the class class Ef E~ (respectively, (respectively, Ilf II~)) from from the the polynomial polynomial time time hierarchy. hierarchy. This This is is essentially essentially due due to to Wrathall Wrathall [1976] [1976] and and Stockmeyer Stockmeyer [1976] [1976] and and was was first first proved proved P , the in in this this exact exact form form by by Kent Kent and and Hodgson Hodgson [1982]. [1982]. Thus Thus we we have have that that N NP, the set set of of nondeterministic nondeterministic polynomial polynomial time time predicates, predicates, consists consists of of precisely precisely the the predicates predicates expressible -formulas, etc. expressible by by Et E~-formulas, etc. The The theory theory T� T~ will will be be defined defined by by restricting restricting induction induction to to Ef-formulas, E~-formulas, ' where by induction we mean the usual 'IND flavor of induction. For where by induction we mean the usual 'IND' flavor of induction. For S� S~,, we we need need some additional varieties of induction: some additional varieties of induction:

1.3.3.1. 1.3.3.1.

101 101

Proof Theory of Arithmetic

Definition. D e f i n i t i o n . Let Let 9 be be a a set set of of formulas. formulas. The The -PIND O-PIND axioms axioms are are the the formulas formulas

A(O) A(O) 1\ A ((Vx)(A(L D A(x)) A(x)) => D ((Vx)A(x) V'x) (A( l�89xJ ) => V'x)A(x) for for all all formulas formulas A A E E O.. As As usual, usual, A A may may have have other other free free variables variables in in addition addition to to x x that parameters. The that serve serve as as parameters. The length-induction length-induction -LIND O-LIND axioms axioms are are the the formulas formulas A(O) 1\ A ((Vx)(A(x) A(O) V'x)(A(x)

D =>

A(Sx)) A(Sx)) => D ((Vx)A(Ixl) V'x)A(lx l )

for all all A A E O.. The The length-minimization length-minimization axioms, axioms, -LMIN, O-LMIN, are are the the formulas formulas for

(3x)A(x) => D (3x) (3x)(A(x) (Vy)([y < Ix Ixll y)( l y lI < (A(x) 1\A (V' (3x)A(x)

D =>

A (y))) ~A(y))) ...,

for for all all A A E E . O. In In addition addition to to induction induction and and minimization minimization axioms, axioms, there there are are replacement replacement axioms axioms that will be defined below been introduced. that will be defined below after after the the Godel GSdel/3(3 function function has has been introduced. All All of of these schemes are in conjunction with aa set purely universal these axiom axiom schemes are used used in conjunction with set of of purely universal axioms axioms called axioms. The consists of: called the the BASIC BASIC axioms. The set set of of BASIC BASIC axioms axioms consists of: aa s < bb J D aa S < Sb Sb aa =I ~ Sa Sa O0 <Sa a aa <SbbA1\a ra =I b t-+ Sa S a <Sbb aa ~=l: 00 J D 22.· a a =l ~: 00 aa <sbbVVb b<sa a aa <SbbAl\b b<Sa Da aJ=ab = b aa s < bbAl\bb<sc e J Daa s < cc 01 = = 0 0 1101 SO I = = SO SO IISOI S(2 · a) aa =I 2 · al = lal ) l a l ) 1\ =/= 0 0 J ~ 112-al - S( S(lal) A I15'(2. a)l 1 = - S( S(lal) aa S al S bl < bb J ~ llal < IIbl bl ) a#bl = l a l . IIbl) = S( S(lal" lla#bl O#a o#a = = SO so aa =I ~ 00 J D 1#(2 1#(2. · a) a) = = 22.. (l#a) (l#a) 1\1#(S(2 AI#(S(2 .. a)) a)) = 22. . (l#a) (l#a). . a#b a#b = = b#a b#a m

bl J llaal I=- IIbl ~ a#c a#c = = b#c al = bl + ei J llaI id = = IIbI + lIcl :::)aa#d = (b#d) (b#d).. (e#d) (c#d) aa <Saa++b b aa <sbbAl\a a~ b=lDb J ·b S(2 S(2 . 9a) a) S < 22 .· bbA1\SS(2 ( 2 . a. a) ) =1 ~ 22.5 aa ++bb==b b++a a aa + +OO ==aa aa ++SSb b ==S S(a ( a ++ b )b) (a (a + + b) b) + + c c = - a a+ + (b (b + + c) c) aa + t-+ bb s + bb s < aa + + ce +-~ < ec aa . · Oo = - 0 o aa - ·( S(Sb) b) = = (a ( a . · b) b) + + a a aa . · bb = ·a = b b.a aa . · (b (b + + c) c) = = (a ( a . · b) b) + + (a (a.. c) SO SO S < aa J D (a (a.· bb S < aa.· ec t-+ o bb S < e) c) aa =I ~ 00 J Dlal S(IL~aJl)l l a l = S( I L� aJ I ) S(2 . a) aa = = L� L b89 ~ 2 2 . .aa = = b bV V s(2. a) = = b b J t-+

These These BASIC BASIC axioms axioms serve serve the the same same role role for for S� S~ and and T4 T~ that that the the axioms axioms of of Q Q served served for arithmetic. There certain amount for the the fragments f r a g m e n t s /�n E n of of Peano Peano arithmetic. There is is a a certain amount of of flexibility flexibility in in the the choice choice of of BASIC BASIC axioms; axioms; essentially essentially any any finite finite set set of of purely purely universal universal axioms axioms which both are bootstrapping of which both are sufficiently sufficiently strong strong to to carry carry out out the the bootstrapping of Si S~ and and are are BASIC axioms.4 contained contained in in the the theory theory Si S~ would would serve serve as as well well for for the the BASIC axioms. 4 4We have given given the BASIC BASIC axioms axioms as as defined defined by Buss Buss [1986]. [1986]. This choice choice is not entirely optimal, since, (a ) , follows since, for instance, the second second axiom aa s <_S S(a), follows from the first, fourth and sixth axioms. axioms. An An alternative, alternative, and and weaker, weaker, set set of of BASIC BASIC axioms axioms are are given given by by Cook Cook and and Urquhart Urquhart [1993]; [1993]; see BASIC axioms. see Buss Buss [1992] [1992] for for aa discussion discussion of of their their BASIC axioms. Buss Buss and and Ignjatovic Ignjatovid [1995] [1995] propose propose that that BASIC axioms. laI S <_a a should should be be added added to to the the BASIC axioms. lal

102 102

s. S. Buss

Definition. Definition. Let Let ii � _> o0.. S� S~ is is the the theory theory axiomatized axiomatized by by the the BASIC BASIC axioms axioms plus plus l E~-PIND. T4 T~ is is the the theory theory axiomatized axiomatized by by BASIC BASIC plus plus I:�-IND. E~-IND. The The theories theories S� S~--1)) I:�-PIND. - l ) are and T TJ(-1) are equal equal to to just just BASIC. BASIC. and S2 ;::: O S� and T2 is U .3.5 shows $2 is is U Ui>oS~ and T2 is t.li>oT~. Section 11.3.5 shows that that S2 $2 and and T2 T2 are are the the same same i i ;:::o T4 . Section theory. theory. 11.3.3.2. .3.3.2. Bootstrapping -definable functions. The Bootstrapping and I:t E~-definable functions. The bootstrapping bootstrapping of Si S 1 and and Ti T21 is is analogous analogous to to the the bootstrapping bootstrapping of of 1.600 IA0 as as described described in in sections sections of 1.2.6-1.2.8 1.2.6-1.2.8 above. above. There There is is now now the the additional additional difficulty difficulty that that the the induction induction axioms axioms are restricted; but other hand, language of are more more severely severely restricted; but on on the the other hand, the the language of S� S~ and and T4 T~ is is richer richer since since it it contains contains the the function function symbol symbol Ixl Ixl and and its its BASIC BASIC axioms axioms and and this this x essentially trivial, makes nition of makes the the defi definition of the the graph graph of of yy = -- 22x essentially trivial, and and thereby thereby helps helps with with defining Ghdel G6del numbering defining numbering of of sequences. sequences. The The most most outstanding outstanding difference difference between between the the bootstrapping of bootstrapping of bootstrapping of Si S~ and and Ti T~ and and the the above above bootstrapping of 1.600 IA0 is is that that quantifiers quantifiers are counted; namely, could use predicates and are more more carefully carefully counted; namely, whereas whereas 1.600 IA0 could use .6oo-defined Ao-defined predicates and -defined predicates I:l -defined functions, El-defined functions, the the theories theories Si S~ and and Ti T~ can can introduce introduce .6ot A~-defined predicates and I:t E~-defined functions. Accordingly, Accordingly, we we make make the the following following important important definitions: definitions: and -defined functions. Definition. -defined by Definition. A A predicate predicate symbol symbol R(x) R(Z) is is .6o� A~-defined by aa theory theory T T if if there there is is aa and such I:�-formula E~-formula
R(x) ++
yy ==f (f(x) y), ~ ) ++ 4-~
T + k (A ~ A-).

103 103

Proof Proof Theory Theory of o/ Arithmetic

An immediate immediate corollary corollary to to this this theorem theorem is is that, that, for for ii 2: > 11,, theories theories such such as as S� S~ and and T4 T~ An can introduce introduce �� E~-defined function symbols symbols and and .D.. A~-predicate symbols and and use use them them can -defined function � -predicate symbols freely in in induction induction axioms. axioms. freely With the the aid aid of of Theorem Theorem 1.3.3.3, 1.3.3.3, the the bootstrapping bootstrapping for for S� S~ and and Ti T~ is is analogous analogous With to the the bootstrapping bootstrapping for for I.D.. IA0o in in section section 1.2.8; 1.2.8; indeed, indeed, every every single single function function and and to predicate symbol symbol which which was was claimed claimed to to be be � El-definable or A0-definable (respectively) predicate -definable or .D.. o -definable (respectively) 1 in I.D.. IA0o in in section section 1.2.8 1.2.8 is is likewise likewise �� E~-definable or .D.. A~-definable in each each of of the the six six in -definable or � -definable in theories S S~, T~,, BASIC B A S I C + IT� II~-PIND, B A S I C + �� E~-LIND, B A S I C + IT� II~-LIND -PIND, BASIC -LIND, BASIC -LIND theories i , Ti and BASIC B A S I C + IT� II~-IND. We shall shall omit omit the the details details of of this this bootstrapping bootstrapping here; here; they they and -IND. We can be be found found in in Buss Buss [1986,1992) [1986,1992] and and Buss Buss and and Ignjatovic Ignjatovi(~ [1995) [1995].. can One consequence consequence of of the the bootstrapping bootstrapping process process is is that that some some of of the the other other forms forms of of One induction follow follow from from �-PIND E-PIND and and IT-IND: H-IND" induction Theorem. (Buss [1986)) [1986]) Let Let ii 2: >_11.. 11.3.3.4. .3.3.4. Theorem. (Buss (1) T4 T~ proves proves IT�-IND II~-IND and and T4 T~ 1==~ S� S~ .. (1)

(2) S~ proves proves ��-LIND, E~-LIND, ITf-PIND I-Ibi-PIND and and IT�-LIND. IIbi-LIND. (2) S;

1.3.4. Polynomial P o l y n o m i a l time t i m e computable c o m p u t a b l e functions functions in in S S~i 1.3.4. The last last section discussed the the fact fact that that �� ~-definable functions and and .D.. A~-defined The section discussed -definable functions � -defined predicates predicates can can be be introduced introduced into into theories theories of of bounded bounded arithmetic arithmetic and and used used freely freely in induction axioms. Of Of particular particular importance importance is is the the fact fact that that these these include include all all in induction axioms. polynomial time time computable computable functions functions and and predicates. predicates. polynomial A function said to A function or or predicate predicate is is said to be be polynomial polynomial time time computable computable provided provided there there exists 'lUring machine machine M and aa polynomial polynomial p(n), p( n) , such such that the exists a a Turing M and that M M computes computes the function or recognizes the predicate, predicate, and and such function or recognizes the such that that M M runs in time runs in time <::; p(n) p( n) for for all all inputs of of length length n. n. The outputs for coded in binary inputs The inputs inputs and and outputs for M M are are integers integers coded in binary notation, thus the length of total length length of notation, thus the length of an an input input is is proportional proportional to to the the total of its its binary binary representation. representation. For our purposes, it is convenient convenient to to use use an alternative definition definition ofthe For our purposes, it is an alternative of the polynomial polynomial time computable computable functions; of this due to Cobham [1965]. time functions; the the equivalence equivalence of this definition definition is is due to Cobham [1965) .

Definition. The The polynomial polynomial time inductively defined defined by by Definition. time functions functions on on N N are are inductively (1) The following following functions functions are are polynomial polynomial time: time: (1) The •

The nullary constant function function 0. o. 9 The nullary constant

•

9 The The successor successor function function xx ~-+ H S(x). S(x) .

•

The doubling doubling function function xx ~H 2x. 2x . 9 The

•

{;

9 The The conditional conditional function function Cond(x, Cond(x, y, y, z) z) = = { y z

if x = 0

if x = 0 otherwise. otherwise.

(2) The projection projection functions functions are are polynomial polynomial time time functions functions and and the the composition composition of of (2) The polynomial polynomial time time functions functions is is aa polynomial polynomial time time function. function.

s. S.

104 104

Buss

polynomial time (n + 1) -ary polynomial (3) If If 9g is is aa ((nn -- 11)-cry time function function and and hh is is aa (n + 1)-cry polynomial ) -ary polynomial time time function function and and p p is is aa polynomial, polynomial, then then the the following following function function ff, , defined defined by by limited h, is limited iteration iteration on on notation notation from from 9g and and h, is also also polynomial polynomial time: time:

f(O, f (O, x) Y,) -- g(x) g(Y,) f(z, x) = h(z, f(z,Y,) h(z , x, Y,, f( f ( L�zJ 1 # 00 , x)) for zz 7'= p(lzl, Ixl) x. provided provided If(z, I / ( z , ex) )l 1 � <- P(Izl, lel) for all z, z,e.

A A predicate predicate is is polynomial polynomial time time computable computable provided provided its its characteristic characteristic function function is is polynomial time. time. The The class class of of polynomial polynomial time time functions functions is is denoted denoted DD~, and the the i , and polynomial class class of of polynomial polynomial time time predicates predicates is is denoted denoted �i A~.. 1.3.4.1. Buss [1986]) 1.3.4.1. Theorem. T h e o r e m . ((Buss [1986]) ((a) a) Every -definable in Every polynomial polynomial time time function function is is �t ~-definable in Si S 1 .. ((b) b) Every �t -definable in Every polynomial polynomial time time predicate predicate is is A~-definable in Si S 1 .. Once one one has has bootstrapped bootstrapped Si S 1 sufficiently sufficiently to to intensionally introduce sequence sequence coding coding Once intensionally introduce functions, functions, it it is is fairly fairly straightforward straightforward to to prove prove this this theorem theorem using using Cobhams Cobhams's's inductive inductive definition main case the proof definition of of polynomial polynomial time time computability. computability. The The main case in in the proof by by induction induction is is the the case case where where f f is is defined defined from from 9g and and hh by by limited limited iteration iteration on on notation: notation: in in this this case ned case the the predicate predicate f(z, f(z, x) s = : Yy is is defined defined similarly similarly to to the the way way ff ((m, m , x) s = : yy was was defi defined in .2.10; the -PIND is in the the proof proof of of Theorem Theorem 11.2.10; the main main difference difference now now is is that that �t Z;b~-PIND is used used to to prove prove w w exists, exists, and and for for this this it it is is necessary necessary to to bound bound w w with with aa term. term. Fortunately, Fortunately, the the bounding condition f(z, x) 1 � bound the bounding condition I[f(z,~)[ _ p(lzl p([z[,, Ixl) I~[) makes makes it it possible possible to to bound the elements elements of of w w,, and and hence hence w w,, with with aa term. term. We We leave leave the the details details to to the the reader. reader. 0 E] A ning the polynomial time A second second way way to to approach approach defi defining the polynomial time function function in in Si $21 is is to to di directly polynomial time rectly formalize formalize polynomial time computability computability using using Turing Turing machine machine computations, computations, instead instead of of using using Cobham Cobham's's definition. definition. This This can can also also be be formalized formalized in in Si S~;; furthermore furthermore Si S 1 can can prove prove the the equivalence equivalence of of the the two two approaches. approaches. See See Buss Buss [1986] [1986] for for more more details. details. For For ii 2: >_ 11,, S� S~ ;2 _D Si S 1 and and also, also, by by Theorem Theorem 1.3.5 1.3.5 below, below, T4 T~ ;2 _D Si S 1.. Therefore, Therefore, the the above above theorem, theorem, combined combined with with Theorem Theorem 1.3.3.3 gives: gives:

11.3.4.2. .3.4.2. Theorem. Buss [1986]) T h e o r e m . ((Buss [1986]) Let Let ii 2: >_ 11.. The The theories theories S� Si2 and and T4 Ti2 can can introduce introduce symbols for time computable symbols for polynomial polynomial time computable functions functions and and predicates predicates and and use use them them freely freely in in induction induction axioms. axioms.

We Theorem 3.2) 3.2) that .3.4.1 also We shall shall show show later later ((Theorem that the the converse converse to to Theorem Theorem 11.3.4.1 also holds holds -define and -define precisely and and that that Si S 1 can can �t Ebl-define and �t A~-define precisely the the polynomial polynomial time time computable computable functions functions and and predicates, predicates, respectively. respectively. 1.3.5. 1.3.5. Relating Relating S� S~ and T4 T~ It S� ;2 Si1 and we have It is is clear clear that that S~ :D S and T4 T~ :) D Ti T1 ,, for for ii 2: > 11.. In In addition addition we have the the following following relationships relationships among among these these theories: theories:

Proof Theory Theory of ofArithmetic Arithmetic Proof

105 105

Theorem. (Buss (Buss [1986]) [1986]) Let Let ii >� 1. 1. Theorem.

T� ~;2 S~. S� . (1) T~ (1) -l . (2) S~ S� ~;2 T~ T�-~ (2) however open open whether whether the the theories theories ItIt isis however St C_ T~ C_ S~ C T~ C_ Sg C_ ...

are distinct. distinct. are Proof. A A proof proof of of (1) (1) can can be be found found in in Buss Buss [1986,sect [1986,sect 2.6]: 2.6] : this this proof proofmostly mostly involves involves Proof. bootstrapping of of T~, T� , and and we we shall shall not not present present itit here. here. bootstrapping The proof proof of of (2) (2) uses uses aa divide-and-conquer divide-and-conquer method. method. Fix Fix ii >� 11 and and fix fix aa E~_ I:L1The l formula A(x); A(x) ; we we must must prove prove that that S~ S� proves proves the the IND IND axiom axiom for for A. A . We We argue argue formula st assuming informally inside inside S~, assuming (Vx)(A(x) (V'x) (A(x) D A(x + + 1)). 1)) . Let Let B(x,z) B (x, z) be be the the formula formula informally :J A(x

::; zz ++ 1)(A(w(Vw (V'w _< ::; x)(Vy x) (V'y _< l) (A(w -'- y) y) D :J A(w)). A(w)). Clearly B B is is equivalent equivalent to to aa m-formula. By the the definition B, it follows that that Clearly II~-formula. By definition of of B, it follows

(V'x)(V'z) (B(x, l�zJ (W)(Vz)(B(~, [~'zJ)) :JD B(x, z)) , and and hence hence by by m-PIND II~-PIND oonn B(x, B(x, zz)) with with respect respect to to zz,,

(V'x) (B(x, 00)) :J (x, x)). (W)(B(,, ~B B(,,,)). Now, (A(x) :JD A(x Now, (V'x)B(x, (Vx)B(x, 00)) holds holds as as it it is is equivalent equivalent to to the the assumption assumption (V'x) (Vx)(A(x) A(x + + 1)) 1)),, and Finally, and therefore therefore (V'x)B(x, (Vx)B(x, x) x) holds. holds. Finally, (V'x)B(x, (Vx)B(x, x) x) immediately immediately implies implies (x) ) : this (V' x) (A (0) :JD A (Vx)(A(0) A(x))" this completes completes the the proof proof of of the the IND IND axiom axiom for for A A.. The The theorem theorem immediately immediately implies implies the the following following corollary: corollary: Corollary. C o r o l l a r y . (Buss (Buss [1986]) [1986]) S2 $2 = - T2 T2..

In � with In the the proof proof of of the the above above theorem, theorem, itit would would suffice sufficefor for A(x) A(x) to to be be b.. A~ with respect respect � -IND. to to S� S~.. Therefore, Therefore, S� S~ proves proves b.. A~-IND. 1.3.6. 1.3.6. Polynomial Polynomial hierarchy hierarchy functions functions in in bounded b o u n d e d arithmetic arithmetic

The The polynomial polynomial time time hierarchy hierarchy isis aa hierarchy hierarchy of of bounded bounded alternation alternation polynomial polynomial � of time time computability; computability; the the base base classes classes are are the the class class P P =- b.. A~ of polynomial polynomial time time recog recognizable P == D� nizable predicates, predicates, the the class class F FP []~of ofpolynomial polynomial time time computable computable functions, functions, the the class class N NP P =- I:i EVlof of predicates predicates computable computable in in nondeterministic nondeterministic polynomial polynomial time, time, the the P == IIi f , Of , class class coN coNP IIp of of complements complements of of N NP P predicates, predicates, etc. etc. More More generally, generally, b.. A~, [l~, I:f E~ and and IIf II~ are are defined defined as as follows: follows:

106 106

s. S. Buss

Definition. The classes classes �� A~ and and O[]~ have already already been Further define, by Definition. The � have been defined. defined. Further define, by induction induction on on ii,, (1) �f E~ is is the the class class of of predicates predicates R(i) R(s definable definable by by (1)

r (3y (3y :::; _< s(i)) (Q(i, y)) y)) R(i) ¢:} for some some term term ss in in the the language language of of bounded bounded arithmetic, arithmetic, and and some some �f A~ predi predifor cate cate Q Q.. f. (2) IT I]ff is is the the class class of of complements complements of of predicates predicates in in � E~. (2) is class class of of predicates predicates computable computable on on aa Turing Turing in in polynomial polynomial time time using using an an (3) O[li+ fP+ !1 is (3) oracle from �f E~.s oracle from .5 (4) �f P+ l is fP+ !1. Ai+l is the the class class of of predicates predicates which which have have characteristic characteristic function function in in O[1~+

( 4)

The syntactically defined defined by The connection connection between between the the syntactically defined classes classes of of formulas formulas � E~� defined by counting and the the computationally counting alternations alternations of of quantifiers quantifiers and computationally defined defined classes classes �f E~ is is given by the next next theorem. given by the theorem. Theorem. Wrathall [1976] T h e o r e m . ((Wrathall [1976],, Stockmeyer Stockmeyer [1976], [1976], Kent Kent and and Hodgson Hodgson [1982]) [1982]) A predicate predicate is is �f E~ if if and and only only if if there there is is aa � E tb-formula -formula which which defines defines it. it. A Proof. proof is � -formula defines -predicate. Proof. The The easier easier part part of of the the proof is that that every every � Z~-formula defines aa �f E~-predicate. For this, start noting that bounded formula polynomial time For this, start by by noting that aa sharply sharply bounded formula defines defines aa polynomial time predicate, predicate, even even when when the the/~f3 function function and and pairing pairing functions functions are are present. present. Then, Then, given given aa � t -formula, one sharply bounded E~-formula, one can can use use the the quantifier quantifier exchange exchange property property to to push push sharply bounded quantifiers quantifiers inward inward and and can can use use pairing pairing functions functions to to combine combine adjacent adjacent like like quantifiers; quantifiers; this nes aa � this transforms transforms the the formula formula into into an an equivalent equivalent formula formula which which explicitly explicitly defi defines E~f property according to property according to the the above above definition. definition. For For the the reverse reverse inclusion, inclusion, use use induction induction on on ii.. To To start start the the induction, induction, note note that that Theorem Theorem 1.3.4.1 1.3.4.1 already already implies implies that that every every �� A~ predicate predicate is is defined defined by by both both aa � E tb- t -formula. For and and aa IT II~-formula. For the the first first part part of of the the induction induction step, step, assume assume that that every every �f A~ predicate definable by both aa �t � -formula. Then immediate that predicate is is definable by both Ebl and and aa IT II~-formula. Then it it is is immediate that every t -formula. For every � E~f predicate predicate is is definable definable by by aa � E~-formula. For the the second second part part of of the the induction induction P P step, step, we we must must prove prove that that every every �f+ Ai+l-predicate is definable definable by by both both aa �f+ Ei+ and aa l1-- and ! -predicate is this, note f+ ! -formula. For f+ l -function has IT IIiP+l-f~ For this, note that that it it suffices suffices to to prove prove that that every every ODiP+l-function has �b+ ! -formula. To its its graph graph defined defined by by aa � Ei+l-formula. To prove prove this this last last fact, fact, use use induction induction on on the the definition fP+ !1: 9 it it is is necessary necessary to to show show that that this this condition condition is is definition of of the the functions functions in in O[li+ preserved composition as limited iteration preserved by by definition definition using using composition as well well as as by by definition definition by by limited iteration on on notation. notation. The The proofs proofs of of these these facts facts are are fairly fairly straightforward straightforward and and can can even even be be formalized formalized by by S� S~,, which which gives gives the the following following theorem: theorem: 5SAn An oracle oracle from from Ef E~ is is just just a a predicate predicate from from Ef E~.. For For our our purposes, purposes, the the most most convenient convenient way way to the class class of the smallest to define define the of functions functions polynomial polynomial time relative relative to an oracle oracle R R is is as as the smallest class class of of functions time functions and closed functions containing containing all all polynomial polynomial time functions and and the the characteristic characteristicfunction function of of R R and closed under under composition composition and and limited limited iteration iteration on on notation. notation.

Proof Theory Theory of ofArithmetic Arithmetic Proof

107 107

( Buss [1986])) [1986] ) ) Let � 1. 1. Let ii >_ TTheorem. h e o r e m . (Buss Every O~ Of function function isis E~-definable I:t -definable in in S~. S� . ( a) Every (a) Every A~ �f predicate predicate isis A~-definable �� -definable in in S~. S� . ( b) Every (b) Proof. The The proof proof proceeds proceeds by by induction induction on on i.i . The The base base case case has has already already been been done done Proof. 1.3.4.1. Part Part (b) ( b) isis implied implied by by (a), (a) , so so itit suffices suffices to to prove prove (a). ( a) . To To prove prove the the as Theorem Theorem 1.3.4.1. as inductive step, step, we we must must show show the the following following three three things things (and ( and show show they they are are provable provable inductive S� ) : in S~): in then the the characteristic characteristic function function X x(x OPf_ I -function, then (1) ( ~ , yy)) i s is aa 0,_l-function, ( ~)) o of f (1) If If ff(x, (3y To prove prove this, this, we we have have by by the the inducinduc ::; t(2))(f(2, t(x)) ( J (x, yy)) -= 0) 0) isis E~-definable. I:t -definable. To (3y <_ b The hypothesis that that f(2, equivalent to to aa Ei_ I:L formula A(2, tion hypothesis f(x, y) y) -= zz isis equivalent tion ~ formula A(x, y, y, z). z) . The 6 I:� -definition of of X(2) x(x) isis thus thus6 E~-definition

t )A(x, y,y, 0)) 0) ) t )A(x, y, X(Z) 0)) V (z (z == 11 A/\ -~(3y y, 0)) x(x) == zz r{:} (z (3y _::; t)A(Z, ::; t)A(Z, (z == 00 A ..., ( 3y <_ /\ (3y which is is clearly clearly equivalent equivalent to to aa E~-formula I:t -formula by by prenex prenex operations. operations. which If functions functions g9 and and f~ h have graph defnable definable by by E~-formulas, then so so does does their their (2) have graph (2 ) If I:t -formulas, then then composition. As an an example to prove prove this, this, suppose g(x, h(2)); h(x)) ; then f(x) == g(2, composition. As example of of how how to suppose f(2) can be be defined the graph of ff can the graph of defined by by = y th(X)) (h(x) = f(x) = = y) g(x, z) y {:} (3z ~::; th(Z))(h(Z) z) = f(~) ~ (3z = zz /\ A g(Z, y),,

where the function where tthh iiss aa term term bounding bounding the function h h.. (3) (3) If If f f iiss defined defined by by limited limited iteration iteration from from 9g and and hh with with bounding bounding polynomial polynomial p, p, and -definable graphs, and 9g and and h h have have I:� E~-definable graphs, then then so so does does f f. . To To prove prove this, this, show show that that ff (z, (z, x) 3) = - Yy is is expressed expressed by by the the formula formula

( lzl,IXI ) , z) (1, w SqBd(2 Pp(Izl'l~), z))[/~(Iz + 1, 3) = - Yy /\ A ,8 ~(1, w)) = -- g(x g(~)A 1, x) ((3w 3w ::;< SqBd(2 ) [,8 (lzlI + )/\ < Izl Izl)(Z(i 2, w w )) -= min min{h([2.z.~,.,J, ~, ,8 ~(i + 1, 1, w w)), )) , 22P(1'+11'1~)})]. (i + (Vi < { h( l2Izl:;�1 J , x, ) (,8 (i + 2, P( Ii+ I I,IXI ) } ) ). /\A(Vi

. . . ) has Here Here the the term term SqBd( SqBd(...) has been been chosen chosen sufficiently sufficiently large large to to bound bound the the size size of of the the sequence sequence w w encoding encoding the the steps steps in in the the computation computation of of ff (z, (z, x) 3).. The The formula formula is is clearly clearly in in I:� E~,, and and the the theory theory S; S~ can can prove prove the the existence existence and and uniqueness uniqueness of of w w by by PIND PIND 0 induction up to . induction up to zz. C:] A �--1I can A more more complicated complicated proof proof can can establish establish the the stronger stronger result result that that T T~ can also also � -define the -functions: I:E~-define the Of 0~-functions:

Theorem. Buss [1990) Theorem. ((Buss [1990])) Let Let ii > > 11.. �--1I .. -definable in a) Every ((a) Every Of O~function function is is I:� 2~-de.finable in T T~ ((b) b ) Every -definable in �--1I .. Every �f A~ predicate predicate is is �� A~-definable in T T~ 6We 6We use use the the convention convention that that aa characteristic characteristic function function of of aa predicate predicate equals equals zero zero when when the the predicate predicate isis true. true.

108 108

s. S. Buss

It possible collapse It is is aa very very interesting interesting question question whether whether the the possible collapse of of the the polynomial polynomialtime time hierarchy hierarchy is is related related to to the the possible possible collapse collapse of of the the hierarchy hierarchy of of bounded bounded arithmetic arithmetic theories. theories. So So far far what what is is known known is is that that if if S $22 is is finitely finitely axiomatized axiomatized ) , then polynomial time (more (more precisely, precisely, if if T� T~ = = S� S~+ +11 for for some some i 2': > 11), then the the polynomial time hierarchy hierarchy collapses Krajicek, Pudlak collapses provably provably in in T T22 (see (see Kraji~ek, Pudl~k and and Takeuti Takeuti [1991] [1991],' Buss Buss [1995], [1995], Zambella Zambella [1996], [1996], and and section section 3.3.2) 3.3.2).. This This means means that that the the hierarchy hierarchy of of theories theories of of bounded bounded arithmetic arithmetic collapses collapses if if and and only only if if the the polynomial polynomial time time hierarchy hierarchy collapses collapses S S2-provably. 2 -provably. 1.3.7. 1.3.7. The T h e theories theories PVi PVi

l and S� Since T� T~--t and S~ can can �r-defi E~-define the Of 0~ functions, functions, it it is is often often convenient convenient to to Since ne the conservatively conservatively extend extend the the language language of of bounded bounded arithmetic arithmetic with with symbols symbols for for these these l ( Of) and functions. Accordingly, Accordingly, we we define define T�T~-t(0~) and S�( S~(0~) to be be the the (conservative) (conservative) Of) to functions. l extensions containing symbols -functions extensions of of T� T~--t and and S� S~ to to the the language language containing symbols for for the the Of 0~-functions ( Of) has with their their � E~-defining equations as as new new axioms. axioms. For For i == 11,, the the theory theory Tf TO(0~) has � -defining equations with to to be be defined defined slightly slightly differently, differently, since since Tf T~ does does not not have have sufficient sufficient bootstrapping bootstrapping power to to �� E~-define the polynomial polynomial time time functions. functions. Instead, Instead, Tf TO(0~) is defined defined to to power -define the ( On is have first-order first-order language language consisting consisting of of symbols symbols for for all all polynomial polynomial time time functions functions and and have (2) axioms predicates, predicates, and and to to have have as as axioms axioms (1) (1) the the BASIC B A S I C axioms, axioms, (2) axioms that that define define the the non-logical non-logical symbols symbols in in the the spirit spirit of of Cobham Cobham's' s definition definition of of the the polynomial polynomial time time and and (3) (3) IND IND for for all all sharply sharply bounded bounded (equivalently, (equivalently, all all atomic) atomic) formulas.7 formulas. ~ l ( Of) and Of) since, One with T�One must must be be careful careful when when working working with T~-I(0~) and S�( S~(0~) since, for for i > > 11,, the the functions functions symbols symbols for for Of []~ cannot cannot be be used used freely freely in in induction induction axioms axioms (modulo (modulo some some open open questions). questions). l ( Of) is Since PVi instead. Since the the notation notation T�T~-I(0~) is so so atrocious, atrocious, it it is is sometimes sometimes denoted denoted PVi instead. PVi can Krajicek, Krajfhek, Pudlak Pudls and and Takeuti Takeuti [1991] [1991] prove prove that that PVi can be be axiomatized axiomatized by by purely purely universal universal axioms: axioms: to to see see the the main main idea idea of of the the universal universal axiomatization, axiomatization, note note that that if if A A is PVi proves is b.� A~,, then then PVi proves A A is is equivalent equivalent to to aa quantifier-free quantifier-free formula formula via via Skolemization Skolemization and and thus thus induction induction on on A(x, A(x, C) ~, , can can be be obtained obtained from from the the universal universal formula formula

A (t, c-')) A (JA (t, C), C)) J ((V~(Vt)[A(0, ~ 1\ A ..., ~A(t, D A( A(fA(t, ~ = 1, 1, C) c--)1\ A ..., -~A(fA(t, ~, C)] ~] JA (t, C) \iC) (Vt)[A(O, C) -'-

where where fA fA is is computed computed by by aa binary binary search search procedure procedure which which asks asks b.f A~ queries queries to to find find A(b 1, C) aa value is true A(b, C) value bb for for which which A ( b - 1, ~ is true and and A(b, c-*)is is false. false. Of Of course, course, this this f f is is aa Of -function and the language 0~-function and therefore therefore is is aa symbol symbol in in the language of of PVi. PVi. -

1.3.8. 1.3.8. More M o r e axiomatizations a x i o m a t i z a t i o n s of of bounded b o u n d e d arithmetic arithmetic

For -definable, For any any theory theory T T in in which which the the Gi:idel Ghdel (J fl function function is is present present or or is is �� E~-definable, in in particular, particular, for for any any theory theory T T 2 _DSJ S 1 ,, there there are are two two further further possible possible axiomatizations axiomatizations that bounded arithmetic: that are are useful useful for for bounded arithmetic: 7The 7The original original definition definition of a theory of this this type was was the definition definition of equational equational theory PV

of polynomial On can also polynomial time time functions functions by Cook Cook [1975). [1975]. T�( TO(D~) also be defined defined as as the conservative conservative extension extension of PV to first-order first-order logic. logic.

109 109

Proof Theory of Arithmetic

Definition. -replacement axioms Definition. Let Let cp (I) be be aa set set of of formulas. formulas. The The cP (I)-replacement axioms are are the the formulas formulas

s \ ) ( :Jyy :S (x + w) :S t) ((w (A(x, (3 (x + < IIsl)( < t)A(x, t)A(x, y) y) � D (:Jw)( (3w)(Vx ~ Isl) Isl)(A(x, Z(x + 1, w)) w)) 1\ A (3 ~(x + 1, 1, w) _ t) \:Ix :S \:Ix :S for semi) terms ss and for all all formulas formulas A A E cP (I) and and all all appropriate appropriate ((semi)terms and tt.. As As usual usual A A may may have other free have other free variables variables in in addition addition to to x x that that serve serve as as parameters. parameters. -replacement axioms The The strong strong cP O-replacement axioms are are similarly similarly defined defined to to be be the the formulas formulas (3w)(V~ :S <__Isl) 1 l)[(3y _ t)A(x, t)A(x, y) y) ++ ++A(x, A(x, (3 ~(x 1, w)) w)) 1\ A (3 ~(x + 1, 1, w) w) :S _< t] t].. (:Jw)(\:Ix (x + (x + 1, [ ( :Jy :S

The The replacement replacement and and strong strong replacement replacement axioms axioms contain contain an an apparently apparently unbounded unbounded quantifier ((3w); however, Si S~ can always bound bound w w by by aa term term SqBd(t, SqBd(t,s) which quantifier :Jw ) ; however, can always s) which si + is is large large enough enough to to bound bound aa sequence sequence of of lIs[ + 11 values values :S _ tt.. For For example, example, setting setting SqBd(t, s) equal to to (2t + 1)#(2(2s 1)#(2(2s + + 1) will work work for for the the sequence sequence encoding given SqBd(t, s) equal (2t + 1) 22)) will encoding given in section 11.2.8. .2.8. in section It � -replacement axioms � -PIND It is is known known that that the the � E~-replacement axioms are are consequences consequences of of the the � E~-PIND axioms, � -replacement axioms � -PIND axioms, and and that that the the strong strong � E~-replacement axioms are are equivalent equivalent to to the the � E~-PIND axioms ((for >_ 11,, and and over over the base theory $21). Figure 11 shows shows these these and and other other for ii � the base theory Si ). Figure axioms relationships among among the the axiomatizations of bounded arithmetic. relationships axiomatizations of bounded arithmetic. 1.4. Sequent Sequent calculus calculus formulations formulations of of arithmetic arithmetic 1.4.

This This section section discusses discusses the the proof proof theory theory of of theories theories of of arithmetic arithmetic in in the the setting setting of of the the sequent sequent calculus: calculus: this this will will be be an an essential essential tool tool for for our our analysis analysis of of the the proof-theoretic proof-theoretic strengths strengths of of fragments fragments of of arithmetic arithmetic and and of of their their interrelationships. interrelationships. The The sequent sequent calculus calculus used used for for arithmetic arithmetic is is based based on on the the system system LKe LK~ described described in in be enlarged Chapter Chapter II of of this this volume; volume; LKe LK~ will will be enlarged with with additional additional rules rules of of inference inference for bounded arithmetic, arithmetic, LKe is for induction, induction, minimization, minimization, etc., etc., and and for for theories theories of of bounded is enlarged include inference bounded quantifiers. enlarged to to include inference rules rules for for bounded quantifiers. 1.4.1. Definition. LKB ((or or LKBe respectively, 1.4.1. Definition. LKB~)) is is the the sequent sequent calculus calculus LK LK ((respectively, LKe the language language of rst-order arithmetic LK~)) extended extended as as follows: follows: First, First, the of fi first-order arithmetic is is expanded expanded

to bounded quantifi ers as part of to allow allow bounded quantifiers as aa basic basic part of the the syntax. syntax. Second, Second, the the following following new new rules rules of of inference inference are are allowed: allowed: Bounded B o u n d e d quantifier quantifier rules rules V< <_:left :left \:I -

A(t) A(t),, r-+� F--+A tt :S s, (\:Ix < s, (vx :S < s)A(x), s)A(x), r-+� r--+A

~ :right :right \:IV < -

bb :S _ s, s, r-+�, F---+A, A(b) A(b) r-+�, F---+A, ((Vx _< s)A(x) \:Ix :S

<_:left :left :J3 :S

s, A(b) , r-+� bb :S <_s,A(b), F--+A r-+� (:Jx (3x :S _< s)A(x), s)A(x),F--->A

~:right :33 < :right

r-+�, A(t) A(t) tt :S <_ s, s, r-+�, F---+A, ((3x <_s)A(x) s)A(x) :Jx :S

where where the the variable variable bb is is an an eigenvariable eigenvariable and and may may not not occur occur in in in in ss or or in in r, F, � A.. The The Cut Cut Elimination Elimination and and Free-cut Free-cut Elimination Elimination Theorems Theorems still still hold hold for for LKB and and LKBe LKBe,, in in the the exact exact same same form form as as they they were were proved proved to to hold hold for for LK LK and and LKe

110 110

S. Buss Buss s. E~-IND ¢=:} ~ m-IND II~-IND ¢=:} ~ E~-MIN ¢=:} r .6.f+1-IND A~+I-IND I:f-IND I:f-MIN

R

P~-PIND ¢=:} r rI~-PIND ¢=:} ~ E~-LIND ¢=:} ~ II~-LIND I:�-PIND rr�-PIND I:�-LIND m-LIND

�

2~-LMIN , = , (I:r (E~b+~ n rrr+1 II~+~)-PIND IN ¢=:} I:f-L )-PIND +1 n

R

b_cIND I: E~_I-IND i

EL1-MIN ~

II~-MIN

S~ )~- T� T~--1l + +E~-replacement I:r-replacement S� B(r.~) B(E�)

E~-PIND+ E~+l-replacement ==~ I:r-PIND E~-PIND ===> ==v I:r-replacement E~-replacement I:� -PIND + I:r + 1 -replacement ===> E~-PIND ¢=:} ~ E~-PIND E~-replacement I:r-PIND I:� -PIND + strong strong I:r-replacement Figure 11 Figure Relationships among Relationships among axiomatizations axiomatizations for for Bounded Bounded Arithmetic Arithmetic relative relative to the the base base theory theory BASIC BASIC with with ii ;::: _ 11;; T� T~ should be interpreted interpreted as as should be to PV1 . See Buss [1986,1990] [1995] for for proofs. proofs. PV1. See Buss [1986,1990], Buss Buss and and Ignjatovic Ignjatovid [1995]

'

of the the bounded are the the in Chapter Chapter I.I. The in The principal principal formulas formulas of bounded quantifier quantifier inferences inferences are formulas in the lower sequent; and (Qx (Qx < :::; s)A s)A introduced introduced in the lower sequent; as as usual, usual, aa cut cut on on aa formulas tt <:::; ss and direct descendent of is anchored. direct descendent of aa principal principal formula formula is anchored. 1.4.2. R Rule induction. We introduce inference rules which which are are 1.4.2. u l e forms forms of of induction. We next next introduce inference rules equivalent to induction axioms; the the reason reason for using rules rules of induction equivalent to induction axioms; for using of inference inference for for induction in place of of induction that the the use free-cut free in place induction axioms axioms is is that use of of free-cut free proofs proofs provides provides aa powerful powerful proof-theoretic tool tool for for the the analysis analysis of of arithmetic. arithmetic. proof-theoretic of fragments fragments of Definition. Let Let W9 be be aa class class of of formulas. formulas. Then Then ~-IND w-IND induction induction rules rules are are the the Definition. inferences inferences of of the the form form

1) A(b), A, A(b A(b), F r ~� .6., A(b + 1) A(O), F'--~ A, A(t) r � .6., where A E w9 and and where where the the eigenvariable eigenvariable bb does not occur except as as indicated. indicated. where A does not occur except induction rules rules are are the the inferences inferences of of the the form form The ~-PIND w- PIND induction The

Proof Theory Theory of Arithmetic

1111 11

A( A(LL�bJ 89j),) , r F --+ ~ 1l, A, A(b) A(b) A(O),, r F --+ --~ 1l A,, A(t) A(t) A(O) where where again again A A E w9 and and bb occurs occurs only only as as indicated. indicated. We leave leave it it to to the the reader reader to to check check that that the the induction induction rules rules w-IND ~-IND and and w-PIND ~-PIND We respectively) ; in are w-PIND ((respectively); are equivalent equivalent to to the the induction induction axioms axioms w-IND ~-IND and and ~-PIND in fact, fact, induction rules this is is true true for for any any class class W9 of of formulas. formulas. The The fact fact that that the the induction rules are are this equivalent induction axioms equivalent to to the the induction axioms depends depends crucially crucially on on the the presence presence of of the the the the side side formulas r F and and II A in in the the inference; inference; when when side side formulas formulas are are not not allowed, allowed, the the inference inference formulas rules than the the induction axioms; see, see, e.g., Parsons [1972] rules are are often often slightly slightly weaker weaker than induction axioms; e.g., Parsons [1972] and and Sieg Sieg [1985] [1985].. It It follows follows that that theories theories such such as as lllo IA0,, �n /En,' mn /IIn,, S� S~ and and T� T~ can can be induction rules induction axioms. be equivalently equivalently formulated formulated using using induction rules instead instead of of induction axioms. For For the will presume the rest rest of of this this chapter, chapter, we we will presume that that these these theories theories are are formulated formulated with with the the induction rules. induction rules. As the free-cut elimination theorem As was was discussed discussed in in Chapter Chapter I, I, the free-cut elimination theorem holds holds for for theories theories such such as as IL;n IE~,, IIIn IH~,, S� S~ and and T4 T~.. In In particular, particular, we we have have the the following following corollary corollary to to the the free-cut elimination theorem, which which generalizes generalizes the the subformula subformula property property to to fragments fragments free-cut elimination theorem, of this theorem, closed under under of arithmetic. arithmetic. For For this theorem, w9 must must be be a a class class of of formulas formulas which which is is closed the taking subformulas the operations operations of of taking subformulas and and freely freely substituting substituting terms terms for for variables. variables. Strictly speaking, classes Strictly speaking, classes such such as as L;i Ei are are not not closed closed under under subformulas, subformulas, since since a a negated) IIi-subformula; L;i -formula may Ei-formula may contain contain aa ((negated) IIi-subformula; however, however, one one may may instead instead use use the all negation negation signs atomic subformulas. the class class of of L;i-formulas Ei-formulas in in which which all signs are are in in front front of of atomic subformulas. This This can can be be done done without without loss loss of of generality generality and and then then this this class class of of formulas formulas is is closed closed both formulas and both under under sub subformulas and under under term term substitution. substitution.

Theorem. T h e o r e m . Let Let W9 be be aa class class of of formulas formulas closed closed under under subformulas sub formulas and and under under term term substitution substitution and and containing containing the the atomic atomic formulas. formulas. Let Let R R be be aa fragment fragment of of arithmetic arithmetic axiomati axiomatized by W9-IND (or (or by by W9 -PIND) plus plus initial initial sequents sequents containing containing only only formulas formulas zed by from from w ~.. Also Also suppose suppose that that the the sequent sequent r F --+ ~ II A contains contains only only formulas formulas from from w9 and and -proof of that that R R f~- r F --+ ~ ll A.. Then Then there there is is an an R R-proof of r F --+ ~ II A such such that that every every formula formula appearing appearing in in the the proof proof is is aa W9 -formula.

The The proof proof of of this this theorem theorem is is of of course course based based on on the the fact fact that that every every formula formula appearing appearing in in an an R-proof R-proof either either is is aa direct direct descendent descendent of of a a formula formula in in an an initial initial sequent and hence sequent or or is is an an ancestor ancestor ((and hence subformula subformula in in the the wide wide sense sense)) of of either either a a cut cut formula formula or or a a formula formula in in the the endsequent. endsequent. Furthermore, Furthermore, in in aa free-cut free-cut free free R-proof, R-proof, all all cut-formulas --+ II cut-formulas are are in in w ~ and and by by free-cut free-cut elimination, elimination, r F--~ A has has aa free-cut free-cut free free proof. proof. The tool for The above above theorem theorem turns turns out out be be an an extremely extremely powerful powerful tool for the the proof-theoretic proof-theoretic analysis analysis of of fragments fragments of of arithmetic. arithmetic. 1.4.3. 1.4.3. We We now now state state and and prove prove aa generalization generalization of of Theorem Theorem 1.2.7.1 1.2.7.1 which which applies applies to to very R, possibly very general general bounded bounded theories theories R, possibly including including induction induction inferences inferences for for bounded bounded formulas. formulas. Assume Assume that that R R contains contains ::; _ in in its its language language and and that that R R proves proves that that ::; < is is reflexive reflexive and and transitive. transitive. Also Also suppose suppose that that for for all all terms terms rr and and ss,, there there is is a a term term tt so so

1112 12

S. Buss Buss S.

that R R proves proves rr � < tt and and ss � _ tt.. Further Further suppose suppose that that for for all all terms terms t( t(d, b) and and r(ii) r(g),, that a, b) there is is aa term term ss so so that that R R proves proves that that bb � _ rr(~) D t(a, t(g, b) b) � _ s(a) s(~).. there (a) :J P a r i k h ' s Theorem. T h e o r e m . Let Let R R be be aa bounded bounded theory theory satisfying satisfying the the above above conditions conditions and and Parikh's A(Z, y) y) aa bounded bounded formula. formula. Suppose Suppose R R f-F ('v'x) (VZ)(3y)A(Z, y).. Then Then there there is is aa term term tt (3y)A(x, y) A(x, such that that R R also also proves proves ('v'x) (VZ)(3y <_t)A(x, t)A(Z, y) y).. such (3y � 1.4.4. Proof. Proof. By the the free-cut free-cut elimination elimination theorem, theorem, there there is is aa free freeSketch) . By 1.4.4. ((Sketch). ' cut cut free free R-proof R-proof P P of of (3y)A(b, (3y)A(b, y) y),, where where the are new new free free variables. variables. By By the the the bb'ss are subformula property, property, every every sequent sequent f F--+ A in in the the proof proof P P contains contains only only bounded bounded subformula -t � formulas in in its its antecedent antecedent f F and and its its antecedent antecedent � A contains contains only only bounded bounded formulas formulas formulas plus possibly possibly occurrences occurrences of of the the formula formula (3y)A(b, (3y)A(b', y) y).. Given Given such such aa � A and and given given aa plus term tt,, let let A
It the inference It is is not not difficult difficult to to check check that that the inference rule rule for for collection collection ((replacement) is replacement) is equivalent to to the the free-cut equivalent the axiom axiom form form of of collection. collection. Furthermore, Furthermore, the free-cut elimination elimination theorem holds holds as as before; of 'free-cut' changed by by also also declaring theorem before; however, however, the the notion notion of 'free-cut ' is is changed declaring every direct descendent of the the principal principal formula collection inference inference to to be be every direct descendent of formula of of aa collection anchored. anchored. One easy consequence of free-cut elimination elimination for One easy consequence of free-cut for collection collection inferences inferences is is that that Parikh Theorem 11.4.3 .4.3 holds also for for theories R that that contain Parikh's' s Theorem holds also theories R contain '2:. E1-REPL; compare 1 -REPL; compare this to Theorem 3.4.1 3.4.1 about about the the conservativity conservativity of of B'2:.i BEi+I over/~i. IL.; . this to Theorem + l over 2. G h d e l iincompleteness ncompleteness 2 . Godel Ghdel's and Godel ' s incompleteness incompleteness theorems, theorems, on on the the impossibility impossibility of of giving giving an an adequate adequate and complete axiomatization axiomatization for for mathematics, mathematics, were were of of great great philosophical philosophical and and foundafounda complete tional importance importance to to mathematics. mathematics. They They are are arguably arguably the the most most important important results results in in tional mathematical logic logic since since the the development development of of first-order first-order logic. logic. Loosely Loosely speaking, speaking, the the mathematical incompleteness theorems theorems state state that that any any sufficiently sufficiently expressive, expressive, consistent consistent theory theory with with incompleteness aa decidable axiomatization is any such theory, the decidable axiomatization is not not complete; complete; and and furthermore, furthermore, for for any such theory, the second incompleteness incompleteness theorem theorem gives an explicit, explicit, non non self-referential self-referential true true statement second gives an statement which is is not not aa consequence of the the theory. theory. More More generally, the set set of of IIl-sentences III -sentences true true which consequence of generally, the

Proof Theory of Arithmetic

1113 13

about the the integers8 integers s is is not not recursively recursively enumerable, enumerable, so so there there is is no no way way to to generalize generalize or or about replace first-order first-order logic logic with with any any other other kind kind of of formal formal system system which which both both admits admits a a replace decidable notion notion of decidable of 'provability' 'provability' and and is is complete complete in in the the sense sense of of 'proving' 'proving' every every true true II Hi-sentence of the the integers. integers. I -sentence of

2.1. metamathematics 2.1. Arithmetization A r i t h m e t i z a t i o n of ofm etamathematics

The methods of The usual usual methods of proving proving G6del's GSdel's incompleteness incompleteness theorems theorems involve involve coding coding metamathematical coding the metamathematical concepts concepts (i.e., (i.e., coding the syntax syntax of of first-order first-order logic) logic) with with integers integers and then then using using a a self-referential self-referential or or diagonal diagonal construction construction to to obtain non-provable and obtain non-provable true statements; ~ this process of of coding coding syntactic syntactic aspects aspects of of logic logic with integers true statements;9 this process with integers is called 'arithmetization'. There There are are essentially essentially two two different different approaches approaches to to the the is called 'arithmetization'. arithmetization ability as arithmetization of of syntax. syntax. The The first first approach approach uses uses numeralwise numeralwise represent representability as aa means computable functions: means of of representing representing computable functions: a a numeralwise numeralwise representation representation of of aa function characterization of the function's particular choice function gives gives a a characterization of the function's values values for for particular choice of of inputs to the the function. To be be precise, precise, a a formula formula A(x, A(Z, yy)) numeralwise numeralwise represents represents aa inputs to function. To function particular integers function f f (x) (Z) = - y y in in a a theory theory T T if if and and only only if, if, for for every every particular integers n n ll ,, .. ... ., , nk nk with m, the with f f ((ii) g) = -- m, the theory theory T T proves proves (Yy)(A(SnlO,..., S nk, y) ++ y = SmO).

It turns out recursive function It turns out that that every every recursive function is is numeralwise numeralwise representable representable even even in in very only recursive very weak weak theories theories such such as as R R and and Q Q;; and and conversely, conversely, only recursive functions functions are are theory, no numeral wise representable numeralwise representable in in any any axiomatizable axiomatizable theory, no matter matter how how strong. strong. However, However, numeralwise numeralwise representation representation of of f f in in the the theory theory T T only only implies implies that that T T can can 'represent' xed values 'represent' all all particular, particular, fi fixed values of of f f ;; this this in in no no way way implies implies that that T T can can prove prove general properties This in contrast to approach to general properties of of the the function function f f . . This in is is in in contrast to the the second second approach to the which involves giving intensional the arithmetization arithmetization of of syntax syntax which involves giving intensional definitions definitions of of certain certain (but not not all) all) recursive recursive functions. functions. In In the the intensional intensional approach approach to to arithmetization arithmetization (but of ne concepts of metamathematics, metamathematics, one one gives gives formulas formulas which which defi define concepts such such as as "formula" "formula",, "term",, "substitution" "substitution",, "proof" "proof",, "theorem" "theorem",, etc; these definitions are said said to to be be "term" etc; these definitions are intensional provided concepts. For intensional provided the the theory theory T T can can prove prove simple simple properties properties of of these these concepts. For instance, theory T define the term instance, one one wants wants the the theory T to to be be able able to to define the notion notion of of substituting substituting a a term into the result result is into a a formula formula and and prove prove the is a a formula; formula; similarly, similarly, T T should should be be able able to to prove prove that the set that the set of of theorems theorems is is closed closed under under modus modus ponens; ponens; etc. etc. (See (See Feferman Feferman [1960] [1960] for for aa comprehensive discussion of intensionality.) comprehensive discussion of intensionality.) It significantly more intensional arithmetiza It is is significantly more work work to to carry carry out out the the details details an an intensional arithmetization wise representation indeed, an tion of of syntax syntax than than a a numeral numeralwise representation of of recursive recursive functions; functions; indeed, an 8SBy By MatijaceviC's Matijacevi~'s theorem, theorem, the the same same holds holds for for the the true, true, purely purely universal universal sentences sentences (in (in the the language of P A). PA). 9There are approaches to the first ~ first incompleteness incompleteness theorem theorem that avoid avoid this arithmetization of metamathematics; for instance, directly prove III sentences sentences of arithmetic instance, one can directly prove that the true II1 do not form a recursively recursively enumerable set, set, say by encoding encoding Turing Turing machine machine computations with integers. integers. For somewhat somewhat different different approaches based based on Berry's paradox, paradox, see Chaitin [1974] [1974] and Boolos Boolos [1989]. [1989].

114 114

s . Buss Buss S.

intensional definition definition of of aa function function isis typically typically aa numeralwise numeralwise representation representation of of the the intensional same function. function. Furthermore, Furthermore, the the intensional intensional representation representation requires requires the the additional additional same verification that that the the underlying underlying theory theory can can prove prove simple simple facts facts about about the the function. function. verification Nonetheless, the the intensional intensional definition definition has has significant significant advantages, advantages, most most notably, notably, in in Nonetheless, allowing aa smoother smoother treatment treatment of of the the Ghdel Godel incompleteness incompleteness theorem, theorem, especially especially the the allowing second incompleteness incompleteness theorem. theorem. second Since Since many many textbooks textbooks discuss discuss the the first first approach approach based based on on numeralwise numeralwise reprep resentability and and since since we we prefer prefer the the intensional intensional approach, approach, this this article article will will deal deal resentability only with with the the intensional intensional approach. approach. The The reader reader who who wants wants to to see see the the numeralwise numeralwise only representability approach approach can can consult consult Smorynski Smorynski [1977] [1977] and and any any number number of of textbooks textbooks representability such as as Mendelson Mendelson [1987]. [1987] . The The intensional intensional approach is due due to to Feferman Feferman [1960]. [1960]. An An such approach is effective unification unification of of the the two approaches can can be be given using the the fact fact (independently (independently effective two approaches given using due to to Wilkie Wilkie and and to to Nelson Nelson [1986]) [1986]) that that IA0-t-g/~ I �O+OI and and Si are interpretable in Q; Q ; since since due S~ are interpretable in 01 and and S~ Si admit admit aa relatively straightforward intensional intensional arithmetization arithmetization both IA0 I �o ++ f~l both relatively straightforward of metamathematics (see Wilkie Paris [1987] [1987] and of metamathematics (see Wilkie and and Paris and Buss Buss [1986]) [1986]),, this this allows allows strong strong forms incompleteness obtained obtained via via the the intensional approach to to apply apply also also to the forms of of incompleteness intensional approach to the Si in can be be theory Q; Q ; paragraph below sketches of S~ theory paragraph 2.1.4 2.1.4 below sketches how how the the interpretation interpretation of in Q Q can used to intensional arithmetization arithmetization in in Q The book of Smullyan Smullyan [1992] [1992] gives gives used to give give an an intensional Q.. The book of aa modern, modern, in-depth in-depth treatment treatment of of Ghdel's Godel' s incompleteness incompleteness theorems. theorems. 2.1.1. 2.1.1. Overview O v e r v i e w of of an an intensional i n t e n s i o n a l arithmetization a r i t h m e t i z a t i o n of of metamathematics. metamathematics. We We now now sketch sketch some some of of the the details details of of an an arithmetization arithmetization of of metamathematics; metamathematics; this this arithmetization arithmetization can can be be carried carried out out intensionally intensionally in in I�o IA0 + + 01 f~l and and in in Si S~.. Detailed Detailed explanations explanations of of similar similar arithmetizations arithmetizations in in these these theories theories can can be be found found in in Wilkie Wilkie and and Paris Paris [1987] [1987] and and in in Buss Buss [1986]. [1986]. We We shall shall always always work work in in the the (apparently) (apparently) weaker weaker theory theory Si S~.. To To arithmetize arithmetize metamathematics, metamathematics, we we need need to to assign assign Godel Ghdel numbers numbers to to syntactic syntactic objects objects such such as as 'terms', 'terms', 'formulas 'formulas',' , 'proofs', 'proofs', etc. etc. Each Each such such syntactic syntactic object object is is viewed viewed as as an an expression expression consisting consisting of of string string of of symbols symbols from from aa finite finite alphabet. alphabet. This This finite finite " " 1\ " , "V" ..., " , "D", :J , "3" alphabet alphabet contains contains logical logical connective connective symbols symbols ""A", "V",, ""~", "3",, "\I" "V",, etc., etc., and and the the comma comma symbol symbol and and parentheses; parentheses; it it also also contains contains non-logical non-logical connective connective symbols symbols for for the the function function and and relation relation symbols symbols of of arithmetic. arithmetic. The The alphabet alphabet also also needs needs " x symbols (and symbols for for variables: variables: for for this this there there is is aa variable variable symbol symbol ""x" (and possibly possibly "a" for I " used for free free variables) variables) and and there there are are symbols symbols "0" "0" and and ""1" used to to write write the the values values of of subscripts subscripts of of variables variables in in binary binary notation. notation. In In addition, addition, when when first-order first-order proofs proofs are are --t " for formalized formalized in in the the sequent sequent calculus, calculus, it it will will contain contain aa symbol symbol ""---~" for the the sequent sequent connective connective and and the the semicolon semicolon symbol symbol for for separating separating sequents. sequents. At At times, times, it it will will be be convenient convenient to to enlarge enlarge the the finite finite alphabet alphabet with with other other symbols symbols that that can can be be used used for for describing describing the the skeleton skeleton of of aa proof. proof. Since Since the the alphabet alphabet is is finite, finite, we we can can identity identity the the alphabet alphabet with with some some finite finite set set {O, ( 0 , .. .. ..,, s} s) of of integers, integers, thereby thereby giving giving each each alphabet alphabet symbol symbol aa aa G6del Ghdel number number which which a-' . Then, isis denoted denoted rraT. Then, given given an an expression expression Q a which which consists consists of of the the symbols symbols aa1la2 a 2 . . , aam, m, let let nn ll ,, .. .. .., , nnm be their their Godel Ghdel numbers, numbers, then then the the least least Godel Ghdel number number of of the the sequence sequence m be (nl' ( n l , .. .. ..,' nrim) is, by by definition, definition, the the G6del Gh"del number number of of the the expression expression Qa.. The The Godel Ghdel m ) is, . . •

Proof Theory Theory of of Arithmetic Arithmetic Proof

115 115

number of of an an expression expression 0 a is is denoted denoted ro'. ra7. There There is is aa subtle subtle difference difference between between number the Godel Godel number number of of symbol symbol (J a and and the the G6del Gbdel number number of of the the expression expression containing containing the ' and it should always be clear from context which just (J a;; these these are are both both denoted denoted rra~ (J and it should always be clear from context which just 10 one is is intended. intended. 1~ one Since we we have have already already discussed discussed that that intensional intensional definitions definitions of of G6del Gbdel numbers numbers of of Since sequences can can be be given; given; it it is is straightforward straightforward to to further further give give intensional intensional definitions definitions of of sequences (Gbdel numbers numbers of) of) syntactic syntactic objects; objects; in in particular, particular, Si S~ can can A� A~-define the following following (G6del -define the predicates: predicates: F r e e V a r ( w ) -- w w codes codes aa free free variable variable FreeVar(w) BdVar(w) w codes codes aa bound bound variable variable -- w BdVar(w) Term(w) w codes codes aa term term -- w Term(w) Fmla(w) w codes codes aa formula formula -- w Fmla(w) Sequent(w) w codes codes aa sequent sequent Sequent( w) -- w For example, example, the the formula formula FreeVar(w) F r e e Y a r ( w ) asserts asserts that that either either w w = -- (a' (ra7,, ro r07) (which For ' ) (which codes the the free free variable variable "ao "Co") or w w is is of of the the form form (a', (ra7, fl.', rlT, wl w l ,, .. ... ., , Wk) wk> with with k k � > 00 " ) or codes and with with each each Wi wi equal equal to to ro' tO7 or or rl' r17 (this encodes "a; "ai"" where where ii > > 00 has has binary binary (this encodes and representation lW l Wll ". . . ·wWk k )). . The The A� Ab1 definitions of Term, Term, Fmla F m l a and and Sequent Sequent are are definitions of representation somewhat more more complicated: complicated: they they depend crucially on on the the fact fact that that length-bounded length-bounded somewhat depend crucially counting is is �� ~ - d-definable e f i n a b l e and and that that therefore therefore terms terms and and formulas formulas may may be be parsed parsed by by counting means of of counting counting parentheses. parentheses. Counting Counting commas commas also also allows allows notions notions such such as as the the means i-th formula of of a a sequent sequent to to be be � ~-defined. i-th formula � -defined. In keeping keeping with with the the convention convention that all syntactic syntactic objects objects are are coded coded by by expressions, expressions, In that all we let let the the G6del Gbdel number number of of a a proof proof be be defined defined to to the the G6del Gbdel number number of of an an expression expression we consisting of sequence of of sequents sequents separated separated by by semicolons. consisting of aa sequence semicolons. A A proof proof is is intended intended to to be valid that each in the the proof proof can be inferred by aa valid of be valid provided provided ""that each sequent sequent in can be inferred by valid rule rule of inference from sequents appearing appearing earlier earlier in in the the proof. Of course the notion notion of of v valid inference from sequents proof. Of course the alid inference depends on on the in which the proof inference depends the formal formal proof proof system system in which the proof is is being being carried carried out. Accordingly, for for an an appropriate appropriate fixed T,, we we wish $21 to to be able out. Accordingly, fixed formal formal system system T wish Si be able to A~-define the predicate predicate to which states states that codes a a valid valid to A� -define the to define define PrOOfT(W ProofT (w)) which that w w codes T-proof. For this to be must be be a a polynomial polynomial time time procedure procedure which T -proof. For this to be possible, possible, there there must which determines whether whether w w codes in general, general, this based on codes aa valid valid T-proof; T -proof; in this will will be be based on aa determines polynomial time procedure which which checks checks whether polynomial time procedure whether a a given given inference inference is is valid valid for for T T.. we will will consider, consider, such S� , T� The theories theories TT that The that we such as as S~, T~,, � /Ei,i ' B�i B~i,' etc., etc., will will have have only axioms axioms and and unary unary and and binary binary inference inference rules, rules, and and these these will will be be specified specified by by aa only 11 finite finite set set of of schemes. schemes. For For such such schematic schematic theories, theories, SSi1 can can A~-define A� -define the the relation relation11 Validlnferencer(u,, v,v , w) w) -- ww can can be be inferred inferred with with aa single single T-inference T-inference ValidInferenceT(u from zero, one one or or both both of of uu and and v. v. from zero, 1~ rA" in a nonconventional manner: the usual definition is to let rAT rA" represent lOWe have defined rAT whose value is equal to the Gbdel Godel number of A. A . We shall represent this alternative alternative aa closed term whose concept with the notation rA7. that we are using is better better for our intensional intensional rA" . The definition for rA7 rA" that development. 11 11 We discuss the situation situation for non-schematic theories below below in section 2.1.3.

1116 16

s. S. Buss

With this, it -define Proo!T(w) -define With this, it is is easy easy for for Si S~ to to At A~-define PrOOfT(W) .. In In addition, addition, Si S~ can can At A~-define the the predicate predicate Pr!T(w, PrfT(W , u) u) which which states states that that ProO!T(W) PrOOfT(W) and and Fmla(u) Fmla(u) and and that that W w is is aa proof of the the sequent sequent � --~ A A where where u u= - rrA7 (i.e., that that W w is is a aT T-proof of the the formula formula A). A). A' (i.e., -proof of proof of Finally, Finally, the the set set of of theorems theorems of of T T can can be be defined defined by by

Thmr(u) u) . ThmT(u) {:} r (3w)Prfr(w, (3w)PrfT(W,U). However, -definable, since However, Thmr ThmT is is not not generally generally �t A~-definable, since it it not not generally generally even even decidable decidable ' s incompleteness (a consequence consequence of of G6del GSdel's incompleteness theorems, theorems, see see below) below).P 12 (a A particularly important substitution of A particularly important syntactic syntactic operation operation is is the the substitution of a a term term into into formula. Let Let A A be be a a formula formula and and let let tt be be aa term; because of of the the sequent sequent aa formula. term; because calculus bound variables, calculus'' conventions conventions on on free free and and bound variables, one one can can always always form form the the formula formula A(t/ao),, which which is is A A with with tt substituted substituted in in for for the the free free variable variable ao a0,, merely merely by by A(t/ao) replacing each each occurrence occurrence of of ao a0 as as a a sub subexpression of A A with with the the expression expression tt.. replacing expression of This -definable in u , vv)) denotes This is is clearly clearly I:t E~-definable in Si S~ and and Sub( Sub(u, denotes the the function function such such that that Sub(rA7, rt') rtT) = - rrA(t/ao) for all all formulas formulas A A and and terms terms tt.. Sub(A', A(t/ao)'7 for One One final final simple, simple, but but important important formalization, formalization, is is the the definition definition of of closed closed canonical canonical terms terms rr n which which represent represent an an integer integer n n.. The The term term Q 0 is is just just the the constant constant symbol symbol o0.. And m and And inductively, inductively, the the term term 2m 2m is is (SSO) (SSO).m and the the term term 2m 2m + + 11 is is 2m 2m + + SO SO.. Note Note that -define the that the the number number of of symbols symbols in in rr n is is O(Jnl) O(Inl) ;; also, also, Si S~ can can I:t E~-define the map map n nH ~ rrr' rn7.· .

2.1.2. arithmetization. In 2.1.2. Intensionality I n t e n s i o n a l i t y of of the the arithmetization. In order order for for the the above-sketched above-sketched arithmetization considered intensional, intensional, it arithmetization to to be be considered it is is necessary necessary that that Si $21 can can prove prove basic basic facts notation, we shall use facts about about the the arithmetization. arithmetization. To To simplify simplify notation, we shall use abbreviations abbreviations such such · · ·), which · · · uu...)). · · · )) . as ('v'rA,) (VrA7)(..rA7...), which abbreviates abbreviates the the formula formula (Vu) (Vu)(Fmla(u) D ((... (- · · rA' (Fmla(u) J as Some examples Some examples of of what what Si $21 can can prove prove include: include:

1. (Vu, v, w) (Fmla(u) 1\ 1. (Vu, v, w)(Fmla(u) A Term(v) Term(v) J D Fmla(Sub(u, Fmla(Sub(u, v))) v))).. 2. (VrA7)(VrB7)( Thrnc(rA 7) 1\ A Thmr(r ThmT(rA D B') B 7) J D Thmr(rB')). ThmT(rS7)). AJ AI) AI) (vrB') ( Thmr(r 2. (vr

3. (VU) (ProO!T(U) J Thms� (Proo!T(yJ' )). 3. (VrA~)(Vu)(PrfT(U, D Thm Thms~ (rprf(u, A')')). Pr!(g, rrA7)7)). 4. (vr A') (Vu) (Prfr(u, rrAt) AI) J s� ( ' 5. A')) ( Thmr(AI) J Thms� (Thmr(AI) )). 5. (vr (WA

These These five five formulas formulas require require some some explanation. explanation. The The first first just just states states that that when when a a term term is is substituted substituted into into a a formula, formula, a a formula formula is is obtained. obtained. The The second second codes codes the the fact fact that that the ponens. Si the consequences consequences of of the the sequent sequent calculus calculus are are closed closed under under modus modus ponens. S~ proves proves this this by by the the simple simple argument argument that that if if there there are are sequent sequent calculus calculus proofs proofs of of � --~ A A and and � calculus proof --~ A A J D B B,, then then these these can can be be combined combined with with the the simple simple sequent sequent calculus proof of of �B using two cuts to to obtain proof of A, A A, AJ DB B--+ B using two cuts obtain a a sequent sequent calculus calculus proof of B B.. The The third third formula formula states states that that any any u u which which encodes encodes a a T-proof T-proof can can in in fact fact be be proved proved to to be be aa T -proof. The T-proof. The intuitive intuitive idea idea behind behind the the fact fact that that Si S~ can can prove prove this this formula formula is is 12Even lT(u) is 12Even for for decidable decidable theories theories T T,, if if T .It ~ (IIx)(IIy)(x (Vx)(Vy)(x = y) y),, then then the the predicate predicate ThfT ThmT(u) is PSPACE-hard. PSPACE-hard. =

of Arithmetic Arithmetic Proof Theory of

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encoding aa proof proof as as aa string string of symbols, it is possible possible to to construct construct aa that, given given uu encoding that, of symbols, it is S� -proof of of the the statement statement PrOOfT(U Proofr(y,)) that that uu codes codes aa T-proof. T-proof. In In fact, fact, the the S~-proof S� -proof S~-proof of of PrOOfT(U Proofr(y')) ,, proceeds proceeds by by verifying verifying that that u, u , viewed viewed as as aa string string of of symbols, symbols, satisfies satisfies all the all the properties properties of of being being aa valid valid T-proof. T -proof. The The provability provability of of the the fourth fourth and and fifth fifth formulas in in S~ S� is is similar similar to to the the provability provability of of the the third. third. formulas The fact fact that can prove prove the the third third formula formula is is aa special of aa more general The that S� S~ can special case case of more general fact: fact: T h e o r e m . (Buss (Buss [1986,Thm be aa E~ the variable Theorem. [1986,Thm 7.4]) 7.4]) Let Let A(b) A(b) be I:t -formula -formula with with only only the variable bb free. Then, Then, S~ S� can can prove prove free.

(Vu)(A(u) (A(yJ')) . (\iu)(A(u) D :J Thms~ Thms� (rA(u_)7)).

Hilbert-Bernays-Lob The following following three three derivability derivability H i l b e r t - B e r n a y s - L S b dderivability e r i v a b i l i t y cconditions. o n d i t i o n s . The conditions, introduced introduced by by Hilbert Hilbert and and Bernays Bernays [1934-39] [1934-39] and and L6b Lob [1955], [1955], give give suffisuffi conditions, cient conditions on on an an arithmetization arithmetization for the second second incompleteness theorem to to hold hold cient conditions for the incompleteness theorem for T, with with respect respect to to aa given given formalization formalization Thrrvr of provability: provability: for aa theory theory T, ThmT of HBLI: For all all A B,, TT ~f- ThmT(rA ~) /\ A Thrrvr( ThmT(rA D B') B ~) :J D ThmT(rB~). Thrrvr (rA') A :J A and and B Thrrvr(rB') . HBL1: For Thrrvr(A') . HBL2: For A,, if if T T Fthen T HBL2: For all all A f- A, A , then T Ff- ThmT(rA~). :J Thrrvr( T~)) ).) . HBL3: For all all A, A , TT Ff- Thrrvr(r A') HBL3: For ThmT(FA ~) D ThmT( Thrrvr( ThmT(rA

Assuming T2 Si , the the first first and third conditions follow from the fact fact that that formulas Assuming T 2 S~, and third conditions follow from the formulas and 55 above above are provable in in S~. Si . The The second the fact formula 55 22 and are provable second condition condition is is the fact that that formula is true; which which of of course course is an immediate immediate consequence consequence of formula 55 above is true; is an of fact fact that that formula above is is provable provable in in Si S 1 and and hence hence is is true. true. 2.1.3. So 2.1.3. Arithmetization A r i t h m e t i z a t i o n of of syntax s y n t a x for for non-schematic n o n - s c h e m a t i c theories. theories. So far, far, we we have considered is not have considered only only schematically schematically axiomatized axiomatized theories. theories. This This is not unreason unreasonable, arithmetic are able, since since many many of of the the theories theories we we are are interested interested in, in, such such as as Peano Peano arithmetic are schematically theories such etc. are schematically axiomatized. axiomatized. Many Many other other theories such as as S� S~,, 1,6, IA0, /El, are 1 , etc. 0 , II: not schematic but not schematic but are are at at least least nearly nearly schematic schematic in in that that they they are are axiomatized axiomatized by by aa finite substitution restricted finite set set of of schemes schemes with with substitution restricted to to certain certain formula formula classes. classes. The The metamathematics metamathematics for for these these latter latter theories theories can can be be arithmetized arithmetized with with only only a a slight slight modification modification of of the the above above methods. methods. A A theory theory is is said said to to be be axiomatizable axiomatizable provided provided that that it it has has aa recursive recursive (i.e., (i.e., decid decid's this able) able) set set of of axioms. axioms. By By aa theorem theorem of of Craig Craig's this is is equivalent equivalent to to having having aa recursively recursively enumerable enumerable set set of of axioms. axioms. In In general, general, there there are are many many axiomatizable axiomatizable theories theories which which are schematic; nonetheless, arithmetization of are not not schematic; nonetheless, the the arithmetization of metamathematics metamathematics can can be be modified modified to to apply apply to to any any axiomatizable axiomatizable theory theory as as follows. follows. Let Let T T be be an an axiomatizable axiomatizable theory. theory. Since Since the the predicate predicate Validlnferencer ValidInferenceT may may no no -definable in longer longer be be polynomial-time, polynomial-time, it it may may not not be be ,6,t A~-definable in Si S~.. However, However, it it is is possible possible to v, w to express express Validlnferencer(u, ValidInference~(u, v, w)) in in equivalent equivalent form form as as

(:3a) a, u, ), (3a) ValidlnfEvidence( ValidlnfEvidence(a, u, v, v, w w),

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s S..

Buss

13 With where ValidlnfEvidence ValidInfEvidence is is �� A~ w.r.t. w.r.t. Si S~ ..13 With this, this, aa T-proof is then then coded coded as as T-proof is where sequence containing containing the the lines lines of of the the proof, proof, plus plus any any necessary necessary evidence evidence values, values, aa,, aa sequence justifying justifying the the steps steps iinn the the proof. proof. IInn this this way, way, the the predicates predicates Proof ProofT(w and Prfr(w, PrfT(W , uu)) T (W)) and are -definable in are �t A~-definable in Si S~;; likewise, likewise, Thrrltr Thor is is definable definable from from Prf PrfTT as as before. before. It 1-5 are It is is easy easy to to check check that that formulas formulas 1-5 are still still provable provable in in Si S~.. Also, Also, if if T T ;2 _~ Si S~,, the the Hilbert derivability Hilbert derivability conditions conditions hold hold as as well. well.

2.1.4. All 2.1.4. Arithmetization A r i t h m e t i z a t i o n in in theories t h e o r i e s which w h i c h contain c o n t a i n only only Q Q.. All our our proofs proofs of incompleteness of incompleteness theorems theorems will will assume assume that that the the theory theory T T under under consideration consideration con contains all hold tains Si S~.. However, However, the the results results all hold as as well well for for theories theories which which only only contain contain Q Q.. The The intensional intensional arithmetization arithmetization in in Si S~ can can be be extended extended to to an an intensional intensional arithmetization arithmetization in Q Q based based on on the the fact fact that that Si S~ is is interpretable interpretable in in Q Q.. The The interpretation interpretation of of Si S~ in in Q Q in is aa very very special special kind kind based based on on an an ind inductive cut J J;; namely, namely, there there is is aa formula formula J J(a) is uctive cut ( a) such such that that Q Q proves proves J J is is closed closed downwards downwards and and is is closed closed under under 00,, S S,, + + ,, . 9and and # # . . In In J addition, for for 4> r any any sentence, sentence, the the sentence sentence 4> CJ,, 4> r relativized relativized by by J J,, is is obtained obtained from from 4> r addition, by (Qx. J (x )) . Then, Then, we by replacing replacing every every quantifier quantifier (Qx) with with (Qx.J(x)). we have have that that Q Q f~- 4> CJJ for for all theorems 4>r of all theorems of Si S~.. The The first first use use of of inductive inductive cuts cuts for for interpretations interpretations was was by by Solovay. Solovay. The The fact fact that that IA0 can can be interpreted in in Q Q was was first first discovered discovered by by Wilkie; Wilkie; aa local I�o be interpreted local interpretation interpretation of I�o IA0 in in Q Q was was independently discovered by by E. E. Nelson. Nelson. For For more more details details of of inductive inductive independently discovered of below, or Pudlak [1983], cuts cuts and and this this interpretation interpretation of of Si S~ in in Q Q see see Theorem Theorem 4.3.3 4.3.3 below, or Pudl~k [1983], Nelson Nelson [1986] [1986] or or Chapter Chapter VIII VIII of of this this volume. volume. Pudlak Pudls [1983] [1983] gives gives aa very very general general form form of of the the interpretation interpretation of of I�o IA0 in in Q Q.. An An intensional intensional arithmetization arithmetization of of metamathematics metamathematics can can be be given given in in Q Q by by replac replacing (W)V .. The ing predicates predicates such such as as Proof(w) Proof(w) with with their their relativizations relativizations J(w) J(w) 1\ A (ProOf (PrOOfT(w)) J The T reader reader can can check check that that the the proofs proofs given given in in the the next next sections sections all all still still work work with with these these relativized predicates. relativized predicates. 2.2. 2.2. The T h e Godel G h d e l incompleteness i n c o m p l e t e n e s s theorems theorems

In this section, diagonal, or fixpoint, lemma, lemma, the In this section, we we discuss discuss the the diagonal, or fixpoint, the first first and and second second incompleteness theorem. incompleteness theorems, theorems, and and Lob's Lhb's theorem. 2.2.1. The fixpoint, lemma 2.2.1. The T h e Godel G h d e l diagonal d i a g o n a l lemma. lemma. The Godel Ghdel diagonal, diagonal, or or fixpoint, lemma is is aa crucial crucial ingredient ingredient in in the the proof proof of of the the incompleteness incompleteness theorems. theorems. This This lemma lemma states states that, that, for for any any first-order first-order property property A A,, there there is is aa formula formula B B that that states states that that the the property property A A holds holds of of the the Godel Ghdel number number of of B B.. Thus, Thus, since since we we know know that that provability provability is is aa first-order will be first-order property, property, it it will be possible possible to to construct construct aa formula formula which which asserts asserts "I "I am am not not provable" provable".. 130ur 13Our formulation formulation works works for any decidable set of axioms axioms and rules rules of inference; inference; we do require always always that all the usual usual logical logical axioms axioms and rules rules of inference inference are present. present. A similar similar construction will will work work for inference inference rules with any finite finite number number of hypotheses.

Proof Proof Theory Theory of of Arithmetic Arithmetic

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Godel's Ghdel's Diagonal Diagonal Lemma. L e m m a . Let Let A(ao) be be aa formula. formula. Then Then there there is is aa formula formula B B

such such that that Si S~ proves proves

+-~ ArB'). A(rB-~). BH

Furthermore, Furthermore, if if A A is is aa �� E~,, II� II~,, � Eii or or IIi Hi formula formula (respectively), (respectively), then then so so is is B B;; and and if if A A involves involves free free variables variables other other than than ao Co,, then then so so does does B B.. Proof. This This proof proof quite quite simple simple but but rather rather tricky tricky and and difficult difficult to to conceptualize. conceptualize. We We Proof. fifirst rst define define aa diagonalization diagonalization function function f f which which satisfies satisfies

for all all formulas formulas e C,, where where erG') C(rC 7) means means erG'lao). C(rC~/ao). To To define define f f , , recall recall that that the the for um(n) = 1lcI is -definable in function function n n H ~ N Num(n) - rrn_n is �t E~-definable in Si S~.. Then Then the the function function f f is is �t -definable by E~-definable by Si S~ since since

Sub(x, Num(x)) Num(x) ).. ff (x) = Sub(x, Next, Next, we we wish wish to to let let the the formula formula e(ao) C(ao) bbee A(J(ao)) A(f(ao));; however, however, since since f f is is not not aa function symbol in the the language language of of Si S 1 , we we must must be be more more careful careful in in defining C.. Let Let function symbol in defining e f (a, b) -defined with (a) = = bb be be �t E~-defined with the the formula formula G G/(a, b) which which defines defines the the graph graph of of f f and and ff(a) (a) :::; f . Now let let tt f/ bbe e aa term term such such that that Si S~ proves proves f f(a) <_tt/. Now the the formula formula e C can can be be taken taken to to be be either either (3x < tf)(Gf(ao, x) A A(x))

or or

(Vx <_ t/)(G/(ao, x) D A(x)).

(With aa little little more more care, care, we we can can choose choose e C to to be be in in the the same same quantifier quantifier complexity complexity (With class as as A.) Finally, Finally, define define B to to be the formula formula erG') C(rCT).. class be the We We claim claim that that Si S 1 proves proves B H ++ A(rB') A(rBT).. The The proof proof of of this this claim claim is is almost almost ') ; immediate. First, by the the definitions of f f and and B B,, we we have have rB' rB7 is is equal equal to to frG f(rCT); immediate. First, by definitions of of Second, by definition of -provably of course course Si S~ proves proves this this fact. fact. Second, by the the definition of e C,, B is is Si S~-provably -provably equivalent equivalent to to A(J A(f(rCT)). Therefore, B is is Si S~-provably equivalent to to A(rB') A(rBT).. rG')) . Therefore, equivalent Q.E.D. Q.E.D.

' s first 2.2.2. Godel 2.2.2. The T h e first first incompleteness i n c o m p l e t e n e s s theorem. theorem. Ghdel's first incompleteness incompleteness theorem theorem states states that that there there is is no no complete, complete, axiomatizable, axiomatizable, consistent consistent theory theory T T ex extending shall prove this in this section. tending Q Q.. We We shall prove several several variants variants of of this in this section. Definition. CO'Tlrr ilt -formula Definition. Let Let T T be be an an axiomatizable axiomatizable theory. theory. Cor~ is is the the V VA~-formula ....,-~Thmr(O ThmT(O #~ 0) 0) which which expresses expresses the the condition condition "T "T is is consistent." consistent." T is is said said to to be be w-consistent w-consistent if if there there does does not not exist exist aa formula formula B(a) such such that that T that T ,B(1lc) for T T IF- (3x)B(x) (3x)B(x) and and such such that T IF- ....-~B(n) for all all n n � _ O0.. T T 2 _3 Si S~ is is weakly weakly w -consistent provided no such formula B which which is ilt W.r.t. w-consistent provided there there is is no such formula is A~ w.r.t. Si S~.. Since Since every every -sentence is provable in weakly w-consistent and only true ilt true A~-sentence is provable in SJ S~,, T T is is weakly w-consistent if if and only if if T T is is consistent and proves only only true 3ilt -sentences. consistent and proves true 3A~-sentences.

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Godel's G6del's First First Incompleteness Incompleteness Theorem. T h e o r e m . Let Let T T be be an an consistent, consistent, axiomatizable axiomatizable theory theory containing containing Q Q.. Then Then there there is is aa true true sentence sentence ifJ r such such that that T T }.! }z ifJ r . Further, Further, if if T is weakly -consistent, then T is weakly w w-consistent, then T T }'! }z -.ifJ -1r . The formula formula ifJ r is is explicitly explicitly constructible from the the axiomatization axiomatization of of T T.. The constructible from

' ll prove Proof. Proof. We We'll prove this this theorem theorem under under the the assumption assumption that that T T ;2 _D S� S~.. Choose Choose ifJ r to to formula such such that that aa formula S~ F- r ~-~ThmT(rr

Intuitively, "I am Intuitively, the the formula formula ifJ r is is asserting asserting "I am not not provable provable in in T" T";; the the Diagonal Diagonal Lemma Lemma 2.2.1 2.2.1 guarantees guarantees that that ifJ r exists. exists. 's show Then, by First, let First, let's show that that ifJ r is is true. true. Suppose Suppose ifJ r were were false. false. Then, by the the choice choice of of ifJ r and since since S� S~ is is aa true true theory, theory, T T It-- ifJ. r Therefore, Therefore, Si S 1 IF- Th'TTl ThmT(rr and again, again, by by the the and lr (ifJ') ;; and choice hich contradicts choice of of ifJ r , Si S~ Ib -.ifJ. -~r Since Since T T ;2 2 S� S~,, we we also also have have T T Ib -.ifJ -~r � which contradicts the the consistency of consistency of T T.. So So ifJ r cannot cannot be be false. false. Second, Second, we we show show that that T T }'! V rifJ. Suppose, Suppose, for for sake sake of of aa contradiction, contradiction, T T IF- rifJ. Then Then S~ IF- Th'TTl ThmT(rr By choice choice of of rifJ, S� S~ IF -~r So also also T T IF- -.ifJ, -~r which which again again contradicts contradicts S� lr (ifJ' ) . By -.ifJ. So the the consistency consistency of of T T.. Third, Third, we we assume assume T T is is weakly weakly w-consistent w-consistent and and prove prove that that T T }'! jz -.ifJ 9r . Suppose Suppose T lr (ifJ' ) . But T does does prove prove -.ifJ. 9r Then Then since since T T ;2 _D Si S~ and and by by choice choice of of ifJ r, T T IF Th'TTl ThmT(rr But ThmT(rr is aa false false :l.�t 3A~-sentence, which contradicts contradicts the the weak weak w-consistency w-consistency of of T T.. Th'TTl lr (ifJ'7)) is -sentence, which Q.E.D. Q.E.D. The The obvious obvious question question at at this this point point is is whether whether the the hypothesis hypothesis of of weak weak w-consistency w-consistency can can be be removed removed from from the the First First Incompleteness Incompleteness Theorem; Theorem; i.e., i.e., whether whether there there is is aa consistent, axiomatizable,, axiomatizable,' complete consistent, complete theory theory extending extending Q Q.. It It turns turns out out that that this this hypothesis hypothesis can can be be removed: removed: Rosser's Rosser's Theorem. T h e o r e m . (Rosser (Rosser [1936]) [1936]) There There is is no no consistent, consistent, axiomatizable, axiomatizable, complete complete theory theory T T ;2 D_ Q Q..

The The proof proof of of this this theorem theorem will will give give aa constructive constructive method method of of obtaining obtaining aa formula formula ifJ r from independent of from the the axiomatization axiomatization of of aa consistent consistent theory theory T T such such that that ifJ r is is independent of T T.. Proof. Proof. As As before, before, we we give give the the proof proof assuming assuming T T ;2 2 Si S~.. We We need need to to define define aa modified modified notion called "Rosser notion of of provability provability called "Rosser provability" provability".. Let Let Neg Neg be be the the unary unary function, function, t -definable in � E~-definable in SJ S~,, such such that that Neg(rA') Neg(rA 7) = -- r-.A' r-~A7 for for all all formulas formulas A A.. Then Then we we define define the the predicate predicate R-Prf R-PrfT(w a) as as T (w,, a)

R-Prf (-.PrfT (v, Neg(a))). T (w,, a) R-PrfT(W a) {::> r Prf Pr.fT(w a) 1\ A ('
of Arithmetic Proof Theory of

121 121

14 The is true true exactly exactly when when Prf PrfT(n is; however, however, this this fact fact is is not not provable provable in in Si S~ ..14 The is A') is; T (n,, rrA'~) Thmr(u) is -formula (3w) (R-PrfT (w,, u)) predicate predicate RR-Thm,r(u) is defined defined to to be be the the 3IIt 3II~-formula (3w)(R-PrfT(W u)).. Now use use the the Diagonal Diagonal Lemma Lemma to to choose choose aa sentence sentence r so so that that Now

S� f- ++ -,R- Thm(') . As before, before, we we have have r is is true. true. In In fact, fact, the the same same proof proof works works as as iinn the the proof proof of of the the First First As Incompleteness Theorem, Theorem, since since Thmr ThmT and and RR-ThmT are extensionally extensionally equivalent. equivalent. Incompleteness Thmr are Secondly, T T }£ tz r again again by by the the same same argument argument as as in in the the previous previous theorem; theorem; in in brief: brief: if if Secondly, T fk r , then then r is is false false by by choice of , r but but we we just just claimed claimed r is is true. true. choice of T Thirdly, we we want want to to show show T T y: jz -,. 9r Suppose T T does does prove prove -' 9r ; let let n n be be aa Thirdly, Suppose Ghdel number number of of aa T-proof T-proof of of -, 9r . Since Since T T is is consistent, consistent, we we have have that that there there is is no no G6del T-proof of of r; thus thus SJ S~ proves proves the the true true �o-sentence Ao-sentence ("Iv (Vv :::; _< n)-,Pr!r(v, n)gPrfT(V , rrr,) . And And T-proof since since Prf PrfT(n Neg(r ) is is true, true, Si $21 also also proves proves (Vv)(v (Vv)(v > >n n :J D -,R-Prf 9R-PrfT(v,rr Hence, T (v, r4)f Hence, T (n,, Neg( r and and since since T T 2 _DSi S~,, this this implies implies T T fk r, Si Thmr(' ) . By which contradicts the consistency of T T.. which contradicts the consistency of Q.E.D. Q.E.D.

's second 2.2.3. T h e second second incompleteness incompleteness theorem. theorem. Ghdel's second incompleteness incompleteness 2.2.3. The G6del theorem improves improves on on his his first first incompleteness incompleteness theorem theorem by by giving giving an an example example of of aa true true theorem formula with with an an intuitive intuitive meaning meaning which which is is not not provable provable by by aa decidable, consistent formula decidable, consistent theory T T.. This This formula formula is is the the formula formula COTVr ConT which which expresses expresses the the consistency consistency of of T T.. theory Note, however, however, that unlike the formula r iinn the the first first incompleteness incompleteness theorem, theorem, Note, that unlike the the the formula ConT is is not not necessarily necessarily independent independent of of T T since since there there are are consistent consistent theories theories that that CoTVr prove their their own own inconsistency. inconsistency. An An example example of of such such aa theory theory is is T T + -, 9ConT. C0TVr . prove G6del's n c o m p l e t e n e s s Theorem. T h e o r e m . Let Let T T be Godel's Second Second IIncompleteness be aa decidable, decidable, consistent consistent theory suppose T T 2 D_Q. Then T Q . Then T/}£ ConT. COTVr . theory and and suppose

Proof. As usual, we T2 be the the formula formula from the proof proof of of Ghdel's G6del ' s Proof. As usual, we assume assume T _DSJ S~.. Let Let r be from the First which is S~-provably equivalent to to -~ThmT(rr -, Thmr(') . First Incompleteness Incompleteness Theorem Theorem which is Si -provably equivalent Recall that that we we proved }£ r . We shall prove that Si will Recall proved T T Jz We shall prove that S~ proves proves COTVr ConT D which--will :J r ; which suffice to show show that that T since T suffice to T /y: ConT, COTVr , since T _D 2 S~. Si . By choice of r , Si S~ proves proves 9r by the in secBy choice of -' D :J ThmT(rr Thmr(' ) . Also, Also, by the formula formula 55 in sec tion S~ proves ThmT(rr 7) Also, by tion 2.1.2, 2.1.2, SJ proves Thmr( ') D Thmr(Thmr(')') . Also, by choice choice of of r :J ThmT(rThmT(rr and by formula 11 of of section S~ proves proves Thmr(Thmr(£) ThmT(rThmT(rr '-~)) D and by formula section 2.1.2, 2.1.2, Si Thmr(-'') . :J ThmT(r-~r Putting Putting these these together together shows shows that that S~ Si proves proves that that 9r -' D :J [ThmT(rr [Thmr(7) ') A Thmr( r-,')]. 1\ ThmT(~r

From whence whence 9r -' D :J -~ConT -,COTVr is easily proved. proved. Therefore, Therefore, S~ Si proves proves ConT COTVr D 0 From is easily :J r . [:] 14This example (see definition which 14This is a good good example (see Feferman Feferman [1960]) [1960]) of an extensional extensional definition which is not an provide an extensionally intensional definition. T , R-Pr]T R-Prfr and R-ThmT provide definition. For consistent consistent theories theories T, correct definition provability, since R-PrST(n definition of provability, R-Prfr (n,, rAT) rA') has the correct truth value value for all particular n and rA7. rA'. However, However, they are not intensionally intensionally correct; correct; since, since, in general, general, T cannot cannot prove prove that R-ThmT(rA 7) and R-ThmT(rA 7) implies R-ThmT(rA') R-Thmr(rA D :::> B B') implies R-ThmT(B). R-Thmr(B) .

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The implies COTltr -equivalent to The formula formula r not not only only implies ConT,, but but is is actually actually Si S~-equivalent to COTltr ConT.. For this, note that that since since r implies implies ..., ~ThmT(rr it can can be be proved proved in in Si S 1 that that r For this, note Thmr(' ) , it implies implies ..., -~Thmr(ro ThmT(q) = - P) 1~).. (Since (Since if if aa contradiction contradiction is is provable, provable, then then every every formula formula is is provable.) provable.) 2.2.4. The 2.2.4. Lob's L6b's theorem. theorem. The self-referential self-referential formula formula constructed constructed for for the the proof proof of of the the First First and and Second Second Incompleteness Incompleteness Theorems Theorems asserted asserted "I "I am am not not provable" provable".. A A related problem would related problem would be be to to consider consider formulas formulas which which assert assert "I "I am provable" provable".. As As the the next next theorem theorem shows, shows, such such formulas formulas are are necessarily necessarily provable. provable. In In fact, fact, if if aa formula formula is is implied implied by by its its provability, provability, then then the the formula formula is is already already provable. provable. This This gives gives aa strengthening of Theorem, which that, in strengthening of the the Second Second Incompleteness Incompleteness Theorem, which implies implies that, in order order A , one helped by to to prove prove aa formula formula A, one is is not not substantially substantially helped by the the assuming assuming that that A is is provable. More More precisely, precisely, the the assumption assumption Thmr(r ThmT(rA 7) will will not not significantly significantly aid aid aa A') provable. theory theory T T in in proving proving A A.. Lob's axiomatizable theory and A L6b's Theorem. T h e o r e m . Let T T 2 D Q be aann axiomatizable A be any sentence. sentence. If If T T proves Thmr(rA') ThmT(rA "1) J DA A,, then T T proves A A.. Proof. u { ...,A} . Proof. As As usual, usual, we we assume assume T T 2 _DSi S~.. Let Let T' be be the the axiomatizable axiomatizable theory theory T TU{-~A}. The Lob's Theorem The proof proof of of Lhb's Theorem uses uses the the fact fact that that T' T ~ is is consistent consistent if if and and only only if if T T }L ]z A A;; and ThmTfA') . From and furthermore, furthermore, that that Si S 1 proves proves Con(T') Con(T') is is equivalent equivalent to to ..., -~ThmT(rAT). From these immediate from second incompleteness these considerations, considerations, the the proof proof is is almost almost immediate from the the second incompleteness AJ theorem. theorem. Namely, Namely, since since T T proves proves ..., ~A D ..., -~ThmrfA') ThmT(rA 7) by by choice choice of of A A,, T T also also proves proves ...~A ,A J Deduction Theorem, D Con(T') Con(T').. Therefore, Therefore, by by the the Deduction Theorem, T' T' fF- Con(T') Con(T') so so by by Godel's Ghdel's Second Second Incompleteness Incompleteness Theorem, Theorem, T' T ~ is is inconsistent, inconsistent, i.e., i.e., T T fkA A.. 2.2.5. The 2.2.5. Further F u r t h e r reading. reading. The above above material material gives gives only only an an introduction introduction to to the the incompleteness incompleteness theorems. theorems. Other Other significant significant aspects aspects of of incompleteness incompleteness include: include: (1) (1) the the strength strength of of reflection reflection principles principles which which state state that that the the provability provability of of aa formula formula implies implies (2) provability the the truth truth of of the the formula, formula, see, see, e.g., e.g., Smorynski Smorynski [1977]; [1977]; (2) provability and and interpretabil interpretability Boolos [1993], ity logics, logics, for for which which see see Boolos [1993], Lindstrom Lindstrhm [1997] [1997],, and and Chapter Chapter VII VII of of this this handbook; and (3) concrete, handbook; and (3) concrete, combinatorial combinatorial examples examples of of independence independence statements, statements, such such as as the the Ramsey Ramsey theorems theorems shown shown by by Paris Paris and and Harrington Harrington [1977] [1977] to to be be independent independent of of Peano Peano arithmetic. arithmetic. 3. O On 3. n tthe h e sstrengths t r e n g t h s of o f fragments f r a g m e n t s of o f arithmetic arithmetic

3.1. 3.1. Witnessing W i t n e s s i n g theorems theorems

In .2.10, it 1In section section 11.2.10, it was was shown shown that that every every primitive primitive recursive recursive function function is is � ~ldefinable D:1 . We definable by by the the theory theory/~1. We shall shall next next establish establish the the converse converse which which implies implies that that the �1 -definable functions the ~l-definable functions of o f / D:l ~ 1 are are precisely precisely the the primitive primitive recursive recursive functions. functions. The D:l provides The principal principal method method of of proof proof is is the the 'witnessing 'witnessing theorem theorem method': method': 1511 provides the the simplest simplest and and most most natural natural application application of of the the witnessing witnessing method. method.

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3.1.1. ((Parsons Parsons [1970] 3.1.1. Theorem. Theorem. [1970],, Mints Mints [1973] [19731 and and Takeuti Takeuti [1987]). [1987]). -definable function �l E1-definable function of of rr: IE11 is is primitive primitive recursive. recursive.

Every Every

Parsons G6del Dialectica Dialectica theorem Parsons'' proof proof of of this this theorem theorem was was based based on on the the Ghdel theorem and and aa 's similar proof volume. Takeuti similar proof is is given given by by Avigad Avigad and and Feferman Feferman in in Chapter Chapter V V in in this this volume. Takeuti's proof ordinals to proof was was based based on on aa Gentzen-style Gentzen-style assignment assignment of of ordinals to proofs. proofs. Mints's Mints's proof proof was was essentially essentially the the same same as as the the witness witness function function proof proof presented presented next; next; except except his his proof proof was was presented presented with with a a functional functional language. language. 3.1.2. For -formula A(b) 3.1.2. The T h e Witness W i t n e s s predicate p r e d i c a t e for for �l E l --formulas. formulas. For each each �l El-formula A(b),, we -formula WitnessA we define define a a boo A0-formula WitnessA (w, (w, b) b) which which states states that that w w is is aa witness witness for for the the truth truth of of A A..

. . . ,,x~)B(Xl,... Definition. Definition. Let Let A(b) A(b) bbee aa formula formula of of the the form form (3Xb (3Xl,... where xk)B(Xl, . . . ,,x~,b), Xk, b) , where B is is a a boo-formula. A0-formula. Then Then the the formula formula WitnessA(w, WitnessA(w, b) b) is is is is defined defined to to be be the the formula formula B -.#

B(/~(1, w), .. .. .. ,, f3( /~(k, w), b) b).. B(f3( k , w), l , w), If bo A = = A',A is aa succedent, then Witness Witnessva(w,~ is defined defined to to be be If bo', A is succedent, then V 11 (w, C) is

WitnessA(f3(l, WitnessA(/~(1, w), w), C) c--)V V Witness WitnessvA,(~(2, w), C). c-]. V l1' (13(2, w), ned similarly Dually, Dually, if if r F = = A, A, r' F' is is an an antecedent, antecedent, then then Witnessl\r Witness Ar is is defi defined similarly as as

WitnessA (f3(l, w), (f3(2, w), WitnessA(~(1, w), C) ~ 1\ A Witnessl\r' WitnessAr,(~(2, w), C). ~. Note Note the the different different conventions conventions on on ordering ordering disjunctions disjunctions and and conjunctions; conjunctions; these these are are not not intrinsically intrinsically important, important, but but merely merely reflect reflect the the conventions conventions for for the the sequent sequent calculus calculus are are that that active active formulas formulas of of strong strong inferences inferences are are at at the the beginning beginning of of an an antecedent antecedent and at at the the end end of of a a succedent. succedent. and It It is, is, of of course, course, obvious obvious that that WitnessA WitnessA is is a a boo-formula, A0-formula, and and that that lboo IA 0 can can prove prove

A(b) (w, b) A(b) f-+ ~ (3w) (3w) WitnessA WitnessA(w, b).. 3.1.3. ((Sketch Sketch of Suppose rr: 3.1.3. Proof. Proof. of the the proof proof of of Theorem Theorem 3.1.1.) 3.1.1.) Suppose PE11 proves proves calculus proof (Vx)(3y)A(x, y) where where A A E E �l El.. Then Then there there is is aa sequent sequent calculus proof P P in in the the (3y)A(x, y) (V'x) theory theory rr: /El1 of of the the sequent sequent (3y)A(c, (3y)A(c, y) y).. We We must must prove prove that that there there is is a a primitive primitive (n)) is such that that A(n, the standard all recursive A( n , ff(n)) recursive function function f f such is true, true, in in the standard integers, integers, for for all nn 2 >__ o0.. In In fact, fact, we we shall shall prove prove more more than than this: this: we we will will prove prove that that there there is is a a -definition in rr: such that rr: proves primitive , primitive recursive recursive function function f f , , with with aa � El-definition in /El, such that /El proves l 1 1 (Vx)A(x, f(x)). . This corollary to to the the next next lemma. lemma. x)A(x, f(x)) This will will be be aa corollary (V' Witnessing W i t n e s s i n g Lemma L e m m a for f o r /rr: E l1. • Let Let r F and and bo A be be cedents cedents of of �l E1 -formulas -formulas and and suppose suppose ---t bo Then there rr: IE11 proves proves the the sequent sequent r F---+ A.. Then there is is aa function function hh such such that that the the following following hold: hold:

124 124

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n:: 1 and is primitive primitive recursive, and (1) h is E1-defined �l -defined by IE1 (1) proves (2) n:: (2) /El1

(VCj (Vw)[ Witnessl\ r ( w, Cj

::)

Witness V Do (h( w, Cj , Cj].

Note that that Theorem Theorem 3.1.1 3.1.1 isis an an immediate immediate corollary corollary to to the the lemma, lemma, since since we we may may take take Note L\ to to be be the the empty empty sequent, sequent, A to be be the the sequent sequent containing containing just just (3y)A(c, (:3y)A(c, y), y) , and and let let Fr to f(x) == ~(1, ,8(1, ~(1, ,8(1, h(x))) h(x))) where where hh isis the the function function guaranteed guaranteed to to exist exist by by the the lemma. lemma. f(x) h(x) will will be be aa sequence sequence of of length length one one witnessing witnessing the the cedent (:3y)A, This isis because because h(x) This cedent (3y)A, so its first first and and only only element element is witness for for the the formula formula (3y)A, (:3y)A , and and the the first first element element so its is aa witness of that that is is aa value value for for yy that that makes makes A A true. true. of It remains remains to to prove prove the the Witnessing Witnessing Lemma. Lemma. For For this, this, we we know know by by the the Cut Cut It Elimination Theorem Theorem 1.4.2, 1.4.2, that that there there is is aa free-cut free-cut free free proof proof PP of of the the sequent sequent Elimination r ---+ A L\ in in the the theory theory /El; n::1 ; in in this this proof, proof, every every formula formula in in every every sequent sequent can can be be F--+ assumed to be aa El-formula. �l -formula. Therefore, Therefore, we may prove Witnessing Lemma Lemma by by assumed to be we may prove the the Witnessing P. induction on on the number of of steps steps in in the proof P. induction the number the proof The case iiss where where there are zero zero inferences inferences iinn the ---+ L\ The base base case there are the proof proof P P and and ssoo r F---} A is an an initial initial sequent. sequent. Since Since the the initial initial sequents allowed in in aann /n:: proof contain contain only only is sequents allowed E l 1 proof atomic formulas, the the Witnessing Witnessing Lemma Lemma is is trivial case. atomic formulas, trivial for for this this case. For induction step, final For the the induction step, the the argument argument splits splits into into cases, cases, depending depending on on the the final inference inference of of the the proof. proof. There There are are aa large large number number of of cases, cases, one one for for each each inference inference rule rule of of the the sequent sequent calculus; calculus; for for brevity, brevity, we we present present only only three three cases cases below below and and leave leave the the rest rest for for the the reader. reader. For nal inference For the the first first case, case, suppose suppose the the fi final inference of of the the proof proof P P is is an an :3 3 :right inference, inference, namely, namely,

. ... . . rF---} ---+ L\, A, A(t) rF---} ---+ L\ A,, (:3x)A(x) (3x)A(x) Let Let cc be be the the free free variables variables in in the the upper upper sequent. sequent. The The induction induction hypothesis hypothesis gives gives aa � El-defined, primitive recursive recursive function function g(w, Cj ~ such such that t h a t /n:: E l1 proves proves l -defined, primitive .

.

9

Witnessl\ w, Cj, . r ( w, Cj WitnessAr(W, ~ ---+ ~ Witness Witnessv{A,A(t)} (g(w, ~, Cj C-"). V{ Do,A{t)} (g(

IInn order w, Cj ,8(2, g(w, Cj order for for Witness Witnessv{A,A(t)} (g(w, c-),, Cj c-) to to hold, hold, either either/?(2, ~)) witnesses witnesses V V L\ A V{Do ,A{t)} (g( or or ,8(l, ~(1, g(w, Cj) c-)) witnesses witnesses A(t) A(t).. So So letting letting h(w, Cj c-*)be be � El-defined by l -defined by

h(w, Cj~

= =

((t(Cj) ((t(~) *9,8(1, ~(1, g(w, Cj), ~), ,8(2, /3(2,g(w, Cj», ~)),

where where *9 denotes denotes sequence sequence concatenation. concatenation. It It is is immediate immediate from from the the definition definition of of Witness that that

w, Cj , Cj. (h(w, Witnessl\ r ( w, Cj ---+ Witness V{ Do,( 3x)A(x)} (h( For :left For the the second second case, case, suppose suppose the the final final inference inference of of the the proof proof P P is is an an :33:left inference, inference, namely, namely,

Proof Proof Theory Theory of of Arithmetic Arithmetic 9

.

125

o

A(b),, rF -+ ~ � A A(b) -+ � (3x)A(x) (3x)A(x), , r F---+ A

where where b is is an an eigenvariable eigenvariable which which occurs occurs only only as as indicated. indicated. The The induction induction hypothesis hypothesis g( w, C, gives gives us us aa � El-defined, primitive recursive recursive function function g(w, g, b) such such that t h a t /ll: E l1 proves proves I -defined, primitive Witnessl\ � (g(w,, C, WitnessA{A(b),r} (w, C) c-) -+ ---} Witnessv WitnessvA(g(w ~, b) b),, C) c-).. { A(b) ,r} (w,

-defined function WO , W Let w ) bbee the W I , .. .. .. , Wn Let tail( tail(w) the �I El-defined function so so that that tail( ((wo, wi,I , .. .. .. , Wn wn)))) = = ((wl, wn).) . Letting h(w, h(w, C) c') be be the the function function g((tail(,8(1 g((tail(~(1, , w)), w)),/~(2, w)),, c, ~,/?(1,/~(1, w))),, it it is is easy easy Letting ,8(2, w)) ,8(1 , ,8(1 , w))) to es the conditions of Witnessing Lemma. to check check that that h satisfi satisfies the desired desired conditions of the the Witnessing Lemma. For For the the third third case, case, suppose suppose the the final final inference inference of of P P is is aa � EI-IND inference: I -IND inference: 9

9 .

.

. 9

A(b), -+ �, A(b), r F---+ A, A(Sb) A(Sb) A(O) , r -+ �, A(t) A(0), F ~ A, A(t)

where where b is is the the eigenvariable eigenvariable and and does does not not occur occur in in the the lower lower sequent. sequent. The The induction induction hypothesis primitive recursive hypothesis gives gives aa � El-defined, recursive function function g(w, g(w, C, g, b) such such that that ll: PE11 I -defined, primitive proves proves Witnessl\ Witnessv{ �,A(Sb)} (g( w, C, WitnessA{A(b),r} g, b) -+ --+ Witnessv{A,A(Sb)} (g(w, ~, b) b),, c, ~, b) b).. { A(b) ,r} (w, C, Let ned as Let k(c, k(g, v, w) be be defi defined as

k(e, v, w) k(c,

{:

= { v

=

W

if if Witness Witnessv{ ~ }} (v, C) c') V{ � otherwise otherwise

Since ne the primitive Since Witness Witness is is aa �o-predicate, A0-predicate, kk is is � El-defined by ll: FE1. Now defi define the primitive I -defined by 1 . Now recursive recursive function function f f (w, c, g, b) by by ff(w, (w, c, g, 0) 0) = ff ((w, w , ~c,, bb++ 1) 1) =

(,8(l (fl(1,, w), w), O) 0) c, b) ) , (,8 (1, 9g((~(1, ( (,8 (1, ff(w,g,b)),~(2, ( w, C, b) ), ,8 (2, w)), (~(1, w)),~,b)), ))) ( (,8 (1, ff ((w, w, c, (2, w)), (2, 9g((/~(1, (c, ,8 (2, ff ((w, w, c, kk(g,/~(2, g, b) b)),) , ,8 /~(2, ~, b) b)),) , ,8 fl(2, w)), C, g, b) b))))

By By Theorem Theorem 1.2.10, 1.2.10, f f is is � E1I definable definable by b y /ll: E l1, , and and since since ff may may be be used used in in induction induction formulas, formulas, � E1I can can prove prove Witnessl\ + Witness WitnessA{A(o),r 5) -----} Witnessv{A,A(b)} (f (w, C, g, b) b),, c, g, b) b).. V{ �,A(b)} (f(w, { A(O) ,r }} (w, C)

with respect Setting h(w, (w, c, establishes the using using � EI-IND respect to to bb.. Setting h(w, C) ~ = -- ff(w, g, t) establishes the desired desired I -IND with conditions of conditions of the the Witnessing Witnessing Lemma. Lemma. 1.1. Q.E.D. Q.E.D. Witnessing Witnessing Lemma Lemma and and Theorem Theorem 3. 3.1.1. 3.1.4. Corollary. The 3.1.4. - definable predicates The �I A1-definable predicates of of ll: IE11 are precisely precisely the the primitive primitive recursive predicates. predicates. recursive

126 126

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Proof. Corollary Corollary 11.2.10 already established established that that every every primitive primitive recursive recursive predicate predicate Proof. .2.10 already is is 6. Al-definable by � /El. For the the converse, converse, suppose suppose A(c) A(c) and and B(c) B(c) are are I; El-formulas 1 -definable by l -formulas 1 ' For such that t h a t /� E l1 proves proves (V'x) (Vx)(A(x) ++ -,B(x) ~B(x)).) . Then Then the the characteristic characteristic function function of of the the such (A(x) ++ predicate predicate A(c) A(c) is is I; El-definable i n /� E l1 since since � 1El1 can can prove prove l -definable in

(Vx)(3!y)[(A(x) O) V (B(x) (B(x) /\ A yy = = 1)]. 1)]. (V'x) (:J!y) [(A(x) /\A Yy = 0)

By By Theorem Theorem 3.1.1, 3.1.1, this this characteristic characteristic function function is is primitive primitive recursive, recursive, hence hence so so is is the the predicate predicate A(c) A(c).. 3.1.5. 3.1.5. Total Total functions functions of o f /�n E n . . Theorem Theorem 1.2.1 1.2.1 provided provided aa characterization characterization of of the the I; El-definable functions of o f /� E l1 as as being being precisely precisely the the primitive primitive recursive recursive functions. functions. 1 -definable functions It It is is also also possible possible to to characterize characterize the the I; Ey-definable functions of o f /�n E n for for n n > > 11 in in 1 -definable functions terms terms of of computational computational complexity; complexity; however, however, the the n n > > 11 situation situation is is substantially substantially more more complicated. complicated. This This problem problem of of characterizing characterizing the the provably provably total total functions functions of of fragments central problems fragments of of Peano Peano arithmetic arithmetic is is classically classically one one of of the the central problems of of proof proof theory; and and a a number number of of important important and and elegant elegant methods methods are are available available to to solve solve it. it. theory; Space Space prohibits prohibits us us from from explaining explaining these these methods, methods, so so we we instead instead mention mention only only a a few few references. references. The The first first method method of of analyzing analyzing the the strength strength of of fragments fragments of of Peano Peano is is based based on on 's assignment Gentzen Gentzen's assignment of of ordinals ordinals to to proofs; proofs; Gentzen Gentzen [1936,1938] [1936,1938] used used Cantor Cantor normal normal form form to to represent represent ordinals ordinals less less than than fO e0 and and gave gave aa constructive constructive method method of of assigning assigning ordinals inductions to ordinals to to proofs proofs in in such such aa way way that that allowed allowed cuts cuts and and inductions to be be removed removed from sentences. This from PA-proofs PA-proofs of of sentences. This can can then then be be used used to to characterize characterize the the primitive primitive recursive recursive functions functions of of fragments fragments of of Peano Peano arithmetic arithmetic in in terms terms of of recursion recursion on on ordinals ordinals of Takeuti less less than than fO e0.. The The textbooks textbooks of Takeuti [1987] [1987] and and Girard Girard [1987] [1987] contain contain descriptions descriptions of of this this approach. approach. A A second second version version of of this this method method is is based based on on the the infinitary infinitary proof proof systems systems of of Tait: Tait: Chapter Chapter III III of of this this volume volume describes describes this this for for Peano Peano arithmetic, arithmetic, and and Chapter Chapter IV IV describes describes extensions extensions of of this this ordinal ordinal assignment assignment method method to to much much stronger stronger second-order second-order theories theories of of arithmetic. arithmetic. The The books books of of Schutte Schiitte [1977] [1977] and and Pohlers Pohlers [1980] [1980] also also describe describe ordinal ordinal assignments assignments and and infinitary infinitary proofs proofs for for strong strong theories theories of of arithmetic. arithmetic. A A further further use use of of ordinal ordinal notations notations is is to to characterize characterize natural natural theories theories of of arithmetic arithmetic in in terms terms of of transfinite transfinite induction. induction. A the computational A second second approach approach to to analyzing analyzing the computational strength strength of of theories theories of of arith arithmetic model-theoretic constructions; Harrington [1977], metic is is based based on on model-theoretic constructions; see see Paris Paris and and Harrington [1977], Ketonen Ketonen and and Solovay Solovay [1981], [1981], Sommer Sommer [1990], [1990], and and Avigad Avigad and and Sommer Sommer [1997]. [1997]. A the Dialectica A third third method method is is based based on on the Dialectica interpretation interpretation of of Geidel Ghdel [1958] [1958] and and 's [1970] on ordinals to on Howard Howard's [1970] assignment assignment of of ordinals to terms terms that that arise arise in in the the Dialectica Dialectica interpretation. interpretation. Chapter Chapter V V of of this this volume volume discusses discusses the the Dialectica Dialectica interpretation. interpretation. A A fourth fourth method, method, due due to to Ackermann Ackermann [1941] [1941] uses uses an an ordinal ordinal analysis analysis of of f-calculus e-calculus proofs. proofs. More More recently, recently, Buss Buss [1994] [1994] has has given given aa characterization characterization of of the the provably provably total total functions �n based functions of of the the theories theories/E~ based on on an an extension extension of of the the witness witness function function method method used used above. above.

Proof Theory Theory of Arithmetic

1127 27

3.2. Witnessing 3.2. Witnessing theorem theorem for for S� S~ Theorem 1.3.4.1 1.3.4.1 stated stated that that every every polynomial polynomial time time function function and and every every polynomial polynomial Theorem � -definable (respectively) time -definable or time predicate predicate is is E� E~-definable or � A~-definable (respectively) by by 8} S~.. More More generally, generally, Theorem -function and -predicate is -definable Theorem 1.3.6 1.3.6 stated stated that that every every Of 0~'-function and every every �f A~-predicate is E� E~-definable or -definable by or �� A~-definable by S� S~.. The The next next theorem theorem states states the the converse; converse; this this gives gives aa precise precise characterization of of the the E� E~-definable functions of of Si S 1 and and of of the the E� E~-definable functions characterization -definable functions -definable functions of of S� S~ in in terms terms of of their their complexity complexity in in the the polynomial polynomial hierarchy. hierarchy. The The most most interesting interesting 1 , where case case is is probably probably the the base base case case ii = - 1, where Si S~ is is seen seen to to have have proof-theoretic proof-theoretic strength strength that that corresponds corresponds precisely precisely to to polynomial polynomial time. time.

Theorem. T h e o r e m . (Buss (Buss [1986]) [1986]) (1) Every -definable function (1) Every E� E~-definable function of of Si S~ is is polynomial polynomial time time computable. computable. (2) Let -definable function -th level, (2) Let ii � >_ 11.. Every Every Et E~-definable function of of S� S~ is is in in the the ii-th level, Of 0~,, of of the the polynomial polynomial hierarchy. hierarchy. Corollary. (Buss [1986]) Corollary. (Buss [1986]) -definable predicate polynomial time. (1) Every (1) Every �� A~-definable predicate of of Si S 1 is is polynomial time. (2) Let -definable predicate -th level, Let ii � >_ 11.. Every Every �� A~-definable predicate of of S� S~ is is in in the the ii-th level, �f A~,, of of the the polynomial polynomial hierarchy. hierarchy.

The corollary the theorem exactly the the same same argument argument as used The corollary follows follows from from the theorem by by exactly as was was used to the theorem, to prove prove Corollary Corollary 3.1.4 3.1.4 from from Theorem Theorem 3.1.1. 3.1.1. To To prove prove the theorem, we we shall shall use use aa witnessing lE I above. First, we witnessing argument argument analogous analogous to to the the one one use use for for/El above. First, we need need aa revised revised predicate; unlike definition of form form of of the the Witness Witness predicate; unlike the the usual usual definition of the the Witness Witness predicate predicate for for bounded bounded arithmetic arithmetic formulas, formulas, we we define define the the Witness Witness predicate predicate only only for for prenex prenex formulas, cation. This formulas, since since this this provides provides some some substantial substantial simplifi simplification. This simplification simplification is is obtained without -formula is obtained without loss loss of of generality generality since since every every Et E~-formula is logically logically equivalent equivalent to to aa E� -formula in E~-formula in prenex prenex form. form. 3.2.1. 3.2.1. Definition. Definition. Fix Fix ii � >_ 11.. Let Let A(C) A(~ be be aa E�-formula E~-formula which which is is in in prenex prenex form. form. Then Then Witness�(w, WitnessiA(w, C) ~ is is defined defined by by induction induction on on the the complexity complexity of of A A as as follows: follows: Witness~(w C) ~ is is just just the the formula formula A(C) A(~, , -formula, then (1) (1) If If A A is is aa TIL Hb i_ 1-f~ then Witness�(w, b (w, C) (2) If A(C) A(~ is is not not in in TIL II~_ 1 and and is is of of the the form form (3x (3x :::; <_t)B(c, t)B(g, x) x),, then then Witness� Witness~(w, ~ is is (2) If the the formula formula

1, w) Witnessk(c,b Witness~(e,b)(~(2, w), c, ~, .8(1, ~(1, w)) w)) 1\ A .8( ~(1, w) :::; < t. t. ) C8(2, w), Intuitively, rst element Intuitively, aa witness witness for for (3x (3x :::; <_t)B(x) t)B(x) is is aa pair pair w w,, the the fi first element of of the the pair pair the second (x) for giving giving aa value value for for x x and and the second element element witnessing witnessing the the truth truth of of B B(x) for that that value value of of x x.. (3) (3) If If A(C) A(c') is is not not in in TIL II~_1 and and is is of of the the form form (Vx (Vx :::; <_ ItI)B(c, Itl)B(~, x) x),, then then Witness�(w, Witness~(w, C) is is the the formula formula c,d (.8(x + 1, (Vx It I) Witnessk (Vx :::; <_ Itl) Witness~(e,d)(~(x 1, w), w), c, ~, x) x).. ( )

S. Buss

128

Intuitively, aa witness witness for for (Vx (Yx ::; <_ ItI)B(x) Jtl)B(x) is is aa sequence sequence W w of of length length Itl + 11,, Intuitively, It I + W w = (WO, (wo, W wt,...,wltl) such that that each each Wx wx witnesses witnesses the the truth truth of of B(x) B(x),, for for I , . . . , W ltl ) such o0 ::; _< x z ::; _< Itl Itl..

Lemma. L e m m a . Let Let ii ;::: >_ 1l and and A A E 2:� E~.. Then Then ((a) a) Witness� Withes4 is is � A~� with with respect respect to to Si S~.. If If ii > 11,, then then Witness� Witness~ is is aa ITL II~_I1-formula. -formula. (b) Witness� -predicate. (b) Witness~ defines defines aa �f A~-predicate. (c) S� S~ proves proves (c) A(C) A(~ +-+ ~ (3w) (3~) Witness� w i t ~ 4 ( ~(w, , C). c-).

-definable function (d) (d) There There is is aa term term tA tA and and aa polynomial polynomial time, time, 2:� E~-definable function ggA such that that A such Si S~ proves proves

Withes 4 (w, ~ D gA (w) < tA (~ A Withes 4 (gA (w), ~.

The lemma is induction on The lemma is easily easily proved proved by by induction on the the complexity complexity of of A A.. For For part part (d), (d), the the function function gA(W) gA(w) merely merely computes computes a a succinct succinct G5del GSdel number number of of w w;; this this just just involves involves removing unnecessary removing unnecessary unnecessary elements elements from removing unnecessary leading leading zeros zeros and and removing from the the witness. witness. We We extend extend the the witness witness predicate predicate to to cedents cedents of of prenex prenex form form formulas formulas as as follows. follows. If If �( W, C) is defined to be A = = �' A',, A A is is aa succedent, succedent, then then Witnesi WitnessivA (w, ~ is defined to be � V

i � CB(2, w), Witness~(~(1, w), w), C) 5) V V Witness Witnessi\/~,(fl(2, w), C). ~. Witness�CB(l, v /

Dually, ned similarly Dually, if if r F= - A, A, r' F ~ is is an an antecedent, antecedent, then then Witness� Witness~rr is is defi defined similarly as as

Witness�(,B(l, Wit,~es4(Z(1 , w), w), C) ~ 1\ A Witnesi W i t n ~ ~l\, (pZCB(2, ( 2 , w), w), C). ~. 3.2.2. (Proof 3.2.2. Proof. Proof. (Proof sketch sketch for for Theorem Theorem 3.2.) We We shall shall prove prove Theorem Theorem 3.2 by by proving proving aa slightly slightly more more general general witnessing witnessing lemma lemma that that applies applies to to sequents sequents of of 2:�-formulas. lemma holds holds for general 2:�-formulas, E~-formulas. Although Although the the lemma for sequents sequents of of general E~-formulas, we we state it in prenex this simplifies the Witness state it only only for for formulas formulas in prenex form, form, since since this simplifies the Witness predicate predicate and and the the proof. proof. Witnessing ---+ � W i t n e s s i n g Lemma L e m m a for for S� S~2.. Let Let ii ;::: >_ 11.. Let Let r F---+ A be be aa sequent sequent of of formulas formulas in in 2:� Eb

in ---+ � in prenex prenex form, form, and and suppose suppose S� S~ proves proves r F---} A.. Let Let c6 include include all all free free variables variables in in the the -function h(w, h( w, C) -defined in sequent. Then sequent. Then there there is is aa Of Dr-function ~ which which is is 2:� E~-defined in S� S~ such such that that S� S~ proves proves ---+ Witness\t �(h(w, ~, C), C) Witness� w~t~~,~(~, ~-+ w~t~&~(h(~, ~.. r (w, C)

The The proof proof of of the the Witnessing Witnessing Lemma Lemma is is by by induction induction oonn the the number number of of sequents sequents iinn aa ---+ � -formula is -formula free-cut free-cut free free proof proof P P of of r F---+ A.. Since Since every every 2:� E~-formula is equivalent equivalent to to aa 2:� E~-formula in in prenex prenex form, form, we we may may assume assume w.l.o.g. w.l.o.g, that that every every induction induction formula formula in in the the free-cut free-cut -formula. Then, Then, by proof P free free proof P is is aa prenex prenex form form 2:� E~-formula. by the the subformula subformula property, property, every every formula -formula in formula appearing appearing anywhere anywhere in in the the proof proof is is also also aa 2:� E~-formula in prenex prenex form. form. The The ---+ � base induction proof base case case of of the the induction proof is is when when r F--} A is is an an initial initial sequent; sequent; in in this this case, case,

Proof Theory of of Arithmetic Arithmetic

1299 12

every formula formula in in the the sequent sequent is is atomic, so the the Witnessing Witnessing Lemma Lemma trivially trivially holds. holds. every atomic, so The nal inference The induction induction step step splits splits into into cases cases depending depending on on the the fi final inference of of the the proof. proof. The The structural structural inferences inferences and and the the propositional propositional inferences inferences are are essentially essentially trivial, trivial, the the latter because because of of our our assumption assumption that that all all formulas formulas are are in in prenex prenex form. form. So So it it remains remains latter to to consider consider the the quantifier quantifier inferences inferences and and the the induction induction inferences. inferences 9 The The cases cases where where the the final final inference inference of of P P is is an an :3 3 � ___:right :right inference inference or or an an :33 � _<:left :left inference inference are are similar similar to to the the :3 3 :right and and :33 :left cases cases of of the the proof proof of of the the Witnessing Witnessing Lemma Lemma 3. 3.1.3 f o r /.IT: E ii, , 1 .3 for so so we we omit omit these these cases cases too. too. Now Now suppose suppose the the final final inference inference of of P P is is aa I:�-PIND E~-PIND inference: inference: 9

9

.

9 9

.

.

A( A([l 89 r--+ b., A, A(b A(b)) � bJ ) , r---+ A(0), F---} b., A, A A(t) A(O (t ) ) , r---+ where where A A E I:� E~ \\ ITL II~_I1 and and where where bb is is the the eigenvariable eigenvariable and and does does not not occur occur in in the the lower lower sequent. induction hypothesis -function g(w, sequent. The The induction hypothesis gives gives aa I:�-defined, E~-defined, Of [l~-function g(w, C, g, bb)) such such that that S� S~ proves proves

iV{D.,A (b)} ((g(w, Witness ---+ Witness WitnessiA{A([ ~, b) b)-'+ Witnessiv{A,A(b)} b),, c, g, b). g (w, C,g, b) \{A( l89 ! bJ ),r} (w, c, Let v, w be defined Let k(c, k(5', v, w)) be defined as as

k(c, k(~', v, V,w W))

=

{/ : f v

w

if v, C) if Witnesst; Witnessiv~D. ((v, c') otherwise otherwise

Since Witness Witness i is is aa b.� A~-predicate, is I:� E~-defined by S� S~;; and and since since Witnesi Witness i is is in in b. Afp,, Since -predicate, kk is -defined by in Of []~.. Define Define the the Of-function 0P-function f(w, f(w, c, g, bb)) by by kk isis in

f(w, f(w, c, 5', OO)) f(w, f(w, C, 5', bb))

= =

(l, w) (~(1, w),, OO)) (,8 (w, c,g, l[~� bJ )) ,, ,8 (2, w ) ) ,, C, b)) ,, ( 1, g ( (,8 ( 1, ff(w, (/~(1,g({/~(1, (,8 f ( ww, , C, l[Xbj)), k(c, ,8 (2, f( � bJ )) , ,8 (2 , g ( (,8 ( 1, ff(( ww,, C, l � bJ )) , ,8 (2,, w ) ) ,, C, b))) ) for bb > > O. O. for

Since -defined Since f f is is defined defined by by limited limited recursion recursion on on notation notation from from g g,, and and since since gg is is I:� E~-defined by also I:� -defined by by S� S~,, f f is is also E~-defined by S� S~.. Therefore, Therefore, f f may may be be used used in in induction induction formulas formulas and and S� S~ can can prove prove

Witnesil\{A (O),r} (w, C) ---+ WitnessiV{D.,A(b)} (J(w , C, b) , C, b) . using Setting h(w, using I:�-PIND E~-PIND with with respect respect to to bb.. Setting h(w, C) 5) = = f(w, f (w, c, ~, t) t) establishes establishes the the desired desired conditions conditions of of the the Witnessing Witnessing Lemma. Lemma. Finally, Finally, we we consider consider the the inferences inferences involving involving bounded bounded universal universal quantifiers. quantifiers. The The cases cases where where the the principal principal formula formula of of the the inference inference is is aa ITL II~_l-formula are essentially essentially I -formula are trivial, since trivial, since such such formulas formulas do do not not require require aa witness witness value, value, i.e., i.e., they they are are their their own own witnesses. witnesses. This This includes includes any any inference inference where where the the principal principal connective connective is is aa non-sharply non-sharply

130 130

s. Buss Buss S.

bounded universal universal quantifier. quantifier. A AV \I <:::; :left where the the principal principal formula formula isis in in ~E� must must :lejt where bounded have have aa sharply sharply bounded bounded universal universal quantifier quantifier as as its its principal principal connective; connective; this this case case of of the the Witnessing Witnessing Lemma Lemma isis fairly fairly simple simple and and we we leave leave itit to to the the reader. reader. Finally, Finally, we we \I _:::; :right :right inference inference consider the the case case where where the the last last inference inference of of PP isis aa V consider

Itl, rr--+ �, A(b) A(b) bb <:::; Itl, + zx, r -+ zx, �, ((\Ix It I )A(t) r--+ w <:::; Itl)A(t) where b 1I and where A A EE 2E�~ \\ IIi_ ITL where the the eigenvariable eigenvariable bb occurs occurs only as indicated. indicated. The The and where only as induction hypothesis hypothesis gives gives aa ~-defined, E�-defined, []~-function Of -function g9 such such that that S� induction S~ proves proves

w) , C) -+ Itl ^A Witness\rCB(2, w), :::; Itl bb _<

b)) , c-). C) . g( w, g, WitnessiA(b)(~(1, c, b) b) VV Witness~/~(~(2, Witness�� (,8 (2, g(w, b)), ~, Witness�(b) (,8 (1, g(w, g( w, g, c, b)), c, b)), Let h (w, ~C) bbee defined defined to to equal equal /~(2, g(w, g, Let fl (w, for the the least least value such value bb _:::; Itl It I such b)) for c, b)) ,8 (2, g(w, this value let fl(w, h (w, c--) C) = V� or ifif there there is is no no such such value value of O. :::; tt,, let this value witnesses witnesses V A,, or of bb _< = O. Since the predicate predicate Witnesi Witness~/~(~(2, g(w, c, g, b)) b)),, ~C) is A~ and A~-defined by by S~, Since the is �f and iiss ��-defined sL the the V �(,8 (2, g(w, function in []~ Of and and is is Z~-defined E� -defined by by S~. s� . Also, Also, let h (w, ~C) equal function h fl is is in let f2(w, equal ((~(1, 1)), ~(1, 0)),), ~(1, g, 22)), ) ) , .. ... ., , ~(1, 9 ( w, g, ( w, C, t I) ) . 9 ( w, if, ( w, g, c, IItl))>. ,8 (1, g(w, ,8 (1, 9g(w, ,8 (1, 9g(w, ,8 (1, g(w, C, 1)), C, 0) It is is easy easy to to verify verify that is in in []~ Of and and is is E� -definable by s� . Now h( w, C) It that h f2 also also is ~-definable by S~. Now let let h(w, c') equal w), c), h ((O, w), C) . It It is is easy easy to to check check that that hh satisfies the desired desired equal (h((O, (f2((O,w>,c-),fl((O,w),c-)>. satisfies the conditions the Witnessing Witnessing Lemma. conditions of of the Lemma. Q.E.D. Witnessing Lemma and 3.2. Q.E.D. Witnessing Lemma and Theorem Theorem 3.2. 3.3. 3.3. Witnessing W i t n e s s i n g theorems t h e o r e m s and a n d conservation c o n s e r v a t i o n results results for for T4 T~

This nable functions This section section takes takes up up the the question question of of the the defi definable functions of of T4 T~.. For For these these theories, there -definable, theories, there are are three three witnessing witnessing theorems, theorems, one one for for each each of of the the E� E~-definable, b -definable and the E�b -definable functions. In addition, there is a close the the E� E~+l-definable and the Ei+2-definable functions. In addition, there is a close +l +2 connection connection between between S�+l S~+1 and and T4 T~;; namely, namely, the the former former theory theory is is conservative conservative over over the the ' ll present these results latter. We latter. We'll present most most of of these results without without proof, proof, leaving leaving the the reader reader to to look look up up the the original original sources. sources. The will apply The results results stated stated in in this this section section will apply to to T4 T~ for for ii � _ 00;; however, however, for for ii = - 00,, the the bootstrapping bootstrapping process process does does not not allow allow us us to to introduce introduce many many simple simple functions. functions. Therefore, ( DD as Therefore, when when ii = = 00,, instead instead of of T� T~ , we we must must use use the the theory theory PVI PV1 = - T� TO([]~) as defined defined in in section section 1.3.7. 1.3.7. To To avoid avoid continually continually treating treating ii = = 00 as as aa special special case, case, we we let let T� T ~ denote denote PVI PV1 for for the the rest rest of of this this section. section. 3.3.1. �b+ l -definable functions 3.3.1. The The E ~i+l-definable functions of of T4 T~ Theorem. T h e o r e m . (Buss (Buss [1990]) [1990]) Let Let ii � >_OO.. �+ l -define eve (1) ry Df (1) T4 T~ can can E Eib+l-define every []iP+l-function. +l -function.

Proof Proof Theory Theory of of Arithmetic Arithmetic

131 131

(2) Conversely, (2) Conversely, every every �� Eib+l-definable function of of T� T~ is is aa Df []iP+l -function. + l -definable function + l -function. i +1l Zi8.S Y]ibl-conservative ",b over T, ' over (3) (3) S S~2+ T~. L.. i l -conservatwe 2i ' + (4) S� S~++1l is is conservative conservative over over TJ T~ + + �� ~ib+l-replacement with respect respect to to Boolean Boolean combi combi(4) + I -replacement with nations nations of of �� Ei+ l+I b formulas. formulas.

' ll just theorem, and We'll just state state some some of of the the main main ideas ideas of of the the proof proof of of this this theorem, and let let We the details. Part the reader reader refer refer to to Buss Buss [1990] [1990] or or Krajicek Kraji~ek [1995] [1995] for for the the details. Part (1) (1) with with - 00 is is trivial trivial because because of the temporary temporary convention convention that that T� T~ denotes denotes PVI PV1.. To To ii = of the -define" every prove formula, prove part part (1) (1) for for ii > > 00,, one one shows shows that that T� T~ can can "Qi "Qi-define" every Df []iP+l formula, +l where Qi-definability Qi-definability is is aa strong strong form form of of �� Eib+l-definability. The general general idea idea of of aa Q; Qiwhere + l -definability. The defi nable function -oracle such definable function is is that that it it is is computed computed by by aa Turing Turing machine machine with with aa �f E~-oracle such that witness. In that every every "yes" "yes" answer answer of of the the oracle oracle must must be be supported supported by by aa witness. In the the correct correct computation, computation, the the sequence sequence of of "yes/no" "yes/no" answers answers is is maximum maximum in in aa lexicographical lexicographical ordering, ordering, and and thus thus TJ T~ can can prove prove that that the the correct correct computation computation exists exists using using ��-MAX E~-MAX axioms (which (which can can be be derived derived similarly similarly to to minimization minimization axioms). axioms). This This proof proof of of (1) is axioms (1) is reminiscent reminiscent of of the the theorem theorem of of Krentel Krentel [1988] [1988] that that MINSAT MINSAT is is complete complete for for Df ~. . Part Part (2) of of the the theorem theorem is is immediate immediate from from Theorem Theorem 3.2 and and the the fact fact that that l Part (3) is T� strengthening of Witnessing T~ � C_S� S~+ +1 .. Part is based based on on the the following following strengthening of the the Witnessing l Theorem Theorem 3.2.2 for for S� S~+ +1": Witnessing W i t n e s s i n g Lemma L e m m a for for S� S~+ +1l .. Let Let ii � > 11.. Let Let r F -7 ~ � A be be aa sequent sequent of of formulas formulas in in ++1l proves �� in prenex form, and suppose S� r -7 � ; let include all free variables Eib+l in prenex form, and suppose si2 proves F ~ A; let ~ include all free variables C l + in h( w, C) in the the sequent. sequent. Then Then there there is is aa Df []i+1 P+ l -function -function h(w, ~ which which is is Qi Qi -defined defined in in T� T~2 such such that that T� T~ proves proves WitnessiA� (w, C) -7 Witnessi.;j� (h(w, C) c-),, C) .

The The proof proof of of this this Witnessing Witnessing Lemma Lemma is is almost almost exactly exactly the the same same as as the the proof proof of of the Lemma in section 3.2.2; the only difference the Witnessing Witnessing Lemma in section the only difference is is that that the the witnessing witnessing functions nable in functions are are now now proved proved to to be be Qi-defi Qi-definable in T� T~.. In In fact, fact, (1) (1) implies implies the the necessary necessary l functions functions are are Q;-defined Qi-defined by by T� T~ since since we we already already know know they they are are �� Eib+l-defined by S� S~+ +1 .. + l -defined by So witnessing functions So the the main main new new aspect aspect is is showing showing that that T� T~ can can prove prove that that the the witnessing functions work. work. Part Part (3) (3) of of the the theorem theorem is is an an immediate immediate consequence consequence of of the the Witnessing Witnessing Lemma. Lemma. Part the Witnessing Part (4) (4) can can also also be be obtained obtained from from the Witnessing Lemma Lemma using using the the fact fact that that l (w, C) + b -replacement can prove that A(C) is equivalent to (:3w) Witness� T� for T~9+ + �� Ei+l-replacement can prove that A(~ is equivalent to (3w) Witness~+l(w, c--) for +l I ' any any A AE E �� E~b+l. + 3.3.2. �b+2-definable functions 3.3.2. The The � Ei+2-definable functions of of T� T~

The The �� Eib+2-definable functions of of T� T~ can can be be characterized characterized by by the the following following theorem, theorem, +2 -definable functions due due to to Krajicek, Kraji~ek, Pudlak Pudls and and Takeuti Takeuti [1991]. [1991]. KPT O. Suppose K P T Witnessing W i t n e s s i n g Theorem. T h e o r e m . Let Let ii � >_ O. Suppose T� T~ proves proves

(w)(sy)(Vz _
132 132

s. S. Buss Buss

and there where where A A E II� IIbi .. Then Then there there is is aa kk > 0 and there are are E� Eib+l-definable function symbols symbols + l -definable function ffl(x), l (x) , h(x, f2(x, Zzl),..., f~(x, Zzt,...zk_t) such that that T� Ti2 proves proves l , . . . Zk-l) such I )' . . . ' fk(x, zd , x, Z2) , X, ZI ) V (Vz2 (Vx) (VZ1 ::; (Vx)(VZl _< t)[A(!I(X) t)[A(fl(X),X,Z,)V (Vz2 ::; _< t)[A(h(x, t)lA(f2(x,z,),x, z2) t) [A( h (x, Zl V(Vza ::; _< t)[A(f3(x, z,,, Z2) z2),, x, x, Z3) z3) V(VZ3 , . . . , Zk- l ) , x, Zk)J . . .J ]] V .· ... . V V (VZk (Vzk ::; _< t)[A(fk(X, Zl,...,Z~_l),X,Z~)]...]]] t)[A(fk(x, Zl V Conversely, Conversely, whenever whenever the the above above formula formula is is provable, provable, then then T� T~ can can also also prove prove (Vx) (3y) (By)(Vz _< t)A(y, t)A(y, x, x, z) z).. (Vx) (Vz ::; The The variables variables x x,, yy and and Zz could could just just as as well well have have been been vectors vectors of of variables, variables, since since the coding can the replacement replacement axioms axioms and and sequence sequence coding can be be used used to to combine combine adjacent adjacent like like quantifiers. Also, the quantifiers. Also, the first first half half of of the the theorem theorem holds holds even even if if tt involves involves both both x x and and yy.. The proof proof of of the the KPT KPT Witnessing Witnessing Theorem Theorem is is now now quite quite simple: simple: by by the the discussion discussion The in section section 11.3.7, we can can replace replace each each T� T~ by by its its conservative, conservative, universally universally axiomatized axiomatized in .3.7, we extension corollary to extension PV;+l PV/+I,, and and now now the the theorem theorem is is an an immediate immediate corollary corollary of of the the corollary to the the generalized generalized Herbrand's Herbrand's theorem theorem in in section section 2.5.3 2.5.3 of of Chapter Chapter I. I. 3.3.2.1. 3.3.2.1. Applications Applications to to the the polynomial polynomial hierarchy. hierarchy. The The above above theorem theorem has has had in showing had aa very very important important application application in showing an an equivalence equivalence between between the the collapse collapse of of the hierarchy hierarchy of theories of the the of theories of bounded bounded arithmetic arithmetic and and the the (provable) (provable) collapse collapse of of the polynomial time hierarchy. equivalence was polynomial time hierarchy. This This equivalence was first first proved proved by by Krajicek, Kraji~ek, Pudhik Pudls and and Takeuti Takeuti [1991J; [1991]; we we state state two two improvements improvements to to their their results. results. (We (We continue continue the the convention convention that that Tf T ~ denotes denotes PV1 PV1.).)

O. If Theorem. T h e o r e m . (Buss (Buss [1995J, [1995], Zambella Zambella [1996]) [1996]) Let Let ii 2: >_ O. If T4 Ti2 F ~ 8� si2++tl ,, then then (1) (1) T4 Ti2 = = 82 $2 the polynomial and and therefore therefore 82 $2 is is finitely finitely axiomatized, axiomatized, and and (2) (2) T4 T~ proves proves the polynomial hierarchy hierarchy equivalent to collapses, and and in a) T4 collapses, in fact, .fact, (2. (2.a) T~ proves proves that that every every E� ~ib3-formula is equivalent to +3 -formula is -formulas and (2. b) T4 proves the polynomial time aa Boolean Boolean combination of E� combination of ~+2b -formulas and (2.b) T~ proves the polynomial time 2 + hierarchy hierarchy collapses collapses to to Ef ~+l/poly. + l /poly . Corollary. proves the Corollary. 82 5'2 is is finitely finitely axiomati axiomatized if and and only only if if 82 $2 proves the polynomial polynomial hierar hierarzed if chy chy collapses. collapses.

Let -definable function Let g(x) g(x) be be aa Et ~-definable function of of T4 T~ such such that that for for each each n n > 0 there there is is g(n) = In an an m m > 0 so so that that T4 Ti2 I~- (Vx) (Vx)(x D g(x) g(x) > m m)) (for (for example, example, g(n) Inll or or (x > nn ::> etc.) Let g(n) I l nl l , etc.) g(n) = --]lull, Let g g E�-IND ~ - I N D denote denote the the axioms axioms A(O) 1\ A (Vx) (Vx)(A(x) D A(x A(x + + 1)) 1)) ::> D (Vz (Vz ::; <_g(x))A(z) g(x))A(z). . A(O) (A(x) ::>

Let (in the Let Vm(N) VIIi(N) denote denote the the set set of of all all '1m VII~ sentences sentences (in the language language of of 82 5'2)) true true about about the the standard standard integers. integers.

Proof Theory Theory of Arithmetic

133 133

3.3.2.2. 3.3.2.2. Theorem. T h e o r e m . (essentially (essentially KrajIcek, Krajihek, Pudlcik Pudls and and Takeuti Takeuti [1991]) [1991]) If T� + v'II� {N) 1= g�� -IND, then the polynomial time hierarchy collapses gEi+I-IND , then the polynomial time hierarchy collapses to to +l p /poly . Ai+l/poly. !J.. f+ l Note Note that that second second theorem theorem differs differs from from the the first first in in that that there there is is no no mention mention of of the the provability of the polynomial hierarchy; hand, the provability of the collapse collapse of of the the polynomial hierarchy; on on the the other other hand, the second second theorem theorem states states aa stronger stronger collapse. collapse. KrajIcek, Kraji~ek, Pudlcik Pudl~k and and Takeuti Takeuti [1991] [1991] prove prove the the second nl and second theorem theorem with with g{n) g(n) = = IInl and without without the the presence presence of of VII�{N) VIIi(N)": their their proof proof gives modifications. gives the the stronger stronger form form stated stated here here with with only only minor minor modifications.

3.3.3. -definable functions 3.3.3. The T h e �� E~-definable f u n c t i o n s of of Ti T21 Buss -definable functions Buss and and KrajIcek Kraji6ek [1994] [1994] characterize characterize the the �� E~-definable functions of of Ti T~ as as being being precisely precisely the the functions functions which which are are projections projections of of PLS PLS functions. functions. Polynomial P o l y n o m i a l Local Local Search. Search. Johnson, Johnson, Papadimitriou Papadimitriou and and Yannakakis Yannakakis [1988] [1988] defined defined aa Polynomial Polynomial Local Local Search Search problem problem (PLS-problem (PLS-problem)) L L to to be be aa maximization maximization problem problem satisfying satisfying the the following following conditions: conditions: (we (we have have made made some some inessential inessential simplifications simplifications to to their their definition definition)) set FL{x) 1}* , there (1)) For For every every instance instance x x E {a, {0,1}*, there is is aaset FL(X) of of solutions, solutions, an an integer integer valued valued (I x) and x) , cost cost function function cds, CL(S,X) and aa neighborhood neighborhood function function Nds, NL(S,X), (2) The binary binary predicate predicate ss E FL{X) FL(X) and and the the functions functions cds, CL(S,X) and NL{s, NL(S,X) are (2) The x) and x) are polynomial polynomial time time computable. computable. There There is is aa polynomial polynomial PL PL so so that that for for all all ss E E

FL{X) , ls i � < pd l xl) . Also, A so, °0 E e Fdx) . x) E FL{X) (3) For For all all ss E {a, {0, 1}* 1}*,, Nds, YL(s,x) Fn(x).. (3) (4) For {x) , if x) =1= x) < {s, x), x) . (4) For all all ss E FL Fn(x), if NL{s, Nn(s,x) r ss then then cds, CL(S,X) < cL{NL CL(NL(S,X),X). (5) The finding aa locally locally optimal (5) The problem problem is is solved solved by by finding optimal ss E FL{x) Fn(x),, i.e., i.e., an an ss such such that that NL(s, x) = s . NL{s, It conditions that It follows follows from from these these conditions that all all ss E Fdx) Fn(x) are are polynomial polynomial size. size. -sentence saying A A PLS-problem P LS-problem L L can can be be expressed expressed as as aa II� II~-sentence saying that that the the conditions conditions above hold; if above hold; if these these are are provable provable in in Ti T21 then then we we say say L L is is aa PLS-problem PLS-problem in in Ti T~.- The The OptL{X, s) � -formula NL(s, x) = formula formula OptL(x, s) is is the the !J.. A~-formula Nn(s,z) = ss.. A A multivalued multivalued function function 9g such such called aa PLS function; 9g must total, but that that for for all all x z,, NL NL (g(x), (g(x), x) x) = = g(x) g(z),, is is called PLSfunction; must be be total, but may may be since there optimal cost be multivalued, multivalued, since there may may exist exist more more than than one one optimal cost solution. solution. The The next next theorem states, states, loosely loosely speaking, the (multivalued -definable functions theorem speaking, that that the (multivalued)) �� E~-definable functions of of Ti T2~ are are precisely precisely the the functions functions f f which which can can be be expressed expressed in in the the form form f f = - 1f r Oo gg,, where where 9g is is aa PLS PLS function function and and where where 1f r is is aa polynomial polynomial time time function function (in (in fact, fact, 1f( r(y) =/~(1, y) Y) = ,8 (1, y) ). can always always be be used used). can Theorem. Buss and T h e o r e m . ((Buss and KrajIcek Kraji~ek [1994]) [1994])

(:Jy)OptL(X, (1) For every every PLS PLS problem problem L L,, Ti T~ can can prove prove (Vx) (Vx)(3y)OptL (x, y) y).. (1) For

134 134

S. Buss

proves (V' i) (3y)A(i, y) (2) If I / AA E e �� E~ and and if i/ Ti T~ proves (VZ)(3y)A(Z, y),, then then there there is is aa polynomial polynomial time time (projection) (projection) function function 7l'(y) r(y) and and aa PLS PLS problem problem L L such such that that Ti T~ proves proves ((YZ)(Vy)(OptL(2, y) :J ~ A(i, A(Z, 7l'( r(y)). V'i) (V'y) (OptL (i, y) Y)). for In In other other words, words, if i/g9 is is aa PLS PLS function function solving solving L L,, then then A(i, A(Z, 7l' ~ 0o g(i)) g(Z)) holds holds/or all i and all all values all ~ and values of o/ g(i) g(Z).. Natural N a t u r a l Proofs. Proofs. The The above above theorem theorem characterizing characterizing the the �� IE~ consequences consequences of of Ti T~ in in terms PLS functions terms of of PLS functions was was used used in in an an important important way way to to establish establish the the independence independence of some some computational computational complexity complexity conjectures conjectures from from S�(a) $2(c~).. Razborov Razborov and and Rudich Rudich of [1994] introduced [1994] introduced aa notion notion of of "P-natural "P-natural proofs" proofs" of of P P =I ~ NP NP;; which which intuitively intuitively are are proofs time method proofs which which provide provide aa polynomial polynomial time method of of separating separating out out truth truth tables tables of of Boolean size circuits. then showed Boolean functions functions that that do do not not have have polynomial polynomial size circuits. They They then showed that that under under aa certain certain strong strong pseudo-random pseudo-random number number generator generator conjecture conjecture (henceforth: (henceforth: the the SPRNG conjecture) be P-natural SPRNG conjecture) that that there there cannot cannot be P-natural proofs proofs of of P P =I ~ NP N P . . Razborov Razborov [1995] then then showed showed that that SHa) S~(c~) cannot cannot prove prove superpolynomial superpolynomial lower lower bounds on the the [1995] bounds on size polynomial hierarchy size of of circuits circuits for for predicates predicates in in the the polynomial hierarchy unless unless there there are are P-natural P-natural P . This proofs proofs that that P P =I ~ N NP. This latter latter condition condition of of course course implies implies the the SPRNG SPRNG conjecture conjecture is is false; false; however, however, most most researchers researchers in in cryptography cryptography apparently apparently do do believe believe the the SPRNG SPRNG conjecture. conjecture. Thus Thus commonly commonly believed believed cryptographic cryptographic conjectures conjectures imply imply that that S�(a) S~(c~) cannot cannot prove prove superpolynomial superpolynomial lower lower bounds bounds for for NP NP predicates. predicates. A A further further observation observation SPRNG conjecture. conjecture. Razborov's of Widgerson Widgerson is is that that S� S 2 cannot cannot prove prove the the SPRNG Razborov's proof proof of characterization of used used the the conservativity conservativity of of S� $22 over over Ti T1 ,, and and the the above above characterization of the the �� E~- consequences communication complexity consequences of of Ti T~;; he he then then combined combined this this with with aa communication complexity result result (analogous to (analogous to Craig Craig interpolation) interpolation) to to extract extract aa P-natural P-natural proof proof from from the the resulting resulting PLS PLS function. function. Razborov Razborov [1994] [1994] has has subsequently subsequently given given aa simpler simpler proof proof of of the the above-discussed above-discussed theorem translations from bounded arithmetic arithmetic into theorem which which uses uses the the translations from bounded into propositional propositional logic logic (see (see Chapter Chapter VIII VIII of of this this volume) volume) plus plus interpolation interpolation theorems theorems for for propositional propositional logic. A logic. A complete complete account account of of this this simpler simpler proof proof can can be be found found in in our our survey survey article, article, Buss Buss [1997] [1997].. 3.4. Relationships Relationships between between B�n B ~ and a n d /.rr: ~n

Recall Recall from from section section 1.2 1.2.9, that B�n BEn+l /E~n f~- B�n BEn.' We We show show in in the the next next . 9, that + l f-~- .rr: is conservative over paragraphs that three theories theories are paragraphs that these these three are distinct distinct and and that that B�n BE~+I is conservative over l +

.rr: n· ~n.

outline aa proof 3.4.1. 3.4.1. Conservation C o n s e r v a t i o n of of B�n BE,+1 over .rr: / E n, .. In In this this section section we we outline proof of of + l over the -conservative over .rr: n ; this was the well-known well-known theorem theorem that that the the B�n BE,+I is IIn IIn+2-conservative over /E,; this was 2 + l is + given by Paris and and first proved proved by first by Parsons Parsons [1970] [1970].. A A model-theoretic model-theoretic proof proof was was later later given by Paris and we sketch below Buss [1994]. Kirby Kirby [1978], [1978], and we sketch below aa proof-theoretic proof-theoretic proof proof from from Buss [1994].

135 135

of Arithmetic Arithmetic Proof Theory of

Theorem. BI:n+ 1 is is Hn+2-conservative IIn+2 -conservative over over I~n. lI:n . T h e o r e m . B~n+l

Recall that that BEn+I BI:n+ l isis equivalent equivalent to to the the theory theory BHn, BIIn , which which has has IIn-REPL IIn -REPL axioms axioms Recall of the the form form of (Vx (:Jz) (Vx (Vx _� t)(3y t) (:Jy _� z)A(x, z)A(x, y) y) y) � (3z) (Vx _< � t)(3y)A(x, t) ( :Jy)A(x, y)--~ where where A A EE IIn. IIn . In In the the above above sequent, sequent, there there are are unbounded unbounded quantifiers quantifiers in in the the scope scope of bounded bounded quantifiers, quantifiers, so so the the formula formula in in the the antecedent antecedent is is aa En+l-formula, I:�++ l -formula, not not aa of I:n+ l -formula. En+l-formula.

Definition. Fix Fix nn and and suppose suppose A A EE ~n+l. I:�+ l . Definition. (1) + ,, then ~<s is A ~E 1-I II� then A A";s is defined defined to to be be A. A. (1) If If A +

(2) + ,, then ~<~ is If A A is is (3x)B (:Jx)B and and A II� then A A";s is defined defined to to bbee (3x (:Jx < s)B . A r� H (2) If � s)B.

t)B then then A A";s is defined defined to to be be (Qx (3) ~<~ is (3) If If A A is is (Qx (Qx <_ � t)B (Qx <_ t) (B";S) . � t)(B<~). Let A11,, .. .. .., , A~---} Ak � B B1l l, . . . , B~ of of En+l-formulas. I:�++ l -formulas. Then Let F---} D.. A be be a a sequent sequent A Then F ~<~ is is the formula /\;= 1 A~<8 Af and and D.. "; s is is the the formula formula V~=l B~ ~. This notation should cause the formula Aik__l--i A ~<8 This notation should cause no confusion are always always clearly clearly distinguished. no confusion since since antecedents antecedents and and succedents succedents are distinguished.

r�

. . . , Bl V�=l Bfs.

s

r,,;s

If 5' vector of then 5* abbreviates the e == CCll ,l. ... . . cs Cs is is aa vector of free free variables, variables, then e _< the formula formula If � uu abbreviates 1\ .. ... . 1\ u ) abbreviate abbreviate the corresponding vectors Cl A cs the corresponding vectors C1 _ C _< � u. u . (Vg_< (Ve � u) u) and and (3g_< (:Je � u) � uu A of bounded bounded quantifiers. quantifiers. of

s

r�

Lemma. Let nn _> � 11 .. Suppose D.. isis aa sequent sequent of L e m m a . Let Suppose F - - } A of I:� F.n+ 1++l -formulas -formulas that that is is provable provable include all all the free variables variables occurring occurring in BI:n+ 1 . Let in in BEn+I. Let ~e include the free in F---} D.. A.. Then Then

r�

u)(r

D

Intuitively, Intuitively, this this theorem theorem is is saying saying that that given given aa bound bound u u on on the the sizes sizes of of the the free free variables and on the sizes the witnesses the formulas variables and on the sizes of of the witnesses for for the formulas in in r F,, there there is is aa bound bound vv for for the the values values of of a a witness witness for for a a formula formula in in D.. A.. The The conservation conservation theorem theorem above above is is an an immediate lemma, so immediate corollary corollary of of the the lemma, so it it remains remains to to prove prove the the lemma. lemma. Proof. P r o o f . We We give give only only a a short short sketch sketch of of the the proof proof of of the the lemma lemma here; here; a a more more detailed detailed version version is is given given in in Buss Buss [1994] [1994] although although the the definitions definitions are are slightly slightly different different there. there. Firstly, -proofs in Firstly, formulate formulate BIIn BHn-proofs in the the sequent sequent calculus, calculus, using using the the inference inference rule rule form .4.5. Secondly, form of of the the IIn-REPL 1-In-REPL axioms axioms described described in in section section 11.4.5. Secondly, since since BIIn BIIn = = BI:n P of � D.. , and by the Free-cut , we may assume there is a BIIn -proof BEn+l, we may assume there is a BIIn-proof P of F ~ A, and by the Free-cut l + Elimination Elimination Theorem, Theorem, we we may may assume assume that that every every formula formula appearing appearing in in P P is is aa P to -formula. Thirdly, we shall use induction on the number of sequents in + -formula. Thirdly, we shall use induction on the number of sequents in P to I:� l + En+l prove prove that that the the Lemma Lemma holds holds for for every every sequent sequent in in P P.. The The induction induction step step involves involves aa number number of of cases; cases; we we shall shall do do only only the the two two cases cases where where the the final final inference inference of of P P is is aa replacement nal inference replacement inference inference and and where where the the fi final inference of of P P is is aa V:right V :right inference. inference. The The latter latter case case is is the the hardest hardest of of all all the the cases; cases; the the rest rest of of the the cases cases are are left left to to the the reader. reader. Suppose Suppose the the final final inference inference of of P P is is aa replacement replacement inference: inference:

r

136 136

s. S. Buss 9

.

9

.

9

(3y)A(x, y) A, ((Vx _< t) t)(Sy)A(x, y) rF---+ -t �, V'x ::; F--+ A, (3z) (3z)(Vx _ t)(3 t)(3yy ::; <_z)A(x, z)A(x, y) y) r-t �, (V'x ::; The The induction induction hypothesis hypothesis is is that t h a t / .IT:n E n proves proves u)[F <~ J D� A �<"v V V (V'x (Vx ::; ___t)(3y t)(3y ::; <_v)A(x, v)A(x, y)] y)].. [r�U ((Vu)(3v)(Vg_< V'u)(3v)(V'C' ::; u)

From this, the From this, the desired desired result result that that u)[s �U <~ D A-<'V <_v) v)(Vx _< t) t)(3y _ z)A(x, z)A(x, y)]. y)]. (3y ::; (V'x ::; J� ((Vu)(3v)(V5'_ (3z ::; �v V (9z V'u)(3v)(V'C' ::; u)[r

follows follows immediately. immediately. Now suppose that ends with with a V' Now suppose that P P ends V :right :right inference: inference: , .

9

.

9

A, B(C', S(g, d d)) rF---} -t �, rF---} -t �, A, ((Vx)B(g, x) V'x)B(C', x) -formula. We We reason reason inside inside I�n IEn . Let Let u u be Note ��++ l1-formula. be Note that that B B E II� H + since since ((Vx)B is aa En+ V'x)B is arbitrary. By By strong strong �n En-replacement the end end of of section section 1.2.9) 1.2.9) there there is is aa u' u ~ :::: > uu arbitrary. -replacement ((see see the such such that that ')B(C', u')). C' ::; u) (Vg_ u)((Vx)B(g,x) (Vx ::; _< uu')B(~, u' ) ) . (V' ( (V'x)B(C', x) H++(V'x

Let Let vv :::: > u' u t be be given given by by the the induction hypothesis so so that that induction hypothesis

(w',

_
d)).

Now . We Now let let C' g ::; <_ u u be be arbitrary arbitrary such such that that r�u F ~<~. We need need to to show show ��V A ~<~V ((Vx)B(g, x).. V'x)B(C', x) B (C', dd), Suppose ) , and Suppose that that ((Vx)B(5', x) is is false: false: then then there there is is aa dd ::; < u' u' such such that that ..., ~B(5', and by by V'x)B(C', x) the induction the induction hypothesis, hypothesis, �� A ~<" holds. Thus Thus �� A~<'V holds. v V ((Vx)B(g,x) v holds. V'x)B(C', x) holds. 3.4.2 3.4.2.. .IT:n /En+~ p r o p e r l y contains contains B�n BEn+I +l + l properly

not equal Theorem. Parsons [1970]) T h e o r e m . ((Parsons [1970]) Let Let nn :::: _> 11.. .IT:n /~n+l is not equal to to B�n B~n+l. + l is +1 . 's Proof. Proof. We'll We'll give give only only aa quick quick sketch sketch of of aa proof-theoretic proof-theoretic proof proof based based on on Godel Ghdel's second incompleteness theorem; theorem; see second incompleteness see Paris Paris and and Kirby Kirby [1978] [1978] for for aa model-theoretic model-theoretic proof. proof. The The two two main main steps steps in in our our proof proof are: are: (1) Con(.IT:n) J (1) .IT:l /'21 fF Con(IEn) D Con(B�n Con(BEn+l). This is is proved proved by by formalizing, formalizing, in in .IT: /El, the 1 , the + 1) . This proof proof of of Theorem Theorem 3.4.1 3.4.1 sketched sketched above. above. That That proof proof was was quite quite constructive: constructive: any -formula can any B�n BEn+I proof of of aa �n En-formula can be be transformed transformed into into aa free-cut free-cut free free proof proof + 1 proof by by aa primitive primitive recursive recursive process. process. Then Then the the transformation transformation of of the the free-cut free-cut free free the proof B�n BEn+l-proof into aa .IT:n-proof, /En-proof, as as in in the proof of of Lemma Lemma 3.4.1 3.4.1,, is is primitive primitive + 1 -proof into in fact ). recursive recursive ((in fact it it is is polynomial polynomial time time). Therefore, Therefore, .IT: /Ell proves proves that that if if B� BEn+I proves aa contradiction, contradiction, 00 = = 11,, then then so so n+1 proves does .IT:n ; i.e., .IT: l proves inconsistent, then .IT:n . does/En; i.e.,/El proves that that if if B�n BZn+l is inconsistent, then so so is is/En. +1 is

Proof Proof Theory Theoryof Arithmetic Arithmetic

137

� 1 can (2) �n / ~ ++1l ft- Con(�n) Con(I~n).. To To prove prove this, this, first first note note that, that, since since/~1 can prove prove the the free-cut free-cut

elimination theorem, theorem, it it is is sufficient sufficient to to prove prove that t h a t /�n ~ + 1+ 1 can can prove prove that that there there is is elimination no free-cut free-cut free f r e e �n / ~ - p-proof r o o f of of 0 = - 11;; in it suffices suffices to to show show that t h a t /�n ~ + 1+ 1 no in particular, particular, it can calculus proof can prove prove that that there there is is no n o /�n ~ sequent sequent calculus proof of of 0 = - 11 in in which which every every formula -formula. Second, �n+ 1 has -formulas; formula is is a a En ~-formula. Second,/~+1 has aa truth truth definition definition for for En ~E~-formulas; i.e., i.e., there there is is a a formula formula Tr(x, y) y) such such that that when when x x is is the the Godel Ghdel number number of of aa -formula and encoding values ~n-formula and yy is is aa sequence sequence encoding values for for the the free free variables variables of of the the En nes the formula, formula, then then Tr(x, y) y) defi defines the truth truth of of the the formula formula for for those those values. values. In In addition, �n+ 1 can definition satisfies addition,/~n+l can prove prove that that the the truth truth definition satisfies the the usual usual properties properties of truth, in of truth, in that that it it obeys obeys the the meanings meanings of of the the logical logical connectives. connectives. Chapter Chapter VIII VIII discusses discusses truth truth definitions definitions in in depth, depth, and and the the reader reader should should refer refer to to that that for for more more details. Third, Third, using -formulas, �n details. using the the truth truth definition definition for for En 5]~-formulas, / ~ ++11 can can prove, prove, by by induction induction on on the the number number of of lines lines in in the the free-cut free-cut free f r e e /�n-proof, ~ - p r o o f , that that every every sequent sequent in in the the proof proof is is valid. valid. Therefore, Therefore, it it cannot cannot be be aa proof proof of of 0 = - 11,, since since that that is is not not valid. valid. So So by by this this means, m e a n s , �n / ~ ++11 proves proves the the consistency consistency of o f /�n ~ n .. (1) and immediately that and (2) immediately that �n / ~ ++1 1 proves proves the the consistency consistency of of BEn B ~ ++I 1; ; therefore, therefore, by by 's incompleteness Godel is not equal to Ghdel's incompleteness theorem, t h e o r e m , �n / ~ ++11 is not equal to BEn B~n+l. +1 .

3.4.3. The 3.4.3. BEn B~n+l p r o p e r l y contains contains �n /~n.. The fact fact that that �n / ~ . does does not not prove prove the the + 1 properly [1978] and -replacement axioms was first established independently by Lessan En+l-replacement axioms was first established independently by Lessan [1978] and En +1 Paris Paris and and Kirby Kirby [1978]. [1978]. Their Their proofs proofs were were model-theoretic; model-theoretic; Kaye Kaye [1993] [1993] gave gave aa proof-theoretic proof-theoretic proof proof based based on on an an argument argument analogous analogous to to the the proof proof of of Theorem Theorem 3.3.2.1 using using aa Herbrand-style Herbrand-style nocounterexample nocounterexample interpretation. interpretation. 4. Strong theorems 4. S t r o n g incompleteness incompleteness t h e o r e m s for f o r I�o IA0 + + exp exp

's Second 4.1. Godel that aa sufficiently 4.1. Ghdel's Second Incompleteness Incompleteness Theorem Theorem states states that sufficiently strong, strong, axiomatized, axiomatized, consistent consistent theory theory T T cannot cannot prove prove its its own own consistency. consistency. One One way way to to strengthen strengthen this this incompleteness incompleteness theorem theorem is is by by working working with with two two theories, theories, S S and and T, T, such such that that S S is is a a subtheory subtheory of of T T:: in in some some cases, cases, one one can can establish establish that that T T cannot cannot prove its subtheory prove the the consistency consistency of of its subtheory S S.. There There are are many many cases cases iinn which which this this strengthening strengthening of of the the second second incompleteness incompleteness theorem theorem can can be be achieved. achieved. One One important important situation situation is is where where T T is is conservative conservative over over S S;; for the consistency �n ' since for example, example, BEn B2n+l cannot prove prove the consistency of of/~n, since BEn B2~+l is conservative conservative + 1 is + 1 cannot over latter theory theory cannot consistency. A second important o v e r�n / ~ and and the the latter cannot prove prove its its own own consistency. A second important example, it example interpretable in Con(S) � example is is where where S S is is interpretable in T T and and thus thus Con(S) ~ Con(T) Con(T);; for for example, it is is known known that that S 5'22 is is interpretable interpretable in in Q Q (see (see Wilkie Wilkie and and Paris Paris [1987] [1987] and and Nelson Nelson [1986]) [1986]) and Q) . and therefore therefore S 5'22 cannot cannot prove prove Con( Con(Q). A third example, A third example, and and the the one one that that is is the the main main subject subject of of this this section, section, is is the the theorem theorem of of Wilkie Wilkie and and Paris Paris [1987] [1987] that that I.6.o IA 0 + + exp exp cannot cannot prove prove the the consistency consistency of of Q Q.. This This example example does does not not fall fall into into either either of of the the above above examples, examples, since since I.6.o IA0 + + exp exp is not not interpretable interpretable in in Q Q.. is

138 138

s. S. Buss

4.2. 4.2. Before Before beginning beginning aa discussion discussion of of the the proof proof that that It.. IA0o + + exp exp does does not not prove prove Con(Q), we discuss few other of the second incompleteness theorem. theorem. The The Con( Q) , we discuss aa few other extensions extensions ofthe second incompleteness first extension is second incompleteness theorem applies first extension is that that the the second incompleteness theorem applies also also to to restricted restricted notions of "bounded consistency" -consistency" . notions of provability, provability, such such as as "bounded consistency" and and "�k "Ek-consistency". Definition. formalized in Definition. Let Let S S be be aa theory, theory, formalized in the the sequent sequent calculus. calculus. We We say say that that there is S is is bounded bounded consistent consistent if if there is no no S-proof S-proof of of the the empty empty sequent sequent � ---} in in which which S -consistent provided only only bounded bounded formulas formulas appear. appear. For For kk � >_ 0, 0, S S is is �k Ek-consistent provided there there is is no no S-proof of -formulas appear. S-proof of the the the the empty empty sequent sequent in in which which only only �k E~-formulas appear. S S is is free-cut free-cut free consistent consistent if if there there is is no no formula formula A A such such that that S S has has free-cut free-cut free free proofs proofs of of both both free � A and �. ---~A and A A---}. The (S)) and -formulas which The formulas formulas BdCon(S) BdCon(S),, ConEk Con~k(S and FCFCon(S) FCFCon(S) are are 'v'IIt VIii-formulas which express bounded consistency, express the the bounded consistency, the the �k-consistency Ek-consistency and and the the free-cut free-cut free free consistency consistency of of S S,, respectively. respectively. Of course, the cut-elimination cut-elimination theorem theorem implies implies that that aa bounded theory S S satisfies satisfies Of course, the bounded theory these these three three notions notions of of consistency consistency if if and and only only if if S S is is consistent consistent in in the the usual usual sense; sense; however, however, since since the the cut-elimination cut-elimination theorem theorem is is not not provable provable in in weak weak theories theories where where the the superexponentiation superexponentiation function function is is not not provably provably total, total, these these three three new new notions notions of of consistency will not generally be other or consistency will not generally be provably provably equivalent equivalent to to each each other or to to Con(S) Con(S)..

Definition. We is bounded Definition. We say say that that aa proof proof is bounded provided provided every every formula formula in in the the proof proof o . Similarly is is t.. A0. Similarly aa proof proof is is �k Ek,' if if every every formula formula in in the the proof proof is is in in �k Ek.. We We write write 8 S � ~ A A and and 8 S � ~ A A to to denote denote the the condition condition that that A A has has aa sequent sequent calculus calculus 8-proof S-proof respectively. in in which which every every formula formula is is in in t.. A0o or or �k Ek,, respectively.

bounded theory theory (such o , 8� Buss Buss [1986] [19861 proved proved that that if if 8 S is is aa bounded (such as as It.. IA0, S~,, T� T~,, 8 $2, 2, etc) consistency; Le., etc) then then S S cannot cannot prove prove its its own own free-cut free-cut free free consistency; i.e., S S cannot cannot prove prove FCFCon(S) hence S Con for all kk > O0.. This This was FCFCon(S) and and hence S j.l ~ BdCon(8) BdCon(S) and and S S j.l ~ �kE~-Con for all was later show that that 8 does not the bounded later strengthened strengthened to to show $22 does not prove prove BdCon(BASIC) BdCon(BASIC),, the bounded consistency consistency of of its its induction-free induction-free base base theory: theory: this this result result first first appeared appeared in in Takeuti Takeuti [1990] [1990] and and Buss Buss and and Ignjatovic Ignjatovid [1995] [1995] building building on on the the earlier earlier work work of of Pudllik Pudl~k [1990]. [1990]. A A related related result, result, proved proved by by Buss Buss and and Ignjatovic Ignjatovid [1995], [1995], is is that that the the theory theory PV PV (and (and hence hence S� S~)) does does not not prove prove the the consistency consistency of of aa finitely finitely axiomatized, axiomatized, induction-free induction-free fragment fragment PVPV- of of PV. PV. Like Like the the theorem theorem of of Wilkie Wilkie and and Paris Paris [1987] [1987] that that we we discuss discuss below, below, these these indepen independence dence results results provide provide situations situations where where aa stronger stronger theory theory cannot cannot prove prove aa consistency consistency statement statement about about aa weaker weaker theory. theory. These These results results are are interesting interesting in in their their own own right right of of course; course; but but in in addition, addition, they they are are motivated motivated by by aa yet-unfulfilled yet-unfulfilled hope hope that that indepen independence P . This dence results results of of these these kinds kinds could could lead lead to to aa proof proof that that P P f:. ~ N NP. This wistful wistful hope hope is is based based on on the the intuitive intuitive idea idea that that P P f:. ~ N NPP is is analogous analogous to to aa finitary finitary incompleteness incompleteness theorem. theorem. 4.2.1. 4.2.1. More More strengthenings strengthenings of of Godel's Ghdel's second second incompleteness incompleteness theorem theorem can can be be found found in VIII of in Chapters Chapters VII VII and and VIII of this this volume. volume.

of Arithmetic Arithmetic Proof Theory o]

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4.3. A A ttheorem of W Wilkie 4.3. h e o r e m of i l k i e aand n d PParis aris

This section section outlines outlines aa proof proof of of the the theorem theorem that that IA0 I1::.o ++ exp exp cannot cannot prove prove the the This consistency of of IA0. I1::.o . Earlier, Earlier, we we used used the the notation notation exp(x, exp(x, y,y, z) z) as as aa Ao-predicate 1::.o -predicate consistency z . We We also also define define "exp" "exp" to to be be the the sentence sentence expressing the the condition condition that that xxyY == z. expressing (W) e~p(x, y, (\fx) (Vy)(3z) (\fy) (3z)exp(x, y, z) z)

stating that that the the exponentiation exponentiation function function isis total. total. stating Theorem. (Wilkie and and Paris Paris [1987]) [1987]) IA0 I1::.o ++ exp cannot prove prove Con(IAo). Con(I1::.o ) . Therefore, Therefore, T h e o r e m . (Wilkie exp cannot I1::.o ++ exp exp cannot cannot prove prove Con(Q). Con( Q) . IA0 It prove the It is is worth worth noting noting some some theories theories that that are are sufficiently sufficiently strong strong to to prove the consisconsis tency of of IA0. I1::.o . First, First, ifif one one considers considers bounded bounded consistency, consistency, we we have that tency have that

I1::.o + IA0 + exp exp r ~ BdCon(I1::. BdCon(IA).) . To prove this fact, fact, one shows that that IA0 I1::.o ++exp can define bounded To prove this one shows exp can define a a truth truth definition definition for for bounded formulas which to allow I1::.o ++ exp to prove the validity every sequent sequent formulas which is is sufficient sufficient to allow IA0 exp to prove the validity of of every which has bounded A0-proof. 1::.o -proof. which has aa bounded Second, let 2% 2~ be the superexponentiation function defined by 20 2~ = x and Second, let be the superexponentiation function defined by = x and 2f . By 2i~+1 - 222f. By the the bootstrapping used earlier, there is is a a 1::. A0-formula 2; +1 = bootstrapping techniques techniques used earlier, there o -formula superexp(i, superexp(i, x, x, z) z) which which intensionally intensionally expresses expresses 2; 2~ = = zz.. Similarly, Similarly, superexp(i, superexp(i, x, x, z) z) is t -definable with is 1::. A~-definable with respect respect to to Si S~.. We We let let "superexp" "superexp" be be the the axiom axiom stating stating

(\fx) y, z) (w) (\fy) (3z)superexp(x, y, z).. Since Since I1::. IA0o + + superexp superexp can can prove prove the the free-cut free-cut elimination elimination theorem, theorem, it it can can also also prove prove that o ) implies Con(I1::. o ) . Therefore, o) . that BdCon(I1::. BdCon(IAo)implies Con(IAo). Therefore, I1::. IA0o + + superexp superexp r F- Con(I1::. Con(IAo). 4.3.1. 4.3.1. We We are are now now ready ready to to outline outline the the proof proof of of Theorem Theorem 4.3. 4.3. It It is is more more convenient convenient to to work work with with S $22 instead instead of of I1::. IA0o and and so so we we shall shall prove prove that that S $22 + + exp exp j.t V Con(S Con(S2). 2) ' Note and I1::. Note that that we we still still have have S $22 + + exp exp r ~- BdCon(S BdCon(S2). Also, Si S~ ((and IA0o + + D ~1) proves 2 ) . Also, 1 ) proves Con( o ) are equivalent. 5 We Q) J Con(Q) D Con(S Con(S2), so Con(S Con(S2)and Con(IAo)are equivalent. 115 We shall shall prove prove 2 ) , so 2 ) and Con(I1::. 4.3, by proving a series of lemmas, theorems Theorem Theorem 4.3, by proving a series of lemmas, theorems and and corollaries, corollaries, namely, namely, Lemma Lemma 4.3.2 4.3.2 through through Theorem Theorem 4.3.10. 4.3.10. The The proof proof is is aa modified modified version version of of the the original original [1987]. proof of Wilkie and Paris proof of Wilkie and Paris [1987]. 4.3.2. Let be . 4.3.2. Lemma. Lemma. Let <jJ(x) r be aa � S 11-formula, -formula, and and suppose suppose S$22 + + exp exp r F- (\fx)<jJ(x) (Vx)r Then Then there there is is aa constant constant kk > 00 such such that that

S2 r (\fx) (2% exists exi t J <jJ(x)) . To To improve improve readability, readability, we we shall shall often often write, write, as as above, above, aa shorthand shorthand notation notation such such as as "2% "2~ exists" exists" as as an an abbreviation abbreviation for for (3y)superexp(k, (3y)superexp(k, x, x, y) y).. 15The 15The proof proof below belowthat that S2 $2 + + exp exp¥Y Can(S2) Con(S2) can can be be understood understood without without knowing knowinghow how to to prove prove in in Si S~ that that I�o IAo and and S2 $2 are are equiconsistent. equiconsistent.

140 140

S. Buss Buss S.

Without loss loss of of generality, generality, r¢>(x) isis of of the the form form (3U)r (3U)¢>M(X, u) U) with with CM ¢>M eE A0. �o . PProof. r o o f . Without proves (Vy)(3z)(2 ('v'y) (3z) (2vY == z)z) D:J (Vx)r The hypothesis hypothesis implies implies that that $2 ('v'x)¢>(x) , which which can can be be The 82 proves put in in prenex prenex form form as as put

('v'x)(3u) (3u)(~y) (3y)(W) ('v'z)[2' [2Y #=I zZ VV CM(X, ¢>M(X, u)]. u) ] . (W) Now, momentarily momentarily enlarge enlarge $2 82 to to have have Skolem Skolem functions functions for for all all A0 �o formulas, formulas, thereby thereby Now, 82 axiomatized axiomatized by by purely purely universal universal formulas. formulas. The The strong strong form form of of Herbrand's Herbrand' s making $2 making ( section 2.5.3 2.5.3 of of Chapter Chapter I)I) implies implies that that there there isis an an [£ >> 00 and and there there are are theorem (section theorem St(X, ul,...,Ut-l,Yl,...,yl-1), Ub · · · , Ut- b Yl , · · · , Yt- l ) , t2 (X, ul,yl),..., Ub Yl ) , . ' " s~(x, tl (X) , s2(x, S2(X, ul, Yl) , t2(x, Ul, yl), terms Sl(U), Sl (U) , tl(X), terms tt (x, uUll,, ... .. ,. , ul-t, Ut- l , YI,..., that $2 so that proves 82 proves t,(x, Yt- l ) so Yb " . , Yl-1)

t 1(x) #=I ~,Zl vV r¢>M(X, ~,(x)) t2 (x,zt l #=I z2 S l (X)) vV (Vz~)[2 Z2 vV r¢>M(X, s~(x, S2(X, ~,))v Zl ))V (w)[(Vz,)[2 ('v'X) [('v'zt } [2''~1 ('v'z2 ) [2'~,'~'~ , , x . ...9. vV (w~)[2,2~-,z, Zl-tl :/:=I z~ . .. ",-,~ , ) ) ])]. . .· ].] .]]]]. . ('v'zt ) [2t2 ( ,Zl ..... S2(X, zZl, ,,. ·. ..,. , ~Zl_-t} Zt vV r¢>M(X, s~.(x,

Since 9r -,¢> D :J -~r -'¢>M(X, s;(x, Z)) , we we immediately immediately have have that that $2 82 also also proves proves Since S,(X, ~),

t 1(x) #=I Zl t2 (x ,Zl ) #=I Z2V ('v'X) [¢>(X) vV (vZl)(2 ('v'zl ) (2'1(~) (w)[r ~v Zl vV (w~)(2 ('v'z2 ) (2'~(~,~1)

. ...9. vV ('v'z l (x ,Zl"....." ,Zl-tl . . . ))] (Vz~)(2 ",-,~ =I # Zt) z~)...))] t) (2t',(~,''

where each t; with polynomial polynomial growth growth rate rate with defined by where each ti is is aa function function with with graph graph defined by aa 1 (x) , holds, provided �o-formula. A0-formula. Thus, Thus, 82 $2 proves proves that that ¢>(x) r holds, provided there there exists exists Zl Zl = = 22ttl(~), z x x, t l , l Z " ,Zl-d z2 = - 2h( 2t2(x'zl),..., - 22te(~'zl ze-~).. Since Since each each t; ti has has polynomial polynomial growth growth rate, rate, d, . . . , Zzlt = ( "..... Z2 ; ' s are the bounded by the values values of of the the zzi's are bounded by 2: 2~+ for sufficiently sufficiently large large x x E E N; N; therefore, therefore, + 11 for holds. 82 $2 proves proves that that if if 2: 2~+ exists, then then ¢>(x) r holds. Taking Taking kk = = £[ + + 11,, Lemma Lemma 4.3.2 4.3.2 is is +11 exists, proved. proved. ((Solovay Solovay [1976]) [1976]) For For each each n, n, kk � >_ 0O,, there there is is aa 8 S~-proof P of of i -proof P n l and (3x)(superexp(k,n,x)) with si size polynomiaUy bounded bounded in in terms terms of of IInl and kk.. In In (3x) ( superexp(k, 11, x)) with ze polynomially addition, addition, P P is is aa �2k E2k+l-proof. + l -proof.

4.3.3. 4.3.3. Theorem. Theorem.

The Proof. P r o o f . The The proof proof is is based based on on using using formulas formulas that that define define inductive inductive cuts. cuts. The particular (x) defined particular ones ones we we need need are are formulas formulas J;(x) Ji(x) and and K; Ki(x) defined as: as:

always true o (x) {::} r 00 = = 00 ((always true)) JJo(x)

Ko (x) {::} (2X = y) Ko(~) ,~ (3y) (3v)(2~=v)

(K; (z) :J~ K;(z ,~ ('v'z) (W)(K,(,) K,(z + + x)) ~)) JJ,+,(~) i+ l (X) {::}

Ki Ki+l(X) r (3y)(2 (3y)(2X~ = = Yy /\A Ji J,+l(y)) + 1 (y)) + 1 (X) {::}

Lemma. Lemma.

((~) a) 8s~i I-F JJ~(0) k (O) ((b) b ) 8i S~ I-k- Jk(x) Jk(x) :JD Jk(x Jk(x + + 1) 1)

Proof Proof Theory Theory of of Arithmetic Arithmetic

141 141

u) (c) si (X) 1\ (c) s~ f-e Jk J~(~) ^ Uu < < x9 :J ~ Jk( J~(u) (d) S~ fF- Jk(x) Jk(x) :J D Jk(x J~(x + + x) x) (d) Si

s~ f-~ Kk g~(o) (e) Si (O)

(U) (f) Si S~ fF- Kk Kk(x) Au u< <x x :J D Kk gk(u) (x) 1\ (f) s] f-~ Kk g~(~) ~ Kk g~(~ + 1) (g) Si (X) :J (X +

(h) , x, z) . (h) Si S~ fR Kk(X) Kk(x) :J D (3z)superexp(k (3z)superexp(k + + 11,x,z). Parts (a)-(g) (a)-(g) are are proved proved simultaneously simultaneously by by induction induction on on k. Part (h) (h) is is likewise likewise proved proved Parts k. Part using using induction induction on on kk.. Moreover, Moreover, it it is is easy easy to to verify verify that that the the S S2-proofs of formulas formulas 2 -proofs of (a)-(g) polynomial size (a)-(g) are are polynomial size in in kk,, and and involve involve only only � E2k+l-formulas. 2k+ 1 -formulas. By using using (d) and (c) (c) of of the the lemma, lemma, it it is is straightforward straightforward now now to to give give find find an an By (d) and Si -proof of (h) give S~-proof of Jk(rr) Jk(n) of of size size polynomial polynomial in in Inl In I and and kk;; from from this, this, (e) (e) and and (h) give the the (3z )superexp(l£, rr, z) . desired desired proof proof of of P P of of (3z)superexp(k, n, z).

4.3.4. Lemma. 4.3.4. L emma. kk � > 00 such such that that

Suppose . Then, Suppose r¢>(x) Ee �l ~]1 and and S $22 + + exp exp fF- (V'x)¢>(x) (Vx)r Then, there there is is aa s~ r

(w)(& %

r

Lemma Lemma 4.3.4 4.3.4 is is proved proved from from Lemma Lemma 4.3.2 4.3.2 by by formalizing formalizing the the argument argument of of Lemma Lemma 4.3.3 4.3.3 in in Si S~..

4.3.5. 4.3.5. Lemma. Lemma. Let Let ¢>( r x ) be be aa V'ITt VII~ -formula, -formula, which which is is without without loss loss of of generality generality of of ¢>M Ee ITt the the form form (V' (Vy)r y) where where CM II~.. Then Then there there is is aa term term tt such such that that Y )¢>M(X, y)

This special case This lemma lemma is is aa special case of of Theorem Theorem 2.1.2. 2.1.2.

Let -sentence such 4.3.6. 4.3.6. Lemma. Lemma. Let ¢> r be be aa V'II� VH~-sentence such that that S $22 + + exp exp fR ¢> r . Then Then there there is is aa kk � that > 00 such such that s~ ~ ~r --+ = Co~.~ ( & ) .

Proof. with Proof. Without Without loss loss of of generality, generality, ¢> r is is of of the the form form (V'X)¢>M(X) (Vx)r with ¢>M CM aa II� 1-It- formula. . On formula. By By Lemma Lemma 4.3.4, 4.3.4, S $22 proves proves ((Vx)(S2 ~ ¢>M(;[)) CM(X__)). On the the other other hand, hand, V'X) (S2 � Lemma Lemma 4.3.5 4.3.5 implies implies that that S $22 proves proves ~r

D (& ~ - r

These two suffice to These two facts facts suffice to prove prove Lemma Lemma 4.3.6. 4.3.6. o nEk (S 4.3.7. 4.3.7. Lemma. L e m m a . Let Let kk > > OO.. Then Then S $22 + + exp exp proves proves C Conr.~ ($2). 2) .

142 142

s. S. Buss Buss

Sketch) . The Proof. Proof. ((Sketch). The proof proof of of this this has has two two main main steps: steps: (82 ) J~ GonEk ( 82 ) . This (1) Firstly, Firstly, one one shows shows that that 8 $22 + + exp exp proves proves BdCon BdCon(S2) Conr.k(S2). This is is (1)

a) Assume done, done, by by formalizing formalizing the the following following argument: argument: ((a) Assume that that P P is is aa �k-proof E~-proof of b) By of 00 = - 11 in in the the theory theory 8 $2. By using using sequence sequence encoding encoding to to collapse collapse 2 • ((b) adjacent adjacent like like quantifiers, quantifiers, we we may may assume assume w.l.o.g. w.l.o.g, that that each each formula formula in in P P has has c) By at at most most k + + 11 unbounded unbounded quantifiers. quantifiers. ((c) By applying applying the the process process used used to to prove prove the the Cut-Elimination Cut-Elimination Theorem Theorem 2.4.2 2.4.2 of of Chapter Chapter I, I, there there is is aa bounded bounded S2-proof of 00 = - 11 of of size size at at most most 2911PII Since only only finitely finitely many many iterations iterations of of �:!4 . Since 8 2 -proof of "2k+4" exponentiation exponentiation are are needed, needed, the the last last step step can can be be formalized formalized in in 8 $22 + + exp exp.. bounded consistency one shows shows that that 8 $22 + § exp can can prove prove the the bounded consistency of of 8 $2. The ((2) 2 ) Secondly, Secondly, one 2 • The general -proof P of 0 = 1 , then there is general idea idea is is that that if if there there is is a a bounded bounded 8 S2-proof P of 0 1, then there is aa 2 aa fixed fixed value value £g so so that that all all variables variables appearing appearing in in P P can can be be implicitly implicitly bounded bounded ize(p) where size(P) is the number of symbols in P. (In fact, g - 3 by L L = - 22~;ize(P) by where size (P) is the number of symbols in P . (In fact, £ = 3 works. bounded by definition can works.)) Once Once all all variables variables are are bounded by L, L, aa truth truth definition can be be given given based on on the the fact fact that that 22 LLs~z~(P) exists. With With this this truth truth definition, definition, 8 $22 + § exp exp can can based 'iz« P) exists. prove prove that that every every sequent sequent in in the the 8 S2-proof is valid. valid. 2 -proof is

4.3.8. The 4.3.8. Corollary. Corollary. The theory theory 8 $22 + + exp exp is is conservative conservative over over the the theory theory 8 $22 U (J {{ GonEk 82 ) : k 2: Con~ k (($2)" > O} 0} with with respect respect to to \lIlt VHb -consequences. -consequences. Proof. P r o o f . The The fact fact that that the the first first theory theory includes includes the the second second theory theory is is immediate immediate from from The conservativity is immediate Theorem Theorem 4.3.7. 4.3.7. The conservativity is immediate from from Lemma Lemma 4.3.6. 4.3.6. Incidentally, Q, we Incidentally, since since 8 $22 is is globally globally interpretable interpretable in in Q, we also also have have that that the the theories theories 82 ) : kk 2: (Q) : kk 2: 8 $22 + + {{ GonEk Con~ k (($2): _ O} 0} and and 8 $22 + + {{ GonEk Con~.~(Q): >_ O} 0} are are equivalent. equivalent. 4.3.9. T h e o r e m .

$2 U {Con~ k(S2): k > 0} l/Con(S2).

It It is is an an immediate immediate consequence consequence of of Theorem Theorem 4.3.9 4.3.9 and and Corollary Corollary 4.3.8 4.3.8 that that 8 $22 + + exp establish. So exp j.t ~ Gon(82) Con(S2),, which which is is the the main main result result we we are are trying trying to to establish. So it it remains remains to to prove prove Theorem Theorem 4.3.9: 4.3.9: 's Diagonal Proof. � -sentence P r o o f . Let Let k > > 00 be be fixed. fixed. Use Use Godel Ghdel's Diagonal Lemma Lemma to to choose choose an an 3� 32~-sentence

( + + GonEk (82) ( ) � -'
Now 82 ) proves Now > 00 82 ) f-k -,
Proof rithmetic Proof Theory of of A Arithmetic

143 143

$22 + + COnEk C o n ~ (S ($2) does not not prove prove -,cPk -~r . Therefore Therefore S $22 + + ConEk Con~ k (S ($2) also cannot cannot prove prove S 2 ) also 2 ) does C o ~ ( S ~ )2 ) .. Con(S Since this this argument argument works works for for arbitrary arbitrary kk,, the the Compactness Compactness Theorem Theorem implies Since implies O} cannot cannot prove that that S $22 + + {{ ConEk Con~ k (Q) (Q):: k k � > 0} prove Con(S Con(S2). 2) '

The The proofs proofs above above actually actually proved proved something something slightly slightly stronger stronger than than Theorem Theorem 4.3.9; 4.3.9; namely, namely, 4.3.10. T h e o r e m .

There is an m = O(k) such that S2+ Con~ (S2) jz Con~m (S2).

We conjecture that proof at We conjecture that m m = -- kk + + 11 also also works, works, but but do do not not have have aa proof at hand. hand.

4.3.11. A 4.3.11. A related related result, result, which which was was stated stated as as an an open open problem problem by by Wilkie Wilkie and and 1987] and 1993,Coro. 5.34 Paris Paris [[1987] and was was later later proved proved by by Hajek H~jek and and Pudlak Pudl~k [[1993,Coro. 5.34],] , is is the the fact fact that there is is aa 'v'm VH~-sentence r such S1 + + exp exp fF- cP r but that Sl S~ J.L jz cP r for for all all but such such that that there -sentence cP, such that that Si kk >�0 O. .

Acknowledgements. A c k n o w l e d g e m e n t s . We We are are grateful grateful to to JJ.. Avigad, Avigad, C C.. Pollett, Pollett, and and JJ.. Krajicek Krajf~ek

for corrections to preliminary versions for suggesting suggesting corrections to preliminary versions of of this this chapter. chapter. Preparation Preparation of of this article was was partially partially supported supported by by NSF grant DMS-9503247 DMS-9503247 and and by by cooperative cooperative this article NSF grant research NSF and Republic Ministry research grant grant INT-9600919/ME-103 INT-9600919/ME-103 of of the the NSF and the the Czech Czech Republic Ministry of Education. Education. of References References W. ACKERMANN W. ACKERMANN (1941) Zur 17, pp. [1941] Zur Widerspruchsfreiheit Widerspruchsfreiheit der der Zahlentheorie, Zahlentheorie, Mathematische Mathematische Annallen, 1117, pp. 162-194. 162-194. R. SOMMER J. AVIGAD AVIGAD AND AND R. SOMMER ordinal analysis, 3, pp. [1997] A model-theoretic model-theoretic approach approach to to ordinal analysis, Bulletin of Symbolic Logic, Logic, 3, pp. 17-52. 17-52. (1997)

J. BARWISE BARWISE [1977] Handbookof of Mathematical Logic, Logic, North-Holland, North-Holland, Amsterdam. Amsterdam. (1977) Handbook J. H H.. BENNETT BENNETT [1962] On Spectra, PhD PhD thesis, thesis, Princeton Princeton University. University. (1962) G G.. BOOLOS BOOLOS 1989) A A new new proof proof of of the the G6del Ghdel incompleteness incompleteness theorem, theorem, Notices of the American Mathematical ([1989] Society, 36, 36, pp. pp. 388-390. 388-390. [1993] The Logic Logic of Provability, Cambridge Cambridge University University Press. Press. (1993) S. R. R. Buss S. Buss

Bibliopolis, Napoli. 1985 Princeton (1986) [1986] Bounded Arithmetic, Bibliopolis, Napoli. Revision Revision of of 1985 Princeton University University Ph.D. Ph.D. thesis. in: Logic (1990) [1990] Axiomatizations Axiomatizations and and conservation conservation results results for for fragments fragments of of bounded bounded arithmetic, arithmetic, in: Logic and Computation, Computation, proceedings proceedings of a Workshop Workshop held held Carnegie-Mellon Carnegie-Mellon University, 1987, 1987, W. Sieg, Sieg, ed., vol. 106 Contemporary Mathematics, American Mathematical Mathematical Society, W. ed., vol. 106 of of Contemporary Mathematics, American Society, Providence, Rhode Island, pp. 57-84. [1992] A A note note on on bootstrapping bootstrapping intuitionistic intuitionistic bounded bounded arithmetic, arithmetic, in: in: Proof Theory: A (1992) selection of H. G. G. Aczel, of papers papers from the Leeds Leeds Proof Theory Programme 1990, 1990, P. P. H. Aczel, H. Simmons, S. Wainer, eds., Cambridge H. Simmons, and and S. S. S. Whiner, eds., Cambridge University University Press, Press, pp. pp. 149-169. 149-169.

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S. s. Buss Buss

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CHAPTER III III CHAPTER

Hierarchies of of Provably Provably Recursive Recursive Functions Functions Hierarchies Matt Fairtlough Fairtlough Matt Department of Computer Science, University of Sheffield, She]field, Sheffield She]field Sl $1 4DP, ~DP, England

Stanley S. Wainerl Wainer 1 Stanley of Pure Mathematics, University of Leeds, Leeds LS2 9JT, England Department of

Contents Contents

1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1. Introduction S t r u c t u r e d ordinals ordinals and a n d associated associated hierarchies hierarchies . . . . . . . . . . . . . . . . . . . 2. Structured 3. C o m p l e t e w-arithmetic w - a r i t h m e t i c .. .. .. .. .. . . . . . . . . . . . . . . . . . . . . . . . . . . . 3. Complete 4. Provably P r o v a b l y recursive recursive functions functions of of PA PA . . . . . . . . . . . . . . . . . . . . . . . . . 4. I n d e p e n d e n c e results results for for PA PA .. .. .. .. . . . . . . . . . . . . . . . . . . . . . . . . . 5. Independence 6. T h e "true" "true" ordinal ordinal of of PA PA .. .. .. .. . . . . . . . . . . . . . . . . . . . . . . . . . . . 6. The Theories with with transfinite transfinite induction induction . . . . . . . . . . . . . . . . . . . . . . . . . 7. Theories References References .. .. .. .. .. .. . . . . . . . . . . . .. . .. .. . . . . . . . . . . . . . . . . . . . . . . . . .

150 153 164 175 190 193 199 203

11The second author thanks the Department Department of of Philosophy at Carnegie Mellon University for generous hospitality and and the opportunity to teach some some of this material, during his year as a Fulbright Scholar 1992-93. HANDBOOK H A N D B O O K OF O F PROOF P R O O F THEORY THEORY Edited S. R. E d i t e d by by S. R. Buss Buss Elsevier Science Science B.V. B.V. All All rights rights reserved reserved © 1998 Elsevier

M. Fairtlough and S. Wainer Wainer M.

150 150 1 1.. Introduction Introduction

Since the the recursive recursive functions functions are are of of fundamental fundamental importance importance in in logic logic and and computer computer Since science, science, it it is is aa natural natural pure-mathematical pure-mathematical exercise exercise to to attempt attempt to to classify classify them them in in some some way logical and way according according to to their their logical and computational computational complexity. complexity. We We hope hope to to convince convince the do: interesting the reader reader that that this this is is also also an an interesting interesting and and aa useful useful thing thing to to do: interesting because it it brings brings to to bear, bear, in in aa clear clear and and simple simple context, context, some some of of the the most most basic basic because techniques theory such cut-elimination and techniques of of proof proof theory such as as cut-elimination and ordinal ordinal assignments; assignments; and and useful because it connections with useful because it brings brings out out deep deep theoretical theoretical connections with program-verification, program-verification, program program complexity complexity and and finite finite combinatorics. combinatorics. One One might might wonder wonder why why this this branch branch of of recursive function theory should should most recursive function theory most appropriately appropriately be be viewed viewed in in aa proof-theoretic proof-theoretic light, light, but but this this is is simply simply because because the the underlying underlying concerns concerns are are of of an an intensional intensional character, to do derivations of character, to do with with computations computations or or derivations of functions functions according according to to given given programs rather than than merely definitions in in extenso extenso as as sets sets of of ordered pairs. programs rather merely their their definitions ordered pairs. The The proof-theoretic proof-theoretic connection connection is is immediately immediately observable observable by by considering considering the the most most basic basic recursive recursive operation operation of of all, all, namely namely composition: composition: given given functions functions f f and and gg define define hh := ff oo gg by by the the rule rule

(g(x) -+ (f(y) (g(x) = = y) y)--+ ( f ( y ) == z) z) -+ --+ (h(x) (h(x)

= =

z) z)..

Then the Then the usual usual quantifier quantifier rules rules of of logic logic yield yield (x) = = y) y) -+ --+ \/ V yy .. 3:Jzz. (. f(f(y) (y) = - z) z) \/V xx ..3:Jyy. (. g(g(x)

-+ --~

(x) = - z) z) \/Vxx ..3:Jzz. (. h(h(x)

and and so so the the totality/termination totality/termination of of h h follows follows from from that that of of gg and and f f respectively respectively by by means means of of two two applications applications of of Cut. Cut. As As we we shall shall see, see, cut-elimination cut-elimination then then yields yields aa "direct" "direct" proof proof from from which which the the complexity complexity of of hh can can be be read read off. off. It relationship between hand, and It is is the the relationship between computational computational complexity complexity on on the the one one hand, and logical logical complexity complexity (of (of termination termination proofs) proofs) on on the the other, other, which which forms forms our our principal principal theme theme here. here. Put Put simply, simply, aa program program satisfies satisfies aa specification specification

\/input. (input, output) Vinput. :Joutput. 3output. Spec Spec(input, output) if computes an y) holds. if for for each each input input x x it it computes an output output y y such such that that Spec(x, Spec(x, y) holds. Mere Mere knowledge knowledge that that the the specification specification is is true true tells tells us us only only that that there there exists exists aa while whileprogram it. But possible structure program satisfying satisfying it. But to to gain gain information information about about the the possible structure and and specification is complexity complexity of of such such a a program program we we need need to to know know why why the the specification is true, true, in in other other words words we we need need to to be be given given aa proof. proof. Thus Thus our our primary primary interest interest will will be be with with those those (recursively (recursively enumerable) enumerable) classes classes of of functions functions which which are are "verifiably "verifiably computable" computable" in in given given subsystems subsystems of of arithmetic arithmetic and and analysis analysis whose whose proof-theoretic proof-theoretic strength strength is is well-understood. is not well-understood. This This is not to to say say that that the the problem problem of of classifying classifying all all recursive recursive functions functions "in "in one one go" go" is is uninteresting-far uninteresting--far from from it. it. The The known known general general results results of of Feferman Feferman [196 [1962] in that that direction-on direction--on completeness completeness and and incompleteness incompleteness of of hierarchies hierarchies 2] in ' s O-raise generated generated along along paths paths in in Kleene Kleene's O--raise further further deep deep questions questions which which remain remain unanswered, unanswered, e.g. e.g. "what "what is is a a natural natural well-ordering?" well-ordering?"

151 151

Provably Recursive Functions

Our Our aim aim then then is is to to find find uniform uniform scales scales against against which which we we can can measure measure the the computational computational complexity complexity of of functions functions verifiably verifiably computable computable in in "known" "known" theories. theories. By By "complexity" "complexity" we we mean mean "complexity "complexity in in the the large" large",, as as measured measured by by the the rates rates of of growth irrespective of they be be polynomial, growth of of resource-bounding resource-bounding functions functions irrespective of whether whether they polynomial, exponential exponential or or much much worse. worse. We We do do not not wish wish to to place place prior prior restrictions restrictions on on their their size, size, but but rather rather to to have have the the means means of of comparing comparing one one with with another. another. How How might might this this be be achieved? achieved? What What form form should should a a "subrecursive "subrecursive scale" scale" take? take? To To answer answer this this we we need need first definitions we first to to ask ask what what kind kind of of features features of of recursive recursive definitions we are are actually actually trying trying to to measure and measure and compare. compare. Suppose kind, together Suppose given given a a number-theoretic number-theoretic program program of of some some kind, together with with an an operational semantics operational semantics determining determining for for each each number number n n aa space space C(n) C(n) consist consisting of of all all computations and sub-computations of the the program, on in ining computations and sub-computations of program, starting starting on puts The sub-computation relation puts � _ n n.. The sub-computation relation induces induces aa tree tree structure structure on on C(n) C(n) and and we we will will assume assume further further that that it it has has been been linearly linearly ordered ordered by by a a suit suitThen able ordering -
.

j,~. N N ~

NN

M. M. Fairtlough and S. Wainer

152 152

as as follows: follows:

{

if j~(g) .= { 9g ifa a ==O 0 Jo. (g) .. = An.J(JPn(a)(g))(n) i f aa ;i # O0 >'n . J(jPn(o.} (g)) (n) if Two Two simple simple examples examples spring spring immediately immediately to to mind, mind, but but the the reader reader should should not not be be deceived deceived by by their their apparent apparent simplicity simplicity since since they they turn turn out out to to be be fundamental fundamental tools tools of of the the sub-recursive sub-recursive hierarchy hierarchy trade. trade. Both Both are are obtained obtained as as iterates iterates of of the the composition composition := g h, the rst with xed and operator operator J(g, J(g, h) h) "= g oo h, the fi first with 9 g fi fixed and h h varying, varying, the the second second with with h h fifixed xed and and 9 g varying. varying. Fix 9 gE E iN and define define Jg(h) Jg(h) = = ggoh. Then for for every every h h E E NN iNs and and every every n n E EN iN,, II Fix NNs and o h. Then

JJ~(h)(n) g~(h(n)) ; (h) (n) = l(h(n)) where where kk is is the the length length of of the the descending descending chain chain ...

o.

In In particular particular with with 9 g the the successor successor function function and and h h constantly constantly zero zero we we obtain obtain for for each each a a :; _ , 7 the the functions functions Ga(n) least kk.. (P (P~(a) O) := least G ; (a) = 0) o. (n) "=

constituting constituting the the so-called so-called Slow-Growing Slow-Growing Hierarchy. Hierarchy. Notice Notice that that G-y(n) G~(n) mea measures sures the the size size of of ,[nJ 7[n].. II II Fix Fix h hE E iNs NN and and define define Jh(g) Jh (g) = = gg oo h. h. Then Then for for every every 9 gE E iNs NN and and every every n n E EN iN,, J~(g)(n) = g(hk(n))

where where kk is is the the length length of of the the descending descending chain chain O.

In In particular particular with with h h the the successor successor and and 9 g the the identity identity function function we we obtain obtain for for the functions every a every a :; ~ , 7 the functions

H Ha(n) least m m.. (P (Pm-lPm-2"'" Pn+lPn(a) = 0) 0) := least m- 1 Pm -2 · · · Pn o. (n) "= + 1Pn(a) = constituting so called constituting the the Hardy Hardy Hierarchy Hierarchy ((so called because because Hardy Hardy [1904J [1904] was was the the first first to to make make use use of of them, them, in in "exhibiting" "exhibiting" aa set set of of reals reals with with cardinality cardinality Nd. R1). As As we we shall shall see see however, however, this this hierarchy hierarchy also also contains, contains, embedded embedded within within it, it, the the Fast-Growing Fast-Growing Hierarchies Hierarchies Bo. Ba and and Fo. Fa which which provide provide crucial crucial links links between between proof proof theory theory on on the the one one hand hand and and recursive recursive function function theory theory on on the the other. other. This This paper paper is is about about these these hierarchies, hierarchies, their their interconnections, interconnections, and and their their relation relationships ships with with proof proof theory theory as as displayed displayed in in "subrecursive "subrecursive classification classification theorems" theorems" of of the the following following form form Given an I TIl , then then the the Given an arithmetical arithmetical theory theory T T with with proof-theoretic proof-theoretic ordinal ordinal IIITII, provably those functions provably recursive recursive functions functions of of T T are are exactly exactly those ]unctions computable computable within within complexity-bounds complexity-bounds HOI. Ha for .for a a � -~ IITII IITll..

Provably Recursive Functions

153 153

We shall shall concentrate concentrate on on the the cases cases where where T T is is Peano Peano Arithmetic Arithmetic (PA) (PA),, aa fragment fragment We of it it or or an an extension extension of of it it by by some some axiom axiom of of transfinite transfinite induction. induction. Pohlers, Pohlers, in in this this of volume, provides provides an an ordinal ordinal analysis analysis for for many many richer richer theories, theories, but but once once this this is is done done volume, for aa theory theory T T its its subrecursive subrecursive classification classification may may then then be be reduced reduced to to that that of of aa for corresponding theory theory of of transfinite transfinite induction induction over over order order types types -< -~ IITII IITII. Thus the the corresponding . Thus two papers papers together together supply supply the the methods methods one one needs needs to to classify classify the the provably provably recursive recursive two functions of of aa wide wide spectrum spectrum of of theories. theories. functions Throughout this this paper paper we we shall shall only only consider consider theories theories based based on on classical classical logic. logic. It It Throughout is known known by by e.g. e.g. Friedman Friedman [1978] [1978] that that in in general, general, the the provably provably recursive recursive functions functions is will be be the the same same whether whether the the underlying underlying logic logic is is intuitionistic intuitionistic or or classical. classical. will We view view proof proof theory theory as as an an attempt attempt to to analyse analyse the the truth truth definition definition and and to to We classify truths. truths. Consequently Consequently we we here here use use proof proof theory theory to to measure measure the the complexity complexity of of classify recursive functions functions according according to to the the logical logical complexity complexity of of the the judgements judgements that that they they recursive terminate. We We begin begin by by developing developing in in section inductive class class OS ~s of of ordinal ordinal general inductive terminate. section 22 aa general presentations, together together with with their their arithmetic arithmetic and and associated associated function function hierarchies. hierarchies. presentations, Section 33 presents presents arithmetic arithmetic truth truth as as a a cut-free cut-free infi infinitary calculus in in the the style style of of Tait Tait Section nitary calculus [1968],, with with an an assignment assignment of of ordinal ordinal bounds bounds based based on on Buchholz Buchholz [1987] [1987] but but modified modified [1968] and simplified simplified here here following following Fairtlough Fairtlough [1991]. [1991]. In In this this presentation presentation the the slow-growing slow-growing and hierarchy arises arises naturally naturally as as the the class class of of bounding functions for for the the truth truth of of existential existential hierarchy bounding functions statements. Addition of the the Cut Cut rule rule then then yields yields an an infinitary proof theory theory for for which which statements. Addition of infinitary proof the fast-growing fast-growing hierarchies hierarchies now now supply supply the the bounding functions. In In section section 44 we we the bounding functions. embedding them read off off classification classification theorems theorems for for PA PA and and its its fragments fragments by by embedding them into into read infinitary framework. the infinitary framework. These These results results are are subsequently subsequently extended extended in in section section 77 to to the theories of of transfinite transfinite induction. induction. Section Section 5 gives gives an an application application to to aa well-known well-known theories mathematical independence result for for PA PA due due originally originally to to Kirby Kirby and and Paris Paris [1982] [1982].. mathematical independence result However the the proof proof given given here [1983] and displays a a clear However here is is due due to to Cichon Cichon [1983] and displays clear connection connection with reduces fastto slow-growing slow-growing by computing aa fast- to by computing with the the Hardy Hardy hierarchy. hierarchy. Section Section 66 reduces such that that Bo. G0.+ ' This This amounts amounts to to aa complete complete cut-elimination I-t c~ a+ map map a c~ ~ + such B~ -= Go+. cut-elimination since proofs of of II rrg~ sentences now reduced cut-free truth-definitions. The since proofs sentences are are now reduced to to their their cut-free truth-definitions. The result is a a new more subtle assignment to theories T T which first result is new and and more subtle ordinal ordinal assignment to theories which was was first discovered by Girard Only the initial stages stages of are needed needed [1981] . Only the initial of this this reduction reduction are discovered by Girard [1981]. appropriate for for PA) they already here (those appropriate PA) but but they already serve serve to to illustrate illustrate the the subtlety subtlety and and here (those power of of ordinal ordinal assignments assignments in power in providing providing uniform uniform measurements measurements of of proof-theoretic proof-theoretic complexity. complexity. 2. S Structured 2. tructured oordinals r d i n a l s aand n d aassociated s s o c i a t e d hhierarchies ierarchies

A "structured" "structured" countable ordinal is is one one for for which an arbitrary but fixed "fun A countable ordinal which an arbitrary but fixed "fundamental sequence" sequence" has has been been assigned assigned to to each each limit limit below below it. it. The The most most convenient convenient damental of abstract abstract "tree"tree way to introduce introduce such such ordinals ordinals is is in in two two stages. stages. First First the the set set ~0 of way to ordinals" is is defined defined inductively, inductively, in in aa manner manner reminiscent reminiscent of of Kleene's Kleene 's O 0 but but withwith ordinals" out any any conditions conditions (effectivity (effectivity or or otherwise) otherwise) being being placed placed on on sequences. sequences. Then Then the the out subset subset ~s OS of of "structured "structured tree-ordinals" tree-ordinals" is is obtained obtained by by requiring requiring that that fundamental fundamental

M. Fairtlough Fairt/otlgh and S. Wainer M.

154 154

sequences sequences mesh mesh together together in in an an appropriate appropriate way, way, as as in in Schmidt Schmidt [1976] [1976] and and Ketonen Ketonen and Solovay Solovay [1981]. [1981] . and It will will be be intuitively intuitively clear clear from from their their uniform uniform construction construction that that the the particular particular It recursive ones. ones. tree-ordinals named named and and used used in in what what follows follows are are all all recursive tree-ordinals

2.1. DDefinition. The set set ~n of of countable countable tree-ordinals tree-ordinals a, a, /~, {3, 7, 'Y, .. .. .., , A, A, ... . . . is is 2.1. efinition. The generated inductively inductively according according to to the the rules: rules: generated • 0En 9 OEf~ •

9 aaEE~n ~::::} aa++l :1=:= a Ua{Ua }{a} E~E

•

n

Vx eE NN ((a n. := ((a~ )x~) NxE N eE n. 9 w ~ x eE ~n)) ~::::} ~a :=

2.2. Note. A will will always always denote denote aa "limit" "limit" A A == {Ax/xeN. (Ax ) E N . We usually write, write, more more 2.2. N ote. A We usually X suggestively, A A == sup sup Ax. Ax . suggestively, 2.3. DDefinition. 2.3. efinition. •

The sub-tree sub-tree ordering ordering .< -< is is the the transitive transitive closure closure of of the the rules rules The

9a a . -< < aa++l f1o for r a lall l a Eaf tE

•

n

am .< all aa EE ~n and and m m EE N. N. -< (a~) (ax ) for 9 a.~ for all

2.4. RRemark. One of of the the immediate immediate consequences of this intensional approach to 2.4. emark. One consequences of this intensional approach to n. For example, if if we identify xx EE N ordinals that .< -< is is only ordinals is is that only aa partial partial ordering ordering on on ft. For example, we identify N with + 11 + + ll(x (x times) times) EE f~ n then then we we can can define define w0 Wo and and w w EE fln by Wo "= := (x) and with 00 + + . .. .. . + by w0 (x) and := (1 (1 ++ x) Clearly w0 Wo and and w w are are incomparable incomparable under w := x).. Clearly under -< .<.. 2.5. fn-induction Principle. 2.5. ~-induction P rinciple. •

9 Q(O) Q(0)

If Q isis aa property that If Q property of of tree-ordinals tree-ordinals such such that

•

9 Q(a) Q(a) ::::} =~ Q(a Q ( a ++l )1)

•

. Q(Ax ) ::::} 9 Vx VxEE N N.Q(A~) ~ Q(A) Q(A)..

then holds for n. then Q(a) Q(a) holds for every every a a E E ft. For well-founded, where For each each a a E E n, fl, the the ordering ordering -< -%a is is well-founded, where -< -%a is is the the restriction restriction of of a. Henceforward, -< .< below below a. Henceforward, Il a aI I denotes denotes the the set-theoretic set-theoretic ordinal ordinal height height of of -< .<,. a·

2.6. For 2.6. Definitions. Definitions. For each each n n E E N N and and each each a a E E n fl the the finite finite set set a[n] a[n] of of n n-predecessors predecessors of of a a is is defined defined recursively recursively by by •9 O[n] 0[n] := := 0 z •9 (a (a + + l)[n] 1)[n] := := a[n] a[n] U U {a} {a} ~[~] := .-- An[n] ~.[~1. . •9 A[n] The -predecessor of The immediate immediate n n-predecessor of a a is is Pn(a) P ~ ( a ) ":= = the the -<-maximum -<-maximum element element of of a[n] a[n] if if a[n] 0, and a[n] =f. # 0, and 00 otherwise. otherwise. 2.7. 2.7. Definition. Definition.

such such that that

The The set set ns fls of of structured structured tree-ordinals tree-ordinals consists consists of of those those a a E E n gt

. Vx Ee N VA � _ a a.Vx N.. (A (A,x E A[X A[x + + 1]). Note Note that that if if a a E E ns f~s and and/~{3 -< -< a a then then/?{3 E E nS f~s..

Provably Provably Recursive Recursive Functions Functions

155 155

Theorem. For 2.8. 2.8. T heorem. For every every a a E E ns f~s we we have have ..,� 1. C_ a[l] a[1] � C_... C_ a[n] C_ a[n + 1] 1] � c _ .. ... . a[n] � a[n + 1. a[0] a[O] � 2. 2. {3 ~ -< -< a a if if and and only only if if {3 ~ E E a[n] a[n] for for some some n n.. Proof. P r o o f . We We proceed proceed by by n-inductions ~-inductions oonn a a": 1. If If a a = - °0 then then a[n] a[n] = = 0 0 = = a[n a[n + + 1] 1].. For For the the successor successor case case assume assume a[n] a[n] � C_ 1. a[n + + l] 1].. Then Then (a (a + + l) 1)[n] = a[n] a[n] U U {a} C_ a[n a[n + + l] 1] U U {a} {a} = = (a (a + + l)[n 1)In + + l] 1].. For For a[n [n] = {a} � the limit case the limit case a a = = (a (ax}, assume that that an[n] an[n] � C_ an[n + 1] 1].. The The structuredness structuredness an[n + x ) , assume condition in condition in 2.7 2.7 gives gives an an E E a[n a[n + + 1] 1] and and it it is is easy easy to to see see that that from from definition definition 2.6 2.6 that that whenever whenever 'Y ")' E E a[x] a[x] then then 'Y[x] "y[x] C C a[x] a[x].. Therefore Therefore an[n an[n + + 1] 1] C C a[n a[n + + 1] 1] and an[n + and hence hence a[n] a[n] = - an[n] an[n] � C_ an[n + 1] 1] C C a[n a[n + + 1] 1] as as required. required. 2. The implication implication from from right right to to left left is is immediate. immediate. The The converse converse is is vacuously vacuously true true 2. The when a a = = 0, and when when a a is is a a successor, successor, say say a' a ~+ + 11 then then {3 fl -< -< a a implies implies {3 fl � ~ a' a~ 0, and when so so the the induction induction hypothesis hypothesis gives gives an an n n E E N IN such such that that {3 fl E E a'[n] a'[n] U U {a'} {a'} = = a[n] a[n].. m, so Finally Finally if if {3 fl -< a a where where a a = : (a (ax) f~s then then {3 fl � _~ a am -< a O/rnT1 for some some m, so m -< m+ 1 for x ) EE ns for some n. By part 1 we may take l [n] by by the the induction induction hypothesis hypothesis/~{3 E E am am+l[n] for some n. By part 1 we may take + structuredness of a, aam+l > m m + + 11 so so that t h a t by by the the structuredness of a, + 2] 2] � C_ a[n] a[n].. Then Then rm + nn > m+ 1 EE aaim a[n] as since 'Y?' EE a[n] implies 'Y[n] a[n] we obtain {3 since din] implies ?'[n] C C a[n] we obtain fl E Ea am+l[n] C a[n] as desired. desired. m+ l [n] C 2.9. Corollary. Corollary. For each each non-zero non-zero a a E E ns f~s the the set set {{3 {~ :" {3 ~ -< -< a} a} is is linearly linearly and and 2.9. For hence hence well-ordered well-ordered by by -< -<,, with with least least element element 0 and and such such that that {3 fl -< -< a a implies implies {3 13+ + 11 � -< a a.. This This well-ordering well-ordering is is the the direct direct union union of of its its finite finite sub-orderings sub-orderings a[n] a[n] for for n n E EN IN..

-< a for some a then then for some n n we we have have {30, ri0, {31 fl: E E a[n] a[n].. But But it it is is a[n] is immediate immediate from from definition definition 2.6 2.6 that that a[n] is linearly linearly ordered ordered by by -< -<,, so so {30 fl0 -< -< {31 ~: or or simple induction {3fl00 = = {31 ~1 or or {31 fll -< {3 rio. A simple induction on on a a E E n f~ss shows shows that t h a t {3 fl + + 11 � _ a a whenever whenever 0· A {3 -< a. The final sentence -< a. The final sentence is is just just a a restatement restatement of of theorem theorem 2.8.1. 2.8.1.

Proof. P r o o f . By By 2.8 2.8 if i f f {30, l 0 , ~{31 l

There 2.10. 2.10. Note. Note. There are are continuum-many continuum-many -<-incomparable -<-incomparable a a E E ns f~s each each with with ust take := (J(x)) a I] == w (j (just take a, a I := (f(x)) for for each each strictly strictly increasing increasing function function f f: : N 1N -+ --+ N IN;; I] a then then GOtf Gas = - f f and and hence hence every every such such function function already already crops crops up up at at the the first first limit limit level level

natural being in n). f~). Some Some of of these these are are more more "natural" "natural" than than others others however, however, the the most most natural being in Wo w0 and and w as as in in 2.4. 2.4. We We will will be be concerned concerned principally principally with with those those tree-ordinals tree-ordinals which which can can be be "constructed" "constructed" from from Wo w0 or or w w..

2.11. Arithmetic 2.11. A r i t h m e t i c on o n n. f~. Any Any primitive primitive recursive recursive definition definition over over N N,, say say f(a, f ( a , O0) ) = f(a, f ( a , bb++ 1) 1) :

fo {a) fo(a) (a, b, f(a, b)) II fl(a,b,f(a,b))

can can be be extended extended straightforwardly straightforwardly to to n ~ by by the the addition addition of of aa third third "continuity" "continuity" clause. clause. Thus, using using Greek Greek letters letters instead, instead, we we have have the the corresponding corresponding definition definition over over n: fl: Thus,

cp(a, 0) cp(a, (3 + 1) + 1)

= =

CPo (a) CP1 (a, (3, cp(a, (3))

156 156

M. Fairtlough and Wainer M. Fairtlough and S. Whiner sup v(., rp(a, ~). v(., Ax). rp(a, ~) A) == sup x X

The and ~rp can can be be expressed expressed simply simply in in terms terms of of the the slow-growing slow-growing The connection connection between between ff and operator G. G. Recall Recall that that G~(n) Go(n) ::= card a[n], a[n] , so so from from definition definition 2.6 2.6 we we immediately immediately operator - card have Go(n) = = O, 0, G~+~ GO+ 1 (n) and Gx(n) GA (n) = Since G~(n) Go(n) is is defined defined Go(n) ++ 11 and have Go(n) (n).) . Since = Gx, GAJn (n) == G~(n) Gn(a) pointwise-at-n, we we shall shall sometimes sometimes find find itit notationally notationally convenient convenient to to write write G,(c~) pointwise-at-n, instead of of G~(n) Go(n) so so that, that, for for each each fixed fixed n, n, G Gn, :: ~n ~-+ 51. N. instead Now suppose suppose we we know know already already that that for for fixed fixed n, n, and and all all ~, a, ~(3 and and "~ , EE ~, n, Now

Gn(rpo (a)) == fo(G.(~)) fo(Gn(a)) G.(~p0(~)) Gn(rpl (a, (3, ,)) = f~(G.(~),Gn(~),G.(~/)). G.(~l(c~,/~,-),)) !I (Gn(a), Gn((3) , Gn(r)).

Then by by aa simple simple gt-induction n-induction on/~ on (3 we we have have for for all all a, Then a, ~(3 EE ~, n, G.(~(c~, $)) = f ( G . ( ~ ) , G.(/~)).

homomorphism, collapsing the arithmetic the arithmetic of Thus Gn Thus G , is is aa homomorphism, collapsing the arithmetic of of n ~ onto onto the arithmetic of N N.. For example, example, defining defining addition, multiplication and exponentiation oonn n For addition, multiplication and exponentiation ~ iinn the the obvious way: way: obvious (3)l ;+ 1 ; a Addition. c ~++ 00: := = a ;a; ca~++ (((3 ~ ++l )1) : = (:= c ~(a +/+ ?)+ ~ ++AA: =:= s u sup(a p(c~++ A xA)x) Addition. a a ((3 + 1) (a · ~) a. · A (3) ++ a; Multiplication. a := 0; 0; a . · (~ + 1) := Multiplication. c~.· 00 := := (~. c~; ~ - sup(a s u p ( a . Ax) Ax) A ::= Az ) l ) ::= a; a Exponentiation. c~~ := (s+l) - af3 c~s ..c~; c~Ax :: = - sup(a sup(c~ x~) := 11;; c~ Exponentiation. aO a({J+ .

we for each each fixed fixed nn EE 51, N, we have have for 1. G Gn(a + (3) Gn((3) 1. .(a + j3) = = Gn(a) G.(a) + + G.(/~) 2. 2. Gn(a C . ( ~ . · (3) ~) = = Gn(a) C . ( ~ ) . . Gn((3) C.(~) f3) . 3. 3. Gn(af3) G.(~) = = Gn(a)Gn( G.(~) ~"(~). What we we need need to to know know is is that that these these operations operations are are well-defined well-defined on on structured structured tree treeWhat ordinals. ordinals.

2.12. For a, (3 2.12. Lemma. Lemma. For all all ~, ~ and and , 7 E En ~ we we have have 1. , [n] ===} + (3) [n] . 1. ~ Ee (3 Z[n] ~. a + +, ~E e (a (~ + Z)[~]. [n] . 2. [n] if 2. , ~ E e (3[n] Z[~] ===} ~ a - . ~· , E e (a ( - . · (3) Z)[~] if 00 E e a -[~]. 3. , E (3 [n] ===} a'Y E af3[n] if 1 E a [n] . Proof. (3. All P r o o f . We We proceed proceed by by n-induction f~-induction on on/~. All three three are are similar similar so so we we just just prove prove (3) (3) assuming assuming (2) (2).. The The case case (3 ~= = 00 is is trivial trivial since since O[n] 0[hi = = 0. O. For + 1, E ((3 + 1) [n] = [n] U {(3} . Then For the the successor successor step step from from (3 fl to to (3 fl+ 1, suppose suppose , ~/E (fl+ 1)[n] = (3 fl[n]U{fl). Then by af3} . Since [n] we by the the induction induction hypothesis, hypothesis, a'Y a~ E e af3[n]U{ a~[n]U{aS}. Since 11 E e a a[n] we also also have have 00 E e af3[ aS[n] n]

1 [n] by induction on by aa simple simple induction on (3 fl using using (2), (2), and and hence hence af3 as = = af3 a s . . 11 E e (af3 (a s . . a) a)[n] -- af3+ aS+l[n] [n] = again again using using (2). (2). For For the the limit limit case case suppose suppose (3 ~ = = sup sup (3x fix and and , ~, E e (3[n] fl[n] = = (3 ~,[n], so that, that, by by the the n [n] , so n n induction hypothesis, hypothesis, a'Y a~ E e a a~"[n]. But a s = sup(aSx), so aS[n] = aS"In] and induction [n] . But af3 = sup(af3z ) , so af3[n] = a [n] and f3 f3 [n] . a'Y ~ Ee af3 ~[~].

157 157

Provably Recursive Recursive Functions Functions Provably

2.13. Theorem. Theorem. 2.13.

1. a, c~, f3 ~ E E OS f~s ==> =:~ a ~ + + f3 ~ E E OS f~ s 1. 2. a, ~ , f3 ~ E OS f~s ==> ==V a c~.· f3 fl E E OS f~s provided provided 00 E a[l] c~[1] 2. 3. a, c~, f3 ~ E E OS f~s ==> = : , afJ c~# E E OS f~s provided provided 11 E a[l] c~[1] 3.

P r o o f . We We proceed proceed by by O-induction fMnduction on on/~. Again, all all three three are are similar similar so so we we prove prove Proof. f3. Again, (3) assuming assuming (2) (2).. If If f3 r = 00 then a# = = 11 and and 11 E Os f~s.. (3) then afJ

+l = For the the successor successor step step from from/~ to f3 r + 11 assume assume afJ a # E Os f~s.. Then Then afJ a #+1 = afJ a # .. a a and and f3 to For this belongs belongs to to OS f~s by by (2), (2), because because the the proviso proviso 11 E a[l] a[1] implies implies 00 E afJ[l] a#[1] from from the the this proof of of Lemma Lemma 2.12. 2.12. proof For the the limit limit case case/~f3 = = sup sup/3x f~s we we must must check check the the structuredness structuredness condition condition f3x EE OS For + 1] 1] for for every every limit limit A A� ~ afJ a #.. This This follows follows immediately immediately from from the the induction induction [n + AA,n EE AA[n hypotheses if if A A -< -~ afJ a # because because in in that that case case we we have have A A� ~ afJz a #x for for some some x. x. It It therefore therefore hypotheses remains to to verify verify it it in in the the case case A A = - afJ a# = - sup(afJz sup(a#x).) . But But then then for for any any fixed fixed n n we we remains have f3 ~,n E E/3[n + 1] f~s,, and and 1 E E a[l] a[1] C C a[n + 1] by the the proviso. proviso. Therefore Therefore 1] by have f3[n + 1] since/3 since f3 EE Os a[n + AAnn = afJ[n + 1] = n+ 1] immediately Lemma 2.12. This = afJn a #" E E a#[n + 1] = A[ A[n + 1] immediately from from part part (3) (3) of of Lemma 2.12. This completes the the proof. proof. completes 2.14. Notation. Notation. In what what follows follows we we shall often drop brackets from from (a+f3) (c~+~)[n] 2.14. In shall often drop round round brackets [n] or (a (c~.· f3) ~)[n] and write write a c~ + + f3[n] ~[n],, a ~./~[n] instead. or [n] and · f3[n] instead. 2.15. Examples. Examples. 2.15.

1. w0 E OS f~s and and w w= = 11 + + w w0o = = sup(1 sup(1 + + x) x) E Os f~s.. 1. Wo

al . n 2. a Olll ,, .. ... ,. , ak Olk E E ~"~S ~ wOOl nll + + ." .. . + -t- W wa akk . 9nk nk E OS ~s.. OS ==> 2. w l n n n = . . • . + [n] == w 3. w ~ [[nn]] = w w"+~[n] w" . n n + + w w"-l 1 . n n + +... . + + W w ·. n n + + n n + + l[n] l[n].. Thus Thus the the elements elements 3. wW of wW [n] are are exactly exactly those those ordinals ordinals of of co'~ of the the form form w w"n . 9mo mo + m ,n_-l l + + m + .. ... . ++ w w . 9m m,n where every ii _< where mmii <_ ::::; nn for for every ::::; n. n. 4. More on c~, of all all tree tree4. More generally, generally, by by induction induction on a, the the finite finite set set w"[n] w a rn] consists consists of o . . . -< . . . . I ordinals the form co#~ 9mo mo + #k . 9mk mk where ~k -< -~ "'" -~ f30 ~0 WfJk +w WfJ#1 .9ml where f3k ordinals of of the form wfJ ml ++ . . . ++ w all lie lie in in c~[n] <_ nn for for every _< k. all a[n] and and mmii ::::; every ii ::::; k. ) n if and only a is expressible E f~s. Os . Thus Thus c~ a -~ -< (CO 5. co 5. eo = = ssup(l, u p ( 1 , ww, , x wW, ~ , w ~w,w. ."'. ,) • • • ) E (e0), if and only if if c~ is expressible -< + wWfJk where f3k wfJ#~o . 9mo in form a in Cantor Cantor normal normal form c~ = = w mo + ml + + .. ... . + #k .9mk m k where ~k -~ +w WfJ#1I . 9ml . 9". -. ~-< f3 -< l . (CO)n 0 #o -~ (~o).-1. 9

· ·a }

a's := and then then the exp� (f3) "= 6. We We define define eXP exp,(~) to be be afJ c~# and the n-th n-th iterate iterate expX(r c~" ll n"~'" 6. a a (f3) to is also provided c~, f~s and and 11 E E a[I] c~[1].. is also structured structured provided a, ~f3 EE OS sUPn (exp�(1 +c~ c(a) := sup,(exp~(1 7. +a+ + 1)) 1)) eE f~s OS isis aa natural natural tree-ordinal tree-ordinal 7. For For each each c~ a EE f~s, Os, e(c~):= notation above I~l. notation for for the the next next e-number c-number above 1 a I. 2.16. he F unction-Hierarchies. 2.16. TThe Function-Hierarchies. N -+ -* N N are are defined defined by by the the recursion recursion The SSlow-Growing functions G~ Ga :: N 1. The 1. l o w - G r o w i n g functions

Go(n) := . - 0, 0, G0(n)

G,a+l G + l ( (n) n)

.- G := n ) ++l , 1, G~a ((n) cGA(n) ~ ( ~ ) := c ~ . ( ~ ). .- GAn (n).

158 158

M. Fairtlough Fairtlough and and S. S. Wainer Wainer M.

2. The The Hardy Hardy functions functions Ha HOI :9NN --+ -+ 1N N are are defined defined by by the the recursion recursion 2. :.--

Ho(n) Ho(n)

n, n,

. - HHa(n Ha+1l (/~) + 1), 1), (n) "--" Ha+ a ( n --~ . - H~o(n). H>.n (n) . H>.(n) .= H~(~)

3. The The Fast-Growing Fast-Growing functions functions Ba Ba ": IN N --+ -+ N N and and Fa Fa :9IN N -+ -+ N N are are defined defined by by the the 3. recursions recursions + l,1, . - nn + Bo(n) := Bo(n) . - B~(Ba(n)), Ba+l(n) Ba+ l (n) := Ba(Ba (n)) , . - B~, B~ (n) B>.Jn) B>. (n) (n) :=

Fo (n) .:=- nn + Fo(n) + 1, l, + l (n) , Fo+l(~) F:+l(~), F: Fa+ l (n) .:= F>.(n) F F~(~) .:= F~o(~). >'n (n) .

2.17. LLemma. 2.17. emma.

For all a, ~ /3 EE n, For all a, gt, H/3 Ha H.+~ go o o H~ +/3 == Ha

Proof. /3. The P r o o f . We We proceed proceed by by n-induction ~-induction on on ~. The case case /3 ~

= - 00 is is obvious obvious because because Ho H0 is the For the is the identity. identity. For the successor successor case case we we have, have, by by the the induction induction hypothesis hypothesis and and definition definition 2.16 2.16 (2), (2),

Ha+(f~+l)(n)

=

=

H H(a+f~)+l(n) (a+/3)+ l (n) Ha+;3(n + 1) 1) Ha +/3(n + Ha(H/3(n Ha(Hz(n + + 1)) 1)) Ha (H/3+ l (n)) . Ha(Hz+l(n)).

For limit case For the the limit case /3 ~ = - sup sup /3x f~ we we have have

Ha g .++~p(n) (~ )

= =

= =

Ha H.+~o(~) +pJn) Ha (HPn (n)) Ho(H~o(n)) H.(H~(~)). Ha (Hp(n)).

2.18. For 2.18. Theorem. Theorem. For all all a a E E n, ~, Ba = H2~

and F a - H ~

Provably Recursive Functions

159 159

Proof. P r o o f . We We proceed proceed by by O-inductions t2-inductions on on a. a . Both B o t h are are similar similar and and we we only only do do the the a - 11 and second. second. If If a c~ = - 0 0 then then w wa = and it it is is clear clear that t h a t both b o t h Fo F0 and and HI H1 are are the the successor successor function. induction hypothesis, hypothesis, function. For For the the successor successor case case a a to to a a + + 11 we we have, have, using using the the induction a w= a+ 1 = 2.17 waa . 9((xx + 2.17 and and the the fact fact that that w w a+l = w w a . 9W = sup supw + 1), 1), Hw<>.(n+ 1 ) (n) ~n+l H;;;-I (n) -- F� F : ++l1 (n) (n) = Fa Fa+l (n).. + I (n) a = sup a and so we have For the the limit limit case case a c~ = = sup sup a c~x, have w wa = sup w w a"• and so by by the the induction induction For x , we =

=

hypothesis, hypothesis,

= =

-

2.19. Theorem. 2.19. Theorem.

n (n)) H H w<>oo( Fan F . o ((n) n) Fa (n). Fa(n).

For 0, define For each each a a E ~, define its its descent descent functional functional Da Da by by

Da(f)(n) = least k . ( P s k - l ( n ) . . . Pp(~)Ps(~)Pn(~) = 0).

Then Then and and

Ga(n) (n) G a (n) = = Da Da (identity) (identity)(n) (n) + Ha(n) Ha(n) = Da(successor) Da(successor)(n) + n. n.

Proof. P r o o f . Note Note that t h a t Da Da satisfies satisfies the the recursion: recursion: Do (f) (n) Do(f)(n) Da 1 Da+l(f)(n) + (f)(n) D>.(f) (n) Dx(f)(n)

= o0 = Da(f) D a ( f ) ((f(n)) f(n)) + + 1l

(n) = D>.JJ) D~ n(f)(n)

because a and because Pn P n ((a a + + 1) 1) = - c~ and Pn(A) Pn(A) = = Pn(An) Pn(A~).. Then T h e n with with f f = = identity identity we we get get exactly the with ff = exactly the definition definition of of Ga Ga,, and and with = successor successor we we get get exactly exactly the the definition definition of of

Ha H a (n) ( n ) --n n. .

2.20. We 2.20. Remark. Remark. We are are going going to to need need various various "majorization "majorization properties" properties" of of the the above above hierarchies, hierarchies, but but these these cannot cannot be be expected expected to to hold hold for for arbitrary a r b i t r a r y tree-ordinals tree-ordinals a EE f/. O. However However for for structured structured a a E E OS gl s they they do. do. 2.21. For 2.21. Theorem. Theorem. For all all a a E E Os, ~s, (n) 1. Ga Ga is is increasing increasing (strictly (strictly if if a ~ is is infinite) infinite) and and f3 ~ E E a[n] ~[n] ==> - - ~ Gf3(n) G~(n) < < Ga Ga(n) 1. 2. 2. Ha Ha is is strictly strictly increasing increasing and and if if f3 fl E E a[n] a[n] then then Hf3(n) H~(n) < < Ha(n) Ha(n) 3. 3. Same Same for for Ea Ba and and for for Fa Fa provided provided n n =I~ oO..

M. M. Fairllough Fairtlough and S. Wainer Wainer

160 160

Proof. obvious since since G,,(n) size of a[n] and a[n] implies P r o o f . (1) (1) is is fairly fairly obvious Ga(n) is is the the size of a[n] and/3j3 E9 a[n] implies j3[n] ~[~] C c a[n] ~[~].. (2) is proved proved by by n-induction. f~-induction. The The case case a ce = = 00 is is trivial. For the the successor successor case case a ce (2) is trivial. For to induction hypothesis to a a + + 11 we we have have by by the the induction hypothesis (n + go+~(,) = H,, H . ((n , + + 11)) < < H,, Ho(, + 2) 2) = = H,,+1 H . + , ((n n + + 11)) H,,+ l (n) =

j3 E9 ac~ + [n] = and and if if/~ + ll[n] -- a[n] c~[n] U U {a} {c~} then then "+ 1 ( n ) . (n) < He(n < H,, Ha(n) < H,, Ha(n + 11)) == H Ha+l(n). (n + H /J (n)) :::;

For limit case a[n + For the the limit case a a = = sup sup a a~x E9 ns ~s we we have have a ~nn E9 a[n + 11]] = = an a,+l[n + 11]] so so by by the the + 1 [n + induction hypothesis, hypothesis, induction "n (n) Ha(n)) = H Ha,,
j3 E9 a[n] and and if if/~ a[n] = = an[n] a,[n],, U~(~) < H.~

= uo(~).

" [n] a[n] implies w" and (3) is immediate from (2) since B" Sa = = H H2~, Fa = = H H~. and/3j3 E9 c~[n] implies 22/Je E 6 22a[n] (3) is immediate from (2) since 2" , F" " [n] , except and O. In and w w/Je E9 w wa[n], except that that the the latter latter only only holds holds when when n n i:# 0. In fact fact F Fa(0) = 11.. ,, (O) = " , B" 2.22. 2.22. Corollary. Corollary. Each Each of of the the hierarchies hierarchies G Ga, Ha, Ba and and F" Fa forms forms aa majorization majorization ", H nS, in sense that hierarchy hierarchy on on f~s, in the the sense that the the functions functions are are strictly strictly increasing increasing (except (except for for the the constant (n) = constant functions functions G,, Ga(n) =a ~ when when a ~ isjinite) is finite) and and each each function function at at level level a c~ eventually eventually dominates dominates the the corresponding corresponding function function at at level level j3 13 whenever whenever j3 13 -< -< a. c~.

Proof. j3 -< P r o o f . Simply Simply note note that that if if/~ -< a c~ E9 ns ~s then then by by 2.8, 2.8, j3 ~ E9 a[n] c~[n] for for all all but but finitely finitely many etc. (n) < many n n,, and and then then by by 2.21, 2.21, G/J(n) Ge(n ) < < G,,(n) Ga(n),, H He(n ) < Ha(n) etc. (n) H,, /J 2.23. 2.23. Comparisons C o m p a r i s o n s Lemma. Lemma.

For For all all non-zero non-zero a ~ E 6 ns ~s and and n n > 11,,

Ga(n) < Ha(n) < Ba(n) < Fa(n) < B~.a(n).

Proof. P r o o f . We We proceed proceed by by simple simple n-inductions ~-inductions on on a r . For For example, example, the the least least straight straightforward final one: forward inequality inequality is is the the final one: F F~(~) l (n) r,+l(n) F"+1 (n) Fx (n) (n) F>.

n+ 1 = Bw(n) n+ F!l' f 2 ++l ~ ((n) ~) = = 22~ + 11 < < n ~+ + 22 "§ = S~(~) + _= F B,+I -.~ B 2"+1 l l F:+I(n) < B�� w.a (n) ('~) < < B��;l ~.a (n) ( n ) == B S~.,+~(n) W.a+w(n) : (n) < = = F>.Jn) F~. (n) < < Bw.>.Jn) S~.~. (n) = Bw.>.(n). S~.~ (n). =

2.24. G, H 2.24. The T h e computational c o m p u t a t i o n a l honesty h o n e s t y of of G, H ,, B B and and F F We call aa computable computable function bounded by We call function "honest" "honest" if if its its complexity complexity is is bounded by some some iterate iterate of of itself. itself. Since Since our our concern concern here here is is with with "large" "large" functions, functions, bigger bigger than than

Provably Recursive Functions

161 161

exponential, it it does does not not matter matter whether whether we we measure measure computational computational complexity complexity in in exponential, terms terms of of space space or or time, time, but but it it will will be be convenient convenient to to think think in in terms terms of of space-complexity. space-complexity. Clearly, the the computability computability and and complexity complexity of of the the functions functions Gc Ga," Hc" Ha, Bo B~ and and F Fa, Clearly, a, for a's a's ranging ranging below below some some fixed fixed tree tree ordinal ordinal , "y depend depend on on the the computability computability and and for complexity of of the the operation operation complexity {(a - 1, 1, x) x) if if a a is is aa successor successor a, xx),) �> (a ((a, ' . 'is llimit. 'f a a IS ax , xx)) Imlt. 1if ((ax, 2.25. Definition. Definition. Let Let qq be be aa strictly strictly increasing increasing number-theoretic number-theoretic function. function. Then Then 2.25. tree-ordinal , ~/is q-space-representable if if there there is is aa uniform uniform method method of of representing representing aa tree-ordinal is q-space-representable each a a -( -~ , ~/as word rr aanI on on aa TUring Turing Machine Machine tape tape ((or more generally generally as as aa term term each as aa word or more in some some finite finite language language), and aa standard standard representation representation of of numbers numbers x x by by words words rrxn, ) , and xI, in such that that the the transition transition such .is a successor a - I' 17,, rrxn) ((rr a a successor x, ) 1'iff aa IS ((ran, raI , rrxn), x, ) �~ (rr aa n-', n) if a a is is limit limit if ( x , rrxx') is computable computable within within space space less less than than q(lr q(/(ran) + / (lr r xx')), n ) ) , where w h e r e /l((rraan,)) denotes denotes the the is a,) + length of of the the word word rran. length aI.

{

{

2.26. Lemma. Lemma. Suppose that that , 7 = sup sup% ~s is is q-space-representable q-space-representable and and let let 2.26. Suppose 'x EE OS ql (n) = ·2 and ·n. Then ql(n) = q(n) q(n).2 and qq~(n) q(n).n. Then for for a a -( -~ , ,~ and and x x E e N N the the space-complexities space-complexities 2 (n) = q(n) (x) are bounded as of computations of of the the computations of Go(x) G~(x),, Ho(x) Ha(x), , Bo(x) Ba(x) and and Fa F~(x) are bounded as follows: follows: ( G +I _< q ,, X ) (Wal) + + Wx')) SG (a, x) ::; ( H � ) SH(a,x) <_ qH~(')(l(ran) +/(rxn)) SH (a, x) < q X (Wal ) + Wx')) <_ qqBl~(')(l(ran +/(rxn)) �� (x) (Wal )) + WxI)) SSB(a,x) B (a, x) < _F2(~) (x � l SF(a, x) < q{ \W a ) + WxI))

Proof. result for for B, B, and and then then indicate indicate briefly briefly what Proof. We We prove prove the the result what modifications modifications are are needed G, H H and F. For For B B and and FF we we cannot cannot prove prove the desired inequality inequality directly; needed for for G, and F. the desired directly; instead need to to prove prove something extract what what we want instead we we need something stronger, stronger, and and then then we we can can extract we want any finite finite sequence easily. First notation: let # l ,, .. ... ,. ~, (3k k - -1 l denote denote any sequence easily. First some some notation: let fl(3 -= #(300,, (31 of tree-ordinals let rrfl7 = r r i(31 l l 7 I,' rrf(3 l 22nI,, . ... . . ', r r /(3k ~ k _-llnI be be the the ((possibly possibly empty) empty) of tree-ordinals -~ -( "y, " and and let (3' = tape-representation of of ft. (3. Define Define tape-representation B((3, x) = B~o Bpo o0 Be, BpI o0 . . . 0o B B(fl, x) = B#k_l (x).. Pk_ 1 (x) • • •

Then Then B(#,x) B ((3, x ) will will bbee computed computed according according to to the the following following tape-transitions tape-transitions where where r#n, r (3', rraanI denotes denotes the the tape-word tape-word r#07, r (30 I, r#17,...,r#k_17 r(31 I, . . . , r(3k - l I,, rraanI with with length length (30 , ) ++ l(W#1 (31 ,) ++. .. .. . ++ W(3k- l ') ++ l(r k: W (3', r a, ) == l(WZ0 Wa, ) ++ k: (<},rxn) (() , rx ' ) :�> rxn rX' (and ( and halt) halt) (r#n, r0n ,rxn ) , > ( r # n , r x + 7) r(3l , rOI, rx') � r(3l , rx +l l')

(r#n, a ++ 17, rx')) ~_+ rx')) r(3l , rra l l , rxn � (r#n, ( r (3l , rraanl ', rraanl ', rxn

r (3', r A I , rx, )

�

( r(3', r Ax I, rX ')

Fairllough and and S.S. Wainer Wainer M. Fairtlough M.

162 162

The space space used used in in the the computation computation of of B(fl, B({3, x) x) will will be be denoted denoted SB(fl, SB ({3, x). x) . The Now let let L~(x) La (x) be be the the step-counting step-counting function function associated associated with with the the recursive recursive definidefini Now La (La (x)) ++ 1;1 ; L~(x) tion of of B~(x), Ba (x) , thus thus Lo(x) Lo(x) == xx ++ 1;1 ; La+l(x) La+1 (x) == L~(La(x)) L;.. (x) == L~=(x) Ldx) ++ 1.l . tion As above, above, define define L(fl, L({3, x) Lf30 o0 L#, x ) == L#o (x) . Then Then what what we we need need to to prove prove isis Lpl o0 .· .· .· o0 L#k_l LPk _ l (x). As the following: following: the

Claim. For every every aa -~-< 7'Y and and every every sequence sequence #,{3, ififfor for every every xx C l a i m . For sB(#,~) <_ q~(~")(l(r# -~) + l(r~-~)) then for for every every xx then

s~(#,~,~) <_q~("~

~) + l(~)).

The desired desired result result follows follows almost almost immediately immediately from from this this because because with with #{3 -= ()() we we The obtain obtain

SB (a, x) x) <_ � ~q�(a,x) (lra"') + Wx..,)) Ss(~,

and it is is an an easy easy matter matter to to check check that that and it

L~(x) < B~+l(x) = S~(x).

We the claim induction on -< 7, 'Y, assuming assuming in each case that {3 is an an We prove prove the claim by by induction on a a -~ in each case that # is arbitrary such that that for arbitrary finite finite sequence sequence such for every every x, x, SB(~,x) < qL(#'Z)(l(r~7) +/(rx'~)).

If If a cr = - 00 then then by by the the second second transition transition above, above,

Ss~(#, o, x) ~) = S s~(#, + 1) B ({3, x 9+ B ({3, 0, +W l(rxx + + 1"')) In)) �(P ,x+1 ) (W{3"') + < q-~L(#'z+l)(l(rflT) x {J,O } < � ( , (W {3"', ro..,) + W < qq~(~,~ l ( ~x..,)) )) since x) = L({3, since L({3, L(fl, 0, O,x) L ( # , xx + + 1) 1) and a n d /Wx (rx + + 1"') 1~) �
paax SSB(#, B ({3, a, ~, a, ~, x) z) <-< q~1�( , , , ) (l(r{3"', ra"', ra"') + l(rx"')) (/(rflT) + + q(Wa q(/(rc~ + + 1"' 1-1) ( r xx..,)) T ) ) .. 2). 2). << q,~,�(L (p# ,,aa ,,aa, ,xx)) (W{3"') ) ++ /W

Provably Recursive Recursive Functions

163 163

Hence Hence L(~,,,,,~) .) + / ( r x T ) ) .92) < 91 SB(fl, 7, rraa ++ 1I') q�(,I3,,,,,,,x) (q(l(rfl (q(W,sI, a ++ 1, SB (,s, (~ + WX')) 2) l , X) x) <_ ~) +/(rXT)) �(,13,,,,,,,x)+ l (W,s', rr~a ++ 1I') q~L(#,-,-,x)+l(/(r#~, + Wx , )) l ) ,X (,I3, <>+ � < ,s', r a q + I') + Wx ')) (W - 91

--

91

since ( # , c~, ( # , 5L� 2 (x) (x) ++ 1) 1) -= L(#, L(,s, ~a ++ 1, l, x). x) . since iL(,s, a, c~, a, x) x) ++ 11 _< ::; iL(,s, fourth transition, transition, the the q-space q-space representabilrepresentabil For the limit limit case sup c~x use the the fourth For the case aa -= sup ax use to obtain obtain in in aa similar similar fashion fashion to to the the ity for c~ ity of of "7 'Y and and the the induction induction hypothesis hypothesis for ax to above, above, max{/(rfl 7) + ax , x)} x)} + q(l(ra q(Wa~) max{W,s') + l(rx-~)),SB(fl, Wx')) , SB (,s, ax, ,) + ,x ) _< qL(#'a"x)(/(r#',rcd~) + / ( r x ~ ) ) ,,. , q�(,13 (W ,s', ra') + Wx '))

SB(fl, SB (,s, a,x) a, x) =

l _< ~ nl)) ++/ (Wx')) rxT)) < qi(#,ax,~)+l(/(r#7, q�(,I3,,,, ,x) + (w,sl , rra

x) + Wx'))' < - q91�(,I3,,, , (w,sl, ral ) +

This now completes the the proof for F F is is almost almost identical except This now completes proof for for B. B. The The proof proof for identical except that the successor-transition is is then then that the successor-transition +

with xx + copies of of a rather than just 2, 2, and is replaced replaced by ql (n) (n) = q( n) . 22 is with + 11 copies c~ rather than just and so so ql = q(n). by q2 (n) = q(n) n . The The results results for for G G and and H can be proved directly directly without without the the q2(n) q(n). . n. H can be proved more complicated complicated claim. claim. The step-counting function G,, (x) is is easily easily seen seen to to be be more The step-counting function for for Ga(x) dominated by (x + 1), and the step-counting function for H (x) is Lo(x) = dominated by G Ga(x 1), and the step-counting function for Ha(x) is io(x) = x, x, ,, ,, 1 , which gives Lo(x) ::; H�(x) by an (x) + L La+l(x) i a ( x + 1) + 1, i~(x) = ixx(x) + 1, which gives La(x) <_ H~(x) by an (x + 1) + 1 , L).. ( x) L).. L = (x) l + . ,, O

easy induction on easy induction on a. a. The The reader reader may may have have noticed noticed that that we we have have made made one one or or two two natural natural assumptions assumptions about encoding rr 7_' in about the the nature nature of of the the encoding in the the proof proof of of this this lemma, which should should strictly strictly lemma, which speaking speaking form form part part of of the the definition definition of of q-space-representability. q-space-representability.

2.27. Theorem. 2.27. Honesty Honesty T h e o r e m . Suppose Suppose 'Y 7E 6 n ~ss is is q-space-representable, q-space-representable, and and suppose suppose ,sfl E6 n ~ss is is such such that that for for all all n n and and m, m, q?(m) q'~(m) ::; <_B B#(max(n, m)). ,I3 (max(n, m)). Then Then for for every every a ~ -< -~ 'Y ~/ we we have have B Ba" E E SPACE(B SPACE(Be 0o Bo Ba), ,, ), ,13 Ba 0o B and and so so B Ba" is is "honest" "honest" when when ,s # j ~_ a ~.. Similar Similar results results holds holds for for G, G, H H and and F. F.

Proof. P r o o f . From From the the above above Lemma Lemma we we have have S s ( a , x ) <_ B#(B~(l(ra n) +/(rxn))),

and and similarly similarly for for G, G, H H and and F. F.

164 164

Fairtlough and and S. S. Wainer Wainer M. Fairtlough

2.28. A in aa function 2.28. Definition. Definition. A function function f f is is elementary elementary ((in function g) g) if if f f is is definable definable explicitly 0, 11,, +, +, - ((and and g), bounded sums products. E(g) explicitly from from 0, g), using using bounded sums and and products. E(g) denotes denotes the class class of of all all such such functions functions f f.. A A relation relation is is elementary elementary if if its its characteristic characteristic function function the is. is. n then E(g) 2.29. If E(g) consists 2.29. Fact. Fact. If 9g is is honest honest and and g(n) g(n) 2: >_ 22~ then consists exactly exactly of of those those .....

functions functions which which are are computable computable within within time time or or space space bounded bounded by by some some fixed fixed iterate iterate g. See 1981 ] . of of g. See e.g. e.g. Cutland Cutland [[1981].

Suppose 2.30. Corollary. Corollary. Suppose w w -< -~ I " 7= - SUP sup% f~ss is is q-space-representable, q-space-representable, and and 2.30. ,x EE n n, m suppose suppose there there is is f3 fl -< -~ I "y such such that that qf(m) q'~(m) :::; <_ B,B(max(n, Be(max(n, m)) m)) for for all all n, m.. Then Then

U SPACE(B~)=U E(B~). a-~7

a-~-),

Similar G, H Similar results results hold hold for for G, H and and F F.. 3. Complete Complete w-arithmetic w-arithmetic

In In this this section section we we formalize formalize the the truth-definition truth-definition for for first-order first-order arithmetic arithmetic as as an an infinitary proof-system; infinitary because infinitary proof-system; infinitary because of of the the "w-rule" "w-rule" which which allows allows us us to to derive derive 'v'xA(x) infinite sequence VxA(x) from from the the infinite sequence of of premises premises A(n) A(n),, n n E EN N.. A A careful careful assignment assignment of of tree-ordinal 1987] , but tree-ordinal bounds bounds to to derivations, derivations, due due originally originally to to Buchholz Buchholz [[1987], but modified modified 1991 ] , then and and somewhat somewhat simplified simplified here here following following Fairtlough Fairtlough [[1991], then provides provides aa direct direct link with bounding functions link with the the fast-growing fast-growing B B~'s, which act act as as bounding functions for for existential existential o 's, which quantifiers. quantifiers. Schutte 1977] was w-rule as Schiitte [[1977] was the the first first to to develop develop the the w-rule as aa systematic systematic basis basis for for proof prooftheoretic analysis, and theoretic ordinal ordinal analysis, and many many others others have have significantly significantly extended extended these these methods methods since. 1981 ] , Girard 1981,1987] ' Howard 1970] , Pohlers since. See See Buchholz Buchholz et et al. al. [[1981], Girard [[1981,1987], Howard [[1970], Pohlers'' article article 1977] and and Tait 1968] . Our in in this this volume, volume, Schwichtenberg Schwichtenberg [[1977] Tait [[1968]. Our development development here here follows follows 1987] . most closely the Buchholz and most closely the ideas ideas of of Buchholz and Tait, Tait, see see also also Buchholz Buchholz and and Wainer Whiner [[1987].

3.1. 3.1. The T h e language language and and rules rules of of w-arithmetic w-arithmetic The A, B, B, C, C, ..... . built The language language of of arithmetic arithmetic consists consists of of those those formulas formulas A, built up up from from A, V 'v' and need to be very atoms the symbols atoms using using only only the symbols A, V,, Y and 3. 3. We We do do not not need to be very specific specific about about the the atomic atomic formulas, formulas, except except to to insist insist that that they they occur occur in in complementary complementary pairs pairs E tn), Ei (tI , " . , tn) tn) and El(t1,..., El(t1,..., and that that in in the the standard standard model model i (t1 , , tn), • • .

(N, .. .. .. ,, Er (N, 0, 0, SUCC, succ, E i N , , .. .. .. )) they relations, in they are are interpreted interpreted as as certain certain elementary elementary relations, in the the sense sense of of 2.28. 2.28. In In particular particular Eo E0 and and Eo E0 will will stand stand for for = = and and r=1-, E E1l and and E E1l for for < < and and 2: _ etc. etc. Note Note that that negation negation ' s laws is is not not included included as as aa logical logical symbol, symbol, instead instead it it is is defined defined by by de de Morgan Morgan's laws thus: thus: ..,E E, ..,E E, -~(A ..,(A A ..,..,A is -~E == - E, ~ E == =_ E, A B) B) == =_ ..,A -~A V V ..,B, -~B, ..,'v'xA ~k/xA == - 3x..,A, 3x-~A, etc. etc. Hence Hence -,-,A is A. A.

Provably Recursive Functions Provably

165 165

In addition addition there there is is to to be be one one further further special special atomic atomic formula formula xx :: N N meaning meaning "x "x is is In natural number" number".. It It will will occur occur only only in in this this atomic atomic state state and and never never negated. negated. Its Its aa natural purpose is is to to keep keep track track of of the the computations computations of of witnesses witnesses for for 3x. 3x. purpose In what what follows follows we we shall shall not not bother bother to to distinguish distinguish between between aa number number m m and and its its In numeral m ~ as as the the context context will will make make it it clear clear which which is is intended. intended. For For formulas formulas A A with with numeral free-variables Xl xl,, ..... . Xk we we will will simply simply write write A(ml A ( m l ,, .. ...., , mk) mk) for for the the instantiation instantiation free-variables

A[ml/x mk/xk] A [ m T / z , , I , .. .. .. ,, m k l~]. .

Our infinitary infinitary proof-system proof-system will will bbee formalized formalized in in the the style style of of Tait Tait [[1968] so as as to to Our 1968] so give an an inductive inductive definition definition of of aa set set of of judgements judgements give ~ AA nn ::NNF I-a where n n :: N N declares declares aa fixed fixed natural natural number number parameter parameter n, n, a a E OS gts bounds bounds the the height height where of the the derivation, derivation, and and A A = = {A { AIx, ,.. .. .., , A Ak} is aa finite finite set set of of closed closed formulas, formulas, possibly possibly of k } is including some some of of the the form form m m :: N N.. The The intended intended meaning meaning of of A A is is the the disjunction disjunction including A1 or or ..... . or or A A~k and and we we write write A, A, B B for A U U {B} {B},, A, A, B B for for A A U UB B,, etc. etc. Thus Thus ..., -~A, B for A A, B Al represents Al A1 -+ -+ (( .. . .. --+ + (Ak (Ak -+ -+ B)) B)).. represents The The rules rules are are given given in in Figure Figure 11 and and fall fall into into two two categories: categories: N-rules N-rules to to do do with with computation, and and L-rules L-rules to to do do with with the the logic. logic. In In the the L-Cut L-Cut rule rule the the right-hand right-hand computation, formula C C is is called called the the cut-formula cut-formula of of that that rule, rule, or or sometimes sometimes the the L-Cut-formula. L-Cut-formula. It It formula must be be an an ordinary ordinary logical logical formula formula and cannot be be of of the the special atomic form form n n :: N N.. must and cannot special atomic 3.2. Definition. Definition. If n n :: N N I-a t-a A A is is derivable derivable without without use use of of either either of of the the cut-rules cut-rules 3.2. If N-Cut or or L-Cut L-Cut we we shall shall write write instead instead n n :: N NF ~a A A or or simply simply F ~aA A if if n n = = O. 0. N-Cut

The in this The reason reason for for this this notation notation is is clear, clear, for for in this case case the the definition definition is is nothing nothing other other than the truth definition definition of arithmetic enhanced enhanced with with two two parameters parameters n n and which a which than the truth of arithmetic and c~ together uniform control over the of sub-derivations sub-derivations as together provide provide aa uniform control over the heights hei�hts of as follows: follows: each n, ~Pn(a) A(n),, •9 if if ~Fa Vx.A(x) then for for each n, nn: : N FPn Vx.A x) then aJ A(n) will be finite 3y.B(y) for for quantifier-free •9 if if nn : 9N N ~F 3y.B(y) quantifier-free B, B, then then the the derivation derivation will be finite with height cardinality of of fl[n] .B[n] and furthermore there there will will be be with height bounded bounded by by the the cardinality and furthermore an bounded by by n n + + card(.B[n]) card(fl[n]).. an existential existential witness witness bounded Thus in particular, since since the the cardinality cardinality of of ,8[n] GiJ(n) , if if F Thus in particular, fl[n] is is G~(n), ~ a Vx.3y.S(x, with Vx.3y.B(x, y) y) with size and + + G (n) will B then G~+~(n) 11 + G~(n) will bound bound the the size and complexity complexity quantifier-free then Gw+ (n) = nn + B quantifier-free a a of recursive function = least m.B(n, m) m).. f(n) = least m.B(n, of the the recursive function f(n) The proof our claim claim that that ~ a is more than The proof of of the the following following result result underlines underlines our Fa is little little more than dressed-up truth aa dressed-up truth definition. definition.

�

3.3. 3.3. Completeness Completeness Theorem. Theorem. for for some some aa E m OSs with with la I a II < <w wW~..

If AA is is true true (in the standard standard model) then ~Fa AA If (in the model) then

Proof. proceed by by induction induction on on the the complexity complexity of of A. A. If If A A isis aa true true atomic atomic P r o o f . We We proceed formula formula the the result result isis immediate immediate by by L-Ax. L-Ax. true then then B(m) B(m) holds holds for for some some m m and and hence hence by by the the induction induction If A A === 3xB(x) 3xB(x) isis true If hypothesis and by by 3.5 3.5 below, below, ~m+~ Fm+iJ S(m) B(m) for for some some ft. ,8. Also Also ~'~ Fm m m :: N N by by N-Ax, N-Ax, hypothesis and N-Succ. Thus ~Fa A A by by the the 3-rule 3-rule with with c~ a -= m m ++ fl,8 ++ 1. N-Succ. Thus 1.

M. Fairtlough Fairtlough and and S. Wainer M. S. Whiner

166 166

(N Ax) (N-Ax) -

n : NN I-'~ 1-0 A, A, mm" : NN n"

(N-Succ) (N-Succ)

n : NN b1-f3 A, mm" : NN n" s A, n : NN F1-0 A,, mm+ + 11 :9NN n" -~ A

(N-Cut) (N-Cut)

A n : NN FI--s~ f3° mm" : NN mm" : NN K1-f31 n" s' A n : N 1-0 A n. NK~A n : NN I1-0 A n" -~ A

(L-Ax) (L-Ax)

(V) (v)

(A) (1\)

N I1-f3 A, Bi B; nn :9N - s A, n : NN I1-0 B1) A, (Bo (Bo VV B1) n" -'~ A, o

n : NN Fl-f3 A, Bo Bo n" n : NN I1-f3 A, B1 Bl n" -s~ A, -s~1 A, n : NN I1-0 A, (Bo (Bo A1\ B1) B1) n" -'~ A,

o : NK 1-f3s 1' AA, : NF l-f3 (3) nn' N - s ~ m : N nn' N , BB(m) (m) (3) nn": NN K1-0~ A, A, 3xB(x) 3xB(x) (V) (v) (L-Cut) (L-Cut)

m <_ � nn ++ 1.l . ifif m if if fl{J EE a[n]. a[n].

if if rio, a[n]. {Jl E a[n]. {Jo , fll if if AA contains contains aa true true atom. atom. iffl if {J E a[n] a[n] and and ii == 00 or or ii == 1. l.

if if rio, {Jo , fll a[n]. {J E a[n].

l

if {Jo, rio, (Jl filCh[n]. if E a[n] .

{max(n, A, B(m)} {max(n, m) m ) ": N N l-f3m Ks~ A, B(m)}meN mEN nn . : NN 1-0 ~-" A, A, VxB(x) VzB(z)

if if (J tim a[max(n, m)] m)].. m EE a[max(n,

o -,C n : N 1-f31 A, C nn": NN l-f3 Ks~ A, A, ~ C n" N Ksl A, C nn . : NN t1-0 - a AA

if {J rio,/31 a [ n ] .. if o , (Jl E a[n]

Figure Figure 11": Inductive Inductive Definition Definition of of nn": N N 1-0 t-" A A

f3m B(m) If If A A == = VxB(x) VxB(x) is is true true the the induction induction hypothesis hypothesis gives gives for for each each m m,, F ~S,, B(m).. Let + 1 and a_l = O. Then by 3.5 below, + (J Let a a = = sup sup a a=x E E OS f2s where where a a=x = = axc~=-1 + fl= + 1 and a-1 = 0. Then by 3.5 below, x l O Om - l +f3m B(m) A by the V-rule. E a[m] for each m . SO F F ~,,-,+S,, B(m) and and a am-1 + {J ~,~ E aim] for each m. So ~ A by the V-rule. m -l + m The The remaining remaining cases cases A A == = B B0o V V Bl B1 and and A A == ~ B B0o 1\ AB B1l are are easy. easy. k where kk Finally Finally notice notice that that in in each each case case the the a a so so generated generated has has rank rank Ila a lI � _< w w ~ where measures measures the the height height of of A. A.

3.4. If 3.4. Corollary. Corollary. If 1-0 F-~ A A then then A A is is true true and and hence hence derivable derivable without without any any cuts cuts of of either either kind. kind. Thus Thus we we have have complete complete cut-elimination cut-elimination for for the the system system 1-0 ~-~.. An An important important question question which which arises arises immediately immediately is is "how "how do do we we measure measure the the increase increase in in ordinal ordinal complexity complexity in in going going from from aa proof proof nn : 9NN 1-0 t-~ A A with with cuts cuts to to aa

Provably Recursive Recursive Functions Functions Provably

167 167

cut-free proof proof nn :9NN ~F' O' A?" A?" This This isis not not at at all all aa trivial trivial question question and and as as the the above above cut-free proof isis non-constructive non-constructive itit gives gives us us no no information information about about the the increase. increase. To To answer answer proof the question question we we must must turn turn to to aa syntactic syntactic treatment treatment of of cut-elimination, cut-elimination, part part of of which which the stems from from Gentzen's Gentzen's elimination elimination of of logical logical cuts cuts and and part part from from the the application application of of stems hierarchy hierarchy theory theory to to the the subsequent subsequent elimination elimination of of N-Cuts. N-Cuts. The The latter latter is is covered covered The rest rest of of this this section section isis devoted devoted to to L-Cut-elimination L-Cut-elimination and and the the in in section section 6. 6. The corresponding bounding bounding results results for for the the full full system system nn :: N N Ff-0 A on on which which the the results results corresponding -~ A in section section 44 are are based. based. in If nn" : NN Ff-0 :::; n', n' , A A C_ C A', A', ~8 E ~s ns and If -~ AA,, nn < and arm] � a/[m] for every m 2: n, then n' : N f-Ho' N .

3.5. W Weakening Lemma. 3.5. eakening L emma.

Proof. We We proceed proceed by by induction induction over over the the derivation derivation of of nn : 9N NK f-0s A A according according to to the the Proof. nine rules. rules. For example, we we consider two of of the the cases: nine For example, consider two cases: m" : N, N, m m" : N N ~1 f-.Bl A A where a[n] . Applying the 1. Assume 1. Assume nn" : N N ~f-.Boo m where /30 fl0,, /3 fll1 E E a[n]. Applying the f-H.Bo m m : 9N N and and induction hypothesis to to each of these these premises induction hypothesis each of premises we we get get n' n ~ : 9N N K6+~o a/[n] C_ + a/[n/] and so so by m By 2.12(1), 2.12(1), ~,+ 8 + ~0, /30 , 8~ + /31 EE 8~ ++ a'[n] m:: N N f-H.Bl F-~+~' N A'.. By + j3~ a'[n'] and by � 8~ + re-applying N-Cut, nn'' : :NN b6+~ f-Ho' A'. A'. re-applying N-Cut, 2. Assume max(n, m)" m) : N A, B(m) B(m) for for every every m, m, where m)] . 2. Assume max(n, N tf-H.Bm -~+~ A, where /3 tim a[max(n, m)]. m EE a[max(n, For For each each m, m, the the induction induction hypothesis hypothesis gives gives

j

A', B(m) max(n',/ , m) m)": N N f-H.Bm ~6+~ A', B(m) max(n

max(n, m)] with with 8+/3 ~+/~m ~+a'[max(n, m)] � C 8+a'[max(n/, ~+a'[max(n', m)] m)].. Therefore Therefore by by re-applying re-applying m Ee 8+a' +0' A', the the V-rule, V-rule, n' n' :: N N f~-~+~' A', VxB(x) VxB(-x),, as as required. required.

3.6. 3.6. II nversion n v e r s i o n Lemma. Lemma.

AA.. If If n n": N N f-0 b-~ A, A, (Bo (Bo A A B1) B1) then n n : 9N N f-0 F-~ A, A, Bi Bi for for each ii = = 0, O, 11.. V. If If n n": N N f-0 ~-~ A, A, VxB(x) VxB(x) then max(n, max(n, m) m)": N N f-0 F-~ A, A, B(m) B(m) for for each m m.. V.

Proof. P r o o f . For For example example we we prove prove V-inversion, V-inversion, the the proof proof of of A-inversion A-inversion being being similar. similar. Proceed Proceed by by induction induction on on the the derivation derivation of of n n :: N N f-0 ~ A, A, VxB(x) VxB(x).. 1.1. Suppose Suppose the the final final rule rule applied applied is is aa V-rule V-rule with with VxB(x) VxB(x) the the "main "main formula" formula" derived. derived. Then Then the the m-th m-th premise premise is is max(n, max(n, m) m)": N N f-.Bm k- ~ A', A', B(m) B(m) where where A' A ' == A, A, VxB(x) VxB(x) or or A' A ' == A A and and where where /3 ~,~ a[max(n, m)] m)] so so /3 ~m[k] C m [k] C m Ee a[max(n, rst case a[k] for for every every kk 2: _> max(n, max(n, m) m).. In In the the fifirst case we we need need to to apply apply the the induction induction ark] hypothesis hypothesis to to obtain obtain max(n, max(n, m) m)": NN f-.Bm ~ A, A, B(m) B(m).. The The desired desired result result then then follows follows immediately immediately by by Weakening. Weakening.

168 168

M. Fairllotlgh Fairtlough and and S. S. Wainer Whiner M.

Otherwise, whatever whatever final final rule rule is is applied applied VxB(x) VxB(x) remains remains as as aa "side "side formula" formula" 2.2. Otherwise,

in the the premises. premises. So So we we can can apply apply the the induction induction hypothesis hypothesis to to each each one, one, in thus replacing replacing nn : : N N by by max(n, max(n, m) m) :: N N and and VxB(x) VxB(x) by by B(m) B(m).. The The desired desired thus result result is is then then obtained obtained by by re-applying re-applying the the same same final final rule, rule, since since f3 /3 E C ark] ~[k] implies/?f3 E e a[max(k, c~[max(k, m)] m)].. Note Note that that if if n n :: N N f-0 ~ A, A, VxB(x) VxB(x) is is an an axiom axiom then then implies F-~ A A is is also also an an axiom, axiom, and and hence hence so so is is max max(n, m) :: N N f-0 ~-~ A, A, B(m) B(m).. nn :: NN f-0 (n , m)

A II of 3.7. Definition. D e f i n i t i o n . The The height height I[A of aa formula formula A A is is defined defined as as follows: follows" 3.7. 1. IIn" n : NNI--0 I=O 1. . . .) I = 2. 2. IIE( E ( .. .. ..))II -= IIE( E(...)I = 11 Al II)) + 3. 3. lI Ao A0 /\ AA A1l II = = II Ao A0 V vA A11 = m max([ A0 [,I, I[A1 + 11 lI = ax (1 Ao A 1 = 4. 3xA(x) Il - II A I1 ++ 1 4. II VxA(x) VxA(x) 1l = - 113xA(x) CI The cut-rank cut-rank of of aa derivation derivation is is then then defined defined as as the the supremum supremum of of all all the the values values of of I[C[ The

where C C is is aa cut-formula cut-formula appearing appearing in in it. it. Thus Thus if if aa derivation derivation has has cut-rank cut-rank 00 then then where there are are no no L-Cuts L-Cuts used used in in it, though there there may may still still be be many many N-Cuts. We need need only only there it, though N-Cuts. We be concerned concerned here here with with derivations derivations of of fi finite cut-rank and and we we denote denote the the fact fact that that a a be nite cut-rank derivation of of n n : 9N N f-0 ~a A A has has cut-rank cut-rank ::; _< rr by by writing writing n n : 9N N f-� t--~ A. A. derivation 3.8. Note. N o t e . Weakening Weakening and and inversion inversion do do not not affect affect cut-rank. cut-rank. 3.8.

Suppose 3.9. Lemma. Lemma. Suppose n n : 9N N f-0 ~a A, A, E E where where E E is is aa false false atom. atom. Then Then n n : 9N N f-0 ~a A. A. 3.9. This lemma lemma is is required required for for the the following following result result (the (the proof proof is is an an easy easy induction induction on on the the This derivation of of n n : 9N N f-0 ~ A, A, E). E). derivation 3.10. C ut-Reduction L emma. Suppose n n : 9N f-� I-~ A, and n n : 9N f-� I-~ A',-~C 3.10. Cut-Reduction Lemma. Suppose A, C C and A/, ...,C where C is is aa formula the form E(( .. ... .)) or = rr + or Bo BI or or 3xB(x), 3xB(x) , and and II C where C formula of of the form E Bo V B1 C II = + 1. l. Suppose that a[n/] Suppose also also that c~[n'] � C_ 7[n'] n.. Then ,[n'l for for every every n' n' > 2: n Then n" N ~-~+= A', A --7'

Proof. by induction induction over over the the derivation derivation of n:N N ~f-� A, A, C. P r o o f . We We proceed proceed by of n" C. If C C is is an an inactive the final rule applied either nn : 9N ~ A A 11.. If inactive side-formula side-formula in in the final rule applied then then either N f-� +o A', is an in which so is is nn :9N A', A; A; or C remains in the the is an axiom axiom in which case case so N ~-~+~ or else else C remains in -f-� -~, premises final rule thus: premises of of that that final rule thus: k:N N F-r f-�~ A", A" , CC k" where we we may may assume, assume, by by weakening weakening ifif necessary, necessary, that that kk _> 2: nn and/3 and f3 E c~[k], ark] , where so/~[n'] C � -y[n'] ,[n'l for for every every n' n' _> 2: k. k . Therefore Therefore by by weakening weakening nn" : N N ~-~ f-� A', A/, ~""C so f3[n/] C_ (if necessary) necessary) to to kk :9N N ~f-� A', A/, ~""C and applying applying the the induction induction hypothesis, hypothesis, we we (if C and obtain obtain k" k:N N ~f-�++~.B A', A/, A A"II o

and hence hence the the final final rule rule can can be be re-applied to obtain obtain nn 9 N and re-applied to N ~-~+~ f-J+ A', as A', A A as --7" required. required.

169 169

Provably Recursive Recursive Functions

If C c == = E is the the "main" "main" formula formula of of n n ": N N I-� F-7 A, A, C C then then this this means means E 22.. If E(( ... .. ). ) is E(( ..... ). ) is is true true and and we we have have an an axiom. axiom. Consequently Consequently -,C ~ C == = E( E ( ... .. ). ) is is false. false. Now Now nn : 9NN 1-;: ~-r7 A' A ~ follows follows from from n n : 9N N 1-;: F-r7 A', A ~, -,C -~C by by Lemma Lemma 33.9 and Weakening Weakening then then . 9 and +Q A', gives --y, gives n n": N N I-;: ~_7+a A ~, A. A 9 3. 3. If If C C == - (Bo (B0 V V B1) B1) is is the the main main formula formula the the proof proof is is similar similar to to the the next next case. case. 4. If If C C == = 3xB(x) is the the main main formula formula then then the the last last rule rule applied applied in in obtaining obtaining 3xB(x) is 4. nn : 9NN I-� F7 A, A, C C is is an an 3-rule 3-rule and and we we may may assume assume that that the the two two premises premises are, are, for for some some m, m, ~_~om m ": N N and and n n": N N 1-� F~11 A, A, B(m), B(m), C C nn": NN I-�o where [n] . [n] � where f3o ~0,131 C_'}' "y[n]. , f31 Ee o:a[n]

Now induction hypothesis Now apply apply the the induction hypothesis to to the the right-hand right-hand premise premise so so as as to to obtain obtain H7mt-~lA i 'I,, A, A, B(m) B(m) n~'~:" N I-;:+Ih

and and apply apply \I-inversion V-inversion and and Weakening Weakening to to the the assumption assumption n n : 9N N 1-;: F-~ A', A', \lx-,B(x) Vx~B(x) to to obtain obtain -,B(m) ((.) *) max(n, m) m)": N N 1-;: t-~ A', A', A, A,-~B(m) max(n, Then N-Cut with n, m) Then one one N-Cut with n n": N N I-�o t-~~ max( max(n, m) : 9N N yields yields H7+-~-~1 A I, A, A,-~B(m) nn": NN I-;: Ih A', -,B(m)

provided 0. For *) provided f3dn] ]~I[T~]=f. ~ O. For if if 65 E e f3dn] j31[n] we we may may weaken weaken the the ordinal ordinal bound bound in in ((.) N-Cut applies both f30 and from f3dn] . from '}' "), to to ,},+ "y+ 65.. Then Then the the N-Cut applies since since both/30 and ,}, -y+ + 65 Iie lie in in '}'+ ~ +/31[n]. Thus Thus if if f3 ~l[n] r 0 o we we can can apply apply an an L-Cut L-Cut with with cut-formula cut-formula B(m) B(m) of of height height rr 1 [n] =f. to A',~ A to obtain obtain n n : 9N N I-rQ ~ +~ A A as as required. required. --7" On On the the other other hand hand if if f3dn] j31[n] = -- 0 o then then n n :: N N 1~ ,811 A, A, B(m), B(m), C C is is an an axiom axiom and and ,8 1 A', A, A, B(m) A', A nn": NN 1-F ~1 A', B(m) is is too. too. Thus Thus either either n n" : N N 1-;: ~ A', A is is already already an an axiom, axiom, or 3.9 and * ) to or else else -,B(m) -~B(m) is is a a false false atom atom in in which which case case we we may may use use 3.9 and ((.) to obtain obtain +Q A', (n, m) A, from max max(n, m ) ": N N 1-;: F~ A', A', A, from which which we we may may deduce deduce n n ": N N I-;: ~-~+" A', A A by by an an N-Cut as N-Cut as before. before.

3.11. (Gentzen 3.11. Cut-Elimination C u t - E l i m i n a t i o n Theorem. Theorem. (Gentzen [1936] [1936] and and Schiitte Schiitte [1977]) [1977]) lf n : N 1-;" If n n": N I-�+l ~-rabl A A then then n" 1-2~ A. A. Hence n n": N N I-� F-7 A A implies implies n n": N N I-f ~* A where 0: a** = = eXP2 exp~(~) Hence A where (0:)

= =

2" } 22",'·2" } rr 2'2's.s .

Proof. induction on nal rule P r o o f . We We proceed proceed by by induction on the the derivation derivation of of n n : 9N N I-� ~-r~_kl A. If If the the fi final rule + l A. applied induction hypothesis applied is is anything anything other other than than an an L-Cut L-Cut of of rank rank rr + + 11,, apply apply the the induction hypothesis o:[n] then to to the the premises premises and and then then re-apply re-apply the the same same rule rule using using the the fact fact that that if if f3 ~ E C a[n] then hand, the final rule 22,8z EE 2Q[n] 2"[n].. If If on on the the other other hand, the final rule is is an an L-Cut L-Cut with with premises premises ~0 ~1 nn " : N I-��l and ~-r+l A, A , - -,C he and n n " : N I-��l ~-r+l A, A, C C

where induction hypothesis possibly aa Weakening where II C CII = - rr + + 11,, then then the the induction hypothesis and and possibly Weakening gives gives n'N 2 ~ A, A , ~-,C C and and n n'N - 2 ~A, A,C n : NF I-;Il : NF I-;Il C

170 170

M. M. Fairllough Fairtlough and S. S. Wainer Whiner

where /3 is /30 , /31 in where/~ is the the greatest greatest of of/~0,/~1 in a[n] a[n].. Now Now the the Cut-Reduction Cut-Reduction lemma lemma 3.10 3.10 applies applies C, ...,~ CC of immediately, immediately, with with 'Y 7 = - a a = - 2{j, 2e, A' A~= = A A and and one one of of C, of the the required required form. form. /l +2/l AA and P [n']'] S;;; Hence Hence n n :: N N r; t-~~+2~ and since since 22Pe + + 22e[n C_ 2°[n'] 2~[n '] when when n' n' � _> n n,, the the desired desired result result follows follows by by a a Weakening. Weakening.

3.12. 3.12. Note. N o t e . It It may may later later be be convenient convenient to to replace replace the the exponential exponential 220~ (in (in 3.1 3.11) by 1 ) by wO w ~.• The The above above proof proof still still works works in in that that case, case, but but with with the the proviso proviso that that the the declared declared P [n']'] S;;; parameter parameter n n should should be be nonzero nonzero since since only only then then do do we we have have w w Pe + +w we[n c_ wO[n'] wa[n '] for for all all n' n' � _> n, n, whenever whenever /3 ~ E E a[n] a[n].. 3.13. is 3.13. Definitions. Definitions. A A E�-formula lE~ is one one of of the the form form 3zl . . . 3zkB(zl, . . . , zk)

where where B B is is "bounded" "bounded" in in the the sense sense that that all all quantifiers quantifiers occurring occurring in in it it are are bounded bounded quantifiers quantifiers 3y(y 3y(y < < tt /\ A . .. .. ). ), , Vy(y Vy(y 1:. ~ tt V V ..... ). ) with with tt either either a a variable variable or or a a closed closed term. term. Note that that this this restriction restriction is is not not a a severe severe one, one, for for if if the the quantifier quantifier bound bound tt were were an an Note open term term then then we we could could rewrite rewrite 3y(y < tt /\ A ....) as 3z(z 3z(z = = tt /\ A 3y(y 3y(y < < z z /\ A ... .. ). )) ) and and open 3y(y < . . ) as vy(y 1:. tt V v . .. .. ). ) as as 3z(z 3z(z = = tt /\ ^ Vy(y vy(y < < zz ^/\. . . ). )) ) .. Vy(y A A closed closed E� E~-formula as as above above is is said said to to be be true true at at m m E E N N if if there there are are m ml, 1 , .. .. ..,, m m, such that B(m (in the mk, all less less than than m, such that B ( m ~1 ,, ... .. ,. , mk) mk) is is true true (in the standard standard model). model). k , all A is A set set A A of of closed closed E�-formulas lE~ is true true at at m m if if at at least least one one Ai Ai E EA A is is true true at at m. m.

3.14. is 3.14. Note. N o t e . If If a a set set A A of of closed closed E�-formulas IE~ is true true at at m m then then it it is is automatically automatically true m'. This Also true at at any any greater greater m'. This property property is is called called E�-persistence. lE~ Also if if A A contains contains a a true bounded then true formula formula all all of of whose whose quantifiers quantifiers are are bounded then A A is is true true at at any any m. m. 3.15. 3.15. Bounding B o u n d i n g Lemma L e m m a for for r. F-. 11.. nn": N r� m : N if and only N ~-~ m " N if and only if if m m ::; <_ Bo(n) B~(n).. 2. 2. Suppose Suppose A A is is aa set set of of closed closed E� E ~-formulas and and n n : 9N N r F-~O A A by by aa derivation derivation in in which which all all the the L-Cut-formulas L-Cut-formulas are are E� E ~ . Suppose Suppose also also that that n n bounds bounds the the value value of of every in A in any every closed closed term term occurring occurring in A or or in any cut-formula cut-formula of of the the derivation. derivation. Then Then A A is is true true at at Bo(n) Ba(n). . 3. 3. If If A A == - 33zZ1l .· .·. · 3zkB(Zb 3 Z k B ( Z l , .. ... ,. Z, Zk) k ) is is aa closed closed E� E~ -formula true true at at Bo(n) Ba(n) and and n n I A I A. bounds bounds the the value value of of every every closed closed term term in in A A then then n n": N N r� }-o+ +1AI A. Proof. Proof. 11.. Suppose no L-cuts Suppose n n : 9N N r� }-~ m m : 9N N.. Since Since no L-cuts are are involved involved only only the the rules rules N-Ax, N-Ax,

N-Succ, N-Succ, N-Cut N-Cut can can have have been been applied applied in in the the derivation. derivation. If If it it arises arises by by N-Ax N-Ax then (n) . If then m m ::; _< n n + + 11 ::; ___Bo B,,(n). If it it arises arises by by N-Succ N-Succ from from n n : 9N N rg t-o~ m m - 11 : 9N N then then inductively inductively we we have have m-l m - 1 ::; _< Bp(n) B e ( n ) and and since since /3 fl E E a[nJ c~[n],, m m ::; _< Bp(n)+l B e ( n ) + 1 ::; _< Bo(n) B~(n). . l m : 9NN then o kk : 9NN and If N-Cut from then If it it arises arises by by N-Cut from the the premises premises n n": N N rg }-o ~~ and k k : 9N N rg t-o~1 m by (k) so by the the induction induction hypothesis hypothesis k k ::; <_ B{j Seo(n and m m ::; _< Bp Be1I (k) so as as /30 ~o,, /3 ~11 E a[n] o/In] o (n)) and we we have have m m ::; <_ Bp Be~(Beo(n)) <_ Bo(n) S~(n). . I (Bpo (n)) ::; -

Provably Provably Recursive Functions

171 171

Conversely suppose suppose m rn � <_ B B~(n). We show show n n : 9N N I-� I-~ m m : 9N N by by induction induction on on Conversely o (n) . We a. I f aa = = 00 then then m m_� n n ++l 1 so so n n": N N I-� F-~ m m" : N hi by by N-Ax. N-Ax. If I f aa = = (3 t3+ + 11 a. If then induction hypothesis then m m � <_ B Be(k where k k = = B Be(n so by by the the induction hypothesis we we have have fj (k)) where fj (n)) so I-� m nn": Nhi I-g F-o ~k k": N N and and k k": N hi I-g F-0 ~m m ": N hi and and hence hence n n": N hi ~-~ m ": N hi by by N-Cut. hi-Cut. Finally Finally if if a a = = sup sup ax ax then then m rn � <_B B~(n) =B B,,,,(n) so n n ": N hi I-�n F-~n m m ": N N by by the the induction induction oJn) so o (n) = n. Therefore hypothesis. hypothesis. But But a a is is structured structured so so an[n/] an In'] � C_ a[n/] a[n'] for for every every n' n' � >__n. Therefore nn": NN I-� ~-~ m m ": N hi by by Weakening. Weakening. 2. Suppose of closed and 2. Suppose A A is is a a set set of closed ��-formulas E~ and n n : 9N hi 1-0 F-~ A A by by a a derivation derivation involving Suppose also involving only only �� E ~ cut-formulas. cut-formulas. Suppose also that that n n is is large large enough enough to to bound bound the the values values of of all all closed closed terms terms in in A A and and all all closed closed terms terms occurring occurring in in the the cut-formulas cut-formulas used used (note (note that that in in general general no no such such bound bound need need exist) exist).. We induction on a, with We prove prove that that A A is is true true at at B B~(n) by induction on a, with cases cases according according to to o (n) by the the last last rule rule applied applied in in deriving deriving n n": N N 1-0 t-~ A. A. L-Ax. k-Ax. Then Then A A contains contains a a true true atom atom so so A A is is automatically automatically true true at at B B~(n). o (n) . 1\ disjunctions or A,, V V.. These These cases cases are are also also immediate immediate since since we we only only need need take take disjunctions or conjunctions conjunctions of of "bounded" "bounded" formulas formulas in in this this context, context, and and so so it it is is only only necessary necessary that that the the rules rules are are sound. sound. fjo m fjl A', 3. Suppose A A = - A', A', 3xD(x) 3xD(x) and and n n ": N hi I~eo m ": N hi and and n n ": N hi I~el A', D(m) D(m) 3. Suppose where where {30 130,, (31 ~1 E e a[n] din].. Then Then by by the the induction induction hypothesis hypothesis A', A', D(m) D(m) is is true true at part (1) at B Be~ (max(n, m)) m)),, and and by by part (1) above, above, fjl (max(n,

m m<
where where {30 fl0,, (3 f l1l eE a[n] a[n],, so so both both B 8eo (n) and and B Be, (n) are are less less than than B Ba(n). The o (n) . The fjo (n) fjl (n) standing standing assumption assumption on on n n means means that that n n bounds bounds the the values values of of closed closed terms terms

172 172

M. M. Fairtlough and S. Wainer Whiner

's true in C. C. Now Now if if A A is is true true at at Bp B/3o(n) then it it's true at at Ba Ba(n) by persistence, persistence, (n ) by in o (n) then and we're we're done. done. Otherwise Otherwise by by the the induction induction hypothesis, hypothesis, 3Z'D(Z) 3s is true true is and so there are m Bp such that the bounded formula (n) at Bp B~o(n), so there are rfi < Bzo(n ) such that the bounded formula (n) , at < o o D(rfi) is is true. true. But But 'I-inversion V-inversion applied applied to to the the second second premise premise of of the the cut cut D(m) P1 A, .D(m) and gives max max(n, nh) :: N N I~_~1 A,-~D(rfi) and we we can can now now apply apply the the induction induction gives ( n, m) hypothesis again again to to obtain obtain A A true true at at Bp BZl1 ((max(n, rfi)),, because because .D(m) -~D(rfi) is is hypothesis max ( n, m)) (Bpo (n)) :::; false. Since Since Bp B~,(max(n, rfi)) < < Bp1 Bt3,(B~o(n)) < B Ba(n) we again again conclude conclude false. a (n) we 1 ( max (n , m)) A true true at at Ba Ba(n) by persistence, persistence, and and this this completes completes part part 2. 2. A (n) by 3. Suppose Suppose A A == - 3z1 3 z l .• .•. • 3zkD(Zb 3z~D(zl,...,Zk) is aa closed closed � E~� -formula true true at at Ba(n) B~(n).. 3. . . . , Zk) is Then we we have have m rh < < Ba(n) B~(n) such such that that the the bounded bounded formula formula D(m) D(rfi) is is true. true. It It is is Then +b D(m) an an easy easy exercise exercise to to check check that that n n ": N N I-� ~_~+b D(nh) where where bb = = II D D jI,, as as long long as as n n a m / :: N bounds the the value value of of any any closed closed term term in in A. A. Then Then since since n n :: N N Ik-a m N for for each each bounds i IAI A by part part 11,, we we obtain obtain n n : 9N N I-�+ ~o +1AI A by by kk applications applications of of the the 3-rule. 3-rule. ii by Bounding Lemma Lemma for F ~ .. 3.16. Bounding a NF ~ m m:: N N if if and and only only if if m m :::;
� nn � 3z . C,(m, ff(m) (rh) ~_ ~ 3z. Cf(T~, n, n, z) z)

is true is true in in N hi..

Also iff it it is is partial partial recursive recursive and i.e. if if the Also ff is is recursive recursive iff and totally-defined, totally-defined, i.e. the II~ rrg -formula

'V'x 3y 3z . C,(x, y, z) v 3v3z.Cf(Z,v,z)

is true ((in in the the standard standard model Roughly speaking, speaking, the "computation formula" is true model hi). the "computation formula" N). Roughly C,(X, y, z) that zz encodes encodes aa terminating terminating computation computation of f(X) with with output Cf(~., y, z) expresses expresses that of f(Z) output value y. y. There are canonical canonical choices of such such computation computation formulas, formulas, depending depending on on value There are choices of the underlying underlying model model of of computation. computation. With With this this in in mind, mind, we make the following the we make the following definitions. definitions. ,-recursive iff iff for for some some computation formula C, and some some c~ a -< -< 7, " 1 . Call Call ff 7-recursive 1. computation formula Cf and I-� V~3y 'V'x 3y 3z. 3z . Cf(~, C,(x, Y, y, z) z) ~

and let let REC(~') REC (-y) denote the class of all all "y-recursive ,-recursive functions. functions. and denote the class of 2. Call Call ff ~/-definable ,-definable iff iff for for some some computation formula C, and some some c~ a -< -< 7, " 2. computation formula CI and a F vz 'V'x3y C,(X, v, y, z) 3v 3z . c:(z,

and let let ~~ �� -DEF (,) denote denote the the class class of of all all "r-definable ,-definable functions. functions. and

Provably Recursive Functions

173 173

3. Call Call f f provably provably recursive recursive in in a a given given arithmetical arithmetical theory theory T T iff iff for for some some compu compu3. tation tation formula formula C Cff

T Ui, y, T IF Vx V~ 3y 3y 3z 3z.. Cf Cf(~, y, z) z)

and and let let PRovREC(T) PrtovREc(T) denote denote the the class class of of all all functions functions provably provably recursive recursive in in T. T.

3.18. Normally 3.18. Remark. Remark. Normally definitions definitions of of provable provable recursiveness recursiveness require require not not only only aa proof proof of of the the existence existence of of the the output output value value but but in in addition addition aa proof proof of of its its uniqueness. uniqueness. However, However, for for standard standard choices choices of of computation computation formula, formula, such such as as those those we we shall shall con construct later, uniqueness will be a a corollary corollary of of existence. existence. Alternatively, Alternatively, in in the the presence presence struct later, uniqueness will be principle would of of � S �~ induction, induction, the the least least number number principle would allow allow any any computation computation formula formula to to be be modified modified to to satisfy satisfy the the stronger stronger definition. definition. 3.19. 3.19. Hierarchy Hierarchy Theorem. Theorem. ,,,, E OS 1. Suppose , 7 = = sup sup')'x f~s is is q-space-representable q-space-representable with with w � -~ " "y, and and suppose suppose 1. Suppose qi'(m) ::; /3 ( max ( n , m )) for there there is is f3 fl � -~ , 7 such such that that q'~(m) <_ B Sa(max(n, m)) for all all n n,, m. m. Then Then REC({) REC(7) = =

U = U U E(Ba) E(B~)= U SPACE(Ba) SPACE(B~)..

a -<'Y

a-<'Y

-space-representable with Suppose , 7 = = sup sup ' 7xx E OS f~s is is qq-space-representable with 22w~ � -~ " 7, and and suppose suppose 22.. Suppose n there such that (m) ::; )) for n, m. lso suppose there is is f3 i~ � -~ , 9/such that qqn(m) <_ G/3(max(n, Ga(max(n, m m)) for all all n, m. A Also suppose 30'. � that that V Vce0, c~1 � -~ , 73c~ -~ , 7 (Ga (Gaoo 0o Ga Gall ::; < Ga) G~).. Then Then 0'.0 , 0'.1 ��-DEF(,) = E~ =

= U U U E(Ga) E(G~)= U SPACE(Ga) SPACE(Ca)..

aa-~7 -<'Y

aa-~7 -<'Y

Proof. Proof. k 1. Suppose 1. Suppose f f : 9N Nk

� --~ N N is is ,-recursive. 7-recursive. Then Then the the graph graph of of f f is is defined defined by by aa (x, y, y, z) � S~� -formula 3z 3z.. Cf Cf(~, z) and and for for some some a c~ � -~ " "y,

I-g Vx3y ve3y 3z 3z.. Cf(x, cf(e, y, y, z) z)..

k For For any any m r5 E N 1N~,, V-inversion V-inversion gives gives max (m, y, max m rh :" N I-g F~ 3y 3z 3z.. Cf Cf(rfi, y, z). z).

The 3.15 (2) The Bounding Bounding Lemma Lemma 3.15 (2) then then shows shows that that the the � S~� -formula 3y J (m, y, y, z) ( max m) 3y 3z 3z.. C Cf(rh, z) is is true true at at Ba B~(max rh) provided provided max max m r5 is is at at least least as as big big there are as as any any number-parameter number-parameter occurring occurring in in Cf Cf ((there are no no cut-formulas cut-formulas since since the the proof proof above above has has cut-rank cut-rank 0). 0). Thus Thus for for all all but but finitely-many finitely-many inputs inputs m r5 there there are are = f (m) ) and numbers (max m) , such numbers no n0 ((= f(rfi)) and ~ = = nl, n l , .. ... ., n, nl t , , all all less less than than Ba B~(maxrfi), such that that Cf(m, Cf(rfi, no,ri) no, ~) holds. holds. Now Now let let ((no, n o , .. .. .., , nl) nt) be be any any standard standard polynomial polynomial coding coding of of Il + + I-tuples 1-tuples into into N, N, . . . , nl) with with inverses inverses (n) (n)ii so so that that if if n n = = (no, (n0,..., nl) then then (n) (n)ii = = n nii for for each each ii ::; _< I. 1.

ii

174 174

Wainer M. Fairtlough and S. Whiner M.

Furthermore let let bb(m) (Ba(max m), . . . , Bo(maxrh)). Then we we can can compute compute Furthermore ( ~ ) : := - (Bo(maxff~),..., Ba(max m)) . Then as follows: follows: ff as f(r~) = (least n < b(~).Cf(r5, (n)o, (n)l,..., (n),)0

built up up from from elementary elementary relations relations using using only only propositional propositional connecconnec But C, But C / i sis built tives and bounded bounded quantifiers, quantifiers, so so C, itself defines defines an an elementary elementary relation. relation. Also Also tives and C/itself ( . . - ) and and decoding decoding ()i Oi functions functions are are elementary elementary and and elementary elementary the coding coding (...) the Ba . functions are are closed closed under under bounded bounded minimisation. minimisation. So So ff is is elementary elementary in in Bo. functions Conversely suppose suppose ff E E(Bo) E{Ba) for for some some ~a -~ -< 7. "I. Since Since E(Bo) E{Ba) expands expands as as Conversely a increases, increases, and and since since w W -~ -< -/, "I, we we may may as as well well suppose suppose also also that that w W 5� c~. a. So So + and by by 2.29, 2.29, ff is is computable computable within within time time or or Bo(m) m+l! and Ba{m) >_ ;::: B~(m) Bw {m) = = m m+ + 22m space bounded bounded by by some some fixed fixed iterate iterate of of Bo, Ba , say Ba+i . But But for any space say B�i B~' -= Bo+i. for any reasonable model model of of computation computation there there is is aa bounded bounded "computation "computation formula" formula" reasonable C, {X,, y,z) y, z) which which expresses the fact fact that that f{x) is computable computable within within resource resource CI(Z expresses the f(Z) is and with with output-value Since we we know that our our ff is is computable computable within within zz and output-value y y.. Since know that resource Ba+i , it follows that that the the formula formula 3y3z C, {m , y, y, z) z) is is true true resource bounded bounded by by Bo+i, it follows 3y3z.. Cf(rh, at Ba+i(max Bo+~(max rh), by Bounding (3),, we we at m) , for for all all inputs inputs ff~. m. Therefore Therefore by Bounding Lemma Lemma 3.15 3.15 (3) have 2, have for for jj = = II c, C f lI ++ 2, a+ + L �:N N -o~-~ ::J .. C � , y, zz)) max m max nh" Ci(rh ::JY-.:Jz , (m, '0 i i ::J3y3z and then then by applications of the V-rule, 'V-rule, and by k k applications of the k \.I�::J ::J a+ � L ~+'+J+k Cf, ((,~, y, zz). vX -.:JY-.:JZ . C m, y, '0 i+i+ V~3y3z. ).

Since limit and Since "I "y is is aa limit and a c~ -< -~ "I "y we we also also have have a c~ + § ii + -t- jj + § kk -< -~ "I "y and and hence hence f f is is "I-recursive. "y-recursive. The The second second equality equality was was previously previously established established at at the the end end of of section section 2. 2. 22.. This This follows follows exactly exactly the the same same lines lines as as (I) (1) but but using using Bounding Bounding Lemma Lemma 3.16 3.16 O for conditions imposed for p ~o.. The The closure closure conditions imposed on on "I ~' ensure ensure that that if if f f E E{Ga) E(Go) + I , and for some some a (~ -< ~ "I "y we we may may assume assume 22wW � _'~ a c~ so so that that Ga{m) Go(m) ;::: _> 2m 2re+l, and then then for ' find find a (~ -< -~ "I ")' such such that that Gal Go, bounds bounds aa fixed fixed iterate iterate of of Ga Go.. Hence Hence f f will will be be computable computable within within resource resource bounded bounded by by Gal Go, and and for for each each input input m, r5, the the �� E]~ al formula {m, y, true at formula 3y3z 3y3z.. C, Cf(ff~, y, zz)) will will be be true at Ga/ Go, {max (max m) r5),, and and so so max max m r5 p ~o' 3y3z 3y3z.. C, {m, y, y, zz)) and C/(r5, and f f is is therefore therefore "I-definable. "),-definable.

3.20. 3.20. Example. E x a m p l e . First First notice notice that that for for each each integer integer kk ;::: >_ 0, 0, B Bo+~ is just just Ba Bo iterated iterated a+k is

m+l {m) and times, so so E{Ba) E ( S o ) == E{Ba+k) E(Bo+k).. However However Ba B o+w + ~{m) ( m )= - B� S2~+l(m) and so so B Bo+~ 22k~ times, ow eventually eventually dominates dominates every every function function in in E{Ba) E(Bo) if if w w � ~ a. a. Thus Thus B Bo+~ E(So). a+w �r E{B a) . In particular then, then, the k with k = 1 , 2, . . . play the role of the Ack In particular the functions functions B Bw.k with k = 1, 2,... play the role of the Ackw. k+2 ermann/Grzegorczyk ermann/Grzegorczyk functions functions in in such such aa way way that that E{B E(B~.k) = Grzegorczyk's Grzegorczyk's £ E k+2. . w .k) = Therefore Therefore for for each each kk ;::: _ 22 we we have have by by 33.19 (1), since since w w.. kk is is q-space q-space representable representable for for . 1 9 (I), some some polynomial polynomial q, q, HI REC(w.. k) k) = = £ ~k--t--1 REC{W

Provably Provably Recursive Recursive Functions Functions and hence hence and

175 175

REC(w 22)) = -- PRIMITIVE PRIMITIVE RECURSIVE. RECURSIVE. REC(W

Note hand, Note that that on on the the other other hand,

E~

= t; 3.

3.21. Example. Example. It should should be be fairly fairly clear clear that that for for limits limits , 7 satisfying satisfying mild mild repre repre3.21. It sent ability and sentability and closure closure conditions, conditions, that that the the ,-recursive 7-recursive functions functions are are exactly exactly those those defined recursions over initial segments ,. For recursion f, defined by by nested nested recursions over initial segments of of 7. For any any such such recursion f, computation formula formula Cf Cf would would be be definable definable and and an an informal informal termination termination proof proof aa computation by transfinite transfinite induction induction could could be be translated translated into into aa formal formal proof proof within within the the system system by f"( . Conversely, F-~. Conversely, the the Hierarchy Hierarchy Theorem Theorem would would show show that that any any ,-recursive 7-recursive function function is is elementary elementary in in some some BOt B, for for a ~ � -~ " 7, and and so so definable definable by by nested nested recursions recursions because because the the B, functions functions are. are. BOt

3.22. 3.22. Note. Note. From From the the example example on on primitive primitive recursion, recursion, we we see see that that part part 11 of of the the Hierarchy Hierarchy Theorem Theorem applies applies to to any any , 7 which which is is q-space q-space representable representable for for some some primitive primitive recursive q, and such such that that w2 w2 :s _ , 7.. recursive q, and Similarly Similarly part part 22 of of the the Hierarchy Hierarchy Theorem Theorem applies applies to to any any , 7 which which is is q-space q-space representable q, provided representable for for some some primitive primitive recursive recursive q, provided , 7 is is at at least least as as big big as as the the fifirst rst primitive primitive recursively recursively closed closed ordinal ordinal and and also also satisfies satisfies the the stated stated compositionality compositionality requirement. requirement. These conditions will will indeed These conditions indeed be be satisfied satisfied by by any any , 7 to to which which we we later later apply apply the the Hierarchy Hierarchy Theorem, Theorem, but but we we leave leave the the reader reader to to convince convince him him or or herself herself of of this this fact. fact. In particular, it In particular, it is is quite quite easy easy to to see see that that C:o ~0 is is polynomial-space polynomial-space representable. representable. See See e.g. Sommer e.g. Sommer [1992] [1992].. 4. Provably 4. P r o v a b l y recursive r e c u r s i v e functions f u n c t i o n s of o f PA PA

Here recursive functions (PA) Here we we characterize characterize the the provably provably recursive functions of of Peano Peano Arithmetic Arithmetic (PA) and its its fragments: fragments: and PRovREC(PA) PRovREc(PA) = = REC(C:o) REC(e0) PRovREC(L:�-IND ) = PROvREc(E~ = REC((C:O)n) REC((e0),) if if n n >� 22 PROVREC(~0_IND)- REC(W REC(w22)) PRovREC(L:�-IND)

These These results results go go back back to to Kreisel Kreisel [1952] [1952] for for PA, PA, Parsons Parsons [1966] [1966] for for the the fragments, fragments, and and to to Wainer Wainer [1970] [1970] and and Schwichtenberg Schwichtenberg [1971] [1971] for for their their corresponding corresponding subrecursive subrecursive hierarchy classifications. hierarchy classifications.

4.1. 4.1. Peano P e a n o Arithmetic A r i t h m e t i c (PA) (PA).. Our Our version version of of PA PA is is formalized formalized classically classically in in aa Tait-style calculus PA built out Tait-style calculus PA f~- A A where where as as before, before, A A is is aa finite finite set set of of formulas formulas built out

E( . . . ) , E( . . . ) using of of atoms atoms E(...), E(...) using V, V, 1\ A,, 3 3,, V; V; but but now now the the formulas formulas may may contain contain free free variables. the free variables. It It is is sometimes sometimes convenient convenient to to display display the free variables variables occurring occurring in in A A by by

176 176

M. Fairtlo'tlgh Fairtlough and S. Wainer M.

L-Ax L-Ax fF- A, A, E(tb E ( t l , .. ... ., , tk) tk),, E(tb E ( t l , .' ... ., , tk) tk) V V

f-F- A, A , Bi Bi A, (Bo (B0 V B B1) f- A, 1)

1\

A, Bl f-t-A, A, Bo Bo fF-A, B1 f-~- A, A, (Bo (Bo 1\ A B1) B1)

3

A, B(t) B(t) f-~ A, 3zB(z) f-~- A, 3xB(x)

f-F- A, A, B(y) B(y) VxB(x) f-F- A, A, VxB(x)

V

= 0 0 or or ii = = 11 ii =

not free free in in A A y not

A , ~...,Ce ft- A, A,C e f-F- A, f-t - AA

L-Cut L-Cut

Figure rules of 2: Logical Figure 2: Logical axioms axioms and and rules of PA PA writing The special not occur in the writing A(Xb A ( x l , ... .. . ,,Xk) etc. The special atoms atoms x x : 9N N do do not occur in the language language of of Xk) etc. PA. PA. 1. logical axioms are as as in 2. The logical axioms and and rules rules are in figure figure 2. 1. The The principle principle of induction is formulated here rule (but (but see Note 4.3 below) : of induction is formulated here as as aa rule see Note 4.3 below)" 22.. The Ind Ind

f- A,B(0) A, B(O) t-

f-~ A, + 1) A , -...,, BB( (x) x ) , ,BB(x (x + 1) f- A, A, B(t) B(t)

where the the variable not free free in any term. where variable xx is is not in A, A, and and tt is is any term. 3. To ground, we to add certain arithmetical 3. To get get the the theory theory off off the the ground, we also also need need to add certain arithmetical axioms defining the the atomic relations between basic terms. Among these these will will axioms defining atomic relations between basic terms. Among the successor successor function function ++ 11 and and the the equality equality and and inequality inequality be 0, the be the the constant constant 0, y) relations Eo(x, y) -== (x (x == y) y) and and Eo(x,y) Eo(x, y) -== (x (x #=f. y), y) , El(X,y) E1 (x, y) === (x (x << y) relations Eo(x,y) and E1 E1 (x, (x, y) y) === (x (x ~
(~0,..., ~-~) - p(...p(p(0, ~0), ~ ) , . . . , ~ _ , )

Provably Recursive Functions

ItII~ItIIFIFIFIIII~I-

177 177

A , xx++l 1 i= r 0 A, A r y + 11,,xx ==y Y A,, xx++ 11 i= A , xx==x x A, A , xx~ i= y , yy,=yx = x A, A, A , xxr 1i=6y, 2 y i= z, x = z A , xx~f:.0 0 A, A , xx < < xx++ l 1 A, A , xx f:. ~ yy ++ l, l , xx < < y, y , xx==y y A, A, A , xx~ f:. x x A, A , xx~ f:. y , yy,~yz ,f:.x
Figure 3: Arithmetical Arithmetical axioms Figure 3" axioms of of PA PA I~- A, A , pp(x, ( ~ , vy) ) i= # O0

~I-

A, A, u(p(x, u(p(~, y)) y)) = = x

A, vv(p(~, y)) = = Yy (p(x, y)) I-~- A,

A, x ~ = = O, 0, x9 = = p(u(x) p ( u ( ~ ) ,, ,v( ~(x)) )) I- A,

axioms of Pairing axioms Figure 4: Pairing Figure 4: of PA PA where thus we p(x, y) where 00 denotes denotes the the empty empty list list ((thus we can can think think of of p(x, y) as as representing representing the the list list with with head head y y and and tail tail x). x). Together Together with with this this list-constructor list-constructor we we require require three three . . . , Xkfurther further terms terms lh(x) lh(x),, in(x, in(x, j), j), (x) (x)jj such such that that if if x x= = (xo, (x0,..., Xk-1) as above, above, then then I ) as lh(x) k, in(x, lh(x) = = k, in(x, j) j) = = (xo, (Xo,... and (x) (x)jj = = xi xj,> provided provided jj < < lh(x) lh(x).. Xj - I ) and . . . ,,xj-1) We shall not We shall not write write down down all all the the axioms axioms needed needed to to define define these these term-constructors term-constructors to say that in model IN, N, they p, uu,, vv,, lh, lh, in, in, (k ()j. It It suffices suffices to say that in the the standard standard model they can can all all p, be be interpreted interpreted as as elementary elementary functions, functions, for for example example take: take: 1

m ++n )n) ( m(m + n++ n + l) 1) + + m m ++ l1 p(m, n) = 2l ((m i lh(n) = least least ii < < n n+ + 22.. (u (ui(n) = 0) O) lh(n) (n) = in(n, j) = uu 1h 'h(n)-j (n) (nH (n) in(n,j) (n) (n)jj = vv(in(n, + 1)) 1)) (in(n, jj +

178 178

M. Fairtlough and and S. Wainer M. Thus they they can can all all be be introduced introduced in in PA PA by by sets sets of of axioms axioms corresponding corresponding to to their their Thus elementary defining defining equations. equations. Further Further terms terms denoting denoting primitive primitive recursive recursive elementary functions could could be be added added at at will. will. functions

Definition. The The fragments fragments ��-IND ~0-IND and and II�-IND II~ of arithmetic arithmetic are are obtained obtained 4.2. Definition. of

simply by by restricting restricting the the logical logical complexity complexity of of the the induction-formula induction-formula B B in in the the Ind Ind simply rule to to B S E e �� T ~, B B E e II� II ~ respectively. respectively. Recall Recall that that aa �� S ~ (respectively (respectively II�) II ~ formula formula rule is one one obtained obtained from from aa bounded bounded formula formula by by prefixing prefixing n n alternating alternating quantifier quantifier blocks, blocks, is starting with with existentials existentials (respectively (respectively universals) universals).. Note Note however however that that the the sequence sequence starting coding coding machinery machinery allows allows each each block block of of like like quantifiers quantifiers to to be be contracted contracted to to aa single single quantifier quantifier and and any any �� T ~ or or II� H ~ formula formula is is provably provably equivalent equivalent to to its its contracted contracted form form without the the use use of of induction. induction. without Note. We do do not not place place any any restrictions restrictions on on the the logical logical complexity complexity of of the the 4.3. Note. We side-formulas A A in in the the induction induction rule rule (but (but compare compare Sieg Sieg [1991]) [1991]).. Consequently Consequently it it is is side-formulas very easy easy to to see see that that in in PA PA or or any any of of the the above above fragments, fragments, the the induction induction rule rule may may very equivalently be re-formulated as as an an axiom axiom scheme scheme equivalently be re-formulated

A,-,B(O), 3x(B(x) 1\ A -,B(x + 1)), 1)), B(t) B(t).. -,B(x + I-F- A, -,B(O), 3x(B(x)

presence of modified subtraction subtraction function Note also also that that in in the the presence of the the modified function -'- , , the the frag fragNote and are ments ments II�-IND II~ and ��-IND T~ are equivalent. equivalent. For For it it is is easy easy to to see see that that the the above above induction axiom can can be be derived derived from from its its dual dual form form induction axiom - ,B(t -'A, B(t B ( t --'- 0) 0),, 3x( 3x(-~B(t "- x) x) 1\ A B(t "- (x (x + + 1))), 1))),-,B(t - ss).) . -,B(t -'B(t -'I-~- A, The rank rank of of aa PA-derivation PA-derivation is maximum of all the heights The is the the maximum of all the heights of cut-formulas and induction-formulas induction-formulas occurring occurring in ]I C e l ,I , II B cut-formulas and in it. it. B ]I of

4.4. Definition. 4.4. Definition.

4.5. Definition. For each PA-derivation 4.5. Definition. For each PA-derivation D D define define its its height height dd by by induction induction as as follows (so as to to incorporate incorporate an estimate of complexity of of witnessing witnessing terms terms in follows (so as an estimate of the the complexity in addition to the the derivation): addition to the depth depth of of the derivation) : 1. Any axiom axiom has height dd = 1 . Any has height = 1. 1. 2. D is Di of heights di (i < If D is constructed constructed from from sub-derivations sub-derivations D; of heights d; (i � 1) 1 ) by by an an 2. If application of of any any of rules V, V, A, 1\ , V, V, L-Cut, max d; + application of the the rules L-Cut, then then d d= -- maxdi + 1. 1.

of A B(t) by by applying applying the the 3-rule 3-rule A,, B(t) If D D is is constructed constructed from from aa sub-derivation sub-derivation Do Do of 33.. If then dd -= max(d0, m ax( do , ItI t [)I) ++ 11 where I t II is is the the least that the the term term then where It least number number kk such such that (regarded as as an an elementary elementary function function of of its its number number variables) variables) is is bounded bounded by by tt (regarded B~.k 3.20 that that such such aa kk exists). exists). (it follows follows from from 3.20 Bw .k (it 4. If If D D is derivation of A, B(t) B(t) by by the the induction induction rule rule from from subderivations subderivations Do, Do , 4. is aa derivation of A, 2. O1, then dd == max(d0, max(do, dl, db [t I t I)I) ++ 2. D 1 , then PA ~d A with with is aa PA-derivation PA-derivation of of A 4.6. NNotation. 4.6. otation. PA I-� A A indicates indicates that that there there is height height dd and and rank rank r.r .

Provably Recursive Recursive Functions Provably Functions

179 179

If then If PA PA f-� F-d A(t A ( t ll ((x) s , . . . , tk(X)) tk(s then for for all all assignments a s s i g n m e n t s of of nl to we have numbers numbers fi 4 = = nl, n l , .. ... ., , nt to the the free free variables variables x ~ = = XI X l,, .. .. .., , Xl xt we have max fi 4": N 51 f-� ~ . d.d A(m A ( m ll ,, .. ... ., , mk) mk) max

4.7. 4.7. Embedding E m b e d d i n g of of PA. PA.

where where ml m l , , .. .. .. ,, mk mk are are the the numerical numerical values values of of the the terms terms tl tl (fi) (4),, .. .. .. ,, tk(fi) tk(4). .

Proof. induction over Proof. By By induction over the the PA-derivation PA-derivation of of A(tl A ( t l ,, .. ... ., , tk) tk).. Let Let n n= = max max~fi where where fi4 is xed assignment 1) E is aa fi fixed assignment and and note note that that w w.· (d (d - 1) E w w . · d[nJ d[n].. either aa logical 1. If A(tl A ( t l ,, .. ... ., , tk) t~) is is either logical or or arithmetical arithmetical axiom axiom then then on on the the assignment assignment 1. If x := fi, the resulting set mk) must 9- 4, the resulting set of of closed closed formulas formulas A(mJ, A ( m l , .. ... ., , mk) must contain contain aa true true w A(ml atom, so so automatically automatically we we have have n n : 9N N fFw A ( m l ,, .. ... ., , mk) mk).. atom,

2. The V V,, /\ A,, and and L-Cut L-Cut cases cases are are immediate immediate since since the the corresponding corresponding rules rules in in the the 2. The infinitary infinitary system system are are virtually virtually the the same. same. 3. Suppose Suppose PA PA f-� Fd A(tl' A ( t l , .. ... ., ' tk), t ~ ) , VVzB(z, z B ( z , tl' t l , .. ... ., ' tk) t~) follows follows by by an an application application of of 3. - I A(tJ the V-rule from PA PA f-� Fd-1 A ( t l ,, .. ... ., , tt~), S ( z , tl, t l , . .. .., . , tk). t~). Then Then the the induction induction the V-rule from k ) , B(z, hypothesis hypothesis gives gives for for every every n' n' E E N, N,

. (d- l ) A(ni), max(n, n') n')": N N f-� t--7"(d-1) A(fit), B(n', B ( n ' , ni). fit). max(n, But But then then the the infinitary infinitary V-rule V-rule applies applies immediately immediately to to give give

d A(ni), F-7."d A(fit), VzB(z, V z B ( z , ni). fit). nn": NN f-� Suppose PA PA f-� ~d A(tJ A ( t l ,, .. ... ., , tk)' t~), 3zB(z, 3 z B ( z , tl t l ,'. .. .., . ' tk) t~) follows follows by by an an application application of of 44.. Suppose the the 3-rule 3-rule from from PA PA f-� F-d~ A(tJ A ( t l ,, ... .. ,. t,~tk)' ) , B(ta, S ( t o , tJ, t l , . .. .., .t ~, tk) ) . . Then Then the the induction induction

hypothesis hypothesis gives gives

.(d- l ) A(ni F-7"(d-1) A(fit), S ( m o a, , ni) fit) nn": NN f-� ), B(m where ma where m0 is is the the value value of of to to after after assigning assigning fi g to to the the variables variables x ~ and and 00 to to any any other other variables variables in in to to.. Now Now the the derivation derivation height height d d is is defined defined so so that that the the witnessing witnessing here B term term to to has has value value ma mo ::; <_ B Bw.(d-1)(n) B is is the the fast-growing fast-growing hierarchy, hierarchy, not not W . (d- I ) (n) ((here (d- l ) ma the the formula formula).) . Therefore Therefore by by the the Bounding Bounding Lemma Lemma 33.15, n : 9N N f-� t-o."(d-l) m0 : 9N N . 15, n and 3-rule in infinitary system immediately gives and so so the the 3-rule in the the infinitary system immediately gives

d A(ni) F-7."d A(r5),, 3zB(z, 3 z S ( z , ni). fit). nn": NN f-� 55.. Suppose Suppose PA PA f-� F-d A(tl A ( t l ,, .. ... ., , tk) tk),, B(ta, B(to, tl t l ,,. .. .., . , tk) tk) follows follows by by the the induction induction rule rule from from PA f-� F-d~ A(tJ A ( t l ,, .. ... ., , tk) tk),' B(O, S(0, tJ, t l , .. ... ., , tk) tk) PA

and and

PA PA f-� F-~l1 A(t A ( t ll ,, .. ... ., , tk), t~), -,B(x, - , B ( x , tJ tl,, .. .. .. ,, tk)' t~), B(x S(x + + 1, 1, tJ, tl, .. .. .. ,, tk) tk)

where A(tJ where II B BII ::; _ rr and and x x is is not not free free in in A ( t l ,, .. ... ., , tk) t~).. Let Let ma m0 be be the the value value of of ta(fi) t0(g).. l) . (d Then Then as as before before n n : 9N 51 f-� F-o"(d-l) ma m0 : 9N 51.. By By the the induction induction hypothesis, hypothesis,

do A(ni), F-~"d~ A(r~), B(O, B(0, ni) fit) nn": NN f-�·

180 180

M. M. Fairtlo'llgh Fairtlough and S. Wainer

n'

and and for for every every n' E N N,,

n')

max(n, "dll A(rii A(r~),-,B(n', rS), S(n'' + + 1, rS). max(n, n') IF-7 1, rii ). ) , B(n ) , -,B(n', rii -�.d

SSoo by ) , B(I, ), by mo m0 successive successive applications applications of of L-Cut L-Cut with with cut-formulas cut-formulas B(O, B(0, rii rS), B(1, rii rS), ...., . . , B(mo B ( m o -- 11,, rii rS)) we we obtain obtain

(d

(d-2)+mo A(rii max(n, mo)" ~-7"(d-2)+m~ A(rS),) , B(mo, rii rS). max(n, mo) : NN 1-�. ).

(d

Since [max(n, mo)] Since w w.· (d - 2) 2) + + mo mo E w w.· (d - 1) 1)[max(n, mo)] we we thus thus obtain obtain by by Weakening, Weakening, l A(rS),) , B(mo, max(n, mo) mo)": N N I~_~,.(d-1) B(mo, rii ff~). ). max(n, -�. (d- ) A(rii

- l) max(n, But n": N N I-�.(d t-o "(d-l) max(n, mo) mo) :" N N from from above, above, so so aa final final N-Cut N-Cut gives gives the the desired desired But result: result: . A(rii n" N N IF~�''d B(mo, rii ff~)) d A(rS),) , B(mo, and this this completes completes the the proof. proof. and

n

n:

all numerical numerical Suppose (X) , . . . , tk (X)). Then Suppose PA PA I-� ~a A(tl A(tl(~),...,t,(~)). Then for for all assignments limination assignments ii ~ to to the the free free variables variables x ~ we we have, have, by by the the Embedding Embedding and and CutCut-Elimination E Theorems 11, Theorems 4~.. 7, 7, 3. 3.11, -g A(mb max max ii g" N N IF~ A ( m l , .. ... ., , m ink) k) where ti(ii) and where as as before before mi m, is is the the value value of of term term t,(g) and where where a a = = eXP2(w exp~(w,. d) d) or or exp�(w exp~(w . 9d) d)..

4.8. 4.8. Corollary. Corollary.

:

4.9. 4.9. Theorem. Theorem. PRovREc(PA) PROVREC(PA) C C_ REC(co) REC(60) PROVREC(E~ C_ REC(W REC(w22)) P ROVREC(��-IND) C C PRovREc(��-IND) PROVREC(Z~ C_ REC((cO)n) REC((eo),)

Proof. Proof.

f \fx 3y3z . C,(x, y, z).

ifn if n >;:::22

3y3z . C,(x, y , z) \fx 3y3z.C,(x, y , z).

��-defined by 11.. Immediate Immediate from from 44.8, for if if f is is E~ by 3y3z. Cf(~, y,z) and and PA PA I . 8, for then for v~ 3v3z. c~(~, v, z) then ~or some ~orne a ~ -< -~ co ~0 we we have h~ve I-g ~-~ V~ 3y~z. C~(~, V, z). 2. Suppose I2. Suppose ��-IND E~ }-- ~'~ 3y3z. Cf(~, y, z). The The cut-reduction cut-reduction method method of of 33.10 . 10 applies applies to to ��-IND E0-IND in in much much the the same same way way as as for for the the infinitary infinitary system, system, so so that that cuts cuts of of (disjunctive (disjunctive or) or) existential existential form: form:

\fx 3y3z.C,(x, y , z)

A, A, B(t) B(t) A, A, \fx-,B(x) Vx-,B(x) A, A, 3xB(x) 3xB(x) A A get get reduced reduced (by (by inverting inverting the the left left premise) premise) to to

A, A, -,B(t) ~ B ( t ) A ,A, B(t) B(t) A A

Provably Provably Recursive Recursive FUnctions Functions

181 181

However � -induction instead However if ifthe the right right premise premise A, A, 33xx B B ((xx)) comes comes about about by by aa � T~ instead of -rule, the of an an 33-rule, the method method comes comes unstuck unstuck and and we we can can reduce reduce the the cuts cuts no no further. further. � -IND derivation Nevertheless, Nevertheless, this this means means that that any any � ~E~ derivation can can be be transformed transformed into into one one in in which which all all cut-formulas cut-formulas are are (at (at worst) worst) � T�~. Then Then applying applying the the Embedding Embedding Theorem Theorem 4.7 4.7 to to the the given given proof proof of of VX VZ 3y3z 3 y 3 z . . C,(x, CI(Z , y, y, z) z) and and inverting inverting the the .d 3y3z yields an an infinitary infinitary derivation derivation max max n 4": N N f-w F-~'d 3 y 3 z .. C,(n, C l ( g , y, y, z) z) in in which which all all VV yields cut-formulas cut-formulas are are � S �~ and and in in which which there there is is aa finite finite bound bound kk on on any any additional additional nu numerical merical constants constants appearing appearing in in them. them. Note Note that that embedding embedding replaces replaces induction induction by by cuts. cuts. The The Bounding Bounding Lemma Lemma 3.15 3.15 (2) (2) then then applies applies to to give give 3y3z 3 y 3 z .. C,(n, Cf(~, y, y, z) z) (max(n, k)) true true at at BW' B~.d(max(~, k)),, for for all all n. 4. As As before before f f is is therefore therefore elementary elementary in in B B~.d dHierarchy Theorem w .d and by by the the Hierarchy Theorem 3.19 3.19 and and 3.20, 3.20, ff E E REC(W2) REC(w2).. and d VX Suppose � ~~� -IND f~_d V~ 3y3z 3 y 3 z . . C,(X, C l ( ~ , y, y, z). z). As As above above we we may may assume assume that that the the 3.3. Suppose proof proof is is reduced reduced to to one one in in which which all all cut-formulas cut-formulas are are in in prenex prenex form form with with at at most most n n alternating alternating quantifiers. quantifiers. Recall Recall that that the the sequence-coding sequence-coding apparatus apparatus allows allows us us to to assume assume further further that that blocks blocks of of like like quantifiers quantifiers are are contracted contracted to to one one such. such. Thus Embedding Theorem Thus by by the the Embedding Theorem and and Inversion Inversion we we have, have, for for all all assignments assignments X := " - - m, T/~, x

.d 3y3z max rh :" N N f-w N~.d 3y3z. . C,(m, Cl(ff~ , y, y, z) z) maxm

by by aa derivation derivation in in which which all all cut-formulas cut-formulas have have at at most most n n -- 11 alternating alternating quantifi ers prefixed quantifiers prefixed to to a a� T~� -formula or or its its negation, negation, and and in in which which there there is is aa finite finite bound bound k k on on the the numerical numerical constants constants originating originating from from the the given given (� (~~� -IND)-proof. The Cut-Elimination the quantifier-complexity The Cut-Elimination method method then then allows allows us us to to reduce reduce the quantifier-complexity of time, to of cut-formulas cut-formulas one one step step at at a a time, to obtain obtain aa derivation derivation of of

" 3y3z m:N y, z) z) max max r~" N fNa 3y3z. . C,(m, Cl(r~ , y, in which only only � By the the Bounding Bounding Lemma Lemma we we then then have have for for every in which T~� -cuts remain. remain. By every m, m, 3y3z . Cl(r~ C,(m,, y, y, z) z) true true at (max(m, k)) k)) 3y3z. at B,, B~(max(rh,

and therefore in Ba, B" , as as before. before. The The desired desired result result then then follows and therefore ff is is elementary elementary in follows immediately provided we immediately provided we can can ensure ensure that that c~ a -~ -< (~0)n. ( CO ) n . Unfortunately this this is is not not straightforward. straightforward. If If we we were were to to apply apply step-by-step step-by-step Unfortunately cut reduction reduction in in the the most most obvious obvious way, way, the the final final bound bound c~ a would would be be too too large. large. cut Instead we we need need to to insert insert aa further further "refining" step, which which is is step step 22 below. below. Instead "refining" step, We We begin begin with with aa derivation derivation of of y, z) z) max 3y3z . Cl(ffa C, (m,, y, N ~_w.d f-w.d 3y3z. max rh" m:N

in which which the the cut-formulas cut-formulas have have at at most most nn- - 11 alternating alternating quantifiers quantifiers prefixed prefixed in to or its its negation. negation. to aa S~ �� -formula or Step 1" 1: Remove the outermost outermost quantifier quantifier from from all all cut-formulas cut-formulas by by cut cut elimination. elimination. Step Remove the This w .d d to to 22w'd. This increases increases the the bound bound from from w. w.d • Step d+l + 2. Step 2: 2: Transform Transform the the resulting resulting derivation derivation into into one one with with bound bound wwd+1 + wwdd •933 §+ 2. We show show how how this this isis done done afterwards. afterwards. We

182 182

M. Fairtlough Fairtlough and and S. S. Whiner Wainer M.

33 to to n: n: Successively Successively remove remove outermost outermost quantifiers quantifiers from from all all cut-formulas, cut-formulas, each each time time as in in Note Note 3.12. 3 . 12 . increasing the the bound bound by by an an exponential exponential to to the the baSe base ww as increasing What remains isis aa derivation derivation of of W h a t remains

:3y:3z.Cf(m, y, z)

m a x ffz m ": NN ~-~ f-0 3 y 3 z . C l ( r h , y, z) max

with only only ~~ E� -cuts, and and where where with c~ = ex-~-2 p~ (w d+l + w d 93 + 2) -~ exp~ (1) = (~0)n

as required. required. as It remains remains to to justify justify step step 2. 2. Given Given any tree-ordinal ~/of , of the the form form It any tree-ordinal ~/ : 2 w'dx " e l + "'" + 2 w'dk " Ck

and c~ Ci > > 0, 0, define define "7' by where where d dl1 > > ". ." . > > dk dk and by ...+w dk .9(3Ok). ,-)1 w ddl1 . . (3cd (3C1) + -~-"""-~Wdk (3Ck). "! --=W

Then by induction on 2w ·w we we have have Then by induction on 7, ~� 2~'~ implies (3 ~'' + + 22 E e , 7'[e(m)] (3 Ee 7[m] ,[m] implies ' [e(m)]

(t) (t)

where where e(m} e(m) = - 33 ..22m+! m+l - l1.. w .d • c. The holds vacuously. c. If The case caSe , 7 = - 00 holds vacuously. Otherwise Otherwise we we can can write write , 7 = - 0~ + + 22~'d" If , 7 is is )}. aa successor [m] U successor then then d d= = 00 and and (3 ~ E e ,[m] 7Ira] implies implies (3 ~ E e 0~ + + (c ( c -- l} 1)[m] U {o (~ + + (c (c- I1)}. By By the the induction induction hypothesis hypothesis we we then then have have ~'' + + 22 E C 0~'' + + 3(c 3 ( c -- l1)[e(m)] U {o' {~' + + 3c 3 c -- I} 1} C C, 7'[e(,~)]. ) [e(m}] U (3 ' [e(m}].

Finally (3 Ee '%~[m]. Finally if if , ")' is is aa limit limit then then d d> > 00 and and/~(3 E E ,[m] "),Ira] implies implies/~ So by by the the m [m]w. .(So +! induction induction hypothesis hypothesis (3 fl'+2 7~[e(m)].. But But 'm ~/,~ = = 0+2 5 + 2 ~w'.dd .. ((c c -_- 11}+2 ) + 2 ~ ' ( dd- 1l))' .22m m+l ' +2 Ee ,:"[e(m)] and and so so 'Tim = = 0(~!' + -~- w ~ddd"• (3c (3C -- 3) 3) + -~- W 5ddd-- l1" (e(m) (e(m) + -~- I}. 1). ,:" Therefore Therefore •

[e(m)] C + 22 E e ,:"[e(m)] 7~[e(m)] = = 05'' + +w w dd" (3c ( 3 c- 2) 2)[e(m)] C, 7'[e(m)]. (3~'' + ' [e(m)]. •

This This completes completes the the proof proof of of (t) (t).. We We now now use use this this to to prove prove n :

n 9N N f-'Y t-~ A A implies implies n n : 9N N f-'Y" b~" A A

+1 + . d1 ."el ..+ . dk ."Ck where if if ,~ / = = 22w~'dl "~"."'" "~-22ww'dk aS above, above, then then ,7"" = = wd1 (-ddl-F1 "~- " ~l + + 22.. Cl + where Ck as Furthermore Furthermore the the transformation transformation from from bound bound ,~/ to to bound bound ,7"" does does not not alter alter any of of the the cut-formulas. cut-formulaS. The The proof proof is is by by induction induction over over derivations. derivations. The The any axiom axiom case caSe is is immediate immediate and and all all other other cases caSes have have the the same same form. form. For For suppose suppose

183 183

Provably Recursive Functions n

-11 A, is derived premises m n :: N N IF-"I~ A A is derived from from premises m :: N N 1kZ A , BB by by any any rule, rule, where where ~/[m].. By By the the induction hypothesis we we have have m m :: N N 1k--~'' B and and by by (3 EE ,[m] induction hypothesis 11" A, A, B 2 A,, BB since Weakening, ee(m) N 1~-"1" " --2 A since by by (t) (t),, (3' f l ' ++ 22 E e ,'[e(m)] ~'[e(m)] and and hence hence Weakening, (m) :: N +1 on (3" fl" E9 ," 9 ' "- 2[e(m)] 2[e(m)].. The The only only reason reason for for the the extra extra wd1 w d1+1 on the the front front of of , 9'"" is is -2 ee(m)" (m) . to _2 (m) � (m) � Hence : N 1 "1" : N by (m) to ensure ensure that that B B~,,_2(m) >_ B B~+,(m) >_ e(m). Hence m " N k~"-2 N by m e W+1 "I" " 1 Bounding A, B Bounding Lemma Lemma 33.15 (1). Therefore Therefore by by an an N-cut, N-cut, m m : 9N N IF-"I~''-1 A, B.. We We can can .15 (1). "I" A, A, B now now re-apply re-apply the the rule rule in in question question to to obtain obtain n n :: N N Ik--~'' B as as desired. desired. +l + Step 22 now now follows follows by by putting putting , -y = 22ww'd SO that that ,,/" ~" = - Wd w d+l + wd w d.. 33 + + 2. This Step .d so 2. This completes completes 4.9. 4.9. -

-

4.10. 4.10. Theorem. Theorem. REC(w 22)) � C_ PRovREC(��-IND) PROVREC(E~ REC(W Proof. By By 3.20 3.20 B B~.k is primitive primitive recursive, recursive, for for each each kk 6E N N and and so so by by the the Hierarchy Hierarchy Proof. w .k is Theorem, Theorem, every every function function in in REC(W REC(w 22)) is is primitive primitive recursive. recursive. We We therefore therefore only only need show show that that every every primitive primitive recursive recursive function function is is provably provably recursive recursive in in ��-IND. 9 need This This is is done done by by assigning assigning to to each each primitive primitive recursive recursive definition definition of of aa function function f, f, aa i, y, with the intuitive meaning: "z is a sequence code which bounded formula bounded formula C, CI(2, y, z) with the intuitive meaning: "z is a sequence code which z) ( describes describes the the step-by-step step-by-step computation computation of of f(i f (~), ending with with output output y" y".. The The formula formula ) , ending 3y 3 z . C, 3y3z. CI(s(i, y, y, zz)) then then � E~� -defines f(i f(E)) and and we we merely merely have have to to show show it it to to be be provable provable in in ��-IND. ~~ If f f is is defined defined by by one one of of the the initial initial schemes: schemes: 1. If

ff ((i M)) -=- 00 or or f(i f ( ~ )) -= x , Xl + l + 1 or or f f ((~i')) = - x Xi i

to be then y, zz)) to then take take C, Cf(s( i, y, be the the conjunction conjunction of of zz = - 00 (the (the empty empty sequence) sequence) with with y -= - 00 or or y y -= X Xl l + l+ 1 or or yy -=x iXi respectively. respectively. Then Then in in each each case case we we have have C, 0) or or c, Cf(E, O, O) Cf(~,x, + 1, l, 0) O) or or C, Cf(~,xi, O) (i, Xi, 0) (i, Xl + ( i, 0,

provable provable immediately immediately by by identity identity axioms, axioms, hence hence IF- 3y3 3y3z. CI(E y, zz)) by by the the z . C, (i,, y, 3-rule (i,, y, 3-rule and and hence hence IFV Vsi 3y3z 3y3z. . C, CI(Y. y, zz)) by by the the V-rule. V-rule. 2. 2. Suppose Suppose f f is is defined defined from from go go,, gl g, and and g g22 by by the the substitution substitution scheme: scheme: i) , gg2(x)) ff(x) (i) = 2 (i)) - g g0(g, O ( gl ((x),

and and assume assume inductively inductively that that g go, g, and and g g22 have have already already been been assigned assigned "compu "compuo , gl tation tation formulas" formulas" C Co, C1 and and C C~2 so so that that in in ��-IND X]~ o , Cl

I-F VVZi 3y3z 3y3z. . Ci Ci(Y., y, zz).) . (i, y, Then to be Then take take C, Cf(s( i, y, y, zz)) to be the the formula formula lh(z)) = = 33 /\ A (z)o (Z)o =f:. ~ 00 /\ A (zh (z), =f:. r 00 /\ A (z (z)~ r 00 /\ A Yy = = u((z)o) u((z)o) lh(z h =f:. ^/\ Cl c1((i,, u((zh ) , v ((zh)) /\ ^ C2 (i, u((zh) , v ((zh)) ( (z)o)) . ^C Co(u((z)l), u((z)o),, vv((z)0)). o (u((zh ) , u((z h) , u((z)o) /\

Wainer Fairtlotlgh and M. Fairtlough M. and S. S. Whiner

184 184

derive can derive that we to see easy to it's easy PA it's in PA arithmetical axioms the arithmetical Now from Now from the axioms in see that we can

",C1 (x, yl, Zl ) , ..,C2(x, Y2, Z2) , ",Co(yl, Yo, zo) , p(yl, z~Zl ),) , p(y~, zo) , p(y~, (P(Yo , zo), Yo , (p(yo, C,(x, yo, c~(~, P (Y2, z~) Z2)))>)..

I- ~c, (z, y~, z,), ~c~(z, y~, z~), ~Co(y~, yo, zo),

Then by quantifier rules in the the correct correct order order we we obtain obtain rules in the quantifier applying the by applying Then I-

3xvyVz . ..,C1(X, y , z) , 3xvyVz . ..,C2(X, y , z) , 3xvyVz . ..,Co(x, y , z) , VX 3y3z .C,(X, y, z) ,

3~yv~. ~c~ (~, y, z), 3~yv~. ~c~(~, y, ~), ~ y v ~ . - C o ( ~ , y, z), v~ 3y3~. c~(~, y, z),

cuts, successive cuts, three successive by three this by and and from from this

VX 3y3z .C,(X, y , z). 91 by and gl 90 and defined from is defined Suppose ff is 3. Suppose 3. from go by primitive primitive recursion: recursion: 1) = )). f(x f(~,, 0) 0) - 90(x) go(~) ,, f(x f(~,, w w+ + 1) -- 9gl(x, f(~,, w w)). 1 (X, w,w, f(x been assigned already been 91 have Assume Assume 90 go and and gl have already assigned formulas formulas Co Co,, C1 C1 such such that that in in I- V~ 3y3z. CI(~. , y, z). ~-

=

have we have �� ~]~ IND we -

VX 3y3z. Co (x, y , z) VX VwVw'3y3z. C1(X, w, w' , y, z).

IF- V~. 3y3z. Co(F., y, z) ~- V~ VwVw'gygz. C1 (~,, w, w', y, z). I-

C,(x, w, y , z) 0) /\ lh(z) = w w+ + 11 A ((z),; =j:. # O) AY y = = u((z)w) u((z)w) Vi << ww ++ 11.. ((z) /\ Vi lh(z) (x, u(( ^ Co Co(~, u((~)o), ~((z)o)) z)o) , v((z)o)) /\ ^ Vi vi < <w ~.c~(~,i,u((z),),u((z),+~),~((z),+~)). .C1 (x, i , u((Z) i ) , U((Z) i+l ) , V ((Z) i+l )). /\

Then Then take take CI(~, w, y, z) to to be be the the formula formula

Using Using the the arithmetical arithmetical axioms axioms we we obtain obtain

..,Co(x, y, z) , C,(x, u( (t)o) , t) by by choosing choosing the the term term tt == ((p(y, z)),, and and P(y, z)) I- ..,C,(X, ~c~(~, ~, c~(~, w w+ + 11,, y' y',, t') e) C1(x, ~,w, yy,, y'y',, z~'),') , C,(x, w, yy,, z)z),, ..,~c~(~, by by choosing choosing t' t ' - p (p(z z , p,(p(y' y ' , z,' z')). ) ) . Thus Thus by by appropriate appropriate quantifier quantifier rules, rules, I- 3xvyVz . ..,Co(x, y , z) , 3y3z . C,(X, 0, y , z) I- ~Co (~, y, z), C~(~, 0, 0, u((t)o), t)

=

and and

VyVz . ..,C,(x, w, y, z) , 3~3w3w'VyVz. ~C1 (x (~,, w, w w',' , yy,, z) z),, 'VyVz. ",C1 3X3w3w 3y3z. CI(~, , w w+ + 11,, yy,, z) z) 3y3z.C,(x,

I-~- VyVz. ~CI(~. , w, y, z),

W,

185

Provably Recursive Functions and hence hence by by applying applying cut cut to to each each one: one: and

:3y:3z . Cf(.x, y, z)

O, y, z) and and I-~- 3 y 3 z . C f ( ~ , 0,

I-

VyVz . -,Cf(x -~CI(~ ,, w, w, y, y, z), z), :3y:3z. 3y3z .Cf(x, CI(~. , w w + 1, 1, y, y, z). z). VyVz.

Then by by ��-IND E0-IND we we obtain obtain Then

:3y:3z.Cf(x, w , y, z)

I-~- 3y3z. C I (~,, w, y, z) and by by V-rules V-rules and

Vw:3y:3z. Cf(x, w , y, z).

Y2 Yw3y3z. CI(~, , w, y, z). I-~- VX

This completes completes the the proof proof since since every every primitive primitive recursive recursive function function is is definable definable This by a a sequence sequence of of the the above above schemes. schemes. by 4.11. Theorem. Theorem. 4.11. REC((S0)n) C_ PrtOvREc(H~

> 2) 2) ((nn �

P r o o f . By By the the Hierarchy Hierarchy Theorem Theorem and and the the Comparisons Comparisons Lemma Lemma it it suffices suffices to to show show Proof.

to )n, the that for for every every a ce -< -~ ((So)n, the fast-growing fast-growing F F~a is is provably provably recursive recursive in in II�-IND II~ For that . For since B if f f E E(Ba) E(B~) then then f f E E(Fa) E(F~) since B~a � _< F F~a on on all all non-zero Thus f f is is non-zero arguments. arguments. Thus if primitive recursively recursively definable definable from from Fa F~ and and so so by by the the proof proof of of 4.10, f f will also be be will also primitive provably recursive recursive by by some some further further � E~� -inductions. provably We need need to to construct construct an an appropriate appropriate "computation "computation formula" formula" CF for for the the F FWe functions. To To do do this this we we must must first first define define a a coding coding of of tree-ordinals tree-ordinals or, fl, I, 7 , .. . - , , -< -~ tSoo functions. a, /3, as numbers numbers a a = = rraT, = rr/~7, so that that the the basic basic ordinal ordinal as a '1, bb = /3'1, Cc == rr77 1 '1,, ....respectively, . . respectively, so operations can can be be simulated simulated by by the the arithmetical arithmetical term-structure term-structure of of PA. PA. This This is is easy easy operations a -< -< (So)n can be be represented represented in Cantor normal normal form form thus thus to do because each to do because each ce in Cantor ( to)n can

CF

0[, =

" ml

W OtO " ~'}~0 3t- ~JOtl

Jr" 9 9 9 -4-(.l) O t k - 1

..

" mk-1

l

exp�- (l) and -< exp$-l(1) are positive positive integers. integers. and m ak - l1 -~ -< ..... . -< al "{ -< O/0 ao "~ where O/k_ where -~ al m oo ,, . .. .. ,. , m m k -k 1 - l are So is coded coded as the number number ai ai we we can can take take as for a: So if if ai cei is as the as code code for c~: aa -= (p o), p ( at , m ( ao, m k- l ) ) (p(ao, mo),p(al, ink-l)) mll )) ," . .". ,,P(ak-1, p ( ak - l , m

the The stands for The ordinal ordinal 00 will will be be coded coded as as the the empty empty sequence sequence 0. O. Succ(a) Succ(a) then then stands for the /\ v(v(a)) v (v (a)) #=I 0) 0) and and Lim(a) Lim (a) stands stands for for the formula (u( formula the formula (u(v(a)) formula (u(v(a)) v ( a)) =I =0A (u(v ( a)) = v ( v ( a)) # =I o). 0) . 0o A /\ v(v(a))

We can then then define define elementary elementary number number theoretic theoretic functions functions We can

1. (a,b) t--+ a@b a EB b ( a, b) ~-~ 2. wbb.• (n (n + + l) 1) 2. (b,n) ( b, n) ~t--+ w 3. x) ~t--+ l(a, x) 3. (a, l(a, x) ( a, x) such such that, that, ifif aa encodes encodes aa and and bb encodes encodes fl/3 then then EB bb encodes encodes aa ++ fl/3 (provided (provided the the lowest lowest exponent exponent of of w w in in aa is is no no smaller smaller than than 1. 1. aa @ the w in in fl) /3) the highest highest exponent exponent of of w 2. 2. w wbb.• (n ( n ++ 1) encodes encodes w wfJz. ((nn ++ 1)

.

M. M. Fairtlough Fairtlough and S. Wainer Whiner

186 186

3. ll(a, (a, x) 3. x) encodes encodes a a ,x if if a a is is a a limit. limit. Since these functions are primitive Since these functions are primitive recursive, recursive, they they are are provably provably recursive recursive in in ��-IND E~ by by the the foregoing foregoing theorem theorem and and we we may may therefore therefore use use them t h e m freely freely to to form form new new terms, terms, in in such such a a way way as as to to render render the the equation equation (a~gw b . ( u + l))~gw b =

a~w b.(u+2)

provable We write wa provable in in ��-IND. E~ We sometimes sometimes write w a for for wa w a. l1.. Now , z) for Now we we can can easily easily define define a a computation c o m p u t a t i o n formula formula C CH(a,x, for the the Hardy Hardy H (a, x , yy,z) functions functions Ho.(x) Ha(x) = - yy,, by by formalizing formalizing the the statement: statement: 0, Xo is a a sequence sequence (p(ao, ~(ao, xo), xo),...,p(ak-l,xk-1)) where ao ao = = 0, Xo = = y y,, zz is . . . , p(ak-b xk-l)) where a~-i = = a, a, Xk-l xk-1 = = x x and and for for each each ii < < k k- 11;; if if Succ(ai+l) Succ(ai+l) then then ai ai EEl @ rr l1~'" = = ak-l Xi+! + ai+l ai+x and and Xi xi = = xi+~ + 11,, and and if if Lim(ai+l) Lim(ai+~) then then ai ai = = l(ai+b l(ai+~,xi+~) and Xi+!) and •

Xi = m Xi+l X i + l ·" Xi

But B u t since since Fo. Fa = - H H ~wo< we we can can therefore therefore take take as as the the ccomputation o m p u t a t i o n formula formula for for F: F" CF(a, x, y, z) -- CH(wa, x, y, z).

for The The following following are are derivable derivable in in ��-IND E~ for all all terms terms a a and and b: b: 3y3z.CH(rl7, x,y,z) since r F c CH(rl7, x,x+l,p(p(O,p(O,x+l)),p(rl7, x))).) . x + 1 , p(p(0, p(0, X+ l)), p(rl', x)) y, z) since 1. rF 3y3z . CH (l"', x, H (rl', x, b , x, y, z) . ( (b, x, y, z), 3y3z . C a EEl (a, x, y, z) , -,3y3z . C F -,Vx3y3z -,Yx3y3z. . C CH(a, x, y, z), -,3y3z. CH(b, x, y, z), 3y3z . CH(a @ b, x, y, z). 22.. r H H H For y), . . . , p(b, x)) codes For if if z' z' = - (P(O, (p(O,y),...,p(b,x)) codes the the computation c o m p u t a t i o n of of H He(x = yy p (x)) = and (P(O, y') , . . . , p(a, y)) codes (Y ) , then and z" z" = - (p(O,y'),...,p(a,y)) codes the the computation c o m p u t a t i o n of of Ho. Ha(y), then zz = (p(O, (P(O, y'), . . . , p(a, y), y'),..., y),..., @ b, b, x)) x)) codes codes the the computation c o m p u t a t i o n of of Ho. Ha+~(x) = . . . , p(a EEl +p (x) = Ho.(H and so H~,(H~(x)) - y' y'.. Clearly Clearly z z is is primitive primitive recursively recursively ((and so provably provably recursively recursively p (x)) = in definable we in ��-IND) E~ definable from from z' z' and and z" z".. Therefore Therefore in in ��-IND E~ we have have

" ) , -,C r --~CH(a, y, y', y, z') b, ~, x, yy',' , z) y', z~"), -~C~(b, ~, y, ~'),, 3z 3z. . C C,,(a ~) ,CH (a, y, H (a EEl9 b, H (b, x, and and (2) (2) follows follows by by the the quantifier quantifier rules. rules. (a, x, x, y, y, z) x, y, y, z) 3. ~ L i m ( a ) ,, --,3y3z ,3y3z. . C CH(ax, z),, 3y3z 3y3z.. CH CH(a, z) 3. rF -,Lim(a) H (ax , x, x, y, x, y, y, z) since r F -,Lim(a), ~ L i m ( a ) , --,CH(ax , C H ( a x , , x, z),, C Cg(a, y, p(z, p(z, p(a, p(a, x)) x))).) . since H (a, x, 44.. r z) by 1 ) since ~- 3y3z 3y3z.. CF(O, CF(O, x, x, y, y, z) by ((1) since WO w~ = = 11.. , y , z) . ~Vx3y3z.CF(a,x,y,z),3y3z.CF(a ~ rl', rl7, Xx,y,z). . CF(a, x , y, z), 3y3z . CF (a EEl 55.. rF -,Vx3y3z First (u + First substitute substitute wa w ~ for for a a and and wa w ~.• (u + 11)) for for bb throughout t h r o u g h o u t the the derivation derivation of of (2) (2) to to obtain obtain F -,Vx3y3z ~Vx3y3z. . C CH (wa, a, x, x, y, y, z) z),, r H (w • (wa (u + -,3y3z -~3y3z. . C c ,H( ~ ~ 9(u + l), 1), x, ~, y, y, z) ~),, 3 y 3 z . C ,H( ~(wa~ • 9((uu + + 2) 2),, x, ~, yy,, z) z) 3y3z.C

using using r ~- wa w a EEl @ wa w a.• (u (u + + 11)) = = wa w ~.• (u (u + + 2) 2).. Since Since the the base base case case ~w3y3z. c~(~ o, ~, y, z), ~y3z. cH(~ o, ~, y, z)

187 187

Provably Recursive Functions follows by by logic, logic, we we can can now now apply apply � E~ over uu to to obtain obtain follows � -induction over

y, z) z),, :3y:3z 3y3z ..C CH(W (x + 1) 1),, x, x, y, y, z) z).. CH (wa , y, H (wa~ .. (x Hence by by (3) (3) with with aa replaced replaced by by w wa~a1n"", , Hence ~- -,'v'x:3y:3z -~Vx3y3z .. CH(W~, x, x, I--

-w

y3z.

o,

y, z), 3y3z.

y, z)

which is is what what was was required required since since CF(a, CF(a, xx,, yy,, z) is is C CH(W z).. which H (waa,, xx,, yy,, z)

6. I-[- -'Lim(a) --Lim(a),, -,:3y:3z ~3y3z. . CF(a CF(a~, x, y, y, z) z),, :3y:3z 3y3z.. CF(a, Cf(a, x, x, y, y, zz)) 6. x , x, again by by (3) (3) with with aa replaced replaced by by w waa.. again Now in in order order to to prove prove Fa F~ totally-defined totally-defined we we clearly clearly need need the the principle principle of of transfinite transfinite Now induction up to to a. c~. This This can can be be formalized-and formalized--and appropriate appropriate instances instances of of it it proved proved-induction up in PA, PA, using neat method method of of Gentzen [1943]. See See also also Schiitte Schiitte [1977], [1977], Feferman Feferman in using aa neat Gentzen [1943]. [1968] and and Takeuti Takeuti [1987]. [1987]. Let Let [1968]

TI(a,A) =_ Prog(A) Frog(A) � --+ 'v'c Vc.. (c (c :::5 -< aa � -+ A(c)) A(c)) TI(a, A) == where Prog(A) Frog(A) meaning meaning A A is is "progressive" "progressive",, is is the the formula formula where Vd(A(0) --+ A(d A(d EB @ rl')) c17)) 1\ A (Lim(d) (Lim(d) 1\ A 'v'x Vx.A(d,) -+ A(d))) A(d))) . A(dx ) � 'v' d(A(O) 1\A (A(d) � and where where (c (c -< -~ a) a) is is a a E~ defining the the partial partial ordering ordering 'Y 7 -< "~ a a on on tree treeand �� -formula defining p ( co, zo ) , . . . ,,p(ck, P ( Ck, Zk )) with with Co C, Ck ordinals, ordinals, i.e, i.e, "there "there is is a a sequence sequence zz = = ((p(co, zo),... Zk)) co = = c, ck = = a, a, and such such that that for for each each ii < < k, k, if if SUCC Succ(ci+l) then Ccii EB @ rl' FlU = = C;+l C~+l,, and and if if L Lim(c~+l) and ( CHl ) then im (cH l ) then ( C;+l , ZHl )" · then Ccii = ll(Ci+l, Zi+l)". For formula A(c) A(c) with with aa distinguished define A'(b) be the For any any formula distinguished free free variable variable Cc define A' (b) to to be the n+ l) _-formula 'v'a(A(a) Va(A(a) ~� A(a A is Note that that ifif A is II~ rr� then then and let let A A((n+l) formula A(a @ ffi wb)), wb )), and == (A(n)) (A(n)),.'. Note o A' is when brought brought to to prenex prenex form. form. is Hm+ rr� A' +l1 when We have have the the following three lemmas: We following three lemmas: 7. 7. If If A A isis aa H~ rr� -formula then then

rr�-IND I-- -,Frog(A), -,Prog(A) , Prog(A') II~ FFrog(X) We argue informally. First First assume The first first conjunct Prog(A') is is We argue informally. assume Frog(A). Prog(A) . The conjunct of of Prog(A') A'(0) follows straight A(a @ r 17).. The The second conjunct is is which follows straight from from A(a) A(a) --+ � A(a EB rl') second conjunct A'(O) which l ") A'(d) ~ A'(d@rl7). To prove and A(a). Then A(a@w dCrl~) A'(dEBrl') . To prove this this assume assume A'(d) A'(d) and A(a) . Then A(aEBwdEV A'(d) � comes by aa H~ comes about about by rr� -induction on on the the formula formula A(a A(a ~EB wwdd.• (u (u + + 1)). 1 ) ) . The The base base comes from A'(d) and and A(a); A(a) ; the the induction induction step step comes comes from from case d) comes from A'(d) case A(a A(a @ EB wwd) A'(d) A'(d) with with aa instantiated instantiated by by aa @ EB wwdd.. (u (u + + 1). 1 ) . Hence Hence VxA(a 'v'xA(a @ EB wwdd.. (x + 1)) 1)) (x + and derl~) follows and then then A(a A(a @ EB wwdan") follows from from Frog(A). Prog(A) . Therefore Therefore we we have have proved proved -,A'(d) ~A' (d),, A(a) d~l~) and � A(a@w A(aEBwdan") and hence hence by by V-introduction 'v'-introduction and and V-introduction, V-introduction, A(a) -+ � � A'(d@rlT). A'(dEBrl') . The The third third conjunct of Prog(A') Prog(A') isis (Lim(d)AVx.A'(dx)) (Lim(d)I\'v'x . A'(dx )) --+ A'(d) conjunct of A'(d) --+ A'(d). then for for any any a, a, A(a) A(a) ~� Vx. 'v'x . A(a A(a @ EB wdx), wd. ) , A'(d) . But But ifif Lim(d) Lim(d) and and Vx. 'v'x . A'(dx) A'(dx ) then so � so by by Frog(A) Prog(A) yet yet again, again, A(a) A(a) -+ � A(a A(a ~EB wd). wd) . Hence Hence (Lim(d) (Lim(d) A 'v'x . A'(d~)) A'(dx )) --+ 1\ Vx. A'(d). Putting these these cases cases together together gives gives Prog(A'). Prog(A') . A'(d) . Putting

M. M. Fairtlough and S. Wainer Whiner

188 188

8. If If A A is is aa II� H~-formula then then 8. IIO-IND fk- -,T "~TI(b, A'),, TI(w TI(w bb,, A). A). I(b, A') II�-IND

For, I(b, A') For, assuming assuming T TI(b, A') and and Prog(A) Prog(A), , we we have have by by (7) (7) Prog(A') Prog(A') and and therefore therefore b . A(d) Vc . Va . (A(a) -+ EEl we) ) . Using Using this this we Vc � ~_ bb.Va.(A(a) --+ A(a A(a~wC)). we can can then then prove prove Vd Vd � ~_ w wb.A(d) b as i, in(d, as follows. follows. Suppose Suppose d d � ~ w wb.. Recall Recall that that for for fixed fixed dd and and i, in(d, i) i) is is the the initial d, with initial part part of of sequence sequence d, with length length i, i, so so if if ii � < lh(d) lh(d) then then in(d, in(d,i)i) = o 9mo EEl9 "" EEl9 wdi - 1 9m w d~ w d~-I mi-1 where do,..., di-1 ~ b. However, in the formal wd i l where � b. However, in the formal d o , . . . , d 1 i theory d)) -+ n(d, ii + 1) = n(d, i)i) EEl theory we we can can actually actually prove prove (i (i � <_ lh( lh(d)) --+ iin(d, + 1) - iin(d, @ wdi w d' . m mi.i . We on We are are going going to to prove prove A(in(d, A(in(d, i)) i)) by by aa II�-induction ri~ on ii.. The The base base case case is is just just A(O) A(0) which which follows follows immediately immediately from from Prog(A) Prog(A). . The The induction induction step step is is A(in(d, i)) -+ ~ A(in(d, + 1)) 1)) which which we we prove prove by by another another II� ri~ -induction. For For A( in( d, i)) A( in(d, ii + A( in(d, i)) , ii � d) and b. For informally, informally, assume assume A(in(d,i)), <_ lh( lh(d) and d; d, � ~ b. For any any aa we we have have A(a) -+ --+ A(a A(a EEl @ wdi wd').) . Therefore Therefore putting putting aa equal to in(d, in(d, i) gives A(a) equal to i) gives •

. . •

•

•

A(in(d, i) ~ w d') and putting putting aa equal equal to to in(d, in(d, i) ~ wdi w d' . (m (m + + 1) 1) gives gives and i) EEl •

A(in(d, i) i ) ~EEl w w di d' . (m (m + + 1)) 1)) -+ --+ A(in(d, A(in(d, i)~9 w di d' . (m (m + + 2)) 2)).. A(in(d, i) EEl w •

•

Hence by Hence by II� ri~ -induction on on m m we we obtain obtain A(in(d, w d'. m;) mi).. Consequently Consequently A( in(d, i)i)@EEl wdi A( in(d, ii + A( in(d, i)) A(in(d, + 1)) 1)) and and this this completes completes the the inductive inductive proof proof of of A(in(d, i)).. If If we we now now A') put put ii = - l hlh(d) ( d ) and and use use d d -= in(d, in(d, lh(d)) lh(d)) we we finally finally obtain obtain A(d) A(d).. Hence Hence TI( TI(b,b, A') implies implies TI(w TI(w bb,, A) A).. •

99.. If E N, If A A is is aa IIg rl~-formula then then for for any any k k E N, ri0n-IND TI(rexp~-l(k) A) II� -IND f-k- T I( exp�- l (k) "7, A)

for for n n � > 22..

n -2 ) This iterating (8) This is is proved proved by by iterating (8) above. above. Note Note that that A' A' is is IIg rio,, A" A" is is II� r i ~, . . . , A A ((n-2) n l ( - ) ) , hence is f-~- TI(k', is II� rio.. For For any any fixed fixed kk E e N N we we have have in II�-IND, ri~ TI(rk7, A A(~-l)), hence " ' in n -3) ) etc. k TI(wr TILT k',,A(~-2)), hence fk TI(w TI(w w~ ,,AA((~-~)) etc. until until finally finally fk TI(a, T I ( a , AA) ) fA (n -2) ) , hence r where w h e r e aa -= rexp� exp~n --1l (k) (k)7'. Now Now we we can can complete complete the the proof proof of of the the theorem theorem by by choosing choosing A A to to be be the the formula formula Vx3y3z x, y, y, z) ) , (5) (5) and (6) this Vx3y3z. . CF(a, CF(a, x, z).. By By (4 (4), and (6) this formula formula is is progressive. progressive. So So by by (9) (9),, /

"

rk7

rk~

.

y, z) riOn-IND fk- Va -( -~ rrexp~,-l(k) Cf(a, x, x, y, z) . Vx3y3z . CF(a, exp�- l (k) '7.Vx3y3z. II�-IND for E N. for every every fixed fixed kk E N. Therefore Therefore if if aa is is a a fixed fixed code code for for aa tree-ordinal tree-ordinal a a -( -~ (C: (E0)n, the O )n, the formula x, y, formula Vx3y3z Vx3y3z. . CF(a, Cf(a, x, y, z) z) defining defining Fa F~ is is provable provable in in II� ri~ -IND. Putting Putting 33.19(1), and 44.11 together and and recalling recalling 22.23 and 4.3, 4.3, we we . 19 ( 1), 33.20, . 20, 44.9, . 9, 44.10 . 10 and . 11 together . 23 and obtain our main main subrecursive obtain our subrecursive classification classification for for arithmetic. arithmetic.

189 189

Provably ProvablyRecursive RecursiveFunctions Functions 4.12. Classifications. Classifications. 4.12.

The primitive recursive functions can be classified as

11.. The primitive recursive functions can be classified as PRovREc(E~ PROV REc(I;� -IND) = REC(w 2)

= =

U E(Ba) UE(F~) = e)-~W

U E(H~). ~-<w w

The multiply-recursive multiply-recursive functions functions of of Peter Pdter [1967} [1967] can can be be classified classified as as 2.2. The

W) PRovREC(I;g-IND) PROVREC(2~-IND) = REC(W REC(w')

=

=

=

U (Ba ) U E E(Ba) U E(F~) = Ot -~ W ~

U

E(H~)

Ot ~ W w ~

More generally, for n � and recalling

3. 2, and recalling (co)o = CO ) m+ l = w« o )m ;~" 3. More generally, for n > 2, (r = 11,, ((r (M(~o)m PRovREC(I;�-IND ) PROVREC(E~

=

REC((co)n) REC((eO).) (Ba ) = U U E E(Ba) =

a-~(~o),,

=

U

U

E(Fa)=

a-~(6o),

E(Ha)

a-~(eo),,+l

4. And finally,

4. And finally,

REC(co) REC(eo) = (Ba ) = U E E(Sa)

PRovREc(PA) = PRovREc(PA)

U

=

U E(F~) = U E(H~) t~ -g e o

~ -~ 6 o

Remarks. Kreisel Kreisel [1952], [1952], using using ideas ideas from from Ackermann Ackermann [1940], [1940] ' was first 4.13. Remarks. 4.13. was first show that that the the provably provably recursive recursive functions functions of of PA PA are are definable definable by by transfinite transfinite to show to -< Co. co . The The refinements refinements of of this this result, result, showing showing the the recursions over over order-types order-types -~ recursions rr�-IND to to be be those those definable definable by by recursion recursion over over provably recursive recursive functions functions of of II~ provably order-types -~ -< (c0),, (CO )n , are are due due to to Parsons Parsons [1966]. [1966]. The The result result that that the the provably provably recursive recursive order-types I;�-IND are are exactly exactly the the primitive primitive recursive recursive functions functions is is due due variously variously to to functions of of ~~ functions Parsons [1970], [1970], Mints Mints [1973] [1973] and and Takeuti Takeuti [1987]. [1987]. The The corresponding corresponding subrecursive subrecursive Parsons [1953] classifications in in terms terms of of the the fast-growing fast-growing Fas Pas follow follow from from Grzegorczyk Grzegorczyk [1953] classifications -< w; w ; from from Robbin Robbin [1965] [1965] for for c~ a -~ -< w~; wW ; and and from from Lbb Lob and and Wainer Wainer [1970], [1970] , for for c~ a -~ Wainer [1970] [1970] and and Schwichtenberg Schwichtenberg [1971] [1971] for for aa -~ -< e0. co . See See also also Constable Constable [1971], [1971] , Wainer Buchholz Buchholz and and Wainer Wainer [1987] [1987] and and Rose Rose [1984]. [1984] .

190 190

M. Fairtlough and and S. S. Wainer Wainer M.

The The Ha's HOI. 's first first appeared appeared in in aa subrecursive subrecursive context context in in Wainer Wainer [1972], [1972]' though though their their definition and and first first use use (exhibiting (exhibiting aa set set of of reals reals with with cardinality cardinality R1) Nd appears appears very very definition early in in Hardy Hardy [1904]. [1904]. The The development development of ofthe the REC(ce) REC(a) classes classes and and corresponding corresponding B~BOI. early functions stems stems from from Fairtlough Fairtlough [1991]. [1991]. See See also also Fairtlough Fairtlough and and Wainer Wainer [1992]. [1992]. Many Many functions other significant significant contributions contributions to to the the theory theory of of provably provably recursive recursive functions functions have have other been (and (and are are still still being) being) made--see made-see for for instance instance Buchholz Buchholz [1987], [1987] , Buchholz, Buchholz, Cichon Cichon been and Weiermann Weiermann [1994], [1994], Buss Buss [1994], [1994]' Friedman Friedman and and Sheard Sheard [1995], [1995], Girard Girard [1981], [1981]' and Leivant [1995], [1995], Ratajczyk Ratajczyk [1993], [1993], Schwichtenberg Schwichtenberg [1977], [1977], Sieg Sieg [1985,1991], [1985,1991]' Tucker Tucker and and Leivant [1992] and and Weiermann Weiermann [1996]. [1996]. Furthermore Furthermore there there are are non-standard non-standard modelmodel Zucker [1992] Zucker theoretic methods methods running running parallel parallel to to the the proof-theoretic proof-theoretic ones ones employed employed here--see here-see theoretic [1980] , Hajek Hajek and and Pudlak Pudlak [1991], [1991] ' Avigad Avigad and and Sommer Sommer [1997] [1997] and and for example example Paris Paris [1980], for [1995] . The The field field is is now now quite quite broad, broad, and and we we can can only only apologise apologise to to those those Sommer [1995]. Sommer many contributors contributors whom whom we we have have not not mentioned. mentioned. many 5 5.. IIndependence ndependence rresults e s u l t s ffor o r PA PA

From 4.12(4) we see immediately immediately that that B~o, Beo , F~ Feoo and dominate all all From 4.12(4) we see and Heo H~o eventually eventually dominate the functions of of PA, PA, and and so so they cannot themselves themselves be be provably provably the provably provably recursive recursive functions they cannot recursive in Consequently recursive in PA. PA. Consequently PA Y VnVx3y3z. CH((co)n, x, y, z).

In In other other words, words, the the proof proof of of 4.11 4.11 shows shows that that although although for for any any fixed fixed n n we we have have PA F- TI((co)n,A)

there of there are are rrg-instances II~ of A A for for which which fl T I(co, A) PA PAY TI(co, A)..

Another Another way way to to see see this this would would be be to to note note that, that, again again for for an an appropriate appropriate formula formula A A PA PA + + TI(co, TI(c0, A) A) fF- "PA "PA is is consistent" consistent".. For, For, by by Gentzen, Gentzen, if if aa false false atom atom were were derivable derivable in in PA, PA, then then by by the the Embedding Embedding and and Cut-Elimination Cut-Elimination theorems, theorems, it it would would be be derivable derivable in in the the infinitary infinitary system system with with cut-rank cut-rank 00 and and ordinal ordinal bound bound a c~ -< -< co c0.. By By induction induction up up to to co Co,, this this is is impossible impossible since since an an infinitary infinitary rank-O rank-0 proof proof of of an an atom atom cannot cannot use use logic. logic. Formalisation Formalisation of of this this argument argument would would lead lead to to aa proof proof of of TI(co, TI(co, A) A) f-F- Con(PA) Con(PA) in in Primitive Primitive Recursive aecursive 's second Arithmetic. Arithmetic. Of Of course, course, G6del G5del's second theorem theorem tells tells us us that that there there is is no no proof proof of of I(co, A) Con(PA) Con(PA) inside inside PA, PA, hence hence no no proof proof of of T TI(co, A).. Thus Thus co Co is is the the least least upper upper bound bound of of the the "provable "provable ordinals" ordinals" of of PA. PA. See See Gentzen Gentzen [1943]. [1943]. These These may may be be termed termed "logic" "logic" independence independence results. results. The The question question then then was was whether whether they they might might be be equivalent equivalent to to other other independence independence results results of of aa more more clear clear

191 191

Provably Recursive Recursive Functions Functions Provably

mathematical character. character. This This was was shown shown to to be be the the case case by by Paris Paris and and HarringHarring mathematical [1977] who who proved proved the the independence independence (from (from PA) PA) of of aa certain certain finite finite version version of of ton [1977] ton Ramsey's theorem, theorem, and and by by Ketonen Ketonen and and Solovay Solovay [1981] [1981] who who then then gave gave aa "rate-of"rate-of Ramsey's growth" growth" analysis analysis of of itit in in terms terms of of the the fast-growing fast-growing hierarchy. hierarchy. In In this this section section we we treat treat [1982] concerning concerning so-called so-called another simpler simpler independence independence result result of of Kirby Kirby and and Paris Paris [1982] another [1983] since since itit isis "Goodstein sequences", sequences" , but but the the proof proof we we give give isis due due to to Cichon Cichon [1983] "Goodstein directly related related to to the the H-functions. H -functions. directly exp;x + 1 (1) 5.1. DDefinition. Given numbers numbers a,a, xx >2:: 11 with with aa <� expx+l (1) the the "complete "complete 5.1. efinition. Given 1) form" form" of of aa isis base-(x ++ 1) base-(x

a--(x+l)a~.ml+(X+l)a2"m2+'''+(x+l)ak'mk ak , and and their their exponents exponents wherein ml, ml , m2, m2 , .. ...., , mmk and all all exponents exponents al al >> .. ". . >> ak, wherein k <_ � xx and etc, are are again again written written in in base-(x base- (x ++ 1) 1) form. form. etc, Let g(x, be the by writing writing a a -- 11 in in complete base-(x g(x, a) a) be the number number which which results results by complete base(x ++ 1) Let 1) + + form, and the base base in this expression expression from from (x 1) to 2) , leaving leaving form, and then then increasing increasing the in this (x + 1) to (x (x + 2), mi fixed. fixed. all coefficients coefficients mi all The Goodstein on (x, (x, a) a) is is then then the numbers {ai}i>x {ai};>x where The Goodstein sequence sequence on the sequence sequence of of numbers where j, ax ax and ax a~ = = a a and a~+j+l = gg(x (x + + j, a~+j). +H l = +j) ' Given (x + 1) form, Given a a number number a a written written in in complete complete basebase-(x + 1) form, let let ordx(a) ord~(a) be be the the tree-ordinal tree-ordinal obtained obtained by by replacing replacing the the base base throughout throughout by by w w..

5.2. 5.2. Definition. Definition.

5.3. 5.3. Lemma. Lemma. oordx(a rdx(a- 11)) == Px(ordx(a)) Px(ordx(a)) for for a a > > OO.. Proof. P r o o f . The The proof proof is is by by induction induction on on aa.. If If aa = - 11 then then ordx(a o r d x ( a -- 11)) = - 00 = -- Px(I) Px(1).. (x + Suppose Suppose a a > > 11 and and that that the the complete complete basebase-(x + 1) 1) form form of of aa is is a = (x + 1)al 9ml -[- (x -[- 1)a2 9m2 + . . .

+ (x + 1 ) a k . m k .

If ak -- 0 then o r d x ( a - 1) - ord~(a) - 1 -- P~(ordx(a)). If ak > 0 then let b = (x + 1)a' 9ml + (x + 1)a2 9m2 + . . . + (x + 1) ak. (ink -- 1).

Then Then

1

-2 . x + = bb + + (x (x + + I1t ) akk - l . xx + + (x (x + + l)ak 1)ak-2-x + '.". . + + (x (x + + 1)° 1)~. x. aa -- I1 = -

•

Let ) . Then Let c¥ a = - ordx(a) ordx(a), f3 fl = - ordx(b) ordx(b) and and C¥k ak = -- ordx{ak ordx(ak). Then by by the the induction induction hypothesis hypothesis we we have have (Ok ) 9xx + (Ok ) . 9xx + ordx {a - 1) ordx(a 1) = = f3 fl + +W w P. P'(ak) +W w P; P2(ak) + . .. .. . + + x. x. ,

-

•

Therefore Therefore by by the the properties properties of of the the function function Px Px we we obtain obtain

ordx {a 1) k) = {ordx{a)) ord~(a1 ) -= Px(!3 Px(fl + +W wOa~) - Px Pz(ordx(a)) -

5.4. 5.4. Lemma. L e m m a . g{x, g ( x , aa) ) = = G G Xz + P z ( o r d z ( (a))) a))). . + 1l ((Px{ordx

M. Fairtlough and and S. S. Wainer Whiner M.

192 192

P r o o f . By By the the definitions, definitions, note note that that Proof. g(x,a) g(x, a)

=

=G Gx+~(ordx(a1)) X+ 1 (ordx(a - 1))

since Gx Gx+l replaces base base w by by (x (x + + 2) 2),, as as in in 2.11. 2.11. The The result result then then follows follows from from 5.3. 5.3. since +1 replaces 5.5. Lemma. L e m m a . Let Let aax, ax+l, ax+2,.., be the the Goodstein Goodstein sequence sequence on on (x, (x, a) a).. Then Then for for x , ax 5.5. +2 , " . be +b ax each jj each 1. ordx+j(ax+j) - Px Px+j-1 Px+j-2 Px(ordx(a)). 1. ordx x (ordx (a) ) . +j(ax+j) = +j - 1 Px +j -2 '" "" "P 2. ax ax+j = Gx Gx+j(ordx+j(ax+j)). 2. +j(ordx+j(ax+j)) . +j = Proof. Proof. By induction induction on on j. j. The The base base case is trivial trivial and and for for the the induction induction step step we we have have 11.. By case is by 5.3, 5.3, by ordx+j+l(g(x + j, ax+j))

= -

ordx ordx+j(ax+j - 1) 1) +j(ax+j Px+j(ordx+j(ax+j)). Px +j (ordx+j (ax+j)).

Hence ordx ordx+j+l(ax+j+l) -- Px Px+j(ordx+j(ax+j)) and the the result result follows follows immedi immediHence +j (ordx+j (ax+j)) and +i+ l (aX+i+l ) = ately from from the the induction induction hypothesis. hypothesis. ately 2. This is is immediate immediate by by iterating iterating 5.4. 5.4. 2. This

5.6. (Cichon 5.6. Theorem. Theorem. (Cichon [1983]) [1983]) Let Let {a {a,},>x be the the Goodstein Goodstein sequence sequence on on (x, (x, a) a).. i };�x be Then 0, and Then there there is is aa y y such such that that ay ay = = O, and the the least least such such yy is is given given by by yy = = H Horde(a)(x). ord. (a) (x) . Proof. 5.5, ordx+j+l(ax+j+l) ordx+j + 1 (ax+j + 1 ) -~ -< ordx+j(ax+j) ordx+j (ax+j) if if ordx+j(ax+j) ordx+j (ax+j) :fl P r o o f . By By 5.5, By welli= 0. O. By well foundedness there must must be stage kk at at which which ordx+k(ax+k) ordx+k (ax+k) = and hence hence foundedness there be aa first first stage -- 00 and = O. O. By By Theorem Theorem 2.19 2.19 we we can can express express this this kk as ax+k as ax +k -kk

= -

least k. k . (Px+k-1 (Px+k- l Px+k-2""Px(ordx(a)) 0) Px+k -2 ' . . Px(ordx(a)) == 0) least D Dord~(a)(succ)(x) ord. (a) (succ) (x)

Hord.(a) (x)).. and therefore therefore xx §+ kk = and ---- Uordz(a)(X 5.7. heorem. (Kirby 5.7. TTheorem. (Kirby and and Paris Paris [1982]) [1982]) Let Let Good(a,x, Good(a, x, y) y) be be aa E~ '£� -formula of of arithmetic arithmetic expressing expressing the the fact fact that that the the Goodstein Goodstein sequence sequence on on (x, (x, a) a) terminates terminates at at y, y, i.e. ay ay == O. O . Then Then VaVx3y. VaVx3y . Good(a, Good(a, x, x, y) y) isis true true by by 5.6, 5. 6, but but not not provable provable in PA. i.e. in PA. PProof. r o o f . If were aa theorem If itit were theorem of of PA, PA, the the function function h(a,x) h(a, x) == least least yy . . Good(a,x, Good(a, x, y) y) would be be provably provably recursive recursive in in PA. PA. For x, set set a(x) a(x) == exp~+l exp;+ 1 (1). (1) . Then Then a(x) would For each each x, a(x) isis primitive would also also be primitive recursive recursive and and so so the the function function h(a(x), h(a(x) , x) x) would be provably provably recursive recursive (co)x, we we have Hco (x) . in PA. PA. However, However, by by 5.6 5.6 and and since since ordx(a(x)) ordx(a(x)) -= (c0)x, in have hh(a(x), ( a ( x ) , xx) ) -= H6o(X). This This contradicts contradicts 4.12(4). 4.12(4) .

Provably Recursive Functions

193 193

6. The T h e "true" " t r u e " ordinal o r d i n a l of o f PA PA 6.

Section Section 44 characterizes characterizes the the provably provably recursive recursive functions functions of of PA PA in in terms terms of of co c0recursiveness but, but, recalling recalling definitions definitions 3. 3.17, it still still remains remains to to characterize characterize them them in in recursiveness 17, it terms of of 'Y-definability. 7-definability. We We shall shall now now "compute" "compute" the the appropriate appropriate 'Y 7 by by appealing appealing terms to the the Hierarchy Hierarchy Theorems Theorems 3.19 3.19 and and finding finding an an ordinal ordinal map map a a 1-7 ~ a+ a + such such that that for for to ' s, a � -< co e0,, and and even even much much larger larger a a's, a B~ = Go+ We then then have have We

P P RRovREC(PA) ovREc(PA) = = REC(co) REC(c0)= = ��-DEF(c:t). E~ 's system For related related results results and and an an alternative alternative treatment treatment in in terms terms of of G6del GSdel's system T T of of For primitive recursive recursive functionals, functionals, see see Schwichtenberg Schwichtenberg and and Wainer Wainer [1995] [1995].. ct c + is is the the primitive proof-theoretic ordinal ordinal of of the the theory theory of of one one inductive inductive definition definition and and is is usually usually referred referred proof-theoretic to as as the the Bachmann-Howard Bachmann-Howard ordinal ordinal ((see Howard [1970]) [1970]).. Girard Girard [1981] [1981] was was the the first first see Howard to to give give a a detailed detailed analysis analysis of of the the relationship relationship between between the the fast-growing fast-growing and and the the to slow-growing hierarchies hierarchies and and once once the the correct correct result result was was known, known, many many others others gave gave slow-growing more direct and simpler analyses. analyses. We We shall shall follow follow the treatment in Cichon and and more direct and simpler the treatment in Cichon Wainer [1983] [1983] and and more more generally, generally, Wainer Wainer [1989]. [1989]. The The main main point point is is that, in order order that, in Wainer to describe describe the the map map a a 1-7 ~+ a+ a +,, one one needs needs to to make make use use of of "higher "higher number number classes" classes" of of to uncountable tree-ordinals. tree-ordinals. However, However, since we are are only only concerned concerned here here with "small" uncountable since we with "small" a a ' 'ss below below co e0,, we we only only need need to to go go to to the the "next" "next" number number class class over over fl. f~.

Let Let flo ~0 = - N N and and flI f~l = - fl. ~. Then Then the the set set fl2 f~2 is is generated generated inductively according according to to the the four four rules: rules: inductively Zero. 0 0 E f~2 E fl2 Zero. Succ. a EE~ 2fl2 = ====} ~ a +al E+~ 21 E fl2 Succ. a = (a~) (ax) EE fl2 flo(ax EE ~2) fl2) ====} Lim0. Vx E E ~0(a~ ~ aa = f~2 Limo. Vx fl2) ==~ ===} aa == (a~) (a� ) EE ~2 Lim l . V� fl2 flI (a� E ~2) Liml. V~ EE ~l(a~

6.1. Definition. 6.1. D efinition.

Note: we we sometimes sometimes write = S U P a e� according a = Note: write aa == sup sup a~ ax or or aa = SUPa according to to whether whether a = (a~) (ax) or a = (ae) in f~2. or a = (a fl2 • � ) in 6.2. D e f i n i t i o n . The The (well-founded) partial ordering defined 6.2. Definition. (well-founded) "subtree" "subtree" partial ordering -< � on on f~2 fl2 iiss defined as the transitive closure of the rules as the transitive closure of the rules a -� . < a ++l l .9 a • 9Vx sup az) ax) Vx EE ~0(az flo (ax -< � sup • 9 V� V~ E E fh flI (ar (a{ -< � SUPar SUPa{) 6.3. r i t h m e t i c on Addition, multiplication and exponentiation Arithmetic on f~2. fl2 . Addition, multiplication and exponentiation of of ~2 fl2 are are 6.3. A for ~1, flI ' but but with with an an extra extra limit limit clause clause in in each each case, case, viz. viz. defined defined exactly exactly as as in in 2.11 2.1 1 for

SUP,B{ = SSUP(a aa++ SU P& = U P ( a ++&,B{) ) SUP,B{ = aa. · SUPflr

SUP(a · ,Bd SUP(a.fl~) SUP /Je = SUP(a&). aaSUP& SUP(a/Je ) .

M. M. Fairtlough Fairtlough and and S. S.

194 194

Wainer Wainer

6.4. 6.4. Examples. Examples. 11.. wo w0 = = sup sup x, x, wl t~ 1 = --- SUP(l SUP(1 + + �) ~c) ~,1 �l W W l ~,+~ = SUP sup(l, Wl, w~l'1,, W WlI ,, .. ... ). ). . 22.. CW (1 , W I,W l +I 6.5. Definition. The 6.5. D efinition. The slow slow growing growing function function G G :: 01 ~1 X • 00 ~0 -+ --+ 00 ~0 is is now now extended extended to notationally convenient to aa map map G G :: O2 f12 x • 00 ~0 -+ --+ 01 ~1.. As As in in 22.11 it will will be be notationally convenient to to swap swap . 11 it the write, for the arguments arguments and and write, for each each fixed fixed n E E N IN,, Gn{a) G , ( a ) instead instead of of G",(n) G , ( n ) . . Thus Thus for for each n we O2 -+ 01 by the following each we define define Gn G , :: ~2 --+ ~1 by the following recursion: recursion:

n

n

Gn(O) G.(0) = = 00 Gn(a G . ( a ++ l ) 1) = = Gn{a) G . ( a ) ++l 1 Gn(sup C.(sup ax) ) = Gn(an) Gn(SUPae) Gn{ax) G.(SUPa~) = = sup supG .(a~) Note Note that that we we immediately immediately have, have, for for every every

n,

n,

Gn(wd an(W1) = : 11 + + wo W0 = - - ww E E 01• ~'~1. 6.6. 6.6. Lemma. Lemma.

For For each each fixed fixed n n E E N IN and and all all a, a, /3 13 E E O2, i22, Gn{a G.(~ + + /3) ~) = Gn(a) G.(a) + + Gn(/3) G.(~) . . /3) Gn(a Gn(a) Gn(/3) a . ( a . fl) = G . ( a ) . G.(fl) Gn(a.8) Gn(a) Gn (lJ) =

Proof. inductions on P r o o f . This This is is by by easy easy inductions on /3 ~ E E O2 f~2..

rules: 6.7. Definition. 6.7. D e f i n i t i o n . Let Let EXP EXP � c_ O2 ~2 be be generated generated inductively inductively according according to to the the rules: • 01 U 9 ~'~1 U {W {~dl} C EXP EXP I} � •9 a, /3, a a , / ~/3 E E EXP EXP � ==a a a + +/~, ~./3, ~ E E EXP EXP.. · /3, a.8 6.8. 6.8. Lemma. Lemma.

have have

Fix Fix n n E E N IN.. Then Then for for every every 'Y 3' E E EXP E X P of of the the form form 'Y 3" = = SUP'Ye SUP3'~ we we V~ 6 f~l(G.(3'~)= G.(3')c.(~))

Proof. P r o o f . This This is is by by induction induction on on the the generation generation of of 'Y 3' E E EXP E X P . . The The base base case case 1I is is easy easy because Gn(re) = Gn{�) = then 'Ye because the the only only possibility possibility is is 'Y 3' = = W Wl, so then 3'~ = = 11 + + �~ and and G,(3'~) = 11 + + G,(~) = I , so . /3 where w wc.(~) = Gn(r) G,(3')c.(~ In case case 22 suppose suppose for for example example that that 'Y 3' = = a c~./3 where a a E E EXP EXP Gn (e) = Gn (e)).. In and a n d / 3/3 E E EXP E X P . . Then Then there there are are two two sub-cases. sub-cases. Either Either/~/3 = - / 3/3' ' + + 11,, in in which which case case induction hypothesis 'Y3' = - a a . /. 3/3'' + + a ~ and and hence hence by by the the induction hypothesis applied applied to to a, a, G.(7~) = Gn{a G . ( ~ . /. 3/3'' + + a ~d ) = = Gn(a G . ( a . /.~/3') ') + + Gn(a) G.(a)a.(~) = Gn(r) G.(7)a.(~). (re) = Gn Gn (e)· Gn ( e ) =

Or induction hypothesis Or/~/3 = = SUP/3e SUP/3~ in in which which case case 'Ye 3'~ = = a a . /· ~/3e, , and and hence hence by by induction hypothesis applied applied to /3, to/3, G.(7~) - G.(c~). G.(/~) = G.(c~).G.(~)G.(~) = G.(7)c.(~).

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Provably Recursive Recursive Functions Functions Provably

6.9. DDefinition. The fast-growing fast-growing hierarchy hierarchy BB :: ~011 •x 9to 00 -+ -+ ~o 00 isis "lifted" "lifted" to to aa 6.9. efinition. The hierarchy ~o:

O2

<po(f3) = Zf3 ++I 1 ~0(/~) <Pa (<Pa (f3)) <Pa v.+,(Z) + 1 CB) = vo(v.(Z)) ~0~(/3) sup <Paz ~o~x(/3) if c~ sup c~ (f3) if a == sup ax <Pa( 3) -- sup v.(/~) Par 'Pa{3 (f3) iiff aa==S USUPae <Pa(3) = ~.,(/~) Then .for for every every aa EE EEXP and all all Fix nn EE N. N . Then 6.10. ollapsing T heorem. Fix X P and 6.10. CCollapsing Theorem. we have have f3 EE ~"~1 0 1 we t~ Gn(~o~(/3)) = Ba,,(,~)(Gn(13)). note tthat PProof. r o o f . First First note hat • E PEXP 9 iiff aa+ + l E 1E X t h e n athen E E X Pa

E EXP • E EEXP then aax. EE EXP for every every xx EE 5I 9 if if aa = = sup sup ax a. E X P then E X P for N •9 if if aa == SUPar SUPae EE EEXP E EEXP X P then then ae ar E X P for for every every � ~E 0 9tl. 1. The proof proof of facts is is by inductive The of these these facts by an an exhaustive exhaustive case-analysis case-analysis according according to to the the inductive ( 1 ) , (2) and (3) (3) mean mean that definition leave itit as as an an exercise. exercise. However, However, (1), definition of of EXP. E X P . We We leave (2) and that we can proceed proceed by to show show tthat E f12, we can by -<-induction -<-induction to h a t for for every every a a E

O2,

a E E X P ==~ Vfl E f~x(Gn(~o~(~))=

Ba.(.)(G.(Z))).

The cases are The cases are easy: easy: therefore we have If a If c~ = = 0 0 then then Gn(a) Gn(c~) = = 0 0 and and therefore we have Gn((Po(/3)) = Gn(j3 + 1) = an(/3) + 1 = Bo(Gn(13)).

then Gn(a) If a a = = a' a' + + 11 then Gn(c 0 If hypothesis hypothesis (twice) (twice) we we have have

= =

Gn(a') and <Pa <Pa' 0o <Pa' Gn(a') + + 11 and ~oa = = ~oa, ~o~,,, so so by by the the induction induction

Gn(~O,~(fl)) = Ba,,(,~,) o Ba.(,~,)(Gn(/3)) = Ba,(,~)(Gn(/3)).

If then Gn(a) and <Pa 'Paz ' so the induction If a a = = ssup u p aax ~ then Gn(a) = = Gn(an) Gn(an) and ~o. = = sup sup~o~x, so by by the induction hypothesis hypothesis applied applied to to an an we we have have Gn(~o,(/3)) = Gn(~o,.(/3))= BG.(,~,,)(Gn(13))= BG,,(,~)(Gn(~)).

If then Gn(a) Gn(ax) (f3) , so If a a = = SUPae SUPa~ then Gn(a) = = sup supG n ( a x ) and and <Pa(f3) ~o~(/3) = = 'Pa{3 ~o~(/3), so by by the the induction induction hypothesis hypothesis applied applied to to ap aS,, and and since since Gn(ap) Gn(a~) = Gn(a) Gn(~)c.(~) by Lemma L e m m a 6.8, 6.8, we we have have Gn (P) by Gn(~o~(/3)) = Gn(~o~(/3))= Be,,(,~)e,,(~)(Gn(~))= BG,(,)(Gn(/3)).

O2

Note that this proof Note t h a t this proof will will go go through through for for any any -<-closed -<-closed subset subset of of ~2 satisfying satisfying 6.8. 6.8. We We did did it it for for EXP E X P since since that t h a t is is all all we we need need here. here.

196 196

M. Fairtlough Fairtlough and and S. S. Wainer Whiner M.

6.11. 6.11. Corollary. Corollary. Suppose a ~ -< -4 co Eo.. Then Then a c~ can can be be written written in in exponential exponential "Cantor "Cantor Suppose normal form" normal form" a oL = = WO w ~o . 9mo mo + -5 WO w ~I 9m m ll + -5 . 9. 9. 9+ -5 W w O~ k . 9mk mk . . where their exponents, where ak ~k -< -4 ak ~k-1 -4 ."'" -< -4 a all -< -4 ao C~O are are also also of of this this form, form, and and their exponents, etc. etc. - l -< Let Let a & E 6 EXP E X P be be the the result result of of replacing replacing w w by by WI Wl throughout throughout this this normal normal form. form. &(wo) E Now define define a ~++ = = CP ~oa(Wo) 6 n1 f~l.. Then Then by by 6. 6.66 and and the the facts facts that that Gn(W G,(wl)l ) = = w w and and Now Gn(m) G , (m) = = m m for for m m 6E N, N, we we have have •

Gn(a) = a

and 10 we n, and therefore therefore by by 6. 6.10 we have have for for every every n,

Gn(a+) G , ( ~ +) = = Bo(n) Bo(n)..

Thus, notation for Thus, reverting reverting to to the the old old notation for G, G, Bo Bo = = Go Go+. +. 6.12. Examples. 6.12. E xamples.

W ) ++ = 11.. (W (ww) = CP ~w,~, (Wo) WI" I (wo) 2. The 2. The obvious obvious and and natural natural extension extension of of a & to to the the case case a a = - co ao is is to go = - cao,~+l, since WI + 1 , since for obtain, for for every every n n,, (cw ( ~ +I + 1 l) )n , = = (to)n (go),.. In In this this case case we we obtain, for each each n n,, Gn(ct) G , ( e +) = = = = =

Gn( a,(~o~,+l(WO)) cp''' I +I (wo)) Gn( CP(''' I + l ln (wO )) B B (('~oo) Jn) ) . ( n )= = BB' ~ oo((n) n )..

= CP''' Thus Thus B B~,oo = = G, G~o+ where ct e+ = ~o~.1+~ (Wo).. I + I (wo) t where

6.13. ce t+ = ''I + i (wo) tree-ordinal representation 6.13. Remark. Remark. = CP' ~o~+~ (Wo) is is our our tree-ordinal representation of of the the Bach Bachmann-Howard will note this mann-Howard ordinal. ordinal. The The reader reader will note that that we we have have not not yet yet proved proved that that this ordinal so, but ordinal is is structured. structured. We We will will do do so, but not not until until after after the the main main result result below: below: 6.14. 6.14. Theorem. Theorem. 11.. U o -<,o EE(Bo) Uo-<~o (Bo) = = U Uo.<~+o ( G o ) and and hence hence o -<,t EE(Go) REC(co) 2. 2. P P RRovREC(PA) o v R E c ( P A ) == R E C ( e 0 ) == ��-DEF(ct) ~-DEF(e+). . Proof. Proof. co then some nn,, and and we 11.. If If a a -< -4 60 then a ~ � ~ (co)n (Co), for for some we then then have have E(Bo) E(Bo) � C_ E(B E(B(6o).). ('o) J . But o ;t = G and hence , But by by 66.11 B(~o) = G G(eo)$ G(e+). and hence . 11 B ('o )n. = n ' ( t) ) (

E(Bo) J· E (Bo) � C_ E(G E(G(~+o (,t)),).

This This proves proves the the containment containment from from left left to to right. right. Conversely Conversely if if a ~ -< -4 ct 6 + then then a ~ � ~ (ct)n (6+)n for for some some n n and and so, so, assuming assuming the the structuredness structuredness of of ct 6 + ,, we we have have

E(Go) E(Go) � C_ E(G E(G(~+o).) = E(B E(B(Eo).) ('o ) J (,t ) J = and and this this proves proves the the containment containment from from right right to to left. left.

Provably Recursive Functions

197 197

2. This This follows follows almost almost immediately immediately from from (1) (1) by by means means of of the the Hierarchy Hierarchy Theorems Theorems 2. 3.19. The The only only necessity necessity is is to to check check the the conditions conditions on on the the second second Hierarchy Hierarchy 3.19. Theorem in in the the case case where where a a = -- c:t ~+.. An An easy easy computation computation gives gives Theorem

~o + 2~ = v~(~o) = v~, (~0) -< ~o+,

and and if if a a0, OL1 -< "~ c:t 60+ then then a a0, al � _~ ((rc:t)n for for some some fixed fixed n; n; so so again again assuming assuming o , a1 o , a1 the structuredness structuredness of of c:t ~+,, both both G G~ooo and and G G~O I1 are are eventually eventually dominated dominated by by the is eventually dominated by B(co)n+1 + I G B(co)n ' and therefore G o o 0 G G(~+). B(6o)., and therefore G~ o o G~I is eventually dominated by B(~o).+l = (c )n O an d hence an(f hence by by B(c B(~o).+ - G G(~o+).+ We can can then then ensure ensure that that the the composition composition (ct )n+ l '. We O )n+l~ = G~o 0o G G O~I is is completely completely dominated dominated by by some some Go Go with with a a -< -K c:t c + ,, by by choosing choosing a a Goo to be be ((c+)~+1 + kk for for some some large large enough enough kk.. to c:t)n+ 1 + 6.15. One 6.15. Note. Note. One can can easily easily read read off off corresponding corresponding results results for for the the fragments fragments of of PA. For For example, example, for for n n > 11,, PA. PrtOVREC(II~ 6.16. 6.16.

= REC((e0)~)= Z~

It now now remains remains to to prove prove that that It

c"' l + 1 (wo) c:~0t+ = = CP v~,+~(~0)

is structured. This is structured. This has has been been done done by by Kadota Kadota [1993] [1993] and and independently independently (though (though pre previously unpublished second author. author. We develop an viously unpublished)) by by the the second We first first need need to to develop an appropriate appropriate notion of of structuredness structuredness for for O f~2. notion 2, Definition. ((cf. cf. 2.6) 2.6) For E f22 O2 and and all N, flf3 EE ~'~1, 01 , define the finite finite set Definition. For each each aa E all nn E E 51, define the set a[n,/~] of aa by a[n, f3] of of -K-predecessors -<-predecessors of by O[n,Z] O[n, f3] a l [n, t~] f3] a + + l[n, a[n, f3] a[n, /~] a[n, fl] f3] a[n,

6.17. Lemma. Lemma. 6.17.

= = Zo

a[n, f3] U = a[n,/~] U {a} {a} an[n, f3] - an[n, /~] if i f aa- -=s usup p a =ax a/J[n, f3] = a~[n, /~] if i f aa = = SUPae SUPae. ·

d

all flf3 EE 01 we have For all all a, a, 7'Y EE f~2, O2 , all all nn EE NN aand n all For f~l we have o (f3) [n]. 'Y eE a[n, -y(f3) E .[~, f3] Z] ~==> cP v~(Z) e cp vo(Z)[~].

Proof. We proceed by by induction induction over over aa EE f~2. O2 • The The zero zero case case is is trivial trivial and and both both Proof. We proceed limit cases cases follow follow immediately immediately from the induction induction hypothesis hypothesis and and the the definitions definitions of of limit from the and a[n, a[n, ~]. f3] . For For the the successor successor case case suppose suppose 7'Y EE aa ++ l[n, U {a}. {a}. f3] -= a[n, a[n, fl] f3] U ~a(~) CPo (f3) and l [n, fl] Then qp~(fl) since 68 ++ l[n] for E qoa(/3)+ Then � qO,+x(~)[n] � tPa(6)[n] cP-y(f3) E CPo (f3) + l[n] cpo ( 8) [n] C CPO+ 1 (f3) [n] since l [n] == qa0((f)[n] cpo (8 ) [nJ for l [n] C_ any any 6. 8. Let be the of the Let -K -< s8 be the transitive transitive closure closure of the rules rules aa -K -<s8 aa ++ 1; 1; 8 8 Vn(an Vn(an -K -<s sup sup a=); ax) ; V7 V'Y EE f~S(c~ -<s SUPa~). SUPae ) . Of(a-y -K 66.18. .18. D efinition. Definition.

198 198

M. Fairtlough and and S. S. Wainer Whiner M.

6.19. Call 6.19. Definition. Definition. Call a a E E n2 f~2 structured structured if if for for all all "small "small limits" limits" A A -:�/ _ s a, a, . Vn Ee N V7 v~ Ee nf af - {O} {0}.w ~.. (An ( ~ Ee A[n ~[~ + + 1~,, 7D ~]).· Let n� f2s denote denote the the set set of of all all structured structured a cz E E n2 f22 and and note note that that a c~ E E n� f2s and a n d / ~fJ -<s _<s a a Let imply t h a t / ~fJ E E n� f2s.. imply that

6.20. Lemma. For have: 6.20. L emma. For every every a c~ E E n� f2 s we we have: 'Po -+ (nf ~o.: (nf C a r- {O}) {0})-~ ( a f -- {O}). {0}).

Proof. P r o o f . We We proceed proceed by by induction induction on on a cz E E nq f2s.. Let Let fJ /~ E E nf f21s - {O}. {0}. If If a c~ = = 00 then then ~o~(/~) = / 3fJ + + 11 E E nf f21s - {O}. {0}. For For the the successor successor case case a c~ -+ --+ a c~ + + 11 we we have have 'Po ~P~+z (/~) = = 'Po (fJ) = +! (fJ) ~o~(~o~(/~)) E E nf f~s _ {O} {0} by by two two applications of the the induction induction hypothesis. If a c~ = = sup sup ax c~ 'Po('Po(fJ)) applications of hypothesis. If then ~ (fJ) (/~) = = sup sup 'Po. ~a~ (fJ) (/~) and and by by the the induction induction hypothesis hypothesis 'Po. ~pa~(fJ) (/~) E E nf f 2 s- {{O} 0 } for for each each then 'Po x. Thus only need structuredness condition x. Thus we we only need check check the the structuredness condition 6.19 6.19 for for 'Po(fJ) ~o~(/~) itself. itself. Now Now ax a[x + , fJl for x, because structured, and czx E E c~[x + 11,/3] for each each x, because a cz is is structured, and therefore therefore by by Lemma L e m m a 6.17, 6.17, ~o~(/3) E E 'Po(fJ)[x ~(/~)[x + + 11 1].· If If a c~ = = SUPae SUPc~ then then 'Po(fJ) ~o~(/~) = = 'Po� ~o~ (fJ) (/~) and and a c~s -<s _<s a c~ and and this 'Po.(fJ) this fl case case is is then then immediate immediate by by the the induction induction hypothesis. hypothesis.

6.21. Lemma. (cf. a, ~, fJ, 85 E 6.21. L emma. (cf. 2.12) 2.12) For For all all c~, E n2 f22,, all all 7 7 E E n1 ~1 and and all all n n E E N N we we have have fJ[n, 71 1. 85 EE fJ[n, 1. t3[n, 71 V] => ==~ a a + + 85 E E a c~ + +/~[n, V] 2. fJ[n, ,1 a 2. 8~ E e ~[~, ~] => ~ ~ ·. 8~ E e a ~ ·. fJ[n, ~[~, 71 ~] if iS 0o E e a[n, ~[~, ,1 ~] afl[n, 71 3. 3. 8~ E E fJ[n, ~[n, 71 V] => ==~ ad c~6 E E c~S[n, V] if if 1I E E a[n, c~[n, 71 V].. P r o o f . This This is is almost almost identical to that that of of 2.12 2.12 but but with, in each each case, case, an an additional additional Proof. identical to with, in trivial trivial step step corresponding corresponding to to "big "big limits" limits"/3fJ = = SUPfJe SUPfl~..

6.22. (cf. 6.22. Theorem. Theorem. (cf. 22 .. 13) 1 3 ) /If f aa,, fJ fl E E n� f2s then then 1. a + fJ E n� /. ~ + # e ~ 2. Vn > 2 a ~. · fJ # E e n� a~ provided , ~ o , i ~ V7 v~ E e nf ~f - {{O} 0 } .. w > O0.. (0 (0 E e a[n, ~[n, 7D ~]) 3. afl E n� provided V, E nf - {O} . Vn > O . ( 1 E a[n, ,D . Proof. All parts induction on 3) assuming P r o o f . All parts are are similar, similar, by by induction o n / ~fJ E E nq f2s.. We We just just do do ((3) assuming -+ fJ 2). If I f / 3fJ = = 00 then then aO c~~ = = 11 E E nq f2s.. For For the the successor successor case case fJ /3--+ /~+ + 1 1 we we have have ((2). afl part (2), the proviso c~s+z - afl c~s . 9a c~ E E n� f~s by by the the induction induction hypothesis hypothesis and and part (2), since since the proviso on on +! = fJx then a a ensures ensures that that afl c~s satisfies satisfies the the proviso proviso in in (2). (2). If If fJ j3 = - sup sup/~x then afl c~s = - sup sup afl• c~sx and and fJx E fJ[x + N . Also by by the the induction induction hypothesis hypothesis afl• a sx E E n� f~s for for each each x x E E N. Also/~x E/~[x + 11,, 71 V] for for all all VE E nf f2s - {0} and all all x x E E N, so afl• a s~ E aS[x + + 1, 7] by by 6.21 6.21 (3) (3).. Therefore Therefore afl as E E nq f2s.. 7 {O} and N , so E afl[x 1, ,1 If/3fJ = -- SUP/3e then afl as = - SUPafle SUPaSe and and since, by the the induction induction hypothesis, aSe E E n� f2s If SUPfJe then since, by hypothesis, afle whenever whenever � ~E E nf f2s,, we we automatically automatically have have afl as E E n� f~g.. 6.23. T h e o r e m .

~o+ = ~E~l+l (W0) E f2s.

Provably Recursive Functions

199 199

Proof. P r o o f . First First note note that that WI Wl = = SUP(l SUP(1 + + �) ~) E E 0,� f~2s,, since since if if A A= = sup sup A Ax ~ s WI wl then then x �s _ 11 + -t- "1 V for for some some "1 VE e 0,r f~ls.. Therefore Therefore A AE e 0,r f~s and and so so An An E E A A[n q- 1] 1] for for all all n n E e N N.. AA � [n + But countable, A[n A[n + , �] But since since A A is is countable, + 1] 1] = = A[n A[n + q- 11,~ c] for for all all �~c E e 0, f~l. Hence we we have have 1 . Hence V~ E e 0,r f~s _ {O} {0}.. Vn Vn E e N N.. ((An e A[n -t- 1, 1, W ~c]).. Note wl[n, �] ~] whenever whenever V� An E A[n + Note also also that that 11 Ee wdn, e 0,r f~ls - {O} {0} and and n n > > 0. Therefore by by 6.22 6.22 (3), (3), wf Wl~ E e 0,� f~2S whenever whenever /3 ZE e 0,� f22s.. Hence Hence �E O. Therefore "' 1 U)l •11 I 11,, WI belong to Wl,, wrl wl , W Wl� ,, . . . aall l l belong to 0,� f~s.. Now Now let let A A= =C~l+l. Then A A= = s usUP p AAxx where where AO A0 = = 11 and and An An+l=W~". Fixing CWI + l ' Then + ! = wtn • Fixing n [n + 1, "1] n E e N N and and "1 7 E e 0,r f~ls - {O} {0} we we have have 11 E e WI wl[n + 1, 7],, and and therefore therefore by by n n successive successive n + dn + 1, "1] A[n + Hence c~w1+1 applications applications of of 6.21 6.21 (3) (3),, An An E e A An+l[n + 1, V] = - A[n h- 11,, "1] V].. Hence f~2s. Thus Thus WI+ ! Ee 0,r (wo) E "' 1 +1 (w0) E 0,r f~ls by by Lemma Lemma 6.20. 6.20. 4"qoE~+l • • •

The The results results of of this this section section suggest suggest that that Peano Peano Arithmetic Arithmetic could could be be reformulated reformulated with a a weaker weaker "pointwise" "pointwise" induction induction scheme, scheme, sufficient sufficient only only to to prove prove termination termination of of with in order order to capture all all the the the slow-growing slow-growing G G functions. functions. But But then, then, in to capture the provably provably recursive recursive functions these inductions initial segments functions of of PA, PA, these inductions would would have have to to extend extend over over all all initial segments of of the Howard Howard ordinal. ordinal. In the Howard Howard ordinal ordinal becomes becomes the the ordinal ordinal of of PA PA the In this this way, way, the with pointwise this idea, idea, and with pointwise induction. induction. The The realisation realisation of of this and appropriate appropriate formulation formulation of of pointwise pointwise induction induction schemes, schemes, is is due due to to Schmerl Schmerl [1982]. [1982]. with transfinite 7. Theories T h e o r i e s with transfinite induction induction

This This final final section section shows shows how how the the foregoing foregoing subrecursive subrecursive classifications classifications for for PA PA can be extended from itit by can be extended quite quite easily easily to to theories theories obtained obtained from by adding adding the the Principle Principle o . Since of nite Induction of Transfi Transfinite Induction over over given given well-orderings well-orderings of of order order type type � >- c~0. Since proof prooftheoretic theoretic ordinal ordinal analysis analysis seeks seeks to to compute compute for for aa given given theory theory T T,, the the least least upper upper see Pohlers bound bound r T of of its its "provable "provable ordinals" ordinals" ((see Pohlers in in this this volume) volume),, the the results results here here will will then viz. then immediately immediately give give a a classification classification of of the the provably provably recursive recursive functions functions of of T T viz. PRovREC(T) PrtovREc(T) = = REc(r) R E C ( T ) ==

U E(Fa) E(F~)..

a -
Of Of course, course, r T must must be be shown shown to to satisfy satisfy the the conditions conditions of of our our Hierarchy Hierarchy Theorem, Theorem, or or something something like like them, them, and and this this often often requires requires some some checking! checking! See See Buchholz, Buchholz, Cichon Cichon and conditions. See and Weiermann Weiermann [1994] [1994] for for related related work work involving involving similar similar kinds kinds of of conditions. See also also Weiermann Weiermann [1996] [1996] for for an an alternative alternative treatment treatment of of PA PA and and transfinite transfinite induction induction in in terms terms of of "ordinal "ordinal majorisation" majorisation" relations. relations. We We shall shall assume assume henceforth henceforth that that "1 3' = = sup sup% >- w w is is aa structured, structured, countable countable 'Yx � tree-ordinal tree-ordinal which which is is "primitive "primitive recursively recursively representable" representable",, i.e. i.e. q-space q-space representable representable for q. The for some some primitive primitive recursive recursive q. The representability representability of of "1 7 ensures ensures that that it it will will be be possible to a, /3 "1, by possible to code code the the well-ordering well-ordering relation relation a c~ -< -~ /3 ~ for for c~, ~ � ~ ~,, by a a �� ~~-formula of of arithmetic "a -< ust as proof of Theorem 4.12 arithmetic "a -~ b" b" (j (just as was was done done for for co Co in in the the proof of Theorem 4.12 and and for for will be quite ordinal notations quite general general systems systems of of ordinal notations in in Sommer Sommer [1992]). [1992]). There There will be primitive primitive successor, and x) H x) recursive recursive functions functions aa H ~-~ aa EB @ rr ll~' representing representing the the successor, and (a, (a,x) ~ l(a, l(a,x) such such that that if if aa encodes encodes a c~ then then l(a, l(a, x) x) encodes encodes a cex when a ~ is is aa limit, limit, and and l(a, l(a, x) x) = = 00 x when

M. Fairllough Fairtlough and S. S. Wainer

200 200

otherwise. otherwise. For For simplicity simplicity we we shall shall assume assume that that PA PA is is extended extended to to include include them them as as new and l(a, x) , with new terms, terms, also also denoted denoted a a ffi (9 rl' r17 and l(a,x), with appropriate appropriate defining defining axioms axioms for for them. We shall also them. We shall also assume assume that that 00 is is the the least least element element of of the the well-ordering well-ordering and and that that the value value of of aa ffi (9 rIo F17 is is always always numerically larger than than aa.. The The top top element element of of the the the numerically larger well-ordering, representing will be a) well-ordering, representing , 7 itself, itself, will be denoted denoted by by cc and and we we shall shall write write Lim( Lim(a) for 1) =f. for "l(a, "l(a, 1) ~- 0" 0"..

7.1. Definition. PA TI(r) is adding to 7.1. Definition. PA + + TI(-),) is the the theory theory obtained obtained by by adding to PA PA the the Principle Principle of Transfinite Transfinite Induction Induction up up to to " -)', formulated formulated either either as as an an axiom-scheme axiom-scheme or or as as aa rule. rule. of We We choose choose the the rule rule TI(r) TI(7):: A,B(0) f-1- A, B(O)

, B(a) im(a) A , -...,LLi m ( a ) , ,33x x - ..., , BB( l(l(a, ( a , xx)) )),B (a) f-1- A, fA, Va(a a )) 1- A, Va(a -/<. ~ cc V B( B(a))

1- A, A , ~...,BB((a), a ) , BB(a ( a (ffi g rrIo) l 7) f-

with with the the restriction restriction that that a a is is not not free free in in A. A. The The embedding embedding of of PA PA + + TI(r) TI(7) into into w-Arithmetic w-Arithmetic follows follows the the same same lines lines as as the the embedding of TI(,) rule embedding of PA PA in in 4.7. 4.7. Applications Applications of of the the TI(3') rule are are dealt dealt with with as as follows: follows:

7.2. Suppose 7.2. Lemma. Lemma. Suppose the the three three premises premises of of the the TI(r) TI(-),) rule rule have have been been embedded embedded 6 A, into -Arithmetic with into w w-Arithmetic with aa fixed fixed ordinal ordinal bound bound 05 E E Os, f~s, so so that that f1-6 A, B(O) B(0) and and for for every every aa E E N, N, B ((a) a ) ,, BB(a ( a ffi (9 rIo) r-17) aa:: N f-t-66 AA,, - ...,- ,B 6 ( a ) , ,33x..., x ~ BB( l(l(a, ( a , xx)), ) ) , BB(a). (a). aa:: N f-1-6 AA,, -...,LLi mim(a)

Furthermore Furthermore suppose suppose that that 05 is is so so chosen chosen that that for for all all a, a, n n E E N, N, 1. the numerical value 1. the term term l(a, l(a, n n --'- 1) 1) has has numerical value bounded bounded by by B6(max(a, B6(max(a, n)) n)),, [a] is B II,, the 2. 2. the the cardinality cardinality of of o5In] is at at least least I[B the height height of of the the induction induction formula. formula. Then number encoding Then for for every every 0: a -< -~ " 7, if if aa is is the the number encoding 0: a we we have have 6+5'0+3 A A,, B( B(a). a ). aa ::NN f-1-6+~.,,+3 Proof. P r o o f . Proceed Proceed by by induction induction on on 0: a -< -~ , -y (for (for notational notational simplicity simplicity we we shall shall suppress suppress case 0: the the side side formulas formulas A). A). The The case a = = 00 is is immediate immediate and and the the successor successor case case from from 0: a to to 6 ..., B (a) , B(a a + + 11 is is also also straightforward, straightforward, since since from from the the assumption assumption a :: N N f1-6 -,S(a), S(a ffi (9 rIo) r17) 0: 6+5.aH B(affir1°) 6+5.a+3 B(a) and the the induction induction hypothesis hypothesis a a :: N N f1-6+5.~+a B(a),, we we obtain obtain a a :: N N f1-6+5.~+4 B(a(gF17) and l a + + 5 .( 6 3 ) + B( by Cut Cut and and hence hence aa ffi (9 rr17 N f1-6+5.(~+z)+a B(aa ffi (9 rIo) r17) by by Weakening. Weakening. Io :: N by Now Now suppose suppose 0: a = = sup sup O: axx and and choose choose any any n n E E N N.. Then Then letting letting m m denote denote the the 6+5.an�1 +3 B(l(a, numerical numerical value value of of l(a, l ( a ,nn --'- 1) 1),, we we have have m m :: N N f1-6+5.~,.1+a B ( l ( a ,nn --'- 1)) 1)) by by the the Bounding Lemma induction hypothesis, max(a, n) n) :: NN f-1-~6 m induction hypothesis, and and max(a, m :: N N by by the the Bounding Lemma 3.15 3.15 and and condition condition 11.. Therefore Therefore by by an an N-Cut N-Cut we we obtain obtain for for every every n n,,

6+5.an�l H B(l(a, max(a,n) B ( l ( a ,nn --'- 1)). 1)). max(a, n ) :: NN f-1-6+5.~,-1+4

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The The structuredness structuredness of of a a gives gives 65 + + 55 . 9an� c~n-1 + 44 E E 65 + + 55 .. a[max(a, a[max(a, n)] n)] and and so so an an l+ application the 'v' -rule yields application of of the V-rule yields +5.a 'v'xB( N fK0~+5"~ V x B ( ll(a, ( a , xx -'- 1)). 1)). aa : : N The The reason reason for for the the second second condition condition on on 05 is is that that it it ensures ensures (we (we leave leave the the reader reader to to check check it) it),, 0+5.a :Jx..., ) ) , 'v'xB(I (a, x)). 3x~B l ( a ,x x --'- 11)),VxB(l(a,x)). B((I(a, aa : : NN f-k-~+5"~ 5 0 a l . + + Hence x)) . Therefore Hence by by Cut Cut we we obtain obtain a a :: N N fK6+~~+1 'v'xB(I(a, VxB(l(a,x)). Therefore from from the the assumption assumption o -~Lim(a),3x-~S(l(a,x)),B(a) L im(a), :Jx...,B (I(a, x)) , B(a) aa:: NN f-K6 ..., o and and since since fF ~ Lim(a) Lim(a) is is an an axiom, axiom, we we obtain obtain by by two two further further Cuts, Cuts, +5.a+3 B(a) B(a) aa ::NN f-K0~+5"a+3

and and this this completes completes the the proof. proof. Now Embedding Theorem Now in in order order to to prove prove the the Embedding Theorem for for PA+TI(,) PA + TI(7) there there is is one one further further crucial requirement requirement to to be be placed placed on on ~'. Clearly if if , -y is is coded coded as as aa number-theoretic number-theoretic crucial ,. Clearly well-ordering well-ordering there there must must be be aa "norm" "norm" function function with with the the property property that that whenever whenever c~ -< -~ , -y is is encoded by the number a, have a c~ E e ,[norm(a)] ~[norm(a)].. See See Buchholz, Buchholz, Cichon a encoded by the number a, we we have Cichon and codings of ordinals this and Weiermann Weiermann [1994]. [1994]. For For "standard" "standard" codings of proof-theoretic proof-theoretic ordinals this norm norm function will often shall merely function will often be be just just the the identity, identity, but but we we shall merely require require that that it it be be primitive primitive recursive. recursive. 7.3. 7.3. Embedding E m b e d d i n g of of PA PA + + TI(!) TI(7).. Suppose Suppose , "7 is is primitive primitive recursively recursively representable, representable, with with aa primitive primitive recursive recursive norm norm as as above. above. Suppose Suppose PA + TI(7) K A ( h ( 2 ) , . . . , tk(E)).

Then d, measuring Then there there is is aa number number d, measuring the the "size" "size" of of this this proof, proof, such such that that for for every every assignment variables fl, assignment of of numbers numbers ii fi to to the the variables ~, we we have have maxfi" A ( mb m l , .. ... . ,,m~) · 'Y ·d A( mk ) max n : NN .-LK5s'~'d �

where numerical values tl (it) , .. .. .., , tk (ii). Further where m ml,l , .. .. .., , mk mk are are the the numerical values of of the the terms terms tl(fi), tk(fi). Furthermore finite cut-rank. more this this infinitary infinitary derivation derivation has has finite cut-rank. Proof. just as P r o o f . All All the the cases cases of of PA-rules PA-rules carry carry over over straightforwardly straightforwardly just as in in 4.8, 4.8, but but with with ,. The w w now now replaced replaced by by 55 ."9'. The only only case case we we need need worry worry about about is is the the application application of of the the TI(!) rule. Assume TI(~/) rule. Assume inductively inductively that that its its premises premises are are all all embedded embedded in in w-Arithmetic w-Arithmetic with , . d. with ordinal ordinal bound bound 05 = = 55 ·"7" d. Assume Assume also also that that d d is is chosen chosen large large enough enough so so that that conditions O. Then conditions 11 and and 22 of of the the previous previous Lemma Lemma are are satisfied satisfied by by 5. Then what what we we need need to to prove prove is is 5 ''Y.(d+2) A, Va(a ~ c V B(a)) fK s'~'(d+2) A , 'v'a (a f< c V B(a))

202 202

M. Fairtlough Fairtlo!.lgh and and S. S. Whiner Wainer M.

(we suppress suppress the the parameters parameters occurring occurring in in the the side-formulas side-formulas AA since since they they play play no no (we -< ~' , with with code code active part part in in this this case). case). Now Now the the previous previous lemma lemma gives, gives, for for every every aa -~ active

a, a,

a: a : N ~_6+5.~+3 f-6+5.0+3 A, A, B(a). B(a) .

Let M(a, M(a, n) n) be be aa ~E~ �� -formula expressing expressing the the relation relation "norm(a) "norm (a) -= n", n" , and and recall recall that that Let , itself itself isis coded coded by by the the top top element element cc in in its its number-theoretic number-theoretic well-ordering. well-ordering. Then Then for for we have, have, for for all all a, a, nn EE IN, N, sufficiently large large dd we aa sufficiently Pa A,-~M(a, A, -,M(a, n) n) VV aa ;d -A cc V V B(a) B(a) max(n, max(n, aa)) : : NN Ff--~a

where ~a f3a == (f0 ++ 5. 5 . aa ++ 44 if -< cc and and norm(a) norm(a) == n, n , and and fla f3a == 50 otherwise. otherwise. Hence Hence where if aa -4 . f3a 5 ~[max(n, a)] for for every every aa and and so so by by the the V-rule, V-rule, ,[max(n, a)] & eE 50 ++ 5. 6+5.'Y A, n) VV aa 7~ -A cc VV B(a)) B(a)) A, Va(-~M(a, Va(-,M(a, n) n: n : NN If--'~+5"'~ for every the V-rule V-rule again, again, every n, for n, so so by by the 6+5.'Y+l tf-6+5"'y+' The The desired desired result result

-A cc V A, V B(a)). B(a)). A, VxVa(-~M(a,x) VxVa(-,M(a, x) VV aa 7~

5.'Y. (d+2) A, A, Va(a Va(a -A 7~ cc V V B(a)) B(a)) f-}_5.~.(d+2)

will will then then follow follow by by a a Cut Cut if if we we can can derive derive +5 .'Y+ l 3x3a(M(a, 3 x 3 a ( M ( a , xx)) A1\ aa -< -~ ccA-~B(a)),Va(a ~ c V B(a)). B(a)). 1\ -,B(a)), Va(a -A f-F 66+5"''/+1 This This is is done done as as follows: follows: for for each each aa E E N IN with with norm(a) norm(a) = = m m we we have have (again (again for for aa o M(a, m o -~B(a), B(a) o -~ c, large large enough enough d) d) f~_6 M(a, m)) and and f~_6 aa -< c, aa -A ~ cc and and ft-~ -'B(a), B(a).. Hence Hence 6+3 (M(a, f-~6+3 -,B(a)), (a (M(a, m m)) 1\ A aa -< -4 cc 1\ A-~B(a)), (a -A ~ cc V V B(a) B(a)).) . But But since since the the norm norm function function is is primitive primitive recursive recursive and and , 7 !: ~ w w,, we we can can assume assume that that the the chosen chosen dd is is large large enough enough to to o m :: NN by ensure ensure aa :: N N f~6 m by the the Bounding Bounding Lemma Lemma 3.15. 3.15. Hence Hence 0+5 3x3a(M(a, B(a)) , (a 3x3a(M(a,x) (a -A ~ cc V V B(a)) B(a)) x) 1\A aa -<-4 cc 1\A -,-~B(a)), aa:: NN f-F-~+5 follows follows by by two two applications applications of of the the 3-rule, 3-rule, and and aa final final application application of of the the V-rule V-rule yields yields 6+5.'Y+ 1 3x3a(M(a, -,B(a)) , Va(a 3x3a(M(a,x) Va(a -A ~ cc V V B(a)). B(a)). x) 1\A aa -<-4 cc 1\A-~B(a)), f-i-6+5"'Y+' This This completes completes the the proof. proof.

7.4. Suppose 7.4. Classification Classification Theorem. Theorem. Suppose ,7 = WO w~ is is primitive primitive recursively recursively rep repa) := sUP (1 + ). resentable resentable with with aa primitive primitive recursive recursive norm. norm. Recall Recall €( c(~) sup xx exp� exp,(1 + a ~+ + 11). Then Then PRovREC(PA (,)) = c (€ ( a )) = PaovREc(PA + + TI TI(~/)) = RE REC(e(c~)) = U U E(Fp). E(F~). ~-~(a) P-« (o)

Provably Recursive Functions

203 203

For the the first first containment containment C_ � suppose suppose ff isis provably provably recursive recursive in in PA PA ++ TI(7). TI(r) . PProof. r o o f . For Then for for an an appropriate appropriate computation computation formula formula C CJI we we have have Then CJ ( x, y, y, z). PA z) . TI(r) Ff- Vx3y3z. 'v'x3y3 z . Cl(x, PA ++ TI(7)

The Embedding Embedding Theorem Theorem gives gives (for (for appropriate appropriate r,r, dd EE IN) N) The

f-�''Y'd w'v'x3y3 y3z, z . CJ ( x, y, z) z)

and, and, with with only only minor minor modifications modifications to to the the proof, proof, the the 55 could could be be replaced replaced by by w thus: thus:

w3y3z, c (z, y, z).

By the the Cut-Elimination Cut-Elimination Theorem Theorem 3.11 3.11 we we then then obtain obtain By

F~ Vx3y3z. Ci(x, y, z) f-� 'v'x3y3z . CJ(x, z) (w '.'y. " d) where 58 == exp� -( expr+l(1 exp�+ ! (1 ++ c~ a ++ 1) 1) -( c ( a) , and and hence hence ff eE REc(c(a)) where exp~(w d) -~ -~ e(~), REc(e(c~)).. The second containment follows from the the Hierarchy B(Je <� F~. F(J . The second containment follows from Hierarchy Theorem Theorem 3.19 3.19 since since B For the the third third containment, containment, suppose some ~(3 -( c(a ) . Then Then ff will will For suppose f f Ee E E(F~) for some -~ e(c~). ( F(J ) for be provably recursive PA + and for for some fixed m m E be provably recursive in in PA + TI(~') TI(r) ifif F~ F(J is is and some fixed E N N we we have have exp;:(1 ++ a + 1). 1) . (3 /~ -( -~ exp.(1 c~ + TI(r) we we can can prove TI(a) and TI(1 ++ aa ++ 1). 1 ) . Then Then by But in in PA PA + But + TI('y) prove TI(a) and hence hence TI(1 by iterating part part 88 of the proof proof of of Theorem 4.11 we up iterating of the Theorem 4.11 we can can prove prove transfinite transfinite induction induction up to exp;:(1 + 1) 1) and and hence hence up up to to ~. As before before FZ F(J is then provably provably recursive recursive in to exp.(1 + + aa + is then in (3. As + TI(~'), TI(,) , and proof. PA PA + and this this completes completes the the proof. As example, let n-times iterated iterated inductive As a a concluding concluding example, let 1Dn ID~ be be the the theory theory of of an an n-times inductive Buchholz et definition definition and and let let Tn T~ denote denote its its proof-theoretic proof-theoretic ordinal ordinal (see (see Buchholz et al. al. [1981] [1981] for for detailed detailed analyses analyses of of these these and and other other theories) theories).. Then Then the the T T~'S have structured structured n 'S have tree-ordinal tree-ordinal representations representations and and by by Wainer Whiner [1989] [1989] we we have have T T;; + = -- T Tn+l. Therefore n+ ! ' Therefore PROVREC(IDn) = REC(~'n)= E~

and and letting letting T r = - sup sup Tx T~,, PaOVREC(II~ - CA0) - REC(T) = g0-DEE(T).

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M. H. H. L6B LOB AND AND S. S. SS.. WAINER M. WAINER number theoretic functions II and II, Arkiv fUr [1970] Hierarchies Hierarchies of of number theoretic functions and II, flitmathematische Logik Logik und [1970] Grundlagenforschung, pp. 39-51 vol. 14, Grundlagenforschung, 14, 14, pp. 39-51 and and 97-113. 97-113. Correction Correction in in vol. 14, pages pages 198-199. 198-199. G. E. E. MINTS G. MINTS [1973] 1, pp. [1973] Quantifier-free Quantifier-free and and one one quantifier quantifier systems, systems, Journal of Soviet Mathematics, 1, pp. 71-84. 71-84.

B. PARIS J. B. PARIS [1980] [1980] A A hierarchy hierarchy of of cuts cuts in in models models of of arithmetic, arithmetic, in: in: Model Model Theory of Algebra Algebra and Arithmetic, L. eds., Lecture #834, Springer-Verlag, L. Pacholski Pacholski and and et et al., al., eds., Lecture Notes Notes in in Mathematics Mathematics #834, Springer-Verlag, Berlin, pp. pp. 312-337. Berlin, 312-337. L. HARRINGTON J. B. B. PARIS J. PARIS AND AND L. HARRINGTON [1977] [1977] A A mathematical mathematical incompleteness incompleteness in in Peano Peano Arithmetic, Arithmetic, in: in: Handbook Handbook of Mathematical Barwise, ed., pp. 1133-1142. Logic, Logic, J. Barwise, ed., North-Holland, North-Holland, Amsterdam, Amsterdam, pp. 1133-1142.

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R R.. PETER P~TER [1967] Academic Press, Press, New New York, York, 3rd 3rd ed. ed. [1967] Recursive Functions, Academic z. RATAJCZYK Z. RATAJCZYK [1993] [1993] Subsystems Subsystems of of true true arithmetic arithmetic and and hierarchies hierarchies of of functions, functions, Annals of of Pure and Applied Logic, 64, 64, pp. pp. 95-152. 95-152.

J. ROBBIN J. ROBBIN [1965] Subrecursive Subrecursive Hierarchies, PhD PhD thesis, thesis, Princeton Princeton University. University. [1965]

H H.. E E.. ROSE ROSE vol. 99 of [1984] Subrecursion: Subrecursion: Functions and Hierarchies, vol. of Oxford Oxford Logic Logic Guides, Guides, Clarendon Clarendon [1984] Press, Oxford. Oxford. Press, SCHMERL U U.. SCHMERL [1982] Number Number theory theory and and the the Bachmann-Howard Bachmann-Howard ordinal, ordinal, in: in: Logic Logic Colloquium '81, '81, J. J. Stern, Stern, [1982] ed., ed., North-Holland, North-Holland, Amsterdam, Amsterdam, pp. pp. 287-298. 287-298.

D D.. SCHMIDT SCHMIDT [1976] Built-up Built-up systems systems of of fundamental fundamental sequences sequences and and hierarchies hierarchies of of number-theoretic number-theoretic func func[1976] tions, tions, Arkiv fii flitr mathematische Logik und Grundlagenforschung, Grundlagenforschung, 18, 18, pp. pp. 47-53. 47-53. K K.. SCHUTTE SCHUTTE N. Crossley [1977] Proof Theory, Springer-Verlag, Springer-Verlag, Berlin. Berlin. Translation Translation by by J. J. N. Crossley [1977] CHWICHTENBERG H H.. SSCHWICHTENBERG [1971] [1971] Eine Eine Klassifikation Klassifikation der der eo-rekursiven e0-rekursiven Functionen, Functionen, Zeitschrift fii fiirr mathematische Logik pp. 61-74. und Grundlagen der Mathematik, 17, 17, pp. 61-74. [1977 [1977]] Proof Proof theory: theory: Some Some applications applications of of cut-elimination, cut-elimination, in: in: Handbook of of Mathematical Logic, J. J. Barwise, Barwise, ed., ed., North-Holland, North-Holland, Amsterdam, Amsterdam, pp. pp. 867-895. 867-895. AND S. S. S. S. WAINER H H.. SCHWICHTENBERG SCHWICHTENBERGAND WAINER [1995] Ordinal Ordinal bounds bounds for for programs, programs, in: in: Feasible Feasible Mathematics II, II, P. P. Clote Clote and and J. B. B. Remmel, Remmel, [1995] eds., 13 of eds., vol. vol. 13 of Progress Progress in in Computer Computer Science Science and and Applied Applied Logic, Logic, Birkhiiuser, Birkh~iuser, Boston, Boston, pp. 387-406. pp. 387-406.

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R R,.. SOMMER SOMMER [1992] Ordinal Ordinal arithmetic arithmetic in in I�o IA0,, in: in: Arithmetic, Arithmetic, Proof Theory and Computational Complexity, Complexity, [1992] P. P. Clote Clote and and J. Krajicek, Krajicek, eds., eds., Oxford Oxford University University Press. Press. [1995] [1995] Transfinite Transfinite induction induction within within Peano Peano Arithmetic, Arithmetic, Annals of of Pure and Applied Applied Logic, Logic, 76, 76, pp. pp. 231-289. 231-289. W. W. TAIT W. TAIT Normal derivability in: The Syntax and Semantics of Infinitary [1968] [1968] Normal derivability in in classical classical logic, logic, in: Lecture Notes Notes in Languages, J. J. Barwise, Barwise, ed., ed., Lecture in Mathematics Mathematics #72, #72, Springer-Verlag, Springer-Verlag, Berlin, pp. 204-236. Berlin, pp. 204-236.

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CHAPTER IV IV CHAPTER

Subsystems of of Set Set Theory Theory and Subsystems and Second Order Order Number Number Theory Theory Second Wolfram Pohlers Pohlers Wolfram fiir mathematische mathematische Logik Logik und Grundlagenforschung Institut for West f~lische WilhelmsWilhelms-Universit~t, D-~81~9 Munster, Miinster, Germany Westfiilische Universitiit, D-48149

Contents Contents

1. Preliminaries .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1. Preliminaries 1.1. O rdinals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Ordinals 1.2. PPartial theory .. . . . . . . . . . . . . . . . a r t i a l models models for for axiom axiom systems systems of of set set theory Connections to to subsystems s u b s y s t e m s of of second second order order number n u m b e r theory 1.3. Connections theory . . . . . . . . . M e t h o d s .. .. .. .. .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.4. Methods order number n u m b e r theory 2. First First order theory .. .. .. .. .. . . . . . . . . . . . . . .. .. . .. .. .. .. . . . . . . . . . . . . Peano arithmetic a r i t h m e t i c .. .. .. .. .. . . . . . . . . . . . . . .. .. .. .. .. .. .. .. . . . . . . . . . . . . . . 2.1. Peano Peano aarithmetic r i t h m e t i c with with additional additional transfinite transfinite induction induction .. . . . . . . . . . . . 2.2. Peano Impredicative systems systems .. .. .. .. .. . . . . . . . . . . . . . .. .. .. .. .. .. .. .. . . . . . . . . . . . . 3. Impredicative 3.1. Some remarks on Some remarks on predicativity predicativity and and impredicativity impredicativity . . . . . . . . . . . . . A x i o m systems systems for for number n u m b e r theory 3.2. Axiom theory .. .. .. .. .. . . . . . . . . . . . . . . . . . . . . A x i o m systems systems for for set set theory theory . .. .. .. .. . . . . . . . . . .. .. . . . . . . . . . . . . . 3.3. Axiom 3.4. Ordinal Ordinal analysis analysis for for set-theoretic set-theoretic axioms axioms systems systems . . . . . . . . . . . . . . References References .. .. .. .. .. .. . . . . . . . . . . . . . .. .. .. .. .. .. .. .. .. .. . . . . . . . . . . . . . . . . . . . . .

H A N D B O O K OF OF P R O O F THEORY THEORY HANDBOOK PROOF Edited Edited by by S. S. R. R. Buss Buss © 1998 Elsevier RV. All rights reserved reserved 9 1998 Elsevier Science Science B.V. All rights

210 210 215 215 219 230 231 231 261 266 267 268 279 294 333

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W. w. Pohlers

11.. Preliminaries Preliminaries

The The aim aim of of the the following following contribution contribution is is to to present present a a sample sample of of ordinal ordinal analyses analyses of of subsystems subsystems of of Set Set Theory Theory and and Second Second Order Order Number Number Theory. Theory. 11 But But before before we we start start the presentation of think we should enter discussion about the presentation of results results we we think we should enter a a general general discussion about the the type type of of results results we we are are going going to to obtain. obtain. We We want want to to keep keep close close to to Hilbert Hilbert's's program program and and try try to to give give consistency consistency proofs proofs for for axiom well aware the obstacles axiom systems systems as as constructively constructively as as possible possible - being being well aware of of all all the obstacles to to ' s Theorems. this this enterprise enterprise resulting resulting from from Godel Ghdel's Theorems. The will be The emphasis emphasis will be on on impredicative impredicative subsystems subsystems of of Set Set Theory Theory and and Second Second Order Order Number Number Theory. Theory. Here Here we we opt opt for for a a seemingly seemingly unconventional unconventional approach. approach. We We will to construct of theories will try try to construct partial partial models models of theories within within the the constructible constructible sets. sets. The The line. Therefore hierarchy hierarchy L L of of constructible constructible sets sets is is determined determined by by the the ordinal ordinal line. Therefore special special care care will will be be given given to to notations notations for for ordinals. ordinals. In In the the Preliminaries Preliminaries we we will will introduce introduce our our concept concept of of ordinal ordinal analysis. analysis. First First we we introduce introduce some some basic basic facts facts on on ordinals, ordinals, then then introduce our our concept concept of of partial partial models models and and fi finally show how how this this is is connected connected to to the the introduce nally show more more conventional conventional approach approach to to ordinal ordinal analysis. analysis. 1.1. 1.1. Ordinals Ordinals

Ordinals will play Ordinals will play the the crucial crucial role role for for all all what what follows. follows. Therefore Therefore we we start start with with aa short short introduction introduction to to ordinals ordinals as as we we will will use use them them in in the the following following contribution. contribution. Ordinals their set Ordinals are are regarded regarded in in their set theoretic theoretic sense, sense, i.e., i.e., an an ordinal ordinal is is a a hereditarily hereditarily transitive transitive set. set. Assuming Assuming that that the the membership membership relation relation E E is is a a well-founded well-founded relation, relation, this this entails entails that that every every ordinal ordinal is is well-founded well-founded with with respect respect to to the the membership membership relation relation and its E-predecessors. and every every ordinal ordinal is is the the set set of of its E-predecessors. For For the the reader reader who who is is not not so so familiar familiar with with set set theory theory we we give give a a brief brief sketch sketch of of the the theory theory of of ordinals ordinals as as far far as as it it will will be be which may needed needed for for this this paper. paper. Besides Besides transfinite transfinite recursion recursion ((which may be be regarded regarded as as a a generalization of generalization of primitive primitive recursion recursion)) all all we we need need from from Set Set Theory Theory are are the the facts facts (Oni) (On1) -- (On4) below. They (One) below. They may may be be viewed viewed as as axioms axioms for for the the theory. theory. To To make make the the article article not long we here. Detailed not too too long we will will not not give give proofs proofs here. Detailed information information how how to to prove prove the the results results of of this this section section from from (Oni) (On1) - (On4) (One) can can be be found found in in Pohlers Pohlers [1989] [1989].. infinite A A linear linear order order relation relation -< -~ well-orders well-orders its its field field iff iff it it does does not not contain contain infinite -<-descending -~-descending sequences. sequences. A A class class M M is is transitive transitive iff iff aa E EM M =} =~ aa � C_ M. M.

(Oni) (On1) The The class class On O n of of ordinals ordinals is is aa non non void void transitive transitive class, class, which which is is well-ordered well-ordered by by the the membership membership relation relation E E.. We We define define a ~ < < f3 ~ as as a c~ E E On O n /\ A f3 ~ E E On O n /\ Aa c~ E E f3. ~. In In general general we we use use lower lower case case Greek Greek letters letters as as syntactical syntactical variables variables for for ordinals. ordinals. The The well-foundedness well-foundedness of of E E on on the the class class On O n implies implies the the principle principle of of transfinite transfinite induction induction

(� ) ] =} On)[(Vr/< =~ F F(~)] =~ ((V~ O n ) F ( ~�)) ((V~ )F(1]) =} \I� EE On)F( [ (\l1] < �~)F(~) \I� EE On) 11iI am indebted to Dr. Arnold Arnold Beckmann Beckmann for proofreading. proofreading. He not only only detected a series series of errors in the first versions versions but also made many valuable valuable suggestions. suggestions.

2211 11

Set Set Theory Theory and Second Order Number Number Theory

and and transfinite transfinite recursion recursion which, which, for for aa given given function function gg,, allows allows the the defi definition nition of of a a function function f f satisfying satisfying the the recursion recursion equation equation

}). f( {j(�) I �~ < 1]) = f(r/) -- g( g({f(~)[ < 1]~}). J . The (On2) e., (V� (31] EE On))[� (On2) The The class class On O n of of ordinals ordinals is is unbounded, unbounded, i.i.e., (V~ E E On) On)(3r] On))[~ < < 1] r/]. The cardinality cardinality IMI IMI of of aa set set M M is is the the least least ordinal ordinal a ~ such such that that M M can can be be mapped mapped bijectively An ordinal bijectively onto onto a. ~. An ordinal a ~ is is aa cardinal cardinal if if lal Ic~l = - a c~.. (On3) M I EE O (On3) If If M M � C O O nn and and IIMI Onn then then M M is is bounded bounded in in On, On, i.e., i.e., there there is is an an a a E E On On such such that that M M � C a. ~. For every every ordinal ordinal a a we we have have by by (Onl) (On1) and and (On2) (On2) a a least least ordinal ordinal a' c~' which which is is bigger bigger For than a. We a. There than c~. We call call 0:' a' the the successor successor of of c~. There are are three three types types of of ordinals: ordinals: •

9 the the least least ordinal ordinal 0, O,

•

', e., ordinals 9 successor successor ordinals, ordinals, i.i.e., ordinals of of the the form form a c~',

•

9 ordinals ordinals which which are are neither nor successor successor ordinals. ordinals. Such Such ordinals ordinals are are called called limit limit neither 00 nor ordinals. denote the ordinals. We We denote the class class of of limit limit ordinals ordinals by by Lim Lira..

Considering Considering these these three three types types of of ordinals ordinals we we reformulate reformulate transfinite transfinite induction induction and and recursion as recursion as follows: follows:

Transfinite E On)[F(a) => Transfinite induction: induction: If If F(O) F(O) and and (Va (Vc~EOn)[F(c~) =V F(a')J F(c~')] as as well well as as (V� )F(�) => ) for >. E Lim then (V� E On)F(�) . (V~ < < >' ,~)F(~) => F( F(A) for ,~ E Lira then (V~ E On)F(~). >' Transfinite g, hh there Transfinite recursion: recursion: For For given given a c~ E E On On and and functions functions g, there is is aa function function f f satisfying satisfying the the recursion recursion equations equations f(O) f (O) = - a e~ f( = f (~') g(:(~)) O = g(J(�)) >. } ) for >.A Ee Lim f(>.) {j(1]) I 1]rl < f ( A ) - = h( h({f(r/) < ,~}) Lim..

An ordinal If,~ satisfying An ordinal satisfying

(Rl) (R1) (R2) (R2)

If, ~

Ee Lira Lim If in If"~, i.i.e., e., there is an If M M � C_ If, ~ and and IMI [M] < < If, ~ then then M M is is bounded bounded in there is an a a E E that that M M � C a a is is called called regular. regular. The The class class of of regular regular ordinals ordinals is is denoted denoted by by lR. JR.

If,

~ such such

J. (On4) e., (V� (31] E 1R) [� :::; (On4) The The class class IR R is is unbounded, unbounded, i.i.e., (V~ E E On) On)(3r/E R)[~ _ 1] r/].

We We define define sup M M := "= min min {� {~ E E On On]l ((Mr/E M ) ( 1] r / <:::; �)} ~)} '11] E M)( sup

as as the the least least upper upper bound bound for for aa set set M M � C_ On. On. In In set set theoretic theoretic terms terms it it is is sup sup M M = - U U M. M. It M is It follows follows that that sup sup M is either either the the biggest biggest ordinal ordinal in in M M,, i.e., i.e., sup sup M M = - max max M M,, or or sup sup M M E E Lim Lira.. By By W w we we denote denote the the least least limit limit ordinal. ordinal. It It exists exists according according to to (On4) (On4) and (Onl). The ordinal W and (On1). The ordinal WlI denotes denotes the the first first uncountable uncountable ordinal, ordinal, Le., i.e., the the first first ordinal ordinal whose whose cardinality cardinality is is bigger bigger than than that that of of w w.. It It exists exists by by (On3). (On3).

W. Pohlers

212 212

For every every class class M M C_ � OOn there isis aa uniquely uniquely determined determined transitive transitive class class For n there otyp(M) C_ On and an order preserving function enM: otyp(M) onto> M. The function otyp(M) � On and an order preserving function enM: otyp(M) � M . The function enM enumerates enumerates the the elements elements of of M M in in increasing increasing order. order. Since Since otyp(M) otyp(M) isis transitive transitive enM it We call it is is either either otyp(M) otyp(M) == On On or or otyp(M) otyp(M) EE On. On. We call otyp(M) otyp(M) the the order order type type of of M. M. In fact In fact otyp(M) otyp(M) is is the the Mostowski Mostowski collapse collapse of of M M and and enM enM the the inverse inverse of of the the collapsing collapsing 7r ) . By By (On3)we (On3) we have have otyp(M) otyp(M) EE On On iff iff M M is is bounded bounded function (usually (usually denoted denoted by by ~r). function in ordinals have order type type On. in On. On. Unbounded, Unbounded, i.e., i.e., proper proper classes classes of of ordinals have order On. If If M M is is aa set set of ordinals ordinals then then otyp(M) otyp(M) EE On. On. of If M n an an order On order preserving preserving function function then then we we If M is is aa transitive transitive class class and and f: J: M M --+~ O :::; fJ(a) for all all aa EE M. M. have aa <_ have (a) for A every class A class class M M is is closed closed (in (in aa regular regular ordinal ordinal a) K,) iff iff sup sup N N EE M M holds holds for for every class N We call f M �M M such such that that IN INII EE On On (IN[ (INI << a). K,). We call M M club club (in (in aK,)) i fiff M is is closed closed and and N C_ unbounded (in (in a). K,). unbounded On (K,-) continuous continuous iff M is is We call call an order preserving preserving function function f" J: M M --+~ O We an order n (a-) iff M (K,-) closed and suprema, i.e., i.e., sup N } -= Jf(sup(N)) (sup(N)) for for any (a-) closed and fJ preserves preserves suprema, sup {J(�) {f(~)[ I �~ E E N} any N� M such such that that INI INI eE On On (INI ( INI << a). K,). N C_ M A normal (R-normal) is an an order-preserving A normal (K,-normal) function function is order-preserving continuous continuous function function On J: K,a --+~ K,a respectively. respectively. J: On f: O n --+> O n or or f:

�O On (M C_ � a) K,) the enumerating function function enM enM is is aa (K,-)normal For M c_ For M n (M the enumerating (a-)normal function function iff iff M M is club (in (in a). K,). is club Extending their their primitive definitions continuously the transfinite Extending primitive recursive recursive definitions continuously into into the transfinite we the basic functions 4-, +, . 9and and exponentiation all ordinals. ordinals. we obtain obtain the basic arithmetical arithmetical functions exponentiation for for all The ordinal ordinal sum, sum, for example, satisfies the recursion equations The for example, satisfies the recursion equations aa ++OO= a= a = (a (~+/~)' + {3)' a~ ++/ ~{3'' = aa + �) Jor + A A= = sup sup~ < ~(a + + ~) :for AA E e Lim Lira.. { < ).(a It It iiss easy easy to to see see that that the the function function A� A(.. a a + 4- �( is is the the enumerating enumerating function function of of the the class class {{(� EE On] On I a club in Hence A� a :::; < 0 (} which which is is club in all all regular regular K, a > > a. a. Hence A(.. a a + 4- � ( is is aa K,-normal R-normal a. We function function for for all all regular regular K, a > > a. We define define [� + (V'1] < a) H := {~a E e On On II a o~# # 0 1\ ^ (V'� (V~ < a) oL)(Vrl o0[~ + 1]r/<< a] ~]}} := { lHI and call the ordinals in club (in and call the ordinals in lHI H additively additively indecomposable. indecomposable. Then Then lHI H is is club (in any any regular regular lHI = ordinal (O) = ordinal > > w), w), 11 := 0' 0' E E lHI H,, w w E E lHI H and and w w rnq H = {I}. {1}. Hence Hence enl8l end(0) = 11 and and rst two enl8l(l) en~(1) = = w w which which are are the the fi first two examples examples of of the the fact fact that that {]. (V'� [enl8l (�) = (V~c E E On) On)[en~(~) = w w~]. Thus Thus A� A(.. w{ w~ is is aa (K,-)normal (a-)normal function function (for (for all all K, a E E lR R bigger bigger than than w). w). We We have have

lHI ][-][ � c_ Lim Lira U {l} {1} and and obtain obtain

a Ee lHI [� + M / i /iff ( v ~(V'� < < a) o0[,~ + a o~ = = a]. 04.

(1)

213 213

Set Set Theory Theory and and Second Second Order Order Number Number Theory Theory Thus for for aa finite finite set set {at { a l ,, ... .. ,. , an an}} � C_lHl H we we get get Thus

..+ at m al + + . .. .. . + + an an = = ak akll + + ."'" + ak akin

�

k, _> ak for for {{kk1l ,, .. ... ., , kkin} { 1,, .. .. .., , n} n} such such that t h a t ki ki < < ki ki+l and a ak~ ak~+l. By induction induction on on m } �C_ {I i+l . By + ! and aa we we obtain obtain thus thus ordinals ordinals {at { a l ,,... .. ,. , an an }} � C_ lHl H such such that t h a t for for a a =f. r 00 we we have have

(2) (2)

a = al + " " + an and al > _ ' " >_ an.

This is is obvious obvious for for a a E E lHl H and and immediate immediate from from the the induction induction hypothesis hypothesis and and the the This above above remark r e m a r k if if a a = = �~ + + 1]77for for �, ~, 1]77< < a. a. It It follows follows by by induction induction on on nn that t h a t the the ordinals ordinals al a l ,l. .. .. ,. , an an in in (2) (2) are are uniquely uniquely determined. determined. We We therefore therefore define define an an additive additive normal normal

form form

aa ==NF N F aa l +t .+ . . .+.a·+an n'r :

¢:}

aa ==a l a {at + .t .+ . +. a. n·+an, , { a t ,,. .. .. ,.a, nan } C}_� H lHl and and a alt

�>_ .".". �>_an. an.

We We call call {at { a l ,,. .. .. ,. , an an }} the the set set of of additive additive components components of of a a if if a a = =NF al + + ' '.' . . + + an an.. NF at We ne the We use use the the additive additive components components to to defi define the symmetric symmetric sum sum of of ordinals ordinals a a = =NF NF . . . by a + a ! + NF an a lt + + ' '.' . . + + an an and and (3 fl = =NF an+l + " " + am by m + . . . aa #~ (3fl := (m) (t) + "= a,, a~(1) +'" + + a,, a~Cm)

where 11" lr is is aa permutation p e r m u t a t i o n of of the the numbers numbers {I, { 1 , . .. .., , m m}} such such that that where

11 :<::; _< ii < < jj :<::; _< m m => =~ a,, a~((i0)

(j) . a~(j). �___a,..

In In contrast contrast to to the the "ordinary "ordinary ordinal ordinal sum" sum" the the symmetric symmetric sum sum does does not not cancel cancel additive additive components. By By definition definition we we have have components. It is is easy easy to check tthat the symmetric symmetric sum sum is order preserving in its its bboth It to check h a t the is order preserving in o t h arguments. arguments. normal form for ordinals ordinals for As another consequence of (2) we As another consequence of (2) we obtain obtain the the Cantor Cantor normal f o r m for for the basis basis w w,, which which says says that t h a t for every ordinal a r=f. 00 there the for every ordinal a there are are ordinals ordinals ~6,1 , .. .. .., ' ~n such �n such tthat hat 6

{n . w~'~ ++ ". '." . ++ ww~". aa ==NNF F w w{r isis aa normal normal function function we we have :<::; w WO~ for for all all ordinals ordinals a. a. We call aa an an Since ~.. w have aa _< We call Since ).,\� eE-number - n u m b e r if if w WO~ == aa and and define define min {{ aa IJ w" w O == aa} rEo ":= - min }.. E enumerate enumerate the the fixed fixed points points of of ).,\� w{r .. IfIf we we put put more generally generally let let ).,\� more ~.. Q ~.. w {

n+ t (a , (3) := a expn (o, ,B) expo (a, (3) ::= = (3 and exp Z) :=

we we obtain obtain

n

exp (w, 0). O) . EO "= := sup sup expn(w, ~o n <w

n(og

Eo we we have have aa << wWO~ and and obtain obtain by by the the Cantor Cantor Normal Normal Form Form TTheorem For 00 << aa << Co For heorem uniquely l , . .. .. ,. , an h a t aa =NF . . . ++ wwOan.n . uniquely determined determined ordinals ordinals aat, an << aa such such tthat W0al1 ++ "'" =NF w

214 214

W. W. Pohlers Pohlers

For class M ne its For aa class M � c On On we we defi define its derivative derivative

M' M' := := {� {~ E e an On[i

enM(�) enM(~) = = o ~}.·

The = enF The derivative derivative J' f ' of of aa function function J f is is defined defined by by J' f ' ::= enFix(/), where ix(J) , where Fix( Fix(f) : - {� {r I J(�) f(r = = o r · j ) := Thus club (in (in some Thus J' f ' enumerates enumerates the the fixed-points fixed-points of of J f. . If If M M is is club some regular regular ",) ~) then then M' M' is also club (in "'~).) . Thus is also club (in Thus if if J f is is aa normal normal function function l' f' is is aa normal normal function, function, too. too. I} is If If {M. {M, II tt E e I} is aa collections collections of of classes classes club club (in (in some some regular regular ",) ~) and and III III E e On On ). (III ([I[ E e "') ~) then then n ~,e, M~ is is also also club club (in (in "' ~). .EI M. These These facts facts give give raise raise to to aa hierarchy hierarchy of of club club classes. classes. We We define define

Cr(O) c (o) ::= = lHl H : = Cr(a') := Cr(a)' Cr('\) := : = n{ < A Cr(O c r ( r Jor f o r ,\ E e Lim Lira.. If If we we put put ~ ::= encr(~), CP cr(o ) , a - en

then then all all CPo 9~a are are normal normal functions functions and and we we have have by by definition definition a < fl => ~oa(~P~7) = ~ZT.

(3) (3)

The The function function cP ~ is is commonly commonly called called Veblen Veblen function. function. From From (3) (3) we we obtain obtain immediately immediately a O/11 < < a C~22 and and /31 ~1 :::; ~_~CP02/32 (~Ot2~ or or a1 al = = a a22 and and /31 131 :::; < /32 ~2 or a or a22 < < a al1 and and rCPo I /31 :::; < /32 f12.· We We define define the the Veblen Veblen normal normal form form for for ordinals ordinals CP ~ (TI d / bby y CP (PC~I~I ~-~ CP02/3 (~cx2~2 iff ig oJ31 :::;

(4) (4)

a a and 'fJ7"1< < a. a = =NF ~ :: {:} r a = = ~CP('l and a. NF CP('l

and aa = Then J2 => {I 'fJ1 and Then a a = = NNF F CP (P~IT]I = NNFF CP6'f (P~2~2 ~ 6 r = "-- 6 r and and 'fJ1 771 = = 'fJ2 ?72.. Since Since �~ < < a a and and 'fJ < a) implies implies cp a) the < /3 ~ E E Cr( Cr(a) ~ r{'fJ < /3 we we call call Cr( Cr(a) the class class of of a-critical a-critical ordinals.. ordinals.. If If -critical then => cp{'fJ < a. a. Therefore a a is is itself itself a a-critical then �, ~, 'fJ r / << a a =v qoer/< Therefore we we define define the the class class SC SC of of strongly critical ordinals strongly critical ordinals by by SC := := {a SC {a E e ani On I a ~ E e Cr(a) C~(a) }}..

The C is The class class SSC is club club (in (in all all regular regular ordinals ordinals '" ~ > > w). w). Its Its enumerating enumerating function function is is denoted denoted by by ,\� A~.. f{ F~.. Regarding Regarding that that by by (4) (4) ,\� A~.. cp{O ~,0 is is order order preserving preserving one one easily easily proves proves sc =

=

:= 00 and := CP9~."(n 00 then If define 'Yo If we we define 7o := and 'Yn %+1 then we we obtain obtain +1 := fa F0 = = sup sup 'Yn 7-.· nn <<ww

215 215

Second Order Order Number Theory Set Theory and Second

For every every a a < r Fo0 there there are are uniquely uniquely determined determined ordinals ordinals 6 ~1,... and For , . . . ,,~n �n < aa and l , . . . , 'f/n r/n < a a such such that that 'f/~h,...,

--,F (P~lrh + . . . + cP~nrl,~ and rii < ~a~,rli for i E { 1 , . . . , n}.

(5) (5)

This is is all all we we need need to to know know about about ordinals ordinals for for the the moment. moment. We We will will have have to to come come This back to to the the theory theory later. later. back 1.2. Partial P a r t i a l models m o d e l s for for axiom a x i o m systems s y s t e m s of of set set theory theory 1.2.

Let E) denote Let £( /::(E) denote aa language language of of Set Set Theory, Theory, i.e., i.e., aa first first order order language language which which contains aa symbol symbol E E for for the the membership membership relation. relation. Later Later we we will will add add aa unary unary relation relation contains symbol Ad. Ad. symbol We will will distinguish between restricted restricted quantifiers quantifiers ((Vx and (::Ix (3x E E aa)) and and We distinguish between Vx EE aa)) and unrestricted quantifiers quantifiers of of the the form form ((Vx) and (::Ix) (3x).. unrestricted Vx) and Recall the the Levy Levy hierarchy hierarchy of of/::-formulas. We say say that that aa formula formula is is L).o A0 if if it it Recall £-formulas. We contains only only restricted restricted quantifiers. quantifiers. A A formula formula F F of of/:: is aa IIn-formula, Hn-formula, if if contains £ is F = (VXl)(3x2)... (Q~xn)a(x~,... ,x~), where G(xl,... is a a L).oA0- formula formula and and V::I V3..-(Qn) an alternating alternating string string of of un un· . (Qn) an where G(X xn) is l > " " ,xn) restricted quantifiers. Dually, formula is is �n En if if of -~F is is logically logically equivalent equivalent to to aa restricted quantifiers. Dually, aa formula 1-In-formula. IIn -formula. A formula formula F(X F ( x ll,'... .. . ,,xn) is a a L).n-formula An-formula of of an an £( s E)-structure !21 92 iff iff there there are are aa A xn) is xn) such (x l > ' " ,,xn) xn) and , ' " ,,x~) 1-In-formula Frr FH(Xl,... and aa �n-formula En-formula Fdxl Fr.(Xl,... such that that IIn-formula .

x x

(x))

(F(x) Fdx)).

FIT 91 A1\ (VX) (VZ)(F(Z) +4 V ) (F( ) +4 f-+ Fs(.~)). f-+ .Fn(.~)) 2t I= 1= ((VZ)(F(Z)

(6) (6)

We call call F x iff x proves proves the the formula (6).. We F aa An-formula L).n-formula of of aa theory theory A Ax iff A Ax formula in in (6) The class of is the smallest class contains the -formulas and and The class of �-formulas E-formulas is the smallest class which which contains the L).o A0-formulas is closed closed under restricted quantification and and A, 1\ , restricted quantification and is under the the positive positive boolean boolean operations operations V and unrestricted quantification. The The class of II II formulas the dual unrestricted existential existential quantification. class of formulas is is the dual of of the the class class there is is aa E-formula �-formula of x l ,,... .. ,. , xn) 92 if if there of E-formulas. �-formulas. A A formula formula FF((Xl xn) is is A L). in in aa structure structure !21 (V ) [F( ) ++ such that that !21 ( )] Fdx , xn ) and FE(Xl,... 92 1= ~ (VZ)[F(Z) and aa H-formula II-formula Fii(Xl,... (Xl > . . . ,,Xn) xn) such t+ Fr~(Z)] l ' . . . ,xn) and !21 (Vx) [F(x) ++ f-+ Fn(Z)]. Frr(x)] . It is A L). for for aa theory theory Ax iff itit is models of and 91 1= ~ (VZ)[F(Z) It is A x iff is L). A for for all all models of Ax. Ax. Recall that that the constructible hierarchy hierarchy L L is is the the union union of of its its stages L= given Recall the constructible stages La given by by the the following following definition. definition.

Frr

11.2.1. .2.1. D efinition. Definition.

Lo La+1 La+! == Def(L~) Def(La) LL~A == U{L~I U{Ld f� << ,~} .x} for for A.x EE Lim, Lim,

x x

Fr: x

L0 -= @ 0

where iff there an /~(E)-formula F(x, yl,...,yn) where aa EE Def(La) Def(La) iff there is is an £( E)-formula F(x, Yl , ' . . , Yn) with with no no other such that other free free variables variables and and aa tuple tuple (bl,...,bn) (bl , . . . , bn) of of elements elements of of L~ La such that

F(s, bl > " . , bn) } . La k1= F(s,b~,...,bn)}. a -= {seL=l { s E La I L,

216 216

w. Pohlers Pohlers W.

The The formula formula FFzZ isis obtained obtained from from FF by by restricting restricting all all unrestricted unrestricted quantifiers quantifiers in in FF to z, z , i.e., Le., replacing replacing (Vx) (V'x) by and (3x) (3x) by by (3x (3x eE z), z) , respectively. respectively. The The formula formula z) and to by (Vx (V'x eE z) obviously A0. FzZ isis obviously F flo . We say say that that aa formula formula G G isis II~ IT� or or E,~ �� ifif We

G- = FL~ FL� G for ITn-formula FF or or En-formula �n-formula F, F, respectively. respectively. for aa YIn-formula Let Ax Ax be be an an axiom axiom system system formulated formulated in in the the language language s£. We define Let We define min {hi {a l L~ II Ax lloo := := min La ~F Ax} [IAx[Ioo Ax}

(7) (7)

the c~-norm oo-norm of of Ax. Ax. By By Lbwenheim-Skolem Lowenheim-Skolem downwards downwards and and the and the and call call I[Ax[Ioo II Ax lloo the Condensation Lemma for the the constructible constructible hierarchy hierarchy we Condensation Lemma for we know know that that [[Ax[Ioo is aa II Ax lloo is countable for our countable ordinal ordinal which which may may be be quite quite big. big. Much Much too too big big for our purpose purpose as as we we will will in aa moment. To obtain ordinals which be characteristic characteristic for for Ax Ax we see moment. To obtain smaller smaller ordinals which may may be we see in regard partial models for For any any complexity class $" we define define :F of/:-sentences of £-sentences we regard partial models for Ax. Ax. For complexity class /\ Ax ConF(Ax) := { F I FF eE :F ConT(Ax) $" A A x e� - FF} } : = {FI

(8) (8)

min {{a[ a l L~ La ~F Con~(Ax)}. I[Axl[~ := min II Ax II F := ConF(Ax) } .

(9) (9)

and and The complexity of of the the sentences sentences in in the the axiom axiom systems The complexity systems which which we we are are going going to to study study will never never exceed Therefore we we always always have will exceed ITa 113.. Therefore have II Ax l l IIa = II Ax lloo . IIAxll = IIAxlloo. We will only only consider consider axiom axiom systems systems which which comprise basis theory We will comprise the the following following basis theory KP KPwhose whose axioms axioms are: are: The The ontological ontological axioms axioms of of Extensionality Extensionality and and Foundation Foundation

(x Ee v) ((Vu)(Vv)[u (x Ee u)] = vv +-+ ~ (V'x (Vx E e u) u)(x v) /\ A ((Vx v)(x u)] V'x Ee v) V'u) (V'v)[u = (Found) [(3x Ee u) (x Ee u) (V'z Ee x)--,(z (Found) (V'u) (Vu)[(3x u)(x u) -+ --+ (3x (3x E e u) u)(Vz x)~(z E e u)] u)].. The Union The closure closure axioms axioms of of Pairing Pairing and and Union (Pair) (V'v) (3w) (V'x)[x Ee w (V'u) (Pair) (Vu)(Vv)(3w)(Vx)[x w +-+ ~ x x = = u u V V x x = = v] v] (x Ee y)] (Union) (Union) ((Vu)(3w)(Vx)[x w ~+-+ (3y (3y E e u) u)(x y)] V'u) (3w)(V'x)[x Ee w The The set set existence existence axiom axiom schemes schemes of of absolute absolute Separation Separation and and Collection Collection ((A0-Sep) flo-Sep) ((Vg)(Va)(3z)(Vx)[x ~ x x E e aa /\ A F(x, F ( x , v)] ~*)] 'v'V) (V'a) (3z) (V'x)[x Ee zz +-+ ((A0-Col) flo-Col) ((WT)(Vu)[(Vx (V'x Ee u) (3y Ee z)F(x, u ) ((3y)F(x, 3 y ) F ( x , y, y, V) g) -+ --+ (3z) (3z)(Vx u)(3y z ) F ( x , y, y, v)] ~7)].. 'v'V) (V'u)[(V'x Ee u) In o -formulas F(x, In both both schemes schemes we we allow allow only only fl A0-formulas F(x, v) g) or or F(x, F ( x , y, y, V) g) of of the the language language (Ext) (Ext)

s E) , respectively. respectively. This This requirement requirement guarantees guarantees the the absoluteness absoluteness of of the the sets sets whose whose £( existence existence is is postulated. postulated. We E.g. We use use the the abbreviations abbreviations which which are are common common in in Set Set Theory. Theory. E.g. aa = [x Ee uu /\A F(x)] [F(x) -+ = {x {x E eu u[l F(x) F(x)}} stands stands for for ((Vx a)[x F(x)] /\ A ((Vx u)[F(x) ~ x x E e a] a];; V'x Ee a) V'x Ee u)

Set Theory Theory and Second Second Order Order Number Theory

217 217

{ x E9 u lI F(x) F(x)}} E9 bb sstands t a n d s f for o r (3y)[y (3y)[y = = {x { x E9 u lI F(x) F(x)}} /\ A yy E9 b] b],, etc. etc. Lower Lower case case {x Greek Greek letters letters are are supposed supposed to to range range over over ordinals. ordinals. Adding Adding the the ontological ontological axiom axiom of of Infinity Infinity

Inf) ((Inf)

(3u)[0 {x} Ee u)] (3u)[0 E e u ^/\ (Vx (w E e u) (x U u {z} u)]

we obtain obtain the the theory theory KPw K P w -- and and adding adding the the Foundation Foundation Scheme Scheme we

(3x)F(x) -+ ~ (3x) (3x)[F(x) FOUND ) (3x)F(x) [F(x) ((FOUND)

/\ A (Vy (Vy E e x) x)-~F(y)] -,F(y)]

we we obtain obtain the the theories theories KP K P or or KPw K P w respectively. respectively. If If we we restrict restrict the the formula formula F(x) F(x) in ((FOUND) to aa complexity complexity class class F $" we we talk talk about about F-Foundation, S'-Foundation, denoted denoted by by in FOUND ) to ((gv-FOUND). F-FOUND) . An ordinal a is is called admissible if if L L~a F ~ KP K P .. An ordinal a called admissible The theories KP and KPw are profoundly The theories K P and K P w are profoundly studied studied in in Barwise Barwise [1975]. [1975]. They They prove prove �-recursion a very important theorem and �-reflection, i.e., the scheme 2-recursion- a very important t h e o r e m - and E-reflection, i.e., the scheme F -+ (3x)F (3x)F X~ F

for F. As both theories for �-formulas E-formulas F. As a a consequence consequence both theories prove prove the the equivalence equivalence of of any any �-formula E-formula to to aa �l-formula. El-formula. -partial recursive f: L" Recall Recall that that a a partial partial function function f: L~ --+p >p L" L~ is is called called "' a-partial recursive if if its its F(x, y, graph graph is is �-definable E-definable over over L" L~ i.e., i.e., if if we we have have aa �-formula E-formula F(x, y, Z) ~ without without further further free free variables variables and and a a tuple tuple c g of of elements elements of of L" L~ such such that that

ff (a) (a) � "~ bb iff iff L" L,~ F ~ F[a, F[a, b, b, Cj c-]

for a, bb EE L ordinals are generalized recursion for all all a, L~. Admissible ordinals are important important for for generalized recursion theory theory " . Admissible because closed under because L" L~ is is closed under all all ",-recursive a-recursive functions functions if if ", a is is admissible. admissible. Obviously Obviously w w is is an an admissible admissible ordinal ordinal and and it it is is aa folklore folklore result result that that WfK w~K,, i.e., i.e., the the least least countable countable ordinal ordinal which which cannot cannot be be represented represented by by aa recursive recursive well-ordering well-ordering on on the the natural natural numbers, numbers, is is the the next next admissible admissible ordinal. ordinal. Therefore Therefore LwC F L~cK ~ KPw KPw 1 K

and, admissibles between KPw 11 00 = and, since since there there are are no no further further admissibles between w w and and WfK w~K,, even even IIIIKP~I[o~ WfK KPw ll II 2 :::; KPw l l E l :::; w~K.. Hence Hence IIIIgPwtlH2 < WfK w~K as as well well as as IIiigPwll~l < WfK w~K.. For For most most impredicative impredicative introduce the theories Ax ll E , . Therefore theories Ax, Ax, however, however, we we have have WfK w~K < < II[[Ax[l~,. Therefore we we need need to to introduce the following following ordinals. ordinals.

1.2.2. 1.2.2. Definition. Definition. Let Let Ax A x be be a a theory theory which which contains contains KPK P - . . Then Then we we define define Ax llEf := [F is � l-sentence /\ := min min {{aa lI (VF) (VF)[F is a a El-sentence A Ax A x e� - F F LT'�~ =} =~ L L~a F ~ F] F]}} IIiiAxilr.~ and and analogously analogously

� =} a l (VF) [F is Ax IIII� "= := min min {{at (VF)[F is a a II II2-sentence A Ax A x e� - F F LL~ =~ L L~a F ~ F] El}} 2 -sentence /\ IIIHAxlHH~ The notation Ax be read The notation Ax � ~- F F L~'�~ has has to to be read with with the the necessary necessary care. care. It It anticipates anticipates that that L" L~ can can be be defined defined in in some some way way from from the the axioms axioms in in Ax. Ax. We We need need �-recursion E-recursion to to

218 218

W. Pohlers Pohlers W.

define the the function function a a f-t ~ La L~.. To To prove prove the the �-recursion E-recursion theorem theorem it it suffices suffices to to have have define KP-FOUND. So for theories extending KP+ � -FOUND we need only KP- + + � El-FOUND. So for theories extending K P + El-FOUND we need only l l description of of a a in in order order to to characterize characterize La La.. We We say say that that Ax A x believes believes that that K,~ is is aa description admissible iff iff Lit L~ is is definable definable in in Ax A x and and admissible

(10) (10)

Ax

holds -reflection holds for for all all sentences sentences G G E E KPK P - . . We We say say that that Ax A x proves proves the the LL- or or Lit L~-reflection rule for for aa complexity complexity class class F ~" iff iff rule A x f~- G a => ~ Ax A x f~- (3,) (3~)(3u)[u = L-y L~ /\ ^ GU a ~]] (3u) [u = Ax or or

A x ef- GL• a '- => Ax A x ef- ((3,) (3u)[u = = L-y /\ ^ Gu ]) L. , Ax respectively, for for all all formulas formulas G G EF .T.. The The following following lemma lemma iiss aa first first easy easy observation observation respectively, about partial models. models. about partial

Ax lloo] be 1.2.3. Lemma. L e m m a . Let Let K, aE E (w (w,, IIIIAxII~] be an an ordinal ordinal such such that that Ax A x believes believes that that K,a is is 1.2.3. Ax II II� = proves the admissible and Ax admissible and A x proves the Lit-reflection L~-reflection rule rule for for � El-formulas. Then IIIlhxll,~ l -formulas. Then Ax II Efr.· IIIIAxll -

need only show the the converse Ax llEf ::; Ax I l II� and Proof. P r o o f . We We obviously obviously have have IIIIAxII~ ~ IIIIAxllr~ and need only to to show converse Ax II Ef ' let (3y)F(x, y) be aa Ih-sentence inequality. Thus inequality. Thus put put a a := : - II[]hxIl~.?, let ('v'x) (Vx)(3y)F(x, y) be II2-sentence such such that that h x ~f- ('Ix (Vx E E LIt) L~)(3y L~)F(x,y) and choose some aa E E La L,.. We We have have to to show show that that Ax (3y EE LIt)F(x, y) and choose some L~ F ~ (3y)F(a, (3y)F(a, y) y).. Since Since a a is is obviously obviously a a limit limit ordinal ordinal there there is is a a j3 ~ < a a such such that that La L~. By By definition definition of of a a there there is is a a El-sentence G such such that that Ax A x f~- G G L• L~ but but Lfj LZ � ~: G. G. aa EE Lfj. � l -sentence G Since A x proves proves L L~-reflection we obtain obtain Ax A x f~-(3-y L~)(3u L ~ ) ( u -= L-y L~ /\ A GU G u)) Since Ax (3, EE LIt) (3u EE LIt)(u it -reflection we and since Ax A x believes admissible by A0-collection relativized L~ also and since believes that that aK, is is admissible by �o-collection relativized to to Lit also hAx x ~ L~ -+ --+ (3z - uu we get (3z EE L~)(Vx LIt) ('v'x EE v)(3y v) (3y EE z)F(x, z)F(x, y). y). Choosing Choosing vv = we thus thus get f- vv EE Lit

GU~ A A x ~L~)(3u = L~ AG (Vx E y)]. /\ ('Ix E u)(3y u) (3y EE z)F(x, z)F(x, y)]. f- (3~/E (3, E LIt) (3u EE L~)(3z LIt) (3z E L~)[u LIt) [u = L-y /\ Ax Since K, is is admissible admissible this this is equivalent to �l -sentence relativized relativized Since Ax A x believes believes that that ~ is equivalent to a a El-sentence to Lit to L~.. Hence Hence L~ G~ A (Vx eE u)(3y z)F(x, y)]. Because La ~F (37)(3u)(3z)[u (3,) (3u) (3z) [u == L~ L-y A /\ GU /\ ('Ix u) (3y Ee z)F(x, y)] . Because L-y is is absolute absolute for for L~ La we finally get get G GL~ E L-y)(3y EL La)F(x, y) for some uu = = L~ we finally and ('Ix (Vx E L~)(3y E ~ ) f ( x , y) for some L� and , < a. a. Because Lfj ~�GGw ewe Hence aa E Lfj � L-y it follows 7 < Because of ofL~ h a have v e f l j3 < < ,. 7. Hence L~ C_ L~ and and it follows L~ F ~ (3y)f(a, desired. O 0 La (3y)F(a, y)as y) as desired. Ax then w If Pw- + + El-FOUND c_ A x then w < < I[[Ax[l~ and Ax A x proves proves the If K KPw� l -FOUND � the =: aa and I Ax l l oo =" L-reflection rule rule for for E-formulas. �-formulas. Interpreting Interpreting the the provable provable sentences sentences of of A Ax it makes makes L-reflection x it no difference difference ifif we we think think that that every unrestricted quantifier quantifier is is restricted restricted by by L~. La . Since Since no every unrestricted KP �A Ax this has has the the same effect as as ifif A Ax believes that that aa is is admissible. Therefore K P -- c_ x this same effect x believes admissible. Therefore we we obtain obtain as as aa corollary corollary of of Lemma Lemma 1.2.3 1.2.3

1.2.4. Corollary. Corollary. 1.2.4.

If K KPw+ E,-FOUND � 1 - FOUND C_ � Ax then [[Ax[Ig If Pw- + A x then II Ax IIII2 ' II Ax l i E1l == [[Ax[[n2.

Another observation observation is is that that adding adding true true II~-sentences II�-sentences does does not not increase increase the the E~: �� Another ordinal ordinal of of an an axiom axiom system. system.

Set Theory and Second Order Number Theory

1.2.5. 1.2.5. Theorem. Theorem.

219 219

Ax + Let Let G G be be aa true II� H~ sentence. sentence. Then Then IIIIAx +G GII~ = III Ax I A xIII E� I~ ' l b:;� =

Proof. Let G G == - HL� H L~ for for aa II Hi-sentence H. Assume Assume that that Ax Ax + + G G ~� FL� F L~ for for aa Proof. Let I -sentence H. EI E1 sentence sentence F F.. Then Then Ax A x e�- ( (H H -+ --+ F F)) L� L~ and and H H -+ -+ F F is is EI El.. For For a c~ :"=- I III A Ax x [I l]E� ~ we we thus have have La: L~ F ~ H H -+ --+ F F.. From From a ~ � _ K., a, L" L~ F ~ H H and and the the downwards downwards persistency persistency of of thus Il Hi-sentences we get get La: L, F ~ H H which which in in turn turn entails entails La: L~ F ~ F F.. Hence Hence I[lAx + G GII~ _ I Ax + l -sentences we I I E�~ � But the the converse converse inequality inequality holds holds trivially trivially and and we we have have I[lAx + G GII~ Ax I I E� ' But I Ax + III[Axll~?. I I E�~ = 0 Ax II E� . D IIItAxII~r We introduce introduce the the following following notation. notation. We

1.2.6. 1.2.6. Definit Definition. ion.

Ax l l ,, :== I[IAxl[~ IIIIAxll~ I Ax I I E�

AxI I II� for Because I Ax l l " = Because of of Lemma Lemma 1.2.3 1.2.3 we we get get IIIAxII~ - IIIIAx[Inu for theories theories Ax Ax satisfying satisfying the the hypotheses of of the the lemma. lemma. hypotheses We We call call the the computation computation of of the the ordinal ordinal II[IAxll~ a-ordinal analysis analysis for for Ax Ax.. It It Axl l " aa K.-ordinal I Ax l lwcK will turn turn out out that that IIIAx[[~CK will be the most important ordinal. In Section 2.1.4 we will be the most important ordinal. In Section 2.1.4 we will 1 will see see that that there there is is also also something as an an w-ordinal which gives gives aa characterization of will something as w-ordinal which characterization of the Skolem Skolem functions functions of of the the provable provable II� H~'-sentences of an an axiom axiom system system Ax Ax in in terms terms the -sentences of of of aa sub-recursive sub-recursive hierarchy. hierarchy. 1.3. Connections C o n n e c t i o n s to s u b s y s t e m s of of second o r d e r number n u m b e r theory 1.3. to subsystems second order

Let Arithmetic. We Let C� L~ be be the the language language of of Second Second Order Order Arithmetic. We assume assume that that C� L~ contains contains aa constant constant Q 0 for for 00 and and constants constants for for all all primitive primitive recursive recursive functions functions and and predicates. predicates. We predicate variables and talk about set set variables. variables. We restrict restrict the the language language to to unary unary predicate variables and talk about This means real restriction restriction since since we we have have a a primitive primitive recursive recursive coding machinery. This means no no real coding machinery. We capital Latin Latin letters letters as as syntactical syntactical variables variables for X instead instead We use use capital for sets sets and and write write tt EE X of X(t).. We assume familiarity complexity classes classes in the arithmetical of X(t) We assume familiarity with with the the complexity in the arithmetical and and analytical analytical hierarchy. hierarchy. all primitive primitive recursive recursive functions functions and and predicates predicates have -definitions in in Since all have �o A0-definitions Since FOUND , we C� as as aa sublanguage sublanguage of of L(E) C(E) by restricting KPwE l -- FOUND, K Pw- + + E1 we may may regard regard L~ by restricting all quantifiers to w and all second second order quantifiers (VX) all first first order order quantifiers to w and replacing replacing all order quantifiers (\IX) and and by (\I X w) and and (3X (3X C_ � w), w), respectively. respectively. We transfer the (3X) C w) We may may therefore therefore transfer the notions notions (3X) by (VX � the arithmetical and analytical hierarchy to C( E). Whenever of the arithmetical and analytical hierarchy to the the language language of of/:(E). Whenever we we of talk of aa H II�, E�, II� , .. .. ..-- sentences sentences in in the the language Set Theory further talk of ~ E ~ 1-I~, language of of Set Theory without without further comments 2 comments we we think think of of aa translation translation of of the the corresponding corresponding sC�-sentence. facts for for the the things things to to come come is is the the w-Completeness w-Completeness Theorem Theorem One One of of the the basic basic facts II} -sentences. We We will will use use the the w-Completeness w-Completeness Theorem Theorem to to introduce introduce the the notion notion for for H~-sentences. of for II~-sentences. value t N of of truth truth complexity complexity for II}-sentences. The The value of aa closed closed term term tt and and the the truth value value of of an an atomic atomic sentence sentence in in the the standard standard structure structure N N are are primitive primitive recurrecurtruth -sively -sively computable. computable. Since Since there there are are symbols symbols for for all all primitive primitive recursive recursive functions functions and and predicates predicates we we obtain obtain the the diagram diagram of of N N

tN

D(N) D(N) "= := {A { A II AA isis an an atomic atomic sentence sentence and and NN ~F AA}}

220 220

W. Pohlers

as aa recursive recursive set. set. For For arithmetical arithmetical sentences sentences which which are are not not atomic atomic the the truth truth definition definition as is given given inductively inductively by by is

N F ~ A A1l and and N N F ~ A A2=v N F ~ A A 1l A1\ Bl B1 N 2 => N A2 N ~FA ,Ai . f ofor r s osome m e i e {i 1E, 2{I, } 2} => = ~N N ~FAA 1l V VA2 N N F ~ A(!l) A(n) for for all all n n Ee N N => =~ N N F ~ (Vx)A(x) (Vx)A(x) N NF p A(!l) A(~) for yor some ~om~ n ~E e N N => ~ N NF p (:3x)A(x) (3~)A(~).. N To extend extend this this truth truth definition definition to to n Hi-sentences we introduce introduce an an infinitary infinitary calculus. calculus. To t -sentences we For technical technical reasons reasons we we opt opt for for aa one one sided sided sequent sequent calculus calculus itg la la Tait. Tait. First First we we fix fix For the language language of of the the Tait Tait calculus. calculus. the The non non logical logical symbols symbols for for the the Tait-Ianguage Tait-language of of .c s are: are: The

�

•

9 The The constant constant Q 0 as as well well as as constants constants for for all all primitive primitive recursive recursive functions functions and and -relations. relations. The logical logical symbols symbols comprise: comprise: The •

9 Bounded Bounded number-variables, by x, y, z, Xl,... and set set variables, variables, denoted denoted number-variables, denoted denoted by Z, Xl , . . . and x, y, X1,... bbyy XX,, Y, Y, Z, Z, X l,. . .

•

9 The The logical logical connectives connectives A, and the the quantifiers quantifiers V, V, :3. 3. 1\, VV and

•

9 The The membership membership symbol symbol E E and and its its negation negation fl. ~..

Terms Terms are are built built up up from from Q 0 and and function function symbols symbols iinn the the familiar familiar way. way. We We use use 5S as as aa symbol for the the successor function. Terms Terms of of the the shape shape ((� S ~ ~ 0 )O) are are numerals numerals and and symbol for successor function. nn --times times

will be denoted by !l. n_. will be denoted by Atomic Atomic formulas formulas are are tt E E X X,, tt fI. ~ X X and and R(tb R ( t l , ... .. ,. t' ~tn) ) , , where where t, t l , .. ...., ,t tn ~ are are t, tb terms, X X is variable and and R R is is aa symbol symbol for n-ary relation terms, is a a set set variable for an an n-ary relation symbol. symbol. From formulas we in the way. From the the atomic atomic formulas we obtain obtain the the formulas formulas of of s.c in the familiar familiar way. Notice have free Notice that that we we do do not not have free number number variables variables in in the the language. language. The symbol is is not not aa basic basic symbol the Tait-Ianguage. define the the The negation negation symbol symbol of of the Tait-language. We We define negation by de de Morgan's laws.. negation of of aa formula formula by Morgan's laws

�

~(t x);; ~(t X) ..., ( t rfI. xX)) . =: = (t Ee x) ..., ( t eE xX)) . -:= (t rfI. X) := (CRtl is aa symbol symbol for for the complement of ~...,(( Rt R t ll. .. .. t. ~tn) ) "=_ R t l ... .. t. ntn) ) where where R R is the complement of R R ~(A A B) B)"_= (-~A - ( - (..., ~ AA 1\ A ..., -,S) V -~S); ...,B ); -~(A ..., (A V VB B)) ::= ..., ( A 1\ AV B) := (...,

-,(Vx)A(x) "=_ (3x)-~A(x); :- (Vx)-~A(x). (:3x)...,A (x) ; -,(3x)A(x) ..., (Vx)A(x) := ..., ( :3x)A(x) := (Vx)...,A (x) . It is It is obvious obvious that that we we have have

(11) (11)

-,-,A=A.

The semantics semantics for for the the Tait-language Tait-Ianguage is is straightforward. straightforward. We We easily easily check check The

N ~s A[Sl ' . . . ' S~] Snl Snl iff � A[S1,..., N ~F ((...,~ A N A ))[Sb [ S l , .· .· .·, , Sn] iff N

for any any assignment assignment of of sets sets SSl1,, .. .. .., , Sn to the the set set variables variables occurring occurring in in A. A. for S~ to We use use capital capital Greek Greek letters letters A, � , F, ' as syntactical variables for finite finite r, A, A, � We A1, ... as syntactical variables for l sets sets of/:~-formulas. of .c -formulas.

�

• . .

221 221

Set Set Theory Theory and and Second Second Order Order Number Number Theory Theory

1.3.1. 1.3.1. Definition. Definition. We We defi define ~ t. A inductively inductively by by the the following following clauses: clauses: ne p (AxM) If I f t. A n N D(N) D(N) =I~ 0q} then then p ~ t. A for for all all ordinals ordinals a (~.. (AxM)

If tN tN = = Ss N N then then p ~ t., A, ss f/. r X X, , tt E e X X for for all all ordinals ordinals a. ~. AxL) If ((AxL) (A) (1\)

(v) (V) (V)

(3) (3)

If t.,, AA, i and , 2 then t., A If � ~ A a n d ~aii < < a a for f o r ii== 11,2 then p ~ A, A1l 1\ A A A22 ..

I , 2} then If some ii Ee {{1,2} Al VV A If � ~ t., A , AA,i and a~d a ~i < < a ~ for 1o~ ~om~ t~ p ~ t., A,A~ g~. 2. If If � ~ t., A, A(i) A(i) and and a aii < < a a for for all all ii E e N N then then p ~ t., A, (Vx)A(x) (Vx)A(x) ..

If y � ~ t., A, A(i) A(i) and a~d ao ~o <
The relations relations p ~ t. A is is to to be be read read that that there there is is an an infi infinite proof tree tree of of V At. whose whose The nite proof depth bounded by a. It depth is is bounded by the the ordinal ordinal a. It is is obvious obvious from from the the definition definition that that we we have have

V

A,

_< (3, Z, aa "5:.

A C_ F

=~

(12) (12)

~ F

and and aa simple simple induction induction on on a a shows shows F1, . . . , Fn =~ N ~ (F1 V . . . V Fn)[S1, . . . , Sm]

FI V ... V Fn). for assignments 8 to the parameters occurring for all all assignments $ 11 ,, .. .. .., , 8m Sm to the set set parameters occurring in in ((F1 We We thus thus have have

(13) (13)

F (9~) =~ IN ~ (V)~)F()~)

m

for -sentences (VX)F(X) for all all H~-sentences (V)~)F(s . If If F F is is aa true true arithmetical arithmetical sentence, sentence, i.e., i.e., if if F F does does not not contain contain set set parameters, parameters, we we obtain obtain by by aa simple simple induction induction on on the the complexity complexity rk(F) rk(F) which can of F (which of the the formula formula F can be be taken taken to to be be the the number number of of logical logical symbols symbols occurring occurring F) in in F) k (F ) F . (14) l[,rk(F)F.

(

(14)

This indeed an This shows shows that that p ~ F F is is indeed an extension extension of of the the truth truth definition definition for for arithmetical arithmetical sentences. -sentences, i.e., sentences. However, However, to to prove prove that that p ~ F F is is also also complete complete for for II II]-sentences, i.e., the the ' s proof opposite opposite direction direction in in (13), (13), needs needs harder harder work. work. We We follow follow Schiitte Schiitte's proof via via search search trees. trees. Let codes for finite sequences natural numbers. numbers. For Let 8f1j Seq be be the the set set of of (codes for) finite sequences of of natural For t E 8f1j we denote by s, C_ tt that that ss is is an an initial initial segment segment of of tt.. A A tree tree is is aa set set of of finite finite s, t C Seq we denote by ss � number-sequences number-sequences which which is is closed closed under under initial initial segments. segments. The The elements elements of of aa tree tree are are called nodes. A set of nodes in a tree is a thread, if it is linearly ordered by called nodes. A set of nodes in a tree is a thread, if it is linearly ordered by � C_.. A A maximal maximal thread thread in in aa tree tree is is a a path. path. A relation -~ -<� A binary binary relation c_ W w xx W w is is well-founded well-founded iff iff there there are are no no infinite infinite -<-descending -~-descending chains. �W chains. For For well-founded well-founded -< -~C wx x w w we we define define

�

(

...: (n) otyp.~ ( n ) :: -= otyp and and

)

{ WsupI {otyp...: (m) + 1 1 m -< n} sup (otyp.~(m) + 11 m -~ n} wl

if if n n E e ffield(-~) �eld ( -< ) otherwIse otherwise

Pohlers W. Pohlers

222 222

:= sup { oty p � (n ) II n eE field(-<)}. field ( -<) } . otyp() := oty p( -<)

For aa tree tree TT we we define define the the tree-relation tree-relation -
S-A is well-founded iff - := otyp. = otypT(()). Oty PT( () ) . otyp(T) otyp(T) -Recall that that the the first first non non recursive recursive ordinal ordinal isis defined defined by by Recall WfKK "= : = sup w~ sup {otyp(-<)l { oty p( -<) I -< -< } is is aa recursive recursive well-ordering}. well-ordering . We are sequences of Such a a sequence sequence We are going going to to define define search search trees trees for for finite finite sequences of formulas. formulas. Such is called reducible reducible ifif it contains at at least least one one non non atomic formula. The The left is called it contains atomic formula. left most most non non reduced sequence sequence atomic in aa reducible reducible sequence sequence is is called called distinguished. atomic formula formula in distinguished. The The reduced D.T reducible sequence sequence A D. is is obtained obtained by by removing distinguished formula from A r of of aa reducible removing the the distinguished formula from the sequence. the sequence. The tree for for aa finite D. of tree Sf:>. together with The search search tree finite sequence sequence A of s.c -formulas is is aa tree Sz~ together with aa label label function function which which assigns assigns aa finite finite sequence sequence 15(s) 6(s) of of s.c -formulas to to each each node node ss 6E Sz~. Sf:>. . It It is inductively defined defined by the following following clauses: clauses: is inductively by the � � and 15(()) = D. . (So ) <>6S~ (S()) 0 E Sf:>. and6(<>)=A. (SAx) AxM oorr (AxL), AxL , then (S~.) If If ss E9 Sf:>. Sa and and 15(s) 6(s) is is an an axiom axiom according according ttoo (AxM) then s�(i) s-. Sz~ for all for all ii E 6 N. N. (I. (I. e. e. ss is is aa topmost topmost node node of of Sf:>. S~ .) .) ( ) ( ) For 8(s) is For the the following following clauses clauses assume assume ss E 6 Sf:>. Sa such such that that 6(s) is not not an an axiom. axiom. (O) E6 Sf:>. (S (S~a) If 15(s) 5(s) is is not not reducible reducible then then s� s~(O) S~ and and 15(s�(O)) 6(s~(O)) = = 15(s) 6(s).. id ) If (i) E6 Sf:>. (S^) If If Fo Fo 1\ A Fl F~ is is the the distinguished distinguished formula formula in in 15(s) 6(s) then then s� s~(i) S~ for for ii = = 0, O, 11 (S,,) and 15(s� 6(s~ (i)) (i)) := 6(s) r, Fi F~.. and := 15(Sy, (S (Sv) Let Fo Fo V F F~l be be the the distinguished distinguished formula formula in in 15(s) 6(s).. Then Then s�(io) s~. S~ and and v ) Let 15(s�(io)) = 15(sy, 6(s~(io)) ::= 6(s) ~, Fo, F0, Fl F~.. /f the the distinguished distinguished formula formula in in 8(s) 6(s) is is (Vx)F(x) (Vx)F(x),, then then s� s~ (i) (i) E6 Sf:>. S~ for for all all ii E6 N N ((Sv) SI;I) If and F(i) . and 15(s�(i)) 6(s~(i)) = = 15(sy, 6(s)~,F(i). (O) E6 Sf:>. S3) If If the the distinguished distinguished formula formula in in 8(s) 6(s) is is (3x)F(x) (3x)F(x),, then then s� s~(O) Sz~ and and (($3) 8(s� (0)) = 15(sy, F(n,) , (3x)F(x) , where n is the least natural at rat number umb r such 80�8 15(so) that that n n i# tt NN for for all all formulas formulas F(t) F(t) E 6 U Usocs 6(s0).. = Observe Observe that that we we introduced introduced clause clause (SAx) (SAx) only only for for better better readability. readability. It It follows follows from from the the other other clauses clauses and and the the fact fact that that Sf:>. Sz~ is is inductively inductively defined. defined. There There are are two two main main lemmas. lemmas. Set Set Theory Theory and and Second Second Order Number Number Theory 1.3.2. Syntactical Syntactical Main Main Lemma. Lemma. 1.3.2. Sa l °typ( ) A and b.. . and jowp(S~) 223 223 If If Sa Sf),. isis well-founded well-founded then then otyp(Szx) otyp(Sf),.) << w~ WfKK PProof. r o o f . Let Let SLX Sf),. be be well-founded. well-founded. The The tree tree relation relation "<szx -< Sa is is obviously obviously recursive, recursive, hence hence otyp(SLx) Every path otyp(Sf),.) << w~ WfK. path in in SLx Sf),. isis finite. finite. Thus Thus (f(s) 8(s) has has to to be be an an axiom axiom for for every every K . Every and the the second second claim claim follows follows easily easily by by induction induction on on otyp(SLx). otyp(Sf),.) . topmost node node ss and topmost 00 the search search tree tree S~ Sf),. isis not not well-founded well-founded then then 1.3.3. emantical M ain L emma. If 1.3.3. SSemantical Main Lemma. If the there is an an assignment assignment SS1l ,, .. ... ., , Sn subsets of N to to the the set set variables variables in in Ab. such such that that there is S,~ of of subsets of iN N V: � V {FI {F I FF eE A}[S~,...,S,.,]. b. } [Sl , . . . , Sn] . N To sketch the the proof proof let let ff be be an an infinite infinite path path in in Szx. Sf),. . We say sloppily sloppily that that aa formula formula To sketch We say occurs in in ss eE Szx Sf),. ifif FF eE 6(s). 8(s) . Let Let f[n] f[n] := := ((/(0) 1)) and f[n] the the FF occurs f ( 0 ) ,, .. . .. ,. , ff(n ( n - 1)) and call call f[n] course of of values of ff below below n. n. We We observe: observe: course values of (1) If AA is atomic formula occurring in then AA occurs in all that (1) If is an an atomic formula occurring in ss E E Sf),. SLX then occurs in all tt such such that - (2) (2) ss Cc;;,_ tt eES zSf),. x. . If atomic formula formula FF occurs occurs in in some then there is an an m m >� nn such such If aa non non atomic some f[n] f[n] then there is that FF is is distinguished that distinguished in in f[m] f[m].. The proof of of (2) (2) is induction on on the the number formulas occurring The proof is an an easy easy induction number of of non non atomic atomic formulas occurring F in 8(J[n]) . Using Using (2) (2) the the proofs proofs of of the following observations left of left of F in 6(f[n]). the following observations are are almost almost immediate the definition definition of of SLX. Sf),. . immediate from from the If aa formula formula A occurs in in f[n] J[n] then then there is an an m m such A or B B occurs (3) If A 1\ A B there is such that that either either A or B (4) (4) occurs in f[m] . occurs in f[m]. If If aa formula formula A A VB B occurs occurs in in f[n] f[n] then then there there are are m mAA and and m m sB such such that that A A occurs occurs in in f[m f[mA] and B B occurs occurs in in f[m f[ms]. A ] and B] . If in f[n] are numbers (5) (5) If aa formula formula ((Vx)F(x) occurs in f[n] then then there there are numbers ii and and m m such such that that V'x)F(x) occurs F(i) [m] . F(~) occurs o ~ r ~ in ~ ff[m]. If in f[n] (6) (6) If aa formula formula (3x)F(x) (3x)F(x) occurs occurs in f[n] then then for for every every number number ii there there is is aa term term tt and and an an mi mi such such that that eo! t TM = = ii and and F(t) F(t) occurs occurs in in J[mi] f[mi].' To To prove prove fact fact (6) (6) we we assume assume that that (3x)F(x) (3x)F(x) is is distinguished distinguished in in 8(J[n]) 6(f[n]).. By By (2) (2) this this means 1]) = means no no loss loss of of generality. generality. Then Then 8(J[n 6(f[n + + 1]) = 8(J[nlY, 6(f[n]) r, F(j), F(j), (3x)F(x) (3x)F(x) and, and, if if ii < < jj we have have F(t) F(t) in in f[m] f[m] for for some some tt and and m m � _ n n such such that that tt NTM = - i. i. If If jj � < ii then then F(t) F(t) will will we occur occur in in f[m f[m + + 1] 1] for for some some m m � _> n n and and tt with with tt NN = = ii as as soon soon as as (3x)F(x) ( 3 x ) f ( x ) becomes becomes distinguished distinguished in in 8(J[m]) 6(f[m]) and and F(t) F(t) has has occurred occurred for for all all tt NN < < i. i. We We define define an an assignment assignment (X) := r : - {{ttN~ II (t (t i r X) X ) occurs occurs in in J}. f }. Here Here F F occurs occurs iinn f f means means that that F F occurs occurs in in f[n] f[n] for for some some n. n. An An easy easy induction induction on on the the length length of of aa formula formula G, G, using using observations observations (3) - (6) (6) and and the the fact fact that that f f must must not not contain contain an an axiom, axiom, shows shows N iN � ~= G[] G[(I)] for for all all formulas formulas G G occur occurring ring in in f f .. Since Since all all formulas formulas of of b. A occur occur in in frO] f[0] this this yields yields N iN � ~ V V {{ F F II F r E e b. A }} [] [(I)].. D o W. W. Pohlers Pohlers 224 224 The The Syntactical Syntactical Main Main Lemma Lemma together together with with the the Semantical Semantical Main Main Lemma Lemma prove prove the following following theorem. theorem. the (VX)F(X) 1.3.4. -sentence. Then 1.3.4. w-Completeness w-Completeness Theorem. Theorem. Let Let (V)~)F(X) be be aa ITt H~-sentence. Then we we have N N F= ~ (V)~)F()() iff iff there there is is an an a ~ < Wf w~KK such such that that � ~ F (X). have (VX)F(X) F (X). Proof. Proof. The The soundness soundness is is already already stated stated in in (13) (13).. For For the the completeness completeness direction direction K . Then, by the Syntactical we assume assume If ~ F()~) for for all all ordinals ordinals a ~ < Wf w~K. Then, by the Syntactical Main Mainwe Lemma, the the search search tree tree for for F()~) cannot cannot be be well-founded. well-founded. Applying Applying the the Semantical Semantical Lemma, Main-Lemma Hence Main-Lemma we we obtain obtain an an assignment assignment (I) over over N N such such that that N N V= ~ f(s Hence 0 N V= ~= (V)~)F()~). D N F(X) F(X) F(X)[]. (VX)F(X). In .3.4 we In view view of of Theorem Theorem 11.3.4 we call call formulas formulas which which contain contain at at most most free free set set variables variables -sentences. Semantically sloppily -sentences as sloppily ITt II~-sentences. Semantically we we treat treat these these pseudo pseudo ITt II~-sentences as their their universal closure, closure, i.e. i.e.,, universal F(X) (VX)F(X). N F= ~ F()~) ::r<=? N N F= ~ (V)~)F()~). N We use use Theorem Theorem 11.3.4 to defi define the truth truth complexity complexity of of ITt II~-sentences. We .3.4 to ne the -sentences. Definition. 11.3.5. .3.5. Definition. G ::= - F(X) F()~) we we define define G (G) .."=_ { WWlI tc(G) tc - (VX)F(X) Complexity)) For For aa ITt II~-sentence G := (V)~)F()~) or or ((Truth Truth Complexity -sentence G { min min {a (c~ II p ~ F f (X)} ()~)} � if if N N V= ~ G G otherwise. otherwise. F l -sentences F as Using truth complexities we Using truth complexities we may may restate restate Theorem Theorem 1.3.4 1.3.4 for for TI H~-sentences as F K. iff t c ( f ) << Wf iff w~K. tc(F) As we have have seen in (14) As we seen in ( 14) we we have have tc(F) << ww tc(F) N F= N ~ f (15) (15) (16) (16) F. But in in contrast contrast to to that that we we have by CorolCorol for all all true true arithmetical arithmetical sentences sentences F. But for have by lary lary 1.3.9 1 .3.9 and and Theorem Theorem 1.3.10 1 .3.10 below below sup {tc(F) ]I (V.7)F(,7) is is aa H~-sentenee ITt -sentence and and r~ N ~F= F(X)} == w~ Wf~K sup {tc(F) (VX)F(X) F(X)} which shows shows that that for for "real" "real" II~-sentences TI l -sentences truth truth complexity complexity is is aa non-trivial non-trivial notion. notion. which arithmetical ifif there there isis an an arithmetical arithmetical formula formula F(x, y) y) We call call aa binary binary relation relation -< arithmetical We such that such that F(x, F( ) m -< -< nn iff iff NN ~F= F(m, m , n_). ll . m The following following Boundedness Boundedness Theorem Theorem isis one one of of the the most most important important theorems theorems for for this this The contribution and and will will return return in in different different variations. variations. contribution 1.3.6. 1 .3.6. Boundedness Boundedness Theorem. Theorem. ( T! TI (-<, -<, X) X) be be the the formula formula Let ~-< be be an an arithmetical arithmetical definable definable relation relation and and Let 225 225 Set Theory and Second Second Order Number Theory (Vx E E field( field(-~))[(Vy)(y -~ x x -+ --+ y yE e X) --+ x x E e X] X] -+ --+ (V (Vx field(-~))[x X] (Vx -<))[(Vy) (y -< X) -+ -<)) [x Ee X] X Ee field( expressing T/(-< , X) ) . ~=p~,,i.g induction i.du~tio. along .lo.g -< -<.. Then Th~. otyp(-<) ::; < tc((VX) tc((VX)TI(-<.X)). To obtain obtain the the Boundedness Theorem we we prove prove the the more more general general Boundedness Lemma. To Boundedness Theorem Boundedness Lemma. We prepare prepare the the Boundedness Boundedness Lemma Lemma by by aa few few notions notions and and observations. observations. We X. An An An .c�-formula /:~-formula is is X-positive X-positive if if it it does does not not contain contain occurrences occurrences of of tt ¢ ~ X. An obvious -positive formulas obvious property property of of X X-positive formulas is is stated stated in in the the next next lemma. lemma. 1.3.7. Let -positive formula 1.3.7. Monotonicity M o n o t o n i c i t y Lemma. Lemma. Let F F be be an an X X-positive formula not not containing containing further further set set variables variables and and M M � CN g � C_N. N. Then Then N N F ~ F[M] entails N N F ~ F[N] F[g].. F[M] entails The proof is induction on The proof is straightforward straightforward by by induction on the the length length of of the the sentence sentence F. F. By induction on By induction on a a we we obtain obtain the the following following inversion inversion properties: properties: A, Al A1 V A A22 � =~ � ~ �, A, Ab A1, A A22 ,, � �, 0 [3 (17) (17) � �, A for A, Al A1 /\ AA A22 � =~ � ~ �, A,A, for ii == 1, 1,22 (18) (18) W for A , , (Vx)F(x) (Vx)F(x) � =~ � ~ F F (/) for all all numbers numbers i. i. �� (19) (19) and and -<, X) Let Let Prog( Prog(-.<, X) denote denote the the premise premise (Vx -<))[(Vy)(y -< (Vx E e field( field(-~))[(Vy)(y -< xx -+ --+ Y yE e X) X) -+ --+ x x E e X] X] of -<, X) . For of the the ITt-sentence II~-sentence T/( TI(-<,X). For aa fixed fixed well-founded well-founded binary binary relation relation -< -< denote the enumerating function of the complement of the set let en�l1'..... let e--ff~z z.} denote the enumerating function of the complement of the set ...'Zn} {ot {otyp.<(zl),..., otyp~(zn)}, i.e. of of On On \\ {otyp {otyp.<(zl),..., otyp.<(zn)}. Define zn) } . Define -« zd , . . . , otyp-« YP-« Zl) , . . . , otyp -« zn)}, i.e. M ~ , .....z.}(a) := { m l otyp.~(m) _< e-ff~z,.....z,,}(o~)} u { Z l , . . . , Z n } . Let Let -< -< be be aa well-founded well-founded transitive transitive binary binary relation relation and and assume assume that that -~Prog(-~,X),z X , ... .. ,. z' £, n ¢ r X, X , A� Prog( -<, X) ' £l1 ¢r X, �, 1.3.8. 1.3.8. Boundedness B o u n d e d n e s s Lemma. Lemma. -positive ITt -sentences A� = ' . . . ' Fn} not not containing set for for aa set set of of X X-positive II~-sentences = {Fl {F1,...,Fn} containing further further setvariables. variables. Then Then .-9 [M{,, .....,,,} (a)]. The Assume The Boundedness Boundedness Theorem Theorem follows follows from from the Boundedness Boundedness Lemma. Lemma. Assume a. Then -<) an ao ttc( r T/( -<, X)) ::; _ ~. Th~n there the~e is for fo~ every n ~ E e field( field(-<) '~o < < a ~ such that that , I! E X otyp(m) ::; � ,Prog(-<, X) , I! ¢q~field(-<) -~Prog(~,X),n field(-<),n X . . Because of of M0(aO) Mo(ao) = = {m {m] l otyp(m) _< ao} a0} we we obtain obtain by by the the Boundedness Boundedness Lemma Lemma [otyp(n) ::; (Vn E e field(-<))(3ao field(-~))(3a0 < < a) a)[otyp(n) _ ao] a0],, (Vn hence otyp( otyp(-~) _ a. c~. hence -<) ::; 226 226 w. W. Pohlers Pohlers We prove prove the the Boundedness Boundedness Lemma by induction induction on on a a.. Assume Assume We Lemma by -~Prog(-<, X) X),' � z,l ¢. r X, X , ... .. ., ' � znn ¢. r X, X, � A. . � -,Prog(�, (i) (i) The The claim claim is is trivial trivial if if D(N) D(N) n n � A =1= r 0. If If (i) (i) holds holds by by (A (AxL) then there there is is aa formula formula xL ) then ins � = Zzii ffor M{~, z.}(a) this , n} . Since Zi M� t E XX inA u c hsuch t h a tthat t N= o r ssome o m e i E { 1{1, , .... ..,n}. SinceziE "~" ..... ..." n} (a) this yields yields the the claim. claim. If If (i) (i) is is obtained obtained by by the the premises premises t il! tE iE E � -,Prog(�, -,Prog(-<, X) X),' � z__l1 ¢. r X X , , .. .. .. ', � z_,n ¢. r X, X , �i Ai then induction hypothesis then we we obtain obtain the the claim claim by by the the induction hypothesis together together with with the the Monotonicity Monotonicity Lemma Lemma and and the the soundness soundness of of � ~ .. The really interesting (i) follows The really interesting case case is is that that (i) follows from from the the premise premise X, ) [-,y � ield(�) /\A (Vy P ro g ( - < X) ,X ),, z� E E ffield(-<) (Vy)[-~y -< � z V V Yy E EX X]l /\ A� z ¢. r X, � -,-~ Prog(�, . . . , � ¢. z~ ¢.r X, x,...,z, r X, x, � zx �l By By inversion inversion we we get get from from (ii) (ii) r o g ( - < ,X) X ) ,,z� E E ffield(-<) ( V y )) [-,y [W � -< � z V V Yy E E X] X],z X ,, ... .. . ,,z�n ¢. r X, X, � A ield (�) /\A (Vy , �l1 ¢.r X � -,~ PProg(�, (ii) (ii) (iii) (iii) and and (iv) (iv) --,Prog(-~,X),z r X,z~ r X,...,z__, r X,/',. If If N ~ V ~[M~, .....~.~(~o)] (v) (v) N ~ V ~X[M~..... ~.~(~)] (vi) (vi) then then by by Monotonicity. Monotonicity. If If N N � ~= V � AIMS1 (co)] n} (ao)l [M�l '..... ... " z.} (vii) (vii) ield(�) /\A (Vy � z) Zz EE ffield(-<) (Vy-< z)[y ~ , '..... n } (ao)l . M�l [y EE M ..."z.}(ao)]. ((viii) viii) V then then the the induction induction hypothesis hypothesis applied applied to to (iii) (iii) gives gives From (iv) we induction hypothesis From (iv) we get get by by the the induction hypothesis ix) ((ix) N N F ~ V � AIMS, n,.} (ao)l· [M�l '..... ... "~.,~}(ao)]. V We We claim claim ((x) x) otyp otyp.~(z) < ~-~, + 1) 1) ::; < e-rib, -« Z) ::; n} (a) . en�l'..... n} (ao + en�l'..... ... "~.}(ao ... " z.}(a). To x) recall recall that z) := sup otyp-« y) + To prove prove ((x) that otyp-« otyp.~(z) sup {{otyp~(y) + 11[1 y � -< z} z} and and observe observe that that for for en�"..... or y EE {Zb y) ::; zn} by -< Zz we we obtain obtain otyp-« otyp.~(y) _ e-n~l {Zl,... by (viii). (viii). Hence Hence . . . ',zn} y� n} (ao) or .. . "~.}(ao) zn } } ::; otyp-« y) + sup {{otyp~(y) + 1II1 y � -~ Zz ^ y ¢. r {Z {z~,... < ~en, �l'..... + 1) 1) b · . . ,,z~}} 'fJ := sup n} (ao + ..."~.}(ao /\ Y Second Order Number Theory Set Theory and Sewnd 227 227 . . . , Zn (y) < en{zl ,..... 1) since and and for for y y �9 Zz such such that that yy E E {Zb { z l , .. ... . ,,z,} we get get otyp--: otyp.~(y) < e--ff~z, z.}(ce0 + 1) since zn} we } (ao + otyp.~(y) is is omitted omitted in in the the enumeration. enumeration. Because Because of of otyp--:(y) ~-ff'~.,.....z.,.)(ao) _< ~-fi~,~,.....,,.}(ao + 1) _< e-ffS, .....z.}(a) (xi) (xi) ... 'Zn'Z ... ,zn we M(z"~1..... ~.,~}(ao) C_ M�l ~.}(a) This finally fnally yields yields N N F we get get by by (x) (x) and and (xi) (xi) M�" M {~1 "~ ,..... } (a) .. This } (ao) � ... ,zn a) l by V 0 V �[M�l A [ M ~ ,..... z.}} ((a)] by (ix) (ix) and and Monotonicity. Monotonicity. [3 As As aa consequence consequence of of the the Boundedness Boundedness and and the the w-Completeness w-Completeness Theorem Theorem we we obtain obtain 1.3.9. 1.3.9. Corollary. Corollary. 0.) ~ K K. Wf If If �9 is is an an arithmetical arithmetical definable definable well-ordering well-ordering then then otyp(�) otyp( 9 < < It should mentioned that that Corollary 1.3.9 can can - with with just just aa little little effort effort - be be extended extended It should be be mentioned Corollary 1.3.9 to I:t-definable ~-definable well-orderings and thus thus comprises comprises the the well-known well-known Boundedness Boundedness to well-orderings �9 and Theorem Theorem of of recursion recursion theory theory without without referring referring to to the the Analytical Analytical Hierarchy Hierarchy Theorem Theorem (cf. (cf. Beckmann Beckmann and and Pohlers Pohlers [1997] [1997]).) . Conversely Conversely to to the the Boundedness Boundedness Theorem Theorem we we obtain obtain otyp(�) otyp( 9 also also as as an an upper upper bound bound for for tc( tc( TI T l (( �, 9 X) X ) )). . We We show show 5.(otyp-« n) + l ) ~Prog( 9 X) X ) , , 'll. n_ E X X 1I. ~(~ 1= ======= -,Prog(�, (20) (20) by (n) where by induction induction on on otyp--: otyp.~(n) where - for for simplicity simplicity-- we we assume assume that that �9 is is aa primitive primitive recursively recursively definable definable relation relation whose whose field field is is all all of of N N.. We We have have 5.(Otyp-<(m) + 1 ) -~Prog( 9X), -,m �9 'll. n_,m X -,Prog(�, , m EE X I15"(~ either either as as an an instance instance of of (AxM) (AxM) or or by by induction induction hypothesis. hypothesis. Hence Hence (i) (i) (Vy)[y �9 'll. n_ -+ Yy E E Xl X] -,-~Prog( Prog(�,9X) , ('v'y)[y by By (AxL) by two two (V) (V) and and one one ('v') (V) "inference" "inference".. By (AxL) we we have have � -,Prog(�, -~Prog( 9X) , 'll. n_ 1q~ X, X, 'll. n_ E EX X .. (ii) (ii) 5 . 0typ-« n)+3 I15"~ (iii) (iii) By By (ii) (ii) and and (iii) (iii) we we obtain obtain 5.0typ-« n)+4 ~ P r o g ( 9 X) , ('v'y)[y (Vy)[y �9 'll. n_ -+ -+ Yy E Xl X] " A 'll. n_ 1~ X, X,_n X.. 'll. E X I15Otyp'~(n)+4 -,Prog(�, (iv) (iv) One One additional additional "inference" "inference" (3) (3) leads leads to to . (Otyp-« n)+l ) ~Prog( 9 X ) , n E X . 5 X), 'll. E X . I[5.(otyp~(n)+l) -,Prog(�, [3 0 From From (20) (20) we we obtain obtain by by aa clause clause ('v') (V) and and two two clauses clauses (V) (V) the the following following theorem. theorem. 1.3.10. 1.3.10. Theorem. T h e o r e m . If If �9 is is a a primitive primitive recursive recursive well-founded well-founded relation relation whose whose order order ::; ttc( T/(�, type type is is aa limit limit ordinal ordinal then then otyp(�) otyp( 9 <_ c(T l ( 9 X)) ::; <_ otyp(�) otyp( 9 + + 2. 2. 228 228 w. W. Pohlers Pohlers Of course course we we should read Theorem Theorem 1.3.10 1.3.10 as as otyp( otyp(-<) = tc(Tl(-<,X)), the Of should read -<) = tc( T/( -<, X) ) , since since the But, since "§ is just just due due to to the the syntactical syntactical peculiarities peculiarities in in the the definition of � ~ � A . . But, since "+2" is definition of all all important important ordinals ordinals will will be be limits, limits, this this is is of of no no importance. importance. We We define define = {tc(F) Specrr specni(N) {to(F) 1[ F F is is aa m-sentence II~-sentence and and N N F ~ F} F} j (N) ::= and N. Due and call call speCrr specs]j (N) (N) the the m-spectrum II~-spectrum of of N. Due to to Theorems Theorems 1.3.4 1.3.4 and and 1.3.10 1.3.10 we we then then have have ((21) 21) specrq (N) = w?K. More More generally generally we we define define the the m-spectrum II~-spectrum of of aa theory theory Ax Ax in in the the language language of of second second order order arithmetic arithmetic by by ( ) speCrr ev'X)F(X) 1[ (VX)F(X) }. specni(Ax) (tc((V)~)F()()) (V)~)F()~) is is aa ITt-sentence II~-sentence and and Ax A x ~f-- F(X) F()~)}. j (Ax) ::=- {tc specrrj (Ax) llrr j ::= IrAxllni - sup( sup(specnl (Ax)).). IIAx (22) {F 1 Ax F} is For For aa recursively recursively enumerable enumerable theory theory Ax Ax the the set set {F[ Ax f-~ F} is recursively recursively enu enumerable, Since Wf merable, too. too. Since w~KK is is recursively recursively regular regular we we get get by by Theorem Theorem 1.3.4 1.3.4 and and the the fact fact that can be be effectively that for for true true m-sentences IIl-sentences F F an an upper upper bound bound for for tC(F) tc(F) can effectively computed computed from from F F via via the the depth depth of of its its search search tree tree IIAxllrr IIAxll,:j < <W w~fK~ for Ax. The for all all recursively recursively enumerable enumerable theories theories Ax. The proof theoretic ordinal ordinal of of aa theory theory Ax is is commonly defined as as Ax commonly defined otyp( -<)1 -< -< , X) } . ][Ax[] ::= - sup sup {{otyp(-<)] -< is is primitive primitive recursively recursively definable definable and and Ax A x ~f-- T/( TI(-<,X)}. IIAxll From From the the Boundedness Boundedness Theorem Theorem we we get get immediately immediately IIAxl1 Wf~K IIAxII :::; _< IIAxllrr IIAxII.:j < <w~ 23) ((23) for for all all recursively recursively enumerable enumerable theories theories Ax. Ax. By By showing showing that that for for every every ordinal ordinal a a < < I[]Ax[]n~ there is is aa primitive primitive recursive recursive order order relation relation -< -< such such that that a a :::; _< otyp(-<) otyp(-<) I Axllrr j there and -< , X) we .3.10 together and Ax A x ef-- T/( TI(-~,X) we get get from from Theorem Theorem 11.3.10 together with with (23) (23) IIAxll IIAxII = = IIAxllrr liAxJlnlj ((24) 24 ) for will be for all all theories theories which which will be analyzed. analyzed. There (24) which There is, is, however, however, also also aa more more general general argument argument for for (24) which we we are are going going to to sketch sketch roughly. roughly. Assume Assume that that Ax Ax is is aa theory theory comprising comprising PA PA and and let let (VY)F(Y) (VI~)F(]~) be be aa ITt-sentence. the Kleene Brouwer ordering in the search tree II~-sentence. Denote Denote by by -<S -<sF(v) the Kleene-Brouwer ordering in the search tree SF(?) S F(Y) F(Y) 9Jt -< F(Y) and assume that Ax for ( sF(Y) ' X) . Then model ff~ F for F(]~) and assume that Ax JL ~ T/ TI(-<sv(v),X). Then there there is is aa model ~ Ax Ax such that and and an an assignment assignment T T � C_ 9Jt ~ for for X X such that 9Jt 93l � ~ T/(-< T/(-<sv(e),X)[T Therefore sF(y) , X) [T]].. Therefore there is through S which is there is an an infinite infinite path, path, say say P P � C_ 9Jt 93t,, through SF(?) is definable definable by by aa first first F Y) which order formula with parameter parameter T emantical Main-Lemma order formula with T.. According According to to the the Semantical Main-Lemma we we get get all Y; which are assignments assignments �(Y;) (I)(Yi) � C_ 9Jt 93t for for all Yi belonging belonging to to Y Y which are definable definable by by first first order order g 229 Set Theory Theory and Second Order Number Number Theory formulas formulas with with parameter parameter T. T. Since Since we we have have induction induction in in VJ1 ffJ~ for for first first order order formulas formulas Y) [(I)] we obtain obtain VJ1 ~ � ~ F( F(]~) as in in the the proof proof of of the the Semantical Semantical Main Lemma using using aa local local Main Lemma [] as we truth truth predicate. predicate. Hence Hence Ax Ax jL ~ F(] ~) and and we we have have shown shown Y) =} hx � ~ F( F(?) ~ Ax Ax � ~- TI(-<sp TI(-~(~>, X).) . Ax (y ' X F(Y ) ) tc (F(Y ) ) Since Since -< -~sv(~) primitive recursively recursively definable definable and and we we have have tc(F(]?)) � ___ sP(Y) isis primitive if Ax implies IIAxllrr otyp(-~sF(~)) [[Ax[[ if Ax � ~ - F ( ] ?) this this implies [lAX[[HIl � ___ IIAxll [tAxi].. Summarizing Summarizing -<sp(y) �_< IIAxll we we get get F(Y ) otyp( 1.3.11. 1.3.11. Theorem. Theorem. Axl1 = IIAxll Let PA rr l ' Let PA � C_ Ax Ax then then IIIIAxllIIAxll,]. The computation of ordinal IIAxl1 commonly called called the The computation of the the ordinal IIAxil is is commonly the ordinal ordinal analysis analysis of of Ax. In In view view of Theorem 1.3. 1.3.11 we also also talk talk about about aa nt-analysis II~-analysis of Ax. To To explain explain Ax. of Theorem 11 we of Ax. the connection between rrlI and .3.4 which the connection between IIAxll liAxi[n] and IIAxl1 ]lAx[] ",C ~cK we use use again again Theorem Theorem 11.3.4 which l K we E1Ell l says says (V)~)F()() NN F~ (VX)F(X) K ) [ p F(X) (3aa < < Wf w~K)[B F(2)].]. (3 r {:} (25) Assume .c� within Assume that that we we have have coded coded the the language language/::~ within Set Set Theory Theory (c.f. (c.f. Barwise Barwise [1975]). The The fact fact that that "z "z is is an an infinitary infinitary proof proof tree tree for for F()~) of of length length � _< a" a" can can be be expressed expressed l r by by aa � A0o formula, formula, say say G G(a, It is is easy easy to to check check that that for for a a E e On On and and zz E e L L (a , zz,, rF(X)7).) . It r l such that that L L F ~ G G(a,z, we have have zz E e La+ La+nn for for some some n n < w. w. If If we we take take into into such F(X) ) we (a , z, rF()~)7) additional number account account that that F()~) may may contain contain additional number parameters, parameters, say say fL g, (25) turns turns into into F(X) F(X) - - F(X) [N (Vii) [N F k (VX)F(X,ii) {:} ** (3a < wfK ) (3z E LwpK) G (a, zz,, rF(X,ii)l)] . This This is is the the well well known known Hyperarithmetical Hyperarithmetical Quantifier Quantifier Theorem Theorem telling telling that that every every n� -formula is II]-formula is equivalent equivalent to to aa � El-formula over L LucK. If we we put put l -formula over wCI K . If := min [His1 := min {a {c~ lI La L~ F ~ H H}} IHb for -sentences H for �l El-sentences H we we get get (26) (26) r ttc:((V)()F()~)) F (X ) ) bI �_ ttc((V)~)F()~))+ (3�) (3 z) G (� , zz,, rF()~;)[:s c((VX)F(X)) �< 1I(:t[)(:tz)G([, c(ev'X)F(X)) + nn for for some some n n < < w w.. Defining Defining I (3�) (3z)G(�, z, CI K := Ilnxll E",~o~ : = sup sup {{l(3~)(3z)G(~,z, IIAxl1 I l r -) F(X )l ) b lI AX rF(X)')l=, nxl-1r F(X F(X)} } - for for aa /.c�-theory ~ - t h e o r y Ax Ax we we get get CK = (27) IIAxl1 IIAxli E",,c~ = IIAxllrrl IIAxllHiI = = IIAxll IIAxil ·. (27) E1l l It It is is evident evident that that the the ordinal ordinal IIAxl1 ilAx][ ",C ~cK for .c�-theories L~-theories is is the the exact exact counterpart counterpart of of K for I E]Ell I the the ordinal ordinal IIAxl1 ]]Axl[~cK which we we defined defined in in the the previous previous section section for for theories theories in in the the K which I E",C I language .c( E ) of language/::(E) of Set Set Theory. Theory. 230 230 W. W. Pohlers Pohlers Analogously to Theorem Theorem 11.2.5 we get get 22 Analogously to .2.5 we Ax llrrl Ax + ll rr l = + F Fllnl = IIIIAxII,I IIIIAx (28) (28) for for all all true true ��-sentences ~-sentences F F with with the the same same proof. proof. We beginning of We mentioned mentioned already already in in the the beginning of the the section section that that C� s can can be be regarded regarded as as sublanguage of of sC( E ) . Call Gall F F a a Tq-sentence H~-sentence of of C s if if it it is is obtained obtained as as aa translation translation aa sublanguage of C� , i.e., of aa Tq-sentence II~-sentence of of/2~, i.e., if if it it has has the the form form ((Vx)[x C_ w w -t --+ G(x)] G(x)] where where G(x) G ( x ) is is a a V'x) [x � �o-formula numbers. If A0-formula whose whose quantifiers quantifiers are are all all restricted restricted to to w w or or to to natural natural numbers. If Ax A x is is theory extending extending KP K P then then we we can can use use the the familiar familiar unsecured unsecured sequences sequences argument argument aa theory F(x) there -definable order to to show show that that Ax A x proves proves that that for for every every Tq-formula H~-formula F(Z) there is is aa �o A0-definable order relation relation - Ax A x ef- ((VZ)[Wf(-4F(~)) ++ (3� (3~ < < L L~cK)IF(~, Z)] wfK )IF(�' x)] V'x) [WI( - otyp( - (29) (29) for sufficiently strong for sufficiently strong theories theories in in the the language language of of Set Set Theory. Theory. This Ax l1 and Ax l 1 l::w~o~ This rough rough sketch sketch should should suffice suffice to to explain explain that that the the ordinals ordinals III[Axll and IIIIAxl[ C 1K 1 carry carry the the same same information. information. So So Tq-analysis 1-I~-analysis and and wfK-ordinal v~X-ordinal analysis analysis are are the the same same things. will see indeed aa good good deal contained in Ax II . things. We We will see that that indeed deal of of information information is is contained in IIIIAx[I. 1.4. 1.4. Methods Methods Before Before we we come come to to examples examples of of analyzed analyzed theories theories we we want want to to outline outline the the methods methods used used in in ordinal ordinal analyses. analyses. There There are are two two main main steps. steps. The The first first is is to to compute compute upper upper bounds bounds for for the the ordinals ordinals in in specrr specnll (Ax) (Ax),, the the second second to to show show that that these these bounds bounds are are the the best possible ones. We explain best possible ones. We explain the the general general pattern pattern on on the the example example of of an an C�-theory s Ax. Ax. If If Ax A x f~- F F then then there there are are formulas formulas A A I1,, .. .. .., , An An E E Ax A x such such that that A 1t ,, .. . .. ,. A, A n ~nFf- F A (i) (i) in in pure pure predicate predicate logic. logic. By By Gentzen's Gentzen's Hauptsatz Hauptsatz we we may may assume assume that that this this derivation derivation is will be (i) into is cut-free. cut-free. It It will be quite quite easy easy to to transform transform (i) into aa truth truth definition definition ~A1, . . . , - , A n , F (ii) (ii) 2Defining 2Defining Ax Ax + + (3X)H(X) (3X)H(X) I~- G :{:} :r Ax Ax I~ -,H(X) -~H(X) V VG G this makes makes sense sense even even without aa notion of proof for Second Second Order Logic. Logic. 231 231 Set Theory Theory and Second Order Number Theory F. where AI, . . , An where 0:: a will will depend depend mainly mainly on on the the complexity complexity of of the the formulas formulas A1,..., AN and and F. A for Then Then we we have have to to compute compute upper upper bounds bounds for for tc(A) for all all formulas formulas in in Ax. Ax. This This gives gives . tc( ) Ai � A; (iii) (iii) tc( ) A; :S for i Ee {I, . . . , n} if if tc(Ai) _ 0:: a,; for {1,..., n}.. The The problem problem is is now now to to link link (iii) (iii) and and (ii). (ii). This This will will be be achieved achieved by by extending extending the the truth truth definition definition � ~ A into into aa semi-formal calculus calculus � ~ A by adding the by adding the cut cut rule rule (cut) (cut) � ~ A� , F, ;F; � ~ A� , - ,- -,F; F ; /3 /31,/32 /32 < /3 rk(F) � � �. and r k ( F ) < =:~ ~ A . This will of meaning as will still still have This will of course course destroy destroy its its meaning as aa truth truth definition. definition. But But we we will have � FF F {::} � F . (iv) By By (ii) (ii) and and (iii) (iii) we we obtain obtain � FY (v) (v) where where/3/3 and and p p are are computable computable from from 0:: a l1,> . . . , , O:: ann and and 0:: a and and the the problem problem reduces reduces to to the system. Since semi-formal system the elimination elimination of of cut cut in in the the semi-formal semi-formal system. Since the the semi-formal system is is obviously sound we obviously sound we get get . • . � FF =~ N N ~FF => F =~ (35<w~ =V (30 (35<w7 < wrKK))�~ FF => (30 < wrKK))�~ FF => showing showing that that the the Cut-Elimination Cut-Elimination Theorem Theorem holds holds for for the the semi-formal semi-formal calculus. calculus. But But this this is is of of little little help help since since we we do do not not know know how how to to compute compute 05 from from /3 /3 and and p. p. In In aa moment, moment, however, however, we we will will see see that that for for predicative predicative theories theories we we can can sharpen sharpen Cut CutElimination to Elimination to �� => �A.,o�. (vi) (vi) By By (v) (v),, (vi) (vi) and and (iv) (iv) we we get get c(F) :S< CP~p/3 (vii) (vii) ttc(F) p/3 /3 and for for Ax Ax � ~ F F.. Since Since/3 and p p only only depend depend on on the the formulas formulas in in Ax Ax this this will will give give an an possible bound bound it upper Ax ll n Il . To upper bound, bound, say say r10, for for IIIIAxlln~. To show show that that 10r is is the the best best possible it suffices suffices to well-ordering -< to prove prove that that for for every every 0:: c~ < < 10r there there is is aa primitive primitive recursive recursive well-ordering -< such such that that 0:: c~ :S < otyp(-<) and and Ax A x ~� T I ( ~ , X ) . It It will will become become clear clear from from the the following following text text how how this this concept concept has has to to be be modified modified as £( E ) of as to to serve serve also also for for theories theories in in the the language language L:(~) of Set Set Theory Theory and and for for impredicative impredicative theories. But before before that that we the example theories. But we demonstrate demonstrate some some details details on on the example on on Gentzen Gentzen's's result. result. otyp(-<) T/ (-<, X ) . 2. First 2. F i r s t oorder r d e r number n u m b e r theory theory 2.1. 2.1. Peano P e a n o arithmetic arithmetic The The paradigm paradigm for for ordinal ordinal analysis analysis is is still still Gentzen Gentzen's's result result on on Peano Peano Arithmetic. Arithmetic. So So we we opt opt for for it it as as our our first first and and simplest simplest example. example. However, However, rather rather than than to to analyze analyze 232 W. Pohlers Pohlers W. 232 PA itself itself we we will will analyze analyze aa conservative conservative extension extension N NT of PA PA which which allows allows constants constants PA T of NT. for all all primitive primitive recursive recursive functions. functions. We We start start with with an an introduction introduction of ofthe theory NT. for the theory 2.1.1. TThe axiom ssystem NT 2.1.1. h e axiom ystem N T The The language language sCN is is aa first first order order language language which which contains contains set set parameters parameters denoted denoted X , Y, Y , Z, Z, X1, Xl , ... . . . and and constants constants for for 00 and and all all primitive primitive by capital capital Latin Latin letters letters X, by recursive functions functions and and relations. relations. We We assume assume that that the the symbols symbols for for primitive primitive recursive recursive recursive for the the constant constant function, function, P~ for the the functions are are built built up up from from the the symbols symbols C~ functions ar: for pr: for for the the successor successor function function by by aa substitution substitution projection on on the the n-th n-th component, component, SS for projection Sub and and the the recursion recursion operator operator Rec. Rec. operator Sub operator The theory theory N NT comprises the the following following sentences: The T comprises sentences: The successor successor axioms axioms The (V'X) [-,Q == Sx] y] (V'x) (V'y) [S(x) == sS(y) ( y ) =} x 9 == y] The axioms for for function and relation relation symbols The defining defining axioms function and symbols which which are are the the universal universal closures closures of the following formulas of the following formulas ( z , , > ". . ". , xn) z,) = Is. k_ = ar: (Xl pr: ' " , xn) P~ (Xl> (Xl,..., x.) = = Xk x~ Sub(g, ) Sub(g, hl> hl, ". . .. ,, h hm ) ( X l , ". . .. ,, xxn) ,) = = g(h g ( h ll (Xl> (Xl, '. .. .. ,, xxn) , ) ) ) ." ." . (h (hm X ll> , '. .. .. ,, xxn) ,)) ) m((x m (xl> Rec(g, (O, Xl> R e c ( g , h) h)(O, x l , .. .. .. ,, xn) x,) = = g g ((XI, x l , .. .. .. ,, xn) x,) Rec(g, R e c ( g , h) h ) ((Sy, S y , Xl> x l , ". . .. ,, xn) x,) = = hh(y, ( y , Rec(g, Rec(g, h) h ) ((y, y , Xl Xl,> '. .. .. ,, xn) x , ) , , x~, x,) Xl> .. .. .. ,, xn) , (X ( Xll >, .' .". , X, xn) n) E e R R H ~ XR ) ( . R(X ( Xl l ,' . ". . , ,Xxn) n) = = 00 The The scheme scheme of of Mathematical Mathematical Induction Induction F (Q) /\A (V'x) F(O) ( V x ) [[F(x) F(x) F (S(x))] (V'x)F(x) -+ --+ F ( S ( x ) ) ] - +-+ ( V x ) F ( x ) for F(u) for all all CN-formulas s F(u). . 2.1.2. NT) 2.1.2. An A n upper u p p e r bound b o u n d for for spec specrq rr ,, ((NT) Following Following the the general general pattern pattern as as sketched sketched in in the the previous previous section section we we have have first first to to deal deal with with the the truth truth complexity complexity of of logically logically valid valid sentences. sentences. Therefore Therefore we we have have to to fix fix aa calculus calculus for for pure pure predicate predicate logic logic and and we we opt opt for for aa cut cut free free Tait Tait calculus, calculus, i.e., i.e., one one sided sided sequent sequent calculus, calculus, which which is is given given by by the the following following rules: rules: 2.1.2.1. 2.1.2.1. Definition. Definition. ((AxL) AxL) (v (V)) (/\) (A) A for ~ - A, A , A, A , --, ~A f o r any a n y m, m , if if A A is is an an atomic a t o m i c formula formula � If I f p. ~2_ A, A, A A ii for f o r some s o m e ii E E {l, {1, 2}, 2}, then then � ~-- A, A, A A1l V V A A22 for f o r all all m m > > mo mo 2}, then If If P ~-- A, A, A A ii and and m m ii < < m m for f o r all all ii E E {l, (1,2}, then � ~ - A, A, A A1l /\ A A A22 233 233 Set Theory and Second Order Number Theory (B) (3) (V) (V') If If � ~ �, A, A(t) A(t),, then then F ~- �, A, (3x)A(x) (Bx)A(x) for for all all m m > mo If not free If � ~2_ �, A, A(u) A(u) and and uu not free in in � A,, (V'x)A(x) (Vx)A(x),, then then F ~- �, A, (V'x)A(x) (Vx)A(x) for for all all m m >>mmo o. · The The identity identity axioms axioms are are the the following following formulas formulas (w:)[x = = x] ~] (V'X)[X (V'x) ( w ) ( v(V'y)[x y)[~ = = Y y ---+ -+ Y y = = x] ~] ---+ x9 = ( w ) ( v(V'y) y)(V z ) [ [x ~ = = Y y 1\ ^ Y y = = z z -+ = z] z] (V'x) (V'z) (V'X) (V'Y) [XI = (V..~)(Vy~[xl = Yl y , 1\ A .. ... . 1\ A Xn xn = - - Yn ---+ "-'+ f f ( x(xI, t , . . . ,.x.n.) , xn ) = -- f f ( y(y , , I. .,. " " ,Yn)] Yn)] ( v ~ ) ((V' v ~Y) [ x[XI , = = Yl y, 1\ ^ . ... . 1\ ^ Xn x,, = = Yn y,, ---+ -+ (( R R ((~Xl , , ,. .. .. . ,,~,,) ~ R(y R ( y ,I, ,. ... . . ,,y,,))] (V'X) xn) ---+ Yn))] (V'x) (V'y) [x = (W:)(Vy)[~ = Y y ---+ -+ (x (x E e X X ---+ --+ Y y E e X)] X)] ' s Hauptsatz Due to to Gentzen Gentzen's Hauptsatz we we have have the the following following theorem: theorem: Due Let Let � A be be aa finite finite set set of of formulas formulas such such that that V V � A is is valid valid in in the sense of the sense of first first order order predicate predicate logic. logic. Then Then there there are are finitely finitely many many identity identity axioms axioms h ,..·,I F ,1 , ,In ' � I1,..., Inn and and an an m m < <w w such such that that ~--~I1,.. A. 1 , . . ..,-,In, 2.1.2.2. 2.1.2.2. Theorem. Theorem. ' Let containing all occur free Let i1 t7 be be aa list list containing all number number variables variables which which occur free in in �. A. An An easy easy induction using the the fact induction on on m using fact (30) (30) A (~) ~,~d ~ = t ~ =. ~ A (t) shows shows r- A(~) ~ (31) ~ A (~) for for every every tuple tuple ii g of of numerals. numerals. We We have have � (V'x)(V'y)[x (w)(vy)[~ = = y y (x Ee X -~ (~ x ---+ -+ Y y E e X)] x)] ---+ and and all all the the other other identity identity and and defining defining axioms axioms for for primitive primitive recursive recursive functions functions and and relations are arithmetical sentences. relations are true true arithmetical sentences. Thus, Thus, using using also also (16) (16),, we we have have tc(F) tc(F) < < w w (32) (32) for for all all mathematical mathematical and and identity identity axioms axioms except except induction. induction. What What really really needs needs checking checking is is the the truth truth complexity complexity of of the the scheme scheme of of Mathematical Mathematical Induction. Induction. Here Here we we need need the the following following lemma. lemma. 2.1.2.3. 2.1.2.3. Tautology T a u t o l o g y Lemma. Lemma. For For every every .e s N -sentence we we have have 12.rk(F) [2.rk(F) � A, ~F, F F.. , ,F, The induction on The proof proof is is immediate immediate by by induction on rk(F) rk(F).. The Mathematical Induction The truth truth complexity complexity for for all all instances instances of of Mathematical Induction follows follows from from the Induction the Induction Lemma. Lemma. 234 W. Pohlers Pohlers W. 234 2.1.2.4. 2.1.2.4. Induction Induction Lemma. Lemma. F ( n ) we we have have F(ll) For For any any natural natural number number n n and and any any eN s -sentence 2. [rk( F(!!))+n] -+ F(S(x))], F(S(x))], F( F(n_.). ...,~F(0),-~(Vx)[F(x) F (Q.) , ..., (\fx) [F(x) -+ ll) . 1I (33) (33) The 20) For 33) asas The proof proof by by induction induction on on n n is is very very similar similar to to that that of of ((20). For n n= - 00 we we get get ((33) . an the Tautology Lemma. For the induction an instance instance of of the Tautology Lemma. For the induction step step we we have have 2. [rk(F(!!))+n] ~F(0), ( V x ) [[F(x) F ( x ) -+ F(S(x))], F(S(x))], F( F ( nll)) F (Q) , ..~.,(\fx) 112"['k(F(~-))+"l ..., by the the induction hypothesis and and obtain obtain by induction hypothesis 2.rk(F(n)) ~F(0), ~(Vx)[F(x)-+ F ( S ( x ) ) ] ,, ~ ( S ( n ) ) ,,FF(S(n)) (S(n)) F (Q.) , ..., (\fx) [F(x) -+ F(S(x))] ""FF(S(n)) 1j2.~k(F(~_)) ..., (i) (i) (ii) (ii) by by the the Tautology Tautology Lemma. Lemma. From From (i) (i) and and (ii) (ii) we we get get 2 [rk(F(n))+n + ~F(0),--,(Vx)[F(x) ~ F(S(x))], F(S(x))], F( F ( nll)) /\ A ..., ~ FF((S(n)) S ( n ) ) , , F(S(n)) F ( S ( n ) ) .. (iii) (iii) F (Q) , ..., (\fx) [F(x) -+ ' ] 1 ..., 112"[~k(F(~-))+"]+l By By aa clause clause (S) we finally finally obtain obtain (3) we 2.[rk(F(!!)) +n]+2 ~F(O), -~(Vx)[F(x) -+ F ( S ( x ) ) ] , F ( S ( n ) ) . [3 o ..., F (Q) , ..., (\fx) [F(x) -+ F(S(x))], F(S(n)) . 1 ]2.[,k(F(n_))+n]+2 By (G) �_~ w ++ 44 for By Lemma Lemma 2.1.2.4 2.1.2.4 we we have have tc tc(G) for all all instances instances G G of of the the Mathematical Mathematical Induction Scheme. Induction Scheme. Together Together with with (32) (32) we we get get (34) tc(F) (34) tc (F) �g w ++ 44 w w for for all all identity identity and and non-logical non-logical axioms axioms of of NT NT.. If {Fl , . . . , Fn If NT NT � ~- F F then then there there are are eN-sentences s {F1,..., Fn}} and and aa natural natural number number m such such that that (35) (35) ~- -,F~, . . . , ~Fn, F Fi and . . . , n} and for for all all ii E {I, {1,..., n} the the formula formula Fi is is either either an an axiom axiom in in NT N T or or an an identity identity axiom. F axiom. For For every every eN-sentence s F such such that that NT N T ~� F F we we thus thus get get by by (35) and and (31) eN-sentences Fn such such that LN-sentences F F1,..., that l , . . , F~ (35) . (36 (36)) -~F1, . . . , -~Fn, F 34 and and by by ((34)) ~+' IIW+4 (31) 37) ((37) FF/i for . . . , n}. n} . for all all ii E {I, {1,..., As Section 1.4 the will be As sketched sketched in in Section the problem problem of of linking linking ((37)) and and ((36)) will be solved solved by by introducing introducing aa semi-formal semi-formal calculus. calculus. 1.4 37 36 2.1.2.5. which We ne � 2.1.2.5. Definition. Definition. We defi define ~ b. A for for aa finite finite set set b. A of of eN-formulas s which contain contain at at most most free free set-variables set-variables inductively inductively by by the the following following clauses: clauses: (AxM) If (AxM) I f b. An N D(N) D(N) =1= r 0q} then then � ~ b. A for for all all ordinals ordinals Q c~ and and p p.. Set Theory Theory and Second Second Order Number Theory 235 235 N then (AxL) If If tt NN = Ss N ~ �, A, ss ¢ r X, X, tt E X X for for all all ordinals ordinals a a and p. (AxL) then � and p. (A) (1\) Is � $ �, A, Ai A, and a . ~ .ai ,(V) (v) Ai and 2} then Al V If If � ~ �, t,,A, a , ~ .ao o (V) (V) If � and ai all i Ee N ~ �, A, A(i) X(i) a,~ ~, < .a for fo~ aUi N then t~. � ~ �, A, (Vx)A(x) (W)A(~).. (3) If If � ~- �, A, A(i) A(~) and a . ~ .ao o (cut) (cut) If 2} and rk(F) << pp then If � ~ �, A , FF,, � ~ �, A , ~of, F , ai < We We have have to to read read � ~ � A as: as" "There finitary derivation "There is is an an in infinitary derivation of of � A of of height height ::; (38) (3s) ~ {::} ~. � ~ �. ~ �� 3 (38) From 1 ) and From ((31) and (38) we we have have � �(u) '* (39) Fo � m for numbers. Another Another immediate for every every tuple tuple z z7of of numbers. immediate property property is is ~ A�, , � o~<_~, a ::; {3, pp<_a ::; a and and A C�Fr � =~ '* ~ Fr. . � (40) (40) The The calculus calculus � ~ is is obviously obviously sound. sound. By By induction induction on on a a one one proves proves easily easily (41) F I , .. .. .., , Fn Fn '* =~ N N F ~ (F (F1I V V ·...· · V Y Fn) Fn)[(I)] (41) [] � FI, for for every every assignment assignment (I):: set set variables variables --+~ Pow(N) Pow(N).. Soundness Soundness of of � ~ and and Complete Complete- (38) ness ness of of � ~ together together with with (38) show show 2.1.2.6. 2.1.2.6. Cut Cut Elimination E l i m i n a t i o n Theorem. Theorem. �~zx. �. If < Wf If � ~ � A then then there there is is aa 'Y 9/< w~Kx such such that that But, as mentioned before, .2.6 does analysis. The But, as mentioned before, Theorem Theorem 2.1 2.1.2.6 does not not help help us us in in ordinal ordinal analysis. The bound for bound for 'Y 7 is is much much too too large. large. What What we we are are looking looking for for is is aa function function which which computes computes aa value value for for 'Y 7 from from the the data data a a and and p. p. The The key key here here is is the the Reduction Reduction Lemma Lemma which which tells tells us us how how to to avoid avoid aa cut cut of of rank rank p p for for the the costs costs of of an an increasing increasing length length of of the the derivation. derivation. � of as = p. p. 2.1.2.7. 2.1.2.7. Reduction R e d u c t i o n Lemma. L e m m a . Assume Assume � ~ �, A, F F and and ~ r, F,-~F as well well as as rk r k((F) F)= 1"'~ # fjt~ A,F. Then �, r . Then ]pp 2236 36 W. Pohlers Pohlers w. The proof proof is is by by induction induction on on a a #~ (3. ft. Assume Assume first first that that F F is is not not the the main main formula formula The of the the last last clause clause in in the the definition definition of of f; ~ �, A, F. F. Then Then either either f; ~ �, A, and and hence hence also also of A, r, F, holds holds by by (AxM) (AxM) or or (AxL) (AxL) or or we we have have � ~ � Ai, for a aii < < a. a. But But then then I[p; #~ /i~ �, i FF for # # we get get by by induction induction hypothesis hypothesis I[p and obtain obtain I; [p ~ /i~ A by the the same same �,, Fr by we ;~'A //i �i', r,F , and ' O~ clause.The case that that of -~F is is not not the the main main formula formula in in � ~ r, F, of -~F is is symmetrical. symmetrical. clause. The case We may may therefore therefore assume assume that that both, both, F F and and of, -~F, are are main main formulas. formulas. Let Let us us first first We assume that that p p = = 0, i.e., that that F F is is atomic. atomic. Then Then both both f; ~ �, A, F F and and � ~ r, F,-~F are assume 0, i.e., of are axioms whose whose main main formulas formulas are are F F and and of, -~F, respectively. respectively. This This excludes excludes axioms axioms axioms according to to (AxM) because we we cannot cannot have have {F, {F,-~F} D(N).. But But then then we we have have according (AxM) because of} �C_ D(N) axioms according according to to (AxL) (AxL) and and we we may may assume assume that that F F is is aa formula formula tt E EX X.. But But then then axioms -~F == - tt ¢. ~ X X and and we we have have aa formula formula 8Sl1 ¢. ~ X X in in � A and and aa formula formula 8822 E EX X in in r F such such of that 8f SlN = - tt NN = = 8s~. But then then [p ~ ~ �, A, r F holds holds by by (AxL) (AxL) . that � . But Now assume assume 00 < < p. p. We We will will only only treat treat the the more more complicated complicated case case that that F F is is aa formula formula Now (Vx)A(x).. The The remaining remaining cases cases are are either either simpler simpler or or symmetrical. symmetrical. Then Then of -~F is is the the (V'x)A(x) formula (3x)oA(x) (3x)--,A(x) and and we we have have the the premises premises formula I; # /i A, F, F, A(i) � �, A(i) (i) (i) for all all ii E EN N and and for � r,F,-~F,-~A(/_o). f A (io ) o , o . (ii) (ii) From (i) (i) and and � ~ r, F, F F as as well well as as from from (ii) (ii) and and f; ~ �, A, F F we we obtain obtain by by the the induction induction From hypothesis hypothesis and and [1p<>p~,o'0 ## ~/i � A,, F, r , A(/o) A(10· ) (iii) (iii) A(10· ) . �, r, o-~A(/_o) Ip<> ~# 8o/io A,F (iv) (iv) aio # # from (iii) and (iv) (iv) by by cut. �, Fr from (iii) and cut. I;[p ~ ~/i A, a# a# a# rk(A (io ))) << rk(F) rk(F) == pp and and aio ~ fl(3 << a ~/3(3 as as a ~/30 (30 < (3, we obtain Since rk(A(/_o) Since as well well as < a # fl, we obtain [3 0 As the Reduction Lemma. As aa first first consequence consequence of of the Reduction Lemma Lemma we we obtain obtain the the Elimination Elimination Lemma. 2.1.2.8. Lemma. 2.1.2.8. Elimination Elimination L emma. 2" If ~<> � A then ~-~ �. A. If � then � a. The proof proof is is by by induction induction on on a. If If the the last last inference inference is is not not aa cut cut of of complexity complexity pp we we The obtain the the claim claim immediately immediately from from the the induction induction hypothesis hypothesis and and the the fact fact that that A~. obtain A� . 2~ is order order preserving. preserving. The The critical critical case case is is aa cut cut ~+~ � �', FFj; r.-~ � is A A F =~ =} ~.� --~2-7"~ A �', ~of � p+ 1 p+ 1 p+ 1 the Reduction Reduction Lemma Lemma we we obtain obtain with with rk(F) rk(F) -= p.p. By By the the induction induction hypothesis hypothesis and and the A and have 2~1 ~ 2~2 < 2max{~l,~2} 92 < 2~ [::! 0 and we we have p ~ 2~2 � 1I2~1 2"1 # 2"2 2E 2<>1 # 2<>2 :::; 2max{<>I,<>2 } · 2 :::; 2<> . The Elimination Elimination Lemma Lemma provides provides the the first first step step in in the the proof proof of of the the following following more more The general general lemma. lemma. 237 237 Set Theory and Second Order Number Theory 'Ppo �. 2.1.2.9. 2.1.2.9. Predicative P r e d i c a t i v e Elimination E l i m i n a t i o n Lemma. L e m m a . If If 10 18+~. � A then then l~ '8 A. rp+wP � 1 2" The induction on induction on The proof proof is is by by induction on p p with with side side induction on a. For For p p = 00 we we obtain obtain f{j ""2~-� A O by rst Elimination which, since claim. Now by the the fi first Elimination Lemma Lemma which, since 20 2a � < W w" = cpoO!, ~0a, entails entails the the claim. Now O. If assume p p > > 0. If the the last last clause clause was was not not a a cut cut of of rank rank :::: >/3f3 we we obtain obtain the the claim claim from from assume the induction and the the fact the functions the induction hypotheses hypotheses and fact that that the functions qOp are are order order preserving. preserving. Therefore Therefore assume assume that that the the last last inference inference is is O!. = = CPp ~ A,F 18+wp I8+wp ~: ~X,-~F ~ p rk(F) I8+wP ~ A rk(F) = rk(F) = al a aI , rk(F) ...,F . {3 F 'PP .__ ^(n+l) . 'PPO l I + 02 �. vp"i+~pa2 A If Ifwe define cp� ~ )Oc)O! ~ := O! c~ and and cp�n+ ~ then we we obtain obtain from from IIf38_i_wO..(n+l) )O!c~ :== cp~a(~(n)c~) we define u (cp�n)O!) then + wU ·(n+ l) 9 n • 'Pp OI +'Pp(2 ) �. lOn hypothesIs {3'P� +I>t(~Opal+~opa2) aa << p by by nn+ + I-fold 1-fold appilcatlOn application of of the the mam main mduct induction hypothesis 118 Finally Finally we we show show cp�n)(cp ~(~)(~pal ~pa2)) < < CPp ~paO! by by induction induction on on n n.. For For n n = = 00 we we have have pO!I ++ CPp0!2 cp� ) == CPp O!In(n+l)+-{- ~pO/2 CPpO!I ++ CPp0!2 O! since ~(aO0)\(qOpO/1 (/9pO~2) ~pO/1 qOpO/ since O! aii < < O! a and ~pO/ C r ( 0 ) . . For For the the and CPp CPp0!2 << CPp O! E Cr(O) cp�n+ cppO!I ++ CPp0!2 induction induction step step we we have have ~ l\(~pal ~pC~)) = = cp ~ u( ~(cp�n)(cp ( n ) ( ~ ppaO!lI + + CPp0!2 ~pa2)) ~pa since )) << CPp O! since induction hypothesis. O! by a <r such such such that that f3 ~ � _ rk(F) < < f3 ~+wP. But then then there there is is an an ordinal such that that rk(F) = f3 ~ ++rif> which, writing writing if>r in + wp • which, in Cantor Cantor normal normal form, form, means means rk(F) = f3 ~ ++W w U~1I + + ... .. +. +W w U~"n < < f3 ~+wP. := al, we n + Hence Hence al < < P p and, and, putting a := we get get rk(F) < < f3 ~+ + W wU~.• ((n + 1) 1).. By By the the side side .putting . r 11 'PP 2 A, 'PPOI Au, A A Is we y aa cut ows induction hypothesis we have have I~ F and and I~(3 0 u, ~F. B By cut Iitt 10 follows mductlOn hypothes . • '8 p ' • '8 . • • A. • E 0 p By Elimination Lemma By the the Predicative Predicative Elimination Lemma we we obtain obtain the the function function which which computes computes bound for cut free an an upper upper bound for the the height height of of the the cut free derivation. derivation. 2.1.2.10. 2.1.2.10. Elimination E l i m i n a t i o n Theorem. Theorem. tpPll{)P " 'CPPn Q 2 �. Then 0 A Then I PI +-I-"'" ...+ �~p �A such such that that p p ---NF w 0)Pl "~-W pr~• 9 wPn = NF Let Let i i0 F := max{rk(FI), . . . , rk(Fn)} Back NT. 36) , ((37), 38) Back to to the the axiom axiom system system N T . If If NT NT � ~- F then then we we obtain obtain by by ((36), and ((38) 37) , and . I~+4+n F for for m m := max{rk(F1), rk(Fn)} + + 11 < < w w By By the the Elimination Elimination Theorem Theorem . . · · · · l 1 h (or even m-fold application of the Elimination Lemma) we obtain I ~p~(~''~ + 4 + n) F . app Icat On 0 t e ImmatlOn emma we 0 tam 0 b f or even m-lO El L r Id ( m Hence xp w , wco ++ 44 ++ n) n) < Hence tc(F) � _ eexpm(w, < co e0 and and we we have have 1�+4+n F tc(F) " 9 9 ) ( 2.1.2.1 1. Theorem. 2.1.2.11. Theorem. lexpm(w,w + 4 + n) F i 0 9 sspecH~(NT) p ecn l NT � C_ co e0.. ( ) NT) 2.1.3. 2.1.3. Lower Lower bbounds o u n d s for for specn specH~, ((NT) I We bound given We want want to to show show that that the the bound given in in Theorem Theorem 2.1.2.11 2.1.2.11 is is the the best best possible possible one. one. By By Theorem Theorem 1.3.10 1.3.10 it it suffices suffices to to have have Theorem Theorem 2.1.3.1 2.1.3.1 below. below. O! 2.1.3.1. For 2.1.3.1. Theorem. Theorem. For every every ordinal ordinal ~ < < co Co there there is is aa primitive primitive recursive recursive well-order well-order -< -.( on on the the natural natural numbers numbers of of order order type type ~ such such that that NT NT � ~ TI( 71(-K, X ) .. -<, X) O! 238 W. Pohlers W. 238 2.1.3.1 The The first first step step in in proving proving Theorem Theorem 2.1.3.1 isis to to represent represent ordinals ordinals below below 60 co by by primitive primitive recursive well-orders. well-orders. This This isis done done by by an an arithmetization. arithmetization. We We simultaneously simultaneously define define recursive aa set On c_ � 1N N and and aa relation relation aa -< � bb for for a, a, bb EE On On together together with with an an evaluation evaluation map map set On On ----t~ OOn such that that On On and and -~ � become become primitive primitive recursive recursive and and aa -< � b b r¢? la] 1" n such lal << 1 · I"1 : On put Ibl . We ]b]. We put and 101 = 0 9 00 EEOOn n and]O]=O • • On and and ](zt,...,zn)] then (zl,...,zn) (Zl " " , zn) EE On l (zl " " , zn)1 == � � zn Zn then � On On and and zl z1 ~_'" 9 If ~" IJ zl,...,zn zl , . . . , zn C_ ...+ + wlz-I wlzal w 1znl W I Zll ++ ... • • • and and a � b : 4~ ¢? a a EEOOn [(aO=A b0r1\ b =I- 0) 9 a- • Observe that On On and and -< recursion and and Observe that are defined defined by by simultaneous of values values recursion � are simultaneous course course of thence are It is is also also easy easy to to check lal < Ibl. thence are primitive primitive recursive. recursive. It check that that a a � -< bb ¢? r ]a I < ]bI. The (On, �) is well-order of of order may therefore represent The order order (On,-<) is aa well-order order type type cr o . We We may therefore represent Thus we every ordinal < co by an segment - .. ... • X :¢? . ~ (Vm€ ( v ~ ) [ ~ <<~ -a~ e--+ x ] € E X] a EE X] X] Prog(X) (Va) [a c_ �X X -+ Prog(X) :¢? :r (Vt~)[oL --+ oL 9 a ~ c� x • • T/(a, 7 - t ( ~ , X) x) :¢? :~ Prog(X) p r o g ( X ) --+ ~ a ~ � c X x Our Since T/(O, holds trivially trivially and Our aim aim is is to to show show T/(a TI(a,X) for all all a a < < co. Co. Since TI(O,X) and X) holds , X) for n (w, 0) co eo = = sup sup {{ exp ezp"@, 0)]I n n E e w} w},, we we are are done done as as soon soon as as we we succeed succeed in in proving proving " , X) NT X) =} N T ~f- T/(a, Tl(a,X) =~ NT N T ~f- T/(w TI(w",X) (42) (42) 9 because X) and because NT N T f~- T/(a, TI(a,X) and (3 fl < < a a obviously obviously entails entails NT N T f}- T/({3, TI(fl, X) X). . The The first first observation observation is is (43) (43) N T ~- F(X) =~ N T ~- F({x I G(x) }) for }) is for all all CN-formulas /:N-formulas G. G. The The formula formula F({x F ( { x l l G(x) G(x)}) is obtained obtained from from F(X) F(X) by by replacing all occurrences ( t) and those of G (t) . To replacing all occurrences of of tt E EX X by by G G(t) and those of tt ¢. r X X by by ..., -,G(t). To prove prove ((43) assume assume 43) NT N T f}- F�) F(X) and and let let 6 ~ be be an an arbitrary arbitrary Cwstructure /:N-structure and and <1> (I):: set-variables set-variables signment signment such such that that | p ~ NT[ NT[(~]. 6 <1>] . ----t (i) m Pow pow(( ~6) ) an an as as(ii) (ii) Set Theory Theory and Second Second Order Order Number Theory 239 We We have have to to show show } ) [<1>J . 6 F( {x I G(x) r F I= F({xl a(=)})[,I,]. (iii) (iii) Define Define aa new new assignment assignment .= w(Y) ~ ( z ) .:= <1>(Y) {{@(Y) {n E e6 ~11 6 ~ F I= G(x) a(=)[n, ~]}} {n [n, <1>l if Y Y # =~X X if otherwise. otherwise. Then Then 6 [WJ iff F( {x l G(x) } ) [<1>J . | F ~ F(X) F(X)[@] iff 6 | F ~ F({zl a(z)})[,I,]. (iv) (iv) We We claim claim | F ~ NT[WJ NT[~].. 6 (v) (v) Then (v) (v) together together with with (i) (i) and and (iv) (iv) prc..ve prove (iii). (iii). To To check check (v) (v) we we have have only only to to take take Then which contain contain the This can only happen care care of of formulas formulas in in NT N T which the set set variable variable X X.. This can only happen in in instances identity axioms. instances of of the the scheme scheme of of Mathematical Mathematical Induction Induction or or in in identity axioms. Let Let I(X) :r H(X,O) (Vx)[H(X,x) -+ H(X, H(X,S(x))]--+ (Vx)H(X,x) x) [H(X, x) -+ S(x))J -+ ('v'x)H(X, :{:} H(X, J(X) Q) 1\A ('v'x) be Mathematical Induction. be an an instance instance of of Mathematical Induction. We We have have } ) [<1>J . 6F ~ / (Jx(X) ) [ ~ ][wJ iff ~ff 6 ~ F I= J1({~1 C(x)))[~]. ( {x l G(x) (vi) (vi) The } , x) is The right right formula formula in in (vi) (vi),, however, however, holds holds by by (ii) (ii) since since H({x H({x]l G(x) G(x)},x) is also also aa 0 formula formula in in NT. NT. Instances Instances of of identity identity axioms axioms are are treated treated analogously. analogously. O The importance of Mathematical Induction The above above proof proof shows shows the the importance of formulating formulating Mathematical Induction as as aa scheme. scheme. Let Let := {a J(X) s(x) := {o~1l ('v'�)[� (v,~)[,~ � c X x -+ ~ �,~ + + W ~oO � c Xl} x]} denote X. Then, denote the the jump of of X. Then, if if we we assume assume NT N I l -� - Prog(X) Prog(X) -+ Prog(J(X) Prog(J(X)),) , (i) (i) we we obtain obtain NT (WO , X ) . N T ~� - T/(a, Tl(a, J(X)) J ( X ) ) -+ --+ T/ Tl(w",Z). (ii) (ii) To (ii) assume To prove prove (ii) assume (working (working informally informally in in NT) NT) T/(a, Tl(a, J(X)) J ( X ) ) , , i.e. i.e. Prog(J(X))) -+ a a � C_ J(X) J(X) Prog(J(X) (iii) (iii) which which entails entails Prog(J(X) Prog(J(X))) -+ --+ a aE e J(X) J(X).. Choosing �~ = definition of jump turns Choosing - 00 in in the the definition of the the jump turns (iv) (iv) into into Prog(J ( X ) )) -+ -+ W w O'~ � C_ X, X, Prog(J(X) which, which, together together with with (i) (i),, gives gives (iv) (iv) (v) (v) 240 W. Pohlers 240 Prog(X) -+ --+ w'" w~ � c_ X, X, Prog(X) which which is is T/(w"' TI(w~,X). Once we we have (ii) we we also also get get have (ii) , X) . Once Tl(a,,.7(Z)) by (43). (43). T/(a , 3(X)) by (vi) (vi) (42) because because T/(a, T l ( a , XX) ) implies implies (42) It we work It remains remains to to prove prove (i). (i). Again Again we work informally informally in in NT. N T . Assume Assume Prog(X) Prog(X).. (vii) (vii) [a � We We want want to to prove prove Prog(3(X)) Prog(,.7(X)) i.e. i.e. (Va) (Va)[a C_ J3(X) ( X ) -+ -+ a a E 3(X)] fl(X)]. . Thus, Thus, assuming assuming also also aa � c_ 3(X) 3"(X),, (viii) (viii) we have have to to show show a a E 3(X) 3"(X).. i.e. i.e. (V�) (V{)[{ C_ X X -+ + �{ + + w'" w~ � C_ X] X].. That That means means that that we we we [� � have to to prove have prove TJ 77 E X X under under the the additional additional hypotheses hypotheses C_ X (ix) (ix) < < �+ + w"' . (x) (x) and and TJ n If TJ 77 < < � ~ we we obtain obtain TJ r/ E X X by by (ix) (ix).. Let L e t ~�_ � < r TJ/ << ~ +�w+ w"'~.. If I f aa= = 0 t h0e the ~ = ~TJa n=d w If � eand we obtain obtain TJ r/EE X X by by (ix) (ix) and and (vii) (vii).. If If a a > > 0 then then there there is is a aa a < < a a and and a a natural natural number number U U (i.e. W~ + =: W (i.e. a a numeral numeral in in NT), N T ) , such such that that �~ < < w + . .. .. . + + W w% =: w U~. n (c.f. (c.f. the the proof proof of of the the 0 • � n n --f fool l dd n Predicative Predicative Elimination Elimination Lemma). Lemma). We We show show aa < n. n (xi) (xi) n 0 n 1 by := m by induction induction on on n. For For n = - 0 this this is is (ix) (ix).. For For n := m + + 1 we we have have + w w u~.' m m � C_ X X �+ (xii) (xii) by by the the induction induction hypothesis. hypothesis. From From a a < < a a we we obtain obtain a a E 3(X) J ( X ) from from (viii). (viii). This This together together with with (xii) (xii) entails entails � ~+ + W wU~. n = = �( + + W wU~. m m + + W wU~ E X X. . This This finishes finishes the the proof proof 0 of of (i) (i),, hence hence also also that that of of (42) which which in in turn turn implies implies Theorem Theorem 2.1.3.1. [3 (42) • n • 2.1.3.1. Summing up Summing up we we have have shown shown 2.1.3.2. 2.1.3.2. Theorem. Theorem. (Ordinal NT) (Ordinal Analysis Analysis of ofN T ) IINT IINTlln~ ll rr l 2.1.3.2 = - EO co.' (24) The The proof proof theoretic theoretic ordinal ordinal of of the the axiom axiom system system As corollary of obtain As a a corollary of Theorem Theorem 2.1.3.2 and and (24) we we obtain 2.1 .3.3. G Gentzen's 2.1.3.3. e n t z e n ' s Theorem. Theorem. NT N T is is Eo ~o.. 2.1.2.11 2.1.3.1 2.1.3.4. Theorem. There -sentence (VX) (Vx)F(X, x) 2.1.3.4. T heorem. There is is aa m II~-sentence (VX)(Vx)F(X, x) which which is is true true in in the the standard F(X, standard structure structure N N such such that that NT NT � ~- F ( X , 11) n) for for all all n n EN N but but NT N T .JL ~ (Vx)F(X, (Vx)F(Z, x) x) .. As As another another consequence consequence of of Theorem Theorem 2.1.2.11 and and Theorem Theorem 2.1.3.1 is is 241 Set Theory and Second Order Order Number Theory x) x x D[3 To (X, x) :<=> To prove prove the the theorem theorem choose choose F F(X, :ca Prog(X) Prog(X) -+ --+ x E On On -+ --+ x E X. X. ' s Theorem. Theorem 2.1 2.1.3.4 is a a weakened weakened form form of of Godel Ghdel's Theorem. The The general general form form Theorem .3.4 is ' s Theorem Theorem says Theorem 2.1.3.4 already for of of Godel Ghdel's says that that Theorem 2.1.3.4 holds holds already for aa rI?-sentence H~ V ) F( ) . ((Vx)F(x). x x 2.1.4. Computational ITg-sentences 2.1.4. C o m p u t a t i o n a l complexity c o m p l e x i t y of of H ~ ' s result Heuristics. H e u r i s t i c s . From From the the point point of of view view of of minimal minimal models, models, Gentzen Gentzen's result appears appears surprisingly minimal model surprisingly complicated. complicated. Shouldn Shouldn't' t it it be be trivial trivial that that the the minimal model for for PA PA is obtained have been is obtained at at stage stage w? w? Consequently Consequently there there have been comments comments stating s t a t i n g -- faintly faintly jjoking oking- that that Gentzen Gentzen tried tried to to secure secure induction induction up up to to w w by by transfinite transfinite induction induction up up to tells us Since we to Eo. c0. However, However, (27) tells us that that this this is is not not at at all all the the case. case. Since we have have tC(F) tc(F) < < w for arithmetical sentences for all all arithmetical sentences we we get get the the least least ITo-model II~-model for for PA P A at at L L~w for for more more or or less reasons. But it is not at all trivial where to locate the least model for less trivial trivial reasons. But it is not at all trivial where to locate the least model for CK wwCK 1 -sentences of the 2: E11 -sentences which which correspond to the the provable provable IT II~-sentences of PA P A according the correspond to according 1 1 1 -sentences to the the Hyperarithmetical Hyperarithmetical Quantifier Quantifier Theorem. Theorem. All All we we know know without without proof proof theory theory is is to 's Theorem tells us that that it it must must be be at at some some ordinal ordinal :::; < Wf w~K~.. Since Since Gentzen Gentzen's Theorem tells us tc(F) tc(F) < < EO Co for -sentences we for all all provable provable ITt II~-sentences we know know that that this this will will be be at at L L~eoo.. Nevertheless Nevertheless it it is is an an unsatisfactory unsatisfactory situation situation that that the the ITt-spectrum II~-spectrum tells tells us us nothing nothing 's Theorem about the the provable provable arithmetical arithmetical sentences sentences of of Ax. Ax. By By Godel Ghdel's Theorem we we know know that that about there there are are arithmetical arithmetical (even (even IT� H ~ ) sentences sentences which which are are not not provable provable from from Ax A x while while by (14) we we have have tc(A) < w w for for all all true true arithmetical arithmetical sentences. sentences. Thus Thus we we are are far far from from tc(A) < by After all one has has the having having tC(F) tc(F) < < IIAx IIAxlll 1 => =~ Ax A x ~f- F F.. After all this this work, work, however, however, one the feeling feeling to to deserve deserve a a stronger stronger result. result. The The aim aim of of this this section section is is to to show show that that we we can can obtain obtain much much more more information information with with just just a a little little more more effort. effort. This This has has been been quite quite recently recently detected detected by by Weiermann Weiermann [1996] [1996] and and it it is is appealing appealing to to include include his his result result here here because the collapsing because it it is is inspired inspired by by the collapsing techniques techniques of of impredicative impredicative proof proof theory theory on on which put the which we we want want to to put the emphasis emphasis of of this this contribution. contribution. The The basic basic idea idea is is quite quite simple. simple. Recapitulating Recapitulating the the computation computation of of the the upper upper , , -.Fn' bound bound Eo Co for for specn specn~: (Ax) (Ax) we we see see that that we we started started with with a a derivation derivation � 0~ -.F1 -~F1,..., -~Fn, F F infinitary derivation and and ended ended up up with with an an infinitary derivation � ~ F F for for some some 0:: a < < Eo s0.. Call Call this this derivation derivation 7r. If (3y)A(y) ~. If we we assume assume that that F F is is a a 2:�-formula E~ (qy)A(y) then then 7r ~r cannot cannot contain contain applications applications of of the the (V)-rule. (V)-rule. Hence Hence 7r ~ is is finite finite and and we we may may hope hope that that the the depth depth of of 7r r can can tell tell something for the something about about the the size size of of a a witness witness for the existential existential quantifier quantifier in in (qx)F(x). So So there there are are two two things things to to take take care care of: of: - • • • (3x)F(x). Assure that the depth of a derivation majorizes witnesses for existential formulas bound for finite derivation 2. 2. Find Find aa method method to to extract extract aa bound .for the the length length of of the the finite derivation of of aa 2:� E ~-formula from from its its assigned assigned ordinal. ordinal. The possibility to ordinals bigger The realization realization of of 2. 2. requires requires the the possibility to collapse collapse ordinals bigger than than w w into into 1. 1. Assure that the depth of a derivation majorizes witnesses for existential formulas finite finite ones. ones. This This technique, technique, however, however, is is known known from from impredicative impredicative proof proof theory theory where where we we have have to to collapse collapse (recursively) (recursively) uncountable uncountable ordinals ordinals into into recursively recursively countable countable ones. All we ones. Once Once we we have have solved solved 2. 2. is is easy easy to to realize realize also also 1. 1. All we have have to to require require is is that that w. W. Pohlers 242 the the witnesses witnesses in in (3)-clauses (3)-clauses are are majorized majorized by by the the collapse collapse of of the the derivation derivation length. length. Later we we will will see see that that in in more more complex complex systems systems this this becomes becomes an an even even more more natural natural Later requirement. requirement. While nite number While functions functions which which collapse collapse ordinals ordinals of of transfi transfinite number classes classes into into lower lower transfinite number classes classes occur occur quite development of transfinite number quite naturally naturally in in the the development of notations notations for for impredicative ordinals above impredicative ordinals, ordinals, i.e., i.e., ordinals above F0, it it had had for for long long been been unclear unclear which which are are the the right right finitary finitary collapsing collapsing functions, functions, i.e., i.e., functions functions which which collapse collapse transfinite transfinite ordinals finite ones. ordinals into into finite ones. It It follows follows from from work work of of Buchholz Buchholz and and Wainer Wainer that that finitary finitary collapsing collapsing functions functions must must be be connected connected to to functions functions in in the the sub-recursive sub-recursive hierarchy. hierarchy. There There is is aa result result that that for for aa certain certain choice choice of of an an infinitary infinitary calculus calculus we we have have (roughly (roughly speaking) speaking) fo, � ~o (3x)F( (3x)F(n,x) (3x < < H H~(n))F(n,x) ,, (n))F(n, x) n, x) =}=v NN l=~ (3x F(x, H" for for quantifier quantifier free free formulas formulas F(x, y) y),, where where Ha is is the the a-th a-th function function in in the the Hardy Hardy hierarchy hierarchy (c.f. (c.f. the the article article by by Fairtlough-Wainer Fairtlough-Wainer in in this this volume) volume).. The The infinitary infinitary system system is is tailored tailored in in such such a a way way that that it it gives gives a a nice nice characterization characterization of of the the Skolem Skolem functions functions for which for rrg-sentences II~ which are are provable provable in in Peano Peano arithmetic. arithmetic. It It turned turned out, out, however, however, that that the the generalization generalization to to stronger, stronger, especially especially to to impredicative impredicative axiom axiom systems systems is is by by far not straightforward. far not straightforward. But But there there are are finitary finitary collapsing collapsing functions functions which which work work smoothly smoothly together together with with the the impredicative impredicative collapsing collapsing functions. functions. The The basic basic idea idea in in their their definition definition is is strikingly strikingly simple. simple. As As soon soon as as we we have have term term notations notations for for ordinals ordinals we we may may define define the the norm N(a) N ( a ) of of an an ordinal ordinal a c~ as as the the number number of of symbols symbols in in its its term term notation. notation. The The norm norm is is thus thus aa finite finite ordinal ordinal with with the the property property that that for for every every natural natural number number kk the the set set {c~ lI gN(a) (a) < < kk}} is is finite. finite. For For any any starting function G: G: N 1N-+ -~ N IN the the function function {a norm starting function 1 1 (3 r � (a) := := sup sup {{r1/J� U3) + + 11 ~ < < a ~ and and N((3) g(~) ::; <_G(N(a)) G ( g ( a ) ) }) U U {O} {0} 1/J (44) (44) is is then then a a finitary finitary collapsing collapsing function function for for ordinals ordinals possessing possessing aa term term notation. notation. Computational C o m p u t a t i o n a l complexity. c o m p l e x i t y . In In order order to to obtain obtain a a more more refined refined complexity complexity for for arithmetical sentences definition given Definition 1.3.1. arithmetical sentences we we refine refine the the truth truth definition given in in Definition 1.3.1. Here Here aa crucial part starting crucial part is is played played by by the the function function 1/J r � . To To specify specify it it we we have have to to choose choose the the starting function function G. G. There There are are symbols symbols for for all all primitive primitive recursive recursive functions functions and and relations relations ins ins e s N . The The ( tb . . . tn) is truth truth of of any any atomic atomic sentence sentence R R(tl,..., is thus thus decidable decidable in in just just one one step step which which means means that that the the starting starting point point in in the the measurement measurement of of computational computational complexity complexity must must be above recursive functions. be above the the primitive primitive recursive functions. Therefore Therefore we we introduce introduce aa function function P aa variant majorizes all variant of of the the Ackermann-Peter Ackermann-Peter function function - which which majorizes all primitive primitive recursive recursive definition of functions functions and and choose choose P as as the the starting starting function function in in the the definition of the the finitary finitary collapsing function function 1/Jw r := CP. collapsing e. := 1/J , tn ) P P 2.1.4.1. 2.1.4.1. Definition. Definition. We We put put Po(x) := Po(x) := 22Xx X+2) (X) PP.+, := p� (x) "= p(x+2)(x) n+l (x) 243 243 Set Theory and Second Order Number Theory and finally finally and P(x) Px(x). P x(x) . ( x) := P Then Then we we have have the the obvious obvious properties properties () n, (45) (45) (46) (46) xx < P Pn(x) for any any n, n x for x < y =a P,(x) < P,(y) for all n as as well well as as mm << nn =} =~ ( for all (47) (47) Pm x) for all xx.. Pro(x)
Every Every primitive primitive recursive recursive function function f f is is eventually eventually majorized majorized by by P. To see see this this P. To we we assume assume that that f f is is represented represented by by the the constant constant f and and show show by by induction induction on on the the definition definition of of "f is a constant for primitive recursive function" the the existence existence of of aa number number e fI such such that that e f "f is a constant for primitive recursive function" (48) (48) X l , . . . , xx#i). f ) . This holds f-tuples ((Xl,..., holds for for all all # #f-tuples This is is obvious obvious for for f f E {{ CI: C~,, PI: P~,, S} S}.. For For ff = = # eg, eh := max{ )J " . . ,,eh~} ehm} and Sub(g, . . . ,,hm) hm ) we Sub(g, hI, hi,... we put put ee := max{eg, eh,,... and obtain obtain by by the the induction induction hypothesis (45) - (47) hypothesis and and (45)(47) f(x) f(~) <
�l 1 n 1). N(a) = (49) 'TJi) + 1). N(a) ::= �)N(�i) E ( N ( ~ , ) + + N( N(r/,)+ (49) i=l This This defines defines aa norm norm for for all all ordinals ordinals < < r Fo. For For a a < < r Fo the the finitary finitary collapsing collapsing function function f ( x l , . . . , X # f ) < P~i(max{xl,... ,x#f }) . 0 := 71 i=1 is thus thus defined defined by by is o. 0 max {{ '¢w((3) ( N(a)) }} UU {O}, = max + r (a) ::= r + 1111 j3 ~ < < a a and and N(j3) N(~) � <_P P(N(a)) {0}, '¢w i.e., (44). We i.e., '¢w r := := '¢� OR in in the the sense sense of of (44). We introduce introduce the the relation relation < ~ "j3 : aa <«i r {::} a n d gN(a) ( a ) <� _ PP( (N(j3) N ( ~ ) ++g (N(�)) ~)). . aa <<~ j3 and We We obtain obtain the the transitive transitive closure closure «e <<~ of of «� <<} by by putting putting a «{+ l j3 : {::} (3'TJ) [a «{ 'TJ «� j3] and and (50) (50) 244 w. Pohlers Pohlers W. 244 a <<~/~. «{ f3 : ~<=> (3~)[~ (3n)[a <<~ «€ Z]. f3]. Instead of of aa <<0 «0 flf3 we we write write shortly shortly aa << « flf3 and and call call aa collapsibly collapsibly less less than than ft.f3. This This is is Instead justified by by justified (51) ce << fl =~ Cw(a) < r (51) It is is perhaps perhaps noteworthy noteworthy that that It { o «� a}. r'l/Jw ( a) = = max max {k[ k I 0 <<0k a}. We moreover moreover have have N(~) N(f,) == ~f, for for ~f, << w. w . Hence Hence We (52) � (aKKfl (a « f3 r<=> aa< (52) Using the we define of ~� A Using the collapsibly collapsibly less less relation relation we define aa refinement refinement [~ 1� A.6. of .6. for for ordinals ordinals as follows: follows: a, pp << F0 r0 as a, 2.1.4.2. Definition. 2.1.4.2. Definition. If.6. then 1� .6. for all all ordinals ordinals aa and and p.p. If A, s ~fj. X, X, tt e X for all and p. p. Iftt NN = sSNN then X for all ordinals ordinals aa and then I~1� .6., .6., Ai « aa for for ii = 1,1, 22 then then 1� .6., alAl At\ Z2. A2 • I~-p A, ai and and ai a, << I~ A, ifIf 1� If I~p Ai and «a for some 2} then then 1� Al VV A2. A2 • If¥~ .6., If A, Ai and ao ao << a for some ii E {1, {1,2} I~ .6., A, A1 If [~-~ ai < (AxM) i/ A fq n D(N) D(N) #=I- 00 then I~ A (AxM) (AxL) (AxL) (^) (t\) (V) (V) (\7') (V) (3) (cut) = s E = E E E N N then then 1� [~ .6., A, (3x)A(x) (3x)A(:r,).. -,F, ai rk(F) < IfI: 1� I~ .6., a, F, 1f¥ I~ .6., a, -F, ~, « << a ~ for :or ii E ~ {1, (1, 2} 2} and an,t ~k(t) < p p then th~n 1� I~ .6.. a. A Ass aa word word of of warning warning we we want want to to emphasize emphasize that that the the definition definition of of the the « << relation relation and and thus thus also also the the definition definition of of the the relation relation 1� [~ .6. A depends depends on on the the term term notation notation for for ordinals (actually (actually only ( a) ) and ordinals only on on the the norm norm function function N N(a)) and thus thus is is only only defined defined for for ordinals below ro ordinals below F0.o But But it it is is obvious obvious how how to to extend extend 1� [~ .6. A as as soon soon as as we we have have term term notations for notations for larger larger ordinals. ordinals. From From Definition Definition 2.1.4.2 we we obtain obtain 2.1.4.2 II~� .6., A, aa � <__ pp ::; < (1 a and and .6. A� Cr F =~ � I~ F (53) (53) immediately immediately by by induction induction on on a. a. The The next next observations observations will will help help us us to to aa better better understanding understanding of of the the relation relation 1� I~ .6.. A. First First we we observe observe I~A :=> ~ A ~ ~A. (54) (54) following collapsing collapsing In In contrast contrast to to the the calculus calculus lrn ~pO the the refined refined calculus calculus has has the the following property: property: 45 2245 Theory and Second Second Order Order Number Theory Set Theory I~~~ A and A C E ~ =~ II~ (55) (55) A. (55) Observation (55) is is immediate immediate by by induction induction on on a. a. The The crucial crucial point point is is that that aa Observation derivation of of aa set set of of I;� E~-formulas which which is is cut cut free free cannot cannot contain contain applications applications of of an an derivation (V)-rule. Hence Hence all all ordinal ordinal assignments assignments (~ in in this this derivation derivation increase increase in in the the sense of (\f)-rule. sense of the collapsibly collapsibly less less relation relation and and may m a y -- by by (51) and (52) thus be be replaced replaced by by 1/J r w(() ' 1) and 2 -- thus the On the the other other hand hand we we also also have have On (5 l~o (3x)F(x) (3x)F(x) I� for an an I;� E~ -sentence (3x)F(x) (2x)F(x) for (5 ) =} =:~ (3x n)F(x). (56) (56) N N� ~ (3x < < n)F(x). (56) (57) I~ D., A, F F and and N N � ~ F F for .for F F quantifier quantifier free free =} =~ I� [~ D.. A. (57) I� We prove prove (57) (57) by by induction induction on on a. a. If If F F does does not not belong belong to to the the main main formula formula of of the the last last We inference in in I� I~ D., A, F F the the claim claim follows follows immediately immediately from from the the induction induction hypothesis. hypothesis. If If inference the main main formula formula of of the the last last inference inference it it can can neither neither be be an an axiom axiom (AxL) (because FF isis the (AxL) (because sentence) nor nor an an axiom axiom according according to to (AxM) (since F F 1~ D(N) D(N)).). Thus Thus F F is is FF isis aa sentence) (AxM) (since either aa conjunction conjunction Fl Ft 1\ AF F22 and and we we have have the premises I~A, F, F, F; Fi for for ii E {I, { 1, 2} 2} and and either the premises I� D., 1N � ~ F; Fi for for some some ii E {1, {1, 2} 2} or or F disjunction Fl F1 V F F22 and and we we have have the the premise premise N F isis aa disjunction I~2- A, F, F; Fi and and N 1N � ~ Fi. Fi. In In both cases we we obtain obtain I� ]~ D. A by by two two fold fold application of I� both cases application of D., F, the induction induction hypothesis hypothesis and and (53). (53). O the The proof proof of of (56) is is by by induction induction on on n n and and needs needs The 0 It is a good exercise to try to generalize (57) (57) to sentences F F containing containing quantifiers. F contains universal quantifiers? What goes wrong if F Back proof of only possibility possibility to clause Back to to the the proof of (56). (56). The The only to obtain obtain I� ]~ (3x)F(x) (3x)F(x) is is aa clause (3) with the premise with the premise (3) (i) IFa (3x)F(x), F(i. [~o (3x)F(x), F(i)) (i) m << nn and n. IfI f NN ~� F(/) F(i.) we done because because of < n, otherwise otherwise for for m a n d i i++ l 1 < < n. we are are done o f ii < we obtain 10~ (57) and and then then N (3x << m)F(x) m)F(x) by by the the induction induction we obtain (3x)F(x) by by (57) N � ~ (3x IFa (3x)F(x) hypothesis. Since m the claim. O hypothesis. Since m << nn this this yields yields the claim. By for the ~ By ((55) 55) and and ((56) 56) we we see see how how to to obtain obtain upper upper bounds bounds for the witness witness of of aa EI;� sentence F F once we succeed (55)itit is is important replace sentence once we succeed in in getting getting I~ I� F. F. To To obtain obtain (55) important to to replace - as as possiblepossible - the the natural order relation relation on the ordinals by the as far far as natural order on the ordinals by the collapsible collapsible ( 56) is majorize the less the term term notations. notations. The The crucial crucial condition condition for for (56) less relation relation on on the is to to majorize the n, 0 - witnesses of of (3)-clauses. explanation is is the assignment to witnesses (3)-clauses. What What still still needs needs explanation the ordinal ordinal assignment to « a. a. So So we we cannot cannot require the (V)-clauses. many ordinals/3 require the (\f)-clauses. There There are are only only finitely finitely many ordinals (3 << ordinals to sense of <<-relation. We We will ordinals to increase increase in in the the sense of the the «-relation. will come come back back to to that that point. point. In definition of we define define In analogy analogy to to the the definition of the the truth truth complexity complexity for for II~-sentences nt-sentences we the by the computational complexity of of aa sentence sentence FF by computational complexity cc (F) (F) "= min min {c~] { a l I~ I� F}. F} . cc By By (54) ( 54) we we then then have have tc (F) _� cc cc (F) (F) tc(F) := 58 (58) ( ) 59 (59) ( ) 246 W. Pohlers 246 F. for all all sentences sentences F. One One is is tempted tempted to to defi define for ne {cc (F) F is a -sentence and and Ax A x ~f- F} F} iinn analogy analogy to to specrr spec~i(Ax). However, we we easily easily get get II[� Lt-~,0 (3y)F(y) (3y)F(y) if if N N F ~ F(rr) F(n)_ : (Ax) . However, which )F(y)) = min which entails entails cc cc ((3y ((3y)r(y)) min {n {n + + 21 21 N N F ~ F( F(n)} for aa � E~� -sentence (3y (3y)F(y). )F(y). rr)} for spec specro(Ax) E~� E� (Ax) := {cc (F) 1I F is a � = Therefore Therefore we we have have spec specro (Ax) = = w for for any any axiom axiom system system Ax A x which which at at least least contains contains Eo1 (Ax) the � -spectrum spec the successor successor axioms axioms which which shows shows that that the the � E~ specro(Ax) carries no no E� (Ax) carries information information about about Ax. Ax. The The analogy analogy to to speCrrl specn~1 (Ax) (Ax) is is apparently apparently the the wrong wrong one. one. CK wwCK l on But But recall recall that that rrt YI~ corresponds corresponds to to � E11 on the the side side of of sub-systems sub-systems of of Set Set Theory Theory and and l OK OK wOK w wwOK l l � The computational -models are the same as rr -models (cf. Lemma E11 -models are the same as II2 ~ -models (cf. Lemma 1.2.3). The computational l 2 wCK CK aspect �l~ -model aspect of of an an axiom axiom system system is is however however better better reflected reflected in in its its rr YI2 -model because because it it CK wCK says says something something about about its its provably provably total total � E1�l~ -functions. -functions. Pulling Pulling this this down down to to w one one should rather look at g -sentences and something as g -spectrum should rather look at rr H~ and try try to to define define something as the the rr II~ of of an an axiom axiom system. system. As As aa first first observation observation in in that that direction direction we we show show 1.2.3). 1I;+i+l FW for for all all ii E 6 N N by by induction induction on on a. a. I� (\1'x )F(x) (60) (60) L1, F(/) I~ A, =~ II"+'+l,p A, L1, (Vx)F(x) :::} (\1'x) F(x) If If the the main main formula formula of of the the last last inference inference is is different different from from (Vx)F(x) then then the the claim claim follows follows directly directly from from the the induction induction hypothesis hypothesis and and the the fact fact a~<<ja (61) (61) =~ a ~ + i + l < < j a + i + l i. (\1') (\1'x)F(x). for for all all jj and and i. The The crucial crucial case case is is that that of of aa clause clause (V) with with main main formula formula (Vx)F(x). There There we we have have the the premises premises I� L1, (\1'x )F(x), F(D F(j) with with aj a# �j <<j a a for for all all jj E 6 N. N. By By the the induction induction hypothesis hypothesis we we obtain obtain I I;i+i+l L1, F{fJ aj +i+ 1 (i) To To get get the the claim claim from from (i) (i) it it suffices suffices to to check check aj a j <�i < i aa =~ aj a j ++i i++l <1<�c ~a++i +i l+. 1. (62) (62) From From aj ~ �} <<~ a ~ we we get get aj ~j + + ii < < a ~ + + ii and and N(aj) N(~j) :::; _< P(N(a) P(N(~) + + i) i) which which entails entails also also N(aj 1) = N(aj) N(c~j + + ii + + 1) N(c~j) + + ii + + 11 :::; _< P(N(a P(N(oL + + ii + + 1)). 1)). Hence Hence aj ~j + + ii + + 11 � <<1l a c~+ + ii + + 1. 1. Iterating Iterating the the procedure procedure we we get get aj c~j +i+ + i + 11 � < = 0 Ij cc ((\1'x)(3y)F(x, y)) a then, jor every i (3y �w (a + i + 1) )F(i, y). To x)(3y)F(x, y) be g -sentence and To prove prove the the theorem theorem let let (\1' (Vx)(3y)F(x,y) be aa rr II2-sentence and put put x)(3y)F(x, y)y) and � (\1' Then aa .:= cccc ((\1'x)(3y)F(x, ((Vx)(3y)F(x, y)) y)).. Then II~ (Vx)(3y)F(x, and we we get get by by (60) (60) If cc ((Vx)(3y)F(x,y)) = = a then, for every i E6 N, N F (3y < < Cw(a + i + 1))F(i, y). 2.1.4.3. 2.1.4.3. Theorem. Theorem. 0 - 247 247 Order Number Theory Set Theory and Second Order a + ii+ l y) for (3y)F(~_,y) for all a l l /i E N. N. I:+ + l (3y)FCL 1Iio (56). and and (56). 1) (55) Hence < 7fw Hence N 1N F ~ (3y (=ty< r (a + i + + 1 ))F( ) F ( ii ,, yy)) by by (55) 0 U 2.1.4.3 (63) ~ ( a(a + + ii + + 1). 1). (63) WW,~(i) a (i ) := 7fw By Theorem Theorem 2.1.4.3 2.1.4.3 it it follows follows that that W W~a majorizes majorizes aa Skolem Skolem function function of of aa rrg-sentence II~ By According to to Theorem Theorem 2.1.4.3 we we define define According := F if if cc cc (F) (F) � _< a. a. If If we we put put F NF ( i ))F(i, y) y)lno min {{a a I (V'i)[ (Vi)[N ~ (3y (3y < < W~(i))F(/, y)]} (V'x) (3y)F(x, y) = min l rrg ::= ]} II(Yx)(3y)F(x, l Wa (64) (64) for aa rrg-sentence II~ (Vx)(3y)F(z, y) then then we we obtain obtain for for rrg-sentences II~ G for (3y)F(x , y) G (V'x) (65) (65) lal rrgo �_< cc (G) (a). . IGl We We define the rrg-spectrum II~ of aa theory theory Ax A x as as define the of spec specno(Ax) = {{IFIno F is is aa rrg g~ -sentence and and Ax A x ~� F} F} Wlrro2 II F rro2 (Ax) = defined for provided that Wlrr IFInoo is is defined for all all such such F F as as well well as as provided that 2 Ax llrrg ::= up(spec,o(Ax)) IIllAxlE,o = sup(spec rrg (Ax) ) and obtain and obtain I (66) (66) _< sup sup {{ cc cc (F) (F) I Ax Ax � ~ F F }}. . Ax llrrg � IIIIAxl[no We We call call the the computation computation of of IIIAxlJno n~-analysis of of Ax. Ax. I Ax ll rrg a rrg-analysis The is The definition definition of of the the rrg-spectrum H~ is admittedly admittedly less less intrinsic intrinsic than than that that of of the the rrt H~spectrum. We give some some reasons reasons why think that that II~ rrg-spectra do have an an spectrum. We want, want, however, however, give why we we think do have intrinsic on the P which intrinsic meaning. meaning. The The H~ rrg-spectrum depends depends on the function function c7ft: which in in turn turn depends depends on the the starting starting function ordinals in and on on the the term term notation notation of of the the ordinals in the the H~ rrg-spectrum. on function PP and Indeed is set of of ordinal than aa set ordinals. We We have have Indeed the the rrg-spectrum II~ is rather rather aa set ordinal terms terms than set of of ordinals. to already starting function function P P has has to already argued argued that that the the dependence dependence on on aa starting P has has natural natural reasons. reasons. P majorize functions for for which in the (c.f. also majorize all all the the functions which there there are are function function symbols symbols in the language language (c.f. also the the proof of another choice choice of language, e.g. Peano arithmetic arithmetic proof of Lemma Lemma 2.1.4.6 below). below). With With another of the the language, e.g. Peano with the only only function for addition addition and even the the language of Set Set with the function symbols symbols for and multiplication multiplication or or even language of Theory with weaker starting function, e.g. something like like )~x. with no no functions functions symbols, symbols, aa weaker starting function, e.g. something .xx . 3xx ,, Theory will do same job. job. Also of functions WaR "+ ii + not will do the the same Also the the hierarchy hierarchy of functions W� : = ,ki. .xi . CP 7ft: (a (a + + 1) does does not depend too too much much on on the not too too different different starting functions P depend the starting starting function, function, i.e., i.e., for for not starting functions P and G we we will will obtain comparatively small such that WaG which that W P ~� W� and obtain aa comparatively small ordinal ordinal aa such which means means that P is G and that W W� is elementary elementary in in Wa W� and vice vice versa. versa. serious is is the the dependence dependence on on the the term term notations notations for the ordinals. ordinals. However, However, itit is is More for the More serious hard to to imagine imagine an an explicit explicit term term notation notation which which could could alter alter the the II~ rrg-spectrum. Weiermann Weiermann hard has has shown shown that that the the fast fast growing growing subrecursive subrecursive hierarchieshierarchies - and and the the hierarchy hierarchy Wa Wa is is fast fast growing growing -- are are very very stable stable against against alterations alterations of of the the term term notations notations (which (which is is not not true true for for at least least every every term term notation notation usable usable the the so so called called slow slow growing growing hierarchies). hierarchies). We We believe believe that that at ordinal analysis analysis will will lead lead to to the the same same finitary finitary collapsing collapsing function function rtfw and and thus thus to to the the in in ordinal although we we have to admit admit to to see see no no way of proving proving this. this. The The lack lack same have to way of same H~ rrg-spectrum -- although of of aa general general and and "natural" "natural" term term notation notation system system for for all all recursive recursive ordinals ordinals (or (or equivalently equivalently W� w. W. Pohlers Pohlers 248 248 the for all recursive functions), hinders us the lack lack of of aa natural natural subrecursive subrecursive hierarchy hierarchy for all recursive functions), hinders us from from defi ning generally defining generally := { lFl rro2 II F and spec specno(N) {IFIno F aa rrg-sentence n~ and N F ~ F} F} rr2o (N) := although although we we conjecture conjecture that that any any possible possible definition definition should should lead lead to to spec specno(N rrg (N)) = speCrrl1 (N) (Ax) speCrrl (Ax) specrq (N) = = WfK w~K (which (which is is motivated motivated by by the the fact fact that that we we have have specrro specno(Ax ) = specn~ (Ax) = 1 2 for for all all "regular" "regular" axiom axiom systems systems which which are are so so far far analyzed). analyzed). Anyway, of Anyway, the the rrg-spectrum II~ of an an axiom axiom system system has has pleasant pleasant properties. properties. Every Every rrg-sentence defines aa partial partial recursive H~ (Vx)(3y)F(x, y) defines recursive function function fFF := :-- M #Y. Y . F(x, y) and and we we call call f fF provably provably recursive recursive in in Ax A x iff iff Ax Ax � ~ (Vx)(3y)F(x, y), i.e., i.e., iff iff Ax A x proves proves provably recursive recursive in then there there is that that f fFF is is total. total. If If fFF is is provably in Ax A x then is an an a a E specrrg specno(Ax) (Ax) such such that that f fF = -- M #Y < Wa(x). F(x, y). Therefore Therefore all all provably provably recursive recursive functions functions of of Y< rrg Ax . By A x are are primitive primitive recursive (even elementary) elementary) in in Wa for for some some a a E e spec specno(Ax). By Ax recursive (even induction on on a a we we obtain obtain (Vx)(3y)[W~(x) = -- y], i.e., i.e., we we have have induction (\>'x)(3y)F(x, y) f f (\>'x) (3y)F(x, y), f f Wo(x) . F(x, y). Wo (\>'x)(3y)[Wo(x) y], a < < IIIAxll A x ~� (\>'x)(3y)[Wo(x) (W)(3y)[W~(x) = - yl y] I Ax l1 ~ Ax '* F(x, y) ( ) (67) (67) Wo. for nition of for all all axiom axiom systems systems Ax A x which which allow allow the the defi definition of the the functions functions W~. (For (For this this it it certainly certainly suffices suffices that that Ax A x allows allows the the definition definition of of all all primitive primitive recursive recursive functions. functions. In In the the rest rest of of the the paper paper we we tacitly tacitly assume assume that that this this is is true true for for all all axiom axiom systems systems considered. considerations which considered. Weaker Weaker systems systems need need more more subtle subtle considerations which are are outside outside the the scope scope of of this this contribution) contribution). . For For axiom axiom systems systems satisfying satisfying this this assumption assumption we we obtain obtain as aa corollary corollary of of Theorem Theorem 2.1.4.3 as 2.1.4.3 2.1.4.4. Ax l1 then 2.1.4.4. Lemma. Lemma. If If cc cc ((\>'x)(3y)F(x, ((Vx)(3y)F(x, y)) y)) < < IIIIAxll then there there is is aa provably provably recur recurthat N x)F(x, f(x)). sive sive function function f f of of Ax Ax such such that NF ~ (\>' (Vx)F(x, f(x)). Because (\>'x)(3y)[Wo(x) =- yly]l.olrrg == a~ ++ 11 we Because of of II(V~)(3y)[W~(~) we obtain obtain from from (67) (67) also also (68) Axllrrg . (68) Ax l1 ~ a a a < < IIIIAxII a < < IIIlAxll.o. '* Hence Hence Ax llrrl = IIIIAxII Ax l 1 S Ax l i rrg . IIIIAxllnl~ IIIIAxllno. 69 (69 (69)) In proper. For IAx l i rro2 , for In general general the the inequality inequality in in ((69)) is is proper. For a ~ := : - I[IAxllno, for ininl stance, we get by ( ) the inequalities + = yl Ax rr stance, we get by (28) the inequalities = ll 1 = IIIIAx + (w)(3y)[Wo(~) = y]lln~ . Ax = Ax < a + most Ax In . llrrg yl = l l rrl = I[[Ax[[ In most cases, cases, II[[Axl[ul S a < II[ [ A x + (Vx)(3y)[W~(x) = y][[no. i l1 _< however, spending aa little ordinal assign however, we we obtain obtain - spending little more more care care on on the the ordinal assignment This ment - sup sup {tc(F)] Ax A x ~� F F} = = sup sup {cc (F) II Ax A x e� - F } .. This then then entails entails Ax l l rrl = = sup sup {tc(F) II Ax A x e� - F} = = sup sup {cc ( F ) [ I Ax A x e� - F } . Together Together with with (69) IIIIAxlln} we then then obtain obtain we 28 {tc(F) I {tc(F) } F} (\>'x)(3y)[Wo(x) (\>'x)(3y) [Wo(x) {cc (F) F} {cc (F) F}. {cc (F) I Ax llrr l S Ax llrrg S Ax li rr l ' Ii Ax Ii = IIAxII - IIIIAxllrxl _ IIIIAxllno ~ sup sup {cc (F) I Ax Ax � I~ F} = - IIIlAxllrq, F} (69) (70) (70) Ax l l rrg . We i.e., Ax ll rr' = i.e., IIIIAxlln, = IIIIAxllno. We are are going going to to call call axioms axioms systems systems for for which which we we have have Ax llrrl = IAx ll rrg regular. ) is the the following = II]Axllno regular. Another Another consequence consequence of of ((67)is following theorem. theorem. IIIIAxlln] 67 249 249 Theory and Second Order Order Number Theory Set Theory 2.1.4.5. e., let 2.1.4.5. Theorem. T h e o r e m . Let Let Ax A x be be aa regular regular axiom axiom system, system, i.i.e., let IIAxl IIAxll1 = = IIAxll IIAxllno. rrg . Then provably recursive Then the the provably recursive functions functions of of Ax A x are are exactly exactly the the functions functions which which are are primitive primitive recursive recursive (even (even elementary) elementary) in in some some W~0 for for a a < < IIAx IIAxll. ll . W W� Without connected to Without further further hint hint we we just just remark remark that that the the functions functions W P are are closely closely connected to the Hardy-functions . A detailed study is in Buchholz, Cichon and Weiermann the Hardy-functions H Ha. A detailed study is in Buchholz, Cichon and Weiermann o [1994]. [1994]. We We want want to to close close this this section section with with the the remark remark that that there there is is also also aa IT YI�~ ordinal ordinal for for theories, theories, whose whose intention intention is is to to express express the the order order type type of of the the shortest shortest primitive primitive recursive recursive well-ordering well-ordering which which is is needed needed to to prove prove the the consistency consistency of of the the theory theory within within finitistic framework. framework. Due Due to to certain certain pathologies pathologies ((cf. Remark 7.1 7.1.9. in Girard Girard [1987] [1987] aa finitistic cf. Remark .9. in which which exposes exposes an an example example due due to to Kreisel) Kreisel) the the definition definition of of the the IT II �~ ordinal ordinal is is not not completely completely straightforward. straightforward. We We omit omit aa discussion discussion since since we we believe believe that that the the known known concepts still too concepts are are still too far far from from a a final final form form and and need need further further research. research. Computational complexity NT. c o m p l e x i t y of of N T . As As an an example example we we want want to to compute compute NT) . The specrro specno2 ((NT). The first first step step consists consists in in computing computing the the computational computational complexities complexities of of the the axioms axioms of of NT. N T . We We observe observe that that (71) (VXl)""" (VXn)a(Xl,... ,Xn) ('v'Xl) · · · ('v'Xn )G(Xl, . . . , Xn ) G(Ul, . . . , Un) (Zb . . . , zn ) G(Zb . . . , zn ) E D(N). ('v'Xl) · · · ('v'Xn )G(Xb . . . ' xn ) ('v'). (VXl) . . . (VXn )G(Xl' . . . , xn ) G( , Un ) holds holds for for all all true true sentences sentences (VXl)""(VXn)e(Xl,...,Xn) where where G(ul,...,un) is is a a quantifier quantifier free free formula. formula. The is simple. The proof proofis simple. For For every every n-tuple n-tuple (zl,..., zn) we we have have G(zl,..., zn) E D(N). Hence 0 Hence I� [~ (Vxl)... (Vx~)G(xl,... ,x,,) by by n-fold n-fold application of aa clause clause (V). r] application of All mathematical mathematical axioms NT, All axioms of of N T , except except the the induction induction scheme, scheme, are are IT-sentences H-sentences of of the the form form ( V x l ) . . . ( V x , ) G ( x l , . . . , x n ) with with G (11.1 u l ,, . . . , u , ) quantifier-free. quantifier-free. Thus Thus (71) (71) gives gives us us bounds bounds for for the the computational computational complexity complexity of of all all these these axioms. axioms. To To compute compute the the computational computational complexity complexity of of the the scheme scheme of of Mathematical Mathematical Induction Induction we we first first prove prove . • • · rk(F) � I,olw'rk(F) /k,, F, F, -,F ~F 1I� (72) (72) rk(F). by by induction induction on on rk(F). The The proof proof is is essentially essentially that that of of the the Tautology Tautology Lemma Lemma bit more more care (Lemma (Lemma 2.1.2.3). 2.1.2.3). A A bit care is is needed needed for for the the case case that that F is is a a formula formula (Vx)G(x). There There we we have have F ('v'x)G(x). ·rk(G(�)) �, I,ol'''rkcac=-)) A, G(;:), a(z_),--,a(z) 1I� -,G(;:) (i(i)) EN for for all all E N by by the the induction induction hypothesis hypothesis and and obtain obtain ;:), (3x)-,G(x) 1I�·rk(G(�)l+Z �, G( G(z_), (Sx)~G(x) zEN (3). w · rk(G(;:)) Z w · (rk(G(z)) ((ii) ii) for holds for for every every z E N by by a a clause clause (3). But But w. rk(G(z)) + -t- z «z ((z w. (rk(G(z)) + -i- 1) 1) holds for all and we all z E N and we obtain obtain zEN ~.rk((V=)C(=)) 1I�·rk«\fX)G(X)) �, ('v'x )G(x), -,('v'x)G(x) ('v'). by by a a clause clause (V). o 250 250 W. Pohlers Pohlers W. The The computation computation of of the the computational computational complexity complexity of of instances instances of of the the scheme scheme of of Mathematical Mathematical Induction Induction is is obtained obtained as as in in Lemma Lemma 2.1.2.4 with with some some extra extra care care on on the the ordinal assignment. assignment. We We prove prove ordinal 2.1.2.4 ·rk (F(Q))+2.n -.F(Q) , -. (\fx)[F(x) II0W'rk(F(~ 1I� n. n 0 ~ --* (72). F(Sx)],F(n)_ F( Sx)], F(nJ (73) (73) (72) by induction induction on on n. For For n = = 0 this this is is (72). For For the the induction induction step step we we get get by by (72) and and by the induction induction hypothesis hypothesis by by an an inference inference (A) the w. (O))+2n+ l -. I][0w' f(Sx)],F(n_) F(Q) , -. (\fx)[F(x) --*-+ F(Sx)], lo rk(FrkCf(0--))+2n+l--,F(O),~(Vx)[F(x) F(nJ -.F(Sn.), F(Sn). ((i)i) We have have nn «: << w w.rk(F(O))+2.(n+ and w·rk(F(Q))+2·n+1 w.rk(F(O_))+2.n+ 1 «: << w w.rk(F(O))+2.(n+ · rk(F(Q))+2·(n+1)1) We · rk(F(Q))+2· (n+1)1) and - and obtain obtain thus thus from from ((i) and i) ·rk(F(Q))+2(n+ l ) -.-,f(O), ]10~.rk(f(0))+2(n+l) F(Q) , -.-~(Vx)[f(x) 11� (\fx)[F(x) by by aa clause clause (3) (3).. --+ --* A--,F(Sn)_, F(Sn)._ A f(Sx)], F(S f(Sn) F(Sx)], rr) w· rk(F(Q)) +2n W · (rk(F(Q)) + 1) (\f) F(Q) , -. (\fx)[F(x) ~ F(S 1I� x)], (\fx)F(x) IIo·(rk(F(Q))+l) -.~F(0),--,(Vx)[F(x) F(Sx)], (Vx)F(x).. ((ii) ii) n D 0 Noticing Noticing that that w.rk(F(O))+2n «:n < (73) --* (74) (74) This This gives gives an an estimate estimate for for the the computational computational complexities complexities of of instances instances of of Mathemat Mathematical Induction. Induction. From From (72) we we obtain obtain also also ical iw'rk(F)_z1 r I,0 (72) k_l,..., z~ ~ k~,-~F(Zl,... , z~), F ( k l , . . . , k~) , ) (71), (74) (75) (75) for for all all n-tuples n-tuples (Z (Zl,...,Zn) and (k1 ( k i ,, .. ... ., kk~) of natural natural numbers. numbers. This This eventually eventually n of l , " " zn ) and as an an upper upper bound bound for for the the computational computational complexities complexities of all instances instances gives w. rk(G) as gives of all G of identity axioms. Pulling together together (71), (74) and we obtain of identity axioms. Pulling and (75) we obtain G W · rk(G) w.(rk(A)+l) ·(rk(A)+ l) 1I� (75) (76) (76) �, A for and identity axiom A of NT. NT. for every every non-logical non-logical and identity axiom A of The adaption adaption of of (39) needs bit more It will, will, however, however, clarify clarify the the role role of of The needs aa bit more care. care. It the the starting starting function function PP in in the the definition definition of of the the collapsing collapsing function function ~'¢;.w ' We We show show (39) 2.1.4.6. LLemma. If ~--i(Ul,...,Un) F � (uI , . . . , Un) and all all free free number number variables variables of of 2.1.4.6. emma. If � (Ub . . . , un) occur in the list U , . . . , Un then there is a finite ordinal k such that A ( u l , . . . , un) occur in the list ul,..., un then there is a finite ordinal such that I W·k+Zl +'+Zn A(zl, III IW'k+zx+'''+zn . zn) holds for all n-tuples � ) holds for all n -tuples (zl (ZI,~ ' ''' ~ , zn) zn ) of of natural natural numbers. numbers. , ( £n . . £1" 0o "" ~ .. 2.1.4.6 The proof proof of of Lemma Lemma 2.1.4.6 isis by by induction induction on on m. m. As As prerequisite prerequisite itit needs needs The I~ A(s) and s N - t N =~ I~ A(t) (77) (77) which follows follows straightforwardly straightforwardly by by induction induction on on ~. Q. The The only only cases cases in in the the proof proof of of which Lemma 2.1.4.6 which which are are not not immediate immediate from from the the induction induction hypotheses hypotheses are are clauses clauses Lemma (3) and and (V). Let Let us us start start with with the the (3) (3) case. case. ItIt isis of of special special interest interest because because itit needs needs (S) the the starting starting function function P. P. We We have have 2.1.4.6 (\f). Set Theory and Second Second Order Number Theory 251 251 o(iI) , (3x) o(iI) , G (iI, t(t(~)) Ao(~), G(~, ~ � o~ � Ao(~), (3z)a(~,z) iI)) => G (iI, x) �� (i) 11�' C�, t(�)). ,o ko+z � Aoo(�) (~),, G G(~, t(zS)). (ii) (ii) and obtain obtain by by the the induction induction hypothesis hypothesis aa finite finite ordinal ordinal ko ko such such that that and The value t(~) NN is is computed from Z. By (48) (48) we we thus thus have have aa computed primitive primitive recursively recursively from z. By The value t(�) N << P natural number number kt kt such such that that t(g) t(_~)N P~,(z-). Taking kk := kt} we we get get ,(Z). Taking := max{ko max{ko ++ 11,, kd natural (iii) (iii) 1I�'k+Z o(g) , (3x) o Ao(~), (3x)G(~_, x) G (�, x) � (iv) (iv) . Hence Hence k w. ko ko + + z ~"« << w w. k k +z ~ and and t(�) t(g) NN « << w w.. kk + + z ~'.. w . from from (ii), (ii), (77) (77) and and (iii) (iii) by by aa clause clause (3) (3).. In In the the case case of of an an (\I) (V) inference inference v) ~=> o~ x) o(iI) , G (iI, v) � ZXo(~), �o(iI) , (\lx) ZXo(~), a(~, (Vx)a(~, G (iI, ~) �� we have finite ordinal we have aa finite ordinal ko ko such such that that lw. Z i Ao(g), (z, i) (z) G Iii~.ko+~+, ,0o ko+ + � G(g,/) for Putting kk ::=- ko for all all ii E C N. N. Putting ko + + 11 we we get get w w . ko ko + + z ~' + + ii « < E - . . for for all all ii - (v) (v) (vi) (vi) (vii) (vii) N N and and obtain obtain o(g), (\lx) 1II1~'*+~ �'k+Z �Ao(~) (W)G(g, x) G (�, x) I0 U o from (vi) and (vii) (vii) by from (vi)and by aa clause clause (\I) (V).. (76) yield Lemma Lemma 2.1.4.6 2.1.4.6 together together with with (76) yield 2.1.4.7. Lemma. If F natural 2.1.4.7. Lernma. If NT N T ~� - F F for an £wsentence s F then then there there are are natural + m for an lw2 numbers numbers m m and and n n such such that that IIIn~§ F. F What is is still still lacking lacking is is aa proof proof that that the the cut cut elimination elimination procedure procedure also also works works for for the the What � �. refined refined calculus calculus II~ A. We We are are going going to to check check this this step step by by step. step. The The first first step step is is checking the the Reduction Reduction Lemma. We claim claim checking Lemma. We and r, ~-,FF and I~~ �, ~ F F , , I~I� _ F, I� II, # {j A, II; �, rF . rk(F) _ =~ ~ ~ rk (F) � (78) The is by The proof proof is by induction induction on on 0: a # ~ (3 fl and and follows follows the the pattern pattern of of the the proof proof of of the the Reduction 2.1.2.7). There Reduction Lemma Lemma (Lemma (Lemma 2.1.2.7). There is, is, however, however, aa subtle subtle point point in in it it which which will clarify will clarify the the ordinal ordinal assignment assignment in in the the case case of of (\I)-clauses. (\/)-clauses. We We treat treat only only this this case. case. All other other cases cases follow follow easily easily from from the the induction hypotheses and and All induction hypotheses a < Assume Assume that that F F is is aa formula formula (3x) (Bx)G(x) which is is the the main main formula formula of of an an inference inference G (x) which (79) (79) 252 252 W. W. Pohlers Pohlers G(i), (3x)G(x) (3x)G(x) => =~ [~ (3x)G(x). Do, G(i), I� A, Do, (3x)G(x). I[~po � A, ((i)i ) Then + 11 « a. From i) and Then ao « << a a and and ii + << a. From ((i) and the the second second hypothesis hypothesis r, (\fx)...,G(x) I� r, ii) ((ii) we we obtain obtain G(/) Do, F, r, G(i) 1;0 #~ .8~ A, 1][pnO ((iii) iii) by On the obtain by (60) and ii) by the the induction induction hypothesis. hypothesis. On the other other hand hand we we obtain by (60) and (53) (53) from from ((ii) also also 1II1!+i+ 1 Do, r,r, ...,G(i). ((iv) iv) P We We have have ao # ,8 « a # ,8 ( v) 1 and obtain from and obtain from ii + + 1« << a a also also a # ,8. ,8 j 3+ + ii++ 1 « < 1 vi) ((vi) rk(G(i.)) rk(F) Because v) , ((vi) vi) and Because of of ((v), and rk(G(/)) < < rk(F) = = p we we get get # 11; .8 Do, r o [3 from iii ) and ((iv) iv) by from ((iii)and by cut. cut. By By (78) we we obtain obtain (78) i a Do => 112"+1 I r;+l rp--- Do a (80) (80) The Lemma 2.1.2.8). The proof proof is is that that of of the the Elimination Elimination Lemma Lemma ((Lemma 2.1.2.8). "2 a't'l by by induction induction on on a. a. All we All we need need to to adapt adapt the the proof proof are are a < (81) (81) al,a2 << a =~ 2~1+1 ~ 2~2+1 <_<2~+1. (82) (82) and and These properties, however, These properties, however, follow follow easily easily from from the the fact fact that that 2a) 2N(a) (83) (83) a which nearly = w . v + we obtain N(2 which in in turn nearly immediate immediate since for a NN("). nisis. (N(v) n �___22for n)turn (v)+since (a)a. = w . v + n we obtain N(2 ~)) =-[3 _< 22". + 1) N(wVV. 22") � 1) �_< 22NN(~)+n One (82) force One should should observe observe that that (81) (81) and and (82) force us us to to use use aa fixed fixed point point free free formulation formulation +l instead Le., 22a~+t of Elimination Lemma The ). of the the Elimination Lemma ((i.e., instead of of 22a~). The reason reason is is that that in in case case that that a a < < 22aa,, a a « << ,8 /3 and and ,8 /3 = = 22.8z we we do do have have 22a~ < < 22.8z but but not not necessarily necessarily N(2 N(2 aa)) � _< ( ~) � N Y(2 _< 2 g(a) n • 0 P(N(2.8) P(N(2~)).) . g -spectrum of Now Now we we have have all all the the data data for for the the computation computation of of the the rr II~ of NT. N T . From From Lemma e p , w2 Lemma 2.1.4.7, (80) and and the the fact fact that that expn(2, w2 + + m) m) < < co r we we obtain obtain 2 . 1.4. 7, (80) x n (2 253 253 Set Theory and Second Order Number Theory N T f~- F F :::} =~ I� I~ F F NT for .eN-sentences F for all all/:N-sentences F and and some some ordinal ordinal a a < < co e0.. Together Together with with (66) (66) we we have have NT II == IIlINT[In NT l l njI S;__ IlINT[In NT l l ng S;_< coeo == II[[NTI[ IIliNT[[no I NTl l ngo.· (84) (84) (85) (85) Combining (85) and Theorem Theorem 2.1 2.1.4.5 we obtain obtain Combining (85) and .4.5 we specn NT I and co specno(NT) e0 = = III[NTII and the the provably provably recursive recursive func funcg (NT) ==which are tions tions of of NT N T are are exactly exactly the the functions functions which are primitive primitive recursive recursive in in some some function function Wa Wa for for a a < < co eo.. 2.1.4.8. 2.1.4.8. Theorem. Theorem. W(x)o ((xh) cc ((Vx)F(x)) for {:> (3y)[(x)o Taking fTaking F(x) ::ca (3y)[(X)o � 6 On On V V ~i/(x)o((X)I ) = -- y] y] we we get get NT~-F(_n) for all n n E EN N but but N T ~-(Vx)F(x) since since cc ((Vx)F(x)) = = co eo.. This This sharpens sharpens Theo Theoall rem 2.1 2.1.3.4 to .3.4 to rem F(x) NT JL (Vx)F(x) NT F(rr) 2.1.4.9. Theorem. Theorem. There is is aa true true JIg ri~-sentence (Vx (Vx)F(x) that NT N T f[-- F( F(n) 2.1.4.9. There )F(x) such such that rr) for all all n n E EN N but NT N T JL ~ (Vx)F(x) (Vx)F(x) .. for 2.1.5. C o m p u t a t i o n a l complexity c o m p l e x i t y of of rIg-sentences II~ rrevisited evisited 2.1.5. Computational !PI ng cc (F) . I(Vx)(3y)F(x, y)l no (86) 1Iio1: ((gy)F(i,y) (86) 3y)F(i, y) for all ii E but there such that that W~(i) < Heuristics. We We have have seen seen in in (65) (65) that that IFIno S; <_ cc ( F ) . Proving Proving also also the the opposite opposite Heuristics. inequality would would be be aa good good argument argument for for the the naturalness naturalness of of the the computational computational inequality complexity. However, However, if if I(Vx)(3y)F(x, y)lno2 = = a a we we get get easily easily complexity. i w,~(i) ,, ( ) that cannot yet the natural natural one. one. There is aa possibility possibility to redefine that the the «i-relation < < /~ f3 /\ N(o:) S; slightly modified modified the < W� Fa :::} W�+l (O) W*;J+l (O). (87) I� (Vx)(3y)F(x, I#o (Vx)(3y)F(x, y). y). (87) for 0:. Hence Hence w w _S; IFIno !PIng =v :::} cccc (F) (F) S;_< IRis0. !PIng . The The collapsing for infinite infinite a. collapsing technique technique of of the the previous sections sections is is based on the the idea idea of of local local predicativity predicativity which which has has been been originally originally previous based on developed analysis of systems (cf. [1978,1981, developed for for the the analysis of impredicative impredicative axiom axiom systems (cf. Pohlers Pohlers [1978,1981, 1991] et al. al. [1981]). It is is hardly the redefinition 1991] and and Buchholz Buchholz et [1981]). It hardly surprising surprising that that the redefinition of before from of < w. W. Pohlers Pohlers 254 254 [1997] . the more in Beckmann and Pohlers the more statical statical aspect aspect of of cut-freeness cut-freeness as as e.g. e.g. in Beckmann and Pohlers [1997]. The The fact that that it it also also needs needs no no alteration alteration of of the the functions functions W Q, indicates indicates the the naturalness naturalness of of fact Buchholz section we Buchholz'' approach. approach. In In the the following following section we introduce introduce the the concept concept of of operator operator controlled controlled derivations. derivations. We We will will demonstrate demonstrate that that the the computational computational content content of of the the cut-elimination cut-elimination procedure procedure is is measured measured by by the the controlling controlling operator. operator. W Operator O p e r a t o r controlled c o n t r o l l e d derivations. d e r i v a t i o n s . In In this this section section we we concentrate concentrate on on arithmetical arithmetical sentences sentences and and dispense dispense therefore therefore with with set set variables. variables. We We call call this this language language .c� L:~ which which we cf. Section .3). There we assume assume to to be be formulated formulated as as Tait-language Tait-language ((cf. Section 11.3). There are are two two types types of of arithmetical arithmetical sentences sentences 1\ • D(N), sentences of -type, which sentences, i.i.e., e., sentences sentences in 9 sentences of A-type, which are are true true atomic atomic sentences, in D(N), and sentences of and sentences of the the form form (A A B) or or (Vx)F(x) and and • (A I\. B) (Vx)F(x) V sentences of -type, which sentences and sentences of 9 sentences of V-type, which are are false false atomic atomic sentences and sentences of the the form form or (3x)F(x). (A V B) or (A V B) (3x)F(x). C (F) For For every every sentence sentence we we associate associate aa characteristic characteristic set set C (F) of of sentences sentences such such that that NN p~ FF We We define define ¢:} r (VG E e C(F))[N C(F))[N F ~ G] G] if if F F E e I\ A --type type { (VG (3G C(F))[N F ~ G] G] if if F F E9 V V-type. (3G Ee C(F))[N -type. {{~ if if F F is is atomic atomic (88) if CC(F) A ' B} if F F is is of of the the shape shape A A oB B (F) ::== �{A,B} {G(Il) {G(n)lI n n E 6 w} w} if if F F is is aa sentence sentence (Qx)G(x), (Qx)G(x), where {I\., V} where o E 6 {A, V} and and Q QE 6 {V, {V, 3}. 3}. To OF(G) which To each each formula formula G G E E C(F) C(F) we we associate associate aa finite finite ordinal ordinal oF(G) which is is 00 if if F F is is of of the the shape shape A A oB S and and n n if if F F has has the the form form (Qx)H(x) (Qx)H(x) and and G G is is H(Il). g ( n ) . In In order order 0 0 0 to to obtain obtain aa more more refined refined truth truth complexity complexity for for arithmetical arithmetical sentences sentences we we introduce introduce operator finitary derivations. operator controlled controlled in infinitary derivations. F: N N F[M](n) m (M U {n})) F[M](n) := {n})).. := F(F(max(M M N For For a a function function F: N --+~ N and and aa finite finite set set M � C N we we define define the the function function ax U We collection of We may may interpret interpret FF as as an an operator operator with with values values in in the the collection of finite finite sets sets of of ordinals ordinals by by defining defining E F(n) F(n) :r Y(c~) N(a) < < F(n). F(n). Instead F. To Instead of of a a E E F(O) F(0) we we write write shortly shortly a a E E F. To simplify simplify notations notations further further we we write write F[n] F[G] instead F[n] for for F[{n}] F[{n}] and and G G E e F F and and FIG] instead of of OF(G) oF(G) E e FF and and F[OF(G)], FloE(G)], respectively, respectively, whenever whenever it it is is clear clear from from the the context context to to which which characteristic characteristic set set C C (F) (F) the the sentence sentence belongs. GG belongs. aa e :¢:} 255 255 Set Set Theory Theory and and Second Second Order Number Number Theory Theory 2.1.5.1. 2.1.5.1. Definition. Definition. Let Let F: F: NN ---+} NN be be an an increasing increasing function. function. For For aa finite finite inductively by by the the set of set of arithmetical arithmetical sentences sentences we we define define the the proof proof relation relation FF ~ A inductively following clauses. clauses. following �� n 1\ � A, �, aG asas well well Ola QG << OL Q for for all all G G E CC (F) (F) then FF ~� A� then IfF eE a�nn V-type, V - type, ,Q~ ,E F,F, aG eE F,F, rF U� a,�, aG ~and Q Io,-~om~ for some aG eE cC (F) (F) (V) 1IF e ~QG << o, (V) then FF ~� A� then We call FF the the main main part part in in instances instances of of ((1\) and (V)" (V). We call A ) and If aQ E FF and and FF ~v W-~ A,�, AA asas well well as as FF ~po W- A,-~A �, -,A for for some some ao Qo << aQ and and some some AA (cut) (cut ) If such that that rk(A) rk(A) < then FF ~� A� such < p then (/\) (A) IfF -type, aQ E FF and and F[G] F[G] ~ If F E A N A-type, It is from (88) and the of the infinitary It is then then obvious obvious from (88) and the soundness soundness of the cut cut rule rule that that this this infinitary calculus is sound, i.e., we have calculus is sound, i.e., we have F~A =~ N ~ V { F (89) (89) I FEA}. We say G extends F, written as FF � G, if F(n) _:::; G(n) G(n) We say that that an an operator operator G extends the the operator operator F, written as c_ G, if F(n) holds for all n. n. By straightforward induction induction on on Q we easily easily obtain obtain holds for all By aa straightforward c~ we 2.1.5.2. Lemma. 2.1.5.2. Lemma. If :::; fl, (3, p <_ :::; a, a, F F c_ �G G g:;) fl, (3, A� � r and and FF ~� � then GG � r. If aQ <_ C_ F A then ~ F. Another Another easy easy observation observation is is the the following following Detachment Detachment Lemma Lemma whose whose proof proof iiss again again aa straightforward induction on straightforward induction on Q. c~. 2.1.5.3. 2.1.5.3. Detachment D e t a c h m e n t Lemma. Lemma. If If FF � ~ � A,, A A and and -,A -~A E D(N) D(N) then then FF � ~v �. A. One One purpose purpose of of the the operator operator is is to to control control the the witnesses witnesses of of existential existential sentences. sentences. This This is is manifested manifested in in the the following following Witnessing Witnessing Lemma. Lemma. 2.1.5.4. Let 2 . 1 . 5 . 4 . Witnessing W i t n e s s i n g Lemma. Lemma. Let (3y)F(y) (3y)F(y) be be aa �� E~ -sentence such such that that F(O))F(y) . FF � ~o (3y)F(y) (3y)F(y). . Then Then N 1N F ~ (3y (3y < < F(0))F(y). � oto The The proof proof is is by by induction induction on on Q. a. The The only only possible possible premise premise is is FF ~ (3y)F(y), (3y)F(y), F(nJ F(n) with (F(11)) Ee F, ( N ) we with Qo c~0 < < Q c~ and and n = O(3y)F(y) o(3y)F(y)(F(n_)) F, i.e. i.e. n n < < F(O) F(0).. If If F(11) F(n) E e D D(N) we are are done. done. Otherwise Otherwise we we have have FF � ~2_ (3y)F(y) (3y)F(y) by by the the Detachment Detachment Lemma Lemma and and obtain obtain the the claim 0 claim by by the the induction induction hypothesis. hypothesis, n The The more more important important aspect aspect of of operator operator controlled controlled derivations, derivations, however, however, is is the the control .5.6 below. control they they give give on on functions functions as as stated stated in in Theorem Theorem 2.1 2.1.5.6 below. The The key key property property here here is is the the following following Inversion Inversion Lemma. Lemma. 256 256 W. Pohlers Let F sentence of -type such Ll, F. 2.1.5.5. 2.1.5.5. Inversion Inversion Lemma. Lemma. Let F be be aa sentence of 1\ A-type such that that FF � ~ A, F. Then we we get get F[G] F[G] � ~ Ll, A, G C for .for all all G C E EC C(F). Then (F) . The induction on according to V ) we The proof proof is is by by induction on a. In In case case of of an an inference inference according to ((V) we get get F[G] -- the claim immediately - due due to to FF � c_ FIG] the claim immediately from from the the induction induction hypothesis. hypothesis. In In (1\) we case case of of an an inference inference (A) we have have the the premise premise F[G] F[G] � ~2_ Ll, A, F, F, G G for for some some aa aa < < a a and hypothesis, Lemma and get get the the claim claim from from the the induction induction hypothesis, Lemma 2.1.5.2 2.1.5.2 and and the the fact fact that that - F[G] [G] F[GI[G] = = 0 rn F[G] FIG].. Combining Witnessing Lemmas Combining the the Inversion Inversion and and Witnessing Lemmas we we get get the the following following theorem. theorem. Let (3y)F(x, y) be Let (V'x) (Vx)(3y)F(x,y) be aa rrg n~ -sentence such such that that (3y)F(x, y) FF � ~o (V'x) (Vx)(3y)F(x, y).. Then Then the the associated associated recursive recursive function function f(x) J'(x) = = ILY #y.. F(x, F(x, y) y) is F, i.e., is majorized majorized by by F, i.e., we we have have f(n) f(n) < F(n) F(n) for for all all n nE Ew w.. 2.1.5.6. 2.1.5.6. Theorem. Theorem. Proof. Proof. Pick Pick n nE Ew w and and apply apply the the Inversion Inversion Lemma Lemma to to get get F[n] Fin] � ~ (3y)F(!£, (3y)F(n, y) y).. Then use use the the Witnessing Witnessing Lemma Lemma to to see see that that ILY #y.. F(n, y) < F[n](0) - F(n) F(n).. Then F(n, y) F[n](O) = (i) (i) 0 E] We during the We have have to to study study the the behavior behavior of of the the controlling controlling operators operators during the cut cutelimination elimination process. process. Let -type and = rk(F) Let F F be be aa sentence sentence of of V V-type and pp ::= rk(F).. Let (m) � (m) . If r, -.F Let FF and and G G be be increasing increasing functions functions such such that that 22.. m m � < FF(m) < G G(m). If FF � ~ F,-~F and G G� ~ A then (F l,p- ~ I-',A Ll,, FF then (F 0oG) G) I;+/J r, Ll . and 2.1.5.7. 2.1.5.7. Reduction R e d u c t i o n Lemma. Lemma. We j3 . First We prove prove the the lemma lemma by by induction induction on on/3. First we we observe observe that that a c~ E E F and and/3j3 E E G imply a N(j3) � ) . We imply a + +/3j3 E E FF oo G G since since N(a g(a + + j3) fl) � <_ N(a) g(a) + + g(fl) _< 22.· G(O) G(0) � < F(G(O) F(G(0)). We also If the is aa cut an inference also have have G G� c_ FF oo G G.. If the last last inference inference is cut or or an inference according according to to (V) (V), , whose F, we immediately by whose main main part part is is different different from from F, we get get the the claim claim immediately by the the induction induction hypothesis hypothesis and and FF � C_ (F (F 0o G) G).. If If the the last last inference inference is is (1\) (A) G[H] Ll, G G[H]p� ~ - ALl, ' GG, ' HH f ~for all H E EC C (G) (G) '* =~ G G� ~ A, G then induction hypothesis then we we get get by by induction hypothesis I;+/JH r, Ll, H (F G[H])Ipa+~" F,A H (F 0o G[H]) (i) (i) for obtain for all all H H E EC C (G) (G) and and obtain I;+/J Fr, LlA (1\) since (F 0o G[H]) (F 0o G)[H] from (i) by from (i) by aa clause clause (A) since (F G[H]) = = (F G)[H].. The interesting case according to The interesting case is is an an inference inference according to (V) (V) (F 0o G G)) ~ (F (ii) (ii) Set Set Theory Theory and and Second Second Order Order Number Number Theory Theory GG~-r,F, � r, F, G for some G Ee CC(F) (F) * forsomeC ~ GG~r,F � r, F whose main part F. Then whose main part is is F. Then we we have have G GE E G, G, 13 flo0 < < 13 fl and and obtain obtain F, D., A G a +130 r, (F G) 1;~+~o (F 0oG)lp 257 257 (iii) (iii) (iv) (iv) by hypothesis. From by induction induction hypothesis. From the the hypothesis hypothesis (v) (v) F � .6o" F we we obtain obtain F[GJ G FIG]� ~ .6o" A, ~G (vi) (vi) by by the the Inversion Inversion Lemma Lemma (Lemma (Lemma 2.1.5.5). 2.1.5.5). But But G G E EG G means means OF(G) oF(G) < < G(O) G(0) which which in n}) �_< F(G(n)), in turn turn entails entails F[GJ(n) F[G](n) = - F(max{ F ( m a x { ooF(G), f ( G ) , nn}) }) � <_ F(max{G(O), F(max{G(0),n}) F(G(n)), i.e. i.e. F[GJ F[G] � C_ (F (F 0o G). G). Hence Hence (F 0o G) G) 1; ~ -+130 r, F, D., A, ,G ~G (F (vii) (vii) ~p from from (vi) (vi) and and Lemma Lemma 2.1.5.2 2.1.5.2 and and we we obtain obtain (F 0o G) G) 1;~~p-+13 F,r, .60A (F (viii) (viii) D D from (iv) and (vii) (vii) by from (iv)and by cut. cut. The The Reduction Reduction Lemma Lemma illustrates illustrates that that avoiding avoiding one one cut cut means means composing composing the the controlling operator. controlling operator. If If we we have have aa derivation derivation F � ~ l .60 A and and we we want want to to reduce reduce the the p+ cut times. This cut rank rank by by 11 we we have have to to iterate iterate the the Reduction Reduction Lemma Lemma a a times. This causes causes no no nite. For problems a, however, problems as as long long as as a a is is fi finite. For transfinite transfinite ordinals ordinals a, however, we we have have to to decide nite iterations decide what what we we understand understand by by transfi transfinite iterations of of operators. operators. There There is is aa big big variety variety of of possibilities, possibilities, e.g., e.g., attaching attaching limit limit ordinals ordinals with with fundamental fundamental sequences sequences and and diagonalizing at points. However, diagonalizing at limit limit points. However, we we already already defined defined aa hierarchy hierarchy of of strictly strictly increasing introducing the increasing functions functions by by introducing the functions functions W'" W~ which which in in turn turn are are based based on on the the collapsing m� collapsing functions functions 'If;e CP.. First First we we observe observe that that 22m _ W",(m) Wa(m).. We We claim claim that that we we also also have have F (90) (90) W a O W fl C Wa ~: ,8+1" This This turns turns the the members members of of the the hierarchy hierarchy W W~'" into into good good candidates candidates for for the the controlling controlling 90) we operators. operators. To To prove prove ((90) we first first observe observe nn << w Then Then we we show show ((91) 91 ) =~ 'If; CeP (n) ( n ) ==nn. . * ((92) 92) cP (a + cP (/3)) _< cP (a ~/3) by induction on 13. For ) . So 13 = claim by induction on/3. For a a = = 00 this this is is (91 (91). So assume assume a a ir O. 0. For For/3 = 00 the the claim e (/3) (13) = follows 13 ifollows trivially trivially from from (91). (91). If If/3 r 00 we we have have 'If; CP = 'If;e CP (7) + + 11 for for some some 'Y 7 « < (r ) w. W. Pohlers 258 (#)) = = 7jJ r � (a + + 11 + + 7jJ r � (r)) 7jJ~�( ~(a ++ 7jJr � (f3)) (~ + + 11 #~ 'Y ~)) :::; ___7jJ r � (a (~ #~ 13). #). :::;_< 7jJr � (a Hence W W=(W#(n)) (a + 7jJ CP + n+ + 1) 1) + + 1) 1) :::; <__7jJ CP (n) Hence � (a(a #~ 13~ ++ nn ++ 2) == WW=o #~ .B#+,+ (n) � (f3(fl +n o (WB(n)) == 7jJCP� (a+ l and we we have have (90) (90).. and 2.1.5.8. 2.1.5.8. Elimination E l i m i n a t i o n Lemma. Lemma. 0 [3 wo+1 �A.· o A then I / W #.B � ~ � then W W ~W�+#Q+ l lF ~ IfW .wa+l a The induction on and the The proof proof is is by by induction on 0/and the crucial crucial case case is is that that of of aa cut cut of of rank rank p p whose whose premises premises are are A A ,A A and and W W .B# � A , -, ~A WW .B# � p~p~+ p~p~+ +l~ �, +l~ �, for a2 EE W for ii = for some some aI, 0/1,0/2 W#.B such such that that ai 0/i < < a 0/for = 11,, 22.. From From the the induction induction hypothesis hypothesis we we obtain obtain Q2+1 ,' -,A Ol+1+1 �, [W WoK' IwWa2+l -~A (i) (i) A and WWW~f3#Ql+ 1+1 l F ,p A,A and W W~,~2+1 ,a Wf3#Q2+l F which (90) entails which by by the the Reduction Reduction Lemma Lemma and and (90) entails Q2 A. W W w~f3#Ql+l .~+~ # ~ w~f3#Q2+ ~-~+~+1 1 +l II.wCrt+l p +W +1 �. w a l + l _}.Wa 2 + l (ii) (ii) Next Next we we show show W.BE => 0/1, a 0/22 < < a 0/ 1\ h at, 0/1, a 0/22 E E W ::~ W Wwfl~al+l f3#Q2+1 +l �C WWwfl~:c~.~. at, wf3# Q+l.1. Wf3#Ql+l #:H:wwfl~a2+l-t-1 Since Since at, 0/1, a 0/22 E EW W .BE means means N(ai) N(0/i) < < W WE(0) cP (13 (fl + + 1) 1) we we have have .B (O) == 7jJ� l o .B .B ui.B o2 l Ol + # # «<< ww # ~ . . 2 ++ 7jJcP(fl + ##wW#*~2+1 W#*# ~+t � (f3 ++ 1)1).· 2. This This entails entails by by (51 (51)) and and (92) (92) Wwf3#Ql+l # wf3#Q2+1 +2 (n) 'f/w (W.B # Ol +l # W.B # 02+l + n + 2) 1) . 22 + < < 7jJ CP CP (13 (/~ + + 1). + n n+ + 22)) � (w(w.B.B#~# 0~ -. 22 ++ 7jJ� # :::;<_ 7jJCP(w � (w.BE*~O ·.l2 #~ 13~ ·. 22 ++ nn ++ 4)4) � (w# #~ O++1 ++ nn ++ 11)) =- W~]wt~ l (n). :::;_< 7jJCP(w Wf3#Q+~+l(n). ." P - (93) (93) . .. ,, 111 ((iii) (iv) (iv) From (ii) and (93) we finally finally obtain From (ii) and (93)we obtain wo+1 �A.· ~ ,~+~ ~ WW.,~ wf3#Q+ IF .wa+l The The remaining remaining cases cases are are immediate immediate from from the the induction induction hypothesis hypothesis and and the the fact fact that that + l E~ W~,8~=e~-F1. 0 0/~E W E implies implies w W.B ~ a#+Ol 1-'1 To reobtain specno(NT) speCrrO2 (NT) in indices of To reobtain in terms terms of of the the indices of functions functions W~ majorizing majorizing the the rrg controlled derivations, II~-sentences provable provable in in NT N T by by operator operator controlled derivations, we we have have to to check check which operators control control the which operators the provable provable sentences sentences of of NT. N T . We We introduce introduce a W.B Wwf3#Q+l . par(F(Zl,...,zn)) := {Zl,...,Zn} W0 (94) (94) zn ) is if , . . . ,,z~. Zn . if F F ((Zl z t ,, ... . . ,,z~) is a a sentence sentence with with only only the the shown shown number number parameters parameters Zt z~,... Then Then we we prove prove 59 2259 Set Theory Theory and Second Second Order Order Number Number Theory Theory rk(F) A, F,F, -~Ff (95) WW2.rk(F) (95) 2-rk(F) [par(F)] 0202.rk(F) [par(F)] 110 by induction induction on on rk(F). rk(F). First First observe observe that that according according to to (91) (91) we we have have Wo(n) W0(n) - n n+ + 11.. by For that F -type. If For symmetry symmetry reasons reasons we we may may assume assume that F has has I\ A-type. If F F is is atomic, atomic, then then f by Ll., F, 6(F) = = 00 and and we we obtain obtain Wo[par(F)] W0[par(F)] � ~ A, F, ~ F by aa clause clause 1\ A" 0 Otherwise Otherwise it it is is C(F) the second Go 1\ AG G11 or or F F - ((Vx)G(x). second case case we we get get FF ==- Go V'x)G(x). InIn the 20ork(G(z-)) A,Ll., GC�'), G ( z ) , - GG( ( z�:) ) for for all all Zz E e w w ((i)i) [par(G(z))] WW2.rk(G(z_)) 20rk(G(�» [par(G( �:))] 1Io2.rk(O(~_)) Ll., o = o == o by induction Since by induction hypothesis. hypothesis. Since Zz Ee WW2.rk(G(z_))[par(G(_z))] rk(G(z)) + + 11 20rk(G(�» [par(G(�))] �C_ WW2.rk(F)[par(F)][z] 20rk(F) [par(F)][z] 3~ 22.· rk(G(�)) i) we we obtain obtain from from ((i) rk(G(z»- +l A,Ll., G(�), WW2.rk(F)[par(F)][z] a ( z ) , - ~ Ff 20rk(F) [par(F)][zl 1oIo202.,k(C(~_))+1 for ii) finally for all all natural natural numbers numbers zz by by aa clause clause V V and and from from ((ii) finally 20rk(F) A, F,F, ~ Ff WW2.20rrk(F) k(F) [par(F)] [par(F)]lo1 02.rk(F) Ll., ii) ((ii) o o by similar but by aa clause clause 1\. A" The The first first case case is is similar but simplero simpler. DE:] Since Since we we have have W Wo[Zl,...,zn]~,G(Zl,...,zn) for every every true true atomic atomic formula formula O [Zl, o . . , Zn] � G(�l ' . . . ' �n ) for G(� G ( z ll ,' .· .o. ., '� zn) we obtain obtain n ) we . . . ((vx,,)a(Xl,... xn) Wwon �~ (V'Xl) (96) (vx,)... V'xn )G(Xl, . . . ,,x,,) for xn) by for every every mathematical mathematical axiom axiom ((VXl)... by n-fold n-fold applications applications of of V'Xn )G(Xl, . . ,,Xn) V'xd . . . ((Vxn)G(xl,... (1\). (A) From method of From the the method of (95) (95) we we obtain obtain also also the the equality equality axioms axioms 2n 20 r k(F(x» 2 w+ + ((VZ)(V~7)[Z=g-+ F ( Zx-)) - +--. F(y-')] (97) WW2"20rrk(F(~))-'l-2n-l-2 Z:;'IJ 1Io02.,k(F(~))+2n+2 vX)(wot\ vy, [x- = y- -t F( --'- F(y o!\ ] (97) k(F(x»+2n+2 [[Z-1 where the tuple ,,,h~re n is the the length l~ngth of oCthe tuple x ~ and and {z} {~'} := {{Zl' z , , . 0. . ,. , zz,,,} p~r((VZ)F(:0). m } == par(( V'x)F(x)). BByy the Lemma 201.2.4 the same same proof proof as as for for the the Induction Induction Lemma Lemma ((Lemma 2.1.2.4)) we we obtain obtain (rk(F(O»+n) --,F(O), f (Q) -,(Vx)[F(x) (V'x)[F (x) -t-+ F(S(x)) F(S(x))],] , F( F(n_.). (98) WW2.(rk(F(O))+n)[par(F(O))][n] n} (98) 20(rk(F(O»+n) [par(F(O)) ][n] 1I-~2002.(rk(F(0))-i-n) But But (98) (98) in in turn turn gives gives WWw.rk(F(0))[par(F(0))] F(x ++ 1)] F(0) 1\ A ((Vx)[F(x) 1)] -t -+ ((Vx)F(x). (99) work(F(O)) [par(F(0))] I~'0�+3 F(Q) V'x)[F(x) -t~ F(x V'x)F(x). (99) 0 �, o ,o It It remains remains to to adapt adapt Lemma Lemma 201.406, 2.1.4.6, i.eo i.e.,, to to show show that that there there is is aa natural natural number number kk such such that that ~- A ( x , , . . . , x , ) =~ W~.~+m[z,,...,z,] 0~ A ( z , , . . . , z , ) . . , Zn . N(w . m) = . m P(m) = = 'IjJ cP� (P(m)) (P(m)) < < 'IjJ cP� (w (w.· m) m) < <W Ww.m(0). P(m) wom (O). P(m) w . m. 00) ((100) 1 for for all all tuples tuples Zl Z l,, 0. . . , zn. Since Since N(w. m) = 22 . m we we obtain obtain P(m) « << w.m. Hence Hence (101) (101) 260 260 W. Pohlers Pohlers m We induction on We prove prove (100) (100) by by induction on m and, and, as as in in the the proof proof of of Lemma Lemma 2.1.4.6, 2.1.4.6, the the only only critical critical case case is is an an inference inference according according to to (3). We We proceed proceed as as in in that that proof proof and and obtain obtain from from the the induction induction hypothesis hypothesis for for the the premise premise ~ Ao(g), G(~7,t(~7)) aa natural natural number number ko such such that that ko (3). (Z) , G( W~.ko+mo ~ � A(~, G(~', W z, t(t(~). Z! f?Z) ). w.ko+mo [[z-] t(�)N k ko � �o(iI), G(iI, t(iI)) z ((i)i ) Because 48) aa natural Because t(g) N is is computed computed primitive primitive recursively recursively from from ~' we we find find by by ((48) natural number number k � > ko such such that that t(g) N < P(k) < W Ww.k+m[z-](0). Applying an an inference inference (V) (V) w.k+m [Z! (O). Applying to i ) we to ((i) we therefore therefore get get t(�)N P(k) Ww. k+m[Z! � �o(Z) , (3x)G(z, x). The immediately from induction hypothesis. The other other cases cases follow follow immediately from the the induction hypothesis. o O By (99) and By (96), (96), (97), (97), (99) and (100) (100) we we finally finally obtain obtain (102 F (3k)(3m)(3r) (102)) for arithmetical sentences F. F. This This yields yields aa rr H~ as follows: follows: If If for arithmetical sentences g -analysis as ((i)i ) NT � ~- (V'x)(3y)F(x, (Vx)(3y)F(x, y) NT y) for aa rr II~g -sentence (V'x)(3y)F(x, (Vx)(3y)f(x, y) y)then for then ii) ((ii) W~.~ ~ (V'x)(3y)F(x, (Vx)(3y)F(x, y) y) W w.k � by 102 ) . Defining Defining wn by ((102). ~n := (A�. (A[.exP(W,[+l)) and wn(�, &~(~,r/) recursively by by exp(w, � + 1 ) )((~)n) and 7J) recursively n+l(�, 7Jr/)) :"== exp(w,wn(�, wo(�,7J &o([,r/)) := �~ and and W &~+l(~, exp(w,&~([,rl)$Co~(rl)+l) we get get from from ((ii) by ii) by 7J) # wn(7J) + l ) we NT N T ~� F F =~ (3k)(3m)(3r) [WW• [W~.k~ => k � F]] Lemma 2.1.5.8 r-fold the Elimination r-fold application application of of the Elimination Lemma Lemma ((Lemma 2.1.5.8)) "'r(m) (V'x)(3y)F(x, (Vx)(3y)f(x, y). y) W wr (w.k,m) I,o[~,(m) W~r(~'k,m) 0 a wr(w·k, m) co iii) ((iii) Putting a ::= &r(w.k, m) < ~o we iii) using Putting we obtain obtain from from ((iii) using the the Inversion Inversion and and Witnessing Witnessing Lemmas Lemmas 2.1.5.5 and 2.1.5.3 ) ( Lemmas (Lemmas 2.1.5.5 and 2.1.5.3) (V'x (Vx E e N)(3y N)(3y < Wet(x))F(x, W~(x))F(x, y) y) ((iv) iv) which which shows shows 1(V'x)(3y)F(x,y) lrrg :::;<_a.a. I(Vx)(3y)f(x, Y)lno et [i] �~ (3y)F(i,y) On hand, if On the the other other hand, if a a "- 1(V'x)(3y)F(x,y) [(Vx)(3y)F(x, y)lno then we we get get W W~[i] (3y)F(/, y) for for l rrg then all a l l /i eE N N and and thus thus also also W W~et � ~o (V'y)(3y)F(x, (Vy)(3y)f(x, y). y). Hence Hence ]} ((103) 103) min {a {a[l (3,8) (3/3) [W [W~et � ~ F F]} lrrg == min W[F[no for for rr ri~g -sentences F. F. O := o Equation Equation (103) (103) can can be be taken taken as as evidence evidence for for the the naturalness naturalness of of the the concept concept of of operator operator controlled controlled infinitary infinitary derivations. derivations. 261 261 Set Theory and Second Order Number Theory 2.2. 2.2. Peano P e a n o arithmetic a r i t h m e t i c with w i t h additional additional transfinite transfinite induction induction 2.2.1. g -spectra 2.2.1. The T h e theories theories NT NT~t and their their rr H~ -< t and Let Let -< -~ be be a a primitive primitive recursive recursive well-ordering. well-ordering. fm) we -<) = N. By -< [m) field ((-~) = N. By T/ TI ((-~ we denote denote the the scheme scheme field Prog(-~, F) Prog( -<, F) --+ -+ For For simplicity simplicity assume assume that that (Vy -< -~ m)F(y) m)F(y) (\:fy F(y) fm. for .c s � -formulas F(y) expressing expressing transfinite transfinite induction induction along along -< -< [m. Then Then for T TI(<[) U T Tl(- mm EEN N denotes the the scheme scheme of of transfinite transfinite induction along all all proper segments of of -< -~.. denotes induction along proper initial initial segments Let Let NT. The g -spectra of The aim aim of of the the following following section section is is to to compute compute the the rr II~ of the the theories theories NT NT. NN:r: € --+~ ww satisfying: satisfying: (N1) (N1) (N2) (N2) (N3) (N3) (N4) (i4) N(O) N(a N(a # ~ (3) fl) = = N(a) N(a) + + N( N(fl). {3). aa =1=C Ww O~ �=~ N(wO) N(w ~) = = N(a) N(a) + + ll.. For {a << €e II N(a) For all all n n E Ew w the the set set {~ N(a) < < n} n} is is finite. finite. Observe Observe that that conditions conditions (N1) (N1) -- N(4) N(4) are are always always satisfiable satisfiable as as soon soon as as we we have have term term notations notations for for the the ordinals ordinals below below €. c. Using ned in Using the the norm norm N N and and the the starting starting function function P P as as defi defined in Definition Definition 2.1.4.1 2.1.4.1 N(0) = - o0.. we we may may extend extend the the collapsing collapsing function function 1/J r w and and the the collapsibly-less collapsibly-less relation relation « <<~€ to to all all ordinals €. Therefore ordinals below below ~. Therefore Definition Definition 2.1.4.2 2.1.4.2 extends extends to to the the ordinals ordinals below below €e and and we we get T/( -< r)[))) get the the same same results results as as in in Section Section 2.1.4. 2.1.4. Therefore Therefore we we need need only only to to know know cc cc (( T I(-< in this, however, in order order to to compute compute spec specno(NT~r To get get this, however, we we need need to to know know aa little little rrg (NT-< t)) .. To bit bit more more about about the the relation relation -< -<.. Call Call a a well-ordering well-ordering -< -< a a good representation for for €~ if if its its order order type type is is €c and and there there is is an an order order preserving preserving mapping mapping good representation 0: -<) --+> €c o: field( field (-<) satisfying: satisfying: 262 262 (ol) (01) (02) (02) W. Pohlers Pohlers w. (Vm)[o(m)eE Lim] Lim].. (Vm)[o(m) (Vm < w)[N(o(m)) w)[N(o(m)) < P(m) P(m) /\ Am m ::; < P(N(o(m)))]. P(N(o(m)))]. (Vm Let TI ( -< r,r, f ) ) for Let -< -~ be be aa good good representation representation for for cc.. We We want want to to compute compute cc cc ((TI(-~ for an an /:N-formula F. Let Let kk := := rk(F) and and put put .eN-formula . (k + + 1) # o(n). := F)) F. rk(F) o(n). an w Since m m -< -~ n n implies implies am am < a a~n E e LLira and N(am N(am + + 4) 4) = N(am) g(am) + + 44 = 22.. (k (k + + 1) 1) + + Since im and N(o(m)) + + 44 < 22.· (k (k + + 1) + P(m) P(m) + + 44 ::; < P(2 P(2.· (k (k + + 1) 1) + + m) m) ::; < P(N(a P(N(an) m) we we get get N(o(m)) 1) + n ) + m) mm --<~ nn =>=~ am a m ++44< «� < ~ a ~a.n . (104) (104) Because of of w w ::; _ a a ,n and and N(n Y(n + + 1) 1) = n n+ + 11 ::; < P(o(n)) P(o(n)) + + 11 ::; < P(a P(an) we also also have have Because n ) we nn ++ 11 «0 (105) < = = We use use (104) (104) and and (105) (105) in proving We in proving (106) F), (Vy n)F(y) (106) by -<-induction -~-induction on on n. For For any any m m E EN we have have N we by ' 0--e-I~ ~Prog(-~, F), (Vy (Vy -< -~ m)F(y), (i) (i) m)F(y), -~m,m -<~ nn -,Prog( -<, F), If?00 or either either by by (AxM) (AxM) with with a;" a,~ am if if m m -< ~ n. n. or by by induction induction hypothesis hypothesis with with a;" a m = am [~.e_ ~Prog(~, -~ n)F(y) -<, F), (Vy -< If?- -,Prog( t = - ' Byy (72) (72) we we also also have have B I = ~o~ -,F(m), --,F(m), F(m) f (m) I[� and obtain obtain and 1 (ii) (ii) F), (Vy m)F(y) -,F(m) , -,m F(m) /\ ~ f ( m ) , ~m_ -< [[0~+1 -~ m ) f ( y ) A -~ n, _n, f(m)_ II:m+ ~Prog(-~, -, Prog( -<, f ) , (Vy -< (iii) (iii) from (i) and and (ii) (ii) by by an an inference we get from (i) inference A. /\. By By (105) (105) we get -<, F), -< _n, F(m) II0~+2 -,Prog( ~Prog(-~, F), -, ~m _m -~ n, F(m) II:m (iv) (iv) from (iii) (iii) by by an an inference from inference 3:3 and and ~Prog(-~ -<, F) F), ~m -,m -<-~ nn vV F(m) F(m) III1~+4 :m H -,Prog( I0 ~ ~ - - __ (v) (v) from (v) (v) by by two two inferences inferences (V). we finally finally get get from (V) . Using Using (104) (104) we F), (Vy (Vy -< n)F(y) I f?- -,Prog( -<, F), V. from by an from (v) (v) by an inference inference V. (vi) D o Together with with the the previous previous section section (107) (107) yields yields that that for every sentence sentence FF in in the the Together for every theory NT - I � F. Together Together with with Lemma Lemma 2.1.4.6 2.1.4.6 this this means means Set Set Theory Theory and and Second Second Order Number Number Theory Theory f- F NT -< t ~- F NT. w)[ I� FJ 2 63 263 => (3c~ (30 << c)(3m =~ c )(3m << w)[l~-~ F] c from which which we we get get by by cut-elimination cut-elimination and and the the fact fact that that c is is an an e-number c-number from f- F NT -< t ~ F NT. c [ I � FJ. (107) ) => (Sc~ (30 << ~)[l~ =~ El. c. (107) c Hence specno(NT c 10 ( 8) (108) 2.1.4.5 2.2.1.1. heorem. Let good representation 2.2.1.1. TTheorem. Let ~c bebe anan e-number c-number and and -<-< aa good representation for for ~.c. NT-< rt are are exactly exactly the the functions functions which which are are Then the the provably provably recursive recursive functions functions ofof NT.< Then elementary inin Wa Wa -- asas defined defined inin (63) some c~0 << c.c. elementary (63) --for f o r some Applying Theorem Theorem 2.1.4.5 we we get get the the following following theorem. theorem. Applying 2.2.2. Significance Significance of NT -< rt 2.2.2. of the the theories theories NT.< An ordinal analysis analysis of of aa theory theory AAx yields -- among among others others -- the ordinal IIAxl A n ordinal x yields the ordinal llAxlll < < K . We call an an ordinal ordinal analysis analysis for for aa theory theory AAx only w~ W e call x 22 NNT T profound if if it it not not only WfK. primitive recursive computes IllAxll computes but also also provides provides aa primitive recursive well-ordering, well-ordering, say say -< 4,, which which IAxl1 but is aa good good representation for llAxll IIAxl1 such such that that is representation for profound f- F Ax F r{:} NT Ax~ F N T ~-< tr ~- F (109) (109) f- formulas. If have aa profound profound ordinal analysis of of Ax we holds for all all arithmetical arithmetical formulas. If we we have ordinal analysis A x we holds for know by by (108) its II~ fIg-spectrum and and by by Theorem Theorem 2.2.1.1 also its provably recursive know (108) its also its provably recursive functions. functions. All All known known ordinal ordinal analyses analyses are are profound. profound. The The general general reason reason for for that that can can be be roughly roughly sketched. sketched. Recall Recall from from Section Section 1.4 the the main main steps steps in in an an ordinal ordinal analysis analysis which which are: are: 2.2.1.1 1.4 • Designing a semi-formal calculus � .0. which commonly needs a term notation for ordinals. Transforming a formal derivation F into an infinite semi-formal derivation 9 Designing a semi-formal calculus ~ A which commonly needs a term notation for ordinals. •9 Transforming a formal derivation Ax Ax f~ F into an infinite semi-formal derivation � F. • Cut elimination for the semi-formal calculus, yielding � F � F. 9 Cut elimination for the semi-formal calculus, yielding ~ F => => ~o F. Arithmetizing Arithmetizing the the term term notation notation gives gives iinn general general aa primitive primitive recursive recursive well-ordering well-ordering -< -< which which is is aa good good representation representation for for IIAxll Ilhxll.. Unravelling Unravelling aa formal formal derivation derivation into into an an infinite infinite one one results results in in aa recursive recursive infinite infinite tree. tree. Therefore Therefore we we may may restrict restrict the the semi-formal semi-formal calculus calculus to to recursive recursive proof proof trees. trees. Then Then there there is is aa recursive recursive predicate, predicate, say say Proofoo(x, y, z, u), such such that that ' ) expresses Proofoo(e__, r~o,' , rp' ~f , '.0. rA~) expresses that that Proofoo (x, y, u), Proofoo (�, "e"e isis the the code code of of an an infinite infinite recursive recursive tree tree tagged tagged with with ordinal ordinal notations notations (i. (i.e., e., elements elements in in the the field field of of -<,) -<,) and and codes codes for for finite finite formula formula sets sets which which is is locally locally correct correct with with respect respect to to the the axioms axioms and and rules rules of of the the semi-formal semi-formal calculus calculus witnessing witnessing � ~ .0.." A." w. W. Pohlers 264 ProolAx(r,., If then the recursive If we we assume assume ProofAx(e__, rF' rF')) then the embedding embedding procedure procedure yields yields a a recursive function function 9 g such such that that Proof Proofoo(g(e_), rF~)) oo (g(r,.) , n_n_,r, rF' '!l, '[, where computable from within NT. where n n and and rr are are computable from ee.. This This can can be be done done within N T . If If we we secure secure that all the proof tree that all the manipulations manipulations which which are are done done to to an an infinite infinite proof tree during during the the cut cut elimination procedure locally recursive, elimination procedure are are locally recursive, we we can can use use the the Recursion Recursion Lemma Lemma to to obtain that obtain aa recursive recursive function, function, say say h, such such that h, F' ) . Proofoo(h(g(e_.)), {3' , O,0, rrFT). Proof oo (h(g(r,.) ), rr3~, Besides NT transfinite induction fr{3' .. There Besides N T the the Recursion Recursion Lemma Lemma needs needs transfinite induction along along -< -~rr37 Therefore within NT -< r . Using Using the fore this this step step can can be be done done within NT.~r. the sub-formula sub-formula property property of of cut cut free free infinite infinite derivations derivations we we obtain obtain Proofoo(e_, Proof oo (r,., n, Q0_,, rFrFT)' ) '!l, ~ Truek( Truek(rF ~) rF') -+ by by induction induction on on -< -~ fn rn where where Truek Truek denotes denotes aa partial partial truth truth predicate predicate for for formulas formulas of of this can done in in NT- we summing up: up: we get, get, summing ProolAx(r,., F) Proofoo (g(r,.) , Proofoo(h(g(r,.) ), 0, F) Ax hx � ~ F F � ::~ NT NT � ~ ProofAx(e_, F) =~ NT NT � ~- Proofoo (g(e_), n, n, r, r, F) F) � =v NT-< NT.~rr � ~ Proo]oo(h(g(e__)), m, m, O, F) � =~ NT.~r ~ Truek(rF ~) � NT-< r � Truek( rF' ) � F =v NT- ( Since -< f)r) we Since we we have have NT NT � C_ Ax A x and and Ax Ax � ~ TI Tl(-~ we also also have have the the opposite opposite direction direction NT- for for arithmetical arithmetical formulas formulas and and the the ordinal ordinal analysis analysis is is profound. profound. Having Having a a profound profound ordinal ordinal analysis analysis for for aa theory theory Ax, Ax, we we can can try try to to sharpen sharpen (109) by - (109) y))] PRWO( -< fm) ) ((V2)[(Vy)(f(Z,y) (3y)(-~f(2, f(:~,y))] -,f(x, Sy) -<-< f(x, V'x) [(V'y) (f(x, y) -<-< m) -+--+ (3y)( ((PRWO(- expresses expresses that that there there are are no no infinite infinite primitive primitive recursive recursive -<-descendent -<-descendent sequences sequences in in -~ fm. Im. We We put put -< -< f)I) "= U PRWO(-< PRWO( PRWO(-< PRWO(- m m -< ) EE field( field(-<) 265 265 Set Set Theory and Second Order Number Theory f) "e is the Proof/:(e, m, index of a primitive recursive tree tagged with members offield( -<) (serving as ordinal notations), numbers numbers (for (for the the cutcut - rank) rank) and and finite finite formula formula sets, sets, which which isis locally locally notations), correct with respect to the the axioms axioms and and rules rules of the semi-formal semi-formal system system (augmented (augmented byby correct with respect to of the aa replication replication rule rule whose whose premise premise and and conclusion conclusion are are identical) identical) such such that that itsits bottom bottom node isis tagged m (coding (coding the the height of the the tree), tree), rr (coding (coding itsits cut cut rank) rank) and and b,A ".". node tagged with with m height of and want want to to replace replace NT.~ NT-< tf by by E~ I;�-IND ++ PRWO(-~ PRWO( -< r) in in (109). (109) . This This does does not not work work for for and The proof proof needs needs Mints' Mints ' arbitrary arithmetical arithmetical formulas formulas but but only only for for ri~ rrg -sentences. The arbitrary continuous [, rAT) express that that "e is the continuous cut cut elimination elimination theorem. theorem. Let Let Proofer(e, m, r_, rb,') express index of a primitive recursive tree tagged with members of field(-~) (serving as ordinal By Mints' Mints ' continuous continuous cut cut elimination elimination there there exists exists aa primitive primitive recursive recursive function, function, say say By H, such such that that H, Proof!:(H(e), k,/s., O,Q, rA7) -+ Proofs Proof/:(e, m,m, r_r_, rA7) rb,') --~ rb,' ) where kk is is computable computable from from H(e) H(e) inin such such aa way that, provided provided that that -~ -< isis aa wellwell where way that, ordering, -~rk -< fk has has order order type type exp"(2, expr (2, otyp.~(m)). otyp-« m) ) . Giving Giving aa sketch sketch of of the the proof proof of of ordering, [, E~ I;�-IND ~� Proofs Mints' contribution. But we want want Mints ' theorem theorem would would lead lead us us far far outside outside the the scope scope of of this this contribution. But we to give of flow flow chart chart how how to to use use itit in in sharpening (109) . Thus assume that that to give a a kind kind of sharpening (109). Thus assume we ordinal analysis A x and let -~ be for we have have aa profound profound ordinal analysis of of Ax and let be aa good good representation representation for \\Ax\\ be aa ri~ IIg -sentence such that [IAx[[.. Let Let (Vx)(3y)F(x,y) be such that (Vx) (3y)F(x, y) Ax (Vx)(3y)F(x, y). y). A x ~� (Vx)(gy)F(x, -< (i) (i) By obtain By (109) (109) we we thus thus obtain NT - (ii) (ii) F1, , there are are formulas formulas F 1 , . . . , Ft F/ which which either either belong to By Theorem 2.1.2.2 and By Theorem and (39 (39)) there belong to NT -< f or NT.~r or are are identity identity axioms axioms such such that that � ~0 ,F ~ F 1l ,, .. .. .., , ' . • • F/ , (3y)F(k, y) ~Fz, (3y)F(/s., y) (iii) (iii) k. Looking more more carefully for for every every number number k. Looking carefully at at the the embedding embedding procedure procedure we we observe observe that primitive recursive index for that the the resulting resulting infinitary infinitary proof proof tree tree is is primitive recursive and and that that an an index for that that tree Since the tree can can be be computed computed from from the the formal formal proof. proof. Since the provably provably recursive recursive functions functions are of I;�-IND E~ are exactly exactly the the primitive primitive recursive recursive ones, ones, this this embedding embedding procedure procedure can can of In be within I;�-IND. be formalized formalized within E~ In the the next next step step we we observe observe that that all all axioms axioms in in NT-< NT.~rf and and all all identity identity axioms axioms have have primitive primitive recursive recursive proof proof trees trees in in the the semi-formal semi-formal calculus. only case is not instance of calculus. The The only case in in which which this this is not completely completely obvious obvious is is an an instance of proof of -<, G) -+ Prog(-~, --+ (Vy -< -~ n)G(y). The The proof of (106) shows shows how how to to construct construct the the tree. tree. Prog( Instead -< r)r) does Instead of of using using induction induction on on -< -~ m I;�-IND E0_IND + + PRWO( PRW0(-~ does not not know know that that -< -~ is is well-ordered we start with the bottom node and enumerate all possible premises. w e l l - o r d e r e d - we start with the bottom node and enumerate all possible premises. This This gives gives G) (Vy n.)G(y). . ..., -<, G), 9 . , ,Prog( --, Prog(-..<, G ) , 'ill --,m -< -~ n. n V V G(ill G(m_.), ) , . .9.9 9 ,Prog( -~Prog(~,-<, G), G ) , (Vy ( V y -< --< n.)G(y) n__)G(y) ,m Above Above any any of of these these nodes nodes we we decide decide primitive primitive recursively recursively whether whether ~m -< -~ n n.. If If this this is is 266 266 w. W. Pohlers Pohlers true Prog( -<, G) m -< node. Otherwise true then then we we add add ..., ~Prog(-~, G),, ..., -~m -~ !l n as as top top node. Otherwise we we construct construct )G(y) G),, (Vy -< --,:m m)G(y) G(m) ...,-~Prog(-,:, Prog( -<, G) ...,-~G(m), G (m) , G( m) G (m) , G(m) G),, (Vy -< 9m)G(y) m)G(y) 1\ A ..., -~G(m), Prog( -<, G) ...,-~Prog(-<, Prog(-<, G),, G(m) G (m__) ...,-~P rog( -<, G) Prog( -<, G) m -< C),, ..., ~m -,; !l nV v G(m) ...,--,Prog(-.<, and Prog( -<, G) m)G(y) . and repeat repeat the the procedure procedure above above ..., -~Prog(-~, G),, (Vy (Vy -< -~ m)G(y). Summing up Summing up we we obtain obtain aa primitive primitive recursive recursive function function hh such such that that r (3y)F(x , yf)]. r~~ ~ (Vx) (W)[P~oof~x(e, y)')) -+ -~ Proo P~oofs y)')]. (iv) (iv) (3y)F(x , yf ��-IND � [ProolAx (e, ~(3y)F(~, !!:;(h(e) , m, [~,, r~(3y)F(~, Together Together with with Mints Mints'' Theorem Theorem this this yields yields Ax~(Vx)(3y)F(x, =~ �� E~- IND � ~ (Vx) (Vx)[Proofs O, rr(3y)F(~, y)7)] (v) (v) (3y)F(x , yf)] [Proo!!:;(H(h(e)), m, Q, Ax � (Vx) (3y)F(x, y) ::} But But (3y)F(x, y) is is aa � E~� -formula. A A cut cut free free infinitary infinitary proof proof of of (3y)F(x, y) cannot cannot contain contain instances instances of of aa V-rule V-rule and and is is thus thus finite. finite. Every Every path path in in the the proof proof tree tree is is -< fr)) to primitive primitive recursive recursive and and we we may may therefore therefore use use PRWO(-~ to deduce deduce E~ + PRWO(- PRWO( PRWO(-< f) from from (v) (v) which which in in turn turn entails entails (3y)F(x, y). E~- IND + + PRWO( PRWO(-< ~-- (Vx) (Vx)(3y)F(x, y). -< f[)) � �� (vii) (vii) By (i) and we have By (i) and (vii) (vii)we have Ax (3y)F(x, y) {o} + -< f) t(3y)F(x, y) ((110) A x ~� (Vx) (Yx)(3y)F(x,y) r ��-IND E~ + PRWO( PRWO(--~r) i-- (Vx) (Yx)(3y)F(x,y) 1 10) for g -sentences (Vx) (3y)F(x, y) since the opposite for rr H~ (Vx)(3y)F(x, since the opposite implication implication holds holds obviously. obviously. 0 O Call aa function Call function f f -<-descendent -,:-descendent recursive recursive if if it it is is represented represented by by aa function function term term operator J.L.." which built up which is is built up from from C;:. C~,, P;:' P~n and and S S by by Sub, Sub, Rec and and the the search search operator #.~ which which is defined by is defined by min {{yy]I ""! Sy) -~ !f (x, (#. -< It It is is not not very very difficult difficult to to show show that that the the provably provably recursive recursive functions functions of of ��-IND + -< n are recursive functions E~ + PRWO(-~r) are exactly exactly the the -<-descendent -<-descendent recursive functions (cf. (cf. e.g.,Pohlers result this this shows e.g.,Pohlers [1992] [1992] for for aa proof) proof).. Together Together with with Weiermann's Weiermann's result shows that that for good representation Ax l1 aa function for aa good representation -< -~ for for II[[Ax[I function f f is is -<-descendent -<-descendent iff iff it it is is primitive primitive recursive can also also be proved directly, some a in Wa for recursive in for some a < < IIAx [[Ax[[. A result result which which can be proved directly, ll . A even conditions on Buchholz, Cichon even under under weaker weaker conditions on -< -~ (cf. (cf. Buchholz, Cichon and and Weiermann Weiermann [1994]). [1994]). A A comprehensive comprehensive study study on on -<-descendence -~-descendence and and proof proof theory theory can can be be found found in in Fried Friedman man and and Sheard Sheard [1995] [1995].. A A completely completely worked worked out out proof proof of of Mints' Mints' theorem theorem is is in in Buchholz Buchholz [1991]. [1991]. PRWO( WQ 3. systems 3. Impredicative Impredicative systems The The aim aim of of this this chapter chapter is is to to give give upper upper bounds bounds for for the the proof-theoretical proof-theoretical ordinals ordinals of of some some impredicative impredicative axiom axiom systems systems of of Number Number Theory Theory and and Set Set Theory. Theory. We We 267 267 Set Theory Theory and Second Second Order OrderNumber Number Theory Theory will restrict restrict ourselves ourselves to to II~ analyses which which are are already already sufficiently sufficiently complicated. complicated. will TIt analyses Moreover, Moreover, we we will will also also not not demonstrate demonstrate the the latest latest state state of of the the art art but but restrict restrict ourselves ourselves to to three three axiom axiom systems systems for for Set Set Theory, Theory, KPw K P w , , axiomatizing axiomatizing an an admissible admissible universe, universe, KPl, KP1, axiomatizing axiomatizing a a union union of of admissible admissible universes universes and and KPi K P i axiomatizing axiomatizing an admissible admissible union union of of admissible admissible universes, universes, and and the the corresponding corresponding axiom axiom systems systems an for Number Theory. Mahlo for Number Theory. Today Today we we know know also also how how to to analyse analyse axiom-systems axiom-systems for for Mahlouniverses, universes, TIn-reflection IIn-reflection and a n d -- though though II have have not not yet yet seen seen the the proofs proofs- even even for for �l -separation. T. the analysis El-separation. T. Arai Arai has has announced announced the analysis of of even even stronger stronger systems. systems. He He uses, uses, however, however, a a different different technique technique which which is is based based on on G. G. Takeuti's Takeuti's methods. methods. 3.1. 3.1. Some S o m e remarks r e m a r k s on on predicativity p r e d i c a t i v i t y and a n d impredicativity The contribution is The focus focus of of this this contribution is on on the the ordinal ordinal analysis analysis of of impredicative impredicative systems. systems. In order order to to distinguish distinguish impredicative impredicative theories theories from from predicative predicative ones ones we we need need aa short short In discussion Limitations of discussion on on predicativity predicativity and and impredicativity. impredicativity. Limitations of space space force force us us to to be be rather rather sketchy. sketchy. There are which characterize £( E) -structure ffJl. 001. The There are two two ordinals ordinals which characterize aa transitive transitive/:(E)-structure The ordinal := min min {a OO1} and which cannot ordinal 0(001) o(gYt) := {c~ E e On On II a a � r if)t} and the the least least ordinal ordinal which cannot be be pinned down down in in 001 9~t (cf. (cf. Barwise Barwise [1975,III.7 and VII.3]). VII.3]). We We need need not not to to repeat repeat the the pinned [1975,111.7 and definition "Pinning down ne it defnition of of "Pinning down ordinals" ordinals" because because we we are are going going to to refi refine it in in the the following following way. way. Assume Assume that that the the language language £oo s ,w is is coded coded as as sets sets as as e.g. e.g. in in Barwise Barwise [1975] [1975].. Introduce the infinitary proof within aa semi-formal Introduce the notion notion of of an an infinitary proof within semi-formal system system for for Set Set F that Theory Section 1.4. Theory as as sketched sketched in in Section 1.4. Denote Denote by by T T � ~-F that T T is is an an infinitary infinitary proof proof tree tree for for the the formula formula F. F. We We say say that that aa countable countable ordinal ordinal a a is is provably provably pinned down transitive £(E) -structure 001 pinned down in in aa transitive s 93~ if if there there is is a a well-ordering well-ordering -~ on on w of order type 001, aa (possibly of order type a c~ in in 9Yr (possibly infinitary) infinitary) formula formula (cf. (cf. Pohlers Pohlers [1989,§19] [1989,w for for examples which expresses examples of of such such formulas) formulas) Found( Found(-<) in in 001 99~ which expresses the the well-foundedness well-foundedness of of -< -~ and and an an infinitary infinitary proof proof T T in in 001 9Yt such such that that T T � ~ Found( Found(-<). Define Define h(001) h(D~t) := := min } . Of min {a { c~ E E On On]I a c~ cannot cannot be be provably provably pinned pinned down down by by OO1 93~}. Of course course we we always always have have 0(001) let 001 be the the initial o(99l) � _ h(001) h(93~).. Now Now let 99l be initial part part La L~ of of the the constructible constructible hierarchy. hierarchy. h(La) . Then Then h( a) ::= h( a) ordinal a Then o(La) o(L~) = - a c~ and and we we put put h(c~) - h(L~). Then a c~ � _ h(c 0.. We We call call an an ordinal c~ (a) . For ordinal we autonomously inaccessible if if a c~ = = h h(c~). For an an autonomously autonomously accessible accessible ordinal we ( a) which pinned down have have a c~ < < h h(c~) which means means that that a c~ can can be be provably provably pinned down by by La L~.. Then Then we we have have a a formula formula Found(a) Found(c~) EE La L~,, expressing expressing the the well-foundedness well-foundedness of of an an well-ordering well-ordering a) . If of of order order type type a c~ and and a a proof proof T T E La L~ such such that that T T � ~ Found( Found(a). If we we denote denote again again by bound for by T T~ F F that that (J fl is is an an upper upper bound for the the height height of of T T and and the the complexity complexity of of all all formulas occurring in formulas occurring in T T and and p a a strict strict upper upper bound bound for for the the cut cut formulas formulas occurring occurring in in T T then then there there are are ordinals ordinals (J fl and and p less less than than a c~ such such that that T T ~ Found( F o u n d ( o la) ) .. If If we anticipate anticipate that that we can construct construct La L~ whenever we have have the ordinal a e we can interpret interpret autonomously autonomously accessible accessible ordinals as ordinals which can be secured secured by smaller ordinals (cf. Schlfiter Schliiter [1990] [1990] for a fully fully worked out version of these ideas) ideas).. The notion of autonomously accessible and inaccessible ordinals is due to Feferman (cf. Feferman Feferman [1964]). [1964]). (cf. -< -<) -<). autonomously inaccessible � , O � r 268 268 W. Pohlers The The Elimination Elimination Lemma Lemma (Lemma (Lemma 2.1.2.8) 2.1.2.8) and and the the Predicative Predicative Elimination Elimination Lemma Lemma (Lemma (Lemma 2.1.2.9) 2.1.2.9) as as well well as as the the Boundedness Boundedness Theorem Theorem (Theorem (Theorem 1.3.6) 1.3.6) carry over. over. So So we we get get carry expn (2, (3) T~n => otyp(-<)< otyp ( -<) � expn(2,13) Found( -<) =~ T � Found(-<) (111) (111) Found( -<) =~ TT~� Found(-<) => otyp(-<)<~Op/3. otyp ( -<) � <{Jp{3. (112) (112) for nn << w w and and for It follows follows from from (111) (111) that that w w and and from from (112) (112) that that all all strongly strongly critical critical ordinals ordinals are are It autonomously inaccessible. inaccessible. This This has has first first been been observed observed by by Feferman Feferman [1964] [1964] and and autonomously independently by by Schiitte Schutte [1965a] [1965a] who who both both could could also also show show that that these these are are the the only only independently autonomously ordinals (cf. (cf. Schiitte Schutte [1965b]). [1965b]). A A proof proof of of this this fact fact which which autonomously inaccessible inaccessible ordinals is in in the the spirit spirit of of the the above sketch can can be be found found in in Pohlers Pohlers [1989]. [1989] . is above sketch In some some sense sense the the notion notion of of autonomous autonomous accessibility accessibility captures captures the the idea idea of of predpred In icativity. First First we we see see that that without without accepting accepting the the ordinal ordinal w we stay stay within within the the icativity. w we hereditarily finite world. we have accepted w w as we can for the hereditarily finite world. Once Once we have accepted as aa set set we can look look for the are provably provably pinned pinned down down in ' Then we look ordinals a ordinals c~ which which are in Lw L~+I. Then we construct construct L,,, L~, look 1 + for ordinals provably pinned in L" so on. at the for ordinals provably pinned down down in L~ and and so on. This This process process will will stop stop at the exhausted by first strongly critical o. On the other hand Lro Lro isis also first strongly critical ordinal, ordinal, i.e., i.e., at at f F0. On the other hand also exhausted by this this procedure. procedure. In In that that sense sense fo F0 is is known known to to bound bound predicativity. predicativity. We We stick stick to to that that notation notation in in aa very very technical technical manner manner and and call call theories theories whose whose Ill-ordinals II~-ordinals are are below below fo possibly stronger Fo predicative predicative without without further further reflection reflection whether whether there there are are also also possibly stronger principles principles which which can can be be predicatively predicatively justified. justified. But But we we will will see see in in the the following following section section that that there there is is aa completely completely novel novel feature feature in in the the ordinal ordinal analysis analysis of of impredicative impredicative (i.e., (i.e., non non predicative) predicative) systems, systems, collapsing. collapsing. The The simplest simplest theory theory which which needs needs aa collapsing collapsing argument argument in in its its Ill-analysis H~-analysis is is the the theory theory of of non-iterated non-iterated inductive inductive definitions definitions which which is is introduced introduced in in the the next next section. section. Its Its ordinal ordinal is is 7/Jw r l ((~w1+1), an ordinal ordinal which which already already has has been been described described by by H. H. Bachmann. Bachmann. C:W1+l), an There C:W1+ 1 ) , e.g. There are, are, however, however, theories theories whose whose Ill-ordinals H~-ordinals are are between between fo F0 and and 7/Jw r l ((c~1+1), e.g.,, the the theory theory ATR ATR introduced introduced by by Friedman Friedman which which axiomatizes axiomatizes autonomous autonomous transfinite transfinite recursion recursion (which (which is is the the axiom axiom (Aut-m) (Aut-H ~ introduced introduced on on page page 276 276 together together with with the the full full fE Most recently many theories scheme of Mathematical Induction) . Its ordinal is scheme of Mathematical Induction). Its ordinal is F, Oo.' Most recently many theories between between fo F0 and and 7/Jw r l (C:W1+ 1 ) have have been been analyzed. analyzed. G. G. Jager J~iger calls calls these these theories theories meta-predicative. meta-predicative. A A good good summary summary on on predicative predicative theories theories can can be be found found in in the the booklet booklet Jager J~iger [1986] [1986].. A A sample sample of of papers papers treating treating meta-predicative meta-predicative theories theories is is Jager J~iger et et al. al. [n.d.] [n.d.],' Jager J~iger and and Strahm Strahm [n.d.] [n.d.], Jager J~iger [1980] [1980],, Palmgren Palmgren [n.d.] [n.d.], Strahm Strahm [n.d.] [n.d.] and and Kahle Kahle [1997]. [1997]. This This ' ' list list has has been been communicated communicated to to me me by by T. T. Strahm. Strahm. 3.2. 3.2. Axiom Axiom systems systems for for number number theory theory In In the the present present section section we we will will introduce introduce some some impredicative impredicative axiom axiom systems systems for for Number Number Theory. Theory. We We will will not not give give an an ordinal ordinal analysis analysis for for these these systems systems directly directly-which which would would be be possible possible in in all all demonstrated demonstrated cases cases -- but but show show that that all all these these systems systems Set Theory Theory and Second Order Order Number Number Theory Theory 269 can analysis for can be be embedded embedded into into axioms axioms systems systems for for Set Set Theory. Theory. The The ordinal ordinal analysis for the number-theoretic number-theoretic systems systems will then be be obtained obtained via via an an ordinal analysis of of the the the will then ordinal analysis set-theoretic systems. impredicative set-theoretic systems. We We start start with with the the most most simple simple example example of of an an impredicative axiom system. axiom system. 3.2.1. ID! 3.2.1. The T h e theory t h e o r y ID1 By a a monotone inductive definition on natural natural numbers numbers we we usually usually understand understand aa By monotone inductive definition on monotone operator monotone operator r: F: Pow(N) -+> Pow(N) Pow(N),, i.e., i.e., an an operator operator for for which which we we have have r (S) � A set set S S � c_ N N is is called called r-closed F-closed iff iff r F(S) C_ S. S. We We obtain obtain the the least least fixed-point fixed-point A Ir of of r F - often often called the fixed-point fixed-point of as the the intersection intersection of of all all r F-closed Ir called the of rF -- as -closed (S) � subsets subsets of of N, N, i.e., i.e., Ir Ir = = n ~ {{S S II r F(S) c_ S} S}.. A A set set P � C_ 1~ is is inductively inductively definable definable iff it it is is primitive primitive recursive recursive in in the the fixed fixed point point of of some some inductive inductive definition. definition. An An iff operator x) such operator is is arithmetically arithmetically definable definable iff iff there there is is an an LN Z:N formula formula A(X, A(X,x) such that that rF(S) (S) = {x e NN] I N x) } . If x) is X-positive formula, = {x N F ~ A(S, A(S,x)}. If A(X, A(X,x) is an an X-positive formula, i.e., i.e., if if its its A(X, x) translation translation into into the the Tait-Ianguage Tait-language contains contains no no occurrences occurrences of of tt � ~g X X,, then then A(X, x) S � C_ T T =? =~ r(S) F(S) � C_ r(T). (T). S P N" E defines i.e., an defines a a monotonic monotonic operator, operator, i.e., an inductive inductive definition. definition. We We do do not not want want to to go go into into the the theory theory of of inductively inductively defined defined sets sets (cf. (cf. Moschovakis Moschovakis [1974] and Barwise [1975] that and Barwise [1975] for for aa profound profound study) study).. All All we we want want to to say say here here is is that the nition comes which are the fixed-point fixed-point Ir Ir of of an an monotone monotone inductive inductive defi definition comes in in stages stages I� ICr which are (I��~)) where := U, reasons there there is defined by defined by I� I~r := := F(I~ where I�� I~ ~ "= I..Jr<{ I�. ICr 9 By By cardinality cardinality reasons is a a countable ordinal aa such nes countable ordinal such that that I� I~ = = I�C1. I~ ~. One One defi defines r If! = min Irl ::-min {{aa lI I� I~ = - l�C1} 1~" } closure ordinal r. inductive norm and If! the Thus Ir and calls calls Irl the closure ordinal of of F. Thus Ir = = Ifl. II~I. For For every every element element ss E IIr r we we may its inductive norm may introduce introduce its Islr � 1 ss E I�}. lair := : - min min {{~1 I~r}. We f! = We then then obtain obtain IIrl = sup sup {{ Islr I~1~ + + 1111 s E e Ir}. I t } . For For arithmetically arithmetically definable definable operators operators we have have Irl IFI_::; WfK w~K.. rF we The The theory theory IDl ID1 axiomatizes axiomatizes the the existence existence of of least least fixed-points fixed-points for for positively positively definable arithmetical arithmetical inductive definable inductive definitions. definitions. Recall Recall the the language language LN s of of Number Number IA for -positive Theory. the language Theory. To To obtain obtain the language LID Z:lox1 we we add add aa set set constant constant IA for every every X X-positive formula formula A(X, A(X, x) x) in in the the language language LN s which which contains contains only only the the shown shown free free variables. variables. We We extend extend the the scheme scheme of of Mathematical Mathematical Induction Induction to to all all L Llox formulas and and augment augment ID 1 formulas the the axioms axioms of of NT N T by by the the schemes schemes [A(IA, x) (IDd IA] (IDx) 1 (V'x) (Vx)[A(IA, X) -+ --+ xX E elA] and and 270 270 W. Pohlers (ID1) (Vx)[A(B,x) ~ B(x)] B(x)] --+ --+ (\lX) (Vx)[x IA --+ -+ B(x)] B(x)] [A(B, x) --+ [X E6_ IA (10 1 ) 22 (\lx) where arbitrary .e1D! formula. where B(x) B(x) is is an an arbitrary s formula. While While scheme scheme (1 (ID1) expresses that that IA IA is is 0 1 ) 11 expresses r FA A(x,x)-closed, scheme (10 (ID1) expresses that that it it is is the the least least r FA A(x,x)-closed set. The The (x,x) -closed set. (x,x) -closed, scheme 1 ) 22 expresses standard standard semantics semantics for for .e1D s ! is is obtained obtained interpreting interpreting IA IA by by IA IA.. Instead ordinal analysis possible e.g. Instead of of giving giving a a direct direct ordinal analysis for for ID} ID1 - which which is is possible e.g. cf. cf. Pohlers Pohlers [1989] [1989] - we we will will show show that that it it can can be be easily easily embedded embedded into into axiom axiom systems systems for for Set Set Theory. Theory. 3.2.2. 3.2.2. IIterated t e r a t e d iinductive n d u c t i v e definitions definitions The The expressive expressive power power of of first first order order logic logic with with free free set set parameters parameters is is of of course course not not exhausted by exhausted by the the axiom axiom system system ID} ID1.. As As soon soon as as we we have have fixed-points fixed-points of of inductive inductive definitions we definitions we may may use use them them in in the the definition definition of of new new operators. operators. We We are are then then leaving leaving the the realm realm of of 'elementary 'elementary inductive inductive definitions definitions'' on on the the structure structure N N in in the the sense sense of of Moschovakis [1974] Moschovakis [1974].. To To formalize formalize the iteration let let -< -< be be aa well-ordering well-ordering of of order order the iteration type binary predicate constant JA JA to -positive .eN type v v and and associate associate aa binary predicate constant to every every X X-positive s formula formula A(X, Y, Y, x, y) which which contains contains at at most most the the shown shown free free variables. variables. We We are are going going to to write write x, y) A(X, (a, b) E JJA; A ; by (3y -< [a E6_ JJ~]. aa E6_ J� JbA instead instead of of (a, b) 6_ by a a E 6_ J1 j~bb we we abbreviate abbreviate the the formula formula (=ly -< b) b)[a �] . The language language .e1D" s is obtained obtained by by augmenting augmenting the the language language .eN s by by aa constant constant for for The is -<) the is saying is aa -< -< and and all all constants constants JA JA.. Denote Denote by by LO( LO(-<) the formula formula which which is saying that that -< -< is linear ordering. ordering. We linear We obtain obtain the the axiom axiom system system ID" IDv by by taking taking all all the the axioms axioms of of NT NT and and adding adding ((TI,,) T I,, ) (ID,,) 1 (10,,) LO(-<) F) LO(-<) 1\ A T/(-<, TI(-<,F) -<)) [(\lx) [A( J� , JJ~,Y,x, (Vy E6_field( field(-<))[(Vx)[A(JYA, y) --+ --+ xx E6_ J� J~]]]] (\lY �Y , x, y) and and (IDa) 22 (\ly (Vy6-field(-<))[(Vx)[A(B,J~Y,x,y)--+ B(x)] --+ --+ ((Vx)[x -+ B(x)]] B(x)]] E field( -<)) [(\lx) [A(B, J �Y , x, y) --+ B(x)] \lx) [x Ee JJ~. � --+ (10,,) where -formulas. Since where F F and and B(x) B(x) are are arbitrary arbitrary .e1D" s Since one one inductive inductive definition definition corresponds corresponds to to one one hyperjump hyperjump the the axiom axiom system system ID" ID~ may may also also be be interpreted interpreted as as the system for iterated hyperjumps. number classes classes 01' the system for v-fold v-fold iterated hyperjumps. The The constructive constructive number (.9, for for J.L in ID" and their their basic be proved # ::; _< v v can can be be defined defined in ID~ and basic properties properties can can also also be proved there. there. We We define define ID UIDe ~ where where every every �~ < < v v is is represented represented by by a a proper proper initial initial segment segment of of -< -<.. One One may may and the the systems ID" to the system also the axioms also combine combine the axioms ID} ID1 and systems ID~ to obtain obtain the system ID�. ID.<. axiomatizing nitions along axiomatizing the the iteration iteration of of inductive inductive defi definitions along the the accessible accessible part part of of aa linear order taking some arithmetically definable definable order linear order -< -<.. This This is is done done by by taking some arithmetically order relation relation -< choosing the -positive formula -< and and choosing the X X-positive formula A [y -< A.<(X,x) :r (Vy)[y -< x x -+ --+ Y yE e X] X].. (\ly) -« X, x) :{::} Set Theory Theory and and Second Second Order Number Number Theory Theory Set 271 271 accessible part Its fixed-point fixed-point ]A~ IA-< isis called called the the accessible part of of -~ -< and and usually usually denoted denoted by by Acc.~. Its Acc.., . The axioms axioms of of ID.~. ID-<. are are The (ACC) 1l (Acc) (V'x) [A .., (Acc.." x) ~-+ xx eE Acc.~] (Vx)[A.~(Acc.~,x) Acc .., ] 1 ( corresponding to to (ID1)1) (101) ) (corresponding (Ace) (Acc)22 B(x)] --+ (Vx)[A.~(B, B(x)] x) --+ (V'x)[A .., (B, x) (V'x) [x eE Acc..< -+ (Vx)[x -+ B(x)] Acc.., --+ -+ B(x)] (corresponding to to (ID1)2). (101) 2 ) . (corresponding (IDa.)1 (1 0 ..,. ) 1 (V'y y) J� ]] J� Y, x, Acc.., ) [(V'x) [A( J� , J~Y, (Vy eE Acc.~)[(Vx)[A(J~, x, y) --+ -+ xx eE J~]] modifying (IDv) (1011) 11 and and modifying B(x)] ~-+ (Vx)[x (ID.~.)2 B(x)]] (V'x) [x eE Acc.~ Acc .., )[(V'x) [A(B, J~Y,x, J �Y , x, y)--+ (10 ..,. ) 2 (Vy y) -+ B(x)] (V'y eE Acc.~)[(Vz)[A(S, Acc.., --+ -+ S(x)]] 2 these systems systems of iterated inductive inductive definitions definitions have have been been modifying All these of iterated modifying (IDv) (10,,) 2.. All introduced by Feferman. Feferman. introduced by There are even even stronger stronger iterations iterations of inductive defi definitions which, however, can be be There are of inductive nitions which, however, can more elegantly formulated within the framework framework of more elegantly formulated within the of Second Second Order Order Number Number Theory. Theory. 3.2.3. IIterated definitions in in second order 3.2.3. t e r a t e d inductive i n d u c t i v e definitions second order In Second Order Order Logic notion of In full full Second Logic we we do do not not have have aa recursively recursively enumerable enumerable notion of provability. fix aa calculus the proof proof strength provability. Therefore Therefore we we have have to to fix calculus and and regard regard the strength of of an an axiom system that calculus. This means means that that we axiom system relatively relatively to to that calculus. This we rather rather use use two two sorted sorted first order. To first order order logic logic than than full full second second order. To fix fix aa calculus calculus we we assume assume that that we we have have aa second introduced in Section 1.3. We second order order Tait Tait language language as as introduced in Section We extend extend the the calculus calculus of of Definition Definition 2.1.2.1 2.1.2.1 by by the the following following second second order order rules: rules: (32) (V2) IfIf � ~ �, A, A(X), A(X), then then � ~-- �, A, (3Y)A(Y) (3Y)A(Y) for for all all m m > ma mo A(X) and If/f � ~-~ �, A, A(X) and X Z not not free free in in any any of of the the formulas formulas in in �, A, ((VY)A(Y), then V'Y)A(Y) , then � �, A, ((VY)A(Y) for all all m m > ma. mo. V'Y)A(Y) for We We say say that that aa formula formula F F is is provable provable from from an an axiom axiom system system Ax A x iff iff there there are are finitely and fi n itely many sentences finitely many many instances instances of of identity identity axioms axioms G1, G 1 , . . . ,, G Gm and finitely many sentences m ,Gm , ...,-~A1,. {Ab An } �C_Ax F for A 1, . .. ..,, .~..,An G1 , . ...,. , ...~Gm, {A1,.. .. .. ,,An} A x such such that that � ~-- ..., ~G1,. A n ,, F for some some m. The The strongest strongest axiom axiom system system for for Number Number Theory Theory is is NT NT22 which which comprises comprises all all the the axioms axioms of of NT N T together together with with the the axiom axiom schemes schemes . • • (CA) (CA) m. (3X)[(V'x) (x Ee X (3X)[(Vx)(x X H ~ F(x))] F(x))] comprehension of of comprehension and and (AC) (AC) (V'x)F(x, YX) X)) -+ (V'x) ( V x )(3X)F(x, (SX)F(x,X -+ (3Y) (3Y)(Vx)F(x, Yx) of of choice. choice. We We put put yy Ee y yxx :<=> :r (y, (y, x) x / eE Y Y and and assume assume tacitly tacitly that that F(x) F(x) and and F(x, F(x, Y) Y) must must not not contain contain the the variable variable X X.. If If the the formulas formulas in in the the schemes schemes (CA) (CA) or or (AC) (AC) are are restricted restricted ttoo aa complexity complexity class class F jc we we talk talk about about (F-CA) (~'-CA) and and (F-AC) (3v-AC),, respectively. respectively. By By (F-CA) (.T-CA) we we denote denote the the W. Pohlers Pohlers w. 272 axiom systems systems which which comprises comprises all all the the axioms axioms of of NT N T extended extended to to the the second second order order axiom language together together with with the the scheme scheme (F-CA) (Jc-CA).. Analogously Analogously we we denote denote by by (:F-AC) (.T'-AC) language the axioms axioms of of NT N T together together with with (F-AC) (Jc-AC).. the We will will also also regard regard axiom axiom systems systems which which are are closed closed under under rules. rules. A A rule rule has has the the We form form F1 , . . . , Fn � F1,...,Fn ~ FF and we we say say that that aa theory theory Ax Ax is is closed closed under under the the rule rule (R) if if Ax Ax � ~- Fi Fi for for ii = = 1, 1 , .. .. .., , n and implies F F E E Ax. For For aa given given rule rule (R) we we define define implies (R) (R) (R) The least least .c s � -theory which which comprises comprises NT NT + -t-(HI-CA) and is is closed closed under under The ( II6 -CA) and the rule rule (R) (R).. the Observe Rn ) where Observe that that the the theory theory (R) (R) is is the the union union of of the the theories theories ((Rn) where (Ro) (R0) = - NT NT + + ) is obtained by closing (Rn) under all applications of the rule and (Rn (1~+1) is obtained by closing (P~) under all applications of the rule ((II~-CA) II6 -CA) and l + (R). (R). In NT NT22 we we may may replace replace the the scheme scheme of of Mathematical Mathematical Induction Induction by by aa single single axiom axiom In X )Q[ 0Ee X X A1\ ((V'Vy)y (y ) ( yEe X X --+ - + yy ++l e1 XE) X) -+ -+ ((Vx)(xeX)]. ((V'VX)[ V'x)(x E X)]. This is is no no longer longer true true if if we we regard regard axiom axiom systems systems with with restricted restricted comprehension comprehension This scheme. Therefore Therefore we we introduce introduce also also the the axiom axiom systems systems (:F(bw- CA)o CA)o and and (:F(.%'- AC)o AC)o scheme. in which which the the scheme scheme of of Mathematical Mathematical Induction Induction is is replaced replaced by by the the single single axiom. axiom. in The formula formula The Wf(-~) (Vx)(x X)] x) (x Ee X)] ( -<) ::r{o} ((VX)[Prog(-<,X) V'X)[Prog( -<, X) -+ (V' Wf expresses the the well well-foundedness of the the relation relation -~. We We have have expresses -foundedness of -<. -< x) (y tf.r X)]. X)]. Wf(-~) r (VX)[(3x)(x X)(Vy-~ x)(y (V'X)[(3x) (x eE X) X) --+ (3x eE X)(V'y -+ (3x Wf ( -<) {o} (113) (113) -<) In NT2 Wf(-~) the scheme TI(-~,F) s � formulas ( -<, F) for In NT2 the the sentence sentence Wf ( entails entails the scheme T/ for arbitrary arbitrary .c formulas F(x). is not not longer for restricted comprehension. Therefore Therefore we we This, too, too, is longer true true for restricted comprehension. F(x) . This, introduce the introduce the scheme scheme (BI) ( BI ) Wf(-~)-~ T/( -<, F) Wf ( -<) -+ TI(-<,F) Bar Induction y). -< of of Bar Induction for arbitrary s.c� -formulas F(x). for definable definable relations relations -~ and and arbitrary F(x) . AA relation -~ is such that that xx -< -~ y 6+ relation -< is definable definable iff iff there there is is an an s.c� -formula G(x, G(x, y) y) such f+ G(x, formula G(x, may contain parameters. We We will sometimes G(x, y). The The formula G(x, y) may contain additional additional parameters. will sometimes emphasize emphasize this this by by writing writing y y) y y, x, - Roughly speaking speaking Bar Bar Induction Induction says says that that aa relation relation which which is well-founded with with Roughly is well-founded for classes. classes. If If the the defining defining formulas formulas for for the the respect respect to to sets sets is is also also well-founded well-founded for relation complexity class relation -~ -< in in the the scheme scheme (BI) ( BI ) are are restricted restricted to to the the complexity class ~" F then then we we talk talk about (~'-BI). (F-BI) . If If also also the the complexity of the the allowed classes is is restricted restricted to to another another about complexity of allowed classes complexity class class Jc2 F2 we we notate notate that that as (F-BI) fF2 . complexity as (.T'-BI)r.T'2. If X isis aa set set parameter parameter we we may may define define the the binary binary relation relation If X Set Theory Theory and Second Second Order Order Number Number Theory Theory 273 273 -< x xx . < x yy :{:} y) EE XX.. :4=~ (x, (x,y) Sometimes Sometimes Bar Bar Induction Induction is is formulated formulated as as the the single single axiom axiom (Bi) (VX) [ Wf (-< ) ---+ TI(.<x,F)] (VX)[Wf(.<x)-+ x TI( -<x , F) J which which in in the the presence presence of of (l1b-CA) (I]~-CA) has has the the strength strength of of (l1b-BI) (n~-BI).. Weaker Weaker than than (BI) (BI) is is TI( -< , F) which M. Rathjen which is is known known as as Bar Bar Rule. M. Rathjen has has shown shown in in Rathjen Rathjen [1991J [1991] that that the the (B R) Wf ( -<) f-- Bar Bar Rule Rule is is of of the the same same strength strength as as parameter parameter free free Bar Bar Induction, Induction, i.e., i.e., the the axiom axiom of Induction in the defining formula for of Bar Bar Induction in which which the defining formula for the the relation relation -< .< must must not not contain contain parameters parameters (not (not even even individual individual parameters). parameters). As As aa basis basis for for nearly nearly all all our our second second order order axiom axiom systems systems we we will will use use (rr�-CA) (H~-CA), , i.e., i.e., the the scheme scheme for for arithmetical arithmetical comprehension. comprehension. This This system system is is also also known known as as (ACA). (ACA). We use use both both notions notions interchangeable. interchangeable. Observe Observe that that (ACA) (ACA) proves proves that that aa relation relation -< .< We is infinite -<-descending sequence, i.e., is well-founded well-founded iff iff it it does does not not contain contain an an infinite -<-descending sequence, i.e., [ (3X)[(3x)(x Ee X) ( -< rX) (3y Ee X)(y ( -<) {:} Wf(-<) 4=~ ..., -,[(3X)[(3x)(x X) /\ A LO tO(-< rX) /\ A (Vx (Vx E E X) X)(3y X)(y -< -< x)]] x)]].. (114) (114) Wf The (Bi), (I1A-BI) The axiom axiom system system (ACA) (ACA) proves proves also also the the equivalence equivalence of of the the schemes schemes (Bi), (IIoLBI) and and the the following following quantifier quantifier scheme scheme (cf. (cf. Feferman Feferman [1970]). [1970]). (QS) (QS) F for arithmetical and arbitrary Because Because of of this this equivalence equivalence it it has has become become common common to to call call (QS) (QS) also also (I1A-BI) (II~-BI).. (VX)A(X) (VX)A(X) ---+ --4 A( A(F)) .for arithmetical A(X) A(X) and arbitrary F(x) F(x).. The iteration of definitions is The iteration of inductive inductive definitions is quite quite elegantly elegantly expressed expressed in in aa second second order order language. -positive formula language. For For an an X X-positive formula A(X, A(X, Y, Y, x, x, y) y) we we introduce introduce the the abbreviation abbreviation x, yy)) ---+ CIA(X, Y, y) y) :~::~ (Vx)[A(X, Y, x, -+ xx E9 XJ X] :{:} (Vx) [A(X, Y, A (X, Y, CI and and define define - ((IT,,) lTv ) (3X) (3X)ITA(.<,X) I1A (-<, X) and and - W. Pohlers Pohlers W. 274 BID~ := (ACA) (ACA)++ WO WO(-K) (IT.) + (B-ITv) (B-IT.) (-<) + (lT v) + BID� where WO WO(-K) stands for for LLO(-K) and A A varies varies over over all all X X-positive formulas where ( -<) stands -positive formulas O ( -<) /\AWf(--<) Wf( -<) and which contain contain at at most most the the shown shown variables variables free. free. Denote Denote by by A(X, Y, x, y) which (ITA) (ITA) (Vx)[WO(-. F(u, v,x) the scheme scheme in in which which F(u, v,x) and and A(X, Y, x, y) are are supposed supposed to to vary vary over over arith ariththe metical formulas formulas without without further further parameters. parameters. We We define define metical ID 2. := (ACA) ( A C A ) ++ (ITA). (ITA). ID2* Finally put put Finally A(B, X -< F,.Y , y) (B-ITA) ('v'x) (Vx)(Vy)(VX)[WO(-KF,=) A I7A ITA(--KF,=,X) A C1 CIA(B,X' F(u, v, -~ --+ B(x))] ('v'(Vx)(x x)(x Ee XX Yu --+--+B(x))] where F(u, v,x) and A(X, A(X, Y, Y,x,y, are as as above above and and B(x) is is an an arbitrary arbitrary .c� s where x) and x, y, ) are formula. The The schemes schemes (B-IT (B-IT,) and (B-ITA) (B-ITA) have have the the flavor flavor of of Bar Bar Induction; Induction; therefore therefore formula. v ) and the B B in in their their identifiers. identifiers. Let Let the BID 2. := := (ACA) (ACA)+ (ITA) + + (B-ITA). (B-ITA). + (ITA) BID2* One should observe that that BID2* BID 2. allows allows only only iterations iterations along along arithmetically definable One should observe arithmetically definable K f well-orderings, i.e., along well-orderings of length < w~ K. So it may appear weaker So it may appear weaker well-orderings, i.e., along well-orderings of length < W . than ID.~. which allows iterations iterations along accessible parts parts of of arithmetically arithmetically definable definable than ID-<* which allows along accessible orderings, i.e., i.e., along along well-orderings well-orderings of of length length :::; < W w~ However, we we will will sketch sketch in in aa fKK.. However, orderings, moment that that BID2* B I D s* comprises comprises ID-<* ID.~... moment If we we drop drop the the restriction to arithmetically arithmetically definable definable well-orderings well-orderings we we obtain obtain the the If restriction to schemes schemes (IT) (IT) and and (B-IT) (B-IT) (vx) wo(-< -+ (3Y) 11A ( -<x, r)] ( -<x)) --+ Y)] X)[[WO ('v' -< FV , y) y) (Vy)(VY)[WO(- ('v'x)(x F(u, v), and and S(x) B(x) F(u, v), may arbitrary .c�-formulas and respectively, where where respectively, may now now be be arbitrary s and Y, x, y) is is an formula. All G(X, Y,x,y) an X-positive X-positive arithmetical arithmetical formula. All formulas formulas may may contain contain G(X, additional parameters. We put additional parameters. We put A u t - I D := A C A ) ++ (IT) Aut-ID (IT) := ((ACA) and and (IT) ++ (B-IT). (B-IT) . (Aut-BID) A C A ) ++ (IT) := ((ACA) (Aut-BID) := One One easily easily shows shows --+ (Yx eE field(-<))(Y Y) A (YI~-CA) (II� -CA ) ~ WO(-<) (115) 11A ( -<, Y) I1A( -<, Z) WO( -< ) A field( -<) ) ( y=x == Z=). Z ) --+ /\ ITA(-<, ZX) . (115) /\ ITA(-<, f- ('Ix F F There an canonical the language There is is an canonical embedding embedding F ~t-+ F** from from the language of of IDv IDv into into the the language language of We replace of IDZu. ID; . We replace every every occurrence occurrence of of ss eE J~t J� by by (3X)[ITA(-~,X) (3X) [11A ( -<, X) A Xt] . The The /\ sS eE Xt]. Theory and Second Second Order OrderNumber Number Theory Theory Set Theory 275 275 theory ID� ID~ shows shows that that there there is is aa set set WA WA such such that that 11A( ITA(-<, WA) and and by by (115) (115) we we theory -<, WA) obtain that that W1 W~ is is uniquely uniquely defi defined for tt -< -< IIv.. Hence Hence (s (s EE J�)* J~)* iff iff ss EE W W tt for for all all obtain ned for t* = ...:t . Since field(-<) which entails entails J� J~** = = W W tt and and J1 j~t* = W W.~t. Since we we have have ClA( CIA(W~, W~ t, t) tt EE fie W1 , Wi, l d(-<) which we get get (1011) (IDa) 11 ** and and from from B-ITII B-IT~ also also (1011)2 (IDa) 2.. So So we we have have shown shown we ID~ � C_ BID� BID~.. ID" (116) (116) An embedding embedding F F f-+ ~ F F** of of the the language language of of 10...: IDa,* into into the the second second order order language language is is An obtained similarly. similarly. Define Define obtained � y) [ ClA (X X , X "':x , x) IU(-~,x) : ~ (V'X (v=__ y)[clA(x',x~',=) IT l (-<, X) :{::} ))] 1\ ^ (V'Y)( (vY)(clA(y,x~',=) ~ (V' (Vz)(z x x9 -+ -+ zz E e Y Y))] z) (z Ee X ClA(Y, X "':X , x) -+ and let let X X � c_ Y Y stand stand for for (V'x)[x (Vx)[x E EX X -+ ~ x x E e Y] Y].. Define Define and Acc(-<,X) Prog(-<,X) (VZ)[Prog(-<,Z) X � C_Z Z]] -<, X) 1\A (V' ( -<, X) ::r{::} Prog( Acc Z) [ Prog( -<, Z ) -+ X and replace replace all all occurrences of ss E E Acc Acc~...: by by (3X) (3X)[Acc(-<,X) X] and and finally finally [Acc ( -<, X) 1\A ss EE X] and occurrences of replace all all occurrences occurrences of of s E E J� J~4 by by (3X)[ (3X)[IT~(-<,X)A Xt].. We We indicate indicate replace ITl ( -<, X) 1\ ss EE Xt] that the the translations translations (ACC) (Acc) 11., (Acc) 2.,, (10...:* (lb.<,)) 11.0 and and (10...: (ID.~,) 2. are are all all provable provable in in that o , (AccF* * F* 2 2 B I D 2.. We argue argue informally informally in in BID B I D 2.. Let x x -<8 _<8 y :{::} :r -< y � -< s. Put * . We xx -< s. Put * . Let BID Ao(X, Y, Y, x, x, y) y) :{::} :r (Vz)[zz -< -< x x -+ Zz E E X] X].. Then Then there there is is a a set set T T such such that that Ao(X, (V'z)[ T) if if 00 denotes denotes the the least least element element in in -< -<.. We We then then have have Prog(-<, Prog(-<,TO) T ~ and and 1Ao (-< o~, T) lITAo(-< define S S := "- TO T ~. Moreover Moreover we we have have Prog(-<, Prog(-<,X) --+ S � c_ X X and and thus thus Acc(-<, Acc(-<,S). So define S) . So X) -+ 1 by Prog(-<, Acc:<= and we we get get (ACC) (Acc) 1.o by Prog(-<,S) and (ACC) (Acc) 22.* from from (B-ITA) (B-ITA).. To To prove prove S) and = SS and Acc� (10...: * ) 11.0 we assume ss EE Acc� E S. -<8) (lb.<,) we assume Acc:<,, i.e., i.e., ss E S. Then Then we we obtain obtain Wf( Wf(-< 8) because because otherwise otherwise according to to (1 (114) there would would be be aa non non empty empty set set V Y � c_ field field(-< 8) containing containing ss and and ( -<8) according 14) there an infi infinite -<-descending sequence. sequence. Since -< is is a a linear linear order order this this entails entails that that if if all all an nite -<-descending Since -< -<-predecessors an element element x x do do not to V V then then xx cannot That -<-predecessors of of an not belong belong to cannot belong belong to to V. V. That means Prog(-<, denotes the the complement complement of of V. S c_ Prog( -<, ~V) -,v) where where ~V -,V denotes V. But But then then S means -,V � ~V which contradicts ss E choose any any X-positive X-positive formula A(X, Y, x, y) y).. By By which contradicts E V. V. Now Now choose formula A(X, Y, x, uniqueness property property (115) (115) ITA there there exists set W W such such that that 11A(-<8, ITA(-<8, W) W).. By By the the uniqueness exists aa set ITA we obtain (Vy)[y x eE WV]. = W8 W 8 and and WY] . Hence Hence (J~)* (JA)* = we obtain (V'y) [y --< � ss --+ (V'x) (x EE J~4)* J � )* ++ -+ (Vx)(x f-+ X 2* from S )* = 1 0 from and we (10...: * ) 1. (j]8), = W"': W.<8s and we get get (lb.<,) and (10...:* (lb.<,)) 2. (J1 from ITA(- As remarked before schemes (B-IT~), (B-ITII) , (B-ITA) and flavor of Bar As remarked before the the schemes and (B-IT) have have the the flavor Bar Induction. We are are going going to to substantiate substantiate this this remark. remark. First First we show Induction. We we show 2 * , AAut-BID} A x E I D 2., n t - B I D } => x ~Ax E {BID~, {BID� , BBID Ax � Bi. :::} A (118) (118) Wf(-<x) and and Prog(-<x,B) Prog(-<x, B) for for some some formula formula B(x). B(x) . We Assume Wf(-<x) to show show Assume We have have to (V'x)B(x) . For For A(Z, A( Z, Y,x, Y, x, y) y) :r :{::} (Vz-<xx)(z (V'z -< x x) ( z E we obtain obtain by by ITs, ITII, ITA ITA or or IT, IT, (Vx)B(x). E Z) Z) we respectively, set T we get respectively, aa set T such such that that ITA 11A (-<~, ( -<�, T). T) . Since Since _ But the the opposite opposite isis also also true. true. We prove But We prove 276 276 w. Pohlers Pohlers W. + (II~-BI) ( IJt- B I ) I--(B-IT,.,) ID; + ID~ � (B-IT.. ) 2 (B-I TA) ID2*+ ( IJt-B I ) I-ID • + (n~-Bl) � (B-ITA) Aut-ID ++ (II~-BI) (lIA-BI) ~� (B-IT). (B-IT) . Aut-ID (1 19) (119) Assume Wf(-.<), We have have to to show show (VxEXY)B(x). (Vx E XY )B(x) . 11A( -<, X) and and CIA(B,X' (QS) So we we have have the the following following theorem theorem So and where means that both theories prove the same theorems. 3.2.3.2. h e o r e m . BID~ u t - B I D == 3.2.3.2. TTheorem. BID; == ID~ ID; ++ (Bi), (Bi) , BID2* BID2 • == ID2*+ ID2 • + (Bi) (Bi) and A Aut-BID A u t - I D ++ (Bi), Aut-ID (Bi), where = means that both theories prove the same theorems. 3.2.4. InI t{-comprehension 3.2.4. - c o m p r e h e n s i o n aand n d bbeyond eyond We mentioned already that the the fixed-point fixed-point of monotonic operator operator Fr can be obob We mentioned already that of an an monotonic can be r -closed sets. sets. For an operator operator which arithmetically tained as as the of all all F-closed tained the intersection intersection of For an which is is arithmetically defined A(X, x) x) this this means means that that we defined by by aa formula formula A(X, we have have IA x l (VX) [(Vy) (A(X, y) X) --t (120) Xl } IA = : {{=1 (VX)[(Vy)(A(X, y) --t -~ yy E e X) --+ x x E e X]} (120) = = {x {~1l (VX)[C/A ( V X ) [ C t A (X) ( X ) --t -+ x9 E e X X]} l} Vice Vice versa, versa, every every lI II{-set can be be shown shown to to be be an an inductive inductive set, set, Le., i.e., aa set set which which is is t -set can definitions are primitive xed-point. So primitive recursive recursive in in some some fi fixed-point. So ((II{-CA) and inductive inductive definitions are n t-CA) and canonically introduce the notations canonically connected. connected. To To iterate iterate ((II{-CA) we introduce the following following notations nt-CA) we 3.2.4.1. Let 3.2.4.1. Definition. Definition. Let H(X, H(X, x, x, y) y) be be an an C s � -formula which which contains contains only only the the shown shown parameters. parameters. We We define define - jump hierarchy and and call call X X the the jump hierarchy based based on on H(X, H(X, x, x, y) y) along along -< -<.. For For aa primitive primitive recursive recursive well-ordering well-ordering of of order order type type vu we we introduce introduce the the scheme scheme ((H{-CA~) nt-CA.. ) (3 (3Z)&(-<, Z)) for .for H(X, H(X, x, x, y) y) Ee nt II{ Z )JH ( -<, Z of of v-fold u-fold iterated iterated n II{-comprehension. We sometimes sometimes express express this this sloppily sloppily by by t -comprehension. We ) ) if we do not want to emphasize the order-relation but its order type. (v, (~Z)JH(I}, Z) if we do not want to emphasize the order-relation but its order type. (3 Z JH Z Transfinitely Transfinitely iterated iterated n H{-comprehensions are axiomatized axiomatized by by t -comprehensions are (A (Aut-II{) ut-lID )] for (VX)[ 3Z)JH (-<x, Z (VX)[Wf(-Kx) --+ ((3Z)JH(-Kx, Z)] for H(X, H ( X , xx,, yy) ) EE nt H{.. Wf (-<x) --t For For aa primitive primitive recursive recursive ordering ordering -< -< of of order order type type vu we we define define (-<) + ((II{-CA~), (rr�-C (II{-CA,,) "= ((A AC CA A )) ++ WO WO(-<)+ nt -CA.. ) , A.. ) := (rr�-C Ae) (II{-CA<~) "= U U (rr�-C (II{-CA~) A< .. ) := { <.. Set Theory Theory and Second Second Order Number Theory 277 277 and and (Aut-H~) .= A C A ) ++ (Aut-ITD (Aut-I1]). . (Aut-II�) := ((ACA) ., ) have Notice that that (ITt-CA) (Ill-CA) and (ITt-CA (H]-CAv) have different different meanings. Due to the possible presence of set parameters in ITt-comprehension l-/~-comprehension formulas we have (IIx~-CA) = = (II�-CA<w) (II~X-CA<~) (II�-CA) though though notationally notationally this this looks looks strange. strange. Recall Recall that that (II�(H~- CA)o CA)o means means (ITt-CA) (II~-CA) + + Mathematical Induction. (ACA)o,, i.e. i.e.,, (ITt-CA) (1-I}-CA) together together with with the the axiom axiom of of Mathematical Induction. The The (ACA)o theories theories (II�-CAv)o (II~-CA~)o are are defined defined analogously. analogously. We will see We will see that that the the theories theories (II�-CAv) (H~-CA~) and and ID� ID~ are are equivalent. equivalent. In In aa first first step step we we show show (ID�)o (ID~)o � C_ (II�-CAv)o (II~-CA~)o and and (Aut-ID)o (Aut-ID)o � C_ (Aut-II�)o. (Aut-II~)o. (121) (121) -<) . Let Let Let -< -~ be be aa well-ordering well-ordering of of order order type type vu or or assume assume Wf( I/Vf(-~). Let A(X, A(X, Y, Y, x, x, y) y) be be an an X-positive arithmetical formula formula and and put X -positive arithmetical put H(X, H ( X , xx,, yy) ) :r (VZ)[C]A(Z,X,y) --~ xx Ee Zl Z].. :{:} ('v' Z )[ ClA ( Z, X, y) -+ (i) (i) Then Then H(X, g ( X , xx, , y) y) E e ITt II} and and by by (ITt-CA). (II}-CA)v, or or ((Aut-II}) there is is aa set set S S such such that that Aut-ITt) there )H -<, S) Hence JH ((-~, S).. Hence - (ii) (ii) which which in in turn turn implies implies ('v'X)[ (VX)[CIA (X, S S -<" (iii) (iii) From From (iii) (iii) we we obtain obtain A(S (X, S- (iv) (iv) (v) (v) D D Pulling -<, S) Pulling (iii) (iii) and and (v) (v) together together we we obtain obtain 1T.4( ITA(-~, S).. To inclusion in To prove prove also also the the other other inclusion in (121) (121) we we use use the the fact fact that that already already (ACA)o (ACA)o proves proves that that every every ITt-formula II~-formula H(Y, H(Y, y, y, Z) 53 is is equivalent equivalent to to the the well-foundedness well-foundedness of of its its associated i.e., that associated tree tree of of unsecured unsecured sequences, sequences, i.e., that there there is is an an arithmetical arithmetical formula formula TH (Y, (]I, x, X, y, y, Z) ~ such such that that TH a well-founded tree. x l TH(Y, x, y, Z) } is (ACA)o � (ACA)o ~ H(Y, H ( Y , yy,, ~Z) +-+ ++ {{x] TH(Y,x,y,~} is a well-foundedtree. (x) O' (y) [TH(Y, (x) Defining A(X, Y, Y, x, 5) :r:{:} ('v'y) (Vy)[TH(Y, (x)ff" (y),, (xh (X)l,, Z) z-) -+ -4 (((x)ff" (y),, (xh) (x)l) E e X X]l we we x, Z) Defining A(X, O' (y) have an an X X-positive formula such such that that have -positive arithmetical arithmetical formula 278 278 W. Pohlers (122) H ( Y, y, H(Y, (122) y, 5) Z) <-~ ++ {x[ {x I TH(Y, TH(Y, x, x, y, y, z-) Z) }} is is well-founded well-founded ++ <<>, y) eE U(~;~ I A(Y,i) ++ (0, y> Y, z-) Z) -+ y) eE z]. +~ ++ (vz)[c&(z, (VZ) [ C1A ( Z, Y, -+ (<>, (O , y> Z] . (ACA)o. The last last equivalence equivalence is is provable provable iinn (ACA)o. The To obtain be aa II}-formula m -formula obtain therefrom therefrom the opposite inclusion inclusion in in (121) (121) let H (Y, x, y) To the opposite let H(Y, y ) be and the arithmetical formula such that according to (122) ) the arithmetical formula such that according to (122) and A(X, A (X, Y, x, y) y H (Y, x, y ) ++ ++ (VZ)[CIA(Z,Y,y)-+ (VZ) [ C1A ( Z, Y, y ) -+ ((>,x> H(r,x,y) (0, x) 6E Z]. Z] . ( i) (i) A (X, {z Define A'(X,Y,x,y) A' (X, Y, x, y ) :~=> Y } , x, y) andlet Define {z Il ((),z) :{:} A(X, and let -<9 eitherbyaprimitive either by a primitive (O , z) 6E Y},x,y) recursive well-ordering well-ordering of of order order type type u/J or or assume assume WO(-<). recursive Then either either ID~ ID� or or WO( -< ) . Then Aut-ID prove the the existence existence of of aa set set SS such such that that ITA,(-<, TA, ( -<, S) , i.e., A u t - I D prove S), i.e., I CIA(Su, {z I ((>,z> 6 S'~U},y) (ii) (ii) { z l <<>,z> (VZ) [ C1A ( Z, {~1 S- ((iii) iii) - , x) 6E S -+ (VZ)[CIA(Z, (VZ ) [ C1A ( Z, {z { z Il ((),z> ((),x) S yY --+ S',z> S Y } , x, y) -+ H({z (O , z) 6E S' (iv) and and Hence Hence (iv) and and -,z> ( { z lI ((>,z> -+ ((),x> -+ (VZ)[CIA(Z, H({z S~Y},y) Z] (v) (O , z) 6 (O , x) 6 (O , z) 6 (v ) , x~>) eE s,. SY . -+ <<>, -+ (O So So we we have have - and putting T := := {{(x,y>l ((>,x) S ~}} we we obtain obtain JH JH(-'<,T). (-<,T). and putting T , x) E6 SY (x, y ) 1 (O (=~Z)JH(~, Z)) is is aa theorem theorem of of ID ID~� or or Aut-ID, A u t - I D , respectively. respectively. (3 Z )JH( -<, Z ((vi) vi ) Therefore Therefore o [3 So So we we have have together together with with (121) (121) and and Theorem Theorem 3.2.3.2 3.2.3.2 the the following following theorem. theorem. (II� (YI~-CAv)o = (ID�)o (IDa)o,, (Aut-II�)o (Aut-YI~)o = = (Aut-ID)o (Aut-ID)o, , -CA,, )o = (II�-CA (II~-CA~) = ID IDa, (YI~-CA~)+(Bi)= ID~� + + ((B B ii )) == BID� BID~,, Aut-ID Aut-ID = = (Aut-lID (Aut-YI~) � , (II�-CA ,, ) = ,, ) + (Bi ) = ID and and Aut-BID Aut-BID = = Aut-ID Aut-ID + + (Bi) (Bi) = - (Aut-II�) (Aut-n~) ++ (Bi) (Bi).. 3.2.4.2. 3.2.4.2. Theorem. Theorem. Regarding Regarding (116) (116) we we obtain obtain the the following following chain chain ID~ � C_ BID� BID~ = = (II�-CA (YI]-CA~)+ (II01-BI). ID" ,, ) + (II5-BI). (123) (123) Feferman Feferman [1970] [1970] has has shown shown that that for for /Ju = = wP wp with with pp > 0 the the theory theory ID" ID~ proves proves the the (II�-BI) . This existence existence of of an an w-model w-model for for (II�-CA<,,) (II~-CA 279 279 Set Theory Theory and Second Second Order Number Theory w-model II. The w-model for for all all ID( IDa,, �~ < < v. The scheme scheme (IIA-BI) (II~-BI) is is not not needed needed here here since since the the translation L: I D� -formula F is arithmetical in -<, X). The translation of of every every/:lD~-formula is arithmetical in some some X y with with JH( JH(-4, The obtained by IIij -CA ) . Writing proof theoretical translation of translation of B-IT B-IT~ is thus thus obtained by ((II~-CA). Writing � _ for for proof theoretical v is reducibility wP, Pp EE Lira Lim the reducibility we we get get for for IIt, = - coP, the following following chain. chain. XY F X). ID<~v � C_ BID B I D ~� v = (II~-CA<~)+(II~-BI) _ ID< ID<~v � _< (II�-CA This theories are This shows shows that that all all these these theories are proof proof theoretically theoretically equivalent. equivalent. To H. Friedman To close close the the section section we we mention mention the the results results of of H. Friedman [1970] who who showed showed [1970] (125) (125) (A~-CA) = (II~-CA<~o) = (]E21-AC) where where (\fx)[A(x) (Vx)[A(x) ++ B(x)] B(x)] -+ --+ (3X)(\fx)[x (3X)(Vx)[x E e X X ~ A(x)] A(x)] for A(x) A(x) E e m I11 and and B(x) B(x) E E� P~21 is the the scheme scheme of of � A21-comprehension. for � is � -comprehension. f-+ f-+ �� -CA) ((A~-CA) [1970] A simpler simpler argument argument than than that that in in H. H. Friedman's Friedman's results results is is given given in in Feferman Feferman [1970] A to � -comprehension rule. to characterize characterize the the � Al-comprehension rule. To To define define the the rule rule let let the the class class of of 'essen 'essentially' m tially' I1~ formulas formulas be be the the smallest smallest class class of of formulas formulas which which contains contains the the arithmetical arithmetical formulas boolean operations formulas and and is is closed closed under under the the positive positive boolean operations 1\ A and and V V,, first first order order quantification quantification and and second second order order \f-quantification. V-quantification. Dually Dually the the class class of of essentially essentially � -formulas is logically equivalent � P~-formulas is the the class class of of formulas formulas whose whose negation negation is is logically equivalent to to an an essentially -formula. Analogously -formulas essentially m II~-formula. Analogously we we obtain obtain the the class class of of essentially essentially m II~-formulas � -formulas instead when when we we start start with with the the essentially essentially � P~-formulas instead of of arithmetical arithmetical formulas formulas and and � -formulas as class. The comprehension rule the class class of of essentially essentially � ~l-formulas as its its dual dual class. The � A �1 comprehension rule is is the defined defined as as follows. follows. ((A2LCR) ��-CR) (\fx)[A(x) (Vx)[A(x) ++ B(x)] B(x)] � ~- (3X)(\fx)[x (3X)(Vx)[x E X X ++ A(x)] A(x)] E e Ax A x /for o r A(x) A(x) essen essen�. tially tially m II~ and and B(x) B(x) essentially essentially � E~. f-+ f-+ Feferman Feferman shows shows (A� -CR) ==- (II�-CA<ww) (A~-CR) (YI~-CA<w~).. (126) (126) 3.3. Axiom 3.3. A x i o m systems s y s t e m s for for set set theory theory 3.3.1. 3.3.1. The T h e axiom a x i o m system s y s t e m KPw KPw 1.2. We We introduced introduced the the axiom axiom system system KPw K P w already already in in Section Section 1.2. For For ordinal ordinal analysis analysis it restate the KPw it will will be be more more convenient convenient to to restate the axioms axioms of of K P w in in aa more more parsimonious parsimonious Ext ) and pairing axiom Pair) to way. way. We We keep keep ((Ext) and modify modify the the pairing axiom ((Pair) to ((Pair') Pair' ) Y (\fx)(\fy)(3z)[x Ee zz ^1\ y Ee z]. z]. Pair' ) by similar way It Pair) follows It is is obvious obvious that that ((Pair) follows from from ((Pair') by �o-separation. A0-separation. In In aa similar way we we modify union to modify the the axiom axiom of of union to Union' ) ((Union') (\fu) (3w) (\fy Ee u)(\fz (Vu)(Bw)(Vy u)(Vz E y)[z y)[z E Ew w]] 280 280 W. Pohlers an axiom axiom which which requires requires only only the the existence existence of of aa superset superset of of the the union. union. Again Again it it is is an clear that that we we obtain obtain ((Union) form ((Union') by b.o-separation. A0-separation. Similarly Similarly we we modify modify Union) form Union' ) by clear the axiom axiom of of infinity infinity to to the (3u)[u "# (Vx u)(3v u)(x v)J. ((Inf') Inf' ) (3u)[u r 0 A (Vx Ee u)(3v Ee u)(x Ee v)]. For convenience convenience we we introduce introduce an an axiom axiom system system BST B S T for for Basic Basic Set Set Theory. Theory. It It For comprises the the axioms axioms ((Ext), and the the scheme scheme comprises Ext) , ((Pair'), Pair' ) , ((Union'), Union' ) , ((A0-Separation), b.o-Separation) , and of foundation. foundation. Adding Adding ((Inf') to B S T we we obtain obtain the the system system BSTw. B S T w . All All ((FOUND) FOUND) of Inf' ) to BST systems we we will will regard regard here here are are based based on on BST. BST. If If Ax A x is is such such aa system system we we denote denote by by systems A x r the the system system which which is is obtained obtained by by restricting restricting the the foundations foundations scheme Axr scheme ((FOUND) FOUND ) to b.o-formulas. A0-formulas. The The system system W-Ax W - A x is is the the intermediate intermediate system system between between Axr A x ~ and and to A x in in which which we we have have ((A0-FOUND) but allow allow full full Mathematical Mathematical Induction, Induction, i.e., i.e., the the Ax b.o -FOUND ) but scheme F(0) A (Vx E e w)(F(x)--+ --~ (Vx E e w)F(x) for for arbitrary arbitrary formulas formulas --t F(x + 1)) --t scheme F. F. F(O) (Vx w)(F(x) F(x 1)) (Vx w)F(x) Adding the the scheme scheme ((A0-Collection) to the the axioms axioms in in BST B S T we we obtain obtain the the axiom axiom Adding b.o-Collection) to system K P , adding it to to BSTw B S T w the the system system KPw K P w These These systems systems are are thoroughly thoroughly system KP, adding it studied in in Barwise Barwise [1975]. We We list list some some of of the the most most important important properties properties of of the the studied system K P w without without giving giving proofs. proofs. All All proofs proofs can can be be found found in in Barwise Barwise [1975]. KPw system Many of of these these properties properties are are already already provable provable from from axioms axioms which which are are weaker weaker than than Many K P w ~.. However, However, we we do do not not have enough space space to to go go into into more more details details here. here. KPwr have enough We We use use class-terms class-terms of of the the form form {x I A(x)} freely freely though though they they are are not not regarded regarded as as terms terms of of the the language. language. The The formula formula z E e (x I A(x) } is is an an 'abbreviation' 'abbreviation' for for A(z). [1975J. [1975J. {xl A(x)} A(z). z {xl A(x)} 3.3.1.1. �-Persistency E-Persistency Lemma. Lemma. 3.3.1.1. Let Let F F be be aa �-formula. P~-formula. Then Then Kpr K P r proves proves F a AAaaC � b -b+--t F Fbb and andFFaa - + FF. . Fa 3.3.1.2. E-Reflection Theorem. 3.3.1.2. �-Reflection Theorem. For have For every every �-formula E-formula F F we we have K P ~ ~f- F Kpr F -~--t (3a)F (3a)Fa.a. --t As P~-Reflection we obtain that in K P ~ every every P~ formula is is provably As aa consequence consequence of of �-Refiection we obtain that in Kpr � formula provably equivalent to to a a E1 formula. equivalent �l formula. For every every E-formula For �-formula F(x, F(x, y)y) we we have have K P r f~- (Vx (Vx E e a)(3y)F(x, a)(3y eE z)F(x, y). Kpr a)(3y)F(x, y)y) --+ --t (3z)(Vx (3z)(Vx eE a)(3y z)F(x, y). 3.3.1.4. E-Replacement every P~-formula we have 3.3.1.4. �-Replacement Theorem. Theorem. For For every �-formula A(Z, A(x, y)y) we have (3f)[Fun(f) AA dora(f) dom(f) == uu AA (V~ (Vx eE u)A(~, u)A(x, f(~))]. f(x))J. K P rr ~f- (V~ Kp u)(3!y)A(x, y)y) ~--t (3f)[Fun(f) (Vx eE u)(3!y)A(~, 3.3.1.5. A b.-Separation Theorem. Let A(x) A(x) bebe aa IIII and and B(x) B(x) bebe aa E� formula. formula. 3.3.1.5. -Separation T heorem. Let Then Then K KPprr ~f- (Va)[(Vx (Va)[(Vx eE a)[A(x) a)[A(x) ~ B(x)] B(x)J --+ --t (3z)(Vx (3z)(Vx eE a)[x a)[x eE zz ~ xx eE aa A B(x)]]. B(x)JJ. There are are many many basic basic relations relations which which are are b.o-definable (cf. Barwise Barwise [1975] [1975J for for details). details) . There A0-definable (cf. is an an ordinal, ordinal, e.g., e.g., can can be be expressed expressed by by Tran(a) Tran(o:) AA (Vx (Vx eE a)Tran(x). ) Tran(x) . The fact that a is The fact that 3.3.1.3. 3.3.1.3. P~-Collection Theorem. �-Collection Theorem. +-+ 0: +-+ 0: Set Theory Theory and and Second Second Order Order Number Number Theory Theory Set 281 281 Similarly (a) 1\A (3x ) [x E aJ ) [x E Similarly we we can can express express by by On On(a) (3x E a a)[x a] 1\ A ((Vx a ) ((3y 3y E a a)[x E y] yJ Vx E a) that usual basic basic notations (r is that a a is is aa limit limit ordinal. ordinal. The The usual notations as as Rel(r) Rel(r) (r is aa relation) relation),, Fun(J) Fun(f) (J (f is is aa function function )) are are �o A0 definable. definable. See See Barwise Barwise [1975,pp.14-29J [1975,pp.14-29] for for aa more more complete complete list. list. If xn) is If F F ((Xb X l , . .. .. . ,,xn) is aa �-formula A-formula of of KPw K P w then then we we may may introduce introduce aa new new relation relation symbol symbol R R together together with with its its defining defining axiom axiom ((Vxl)... (Xb .. .. .. ,, xn)J. (Vx,)[R(Xl,... , ) ++ ~ F F(Xl, x,)]. VX 1 ) · · · (VXn)[R(Xb · · · ,, xxn) Adding .c(E) will Adding defined defined � A relation-symbols relation-symbols to to the the language language/:(E) will not not alter alter the the class class of of Xn , y) is �E - and and �-formulas. A-formulas. If If F(Xb F ( X l , . .. .. . ,,x~, is aa �-formula E-formula such such that that y) KP � ~ ((Vx~)... x ~ ) (3 ( 3 !!y)F(Xb y ) F ( x ~ , ... .. . ,, xXn n ,,yy) ) KP VX 1 ) . . . ((VVxn) then we symbol FF to then we may may add add an an n-ary n-cry function function symbol to the the language language of of KP K P together together with with its ning axiom its defi defining axiom Xn , y)] ((Vxl) = Y y ++ ~ F(Xb F ( X l , ·. ·. .. ,,xn, y)].. Xy ) = Vxn) (Vy) [F (Xb ·. ·. ·. ,,xy) VX 1 ) ·.-.· · ((Vz,)(Vy)[F(Xl, Extensions by by definitions definitions of of � E function-symbols function-symbols will will also also not not alter alter the the class class of of � AExtensions and and �-formulas E-formulas of of KPw. K P w . Details Details about about 'Adding 'Adding Defined Defined Symbols Symbols to to KP' K P ' can can be be found Barwise [1975J found in in Chapter Chapter I1 55 of of Barwise [1975].. �-Recursion Theo One One of of the the most most important important theorems theorems of of KP K P is is the the following following E-Recursion Theorem. rem. Let by an n -function symbol of function-symbol of such that 3.3.1.6. Theorem. 3.3.1.6. �-Recursion E-Recursion T heorem. Let G G by an n + + 22 --cary ry � E function-symbol o.f Then there is an n + + 1l --ary cry � E - f u n c t i o n symbol FF of KP K P such that KP. KP. Then there is an n KP (a, a) KP � ~- FF(~, a) = = G(a, G(~, a a,, U U F(a, F(~, m ~)).· {< a The �-Recursion Theorem The above above E-Recursion Theorem is is aa special special case case of of the the more more general general �-Recursion ]E-Recursion Theorem Theorem as as stated stated in in Barwise Barwise [1975] [1975].. Because nity we KPw Because of of the the axiom axiom of of infi infinity we obtain obtain K Pw � ~ (:3a)Lim(a) (3a)Lim(a) and and thus thus aa �o-definition A0-definition of of w w as as aa point point by by Lim(w) Lim(w) 1\ A ((Vx w)[-,Lim(x)]. --, Lim(x)] . Vx E w)[ Stronger �-refiection is Stronger than than E-reflection is the the I1 H2-reflection scheme 2 -refiection scheme a F [a =f. (n2-Ref) F -+ --+ (3a) (3a)[a r 00 1\ AF F a]] .for I1 n22 formulas F F. . Observe Observe that that any any model model of of KPw K P w iinn the the constructible constructible hierarchy hierarchy already already satisfies satisfies fh-refiection. II2-reflection. To To see see that that let let F F == - (Vy) ( V y ) ((3y)A(x, 3 y ) A ( x , y) y) be be aa I1rsentence II2-sentence and and assume assume La L~ F ~ KPw K P w . . For For a E La L~ there there is is a a least least 130 fl0 < < a a such such that that aa E L,8o L/~o and and we we be the the least least ordinal such that that La ) (3y E L,8n+l)A(x, define define f3n fln+l to be ordinal such L~ F ~ ((Vx L/~,)(3y L/~,+I)A(x, y). Vx E L,8n + 1 to The ordinal f3n �-Reflection. This The ordinal fln+l exists by by E-Reflection. This defines defines aa sequence sequence (f3n; (/~n;n E w w)) + 1 exists which �-definable in := sup which is is E-definable in La La.. Hence Hence 13 fl := sup {f3n {flnlI n E w w}} < < a a and and we we have have La La F for formulas a n y). ((Vx (3y E L,8)A(x, L~)(3y L~)A(x, y). Vx E L,8) n y). Due �-Recursion Theorem Due to to the the ~-Recursion Theorem we we can can prove prove the the existence existence of of the the stages stages of of inductively ned sets inductively defi defined sets in in KPw K P w . . Let Let S S be be an an additional additional unary unary predicate predicate symbol. symbol. We ( ) . We We write write aa E E S S instead instead of of S S(a). We obtain obtain the the stages stages of of an an inductive inductive definition definition as as stated in stated in the the following following theorem. theorem. a Pohlers 282 282 W. Pohlers W. 3.3.1.7. Let 3.3.1.7. Theorem. Theorem. Let B(i, B(~, y, y, S) S) be be aa /:l.-formula A-formula of of KPw. K P w . Then Then there there is is aa � function-symbol lB Is such such that that function-symbol y, {z E xl i (3� g P w f-t---IB(a, Is(a,:~') B(~.,y,{zExll (3r E a)[z E IB(� Is(~,i')]})}. KPw i) = {y{yEXll E xd B(i, ' i)]})}. Putting Putting G G(~, {y E E xl Xlli B(i, B(~, y, y, S)} S) } we we observe observe that that G G possesses possesses aa � ~ definition definition (i, y,y, S)S) "= {y = := D 0 and get get the the theorem theorem immediately immediately from from the the �-Recursion ~-Recursion Theorem. Theorem. and B(i, y, b A /:l.-formula A-formula B(~, y, S) S) and and aa tuple tuple b of of sets sets induce induce an an operator operator A ) Pow(bl) B,';: rB,b( S) := := {y {y E e bbll lI B(b, B(b', y, S)} Pow(bl) rFs,~: Pow(bd ---+ b. occurs positively which depends depends on on the the parameter parameter list list b. Again Again we we say say that that S S occurs positively in in which formula F(S) F(S) if if the the formula formula corresponding corresponding to to F(S) F(S) in in the the Tait-Ianguage Tait-language (cf. (cf. aa formula Section 1.3) does does not not have have occurrences occurrences of of the the form form t fI.r S. We We sometimes sometimes denote denote Section 1.3) + ) . For this by F(S F(S+). For S-positive S-positive formulas formulas B(~, y,S) the the associated associated operator operator r Fs, ~ is is this by monotonic, i.e., i.e., we we have have S S 5;;; c T � ~ r Fs,~(S) C_ r Fs,~(T). We We say say that that aa set set is is closed monotonic, ,'; S 5;;; under an an operator operator r F if if we we have have r F(S) C_ S S for for all all S. For an an monotonic monotonic operator operator under S 5;;; S. For r: Pow(b) F: PoT(b) ---+> Pow(b) PoT(b) we we obtain obtain its its least least fixed-point fixed-point as as the the intersection intersection of of all all r Fclosed subsets subsets of of bb.. closed The := {x E bll A(x)} E Pow(bd be The fact fact that that aa class class T "= Pow(bl) is is r Fs, ~ -closed -closed can can be expressed by by the the formula formula expressed T () T t S. B(i, y, S) B ( ) B,,;(T). {x E bI ! A(x)} E B ,b closed B ,b (127) CIs(b,T) bl)[S(b, y, (127) C1B(b, T) :_= ((Vy y, T) T) -t-+ yy E TT].j . ' K P w t- CIs(b, A) ~ (Vr E IB(~, b') ~ A(x)]. IB(b) b, c E B(b, c, IB(b)). B(b, c, IB(b)) d B(b, c, {x E bd (3� E d)(x E IB(�, b))}). {J3}. a /:l.o {� E d l � E On} a {� E dl � E On} a. B(b, c, {X E bl l (3� E a)(x E IB(�, b))}) c E IB(a, b), c E IB(b). It follows from Theorem Theorem 3.3.1.8 3.3.1.8 tthat is the the least fixed-point of the operator It follows from h a t IB(b') is least fixed-point of the operator r las,b .. To theorem pick pick aa tuple tuple b', c e bl bl and and assume assume S(b', c, Is(b)). Since Since To prove prove the the theorem is still still aa E-formula �-formula and and by by ~� S(~, y, S) S) is is S-positive S-positive the the formula formula S(b, c, Is(b')) is Reflection we we obtain obtain aa set set d such such that that S(b,c, {x e bll (3~ e d)(x e Is(~, b))}). Now Now Reflection define flJ3 := := U and a := := flU{fl}. J3 U Then a is is aa set set by by A0define Then U{ ~ e d I ~ e O n } and Separation, Union such that that {~ E d I ~ E On} C_ 5;;; a. By By E-Persistency �-Persistency itit Separation, Union and and Pair Pair such follows S(b,c, {x e bll (3~ e a)(x e Is(~r b'))}) and and by by Theorem 3.3.1.7 we we obtain obtain follows Theorem 3.3.1.7 i.e., c e Is(b'). This This proves proves the the first first part part of of the the theorem. theorem. c e Is(a, b), i.e., For For the the second second part part we we show show B ,b . B(i, y, s) c s (i) (i) Set Set Theory Theory and and Second Second Order Order Number Number Theory Theory 283 283 b by induction induction on on �. ~. Assuming Assuming the the hypothesis hypothesis C1 CIs(b, S) we we obtain obtain by by induction induction by B ( , S) hypothesis hypothesis U B (, b) U IIB(r b) � c S. S. (ii) (ii) «r { The monotonicity of induced operator The monotonicity of the the induced operator therefore therefore implies implies (\Ix (Vx E E b1) bl)[B(b', U IIB(r b')) [B(b, x,x, U B ( , b)) «r { =~ =? x, S)] B(b', x, S)].. B(b, (iii) (iii) ClB (b, (i) from D By Theorem Theorem 3.3.1.7 3.3.1.7 and and the the hypothesis hypothesis CIB(b , S) S) we we get get (i) from (iii). (iii). [3 By It It follows follows from from Barwise Barwise [1975] [1975] that that every every primitive primitive recursive recursive function function has has aa A l1-definition in K P w . Therefore Therefore we we may may add add function function symbols symbols for for all all primitive primitive � -definition in recursive functions functions to to the the language language of of K P w . So So we we may may regard regard the the second second order order recursive .3. language language of of NT NT22 as as sublanguage sublanguage of of K P w . We We mentioned mentioned that that already already in in Section Section 11.3. Interpreting the the language language C� /:~ as as aa sublanguage sublanguage of of C(E) s augmented by by � E function function Interpreting augmented . . )) turns turns first symbols symbols (call (call this this language language C(E, s first order order formulas formulas with with set set parameters parameters into into �o A0 formulas formulas (which (which in in turn turn are are � A for for the the theory theory K P w ) . It It is is obvious obvious that that the Mathematical Induction Induction can the scheme scheme of of Mathematical can be be easily easily derived derived from from the the Foundation Foundation Scheme. All All defining defining axioms Scheme. axioms for for primitive primitive recursive recursive functions functions are are provable provable in in K P w . By By Theorem Theorem 3.3.1.8 3.3.1.8 we we may may also also interpret interpret the the additional additional constants constants llB. So we we get get the the B . So following following theorem. theorem. KPw. KPw. KPw. . KPw). KPw. The theory is a subtheory ofKPw. 3.3.2. 3.3.2. The T h e theory t h e o r y KPI KP1 viewed as a theory in the language 3.3.1.9. . 3.3.1.9. Theorem. Theorem. The theory IDI ID1 viewed as a theory in the language C(E, s is a subtheory of K P w . . .) KPI We We are are now now introducing introducing the the theory theory KP1 which which axiomatizes axiomatizes aa set set universe universe which which is is the the union union of of admissible admissible universes. universes. Therefore Therefore we we augment augment the the language language of of Set Set Theory Theory by by an an additional additional constant constant Ad whose whose intended intended interpretation interpretation is is that that of of an an admissibility admissibility predicate. The defining axioms predicate. The defining axioms for for Ad are are (Ad 1) (Vu)[Ad(u) -+ (Adl) --+ w wE e u u /\ A Tran(u)] Ad Ad Tran(u)] (\lu)[Ad(u) (Ad 2) (\lx)(\ly)[Ad(x) (Ad2) (Vx)(Vy)[Ad(x) /\ A Ad(y) Ad(y)--+ x E e Yy V Vx x= = yy V V Yy E e x x]] -+ x (Ad 3) (\lx)[Ad(x) (Ad3) (Vz)[Ad(x) --+ (Pair (Pair')' tx /\ A (Union (Union')' )Xx /\ A (�o-Separation)X (A0-Separation) ~ /\ A (�o-Collectiont] (A0-Collection) ~] 3.3.2.1. The KPI isis the BSTw 3.3.2.1. Definition. Definition. The theory theory KP1 the system system B S T w together together with with the the -+ axioms (Ad (Adl) and the the axiom axiom axioms 1) -- (Ad3) (Ad 3) and (Lim) (Vx)(3u)[Ad(u) /\ (Lim) Ax E e u]. ] (\lx)(:3u)[Ad(u) x u . Let {O} of admissible ordinals ordinals (augmented Let ).� ,~.. n{ gt~ enumerate enumerate the the class class Reg Reg U U {0} of admissible (augmented by by 0) and and their Then the the smallest 0) their limits. limits. Then smallest constructible constructible model model of of KP1 is is Low LN~,, i.e., i.e., IlgPlll~ = = n ~ .w ' Since Since we we have have g P 1 ~ Ad(u) -+ ~ FU F ~ for for every every sentence sentence F r E e gPw we obtain obtain we II KPl ll oo KPI � Ad(u) KPI KPw Pohlers 284 W. Pohlers 284 Ad(u) FU . Also, F 3.3.2.2. 3.3.2.2. Lemma. Lemma. F KPwr KPw r � ~- F � =~ KPlr KPF � [-- Ad(u) -+ --+ F ~. Also, KPw KPw � ~ F � =~ KPI � KP1 ~ A d ( u )-+ ~ F ~. Ad(u) FU . As KPw KPI will will be I;-recursion the As in in K P w the the most most important important theorem theorem in in KP1 be the the E-recursion theKPI. Therefore orem. But �o-collection in orem. But we we do do not not have have full full &0-collection in in in KPI. Therefore we we be aa collection will in aa relativized Let IF will obtain obtain I;-recursion E-recursion only only in relativized version. version. Let F be collection of of new new function function symbols. symbols. Denote Denote by by KPI(IF) KPI(F) the the theory theory KPI K P I formulated formulated in in IF) together the the language language s together with with defining defining axioms axioms for for the the function function symbols symbols in in IF. F. Let Let A(~,y) be be aa I;-formula E-formula such such that that KPI(IF) KPI(F) � ~ (V~)(3!y)A(~,y) and and introduce aa new KPI(F) � ~ (Vu)[Ad(u) -+ --+ (V~ e u)(3y e u)A(~,, y)]. Then Then we we introduce new func funcKPI(IF) tion ning axiom tion symbol symbol G and and its its defi defining axiom .c( E, A(X', y) (V'X')(3!y)A(X', y) (V'u)[Ad(u) (V'X' E u)(3y E u)A(X', y)]. G (D (Da) (VZ)(Vy)[G(Z) y)] A(X', y)] (V'X')( o) V'y)[G(X') == yy ~ A(Z, and and call call G G aa relative relative I;-function E-function symbol symbol of of KPI(IF) KPI(F).. Let Let KPI(G) KPI(G) be be +-t the the ( E, IF U {{G})-theory G})-theory KPI(IF) Do . By L L(E,FU KPI(F) + + DG. By the the common common techniques techniques we we obtain obtain that that KPI(G) definitions of KPI(G) is is an an extension extension by by definitions of KPI(IF) KPI(F) in in the the following following strong strong sense. sense. 3.3.2.3. For E, G) 3.3.2.3. Lemma. Lemma. For every every formula formula F(X') F(~) in in the the language language .c( f_,(E, G) there there is is aa formula E, IF)F) such formula Fo(X') Fo (~) in in the the language language .c( f_,(e, such that that KPI(G) (V'X')[F(X') ~ Fo(X')]. KPI(G) � ~-(VZ)[F(Z) F0(Z)]. IfIf F(X') of .c( F(Z) is is aa I;-formula E-formula of s E, G) G) then then Fo(X') Fo(Z) is is aa I;-formula E-formula of of .c( s E, IF) such such that that KPI(IF) KPI(F) � [--(Vu)[Ad(u)--+ (VZ E e u)(Fo(X') u)(Fo(Z) ~-~ Fo(X')U)]. F0(Z)u)]. (V'u)[Ad(u) (V'X' IfIf F(X') �o-formula ofof f-,(e, .c(E, G) F(Z) is is aa &o-formula G) then then there there is is moreover moreover also also aa IT-formula H-formula F1(X') FI(Z) such that such that KPI(G) KPI(G) � ~-(V~')[F(:~) F~(x)] (V'X')[F(X') 4+ F1(x)] KPI(IF) KPI(F) � ~-(Vu)[Ad(u)~ u)(Fo(Z) ~ Fo(X') Fo(Z) U~ ~ Fl(X') FI(Z)~)]. (V'u)[Ad(u) ((VZ V'X' Ee u)(Fo(X') U)], Iterating Iterating Lemma Lemma 3.3.2.3 3.3.2.3 we we see see that that we we can can identify identify the the theory theory KPI KP1 with with its its closure closure +-t +-t -+ +-t +-t -+ +-t under nitions of under extensions extensions by by defi definitions of relative relative I;-function E-function symbols. symbols. The The I;-Recursion E-Recursion Theorem modified in Theorem can can now now be be modified in the the following following way. way. Let G be an n 2-ary relative -function symbol of Then there exists an n l -ary relative -function symbol symbol FF such such that that (X',�)) G(X', a,a, U (X', a)a) == G(~, KPlr KPF � ~ FF(~, U FF(~, ~)) 3.3.2.4. 3.3.2.4. Relativized Relativized I;-Recursion E - R e c u r s i o n Theorem. Theorem. Let G be an n + 2-ary relative I; E-function symbol of KPlr K P F . . Then there exists an n + 1-ary relative I; E-function e [1975] C(X', a, v, f) (a 1\ f = 1\ v = (a E 1\ Fun(J) 1\ dom(J) = a 1\ (V'� E a)[f(�) = G(X', �, U« e f(()) 1\ v = G (X', a, U(Eo f(())]). To To prove prove the the theorem theorem we we follow follow Barwise Sarwise [1975] as as far far as as possible. possible. Let Let C(~, a, v, f) be be the the �o-formula A0-formula (a i ~ On OnAf = 0O A v = 0) O) V V (a E On OnAFun(f) A We dom(f) -- a A (V~c e a)[f(~r - G(~,r162 f(r A v = G(~,a,(.Jce a f(r We then then show show )C(X', a,a, z,z, 1f)) (3!z)(31f)C(:~, KPl g P l "r � ~ (V'X')(V'a) (V:~)(Va)(3Iz)(3 ((i) i) 285 285 Theory and Second Second Order Order Number Theory Set Theory and and KPlr KP1 ~ � ~ (Vu)[Ad(u) (Vu)[Ad(u) -+ --~ (Vx (V3 Ee u)(Va u)(Va Ee u)(:Jz u)(3z Ee u)(:Jf u)(3f Ee u)C(x, u)C(3, a, a, z, z, J)J f)] (ii) (ii) and introduce introduce aa new new relative relative �>function E-function symbol symbol FF and and its its defining defining axiom axiom and (V3)(Va)(Vz)[F(3, (3f)C(3, a, a, z, z, J)J. f)]. (Vx)(Va) (Vz)[F (x, a)a) == zz ~ (:JJ)C(x, To prove prove (i) (i) and and (ii) (ii)it suffices to to show show that that KPlr g P l r proves proves To it suffices C(3, a, a , zz,, IJ) ) /\ A C(x, C(3, a, a , zz', ' , I1' )') -+ ~ zz = z' z' /\ A fI = l' I' C(x, +-+ (iii) (iii) = (iv) (iv) and and (v) Ad(u) /\ Ax ~E e uu /\ Aa a E e uu -+ --+ (:Jz (3z E e u)(:Jf u ) ( 3 / eE u)C(x, u)C(~,, a, a, z, z, J). I). (v) Ad(u) We prove prove (iv) (iv) and and (v) (v) by by induction induction on on a. a. Since Since C(x, C(~, a, a, z, z, J) f) is is �o A0 we we can can formalize formalize We this in in KPlr KP1 ~ where where we we have have (�o-FOUND) (A0-FOUND).. The The proof proof of of (iv) (iv) is is exactly exactly the the same same this as as in in Barwise Barwise [1975J. [1975]. For For the the proof proof of of (v) (v) we we also also follow follow Barwise Barwise [1975J [1975] but but use use the the additional observation observation that that G(x, G(3, a, a, J) f) E u u whenever whenever u u is is admissible admissible and and x, ~, a, a, f f E Cu u.. additional We do do not not have have full full �-replacement E-replacement in in KPlr KP1 ~ but but we we may may use use its its relativized version relativized version We according to to Theorem Theorem 3.3.1 3.3.1.4 and Lemma Lemma 3.3.2.2 3.3.2.2 whenever whenever it it is is used used in in Barwise Barwise according .4 and [1975]. Once Once we we have have the the relative relative function function symbol symbol FF we we may may proceed proceed literally literally as as in in [1975J. Barwise [1975J [1975].. [3 0 Barwise There is is a a relativized version of of Theorem Theorem 3.3.1 3.3.1.7. Let B(3, v,S) be aa There relativized version .7. Let S) be A0-formula. Then we we obtain obtain a a relative relative �-function E-function symbol symbol G such such that that �o -formula. Then G(3, v, s) = = {y E e xl I B(3, y, s)} and and we we apply apply the the relativized relativized �-recursion E-recursion Theorem Theorem (Theorem 3.3.2.4) 3.3.2.4) to to obtain obtain aa relative relative �-function E-function symbol symbol lIB B such such that that (Theorem � Is(c~,3) = ((:J� 3 ~E e~)(z E e IB(~,3))})} = { Y eEX l l B(3, y , { z eEx l l B(x, v, G G(x, v, s) {y xI I B(x, y, s)} a)(z IB( x))})} IB(a, x) {y xd B(x, y, {z xI I is provable provable iinn KPlr. Defining 1~(3) I�(x) ::== Ur U{EU IB(C we obtain following analogue is g P l r. Defining Is(~, x) 3) we obtain the the following analogue ' of Theorem Theorem 3.3.1.8. of 3.3.1 .8. KPI KPlr) Let B(x, y, be an -positive -formula. Then there is a relative -function symbol lB such that B(X, I�(x)) Ad(u) CIB(3, 1~(3)) -+ C/ Ad(u) A/\ x3 eE uu --+ and and A) --+ -+ I~(~) I�(x) C<; {x{x eE uull A(x)} Ad(u) A ~, eE uu /\ A C/B(X, CIB(~, A) /\ x A(x)} Ad(u) �o -formulas -formulas A(x). are provable inin KP1 KPlfor arbitrary formulas formulas A(x) A(x) and and inin KKPlr for A0 are provable for arbitrary P I r for A(x). The We fix The proof proof is is essentially essentially that that of of Theorem Theorem 3.3.1.8. 3.3.1 .8. We fix an an admissible admissible uu such such that that and repeat repeat the the proof proof substituting substituting E-reflection �-reflection by by its its relativized relativized version. version. If If we we 3x cE uu and only A0-formulas A(x) only consider consider �o-formulas A(x) then then only only (A0-FOUND) (�o-FOUND) is is needed. needed. Otherwise Otherwise we we need [3 need the the full full strength strength of of KP1. KPI. 0 3.3.2.5. T heorem. (Inductive in K P I and and KP1 r) Let B(3, y,S) 3.3.2.5. Theorem. (Inductive Definitions Definitions in S) be an S-positive Ao E-function symbol Is such that S �o -formula. Then there is a relative � Observe that that Theorem Theorem 3.3.2.5 3.3.2.5 does does not not immediately immediately follow follow from Theorem 3.3.1.8 3.3. 1.8 Observe from Theorem by differ slightly. slightly. by Lemma Lemma 3.3.2.2 3.3.2.2 since since the the definitions definitions of of the the function function symbols symbols IB differ IB W. Pohlers 286 286 Let B(x, y, be an -positive -formula. Then proves thatfor every admissible set a containing the parameters b and every admissible set containing a the class I�(b) is a set in which is the least fixed point of the monotone operator induced by B(b, y, Let B ( Z , y , SS) ) be an S S-positive 600 Ao-formula. Then KPlr KP1 r proves that.for every admissible set a containing the parameters b and every admissible set u u containing a the class I~(b) is a set in u u which is the least fixed point of the monotone operator r FB, S).. B ,b~ induced by B(b, y, S) 3.3.2.6. 3.3.2.6. Corollary. Corollary. The symbol. and The function function symbol symbol I]B is a a relative relative �-function C-function symbol, and we we have have B is Proof. Proof. I�(b) by Lemma and 6oo-Separation I~(b) � c_ a a E E c. So So by Lemma 3.3.2.3 and A0-Separation relativized relativized to to u we we obtain obtain I�(b) � (b) isis the I~(b) E E u. That That II~(b') the least least fixed-point fixed-point of of r FB B,,bg follows follows from from Theorem Theorem 3.3.2.5. 3.3.2.5. u. c. u o D 3.3.3. 3.3.3. The The quantifier quantifier theorem theorem and and axiom axiom/3f3 The tool for embedding subsystems The most most important important tool for embedding subsystems of of NT NT22 into into subsystems subsystems of of Set Theory is er Theorem Theorem which going to this section. Set Theory is the the Quantifi Quantifier which we we are are going to present present in in this section. It It is is based based on on a a theorem theorem which which is is commonly commonly known known as as Spector-Gandy Spector-Gandy Theorem. Theorem. First First we we fix fix the the following following notations: notations: T) ("Ix E a)[(Vy E a)(y x y E T) x E TJ Proga( --<, T) ::r<=? (Vx E a)[(Vy E a)(y --< Proga(-<, -< x -+ y e T) -+ --+ x E T] T where where T may may be be a a set set or or aa class class-term, -term, a ( --<, T) T) -+ TP(-<, T) :<=? :r Prog Prog~(-<, --+ ("Ix (Vx E E a)(x a)(x E e T), T), T1 a ( --<, T) a ( --<) :<=? wt"(-<) :r Wf ("Ix) T1a ( --<, x), a (--<) a (--<) :r:<=? aa = WOn(-<) = field field(-.,<)A LO(-.<) 1\A Wf Wfa(-<) (--<) 1\ LO(--<) WO where --<) says x, y)y) we where LO( LO(-~) says that that --< -< is is a a linear linear ordering. ordering. For For aa 6oo-formula A0-formula A(x, A(~,x, we define define aa relation :<=? A(x, x, y)y) and relation x x --<x -~ Y y :r A(Z, x, and call call it it a a 6oo-relation. A0-relation. 3.3.3.1. 3.3.3.1. Lemma. L e m m a . Let Let --<x - ~ be be aa 6oo-relation Ao-relation and and define define A(b, A(b, X, 3, x, x, S) S) as as (Vy (Vy E E b)[y b)[y --<x -~ xx -+ y EE SJ. S]. Then Then b, x))). UP1 rr � ~- Ad(a) Ad(a) 1\ A b, b,Zx E E aa --~ ((~/~/fb(~) ~ (Vy (Vy E E b)(3� b)(3~ E E a)(y a)(y E E IA(�' I~(~,b,Z))). Wfb ( --<x) ++ KPl To ( --<x, x) ++ To sketch sketch the the proof proof we we work work informally informally in in KPlr K P F . . We We have have Progb Progb(-<~,x) admissible set containing bb and CIA (b, x, x). Let CIA(b,~,x). Let a a be be an an admissible set containing and all all members members of of x ~ as as elements. (3� EE a)(y (�, b,b, x))} elements. By By Corollary Corollary 3.3.2.6 3.3.2.6 we we get get d d := {y {y E Ea a II (3~ a)(y E E IA I~A(~, ~))} as as a a -+ Y -+ := set to show set provable provable in in KPlr K P F . . We We have have to show b ( --<x) ++ Wf Wfb(-<~) ++ bb � C_ d. d. (i) (i) d) b ( --<x) implies For --<x, -+ For the the direction direction from from left left to to right right observe observe that that Wf Wfb(-~) implies Progb( Progb(-<~,d) -~ A which b C_ d. But But by by Theorem Theorem 3.3.2.5 we we have have C1 CIA(b,Z,d) which is is equivalent equivalent to to Progb( --<x, d). For Progb(-<~, For the the opposite opposite direction direction we we use use the the second second part part of of Theorem Theorem 3.3.2.5 3.3.2.5 to A -+ d c_ x for to get get C1 CIA(b,Z,x) for any any x. Together Together with with b C_ d this this implies implies b ( --<x) . Progb(-<~,x) ~ b C_ x which is Wf Wfb(-~). [3 Progb( --<x, -+ which is 0 The notion is The following following notion is motivated motivated by by the the n H{-completeness of well-foundedness. well-foundedness. t -completeness of b � d. d) . (b, x, x) d � x x) b � x (b, x, d) x. b�d 287 Set Theory and Second Order Number Theory 3.3.3.2. Definition. Definition. Let Let Ax Ax be be aa theory theory in in the the language language of of Set Set Theory. Theory. We We say say 3.3.3.2. that formula A(x) A(Z) is II ~((Ax) there is is aa �o-relation A0-relation -<x� -<~C w wxw w such that that aa formula is II Ax) iff iff there such that l Ax � ~-(Vs163 Wf(-<~)].] . Ax (\lx) [A(x) ++ Wf(-<x) Observe -formula is Observe that that the the folklore folklore fact fact that that every every m II~-formula is equivalent equivalent to to the the well wellfoundedness associated tree unsecured sequences foundedness of of its its associated tree of of unsecured sequences can can be be proved proved in in BSTr B S T r.. -formula is ) -formula such defining formula Therefore Therefore every every m II~-formula is aa II II II (( BSTr B S T r)-formula such that that in in the the defining formula for for its its corresponding corresponding �o-relation A0-relation all all quantifier quantifier are are restricted restricted to to w w.. For every IIl (KPlr)-formula a C(X) such that KPlr � Ad(a) there is a 2:.-formula 3.3.3.3. 3.3.3.3. Theorem. Theorem. For every IIl(KPF)-formula A(x) A(~.) there is a E-formula . C(s such that K P F ~ Ad(a) 1\ Ax ~E E a -+ --+ (A(x) (A(Z) ++ ~ c(x)a) C(s We indicate indicate the the proof. proof. Let -<~C_ w x x w w the the �o-relation Ao-relation such such that that We Let -<x� w A(s ~ Wf( Wf(-<~). an admissible admissible a a such such that that X, s w E E a a and and define define A(x) ++ -<x) . Choose Choose an W (-<x) ++ B(x, E w) [y -<x X -+ Then B ( s x, S) :{::} :4=~ (\lY (VyEw)[y-<~x --+ Yy E E S] S].. Then A(x) A(s ++ ~ Wf Wf~(-<~) ++ Since 1. (y E ( V yE e ww)) ((3� 3 ~ EE a) a)(y e IB(�' Is(~,sx))) by by Lemma Lemma 3.3.3. 3.3.3.1. Since IB IB is is aa rela rela(\ly ++ C(x)aa for tive tive 2:.-function E-function symbol, symbol, Ad(a) and and x ~ E E a a we we get get A(x) A(s ++ C(s for D (vyeE w) (3�) (y Ee IB(� ' x)) . n C(X) : {::} (\ly Ad(a) For every m -formula in the language .c� there is a 2:.-formula so that Ad(a) is provable in KPlr. For every 2:.� formula there is a 2:.-formula such that is provable in KPlr. Dually for every m-formula there is a II-formula such that is provable in KPlr. KPF. Recall restricting all Recall that that we we regard regard .c� s as as aa sublanguage sublanguage of of .c s ( E, . . . ) by by restricting all first first order order 3.3.3.4. Theorem. 3.3.3.4. Quantifier Quantifier T h e o r e m . For every II~ -formula A(X, A(f,, x) s in the language s -+ (\IX (\lx E ) [A(X, x) ++ C(X, x)a] there is a E-formula C(X, C ( s x) s so that Ad(a)-~ (V)~ E E a) a)(Vs Ew w)[A(.P,,s ~] x) , o,aU gPr. :o mula A(X, A(f,,2) a r,-fo m la C(X, C( 7, x) 2) such that A(X, A(X,sx) ++ ~ C(X, C ( 2 , ~x) ) is provable in K P F . Dually for every YI~-formula A(X, A()~, x) 2) there is a II-formula C(X, C(X, x) ~) such that A(X, A(fi,, x) ~) ++ o C(X, C(X, x) ~) is provable in quantifiers quantifiers to to w w and and all all second second order order quantifiers quantifiers to to subsets subsets of of w. w. As As already already remarked remarked -formula is III1((Kp every every m II~-formula is aa II II l1(( BSTr B S T ~)) and and hence hence also also aa II g P Fn ) - f-formula. o r m u l a . Assume Assume a. By x) such Ad(a) and and X )~ E E a. By Theorem Theorem 3.3.3.3 3.3.3.3 we we obtain obtain aa 2:.-formula E-formula C(X, C(.Y,x) such that x) ++ the second that A(X, A()(,~) ++ c(X, C()~,sx)a. a. For For the second claim claim assume assume A(X, A()~,sx) {::} r c_ ((3Y)[Y Y) [Y � (X, Y, -formula B (X, Y, w 1\ AB B()~, Y, x)] s for for aa II~-formula B()~, Y, x) s . By By the the first first claim claim there there is is aa 2:.-formula E-formula w C'(X, x) such Y E x) ++ C'()~, Y, Y,Z) such that that for for Ad(a) and and X, )~,Y e a a we we get get B(X, B()~, Y, Y,Z) e+ C'(X, C'()~, Y, y,sx)a provable Lim) we provable in in K P F . By By axiom axiom ((Lim) we always always find find such such an an a a and and thus thus obtain obtain the the claim claim Y, x)Z] . defining C(X, bydefining C(.X,sx) :{::} :r (3Z)[Ad(Z) 1\ AX )~ E E Z 1\ A (3Y (3Y E E Z)(Y � C_w w 1\ A C'(X, C'(fi,,Y,s by DEl The The last last claim claim follows follows from from the the second second by by taking taking negations. negations. Ad(a) m KPlr. 3 Ad(a) (3Z)[Ad(Z) Z Z)(Y � KPlr. Let Let H(X, H(.~, x s , x)) be be aa II�-formula. II~-formula. Then Then As corollary of -comprehension in As aa corollary of Theorem Theorem 3.3.3.4 3.3.3.4 we we get get II YI11-comprehension in K P F . x 3.3.3.5. 3.3.3.5. m-Comprehension I I ~ - C o m p r e h e n s i o n Theorem. Theorem. }]. (3y)[y = {z {z E K P F proves (\lX) (V.~)(V~ e w) w)(3y)[y ew w iI H(X, H()(, x, ~, z) z)}]. (\lX E KPlr proves Proof. x, x) such Proof. By By Theorem Theorem 3.3.3.4 3.3.3.4 there there is is aa 2:.-formula Z-formula C(X, C(.,~,Z,x) such that that KPlr Using axiom KP1 r � ~ Ad(a) Ad(a) 1\ AX )~ E E aa 1\ A X, ~, X x E E w w -+ --+ (H(X, (H()~, x, ~, x) x) ++ +-~ C(X, C()~, x, s x)a) x)a).. Using axiom 288 288 W. Pohlers w. (Lim) we we find find such such an an admissible admissible a a and and another another admissible admissible u u such such that that a a E Eu u.. Now Now (Lim) we apply apply �o-Separation A0-Separation relativized relativized to to u u to to obtain obtain d := {x E E w iI C(X, C()~, x, Z, x) a} as as aa we set. Obviously Obviously d is is aa witness witness for for y in in the the claim. claim, r7 D set. The good example analytical ' The Quantifier Quantifier Theorem Theorem is is aa good example for for reducing reducing the the complexity complexity of of ''analytical' formulas by by translating them into into the the language language of of Set Set Theory. Theory. (The (The reason reason for for this this formulas translating them reduction is is the the presence presence of of the the axiom axiom (Found) (Found) in in Set Set Theory) Theory).. Another Another example example is is reduction Axiom/3 which which turns turns the the III H1 notion notion of of well-foundedness well-foundedness into into aa/k-notion. We will will �-notion. We Axiom show that that Axiom Axiom fl is is provable provable in in KPIT K P F . . This This needs needs some some notations. notations. Define Define show Found(a, r) :{:} :r (Vx)[x � C_ a 1\ A x =f. ~ 0 -+ --+ (3z E E x)(Vy E E x)((y, zz)) � r r)]] (128) d d := {x y x)a} fJ fJ (3z x)(\fy x)((y, r) (128) (\fx)[x a x Found(a, r) expressing that that rr is is aa well-founded well-founded relation relation on on the the set set a. a. To To increase increase readability readability expressing we use use the the infix infix notation notation for for relations, relations, i.e., i.e., x x rr yy instead instead of of ((x, y) y) E E r. Recall that that we r. Recall Found(a, r) r) entails entails (\fx (Vx E E a) a)(x Found(a, (x r//x). x). (Axfl) (Vx)(Vr)[ Found(x, Found(x, r) r) -+ --+ (3f)(30:)(Fun(J) (3f)(3a)(Fun(f) 1\ A dom(J) dom(f) = =x x 1\ A rng(J) rug(f) = 0: (AxfJ) (\fx)(\fr)[ 1\ A (\fu (Vu E E x)(\f x)(Vvv E E x)(u x)(u rr vv -+ --4 f(u) f(u) < < f(v))) f(v)))].] . 3.3.3.6. Theorem. T h e o r e m . Axiom Axiom fJ fl is is aa theorem theorem of of KPIT KPF. . 3.3.3.6. x, = P r o o f . We We start start with with the the obvious obvious observation observation Proof. b KPI" � ~- Found(b, Found(b,r)-+ Wfb(r). KPIT r) -+ Wf (r) . (i) (i) Then we we obtain obtain from from Lemma Lemma 3.3.3.1 3.3.3.1 Then Ad(a) 1\ A b, b, rr E E aa -+ --4 [Found(b, [Found(b, r) r) +4 (\fy (Vy E E b)(3� b)(3~ E E a)(y a)(y E E I�(�, I~(~, b, b, r)) r))]] (ii) (ii) �~- Ad(a) given. By for for B(x, S(x, S) S) :r (Vy)[y rr x x -+ --4 yY E E S] S].. Let Let bb and and rr be be given. By (Lim) (Lim) we we choose choose (\fy)[y such that that b, b, rr EE a.a. For there is is by (ii) aa �~ EE a such that an admissible a such an admissible For x E E bb there by (ii) such that b, r)r) and and we we define define f(x) f(x) :=:= min min {~{� eE a[al xx EE I~(~, I�(�, b,b, r)} defines aa xx E E IB(�, IB(~, b, r)}.. This This defines we see see that that ff is is function ff with with dora(f) dom(J) == b.b. Defining Defining a0: := function := sup sup {f(x) {J (x) + 1[1 1 xx EE b}b} we an (r, E)-homomorphisms from from bb onto Q (r, E)-homomorphisms onto a. an 0:. Observe that that the function ff whose whose existence existence is required by by (AxfJ) is Observe the function is required (Axl~) is the property that uniquely determined and and has well-founded rr the uniquely determined has for for transitive transitive well-founded property that otyPr and and define define f (x) = { J (y) I yy rr x}. x} . We f(x) = {f(y)[ We often often denote denote this this function function by by otypr As aa corollary corollary of of the the proof proof of of Theorem we obtain obtain otyp(r) Theorem 3.3.3.6 3.3.3.6 we otyp(r) :=:= rng(otypr). rng( otYPr ). As � Ad(a) K P F ~Ad(a) A Found(b, r) --4 otyPr otyp~ E (129) (129) KPIT 1\ Found(b, r) 1\A b,b, rr EE aa -+ E a.a. As soon soon as we have have (Axfl) (AxfJ) we that well-foundedness well-foundedness for for sets sets is is in in KP1 KPI extensible extensible As as we we see see that +-+ KP1 r KPIT :{:} a x a D to well-foundedness well-foundedness for classes. to for classes. Let be a �o-relation. Then for any class-term For -classes this is already provable in KPI ~� Wf~(-<~) Wfa(-<x) --~ -+ Let --<x ~ be a Ao-relation. Then KP1 Tla( -<x, T) T) .for any class-term T. T. For Ao-classes �o TI~(-.<~, TT this is already provable in K P I r .. KPIT 3.3.3.7. TTheorem. 3.3.3.7. heorem. T) Proof. Assume Wfa(-~). Wfa( -<x) . We have to to show show that that from the hypothesis hypothesis Proga(-~, Proga( -<x, T) P r o o f . Assume We have from the we Define := {(x,y) l x EEa A y1\E a A x - <1\~ y } . Then r is set we oobtain b t a i n a C� _ T . Definer Then is aa set a T. r := {(x, y) 1 x a y E a x -<x y}. r Set Theory Theory and and Second Second Order Order Number Number Theory Theory Set 289 289 (Axt1) B(�) � a ( -<x) . By by flo A0-separation and we we obtain obtain Found(a, Found(a, r) r) from from Wf td/fa(-~). By (Axle) there there is is an an by -separation and onto ordinal a c~ and and an an order order preserving preserving mapping mapping f: aa �) a ~.. Put Put B(~) :"r < a ~ -+ ordinal <=? ~ < (Vy E a)[f(y) = - ~ -+ Yy E T T].. Since Since we we have have the the scheme scheme (FOUND) (FOUND) we we have have induction induction on ordinals, ordinals, i.e. i.e.,, especially especially on (Vy a)[J (y) � f: ] (V{)[(Vr < �)B(() {)B(r -+ --+ B(�)] B({)] -+ + (VOB(O. (V{)B({). (i) (i) (V�)[(V( But from from (V�)B(O (V{)B({) we we immediately immediately obtain obtain (Vy (Vy E a)[y a)[y E T T ]] , , Le., i.e., aa � C_ T. So So But assume (V( (Vr <{�)B(() ) B ( r for for �{ < < a. ce. We We have have to to show show B(�). B({). Let Let x E aa such such that that assume For all all yy E E aa such such that that yy rr x we we get get f(y) f(y) < < �{ and and thus thus B(f(y)). B(f(y)). Hence Hence � = f(f(x).) . For a ( -<x, T) implies (Vy E a)(y a)(y rr x -+ --+ y E T) T) which which by by Prog Proga(-~, implies x E T. Hence Hence B(�) B({) and and we we (Vy are done. done. If If T T is is aa flo-class A0-class then then (flo-FOUND) (A0-FOUND) suffices suffices to to have have (i) (i) which which allows allows to to are formalize the the proof proof in in KP K P Flr. . [:] formalize = x x x x x 0 3.3.3.8. Corollary. Corollary. 3.3.3.8. The schemes schemes (II6-BI) (/]~-BI) as as well (Bi) are are theorems theorems ofKPI. of KP1. The well asas (Bi) P r o o f . Let Let -< -~)?,~ be arithmetically arithmetically definable definable in in .c�. Z:~. Then Then its its defining defining formula formula is is Proof. x x be -< x x) as .c( E, . )-formula becomes A0 in in the the sense sense �off /.c: ((EE,, . .. . ). ). . The The formula formula Wf( Wf(-~Z,~) as/::(E,...)-formula becomes flo � KPI � . . Wf~(-~,~). Hence g P l ~ WfW( Wf~ -< x ,x) -+ -~ TIW TI~(-~)?,~,T) for every every class class term term T T by by WfW( ( -< ,x ' T) for -< x,x) . Hence Theorem 3.3.3.7 which which implies (II~ As aa special case we we obtain the provability provability Theorem implies (m-BI) . As special case obtain the 0 of WfW(-<x) k~/fw(-<x) -+ ~ TIW(-<x TI~(-<x,T)which entails (Bi). (Bi). [:] of , T) which entails Summarizing Summarizing the the work work of of this this section section we we have have the the following following results. results. The theory is a subtheory of W-KPI and is a subtheory of KPlr. The theory a subtheory ofKPI. 3.3.3.9. CA)o is a subtheory of KP1 r. The theory 3.3.3.9. Theorem. Theorem. The theory (rr�(YIx~- CA)o (yI1-CA) + (Bi) (rr�-CA) + (Bi) a subtheory o f g P 1 . (rr�-CA) is a subtheory of w - g P 1 and (YI~X-CA) KPi KP KPI isis the the theory theory K KPi we are are going to Much stronger stronger than than the the theory theory KP1 Much P i which which we going to 3.3.4. The f3 3.3.4. T h e theories t h e o r i e s K P i and a n d KPf~ study in in this study this section. section. 33.3.4.1. . 3 . 4 . 1 . Definition. Definition. KP1. KPI. KPi KPi KPw The theory union of the axioms in K P w and The theory K P i is is the the union of the axioms in and The axioms in in K P i describe describe aa universe universe which which is is admissible admissible and and simultaneously simultaneously the the The axioms where II denotes denotes the the first first union of of admissible admissible universes. universes. So So we we obtain obtain IIKPiI]~ -= II where union recursively recursively inaccessible inaccessible ordinal, ordinal, i.e., Le., the the least least ordinal ordinal which which is is admissible admissible and and the the limit follow from from limit of of admissible admissible ordinals. ordinals. Most Most of of the the properties properties of of the the theory theory K P i follow the the previous previous sections. sections. IIKPilloo KPi KPir) Suppose that A(X, y) is a H~V y E ww)(A(X, ) ( A ( X , ~X, , yy) ) ~ B ( f , , ~X, , yy)) ) ) -+ m - and and B(~',Z',y) B(X, X, y) aa ~l-formula. '2:,�-formula. Then Then ((Vy B(X, (3z)(z Pff. {y E ww II A()(, A(X, ~,X, y)}) y)}) isis provable provable inin gKPir. (3z) (z = (y 33.3.4.2. .3.4.2. T heorem. Theorem. = 1 (A2-comprehension in (fl�-comprehension in K P i r) Suppose that A(X, ~, X, y) is a +-+ 2290 90 W. Pohlers (\ly w)(A(X, x, y) +-+ B(X, y)) A(X, x, y) KPir. Proof. the hypothesis Proof. From From the hypothesis (Vy E w)(A(f,,Y~,y) ++ B(.~,Y.,y)) and the the Quantifier Quantifier X, and Theorem Theorem 3.3.3 is is follows follows that that A()~, ~, y) is is aa �-formula A-formula of of K P F . Therefore Therefore z "= X , x, �-comprehension. D {y E ew w[! A( A()~, ~, y) y)}} is is aa set set by by A-comprehension. [:] {y := As As an an immediate immediate consequence consequence of of Theorem Theorem 3.3.4.2 we we obtain obtain the the following following theorem. theorem. The theory is a subtheory of KPir, the theory is a subtheory of W-KPi and the theories and are subtheories ofKPi. We We have have seen seen in in the the previous previous section section that that KPI KP1 proves proves (Ax,B). (Ax~). We We will will now now show show that that KPw KPw K P w cannot cannot prove prove (Ax.B). (Axfl). It It will will moreover moreover become become clear clear that that augmenting augmenting K P w by by (Axfl) will will give give aa theory theory of of the the strength strength of of KPi. KPi. (Ax,B) 3.3.4.4. 3.3.4.4. Definition. Definition. Let Let KP,a KP/3 be be the the theory theory KPw KPw + + (Ax,B). (Axfl). Since we Since we have have already already shown shown that that KPlr KPF � ~ (Ax.B) (Axfl) we we get get immediately immediately The theory theory 3.3.4.5. The KP,arr isis aa subtheory 3.3.4.5. Theorem. Theorem. The theory theory KP/3 subtheory of of KPir. K P i r. The W-KP,a and KP/3 KP,a isis aa subtheory W-KP/3 is is aa subtheory subtheory of of W-KPi W-KPi and subtheory ofKPi. o/KPi. We We will will show show that that conversely conversely KP,a KP/3 is is of of the the same same proof proof theoretical theoretical strength strength as as KPi. KPi. That, That, however, however, does does not not mean mean that that both both theories theories coincide. coincide. Though Though !IKP,a ]lKP/311oo ll oo == KP,a but ordinals a II = = IIKPilioo []gPilloo there there are are ordinals a such such that that La L~ F ~ KP/3 but La L~ � ~ gKPi P i (e.g., (e.g., R~+ + cf. cf. Platek Platek [1966] [1966] 5.11). 5.11). We We will will first first check check that that KP,a KP/3 allows allows the the embedding embedding aa == Nt of as KPi. Therefore of the the same same .c�-theories s as KPi. Therefore we we need need an an equivalent equivalent for for the the Quantifier Quantifier Theorems KPI. Theorems of of KP1. l (KP,ar)-formula A(x) 3.3.4.6. For 3.3.4.6. Lemma. Lemma. For every every II IIl(KPi3r)-formula A(Z) there there is is aa y:,-formula E-formula such that (\lx)(A(x) +-+ B(2) such that KP,ar KP/3 r � ~ (V2)(A(2) ~ B(x)). B(2)). For For every every y:,�-formula E~-formula A(X, A(.~,Y~) there x) there B(x) isis aa y:,-formula B(X, x) and II -formula such E-formula B(fi,,Y.) and dually dually for for every every m-formula, II~-formula, aa H-formula such that that KP/3 r � ~ (\lX)(\lx (V)~)(V2 eE w w)[)~ C_ w --+ (A()~, ~) ~ B()~, ~))]. KP,ar (A(X, x) +-+ B(X, x))]. )[X � -+ Proof. be aa �o-relation ( -< x x) +-+ Proof. Let Let -< "~2,~ C_ w x x w be A0-relation such such that that KP,ar KP/3 r � ~- Wf Wf(-~:~,~) x ,x � , ++ is a set and we } Y y) ! Xx eEwwA y/\eywEAWx -/\~ 2X, ~- onto <=> :<=> 291 291 Set Set Theory and Second Order Order Number Theory KP,Br A(X, x). which which is is in in KPf~ r equivalent equivalent to to A()~, Z). The The dual dual claim claim follows follows by by taking taking negations. negations. D [3 KP,Br N ow having Quantifier Theorem Now having also also aa Quantifier Theorem in in K P ~ ~ we we get get with with the the same same proof proof the the equivalent equivalent of of Theorem Theorem 3.3.4.2. 3.3.4.2. 3.3.4.7. x, y) bebe aa I-I~m- and 3.3.4.7. Theorem. Theorem. (��-Comprehension (A~-Comprehension in in KPW) KPf~ ~) Let Let A(X, A(f,,Z,y) and B(X, B(:~, x, ~, y) y) aa '2:,�-formula. 2 ~ - f o ~ l a . Then Th~ ~ B(X, B()~,i,y)) {year A()~,i,y)}) E wlI A(X, x, y)}) x, y)) -+ ((3z)(z= ((Vyew)(A(f,,i,y) :lz)(z = {Y 'tIY E W)(A(X, x, y) t+ isis provable provable in in KP,Br KPf~r.. As in in Theorem Theorem 3.3.3.7 3.3.3.7 we we get get from from (Axj3) (Ax;3) the the well-foundedness well-foundedness for for classes classes for for every every As � �o-relation sets. I.e., A0-relation which which is is well-founded well-founded for for sets. I.e., we we have have with with the the same same proof proof Let KP,B f-~ Wr (-<x) --+ Let --<x ~ be be aa �o-relation. Ao-relation. Then Then KP/3 t'Vfaa(-~) 7-1(-~, T) for for any any class-term class-term T. T. For For �o-classes Ao-classes T T this this is is already already provable KPf~r.. T/( -<, T) provable inin KP,Br � 33.3.4.8. .3.4.8. Theorem. Theorem. Summing up obtain Summing up we we obtain The theory is a subtheory of KP,Br, the the is a subtheory of W-KP,B and the theories and are subtheories ofKP,B. It ordinal a It follows follows from from Theorem Theorem 3.3.4.9 3.3.4.9 that that for for every every ordinal c~ for for which which we we have have La F KP,B we KPf~ we also also have have La L~ n N Pow(w) PoT(w) F ~ (a�-CA) ( A ~ - C A ) + (Bi). (Bi). But But again again the the opposite opposite 3.3.4.9. Theorem. 3.3.4.9. T heorem. The theory (a�(A~_ CA)o CA)o is a subtheory of KPf} r, the theory ((A~-CA) (A~-CA) + + (Bi) (Bi) and a�-CA) is a subtheory of W-KP/3 and the theories (a�-CA) (a�-CA) (A~-CA) + + (Bi) (Bi) are subtheories o f g P ~ . ory + claim is is not not true. true. S. Simpson in in aa private private communication communication to to G. J~iger constructed constructed claim S. Simpson G. Jager the the following following counterexample. counterexample. Let Let a c~ be be the the least least admissible admissible ordinal ordinal such such that that La L,~ F ~ (:lm( (3~)[~c is is uncountable] uncountable].. Then Then a c~ = (l{�Q (R~))+. +. While While La L,~ F ~ NT NT22 and and therefore therefore also also La L~ F ~ (a�-CA) (A~-CA) + + (Bi) (Bi) we we have have La L~ � ~= (Axle). = (Axj3). 3.3.5. 3.3.5. Theories Theories of of iterated iterated admissibility admissibility KPI KPir KPi. Within Within this There There is is aa tremendous tremendous gap gap between between KP1-even and KPi. this -even K P i r -- and gap M. Rathjen gap there there is is aa whole whole zoo zoo of of theories. theories. M. Rathjen in in his his thesis thesis [1988] studied studied these these theories English. But theories exhaustively. exhaustively. Unfortunately Unfortunately his his thesis thesis never never appeared appeared in in English. But there there is introduce all theories. As is not not enough enough space space to to introduce all these these theories. As examples, examples, however, however, we we will will introduce introduce some some theories theories for for iterated iterated admissibility, admissibility, i.e., i.e., theories theories which which axiomatize axiomatize universes universes containing containing aa certain certain number number of of admissibles. admissibles. As As in in Barwise Barwise [1975] we we denote denote (th admissible, admissible, i.e., by by T€ T~ the the ~th i.e., A( )~.. T€ T~ enumerates enumerates the the class class of of admissible admissible ordinals ordinals including w. So etc. including So we we have have TO To = w, 1'1 T1 = W~K = n1 f21,, Tw T~ = nw ~ ++11 etc. [1988] [1975] w. = W , = wfK = = 3.3.5.1. 3.3.5.1. Definition. Definition. Let Let J (()) f) :4=> Fun(f) /\A aoL E On O n /\ A dom(J) dom(f) = = a c~ /\ A ((V~ c~)[Ad(f(~c)) Ill:Ad(~, 'tI(c << a)[Ad( :¢:? Fun(J) tAd(a, J) )(x == f(()))] )(J (() EE f(f(~)) (::I(~< < (~)(x f((~)))] 'tIx EE f(f(~c))(Ad(x) /\A (('v'(~ ())(Ad(x) �~ (:I( O) /\A (('v'x 'tI( << (~c)(f((~) 292 292 W. Pohlers w. express that that there there are are at at least least a a many many admissibles. admissibles. We We define define express )(3J)ltAd(f" 1) KPIv := KPI KPI + + ("If, (V~ < < 1Iv)(3f)ltAd(~, f).. KPI/l If --< -< is is aa well-ordering well-ordering we we put put If KPI.~ KPI + + W WO(-<) + ("If, (V~ E E ootyp(-<))(3f)ltAd(~, f).. KPI-< := KPI typ ( --<) )(31)ltAd(f" 1) O( � ) + The theory theory The A u t - K P I := := KPI KP1 + + (Va)(3J)ltAd(a, (Va)(3f)ltAd(a, 1) f) Aut-KPI axiomatizes aa universe universe whicH which contains contains as as many many admissibles admissibles as as ordinals. ordinals. Observe Observe that that axiomatizes we can can define define x x = = Lnt Lnl :{:} :r Ad(x) 1\ A (Vy (Vy E E x)--,Ad(y) x)-~Ad(y).. The The formula formula x x E E Lnt Ln~ is is aa we Ad(x) ] . We A-formula since since x x E E Lnt Ln~ {:} r (Vy)[Ad(y) --+ --+ x xE E y] y] {:} r (3y)[y = = Lnt Lnx 1\ Ax x E E yy]. We (Vy)[Ad(y) �-formula (3y)[y define the the theory theory define KPI* KP1 + + ("If, (V~ E E Lnt Lnx)(3f)ltAd(~, f). KPI* := := KPI ) (31) ItAd(f" 1). It follows follows by by (�o-FOUND) (A0-FOUND) that that the the enumerating enumerating function function of of the the admissibles admissibles is is It uniquely determined, determined, i.e., i.e., we we have have uniquely KP1 r � ~ ItAd(a, 1) f) 1\ A ItAd(a, g) --+ Jf = = g. KPlr g) --+ (130) As As a a consequence consequence of of (130) (130) we we obtain obtain KPI KPI** � ~ (Va (Va E E Lnt)(3!1)ltAd(a, Lnl)(3!f)ltAd(a, 1) f).. The The for formula mula RItAd(a, J, f, a a)) RltAd(a, :r a E E On O n 1\ A Ad( Ad(a) A Fun(J) Fun(f) 1\ A dom(J) dom(f) = a a 1\ A J(O) f(O) = a 1\ A :{:} a a) 1\ ("If, [Ad (J(f,)) 1\A (V( (V~ < < a) a)[Ad(f(~)) (V( < < f,)(J(() ~)(f(() E E J(f,) f(~) 1\ ("Ix E J(f,))(Ad(x) 1\ a E x --+ (3( < f,)(x == f(())))] J(())))] A (Vx E f(~))(Ad(x) A a E x -+ (3( < ~)(x expresses a-fold a-fold iteration to aa start-admissible is easy easy expresses iteration of of admissibility admissibility relative relative to start-admissible a a.. It It is to to show show that that A ut-KP1 r � ~- Ad( Ad(a) -+ (Va)(3!1)RltAd(a, (Va)(3!f)RItAd(a, J, f, a). Aut-KPlr a ) --+ a) . (131) (131) The uniqueness follows again again by (�o-FOUND). To To prove prove the existence of function The uniqueness follows by (A0-FOUND). the existence of the the function fJ we we choose axiom (Lim) two admissible admissible sets sets bb and u such such that By choose by by axiom (Lim) two and u that a, a, aa EE bb EE u.u. By := b b fq n On u. Then Then {3 i bb and and thus �o-separation relativized to u A0-separation relativized to u we we obtain obtain {3 ~ := On E e u. ~ @ thus g).. By By (A0-FOUND) we obtain {3 ir a. There There is is aa function function g9 such such that that ItAd(13 ItAd({3 ++ ~, (3, g) (�o-FOUND) we obtain 1 ) , ~(3 ri aa and and axiom axiom (Ad2) we 1)) . Because Because of of ~(3 EE g(~ g({3 ++ 1), (Ad 2) we (V~ < ("If, < ~(3 ++ ~)(~ (3)(f, eE g(~ g(f, ++ 1)). g(p) . We may now now define define that aa == g(p). get We may g({3 ++ 1). 1) . Therefore Therefore there there is is aa pp <::; ~(3 such such that get aa EE g(~ g(p ++ ~) f,) for for ~f, << aa << Z(3 and and easily easily check check RItAd(a, RltAd(a, f,J, a). a) . fJ(f,) (~) := g(p We will We will now now show show that that the the theories theories for for iterated iterated inductive inductive definitions definitions can can be be embedded embedded into into the the theories theories of of iterated iterated admissibility. admissibility. In In the the following following lemma lemma we we use use the the same same notational notational conventions conventions as as in in Section Section 3.2.3. 3.2.3. So So xx EE X XyY stands stands for for (x, (x, y) y) EE XX and l a t i o n rr we and for for aar erelation we write write xx EE X ry for for (3z)(z (3z)(z rr y A1\x x EE XZ). XZ) . The The capital capital letters letters serve serve only only to to improve improve readability readability and and to to emphasize emphasize the the close close connection connection to to Section Section 3.2.3. 3.2.3. Their Their meaning meaning is is that that of of ordinary ordinary variables variables for for sets. sets. xry 293 293 Set Theory Theory and and Second Second Order Number Number Theory Theory Set be an -positive .6.0 -formula, and w w w) w Let w w and A0-formula, rr C 3.3.5.2. Lemma. Lemma. Let A(S, A(S, T, T, x, x, y, y, ~Z) be an S-positive S � w xx w and 3.3.5.2. C1':4 (S, T, T, y, y, ~Z) :r (Vx E9 w)[A(S, w) [A(S, T, T, x, x, y, y, ~Z) ~-+ xx E9 S] Sl and :{:} (Yx define C]~(S, y, Z) IT% (r, X, Z) :4=~ :{:} rr C (Vy E C1% (XY, XrY, IT~(r,X,z-') -*) � w xx w A1\ (Vyew)CI~(XY, xry , y,z y r ^1\ (vy)(y c ~ -~ a% (y, z ~, y, ~ -~ (VY)(Y � -+ C1% (Y, x , y, Z) -+ z~ � y). Y). x y c_ define Then we obtain Then we obtain KPI-< ~-(3Z)lT~(-~,X,z-') � (:3X) IT%(-<, X, Z) KPI.~ (132) (132) KPI* KPI* ~� WOW(r) WOW (r) A 1\ r,Z r, z E9 La Lnl1 -+ (:3X) IT% (r, X, Z) -+ (3Z)lT~(r,X,z~ (133) A u t - K P 1 ~� WOW(r) Aut-KPI WOW (r) -~ -+ (3X)IT~(r,X,Z). (:3X) IT%(r, X, Z) . (134) (134) and and The proof proof of of all all three three claims claims is is essentially essentially the the same. same. We We sketch sketch the the most most complicated complicated The case of of (134). Assume WOW(r) and chose chose by by (Lim) an admissible a such such that that case (134). Assume WOW (r) and Lim an admissible a and we by (129) (129) otyp~ a. Hence Hence also r , ~z E9 a. we get get by also r, a. Then Then field(r) field(r) N n w E9 aa and otyPr E9 a. Lim c~ otyp(r) E9 a. By (131) (131) there such that a).. By By ((Lim) a ":= otyp(r) a. By there is is function function f such that R/tAd(a, RltAd(a, f, a) we obtain an admissible admissible set that a, a , f,, aa E9 u ( ) w f f {x E w l A(U{hy(r A( {hy (() I «r < n),O,x,y, {~e~,l zD} and and 1J} , 0, x, y, Z)} U h hy(rl) y ( 1J) = - f, � f(�), ) dom(h dom(hy)= O n nnaa y ) = On if if y y i. r fie field (r) ld (r) (i) i A( {hy (() I r« < v),U{g(r Pr ( Y) ) , x, y, Z)} ({x = eE~wl l A(U{h~(r otyp~(y)},x,y,~} 1J }, { g (() 1 «r < oty rng (hy ) and and dom(hy) = : On On n NU Uotypr(u) and g(otyP g(otyp~(y)) : Urng(hy) OtyPr(y) and r (Y)) = if Yy E9 fie field (r). if ld (r). U dom(hy ) U U () Put Put s :u) 1l yu Ee fie field(r) g(otyp~(y)) S := {((~, (x, y) ld(r) 1\A xx E9 g(otYP r ( Y) ) }} u U{{ (x, (x, y) Y) lI yY E9 w w 1\ AY y i. ~ field(r) field(r) 1\A xx E9 Urng(hy)}. rng (hy ) } . U By By construction construction we we have have S, S, h hy, (hy, Y)Il yY E9 w w}} E9 u u.. We We show show y , y) y , gg,, {{ (h ry , y, y, Z) C1%(SY, c/x (s~, s s~, z-) ((ii)ii) and and y , y, ((vx)[x (iii) g ~ -+ -~ C1%(X, a%(x, s s r~, y, Z) ~ -+ -~ s s~y � g X x].l. iii VX) [X � ry = First 0. The �-definable in First assume assume yy i. ~ fie field(r). Then s Sry = 0. The function function h hyy is is E-definable in the the ld (r) . Then y admissible admissible set set a. a. We We have have s S~ = - {{xx lI (:3( (3~ E E a)(x a)(x EE hhy((~))}. Thus s S yy corresponds corresponds y (() ) } . Thus to ii as ii and to the the class class IIA(~ of Section Section 3.3.1 3.3.1 and and we we show show (ii) and (ii) as in in the the proof proof of of A (Z) of w ( ) () Theorem 3.3.1.8. Theorem 3.3.1.8. Now Now assume assume yy E E fie field(r). Let �~ := : - otY otypr(y). We show show ld(r) . Let Pr (Y) . We g r~ e u~ () iv) ((iv) 294 294 Pohlers W. Pohlers �. f( by by induction induction on on ~. From From the the induction induction hypothesis hypothesis we we obtain obtain g9 r( E ur u, E u~ ue for for r The The function function ~ >> ( ~H g9 r( f( isis thus thus E-definable �-definable in in u~ u e and all ( << ~. and because because of of all by E-replacement �-replacement relativized relativized to to u~. u e . This This we get get {{ ((, ( g9 I()l I (« < ~} 0 eE u~ ue by eE aa E u~ ue we proves (iv) (iv) in in case case that that ~� E kim. Lim. IfIf ~� == ( ++ 11 then then ( -= otypr(z) for for some some zz E field(r) field (r) proves and dom(hz) -= ur u, as as well well as as gI( are are elements elements of of u~. u e . By By E-recursion �-recursion relativized relativized to to and we obtain obtain hz eE u~. u e . Therefore This proves proves Therefore gI~ = = gI( U U {((,Urng(hz))} e E u~. ue . This u~ ue we and S y -= [.J rng(hy). For For (iv). we obtain obtain S ~ eE u~ ue and (iv) . By By (iv) (iv) and and the the definition definition of of SS we we have have hy(~?) = u e we = {x e E w Ii A((J~<,hy((),SrY, x , y , ~ } which which shows shows that that hy isis r] eE u~ ue by by E-Recursion. �-Recursion. Therefore Therefore the the formula formula A(SY, S ~y, x, y, z-') is is still still E� definable in in u~ definable by E-reflection �-reflection relativized relativized to to u~. ue . and we we obtain obtain (3r/eE u~)A([.Jr hy((), S ~u, x, y, ~ by and Hence x ~E S ~. This This proves (ii). To prove also also (iii) (iii) we we show show Hence proves (ii). To prove � ( � ( (, f() g f( ( ( otyPr (z) hz gf� gf( {((, U rng(hz))} sy U rng(hy ). sry hy (Tf) {x A(U,<'1 hy ((), sry , x, y, z)} h A(SY, sry , x, y, Z) y (3Tf ue)A(U,<'1 hy ((), sry , x, y, Z) x sy . dom(hz) Tf (v) Cl~(X,S~Y,y,z-') -+ hy(~) C_ X U < h (() A(U,q hy ((), sry , x, y, Z) A(X, 'sr'1y , x,y y, Z) . sry , y, Z) hy (Tf) by induction induction on on r/Tf E u~. u{ . From From the the induction hypothesis we we have have [.Jr hy(() C_ by induction hypothesis � X which implies by Z-positivity X which implies by X -positivity A([.Jr hy((), Sry, x, y, ~ ~-+ A(X, Sry, x, y, ~ . Together with with the the hypothesis eft (X, S ~y, y, ~ this this yields 0 X. Together hypothesis C~(X, yields hy(~) � C_ X. El The following following theorem is aa consequence consequence of of Lemma Lemma 3.3.5.2. The theorem is 3.3.5.2. 3.3.5.3. Theorem. 3.3.5.3. Theorem. (i) (i) (ii) 5i) (iii) (iv) (v) (vi) (vii) (II�-CAv ) o == (ID~)o ( ID�) o c_ � KPlr, KPI� (rt~-CA~)o (II� -CAv ) == ID~ ID� c_ � W-KPI~ W-KPlv (H~-CA~) IDv (Bi) = BID~ BID� � ID~ � C_ ((II H 1�-CAv - C A ~ )) ++ (Bi)= C_ KPlv gPl~ ((Aut-H~)o Aut-lID o = Aut-ID ) o � = ((Aut-ID)o C_ Aut-KPIT Aut-KPF (Aut-II (Aut-YI1�) ~) = = Aut-ID Aut-ID � C_ W-Aut-KPI W-Aut-KPI ((Aut-H~) Aut-lID + + (Bi) (Bi) = = Aut-BID Aut-BID � C_ Aut-KPI Aut-gP1 ID-<* ID.~. � C_ BID· BID* � C_ KPI· KPl* Proof. Proof. Claims Claims (i), (i), (ii), (ii), (iv), (iv), (v) (v) and and (vii) (vii) follow follow immediately immediately from from Lemma Lemma 3.3.5.2 and and (iii) and Theorem Theorem 3.2.4.2. Claims Claims (iii) and (vi) (vi) follow follow from from Lemma Lemma 3.3.5.2, Theorems Theorems 3.2.4.2 0 and and 3.2.3.2 and and Corollary Corollary 3.3.3.8. 3.3.3.8. D The The opposite opposite inclusions inclusions in in claims claims (iv), (iv), (v) (v) and and (vi) (vi) are are also also true. true. In In (i) (i),, (ii) (ii) and and (iii) (iii) this this can can of of course course only only be be true true for for limit limit ordinals ordinals /Ju.. And And this this is is the the case. case. 3.4. 3.4. Ordinal O r d i n a l analysis analysis for for set-theoretic set-theoretic axioms axioms systems systems 3.4.1. 3.4.1. Ramified Ramified set set theory theory Ramified Set Theory. We We introduce introduce the the language language .eRS l:as and and of of Ramified Set Theory. The The basic basic symbols symbols are are E E,, � ~,, Ad, Ad, ,Ad ~Ad and and aa constant constant La [~ for for every every ordinal ordinal a a.. 295 295 Set Theory and Second Order Number Theory 3.4.1.1. Inductive definition 3.4.1.1. Definition. Definition. ((Inductive definition of of the the set set terms terms of of CRS Z:as)) Every Every constant constant In is is an an atomic set term of of stage stage 0:: c~.. If If a l , . . . , an are are set set terms terms of of stages stages < < 0:: c~ and and F(x, x l , . . . ,xn) is is an an sC formula formula without without further further free free variables variables then then Let atomic set term F(x, Xl, . . . , xn ) {xeL, {x E Let I F(x, al, . . . , an)L,,}} aI, . . . , an composed set term s (s) . is is aa composed set term of of stage stage 0:: ~.. We We denote denote the the stage stage of of aa set set term term s by by stg stg (s). There ( E, Ad) There are are only only sentences sentences in in CRS Z:as.. An An CRS s -sentence is is obtained obtained from from an an C Z:(E, Ad)'-style by formula in 'Tait formula ((in 'Tait'-style) by replacing replacing all all free free variables variables by by CRS-terms Z:as-terms and and restricting restricting all to CRS-terms. all unbounded unbounded quantifiers quantifiers to Z:as-terms. To To have have aa uniform uniform notation notation we we refer refer to to CRS -terms and -sentences as Z:as-terms and-sentences as CRS Z:as-expressions. Let Let ) -expressions. {t l (t) o:: } . ((135) 135 ) To g < T~ := {t I st stg(t) < c~}. We but ne We do do not not count count the the equality equality symbol symbol among among the the basic basic symbols symbols of of CRS s but defi define s t = (\Ix E s)[x E t] /\ (\Ix E t)[s E s]. :¢:} We hierarchy to We transfer transfer the the Levy Levy hierarchy to the the language language CRS Z:as.' 3.4.1.2. 3.4.1.2. Definition. Definition. Let Let :F jc be be aa complexity complexity class class in in the the Levy Levy hierarchy. hierarchy. We We call call an an CRS-sentence Z:as-sentence F an an :Fet-sentence ~'~-sentence if if there there is is aa :F-formula ~'-formula G(x) G(Z) in in the the language language C Z: which only the which has has only the shown shown free free variables variables and and aa tuple tuple a ~ of o f /CRS-terms : a s - t e r m s of of stages stages less less than than 0:: c~ such such that that F == - G(a)L" G(g) L,.. it -sentence, we If � is denote by obtained by If F F LL~ is a a :F ~'~-sentence, we denote by F z the the sentence sentence which which is is obtained by replacing all quantifier replacing all quantifier restrictions restrictions E / ~it by by E z. F F EL FZ E The is The standard standard interpretation interpretation of of CRS s is given given by by La LetLaLL = Let {X Ee LetI an)L,,}L} r,, {x{x Ee Letl c,, I F(x, F(x, al, al,..., Lo, I Let Lo, P ~ F(x, F(x, at, a~,..., a n ) ). . . . , an)L" · · · , a�)}. It It is is obvious obvious that that for for every every set set term term ss of of stage stage 0:: a we we have have sL s ~" E E Let L~+t. We have have + !. We LL ~ps EsLE, Let r LL ~Ps -st t for term tt with with stg 0:: , (136) for some some set set term stg (t) (t) < = __ ¢:} ¢:} and and = = LL p~ Ss Ee {X{x eE LetL~]I F(x)} F(x) } t L ¢:} r LL p~ tt = ss /\A F( t) F(t) (138) (138) = for for some some set set term term t with with stg(t) stg(t) < < 0:: ~.. This This motivates motivates the the following following definition. definition. s r, (t) s � t, (t), 3.4.1.3. CRS-sentences of d , AA VV B 3.4.1.3. Definition. Definition. We We say say that that/:as-sentences of the the shape shape s E E r, A Ad(t), B and Dually sentences Ad A and (3x E r)G(x) have have V V - -type. t y p e . Dually sentences of of the the shape shape s ~ t, ..., -~Ad(t), A /\ AB B and G are -type. and (Vx E r)G(x) are said said to to have have I\ A-type. We We put put (3x E r)G(x) (\Ix E r) (x) W. W. Pohlers Pohlers 296 if (t) << a} stg (t) a} if rr = = La La CC(s(s Ee r)r) .:== { {t{t{t{t ==== sss~11I\nstg F(t) 1l stg stg (t (t)) < < a} a} ifr if r = = {X {x E E lkol F(x)}, F(t) a I F(x)}, C(Ad(t)) = ttll a Ee Reg 6 (Ad(t)) := := {l { k,~l< = R e g 1\ A K; � _< stg stg (t) (t) }, }, C( A vv B) C(A B):= := {{A,B} A, B } K, K, and and if {G(t) {a(t) 1l stg stg (t) (t) < < a} a} if rr = = La L~ (( 3x E r)G(X)) :=:= { {F(t) CC((=txEr)G(x)) { F ( t ) A1\ a(t)l a} ifr i f r == {x { x eEkl~al IFF(x)}. (x)}. G(t) 1 ststg(t)(t) << a} This -type. This defines defines the the characteristic characteristic sub-sentences-set sub-sentences-set C 6 (F) (F) for for all all sentences sentences F F of of V V-type. g Dually Dually we we define define Cc (F) ...,G I G (F) := := {{-~al a E e C c (..., ( - FF)) } for 1\-type. for sentences sentences F F of of A-type. If (F) isis ofof the (t) If F F is is not not aa conjunction conjunction or or disjunction disjunction then then every every G G E EC C(F) the form form H H(t) for = tt and OF(G) :=:-- stg(t) for some some characteristic characteristic set set term term t. t. We We define define tF(G) rE(G) ::= and oF(G) stg(t).. For For FF == Ao I\} and and G := liki and and OF(G) := i/ i iff A0 o AI A1,, o E E {V, {V,A} G E E C(F) C(F) we we put put tF(G) tF(G):= oF(G):= GG == AAii for i E {a, I}. for i E {0,1}. 0 0 The following following lemma lemma is is an an immediate immediate consequence consequence of of Definition Definition 3.4.1.3 3.4.1.3 and and its its The preceding preceding remarks. remarks. 3.4.1.4. 3.4.1.4. Lemma. Lemma. F {::} ~ F {::} r162L ~F L L ~FF and and dually dually L ~FF For -type we For every every sentence sentence F F of of V V-type we have have L L ~F V V F Ga GG EE CC(F) () for sentences ofof 1\ -type. for sentences A-type. 1\ A G G GGEE CC(F) (F) Lemma 3.4.1.4 Lemma 3.4.1.4 is is the the basis basis for for the the following following infinitary infinitary calculus. calculus. 3.4.1.5. 3.4.1.5. Definition. Definition. We We define define the the relation relation p ~ � A for for finite finite sets sets of of CRS-sentences s �A inductively inductively by by the the following following two two clauses: clauses: -type and o< for V-,,,,~ o,,,~ � ~ " ,� ~ , G as well as a ~,o ~,a , and a,,d OF(G) o,~(~) < ~ , ,a hold ,,o,dSo,(( VV) ) If,s~F is~ ofos V some GG EE C(F) 6(F) then then p ~ � A ,,FF o~,,,~,,,,~ some and and for all then (1\) (A) IfF isF is ~ of os 1\ A ,-type - ~ and o~ � ~ � ", , G a as o~ well ~ , , as o~ a ~G < < a ,, hold ~o~so~ o. G a~E C(F) c(~),,,o,, p~ A�, F,.F . Set Theory and Second Order Number Theory 297 297 F From .5 we From Definition Definition 3.4.1 3.4.1.5 we get get for for eR L:Rs-sentences s -sentences F F (3a) � F and ordinal -y we and for for aa � El-sentence F and and an an ordinal we have have l -sentence F �~ FFL�L~ =~ LL~a F~ F.F. L FF L~ {::} r (3a)~F 'Y :::} (139) (139) (140 (140)) If If we we put put min {a F F } if L F F := { tc(F to(F)) := { minoo{aIl ~ F } otherwiseifL ~ F otherwise for eRs -sentences F for/:as-sentences F we we obtain obtain from from (140) (140) IFI~,l :::;< tc(FL�) tc(F L~) IFIL for ordinals 7. for all all � El-sentences F and and all all ordinals l -sentences F oo 'Y . (141 (141)) (142)) (142 We are are going going to to define define aa rank rank function function for for eR L:Rs-expressions in such such aa way way that that all all We s -expressions in sentences characteristic sub-sentences sentences in in the the characteristic sub-sentences set set of of aa sentence sentence F get get lower lower rank. rank. 3.4.1.6. 3.4.1.6. Definition. Definition. clauses: clauses: F For For an an eR L:Rs-expression E we we define define rk(E) rk(E) by by the the following following s -expression E rk(L~) rk(L a ) ::=- w.. ao~ max{rk(La ) + 1,I, rk(F(Lo)) rk( {x E LL~Ia l F(x)}) + 2} rk({x F(x)} ) := max{rk(L~)+ rk(F(L0))+ 2} rk(Ad(t)) := rk(-, rk(-~Ad(t)):= rk(t)++ 55 Ad(t)) := rk(t) rk(Ad(t)) rk(s + 6, rk(t) rk(s E e tt)) : = rk(s rk(s (j. ~ t) t ) ::= = max{rk(s) max{rk(s)+ rk(t) + + I} I} rk(A rk(A V B) B ) ::= = rk(A rk(A 1\ A B) B ) ::= = max{rk(A), max{rk(A), rk(B)} rk(B)} + + 11 rk((3x E s)F(x)) s ) F ( x ) ) : = rk(( rk((Vx s ) Y ( x ) ) ::= = max{rk(s), max{rk(s), rk(F(Lo)) rk(F(L0)) + + 2} 2} rk((3x V'x E s)F(x)) w := := := The The crucial crucial property property of of the the rank rank function function is is stated stated in in the the following following theorem. theorem. 3.4.1.7. 3.4.1.7. Theorem. Theorem. For For G G E C(F) C(F) we we have have rk(G) rk(G) < rk(F) rk(F).. a, b The will not The proof proof of of the the theorem theorem is is aa bit bit lengthy lengthy and and we we will not give give all all details. details. Let Let a, b and be eRS -expressions. First and c be/:Rs-expressions. First we we show show c stg(a) << stg(b(Lo)) :::}=~ rk(b(a)) rk(b(a)) < < rk(b(Lo)) by by an an easy easy induction induction on on rk(b(Lo)). rk(b(Lo)). The The next next step step is is to to show show stg(c) { ·.a,a, rk(b(Lo)) + I} stg(c) < a a =~ rk(b(c)) rk(b(c)) < < max max{w rk(b(Lo))+ 1} by ii) we by induction induction on on rk(b(Lo)). rk(b(Lo)). From From ((ii) we get get easily easily stg(c) rk(s E {x stg(c) < < a a :::} =~ rk(F(c)) rk(F(c)) + + 11 < < rk(s {x E LLa F(x)}). a lI F(x)}). :::} w E Now Now we we compute compute ((i)i ) ((ii) ii) ((iii) iii) 298 298 W. Pohlers 4. rk(a = b) b) = max{rk(a), max{rk(a), rk(b)} rk(b)} + + 4. rk(a = = (iv) (iv) Finally Finally we we show show GG E CC (F) (F) '* =~ rk(G) rk(G) < < rk(F) rk(F) (v) (v) F. by by distinguishing distinguishing cases cases on on the the shape shape of of F. We We only only consider consider sentences sentences of of V V - -type. type. The The case case of of 1\ A --type t y p e is is dual. dual. If If F == Ad(t) then then G == - (t = = L~) for for some some ~ E Reg R e g Nns t g ( t ) + + l . 1 . Hence Hence rk(G) = = max{rk(t), ~} + + 44 < < rk(t) + + 55 = = rk(F). a. But If F === (s E La) then then G - == ( t ==s ) for for some some t such such that that stg(t) < < c~. But then then If rk(G) = = max{rk(t), rk(s)} + + 44 < < max{w 9a c~ + + 1, + 6} = = rk(F). 1, rk(s) + If If F == =_ (s E e { x eEk ~ l H ( x ) } ) then then G == = (s = = t /\ A H ( t ) ) for for some some t such a. Hence rk(G) = such that that stg(t) < < a. Hence = max{rk(s = = t),rk(H(t))} + +11 = = max{rk(s) + + 5, But rk(s) + + 55 < < rk(s) + + 6 _ rk(f), 5, r k ( t ) ++ 5, 5, r k ( H ( t ) ) ++ 1}. But also r k ( g ( t ) ) ++ 11 < rk(t) + + 55 < < w . aa :::; _ rk(F) and and by by (iii) (iii) also < rk(F). The The claim claim is is obvious obvious for for F == - (A (A V Y B). B) . If a. Then If F == = (3z E e L~)H(x) then then G == = H(t) for for some some t with with stg(t) < < a. Then by by (ii) (ii) a, rk(H(Lo))++ i} _< max{w .o~, a, rk(H(Lo))++ 2} = rk(G) < < max{w .oL, = rk(r). If If F == =_ ( 3 x eE {yEE k~[H(y)})K(x) then then G == = H ( t ) A/\ K(t) for for some some t such such that (ii) rk(G) = that stg(t) < < a a.. Then Then by by (ii) = max{rk(g(t)),rk(g(t))} + + 11 < max{w. a + 2, rk(K(Lo))+ + 2} = + 2} = a,, rk(H(Lo))+ = max{rk({x EEEL~,IH(x)}), rk(K(Lo))+ = rk(F). F Ad(t) G (t Lit) K, stg(t) rk(G) max{rk(t), K,} rk(t) rk(F). t stg(t) F (s La ) G (t s) rk(G) max{rk(t), rk(s)} max{w · rk(s) 6} rk(F). F (s {x La l H(x)}) G (s t H(t)) t stg(t) rk(G) max{rk(s t), rk(H(t))} rk(s) rk(s) 6 :::; rk(F), max{rk(s) rk(t) rk(H(t)) I}. rk(H(t)) rk(F). rk(t) w · rk(F) F F (3x La)H(x) G H(t) t stg(t) rk(G) max{w · rk(H(Lo)) I} :::; max{w · rk(H(Lo)) 2} rk(F). F (3x {y La l H(y)})K(x) G H(t) K(t) t stg(t) rk(G) max{rk(H(t)), rk(K(t))} max{w · rk(H(Lo)) 2, rk(K(Lo)) 2} max{rk( {x La l H(x)}), rk(K(Lo)) 2} rk(F). Because Because of of rk(G) rk(C) << rk(F) rk(F) for for all all G a e C(F) C (F) we we get get by by induction induction on on rk(F) rk(F) rk(F) F F.. 143) LL ~FF F ~ IIrk(F) ((143) o E '* Hence Hence tC(F) :::; rk(F) for all LRS-sentences F.F. (144) (144) tc(F) < rk(F) for all s 3.4.2. 3.4.2. A A semi-formal s e m i - f o r m a l calculus calculus for for ramified ramified set set theory theory It It follows follows by by (139) (139) that that the the rules rules (cut) (cut) and and ~ � A , ,AA and and � ~ � A , ,~--,A A '* =~ � ~ � A p FL. (3z LIt)[z PZ] for K, FL. -sentence (Ref) (Ref) p ~ � A,, F L~ '* =~ � ~ � A,, (3z E e L~)[z =I ~: 00 /\ A f z] for ~ E e Reg, Reg, F L~ a II� II~-sentence are o. However, are admissible admissible for for some some ordinal ordinal 6. However, we we do do not not yet yet know know how how to to compute compute 06 from from a c~ or or a c~ and and/~f3 respectively. respectively. Therefore Therefore we we design design aa semi-formal semi-formal system system having having these these rules rules as as basic basic inferences. inferences. 3.4.2.1. and 3.4.2.1. Definition. Definition. Let Let � A be be aa finite finite set set of of LRS-sentences s and a c~ and and p ordinals. ordinals. We We define define the the relation relation � ~ � A by by the the following following clauses: clauses: Set Theory Theory and Second Order Number Number Theory 299 299 IfF -type, � If F E E� An n 1\ A-type, ~ �, A, G G and and ac ac < ( A )) (/\ Lrp� � . IfK, FLK (3z LIt)[z =I- FZ] FLK and a a then then � ~ �. A. We (3z EE/~)[z LIt)[z =I---~ 00 1\A PZ] We call call F F the the main main part part in in instances instances of of ((1\) A ) and and ((V) V ) and and (3z F z] (Ref II2, (~z Ee L~)[z # 00 1\ (Rely) l e g , F L~ E e n;. ^ F "] E e ~�,, fT ~ �, A . F L~ a - d ~K". ~ao+ o + l1 < < it ) IS~ Ee Reg, the the main main part part in in an an instance instance of of (Ref (Rely). it ) . If A, -.A for some and some A such that rk(A) then (cut) If fT ~po �, A, A, fT ~pO�, A,-~A for some ao ao < a a and some A such that rk(A) < pp then (cut) � �. We if ordinals a We say say that that � A is is semi-formal semi-formal derivable derivable in in .cRS s if there there are are ordinals a and and p p such such that that � ~ �. A. As As an an immediate immediate consequence consequence of of the the soundness soundness of of all all rules rules we we get get the the soundness soundness of calculus, i.e., of the the semi-formal semi-formal calculus, i.e., A => L ~ V (145) (145) A. Since Since � ~ � A is is aa correct correct calculus calculus which which derives derives sentences sentences we we get get cut cut elimination elimination nearly nearly for for free. free. We We prove: prove: IfIf �~ A,�, rr and and L L � ~ F F for forall all F F E E rr then then � ~ � A (146) (146) by by induction induction on on a. a. The The claim claim is is immediate immediate from from the the induction induction hypothesis hypothesis if if the the belong to last last inference inference is is according according to to ((1\) A ) or or ((V) V ) and and its its main main part part does does not not belong to F. If If the the main main part part is is in in r F we we have have in in the the case case of of an an inference inference according according to to (V) ( V ) an an F E EFn MV V --type t y p e and and the the premise premise fT ~2_ �, A, r, F, G G for for some some G GE E C (F). Then Then L L � ~ G G and and induction hypothesis. hypothesis. In according to �A follows follows by by induction In the the case case of of an an inference inference according to (1\) (A) there therefore aa G there is is an an F E EFn M I\ A --type t y p e and and therefore GE E C (F) such such that that L L IF ~ G. G. But But then then there there is is aa premise premise � ~ �, A, F, G G with with ac c~c < < a c~ and and we we obtain obtain ~ � A by by the the induction induction hypothesis. hypothesis. If If the the last last inference inference is is aa cut cut with with cut-sentence cut-sentence F then then either either L L � ~ F or claim by or L L � ~ ~ F . We We pick pick the the corresponding corresponding premise premise and and obtain obtain the the claim by induction induction hypothesis. hypothesis. If If the the last last inference inference is is according according to to (Ref (Ref~) we distinguish distinguish the the cases cases that that it ) we for for its its main main part part we we have have L L F ~ (3z E E L~)F z or or not. not. In In the the second second case case we we also also have have L the claim the induction the first first case L � ~ F [~ and and obtain obtain the claim by by the induction hypothesis. hypothesis. In In the case we we have (3z E E L~)[z ~: 00 1\ A F z] E E� A and and get get ~ � A from from rk((3z E E L~)F z) = = a < < a a and and have (143). 0 (143). [:] Summing up Summing up and and taking taking r F as as the the empty empty set set in in (146) (146) we we get get the the following following theorem. theorem. r. F r � C(F) . F r r, -.F. FLK (3z LIt)[z =I- (3z LIt)FZ PZ] � C(F) � F F rk((3z LIt)FZ) K, 3.4.2.2. 3.4.2.2. Theorem. T h e o r e m . The The semi-formal semi-formal calculus calculus is is sound sound and and allows allows cut cut elimination. elimination. Especially Especially we we get get � ~ � A from from � ~ �. A. 300 300 W. Pohlers Pohlers w. 3.4.3. Operator-controlled O p e r a t o r - c o n t r o l l e d derivations derivations 3.4.3. It follows follows from from Theorem Theorem 3.4.2.2 3.4.2.2 that that cut-elimination cut-elimination alone alone cannot cannot be be crucial crucial for for It the ordinal ordinal analysis analysis of of theories theories Ax for for which which we we have have the A x ef- Wf(-<) Wf(-~) Ax r {:} Ax AxfA x ~ (3� (3~ E e LL~,xc~)H.~(~ wcK )H� (�),), 1 i.e. for for theories theories in in which which the the well-foundedness well-foundedness of of aa �o-definable A0-definable ordering ordering can can be be i.e. wOK C K expressed by by an an I;�' E11 -sentence. -sentence. This This is is true true for for all all theories theories comprising comprising KPw K P w . . The The expressed main problem problem there there is is to to collapse collapse the the ordinals ordinals which which arise arise canonically canonically in in the the embedembedmain wOK CK -sentences into ordinals below co~KK.. Collapsing ding procedure procedure of of I;�' lC1x -sentences is therefore therefore ding into ordinals below Wf Collapsing is the Leitmotiv Leitmotiv of of Impredicative Impredicative Proof-Theory Proof-Theory (but (but we we will will see see that that cut-elimination cut-elimination the will will be be needed needed for for collapsing) collapsing).. We We will will use use the the technique technique of of local p r e d i c a t i v i t y , first first introduced in in Pohlers Pohlers [1982a,1982b] [1982a,1982b], Buchholz Buchholz et et al. al. [1981] [1981], but but we we are are going going to to introduced ' ' use an an essential essential simplification simplification of of the the original original technique technique which which has has been been introduced introduced use by Buchholz Buchholz [1992] [1992].. We We already already used used collapsing collapsing techniques techniques in in the the rrg-analyses YI~ of by of Sections 2.1.4 2.1.4 and and 2.1.5. 2.1.5. We We will, will, however, however, not not give give rrg-analyses II~ for impredicative impredicative Sections for theories. Already for nt-analyses II~-analyses the the matter matter is is sufficiently sufficiently complicated. complicated. We We just just beg theories. Already for beg the reader to to believe believe that the refinement to rrg-analyses II~ can be by modifying modifying the reader that the refinement to can be done done by the techniques techniques of of Section Section 2.1.5 2.1.5 (cf. (cf. Blankertz Blankertz and and Weiermann Weiermann [1996], [1996], Blankertz Blankertz [1997] [1997] the for more more details). details). Another way to to extend extend the the following following analyses analyses to to rrg-analyses ri~ is to to for Another way is apply Section 2.2.2. 2.2.2. apply Section Our presentation presentation will will follow follow quite quite closely closely that that of of Buchholz Buchholz in in Buchholz Buchholz [1992] [1992].. Our ' s presen (Those (Those who who have have tried tried know know that that it it is is hardly hardly possible possible to to improve improve Buchholz Buchholz's presentations.) tations.) local predicativity, 3.4.3.1. Definition. 3.4.3.1. Definition. An An (ordinal-)operator ( o r d i n a l - ) o p e r a t o r is is a a map map 7/" Pow(On) Pow(On).. 1i: Pow(On) -+~ Pow(On) We introduce We introduce the the abbreviations abbreviations :{:} a a E 1i(0) :r MC_7-/(0) M � 1i(0) (147) (147) : ~ 7/(0) 1i(0) �C_MM :,~ ((vx)[n(x) 1i'(X)] 'v'X )[1i(X) �c n'(x)] and Onn if Onnn -+> O and call call an an operator operator 7/closed 1i closed under under aa function function f" f: On if { 6 , · · · ' �n } C_� 7/(X)]. 1i(X)]. (VX 1i(X) r {r162 ('IX eE Pow(On))(Vr162 Pow(On))('v'6 ) · · · ('v'�n )[J(6 , . . . ' �n ) eE 7/(X) o . . . ##ww'~"n we In case case that that 7t 1i isis closed closed under under f! ((al we call call it In a l , , .. .. .. ,, aann)) := w ~1 ## "'" it CCantorian antorian:= wetl closed. closed. A A set set M and an an operator operator 7t 1i induce induce aa new new operator operator ~1i[M] by by M C_� On On and 1i[M](X) := 7t(M 7t[M](X) (148) 1i(M uU X). X). E 1i MC_7r � 1i M 7/C_M 1i � M n1i c� 1i' n' :{:} :{:} :{:} {:} let For an an sCRS -expression EE let For 301 301 Set Theory and Second Order Order Number Theory occurs in (149) (149) par (E) ::= par(E) - - {a {~1I La L~ o ~ u ~ i~ E E }} . 1l If 8 {3 is is aa set set of of £R s s -expressions and and 7i an an operator operator we we define define 1l ~ [8] := := 1l "//[par(O)]. If [par(8)] . 1l acceptable 3.4.3.2. Definition. Definition. An operator operator ~ is is acceptable if if it it satisfies satisfies the the following following An 3.4.3.2. conditions: conditions: o E~ n(O) 7/isis Cantorian-closed Cantorian-closed (150) (VX E E Pow(On) Pow(On))[X C_ 1l ~ (X)] (150) (' ~/(X) � c_ 1l(Y)]. n(y)]. (' 1l 1l(0) 1l(Y) 1l(X) 1l: 3.4.3.3. Definition. Definition. Let Let 7r Pow(On) Pow(On) --t ---4 Pow(On) Pow(On)be be an an operator. operator. For For aa finite finite 3.4.3.3. set Do A of of £R s s -sentences we we define define 7/ � ~ Do A iff iff par(Do) par(A)t0 {a} � C_ 7t and and one one of of the the set U {a} following conditions conditions is is satisfied: satisfied: following 1l 1l There There is is aa sentence sentence F F E E Don AM /\ A - t-type y p e such such that that 1l[tF(G)] 7-l[tF(G)] � ~ - Do, A, G G and and aa ar < a a (F) . for all all G GE EC d(F). for -type such Do, G (V) There There is is aa sentence sentence F F E E Don AM V V-type such that that 1l 7-I � ~ - A, G and and OF(G), oF(G), aa av < a a (V) for some some G G E EC d(F). for (F) . (Ref~) There There is is aa II2 H~-sentence such that that (3zEl~)[z r 0 1\ A FF'] ~] E E Do A,, (Ref,,) -sentence FFL[~K such (3z E L,,) [z =I~po Do A,, FL F [~K and and a, ao ao + + 11 < a a.. 1l7-l � (cut) There There is is aa sentence sentence A A such such that that rk(A) 7-I � ~po Do A,, A A and and 1l 7-I � ~po Do A,-~A for rk(A) < p, 1l , --.A for (cut) some aoao < a.~. some We say say that that Do controlled derivable derivable ifif there an acceptable We A is is operator operator controlled there are are an acceptable operator operator 1l 7t and and ordinals ordinals aa and and p such such that that 7t 1l ~� A. Do . (A) (/\) K" p, p From now we will will only only regard acceptable operators operators without mentioning it From now on on we regard acceptable without mentioning it explicitly. explicitly. The following of operator-controlled operator-controlled derivations derivations are immediate conse conseThe following properties properties of are immediate quences of of Definition 3.4.3.3. quences Definition 3.4.3.3. If 1l � 1l 1l', 1l', f , aa _< � F, � 7~', A /f 7t ~ Do A,, 7/C_ _< a a and and par(F) C_ 7~' f3 E E 7/', pp s:: par(f) � D. C_ s:: 13 1l' (151) (151) 1l' then f. then 7t' � ~ F. 1l then 1l � If 1l � and If 7/1l ~� A,Do, (Vx (' proved by on a. proved by induction induction on a. The The predicative predicative cut-elimination cut--elimination procedure procedure works works also also for for operator operator controlled controlled derivations. derivations. First First we we prove prove 3302 02 W. Pohlers 3.4.3.4. Lemma. 3.4.3.4. Inversion Inversion L emma. IfIf 1£7-l �~ �, A ,F F and and 1£[t 7t[tF(a)] A, G G for for all all G G E e C(F) C (F).. F (G) ] �~ �, F F E E 1\ -type then then A -type The The proof proof is is aa straightforward straightforward induction induction on on a. c~. IfIf FF E V -type, rk(F) V-type, rk(F) - 3.4.3.5. 3.4.3.5. Reduction R e d u c t i o n Lemma. Lemma. asas well well asas 1£7/ 1 ,8 then = E 1"+,8 A�, rF . rp r, F ~ F F then 1£ 7/ , pP ~ and pp �r Reg R e g and 1£ 7/ � ~ �, A, --' ~F F 9 The Lemma 2.1.5.7) The proof proof is is that that of of the the Reduction Reduction Lemma Lemma ((Lemma 2.1.5.7) of of Section Section 2.1.5. 2.1.5. Since Since we we restrict restrict ourselves ourselves to to IT II~-ordinal analysis we we consider consider finite finite ordinals ordinals as as trivial. trivial. t -ordinal analysis Therefore don't have Therefore we we don't have to to compose compose the the controlling controlling operators. operators. This This makes makes the the proof proof simpler. simpler. The The hypothesis hypothesis p p� ~ Reg R e g is is needed needed to to exclude exclude the the case case that that F F is is the the main main part Ref) . part of of an an inference inference according according to to ((Ref). As in in Section 2.1.2 we we obtain obtain the the Predicative Predicative Elimination Elimination Lemma Lemma as as aa straight straightAs Section 2.1.2 forward forward consequence consequence of of the the Reduction Reduction Lemma. Lemma. Let be an operator which is closed and 3.4.3.6. Predicative P r e d i c a t i v e Elimination E l i m i n a t i o n Lemma. L e m m a . Let 1£ 7-l be an operator which is closed 3.4.3.6. £. p �, [13, 13 + under the function a, ~, 13 t3 � ~ 'P"f3 qo~t3 ,, 1£ 7-l 1I~+~p A, [13, t3 + w Pp)) n fq Reg Reg = -- 00 and pp E E 174. ;+w under the function Then Then 1£ 7-l rp~i~ � A. ICPP" The Elimination Lemma Lemma 2.1 .2.9) with The proof proof is is that that of of the the Predicative Predicative Elimination Lemma ((Lemma 2.1.2.9) with some some extra controlling operator. operator. As extra care care on on the the controlling As an an example example we we treat treat the the case case of of aa cut. cut. There There we we have have the the premises premises 17t£ 1I~+w A, F F ;:wpp �, o~o ~o and £ 1I~+~p and 17i A, --'F. ~F. ;: wp �, ((i)i ) Using Using the the induction induction hypothesis hypothesis we we obtain obtain P"o �, 7/ I[Z~2_ A F F and A --. ~F. £ I;[Z~2Z and 17/ 1£ F. ;P "o �, 't~ ' '8 ' ((ii) ii) If If rk(F) rk(F) < 1n£ ~� A� ,8 Pn < 13 by 13 � by aa cut cut since since 'P qOpC~0 < 'P ~Op~ ~ . If If/3 _ rk(F) rk(F) =NF 13 /~ + + w P1 px + + ' .' ". . + +w wP" fl + +w wPP we we pao < pa EE 1£. first obtain first obtain 1qt~£ p"0'2 A CP~~176 ,iIrk( rkCF) F) � ((iii) iii) p Icp~irk(f) rk("F) �A ((iv) iv) by Lemma 3.4.3.5). £ we by the the Reduction Reduction Lemma Lemma ((Lemma 3.4.3.5). Since Since par(F) par(F) � C_ 17t we also also get get rk(F) rk(F) E E 1£ 7/ and iii) we and therefore therefore {pr { p l ,, .. ...., , Pn p,}} � c_ 1£ 7/.. From From ((iii) we first first get get 17/£ and and finally finally 1£ n ~� A� ,8 303 303 Set Theory and Second Order Number Theory by n-fold n-fold application application of of the the main main induction induction hypothesis hypothesis since since 'P ~om~Opa - 'P ~Opa for by p, 'Pp a = pa for ii -= 1, 1 , .. .. ..nn. . [] 0 Let H Fl. -sentence Fl• . 3.4.3.7. Boundedness Boundedness T heorem. Let 7"L � ~p b. A ,,F L~ for for aa �K ~-sentence F L~. 3.4.3.7. Theorem. Then 7-l � ~ b. A, , FL~ for for all all f3 ~3 E E [a, K~)) n N t£. 7-l. Then H Fl,3 [a, The induction on a. The immediately from The proof proof is is by by induction on a. The claim claim follows follows immediately from the the induction induction hypothesis if if the the main main part part of of the the last last inference inference is is different different from from F L~. So So assume assume hypothesis that F L~ is is the the main main part part of of the the last last inference. inference. If If F L~ E E 1\ A --type t y p e then then every every member member that of C (F L~) is is aa �K-sentence Z;~-sentence of of the the form form G L~ and and we we have have the the premise premise of Fl. C (Fl.) Gl• Fl. . Fl. 7-l[tfL~ (GL~)] U A, F L~, G L~. (i) (i) 7-l[tFL~(GL~)] ~ A, FL~, GL~ for all G L~ E C (F L~) (ii) (ii) Hence Hence Fl. tFL. (Gl.) tFL,3 (Gl,3) If Fl. F L~ E E V V --type t y p e and and FFl. L~ '¥. ~ (3x (3x E E LK)G(X) L~)G(x) then then every every member member of of C C ((Fl.) F k~) is is If again aa �K-sentence 2n-sentence of of the the form form Gl• G L~ and and we we have have the the premise premise again by induction induction hypothesis. hypothesis. But But since since F L~ is is aa �K-sentence 2~-sentence we we have have tfL~ (G L~) = --by (ii) by tEL~ (G L~) and and get get the the claim claim from from (ii) by an an inference inference according according to to (1\). (A)" (iii) (iii) 7t ~2_ A, F L~, G L~ for a. Applying for some some ao a0 < < a. Applying the the induction induction hypothesis hypothesis to to (iii) (iii) we we get get the the claim claim by by an an inference inference (V) (V). . if F L~ == - (3x E E k~)G(x) L~ then are either either in in the the case case of of an an inference inference (V) (V) If then we we are whose whose premise premise is is Fl. (3x LK)G(X)l. ~po A, FL,,G(t)L. b., Fl., G(tl· H74 � with with ao a0 < < a a and and stg(t) stg(t) < < a a or or in in the the case case of of an an inference inference (Ref (aef~) K) ~ b., A, Fl., F k', (\Ix (Vx E E LK)(3y L~)(3y E E LK)G(X, L~)G(x, y) y) H"t/ � =~ H 7/ � ~ A, (3z E E LK)(\lX L,)(Vx E E z)(3y z)(3y EE x)G(x, x)G(x, y) y) b., (3z => (iv) (iv) (v) for a. In In the for ao a0 + + 11 < < a. the first first case case we we get get H7-I � ~o~ b., A, Fl,3, FL,,G(t) G(tl,3L~ (vi) (vi) G(t)l,3 C ((3x Lfj)G(x)lil) stg(t) by induction induction hypothesis. hypothesis. We We have have G(t) L~ E E C ((3x E E [.~)G(x) L~) because because of of stg(t) < by a (iv) by a � _ / 3f3 and and we we get get the the claim claim from from (iv) by an an inference inference (V) (V)" . In In the the second second case case we we get induction hypothesis get by by the the Inversion Inversion Lemma Lemma and and the the induction hypothesis ,3, (3y (3y EE Lk~o)G(t, 7~[t] � ~ b. A,, F f lL~, H[t] oo )G(t, y)y) tg(t) << a. Since for .eRs -terms tt such for all all/:as-terms such that that sstg(t) Since ao a0 < < a a � < f3 fl < < a we we obtain obtain K. K (vii) (vii) 304 304 w. W. Pohlers A FEe 1;0+ 1 �, "Jr/ ~I ~ tI. ' pL{l ,' y) ((Vx oo ) G (x, y) oo )(3y Ee LL,o)G(x, "Ix Ee Lk~o)(3y (viii) (viii) from from (vii) (vii) by by an an inference inference (/\) (A)". But But )(Vx Ee zz)(3y (3z Ee L,B ) G (x, y) Ee CC (((3z ("Ix ) (3y Ee zz)G(x, ) G (x, y) (Vx E e LL~o)(3y L~ola(z, La)(Vx y))) oo) (3y Ee Loo o and (viii) by and we we obtain obtain the the claim claim from from (viii) by an an inference inference (V). (V). 3.4.4. 3.4.4. Collapsing C o l l a p s i n g functions functions In operators which In order order to to define define operators which allow allow the the collapsing collapsing of of derivations derivations we we need need to about ordinals. to know know more more about ordinals. This This theory theory turns turns out out to to be be very very complicated. complicated. To To simplify an abstraction. of using simplify things things we we are are going going to to use use an abstraction. Instead Instead of using admissible, admissible, Le. i.e.,, recursively ordinals, we recursively regular regular ordinals, we will will develop develop the the theory theory on on the the basis basis of of just just regular regular ordinals. bother about ordinals. This This has has the the advantage advantage not not to to have have to to bother about the the complexity complexity of of the ordinal ordinal functions functions which we are are going going to to define. define. Complexity will be the which we Complexity arguments arguments will be replaced arguments. The the replaced by by simple simple cardinality cardinality arguments. The disadvantage, disadvantage, however, however, is is that that the replacement of of regular ordinals by by recursively regular ones ones is is not not at at all all easy. easy. See See replacement regular ordinals recursively regular Rathjen [1995] [1995] and and Schliiter [1993,n.d.] for for details. It is is outside outside the the scope scope of of this this Rathjen Schliiter [1993,n.d.] details. It contribution indicate is contribution even even to to give give a a hint hint how how this this can can be be performed. performed. All All we we can can indicate is that that the the segment segment below below W wl, the first first regular regular ordinal, ordinal, is is recursive recursive and and therefore therefore already already aa I , the wlOK cK segment fKK.. This computation of segment below below W w~ This yields yields at at least least a a correct correct computation of the the E� E 1l -ordinals -ordinals of of the the analyzed analyzed axiom axiom systems. systems. In In this this section section we we denote denote by by Reg R e g the the class class of of regular regular ordinals ordinals above above w w.. By By Reg R e g we we denote denote its its topological topological closure, closure, i.e., i.e., the the class class of of uncountable uncountable cardinals. cardinals. Let Let 0, ftl = ~ . f2~ denote denote the the enumerating enumerating function function of of Reg Reg U U {O} {0}.. Then Then f~0 = = 0, = R1 = = W wl, I, f~ = = R~ etc. etc. We We reserve reserve the the letters letters n, 7r, h i , . . . , 7rl,... to to denote denote members members of of Reg Reg exclusively. exclusively. We We put put A�. n{ nw �w 1'\,, 11", 1'\,1 , , 11"1 , • • • no n1 �1 . • . aa ++ : = mmin i n { ~{I'\,E R eReg g 1I a a <<~ }I'\,.} . := E For For the the following following we we assume assume that that there there exist exist aa weakly weakly inaccessible inaccessible ordinal. ordinal. Let Let II be the ordinal, i.e. Le. II is ordinal for be the least least such such ordinal, is aa regular regular ordinal for which which we we have have f~I = - II and and a we have have I'\,~ = a < < f~,, for for all all a a E Reg Reg n A I I.. Then Then we = f~,,+i for for all all I'\, ~ E Reg Reg n A I. I. nO" n[ nO"+! Cl(a, (3} (3) 1\ min {/51 Cl(a,/3) A Cl(a, Cl(a, (3) ~) n A I'\,~, � C_/~} {(3 I I'\,~ E Cl(a, 'I/Jr "a := min by by recursion recursion on on a. a. The (3) is The set set Cl(a, Cl(a,/~) is the the least least set set which which contains contains/3(3 tJ {O, {0, II}} and and is is closed closed under under ordinal addition +, the Veblen-function A�. ordinal addition +, the Veblen-function A(. A'T}. A~. 'P{'T} ~o~r/,, the the enumerating enumerating function function A� A~.. n{ f~ of of the the class class Reg R e g and and the the function function A� A~ < a. a. ATr. 'I/J r ,,� . (3. There We We call call Cl(a, Cl(a, (3) fl) the the a-iterated a-iterated closure closure of of ft. There are are some some immediate immediate consequences consequences 3.4.4.1. 3.4.4.1. Definition. Definition. We We define define sets sets Cl(a, (3) fl) and and functions functions E U A1r . of of Definition Definition 3.4.4.1 3.4.4.1.. Set Set Theory Theory and and Second Second Order Order Number Number Theory Theory ao C~o� < ac~ and and {30 flo � <_{3fl � =:> Cl(ao, Cl(c~o, {30) ~o) � c Cl(a, Cl(c~, {3) ~) {3 E Lim � Cl(a, {3) = U Cl(a, 1'}) f} E {3 Cl(a,, {3) 1 < r IIcl( cl( , 1/J/ta) n K, = = 1/J/ta. Cl(a, Since a a < nu+1 ft~+l = = K,a � C_ Cl(a, Cl(a, K,) ,~) for for K,~ < JI we we get get Since K,~; EE Cl(a, Cl(a, K,) ~) for for all all a. a. 305 305 (155) (155) (156) (156) (157) (157) (158) (158) (159 (159)) A little little more more effort effort is is needed needed to to show show A (160) 1/J/ta < K,,~ and ¢r Cl(a, r and 1/J/ta r Cl(c~, 1/J/ta). r (160) By ((156) and (159) (159) we we have have 1'}r/00 := "= min min {�I {~1 K, ~E E Cl(a, Cl(a,~)} Putting By 156) and �) } < K,.~. Putting := min min {�I {~l Cl(a, Cl(a, 1'}rln) ~} n ) nM K,,~ �C_0 1'}r/~+l n+1 := we obtain obtain 1'}r/~ from (157) (157) by by induction induction on on n. n. Hence Hence 1'}r/ := := sUP supne~ and we n << K,~ from nEw 1'}r/nn < K,~ and 1'} = 1'} . This shows 1/J/ta � 1'} < K, and, Cl(a, 1'} � sUP ) n K, n +1 Cl(a, rl)M~ U~e~oCl(a, rln)M~ C_ sup~e~ r/~+l = 77. This shows r _< r / < ~ and, nEw n CI(a, 1'}) n K, = UnEw since Cl(a, r M K, ~= = r1/J/ta, also also 1/J/ta r r Cl(a, 1/J/ta) n ¢r Cl(a, Cl(a, 1/J/ta) . since Putting Jr := min 'Y} itit follows {3) �c_ Jr Putting I r "= min b {7 E E SC SC II IJ < < 7} follows from from (160) (160) that that Cl(a, Cl(a,~) I r for for {3/ 3 c�_ IJrr.. ao � < (161) C~o < a a and and ao ao E E Cl(a, Cl(a, 1/J/ta) r => 1/J/tao r < 1/J/ta r (161) follows r E Cl(a, follows since since 1/J/tao E Cl(a, r1/J/ta) M n ~. K, . We We obtain obtain r r (ft~ I a < ft~} ((162) 162) since the assumption assumption 1/J/ta = nu >> aa EE Cl(a, Cl(a, r1/J/ta) leads to 1/J/ta E since the r = ft,, leads to r E Cl(a, Cl(a, r1/J/ta) which which contradicts contradicts (160). (160) . We have We have E SC SC r1/J/ta E since entails ~o~rl since ~, �, r1'}/ << r1/J/ta C_ Cl(a, r1/J/ta) entails CP�1'} EE Cl(a, Cl(a, r1/J/ta) M n ~K, == r1/J/ta. � Cl(a, By definition we have nu E CI(a, {3) for a E CI(a, {3) extends to to By definition we have f ~ E Cl(a, ~) for a E Cl(a, ~). . This This extends Cl(a, ~) {3) r{:} aa EE Cl(a, Cl(a, ~) {3) fnu ~ EE Cl(a, (163) ( 163) ( 164) (164) which isis obvious obvious for for aa == ft~. nu . IfIf we we assume assume ft~ nu >> aa ~¢ Cl(a,~) Cl (a, {3) then then we we get get which some nu r¢ flU{0, {3u {O, I}I} asas well well as as ft~ nu #=I- r1/J/t 1'} for all all ,~ K, and and r/. 1'}. IfIf ft~ nu == ~+r/or �+1'} or ft~ nu == qo~r/for CP�1'} for some ft~ � and and r/then 1'} then gt~ nu EE {~, {�, 77}. 1'}}. This This shows shows that that the the set set Cl(a, Cl(a, Z) {3) \\ {ft~} {nu} contains contains ~U {3u {0, {O, I}I} and and has has the the same same closure closure properties properties as as Cl(a, Cl(a, fl). {3) . Hence Hence Cl(a, Cl(a, fl) {3) == Cl(a, Cl(a, fl) {3) \\ {f~}, {nu}, i.e., i.e., ft~ nu r¢ Cl(a, Cl(a, ~5). {3) . We define define the the set set of of strongly critical components SC(a) SC(a) of of an an ordinal ordinal aa by by We strongly critical components 306 306 W. Pohlers Pohlers W. iiff aa ==00 0 sc(~) .= {~} SC(r U SC(r/) sc(~l) u... sc(~.) if ~a eE sc SC ~f =N F ~('r/ ifif aa =NF 'Pt;'TJ if ~a ==NF an · ~f ~ ~a1 ++. .. .. . ++ ~.. (165) (165) Then obviously obviously SC(a) SC(a) c� SC SC and and SC(a) SC(a) c_ � aa 4+ 1. 1 . Now Now we we get get Then "I eE cl(~, Cl(a, ~) {3) ,~ {:} sc(~) SC('TJ) c� cl(~, Cl(a, ~). {3). (~66) ( 166) The claim claim isis trivial trivial for for 7/ "I E SC. Sc. So So assume assume 77 "I ~� SC. Sc. The The direction direction "r " <=: " is is clear clear The by definition. definition. We show the the opposite opposite direction direction by by induction induction on on 7. "I. Assume Assume SC(r/) SC('TJ) g; by We show or 77 "I == ~1 6 4+ ~2 6 for for ~i �i << r/"I we we get Cl(a, {3) . Then Then r/"I r� /~U {3 U {0, {O, I}. I} . IfIf 77 "I =NF =NF q0~1~2 'Pf;!6 or Cl(a,~). get SC('TJ) C_ SC(6) U SC(~2) SC(6) and and SC(~,) SC(�i) gg; Cl(a, Cl(a, fl) {3) =~ Cl(a, fl) {3) by by induction induction SC(r/) ~ ~� Cl(a, => �i � SC(~Cl)U hypothesis. Therefore Therefore we we can't can't have have r"I/ == qor 'Pf;!6 or or 77 "I == ~1 6 4+ ~2 6 such that ~i �i E E Cl(a, Cl(a, ~) {3) hypothesis. such that for ii -= 1, 1 , 2. 2. Thus Thus Cl(a,/~) Cl(a, {3) \\ {r/} {"I} contains/3 contains {3 U U {0, {O, I} I} and and satisfies satisfies the the same same closure closure for conditions as as el(a, Cl(a, 1~), {3) , i.e., i.e., r/~ "I � Cl(a, Cl(a, 1~). {3) . conditions If = f~+l a E Cl(a, (164).. Hence If ~K, = n",+! then then a Cl(a, r'IjJ/ta) by by the the definition definition of of r'IjJ/ta and and (164) Hence n", E Cl(a, 'IjJ/ta) n = r'IjJ/ta < < ~'~a4-1 n",+! and and we we have have ~'~a Cl(a, ~)~a) N K, ~ = (167) Let �~ << a. a. By obtain ~K, E Cl(~, Cl(�, r'IjJ/ta) . Since Since Cl(~, Cl(�, r'IjJ/ta) n � Cl(a, Let By (167) (167) we we obtain M K, ~ c_ el(a, r'IjJ/ta) n M K, = we and = 'IjJ/ta r we get get 'IjJ/t� r162 ::; <__'IjJ/ta r and therefore therefore =~ 'IjJ/t� r162 ::; < r'IjJ/ta /\ A Cl(�, Cl(~, r'IjJ/t�) � c Cl(a, el(a, 'IjJ/ta). r �r ::; ((168) 168) Together 161 ) this Together with with ((161) this gives gives a a < {3 ~ /\ A a a E E {3 => < Cl( Cl(7, r 'Y ) for for some some 'Y7 ::; <--/~ =~ 'IjJ/ta r < 'IjJ/t{3. r r , 'IjJ/t (169) (169) is It It follows follows from from (167) (167) that that for for K, n < II the the ordinal ordinal 'IjJ/ta r is not not in in Reg. Reg. For For K, n = = II the the situation situation is is different. different. There There we we have have nr ((170) 170) = r showing is 170) choose showing that that 'IjJ[a r is aa limit limit of of regular regular ordinals. ordinals. To To prove prove ((170) choose a a such such that that n", . Hence ~'~a ::; _~ 'IjJ[a ~/3Ia < < n",+! flaW1". Then Then n",+! ~"~a-t-1 < < II which which entails entails n",+! ~'~a+l �r Cl(a, Cl(a, 'IjJ[a) r Hence a � . But ~ Cl(a, Cl(a, 'IjJ[a) r But then then 'IjJ[a e t a ::; <_a a ::; <_n", f]~ ::; <_'IjJ[a. r Since for Since n", f~ E SC SC and and n", f~ =f. ~- 'ljJ1f'TJ r for 7r 7r =f. r II by by (167) (167) we we also also get get n", f~ = =a a E Cl(a, Cl(a, {3) ~) => => a= = II or or a = = 'IjJ['TJ CPl for for some some "I. ~7. (171) (171) are So So II and and the the ordinals ordinals of of the the form form 'ljJI� r are the the only only ordinals ordinals in in Cl(a, Cl(a, {3) ~) which which are are fixed fixed points points of of A� A~.. nt; f~r . [ ~ , ~+~) n c~(~, ~) # 0 ~ ~ e c~(~, ~). ((17~) 172 ) We 172) by {3) . We We prove prove ((172) by contraposition. contraposition. Assume Assume that that n", f~ �r Cl(a, Cl(a,Z). We are are going going to to show {3) \\ [[f~, n"" n",+!) show that that then then M M := :- Cl(a, Cl(a,/~) f~+l) satisfies satisfies the the same same closure closure condition condition as as n"" n",+!) Cl(a, . Then Cl(a, {3) 1~). Then M M = - Cl(a, el(a, {3) ~) which which shows shows Cl(a, el(a, {3) Z) nn [[f~, f~+l) = = 0. O. First First we we have n"" n", have {3/3 U U {O, {0, I} I} � C_ M M.. If If �, ~, "Ir/ �r [n"" [f~, n",+ ~'~a+l) we get get �~ + + "I77 �r [[f~, ~'~a4-1) as well well as as + l) as d we Order Number Theory Set Theory and Second Order 307 307 ncr, ncr t.p~or162 ncr, ncr+l). IfIf 'lf1r 1t� EE [[flo, �, II; } �c_ Cl(a, flo+l) for {{~,~} Cl(a, f3) fl) but but II;tr, � ¢r [ncr, [flo, ncr f~o+l) + ! ) for + !) (fJ ¢ [[f~,f~+l). then we we have have II;~ = - ncr+l f~+l by by (167) (167).. Hence Hence ncr f~o EE Cl(a, Cl(a, (3~)) by by (164). (164). A A contradiction. contradiction. then That M M is is closed closed under under A� ,k~.. ne f~r is is obvious. obvious. That Next we we observe observe Next cl( , = cl( (173) (173) , 1] i 1]~/r¢ Cl(a, } . Then To To prove prove (173) (173) put put �~ := : : min min {{77[ Cl(a, ncr) f~)}. Then �~ � c_ Cl(a, Cl(a, ncr) fl,)Mn ncr+! ~'~a+l which which entails Cl(a, Cl(a, ncr) ~o) = = Cl(a, Cl(a, €) ~).. Therefore Therefore we we have have �~ ¢ r Cl(a, Cl(a, �) ~) and and �~ < ncr+! fl,+l which which entails implies 'lf1 r n,,+,a a � _< �~r because because ncr+! fla+l E E Cl(a, Cl(a,~o) Cl(a,~). Hence Cl(a, Cl(a,f~a) implies ncr) �C_ Cl(a, �) . Hence ncr) �c_ Cl(a,r'lf1n,, +,a) � C_Cl(a, Cl(a,~) Cl(a, ncr) fl~) by(167). by(167). Cl(a, �) == Cl(a, From (173) (173) we we obtain obtain especially especially From Cl(I r, O) 0) n Mnl ~'~1 = ----- 'lf1 r w, (Ir). (ir). Cl(Ir, (174) (174) It follows follows from from (174) (174) that that all all ordinals ordinals below below 'lf1w, r Ir can can be be represented represented by by terms terms It which are are built up from from 0, O, I, the unary unary function function A� ,~(.. n f~r{ and and the the binary functions I, by by the binary functions which built up A1] . 'lfr1,,'T} solely. ~ . . A1]. It/. �~ + + 1] 77,, A� I ( . . A1] Ar].. t.p ~oCr] and A7r 1~r.. )~. solely. We We already already defined defined the the notions notions A� {1] and =NeF al al + + ' '. '. . + + a a~n and and a a =NF =Ne t.pe1]. ~r This This gives gives unique unique term term notations notations for for terms terms not not aa =N in SC. we define define in Sc. IfIf we a--NF ~"~a :r a-- ~'~a A G < a and and =,F :,, = A Cl( , we get get we /\ a : N F ~'~r/ ~=} ~ = 77 a : N F ~"~ /k a = N F n{ a =NF n'1 � = 1] (175) (175) as well as as as well a=NFr162 =~ ( a = f l r ~=77). (176) (176) While (175) obviously we we get get (176) Cl(~, r'lf11t€) implies implies (175) holds holds obviously (176) because because ~� << r/and 1] and ~� EE Cl(�, While � r'lf11t 1] by (168) E Cl(~, Cl(1], r'lf11t 1]) nM 77. (168) and and therefore 1]. Hence f3 by r'lf11t � _< therefore ~� E Hence a a < < fl by (161). (161). This This entails also direction. entails also the the opposite opposite direction. As consequence of of (176), that aa --NF =N F 'lf1,,1] determines As aa consequence (176), (167) (167) and and (170) (170) we we obtain obtain that r determines is in normal form, form, we to In order to decide whether 'lf1lta r and and r/uniquely. order to decide whether r is in normal we have have to 7r 'T} uniquely. In decide E Cl(a, Cl(a, 'lf1lta) . By (173), however, however, it to decide decide a a E r By (173), it suffices suffices to decide a a E E el(a, Ir l ) Cl(a, i'lf1lta where lena] i'lf1ltai denotes denotes the the cardinality cardinality of of r'lf1lta. For For #J.L EE R Reg we get get more more generally generally where e g we Cl(a, #) J.L) ifif and and only only ifif one one of of the the following following conditions conditions is is satisfied: satisfied: flf3 EE Cl(a, #e f3 E J.Lu{0,I} U {O, I} and Z SC(f3) C_ � el(a, Cl(a, #) J.L) f3 r¢ SC SC and SC(fl) and #J.L <_ � flf3 = = r'lf1,,1], 77 1] < aa and {Tr, {7r, 77} 'T}} C_ � Cl(a, Cl(a, #) J.L) Cl(a, J.L). #J.L <_ � flf3 = = flo ncr and aa EE Cl(a,#). This gives raise This gives raise for for the the following following definition. definition. and (177) (177) 308 308 W. Pohlers , E Cl(Ir,O) U {KJ. '(j3) II j3Z Ee SC(a)} {K.(Z) sc(~)} ifi f aa ¢it SC if i f aa E E J.L # UU {I,O} {I, 0} 00 {O �iff J.L# :::;_ aa == 7/Jr 1f � {~} Uu K K,,(r K,,(~) ' (7r) J.' (�) U KJ. 3.4.4.2. Definition. D e f i n i t i o n . For For ordinals ordinals a a, J.L # E Cl(I r, 0) we we define define finite sets KJ. K,(a) by 3.4.4.2. finite sets ' (a) by KK,,(,~) J.' (a) .:=- ._ { . KJ. ' (a) K~,(a) If if J.L # :::; _< a a = = nO". ft~. It does not definition is deterministic in It does not matter matter that that the the above above definition is not not deterministic in the the case case that that a a is an an ordinal below J.L # which which is is not not strongly strongly critical. critical. We We get get KJ. K,(a) regardless of of is ordinal below ' (a) - 0 regardless the clause we the clause we apply. apply. From From (177) (177) and and Definition Definition 3.4.4.2 3.4.4.2 we we get get j3 ~ E e l ((aa, #J.L)) {:} r (V~ E KJ. K~(/?))[~ a].. (V� ' (j3))[� << a] Putting Kt,(a ) < < j3 ~ :{:} :r (V~ E KJ. K~(a))[~ we get get (V� ' (a))[� << j3~]] we Putting KJ.'(a) =0 E E Cl , E and KKIrltPw'lI ('TJ) < 'TJr/.. = 7/J1f'TJ and Observe ordinals a Observe that that for for ordinals a E E Cl(Ir, Cl(I r, 0) O) the the cardinality cardinality is is determined determined by by if a if a a = nO" ~t~ or or a a = 7/J r r'TJ f . . . a + � a + an {la , lan l } = if a . l . , max{l~l,..., I~.1} max d · lal If I~1= IIvl if a a = = ifJ{ ~r 'TJ 'TJ I nO" if if a a = = 7/J r 1f 'TJ and I I > > 7r ~r = nO"H ' a O / -=NF - N F 7/J1f'TJ ~)~r~ = { {:} r a a - ~)lr?~ = a = (178) (178) = a 1 -[-- 9 9 9 -4- a n = (179) (179) ~a+l. All All that that opens opens the the possibility possibility to to define define simultaneously simultaneously aa term-system term-system T T together together with with an an evaluation evaluation function function 1I 1iv: 7" ----+> On O n and and aa "less "less than" than" relation relation < < on on the the 0: T ordinal-terms l < IIbio l and ordinal-terms such such that that a < b {:} r and the the "less-than" "less-than" relation relation on on the the l]alo ordinal terms ordinal terms becomes becomes primitive primitive recursive. recursive. We We will will not not do do this this in in all all details details but but only only indicate indicate the the essential essential steps. steps. There There are are the the following following sets sets of of ordinal-terms ordinal-terms a b • ao bo the set comprising all ordinal terms the set PP of principal the set principal terms terms denoting denoting additively additively indecomposable indecomposable ordinals ordinals the set SC SC the set SC denoting denoting strongly strongly critical critical ordinals ordinals in in SC the set KK ofof cardinal the set cardinal terms terms denoting denoting ordinals ordinals in in Reg Reg the set FF ofof fixed-point the set fixed-point terms terms denoting denoting ordinals ordinals which which are are fixed-points fixed-points of of the the enumerating enumerating function function of of Reg Reg the set RR ofof regular-terms the set regular-terms denoting denoting ordinals ordinals in in Reg Reg 9 the set T T comprising all ordinal terms • 9 • 9 • 9 • 9 • 9 which which are are defined defined by: by: • R � � � P � TT 9 F� FcKK E TT,, IIOlo:-0 9 II O_E llio := 0 IIo ::== Iz 9 I / E E R R nn F F,, IIZlo al �>_ .. ". . �>_ aa,n aa lI,, ... .. ,. a, ,an EE PP and and al . . + . + laI~xlo l a nl o do + ' " + I~.lo 9 RcKK c S C cSC Pc • • • •9 al an E l a l a o := nl := � =~ al + + . .. .. . + + a, E T T,, la~ + + ' .". . + + a.lo Set Set Theory Theory and and Second Second Order Order Number Number Theory Theory • 9 a, a, bb EE T T � =~ -q5 - ~aab bEE PP,, l-q5 ]-~ab[o:-~l~lolb[o a bl o := -q5lalo I b l o 309 309 'P, where -q5 ~ is is the the fixed-point fixed-point free free version version of of the the function function ~, i.e. i.e. where qp~(fl + + 1) 1) if if f3 fl = = ,"),+ + nn for for some some nn < < w w and and ,-), such such that that 'Po, ~o~q,= - , ,), 'Pof3 '' =9= {L~ 'Po(f3 otherwise, ~ f l otherwise, 'Pof3 _ • al o p a Ee SC 'l/Jp a l o := 9 Ifp I.fp EE R, R, aa =J ~ L I, aa EE T T and and K KilCpal (a) < < aa then then 'l/J Cpa SC and and IICpalo "- 'l/J r lp lo llalo I 1PpallI(a) • 9 If If aa EE T T and and K KIICL~II (a) < < aa then then 'l/J CLa F and and I'l/J ICLalv r l al o ra EE F II1P]all (a) ra l o :"-= 'l/Jl • 9 If If aa EE T\ T\ F F then then D f ~a EE K K and and D f~a+l and ID If~lo '- D f~l~lo a l o := a+l EE RR and 1a 1o 'P!& where 1 1 :"=- 9~o0. where The The definition definition of of the the sets sets Kp(a) Kp(a) for for pp EE K K and and aa EE T T should should be be obvious obvious from from Definition ial o l Definition 3.4.4.2. 3.4.4.2. Similarly Similarly obvious obvious is is the the definition definition of of the the "cardinality" "cardinality" Ilicit = Iilalo] i a ll = of an an ordinal ordinal term term aa from from (179). (179). of Finally we we have have ((omitting some obvious obvious cases cases)) Finally omitting some aa = a ll = = 00 1\ A bb =J -7(=0, 0, or or lIlall < Ilibl], or Iilicit - Ilbll ]]bi] and and one one of of the the following following I bll, or I all < conditions conditions is is satisfied satisfied aa = = 'l/J r p c 1\ A bb -=e n'l/Jdpd 1\ A cc < < dd =-~cd 5 C A1\ c, c,dd < < bb aa = -q5cd 1\A bb EE SC aa EE SC (a � S C A1\ bb -= - ~-q5cd c d 1\ A (a < cc V aa � <_ d) d) a < b 4=~ { cc<< e Aedl\ < bd < b aa = -q5cd 1\A bb = -q5e f 1\A cc -=- ee 1\A dd < =-~cd =-~ef < ff ee < { to to realize realize that that Ial o Il aa EE 7-} T} == CCl(Ir, 0). {{ lalo l ( I r, 0). Since r, 0) r, O)Nf~ Cl(Ir, O) Nf~l n Dl is is aa segment segment of of the the ordinals ordinals we we obtain obtain Cl(I Cl(Ir, 0) n Dl C_ WfKK which which Since Cl(I � w~ K. K r entails by (174). ( 174) . This This shows shows that that we we at at least least may may replace replace f~l D1 by by w~ WfK. entails r'l/JwJ << w~ WfK by needs, however, however, considerably considerably more more effort effort to to show show that that we we can can replace replace all all regular regular It It needs, K. r , O) ordinals by by recursively recursively regular regular ones ones without without changing changing the the segment segment CCl(I ordinals l ( I r, 0) Nn w~ WfK. Nevertheless, we we will will pretend pretend that that we we have have done done that that and and interpret interpret the the ordinals ordinals in in Nevertheless, R e g as Reg as recursively recursively regular, regular, i.e. i.e. admissible admissible ordinals. ordinals. 3.4.5. he C ollapsing T heorem 3.4.5. TThe Collapsing Theorem We are are now now prepared prepared to to define define the the controlling controlling operator operator which which allows allows to to collapse collapse We the the controlled controlled derivation. derivation. 310 310 W. Pohlers 3.4.5.1. Definition. Definition. Put Put 3.4.5.1. 1i.., n . ((xX) ) :.= N {{Cl(a cl(., , � Z)) 1l X x � c Cl(a cl(., , � Z)) 1\ ^ 'Y < a}. .} =n We We certainly certainly have have 00 E E 1i.., 7/7 for for all all 'Y 7 and and obtain obtain from from (166) (166) that that all all 1i.., 7/7 are are Cantorian Cantorian closed definition we closed and and closed closed under under ~. By By definition we get get X X � C_ 1i..,(X) 7/7(X) for for all all X X � C_ On. On. If If we we assume and Y (X) and assume X X � C_ 1i..,(Y) 7/~(Y),, �~ E E 1i.., 7/7(X) Y � C_ Cl(a Cl(a, , m fl) for for some some 'Y 7 < < a c~ then then we we obtain obtain also also X X � C_ Cl(a, Cl((~, m ~) and and therefore therefore also also �~ E E Cl(a, el(a, m fl).. Hence Hence �~ E E 1i..,(Y) 7/7(Y) and and we we have have 7/~(X) ~7(Y). Pulling this this together together we we have have the following lemma. lemma. 1i.., (X) �c_ 1i.., (Y). Pulling the following 3.4.5.2. The 3.4.5.2. Lemma. Lemma. The operators operators 1i.., 7-l7 are are all all acceptable acceptable and and closed closed under under the the cpo Veblen Veblen-function and the the function function A� ~. . O f~re .. -function cp~ and 1i.., But But we we also also get get the the closure closure of of the the operators operators 7/~ under under the the functions functions 7/J r e in in the the following following sense. sense. � ::; < 'Y 7 1\ ^ �, /'i, E n 1i..,(X) (x) :::} 7/J1t� E 1i..,(X). (180) (180) From From (172) (172) and and (164) (164) we we get get 1i..,(X) =f:. 0 1i..,(X). ]n [0 [~"~a, ~"~o' N n 7 ( X ) # 0 :::} =:~ {O {~"~o-, ~"~o-+1} C_ n T ( X ) . O'+l } � 0' , 0 O' , O 0'++1 1] (181) (181) The operators are The operators are also also cumulative. cumulative. We We have have 7_<5 ~ 7/~c_7/6. ((182) 1 82 ) The Collapsing Theorem, Theorem, i.e., The aim aim of of this this section section is is to to prove prove the the Collapsing i.e., to to show show that that every every derivation can be be collapsed into aa derivation whose derivation derivation of of a a set set of of �It-sentences E~-sentences can collapsed into derivation whose derivation length length is is less less than than /'i, n.. Obviously Obviously such such an an derivation derivation must must not not contain contain cuts cuts whose whose cut-sentence /'i,. Therefore cut-sentence has has a a complexity complexity above above a. Therefore collapsing collapsing below below /'i, a has has to to come come /'i,. Cut together elimination of together with with the the elimination of all all cuts cuts of of complexities complexities above above n. Cut elimination elimination as doesn't say as stated stated in in Theorem Theorem 3.4.2.2 3.4.2.2 is is of of no no help help since since it it doesn't say anything anything about about the the controlling operators. controlling operators. However, However, we we know know already already from from the the Predicative Predicative Elimination Elimination Lemma Lemma 3.4.3.6 3.4.3.6 that that we we can can eliminate eliminate cuts cuts whose whose cut-ranks cut-ranks are are in in (0)., (F/h, 0>' ~,k-t-1] for + 1 ] for A EE I_im 00'+ 1 ] without Lim or or in in (00' (Q~ + + 1, 1,~+1] without losing losing information information about about the the controlling controlling operator. operator. Therefore Therefore we we introduce introduce the the class class of of left left initial initial points points of of these these intervals, intervals, Le., i.e., we we put put Reg :={ I{I+ l+} Ul}{ fU~ l{OO' I O'a EEI Cl Ink iLim} Lim} Reg:= m } UU{ ~{OO' , , + l+ [ 1 1 O' a EEI \IL\i m } The The crucial crucial situation situation which which is is not not yet yet covered covered by by the the Predicative Predicative Elimination Elimination Lemma Lemma are are derivations derivations of of the the form form 1i f; � for tL E Reg. it -sentences we In In the the case case that that � A is is a a set set of of E E~-sentences we want want to to collapse collapse this this derivation derivation below below /'i,. We a. We are are going going to to show show the the following following theorem. theorem. 3311 11 Theory and Second Second Order Order Number Theory Set Theory A 3.4.5.3. Collapsing TTheorem. Let 3.4.5.3. Collapsing heorem. Let LJ. A be be aa set set of of �I< E~-sentences, Reg, -sentences, j),# EE Reg, {n,.y,#} 7/7 and and assume assume 1i"( 7-l~ f; ~ LJ. A.. Then Then this this derivation derivation is is collapsed collapsed to to " j),} �C_ 1i"( {K, wI'+" ) LJ.A . �("(++wl' 1i,,(+wI'+" 1 ,p,p0~(~+~+~) �("( +" ) To induction work and To make make the the induction work we we assume assume that that 8 e is is aa set set of of .eRS-terms s and prove prove the the more more general general claim claim } 2 ~,, j), # E e Reg Reg LJ.A �C_ �I< )) I 1' �-> K} => 1i"( wl'+,, [8] {Cl(r ++ l1,, 1/1r (r ++ l1))1 par(8) �c n N{cl( [el 1 ,p� ("(+wl'+" ) LJ. A ((i) + K" , j), E 1i"([8] ,p� ("(+wl'+" ) i) 1i"([8] f; LJ. by main main induction on j), # with with side side induction on c~. To simplify simplify notations we abbreviate abbreviate by induction on induction on a. To notations we ) . So i) the first first three three lines lines in in the the assumptions assumptions of of claim claim ((i) by Asm Asmp(A; e; j), #;j Kj a; , 7). So ((i) p(LJ. j 8j i ) by the becomes becomes +wl'+" ) LJ.. ((ii) ii) Asmp(A; O" 7/~[O] Ir.;~£>.. LJ. A => =~ 1i,,( 7/~+~,+~ [O] lI,pr �("( Asmp(LJ.j 8j j),#;j Kto;j ~/) ,) /\A 1i"([8] +wl'+" [8] ,pK ("(+Wl' " ) A ' r + " induction we To prepare prepare the the induction we first first observe observe that that by by Lemma Lemma 3.4.5.2 3.4.5.2 we we have have To ((iii) iii) Asmp(A; 8j O; j), #;j K ~;;j , ')')) /\ Aa c~ E E 7-/.y[O] =~ , 7+ + w w~'+'~ ~ ]. Asmp(LJ.j 1i"([8] => lJ+£> Ee 1i"([8]. + From iii) , ((182), 182) , ,"f' � 180 ) we then From ((iii), _< , ~' + + wlJ w~+~£> and and ((180)we then obtain obtain +£» EE 1i,,( ((iv) iv) + wlJ ) /\A a~ EE 7/~[O] p(LJ. j 8j Asmp(A; O; j), #;j K a;j , ")') =~ 1/11<(r 0~(~/+ w"+~) 7/~+~+. [O]. Asm 1i"([8] => +wI'+" [8]. Most collapsing property Most important important is is the the following following collapsing property of of the the function function 1/11< r (, + + w w~+~). lJ+£» . +£> << (3fl => +£» << 1/11< v) + w ~+a Asmp(A; 8j O; j), #;j Kj a; ,) "7) /\ Aa c~ E 1i"([8] 7-/~[O] /\ A, ~'+w =~ 1/0,(~, w"+a) r {3. ((v) Asmp(LJ.j 11<(r ++ WIJ +£> EE Cl(o, To v) it To obtain obtain ((v) it suffices suffices by by (169) (169) to to find find some some 05 � _ (3 fl such such that that , 7 +wlJ + w"+~ Cl(5, 1/11<0). 0,5). + + )) by iii) 1 , 1<(r � (r + E +wlJ + wlJ 1i"([8] c_ Cl But we we have have , 7+ + 11 � _ , 7+ w"+~£> and and , 7 + w"+~£> e 7/~[O] Cl(7 + 1, 1/01, ( 7 + + 11)) by ((iii) But 1 , 1/11<(r + 1 )) . So we may choose 0 : = , + 1 . + ) � and the the assumption assumption par( par(O) C_ Cl(9/+ 1, 0~(7 + 1)). So we may choose 5 "= ")f + 1. and Cl(r 8 lJ i ) we To To prove prove claim claim ((i) we run run through through the the cases. cases. If If the the last last inference inference was was by by (/\ ( A )) 1\--type then then there there is is aa sentence sentence F F E E LJ. An MA t y p e and and we we have have the the premises premises LJ., GG and ((vi) vi) 7{~[O u U tF(G)] tF(C)] � ~ A, and aa ac < K ~ we we get get 1i"([8] 7/~[O] n MT � C_ Cl( Cl(~/+l,r , + 1, 1/Ir(r + 1)) n Hence par(tF(G)) = 1/I Or(')' par(tf(G)) � C_ 05 < < 1/I r r(, + I1)) � C_ Cl( el(7, + + 11,, 1/I 0r(7 which r (, ++ I1).). Hence l'V = r (r ++ l1)))) which shows + 1,1, 1/Ir r (r ++ 1)) shows par(8 par(O u U tF(G)) tf(G)) � C_ n Nr>, Cl(~/+ 1)) for for all all G G E E C(F) C(F).. So So we we have have r>1< Cl(r for all all G Asmp(A, tf(G);j j), #;j Kj a; ,y) Z/) for GE e C(F) C(F) and and get get As m p(LJ., GG;j 8O UU tF(G) ,pK ("(+wI'++"G ) LJ., ((vii) vii) A G 1i7-l.~+~,+,o[OUtf(G)] "(+wl'+"G [8 U tF(G)] I[0"(~+~"+"~ ,pK ("(+Wl' "G ) G +£>G ) < for induction hypothesis. hypothesis. By v) we ++ wlJ for all all G G E C(F) C(F) by by the the side side induction By ((v) we have have 1/I1<(r r w'+"G) +£» EE 1i,,/+ +£» and by ( iv ) . Therefore we obtain I'+ 1/I1<(r ++ wlJ r w'+~) and r1/I1<(r + + wlJ w'+~) 7/~+~,+~[O] by (iv). Therefore we obtain [8] w " l' l' 312 312 W. Pohlers Pohlers w. (-y+Wl'+a) [0] II1/!,r'/'''K (-yC~§Wl'++''a))) � 1/."H.,.,,+,.,.+,:,, "),+wl'+a [8] 1/!K + from (vii) (vii) by by an an inference inference (A) (A).. from In In the the case case of of an an inference inference ((V) V ) there there is is aa sentence sentence F F EE � An NV V-type, sentence -type, aa sentence C (F) and and some some o! Soo < O! e such such that that GG EE C(F) 3/~[0] ~ - A�, ,GG [8] � 1/.")' and OoF(G)< F (G) < O!.oL. (viii) (viii) By side side induction induction hypothesis hypothesis we we obtain obtain By 1/!K/! (-y+Wl'+ao ) �~ , G. n~+~.+oo [o] IIrr176176 (ix) 1/.")' (ix) +wl'+ao [8] +ao ) ' c 1 K (-y+Wl',+~o) +<>o) < +<>o < 'Y" y++ wwl'u+a +<> and + wl'u+~~ From From 'Y "),+w and o! Soo Ee 1/.")' 7/7[0 we obtain obtain 'l/I"b r + wl'u+a~ [8]] we +<» by r'l/I"b + wl'u+") by (v) (v).. Since Since OF oF(G) par(A,G)Cl ~ C_ 7/~[0] Cl n C_ r (G) Ee par(� , G) n K � 1/.")'[8] n K � 'l/I"b + l)1) �_ +<» we r'l/I,,('Y + + wl' wu+a) we obtain obtain K (-y+Wl'+a) A 1/.~/~+~.+~ "),+wl'+a [8] [0] I,rI1/!1/!r K (-y+W..+a) � from (ix) (ix) by by an an inference inference (V) (V). . from In In the the case case of of an an inference inference (Ref,,) (Ref~) there there is is aa II� II~ sentence sentence ((Vx L~)(3y L~)Y(x, )( 3y EE L,, )F(x , yy)) "Ix EE L,, such ) ] Ee �A,, an such that that ((3z L~)[z (Vx E e z)( z)(3y z)F(x,, yy)] an ordinal aoo such such that that ordinal o! )[z =f.~ 0 /\A ("Ix 3z EE L,, 3y eE z)F(x and Kn,, o!C~oo ++ 11 << a and (x) 7/~[O1 ~2_ � A,, ("Ix (Vx E e L,, L~)(3y L~)F(x, (x) )( 3y Ee L,, )F(x , yy).). 1/.")' [8] � O! By inversion inversion we we obtain obtain from from (x) (x) By 1/.")'[8, t] � � , (3y L,, )F(t , y) 3/~[0, t] ~2_ A, (3y eE k~)f(t, y) (xi) (xi) +<>o . n. By o!o E E 7/7[0 := 'Y wl'+<>o+ l ) and 1/.")'[8]] for and 'Yn 7n := 7 ++ wl' wu+~~ By /-t, #, 'Y, 7, ao ", . Let Let r/:= TJ := r'l/I"b ++ wu+~~ for all all tt eE T~. we we get get 'Yn ~n E e 1/.")'[8] 3/~[0] C_ w.. For For tt E T~ there is an an n n E e w such that for all all nn EE w E 7;, there is w such that � 3/~.[0, 1/.")'n [8, t]t] for 1)) c_� Cl('Yn 1 , 'l/Irbn 1)) and We have par(O) par(8) C_ � Cl(7 Clb ++ 1, 1 , r'l/Irb ++ 1)) stg(t) r We have Cl(Tn ++ 1, Cr(Tn + + 1)) and stg(t) << 'l/I,,'Yn. 1))nT == Cr(Tn 'l/Irbn ++ 1) EE 7"IT,[O]NT obtain stg(t) < < 'l/I,,'Yn r + 1))Fl~1) obtain par(t) � C/(Tn+l, Clbn+1, r'l/Irbn + par(t) �C_ stg(t) 1/.")'n [8]nT C_ for all T >� n.K. Hence Hence par(t) par(t) C_ for all T n. So have 1)) for for all all TT _� K. SO we we have � Cl(% Clbn ++ 1, 1 , r'l/Ir bn ++ 1)) 'Yn) (xii) Asmp(A, r ( t , , y); (xii) 8, t;t; #;/-t; n;K; 7n) y) ; (9, Asm p(� , ((3y 3y EE /L,,~ ))F(t 7;, . Observing that 'Yn + wU+"~ wl'+<>o == 'Yn+ induction hypothesis hypothesis applied for all for all tt E E T~. Observing that 7, + %+1l the the induction applied to (xi) yields to (xi) yields �")'n 1� (xiii) (xiii) �KtnH� (xiv) ( 3y eE L.~)F(t, [8, t]t] I.,rIr11/!/!KK ")'n+H �A,, (3y 1/.")'n+ 1 [e, L,,)F(t , y)y) n..,,,,+, for all tt EE T~. 7;, . Using Using the the Boundedness Boundedness Theorem 3.4.3.7 we we obtain obtain for all Theorem 3.4.3.7 (3y eE t.)F(t, 1/.")'nH [8, t]t] IrIr " "Yn+l ~ ~,� , (3y L'I )F(t, y)y) n~,,,+,[e, +<>o+ l << "y +<> we we get get for for all all t EE T~. 7;, . Since Since ~/. 'Yn << ?'Y ++ wWl'u+~~ 'Y ++ w wl'u+~ t w..+ao + 1 ) 1/.�, +wl'+ao + 1 [8] Ir1/!K1/!K ((W"+aoo +1 ,) �A,, ("Ix E L'I)( 3y E L'I )F(x , y) "H..,,+,.,.,.+..,,o+,[e] (xv) 313 313 Set Set Theory Theory and and Second Second Order OrderNumber Number Theory Theory � +O (cf. by (x EE L1)) by an an inference inference (/\ (A).) . Since Since 1lo no ~0(3x (3x EE L1)) k,)(x In) for for some some 65 << ,¢,,,(-y r ) (cf. ++ wwl'~+(') (191) below) below) we we obtain obtain (191) ,pK (-y+ l'+a ) )F(x, y)] ) (3y Ee zz)F(x, [ # 0O1\A (\Ix A, (3z Ee L~)[z (Vx Ee zz)(3y y)] 1l'Y+wl'+a [8] , r +W ) fl ,pK (-y Wl'+a , (3z L,,) z =J from (xv) (xv) by by inferences inferences (/\ ( A )) aand n d (V). (V). from In In the the case case of of an an inference inference (Ref,.. (Ref~)) with with 7r~T< < K~ we we obtain obtain the the claim claim directly directly from from the side side induction induction hypotheses. hypotheses. the The The real real crucial crucial case case is is aa cut. cut. There There we we have have aa sentence sentence A A with with rk(A) rk(A) ::; _< /l# and and an ordinal ordinal Q a0o such such that that an (xvi) (xvi) ?-/~[e] ~2_ A , A and ?t~[e] ~2_ A,-~A. The The simple simple case case is is rk(A) rk(A) < K. a. Since Since 1l'Y[8] n~[O] is is Cantorian Cantorian closed closed and and par(A) par(A) � C_ 1l'Y[8] 7-/7[0 ] l'+O ) . 9This we + we get get rk(A) rk(A) EE 1l'Y[8] HT[O] n n Ka � c_ '¢' r "b + + 1) 1) ::; _< '¢'"b r +w w~+") This together together with with the the side side induction hypotheses hypotheses applied applied to to (xvi) (xvi) yields yields the the claim claim by by aa cut. cut. induction rk(A) ::; Now Now assume assume Ka ::; __ rk(A) _< /l. #. First First we we consider consider the the sub-case sub-case that that rk(A) rk(A) (j. Reg. Reg. Then Then Kg ::; _< rk(A) rk(A) < rk(A) rk(A) ++ =: ='~T7r ::; _< /l. #. As As before before we we have have rk(A) rk(A) E E 1l'Y[8] ~7[O] -.A} � and thus thus also also 7r r E 7/7[0 by (199) (199) and and trivially trivially � A U U {A} {A} U U {{-~A} C_ �,.. E ' . . Hence Hence and 1l'Y[8]] by -.)A; O; Asmp(A, ((-~)A; #; 7r; ~T;,) 7) and and the the induction induction hypothesis hypothesis applied applied to to (xvi) (xvi) yields yields Asmp(fl, 8; /l; 7-/7+,.,,.+,,,0[0] Jr176 'r ' A and nT+.,,`+o,o[O] 1rr176 -~A. (xvii) In this this situation to make cut, apply apply the the Predicative Predicative Elimination Elimination Lemma Lemma In situation we we want want to make aa cut, (Lemma and then then use the main Since the the same (Lemma 3.4.3.6) 3.4.3.6) and use the main induction induction hypothesis. hypothesis. Since same situation we are state this in aa more situation will will return return we are going going to to state this in more general general form. form. � l'+o , ~l E ?-/,[O], rk(A) << 7r Let "7 , <_ ::; ~1] < "7 ,+ ::; #, /l, 7-/,7[0] 1l1)[8] ~ A, fl , AA and and 7r _< Let + w"+~, 1] 1l1)[8] , rk(A) 1l1)[8] n,[o] �f3f3 fl , -.A for some Z(3 < 7r .9 Then 1l-y+wl'+a [8] ,pK (-y+Wl'+a ) fl . I, r A I r,pI,f'K,,(('Y+ 7 "i- ~Jl "~"0' )) W,~,`l'+a (xviii) (xviii) l'+OO ) to w l'+oo and/? Applying (xviii) (xviii) with with rl1] == ff, ++ w'+~~ and (3 == r'¢',.. b ++ w"+~~ to (xvii) (xvii) will will then then finish finish Applying the the case. case. := max{rk(A),/~} max{rk(A) , (3} ++ 11 << r7r and and choose choose pp eE Reg Reg such such To prove prove (xviii) (xviii) put put 56 := To Defining t~ otherwise we we get get that pp << 56 << p+. p+ . Defining := ppi fifp p r� Reg Reg and and jh:= p := pp++l 1 otherwise p := that ItS, 6) nn Reg [p, ~p ++ wWO) Reg == 0. 0. From From the the hypotheses hypotheses we we obtain obtain ,f3H 1l 1) [8] ',~+J fl ?-/,110] ~ w' A p+ (xix) (xix) by aa cut. cut. Since Since par(A) par (A) C_ � 7-/,7[0] 1l1)[8] and and ~(3 eE 7-/,110] 1l1)[8] we we have have 56 eE n,[O]. 1l1)[8] . Using Using the the by Predicative Predicative Elimination Elimination Lemma Lemma (Lemma (Lemma 3.4.3.6) 3.4.3.6) we we therefore therefore obtain obtain n,,[o] V35(~+1) A . (XX) ( xx) From ff, <::; r]1] we By (199) (199) we we get get ~p EE 7/,[O]. 1l1)[8] . we also also obtain obtain par(O) par(8) _C � By From Cl(,+l , r'¢'T ( ' ++ 1)) 1)) c� Nr>~ Cl( 1]+l, Or(r] '¢'T ( 1] ++ 1)). nT>" C/(-y+l, Sowe wehave have Asmp(A; 1)). So nT>" C/(rl+l, ~r>~_ Asmp(�; O; 8 ; t~; p; ~;K; r/) 1]) and andf~p << #./l. The The main main hypothes]-s hypothesls applied applied to to (xx) (xx) thus thus yields yields 314 314 w. Pohlers Pohlers W. r ~6( )) (xxi) " l ) << co". Since p + ~o6(fl <(J/j((3 ++ 1) 1) << r'Tr <::; #J.L we wJJ. From From r/< 'T/ < 7'Y++wwJJ+<> we either either WP+105(fH we have have w Since/5+ ~+~6(~+1) "+~ we l ::; 7'Y and and thus thus also also r/+w 'T/+WP+105(fH get 77 'T/ <_ 'Y+wJJ _< ::; 7+w 'Y+wJJ+<> get ~+~(~+1)) << 7+w" "+~ or or 77 'T/ == 7+~ with r( << co"+~ 'Y+ ( with l) << 7'Y ++ r( ++ cow l )) _< WP+105(fH which entails entails 77 'T/ ++ w wJJ << 7'Y ++ w"+~. wJJ+<> . Hence WP+105(f1+ which ~+~(~+1) ~+~6(~+1)) Hence r'l/J,,('T/ + w ::; wJJ+<» and and we we obtain obtain the the claim claim from (xxi) by by aa structural structural rule. rule. r'l/J,,(-y ++ w"+~) from (xxi) ::; rk(A) Now we we consider the sub-case sub-case that that ~K, <_ Now consider the rk(A) =: =: r'Tr EE Reg. Reg. Then Then eiei ther A A or or ~A --,A has has the the shape shape ((3x L,,)G(x) . Since Since ~K, _::; ~r 'Tr we we easily check ther 3 x eE L~)G(x). easily check Asmp(A, (3x (3x eE L~)G(x); L,,)G(x) ; O; #; J.L; r;'Tr; 7) 'Y) and and obtain obtain thus thus by by the the induction induction hypothesis hypothesis wJJ+<> Asmp(�, 8; "',,(,+wl'+ao ) A, (3x eE k~)V(x). L,,)G(x). 1i-y+Wl'+ao [el [8] I'rIr176 ?-/.y+.,.+oo ",,,(-y+wl'+ao ) �, (Sx +<>o ) we Defining r( := := r'l/J,,(J.L ++ co w-Y"Y+~~ we have have Defining ( eE 7/~+~,+~o[O] 1i,+Wl'+ao [8] nn lr'Tr (xxii) (xxii) (xxiii) (xxiii) by (iv) and and get by the the Boundedness (Theorem 3.4.3.7) 3.4.3.7) by (iv) get from from (xxii) (xxii) by Boundedness Theorem Theorem (Theorem "'"(-y wl'+ao + ) �, (3= n~+~.+oo[O] Lr G(x). 1i-y+wl'+ao [8] IIr176176 r",,, (-y+wl'+aO ~ ' (3x eE Ld ) (xxiv) (xxiv) Respecting (xxiii) (xxiii) we may apply apply Downward second premise premise Respecting we may Downward Persistency Persistency (153) (153) to to the the second in (xvi) obtain in (xvi) and and obtain 1i-y+wl'+ao [8] �, (xxv) (xxv) +wJJ+<>o) we obtain obtain Asmp(A, Asmp(�, ((Vx Since 'Y 7 < < 'Y ")'+wJJ+<>o + w"+a~ eE 7/~[(~] e Ld--, Lr G(x) ; O; "+e~ 8; J.L#;; r;'Tr; "y'Y +w Since "Ix E 1i-y[8] we and may therefore apply the side side induction This yields yields and may therefore apply the induction hypothesis hypothesis to to (xxv) (xxv).. This "'" (-y+wl'+"o +wl'+ao ) �, e Ld--, Lr G(x) . (xxvi) (xxvi) 1i"/L~+,,.,,,+,,o+,,,,,+,,o "Ix E -y+wl'+aO +Wl'+a0 [8] [~1] I",'~'r176176176176 r ,, (,+wl'+"O +wl'+"0 ) ,a, ((v= +<>o ++ wJJ+<>O) Putting := 'Y7 + := 'l/J obtain from Putting 'T/ 77 "+ wJJ+<>o c~176 + + wJJ+<>o w"+~~ and and (3 fl "= r ,, ('Y + + wJJ co"+~~ w"+~~ we we obtain from ("Ix eE L()~C(x). 7/~+~,+.o[e] ~Ld--,G(x) . � A, (Vx (xxiv) and (xxvi) (xxiv) and (xxvi) 1i~[e] z~, (3x (~ Ee LdG(x) ~)c(~) '1[8] �~ �, and a~ 1i'1[8] ~[e] � ~ �, ~, ((w ~)~a(~).. "Ix Ee Ld--,G(x) (xxvii) (xxvii) We We realize realize that that rk((3x rk((~x E ~ LL~)G(x)) < 'Tr, ~r, (3 fl < < 'Tr, ~r, 'Y ~/ ::; _ 'T/ 77 < < 'Y -~ + + wJJ+o w"+~ and and obtain obtain by by d G(x)) < (xviii) 0 (xviii) the the claim. claim. ~] One should notice procedure not One should notice that that the the collapsing collapsing procedure not only only collapses collapses the the derivations derivations but but also also removes removes applications applications of of the the rules rules (Ref,,) (Ref~).. 3.4.6. 3.4.6. Controlling C o n t r o l l i n g operators o p e r a t o r s for for axiom a x i o m systems s y s t e m s of of set set theory theory The The aim aim of of the the following following section section is is to to determine determine controlling controlling operators operators for for different different axiom axiom systems systems for for Set Set Theory. Theory. We We will will see see that that this this is is aa fairly fairly straightforward straightforward procedure procedure which which parallels parallels the the last last part part of of Section Section 2.1.5. 2.1.5. Due Due to to extensionality extensionality of of sets, sets, however, however, it it will will turn turn out out to to be be more more painstaking. painstaking. We We start start with with pure pure logic. logic. Controlling C o n t r o l l i n g operators o p e r a t o r s for for pure p u r e logic. logic. First First we we show show an an analogue analogue of of (95) (95) 315 315 Set Theory Theory and and Second Second Order Order Number Number Theory Theory Set 3.4.6.1. Lemma. L e m m a . Let Let � A be be aa finite finite set set of of .cRS f~Rs-sentences and F F be be aa sentence sentence such such 3.4.6.1. -sentences and ( r 2 F . k ) 2.rk(F) � A holds holds.for 74.. that -,F} �C_ � [par(�)] 110 for every acceptable operator tl that {F, {F,-~F} A.. Then Then tl 74[par(A)] 0 The proof proof is is by by induction induction on on rk( rk(F). Without loss loss of of generality generality we we assume assume F F EE The F) . Without -type. Then we have V V -type . Then we have 2.2"rk(G) rk(G) � A , G, G,~G -,G tl [par(�, G)] 0 7-/[par(A, G)] 1Io ((i) i) for all all G GE EC C (F) (F) by by the the induction induction hypothesis hypothesis and and obtain obtain first first for k( · r G) + l � -,G A , F, F,-~G [par(�, G)] n[par(A, G)] 1� Io~.rk(a)+l tl ((ii) ii) for (F) by for all all G G E E C C(F) by an an inference inference ((V V )) and, and, since since 22.· rk(G) r k ( G ) ++ 11 < < 22.· rk(F) rk(F) and and par(A, G) G) � c par(� par(A U u tF tF(a)), from ((ii) finally par(�, (G) ) , from ii) finally 2. rk(F) � A , F, F, -,F ~F tl [par(�)] 112"rk(F) , 00 , D [-1 by an an inference inference (/\) (A).. by For aa finite finite set set � A of of .cRS-sentences LRs-sentences we we define define For (183) (183) rk(A) := := max max {{rk(F)] F E E� A}. rk(�) rk(F) I F }. Now recall recall the the cut-free cut-free Tait-calculus Tait-calculus introduced introduced in in Section Section 2.1.2. 2.1.2. We We obtain obtain the the Now following lemma. lemma. following 3.4.6.2. Let (E, Ad) -formulas whose 3.4.6.2. Lemma. Lemma. Let �(XI A ( x l ,, .,. ,. ,, X, Xn) n ) be be aa finite finite set set of of .c f_.(E, Ad)-formulas whose free variables variables occur occur all list {Xl,... such that that ~-( x l ,'. .. .. . ,,Xn). For any p. A�(XI xn) . For any free all in in the the list , xn } such {Xb ' . . ,Xn} A, any any n-tuple n -tuple ((al , an) of of LRs-terms .cRS -terms of ordinal and any than A)~ and ordinal )~, a l ,, ... .. . ,an) of stages stages less less than any acceptable operator acceptable operator 74 tl we we obtain obtain 12.(a+m) A(all ", ..' " , an an)) LLxA ) LA )] Io02.(~+m) �(a tl [par(�(al " ' " an 7-/[par(A(a,,..., an)Lx)] for k ( A (ab ( a l , .. ... ., , an)Lx). := rrk(� an) LA) . for aa :-- The is by induction on We abbreviate abbreviate � A ((aI a l , ,. .. .., a. ,nan) ) LA L~ by by �' A'.. In In the the The proof proof is by induction on m. m. We case of an axiom ( AxL ) we obtain the claim from Lemma 3.4.6.1 . In the case case of an axiom (AxL) we obtain the claim from Lemma 3.4.6.1. In the case of (V) there is a a sentence ( a l,, .. .. .., a, an) n ) LA Lx V V Al A l ((aI a l ,,... .. ,. a, nan) ) LL~ A'. of an an inference inference (V) there is sentence A Ao0(aI A EE �'. Then par(A') and rk(Ai(al,...,an) L~) < rk(A') and we Then par(Ai(al,...,an) par(Ai (ab . . . , an) L~) LA ) C_ par(�') and rk(Ai (al , . . . , an)LA) < rk(�') and we � I�·(a+mo) 12'(a+mo)A', get from from the the induction induction hypothesis hypothesis 7/[par(A')] tl [par(�')] ,0 �', AAii ((aa b get l , .. ... ,. , an) for some some an) L~ LA for (V) I�·(a+m) E {0, {O, 1}. I } . By By an an inference inference ( V ) we we finally finally obtain obtain 7/[par(A')] tl [par(�')] [02.(~+m) A'. ii E �' . The The case case of of an an inference inference (A) (1\) isis treated treated analogously. analogously. In the the case case of of an an inference inference (3) (3) there there are are two two subcases. subcases. First In assume that First assume that we we are are in in the the situation situation of of an an unrestricted unrestricted quantifier. quantifier. Then Then there there is is aa sentence sentence LA EE C((3x (3x L~ E For aa E have F(a,~) L>.)F(x, a)LA E A'. �' . For E T~ T>. we we have F(a, a) L~ C ( (3X EE L~)F(x,~)L~), L>.)F(x, a) LA ) , (3X EE L~)F(x,~) L L stg(a) << A A) == stg(a) (F(a, d) a) L~) A _< and rk(F(a, rk(F(a, d)) a)) << a. O(3xeL~)f(x,~)L 0(3xElA )F(x,ci) ~A (F(a, ::; aa and a. Moreover, Moreover, we we have have 316 w. W. Pohlers Pohlers a)L>.) par(L\'). � L\(x), F(y, x). y E {Xl, . . . , X } par(F(ai, m 2. o a ( ) + y Lo. 1£ [par(L\')] 1 0 L\', F(a, a)- L>. (V ). (::Ix E ai)F(x, a)L>. L\'. ai {x E Lo l G(x, a)} . � L\, y E Xi F(y, x) m o ·(a ) + 1£ [par(L\')] I� L\', a E ai F(a, a) for subcase. By -inversion this implies for aa E E 0. T~ as as in in the the previous previous subcase. By 1\ A-inversion this implies mo mo 2. 2. a a 2.(aTmo) ( ( ) ) + + A' ~aa E e ai ai and and 1£[par "]-/[par(A')] A', F(a, F(a, a) d) . We We easIly easily prove prove L\', L\', 1£"]-/[par(A')] [par(L\')] 112.(~+m~ 00 (L\')] 1100 1£ fi L\, a E {x E Lo l G(x)} 1£ fi L\, G(a) ( 84) (184) l m ·(a+ o )+ L\A'' , G(a, G(a, a) ~) 1\ /X F(a, F(a, a) ~) we obtain [par(L\')] I10�2.(.+mo)+~ by by induction induction on on a. Using Using (184) (184)we obtain 1£ "]-/[par(A')] and obtain the claim by and obtain the claim by an an inference inference ((V). V ). In In the the case case of of an an inference inference (\1') (V) we we have the premise premise � ~ L$x), A(~), F(y, F(y, x) :~) with with yy � r have the ·a+mo L\', F(b, a)L>. ... } The induction hypothesis {Xl, {xl,...,xn}. The induction hypothesis implies implies 1£[par(L\'), 7/[par(A'),b] A',F(b,d) t~ b] IIo�2.a+mo ·a+m L\'. /V. ['7 for £ [par(L\') ] I10�2.a+m for all all bb E E 0. T~.. Using Using an an inference inference (1$ (A) we we obtain obtain 17/[par(A')] then par(F(a,,~) L~) � the premise If y e {xl,...,xn} C_ par(A'). n then the premise ~ A(Z),F(y,Z-). If 2.(a+mo ) Otherwise Otherwise we we replace replace y by by L0. In In both both cases cases we we get get 7/[par(A')] 10 A', F(a, g)L~ by induction hypothesis by the the induction hypothesis and and obtain obtain the the claim claim by by an an inference inference (V). In In the the case case of of an an restricted restricted quantifier quantifier there there is is aa sentence sentence (3z E ai)F(x, d) L~ in in A'. Assume From the Assume that that a, = = { x e L~I G(x,~)}. From the premise premise ~ A,y e ~i 1\ A F(y,~) 2.(~+mo) A', a E ai 1\ A F(a, ~) we induction hypothesis we obtain obtain by by the the induction hypothesis 7/[par(A')] 10 _ . '* ct . , Xn . 0 Regarding Regarding identity identity axioms axioms as as part part of of Pure Pure Logic Logic the the next next step step is is to to deal deal with with these that, because these axioms. axioms. We We already already mentioned mentioned that, because of of the the extensionality extensionality of of sets, sets, this this simple. The is by by far far not not simple. The tedious tedious point point is is the the bookkeeping bookkeeping of of derivation derivation lengths. lengths. is To To obtain obtain precise precise bounds bounds we we are are forced forced to to derive derive all all axioms axioms step step by by step step which which is is aa bore. However, bore. However, we we do do not not need need absolutely absolutely exact exact bounds. bounds. The The collapsing collapsing procedure procedure will will equalize equalize too too precise precise bounds bounds anyway. anyway. We We already already observed observed that that the the rank rank of of aa sentence bound for introduce sentence is is always always an an upper upper bound for its its truth truth complexity. complexity. So So we we will will introduce aa more more liberal liberal derivation derivation calculus calculus � ~ A and and show show afterwards afterwards that that � ~ A entails entails 7/[par(A)] ~ A where where ct a is is computable computable from from rk(A). For be the successor, i.e., For aa set set e (9 � C_ On On we we define define e O to to be the closure closure of of e (9 u tJ {w} under under successor, i.e., �H ~ �~ + + 11 and and regular regular successor, successor, i.e., i.e., �~ H ~ �~+.. We We define define the the relation relation � ~- A by by the the rules rules L\ 1£ [par(L\)] � L\ (1\ (A')' ) L\ rk(L\). + {w} L\ � L\, GGlorallG~C(F) for all G E C(F) '* [-A, ~ � ~ AL\, , FF and and ((V') V') � ~- L\, A, f F, C (F) and and par(L\, par(A, f F)) � C_ par(L\, par(A, F) F) =~ � ~- L\, A, F. F. fF �c_ C(F) Here . . , toto denote multi-sets, i.e., Here we we want want L\, A, ...., denote multi-sets, i.e., sequences sequences which which are are independent independent from but count from the the order order of of their their elements elements but count their their multiplicity. multiplicity. To To avoid avoid distinctions distinctions by by cases cases we we introduce introduce for for aa E E Tstg( Tstg(b) the relation relation b) the G(s) ifif bb == {x{x Ee LL~Ia I F(x)} F(a) 1\ A G(s) F(x)} (185) (185) a € b 1\ G(s) . { F(a) if G(s) if bb = - LL,. a. '* , . Dually we Dually we put put ¢:} 317 317 Set Theory Theory and Second Second Order OrderNumber Number Theory Theory ' s ) .:r¢:> aa rfL bb VY G( G(s) {{ G(s) ,F(a) -~F(a) V V G(s) G(s) G(s) if if bb = = {x {x EE Lo k, lI F(x)} F(x)} if bb = = Lo' k~. if (186) (lS6) For multi-sets multi-sets we we defi define analogously For ne analogously { )} S {{.'.,-~F(a),G(s),...} ifbb = = {x {z E Lo k~[I F(x F(x)} . . . ' 'F ( a) , G ( s ) , . . . } if {( .. ... . ,, aa I-f b,b, G(s G ( s ) ,), .. .. ..}} := : - ~ { {. . . ,,G(s),...} i f bb -=k ~Lo. . if (s ) , . . . } G This has has the the notational notational advantage advantage that that C C ((aa EE b) b) = = {t = aa lI stg(t) stg(b)}} This {t (e bb 1\A tt = stg(t) < stg(b) stg(t) < stg(b) and C C((3x b)F(x)) = - {{tt d e b 1\ A F(t) F(t)l I stg(t) stg(b)}} independent independent of of the the shape shape of of and ( (:lx E b)F(x)) b.b. There is is aa number number of of inference inference rules rules which which are are derivable derivable or or admissible admissible within within the the There calculus � ~ . . We We list list the the most most important important ones ones calculus ((Str) Str) ~ A� and andA C_F =~ � ~-Fr � �� r =} ~ A, A,-~A ((Taut) Taut ) � ,A ~ A�, , AA =} =~ � ~ A�" , ~ BB, , A Sent ) � A AI\BB ((Sent) (E) (E) ((ri ) �, aa EEb b ~-A, f oar sfor o msome e t E TtsEt gTstg( ( b )b) =} =~ � ~--A, � �, tt e(bbA1\t -ta= ~- �, A, tt I-f bb,, tt =I: # aa for for all all tt E Tstg( %tg(b) :=~ � ~- �, A, aa i r bb � b) =} (V0~)) � ~- �, A,F(t) for all t E To. T~ =} =~ � ~ A�, , ( V(\Ix x EEL ~Lo)F(x) )F(x) F(t) for (\1 F(Lo )) and (3 0~)) tt E E To. T~,, par(t) par(t) � C_ par(�, par(A,F(Lo)) and � ~- �, A , FF( ( tt)) =} ::~ (:1 (Vbb)) (\l b) (~b) (:l � ~- �, A, ((3x La)F(x) :Ix E Lo)F(x) ~- A, b, F(t) F(t) for for all all tt E Tstg( Tstg(b) =~ � ~- �, A, (\Ix (Vx E b)F(x) b)F(x) � �, tt/[I- b, b) =} Tstg(b), par(t) � C_ par(�, par(A, F(L F(L0)) and � ~ A, A F(t) F(t) �, tt (e bb 1\ tt E Tstg( o )) and b) , par(t) =} � �, (:Ix E b)F(x) We as Structural as Tautology Tautology Rule, Rule, to We refer refer to to (Str) ( Str) as Structural Rule, Rule, to to (Taut) (Taut ) as to ((Sent) Sent ) as as Sentential Rule to ((E), E ) , (r (i) as as E-rule E -rule or The proofs proofs are all obvious. obvious. or i~-rule, Sentential Rule,, to etc. The are all -rule, etc. For aa multi-set multi-set � A of of/:as-sentences we define define For eRs -sentences we ~:A "~----~Ew,k( Wrk(F) F) #� := FEA FE Ll / (V') and observe observe that according to to (A') V\ ) and we always #�p << #Ac #�c and that in in rules rules according and (V') we always have have #Ap the rule. be the main if premise and and Ac �c the the conclusion conclusion of of the if �p Ap denotes denotes aa premise rule. This This will will be the main argument in showing argument in showing Let Let 74 1-£ be be an an acceptable acceptable operator operator which which is is closed closed under under ~� ~+ f--t ~+. �+ . 1-£ [par(�)] 0 ~ A. �. Then �A � implies implies 74[par(A)] Then ~- 3.4.6.3. emma. 3.4.6.3. LLemma. � proof isis by by induction induction on on the the definition definition of of ~� A. � . In In the the case case of of an an inference inference The The proof I V\ ) we we get get the the claim claim immediately immediately from the induction induction hypothesis, hypothesis, the the previous previous (A') from the In remark have par(G) G E C(F) C (F) we we have par(G) c� par(F par(F U U tf(G)). remark and and the the fact fact that that for for G tF(G) ) . In (V') 1�(Ll ,r) the case case of of an an inference the inference ( V ' ) we have 7/[par(A, ,0 A - C(F) we have 1-£ [par(�, F)] r)] I#(A'r) � ,, Fr ,, Fr C C (F) and and � 318 318 W. Pohlers Pohlers W. the par(A, par(L\, F) par(L\, F). F) . Since Since ?11./ i sis Cantorian Cantorian closed closed and and closed closed under under ~c � ~_~ t-+ ~c+ �+ the f) C_ � par(A, : (A,r) AL\, , Ff _� ?/[par(A,F)]. latter implies implies ?/ 1I. [par(A, [par(L\, F)] r)] C 11. [par(L\, F)] . So So we we get get ?/[par(A, 1I. [par(L\, F)] F)] I,0 I#(~'r) latter : (A rl+ 1r l A F where IFI denotes the cardinality of the L\ , F where If I denotes the cardinality of the which entails entails ?/[par(A,F)] 1I. [par(L\, F)] l0 1#(~,r)+lrl , which finite rk(A) ++ iFI rk(f),, finite set set F. f . Since Since rk(A) rk(A) << rk(F) rk(F) for for all all A AE E CC (f) (F) we we get get ~ w W,k(A) If! << w W,k(F) AE r AEF 0 hence #(A, #(L\, F) f) ++ IFI If I << #(A, #(L\, F) F) and and this this yields yields the the claim. claim. hence D E We are are now now going going to to derive derive aa series series of of sentences sentences which which will will be be needed needed in in the the We computation of of the the truth truth complexities complexities of of identity identity and and non-logical axioms of of Set Set computation non-logical axioms Theory. Theory. � aa ri aa for for all all s£RS - terms terms a. a. (187) (187) The proof proof is is by by induction induction on on rk(a). rk(a) . We obtain ~� bb ~i bb for for all all bb EE Tstg(,) by the the The We obtain Tstg(a) by induction hypothesis. hypothesis. Hence Hence ~-b � b f/. a, a, bb ef aa A /\ bb ~( i bb by in turn gives induction by (Sent) (Sent) which which in turn gives � a, (3x (:Jx E E a) [x ~( i b] b] by by (3b). (:Jb ) . This This implies implies � a, b ~=I- aa for all bb EE Tstg( ~-bb /. X a, a)[x ~-bb f/. a,b for all Tstg(,). a) ' � a i~( aa by by (i) Hence ~-a 0 Hence (r . D � aa � hence also also ~� aa = for all all seRs -terms aa C_ aa hence = aa .for (188) (188) The proof rk(a) . We We get for all E Tstg( The proof is is by by in in induction induction on on rk(a). get � ~-bb � C_ bb for all aa E Tstg(,) by the the a) by induction induction hypothesis. hypothesis. For For symmetry symmetry reasons reasons this this implies implies � ~ bb = = b. b. So So � ~ bb /. f a, a, bb fe ac a/\b -b = bb by (Sent) and by (Sent) and � ~-bb /. t a a, , bb E E aa for for all all bb E E Tstg( T~tg(,) by (E) (E) which which implies implies a) by 0 � [x E a] ~- (\Ix (Vx E a) a)[x a] by by (\la). (V'). D As As aa corollary corollary of of the the proof proof of of (188) (188) we we obtain obtain � F- b /.,~ a, a, bb Ee aa for for all all bb Ee Tstg( Tstg(~). a) ' (189) (189) � b aa =I:fi b, b, bb = = a. a. ((190) 190) By [x ir b], [x i~ a], [x EE a] [x EE By (Sent) (Sent) we we have have � ~- (:Jx (3x E E a) a)[x b], (:Jx (3x EE b) b)[x a], (\Ix (Vx EE b) b)[x a] /\ A (\Ix (Vx EE a) a)[z b] b] which which entails entails Since Since we we have have � ~- bb = bb for for all all bb EE 'To T~ we we get get � b bb E E La L~ for for all all bb EE 'To T~.. (191) (191) Now Now we we show show � [--- 7mn(La). Tran(L,~). (192) (19 2) For For aa EE 'To T~ and and bb EE Tstg( Tstg(a) we get get stg(b) stg(b) < < a. e. Hence Hence by by (191) (191) � ~- bb /.f a, a, bb EE La L~ which which in in a) we turn [y EE La] a ) (\ly EE x) [y EE La] turn gives gives � ~- (\ly (Vy EE a) a)[y L~] for for all all aa EE To T~.. Thus Thus � ~- (\Ix (Vx EE LL~)(Vy x)[y L~].. 0 D We We prove prove the the next next item item stg(b) < 8 =} stg(b)<~ =~ � b aa# b=I-, b, L6 L 6=I-# aa (193) (193) by by induction induction on on 8. &. Put Put (3fl := stg(b) stg(b).. For For tt EE 7r:J 7~ and and ss EE Tstg( ~tg(a) we obtain: obtain: a) we � ~- ss =I:~ t,t, L,(3 L~ =I# ss by by induction induction hypothesis hypothesis (i) (i) Theory and Second Order Number Theory Set Theory 319 319 ~- tt i1[b, b, tt =1= 7s s, s, L,B 1/3 =1= ~ ss for for all all tt Ee T.a T~ from from ((i) by ((Str) i ) by Str) f- ((ii) ii ) ~ - ss ~¢b b, , L,B L z =1= C ss from from(ii) b y ((¢) r f( ii) by iii) ((iii) ~ - Ss e€aaA /\ s Csb ¢, b, ss;ga, ks ~=1=s s from from(iii) by(Sent) ( Sent ) fi a, L,B ( iii) by iv) ((iv) ~- (3x (3x E e a)[x a)[x ¢ r b], b], ss i f a, a, L,B LZ =1= ~ s from from ((iv) by (3a~)) iv) by f- v) ((v) S i n c eaa ==b b- a==C ba A�b bC /\ a - b( V�x ae a==) [ ("Ix x E bE] a) [x E b] /\ A ("Ix (Vx Ee bb)) [x [x Ee a] a] we we may may continue continue Since ' f- aa =1= v) by ~ b, b, ss i f a, a, L,B Le =1= ~ ss for for all all ss E e Tstg( Tstg(a) from ((v) by (V ( V ' )) a) from ((vi) vi ) ~ r ¢ a from from(vi) b y ((¢) r f- aa #=1=b ,b,L L,B (vi) by ((vii) vii) ~- aa =1= ~ b, b, ((3x L~)[x a] from from ((vii) by (3°) (3 ~) f3x EE Lo) vii ) by [x ¢r a] ((viii) viii) ' D r-1 fvii) by o =1= ~- aa =1= -J=b, b, L[_~ :/= aa from from ((vii) by (V (V').). The most most painstaking painstaking part part iiss the the proof proof of of the the following following theorem theorem The 3.4.6.4. Equality Equality Theorem. Theorem. 3.4.6.4. and bb we we have have aa and For -terms For every every £RS s -sentence F(Lo) F(k0) and and all all £RS f-.Rs-terms ~ b, b, , ~F(a), F(b). F ( a) , F (b). f- aa =1= To To prepare prepare the the proof proof we we show show aa more more general general lemma lemma which which entails entails the the Equality Equality Theorem. Let Let aa =1=' -J:' bb denote multi-set ,a --a � C_ b, b,--b C_ a. Theorem. denote the the multi-set ,b � Assume -formula in Assume that that A A ((XI X l ,, .. ... ,. x, xn) n ) is is aa D.o Ao-formula in £(E, /:(E, Ad) Ad) in in which every variable variable X at most an and b l ,, .. ... ., , bn Xll ,, .. ... ,. , Xn occurs at most once once and and let let aa Il ,, ..... , , an which every and bl bn Xn occurs be/:as-terms. be £RS -terms. Then Then 3.4.6.5. 3.4.6.5. Lemma. Lemma. . f- al al =1=' , . . . , an an yk' =1=' bn, bn, ~,A n ) , A(b~,..., A (bI , . . . , bn) ~k' bI bl,..., A ((al a l ,, ... .. ,. , aan), bn).. We prove the lemma by k ( A (al ( a l ,, .. .. . ., ,aann))))~ # rk(A(bl,...,bn)) disWe prove the lemma by induction induction on on rrk(A rk(A (bl , . . . , bn) ) and and dis tinguish following cases: cases: tinguish the the following A (XI , . . . , xn) === Xl E x2. X2 . For stg(a) < and stg(b) stg(b) < stg(b2) we have A(Xl,...,xn) xl E For stg(a) < stg(a2) stg(a2) and < stg(b2) we have rk(a == al) ad < rk(al eE a2) and rk(b rk(b == bl) bl ) << rk(bl rk(bl eE b2). b2 ) . So So we we get get for and rk(a < rk(al a2) and for aa eE %tg(a2) Tstg(a2 ) and E Ts bb E Tstg(b2) tg(b2 ) bl by by induction induction hypothesis hypothesis f- al al 7=1=k'' bl, all aa ~s =1=' b,b, bb -= bl bl, aa ~=1= al, f. =1=' b, b, bb e b2 b2 A /\ bb aI, bb fI b2, b2 , aa ~-' f- elal :/:'=/:-' bl,bl , aa :/:=1= al, = bl from ( i ) by by (Sent) ( Sent) bl from (i) (i) (i) (ii) (ii) =1=' bl, f- al al -J:' aI, bb fi b2, b2, aa r=1= b, b, bl bl eE b2 ( ii ) by by (e) b I , aa -J: =1= al, b2 from ( E ) and and (V') (V' ) from (ii) ( iii) (iii) f- al al ~' =1= ' bl, all aa r¢ b2, b2, bl bl eE b2 bb aa -J: =1= al, b2 from from (iii) (iii) by by (r (¢) (iv) ( iv) a2 ) f- al =1=' bl, al ~' a2, aa -J: =1= al,-,(a2 aI, ,(a2 C_ bl , aa f'i a2, � b2), b2) , b~ bl eE b2 b2 from (v) by by (3 (3a'-) from (v) (vi) (vi) f- al al ~:' bl, aa ti a2, a2, aa r=1= al, all aa ef. a2 a2 A =1=' bl, b2, bl bl eE b2 b2 from ( iv) by by (Sent) ( Sent ) /\ aa r¢ b2, ~from (iv) ( v) (v) 320 320 W. Pohlers from (vi) (vi) by by (~). ( ¢. ) . ~-al a2, bl bl EE b2 b2 from bl , a2 a2 -J:' b2, al al ~¢. a2, I- al r#' bl, #' b2, For ~/'i, <::; min{stg(al),stg(bl)} min{stg(al), stg(bl) } ='1~ =: (3 we we have have A(Xl,...,Xn) For A (XI, ' " , xn) -== Ad(xl). Ad(XI ) ' rk(LIt == al) al) << rk(Ad(al)) rk(Ad(al)) and and rk(L~ rk(LIt == bl) bl) << rk(Ad(bl)). rk(Ad(bl) ) . Therefore Therefore we we obtain: obtain: rk(L~ bl , L~ Lit r# al, all L~ Lit -= 51 bl for for all all ~/'i, <_ ::; ~(3 ~I- al al ~' #' bl, by induction induction hypothesis hypothesis by al ~-' I- al #' bl, bl , L~ Lit r# al, aI , Ad(bl) Ad(bl) from from (vii) (vii) by by (V') (V' ) I- al bl , L~ Lit ~# al, all Ad(bl) Ad(bd ]or for all all ~/'i, < ::; stg(al) stg(al) al -J=' #' bl, (vii) (vii) (viii) from from (viii) (viii) and and (193) (193) (ix) (ix) (/,1) I- al #' bl , -,Ad(al), Ad(bd from (ix) by (A'). For A(xl,...,xn) A (XI , ' " , xn) -== A0(xl,...,Xn) AO (Xb ' " , xn) A /\ Al(Xl,...,Xn) AI (XI , . . . , xn) we we obtain obtain the the claim claim easily easily For from the the induction induction hypothesis. hypothesis. from A A= == (3y (:3y E E Xl)B(x2,...,xn, xI)B(X2, . . . , Xn, y). y) . Putting Putting ~a2 "= := a~,...,an a2, " " an and and defining defining b~ b� analoanalo E Tstg(bl)" Tstg(bd : and bb E gously we obtain for for aa E E Tstg( Tstg(al) we obtain gously ad and ~- a~ -~' b~, a g:' b, ~B(a~, a), S(b~, b) by by induction induction hypothesis hypothesis (x) (x) � , bb ~I bl, ~A B(~, from (x) by (Sent) (Sent) (xi) (xi) b) from a), bbef bl bI /\ (x) by #' b~, b, -,B(a2 , a), I- ~a2 ~' bl , a r#' b,-,B(c~, B(b� , b) � , bb tI bl, y) a), (3y (:3y E E bl)B(b~, bl)B(b� , y) b, -,B(a2 , a), a2 ~:' bl , aa -~ #' bb~, I- a~ # b,--B(a~, from (xi) (xi) by by ((3:3bhi)and (V')' ) from l ) and (V (xii) � , a r¢. bl,-,S(c~,a), a2 r#' b~,a (:3y E E bl)S(~,y) bdB(b� , y) bl , -, B(a2 , a), (3y ~- c~ I- from (xii) by from (xii) by ((r¢. ) (xiii) (xiii) /\ aa r¢. bl, al A b� , aa ef al (:3y E bdB(b� , y) y) I- a42 #' bl , aa tl al,-,S(4, ab -,B(a2 , a) ~#' g, a),, (3y E bl)B(~, from (xiii) (xiii) by by (Sent) (Sent) from (xiv) (xiv) � , -,(al � � , yy)) a), (:3y all -,B(a2 , a), I2 #' ~- aa~ =/='bb~,--(al c_ bl), bl), aa I ,/al,--B(a~, (3y EE bl)B(b bl)B(b~, from (xiv) (xiv) by by ((3:3aa'l ) from (xv) (xv) (xii) � , y) � , al y) ~:' bb~, al #' ~-' bl bl,, ((Vy E al) al)~S(4, y), (:3y (3y EE bdB(b bl)S(~, I- aa~ Vy E 2 #' -,B (a2 , y), a l ) and from from (xv) (xv) by by (v (V~') and (Str) (Str).. Since the , . . . , an) (bl , . . . , bn) Since the claim claim is is symmetrical symmetrical in in -,A(al -,A(al,..., an) and and A A(bl,..., bn) the the distinction distinction by 0 by cases cases is is complete. complete. 0 To , . . . , xn) xn) be To infer infer the the Equality Equality Theorem Theorem from from Lemma Lemma 3.4.6.5 3.4.6.5 let let G(Xl G(Xl,..., be an an .0. A00. . . , a) formula formula in in which which every every variable variable Xxii occurs occurs at at most most once once such such that that F(a) F(a) == =_G(a, G(a,..., a).. Then ' . . ,,xn) xn) and Then apply apply the the lemma lemma to to G(Xb G(xl,... and infer infer the the theorem theorem by by an an inference inference (V\ (V')" Observe 188), (190) Observe that that by by ((188), (190) and and the the Equality Equality Theorem Theorem we we have have ~- A A LA L~ for for all all identity identity axioms axioms A. A. I- ((194) 194) If (XI ' ' ' ' , xn) is Ad) CRS-terms of If FF(xl,...,xn) is an an C(E, s Ad) formula formula and and al, a l , .. ... ., a, an n are are all all/:Rs-terms of stages stages < (al, . . . , an) < oX~ we we obtain obtain that that rk(F rk(F(al,..., an))) < <W w.· oX~ + + nn for for some some nn < <w w.. So So putting putting together together Lemma 194) we Lemma 3.4.6.2 3.4.6.2 and and ((194) we obtain obtain the the following following theorem. theorem. Set Set Theory Theory and and Second Second Order Order Number Number Theory Theory 321 321 3.4.6.6. Let -formula which 3.4.6.6. Theorem. Theorem. Let F F ((x1, x l , .. ... ., ,xxn) n ) be be an an C(E, f_.(e, Ad) Ad)-formula which is is valid valid in in Pure Logic Logic with with identity identity and and 1'7{ an an acceptable acceptable operator. operator. Then Then there there is is an an m m< <w w such such Pure that that n p r(a,) u F(al,...,a,) Controlling Controlling operators o p e r a t o r s for for set-theoretic s e t - t h e o r e t i c axioms. axioms. Having Having established established controlling controlling operators and and derivation derivation lengths lengths for for logically logically valid valid sentences sentences we we will will now now determine determine operators controlling operators operators and and derivation derivation length length for for axioms axioms of of Set Set Theory. Theory. This This is is fairly fairly controlling easy for for the the axiom axiom of of extensionality. extensionality. Since Since we we defined defined aa = = bb to to stand stand for for aa � C bb 1\ A bb � C aa easy we obtain L~)(Vy L~)[x = y ++ 4-+ (Vz Ee x)(z x)(z Ee y) 1\A (Vz Ee y)(z y)(z Ee X)]. x)]. (Vy Ee LA) [x = (Vx Ee LA) (195) remaining axioms in their their modified modified forms as introduced introduced in Section 3.3. We derive the remaining We prove (Pair')fL~ for AA EE Lim. Lira. � (Pair >. for (196) (196) Let a, bb E e T>. T~ and and/3(3 := := max{stg(a) max{stg(a),, stg(b)} stg(b)} + + 1. Then Let 1. Then ~-aa E e LL/~ from(191). (191). � ,8 1\A b ES LL/~ ,8 from ( V') (i) Since U par(b) par(b) and we get get from from (i) by (V') Since/3(3 E9 par(a) par(a) U and/3(3 < < A A we (i) by L~)[a � (3z (3z eE LA) [a eE zZ A1\ bb eE z]. z] . A) Hence Hence by by twofold twofold (V (Vx) 1\ yY E9 z]. z] . � (Vx (Vx eE L~)(Vy LA) (Vy SE L~)(3z LA) (3z E9 L~)[x LA) [X E9 zZ A (ii) (ii) [] D Next Next we we show show , L>. for � (Union') (Union ) L~ for AA eE Lira. Lim. (197) (197) Let aa eE T~ we obtain: obtain: T>. and and c~ a := := stg(a). stg(a) . For For tt eE T~ To. and and ss eE Tstg(t) 7.tg(t) we Let � ss fi t,t, ss eE L~ La from from (190) (190) by by (Str) (Str) (i) (i) � (Vx eE t)[x t) [x eE L~] La] from from (i) (i) by by (~) (VI) ~-(Vx (ii) (ii) � tt ~/a, i a, (Vx (Vx E9 t)[x t)[x eE L~] La] from from (ii) (ii) by by (Str) (Str) ~- (iii) (iii) a) � (Vy (Vy E9a)(Vx a) (Vx eE y)[x y)[x eE L~] La] from from (iii) (iii) by by (V (va) }-- (iv) (iv) (3w eE L~)(Vy LA) (Vy eE a)(Vx a) (Vx eE y)[x y)[x eE w] w] from from (iv) (iv) by by (qLx) (3L>.) F� (3w (v) (v) � (Vu (Vu eE L~)(3w LA) (3w eE L~)(Vy LA) (Vy eE u)(Vx u) (Vx eE y)[x y)[x eE w] w] from from (v) (v) by by (vL~). (VL>.) . ~- DD We prove prove the the set set existence existence axiom-schemes axiom-schemes of of Separation Separation and and Collection Collection in in the the form form We ( L\o-Sep) (Vg)(Va)(3z)[(Vx (W) (Va)(3z) [(Vx eE z)(x (A0-Sep) z) (x eE aa A1\ F(x, F(x, if)) v)) A1\ (Vx (Vx eE a)(F(x, a) (F(x, if) z)] --+ xx eE z)] v) -+ 322 322 W. Pohlers Pohlers W. z)F(x,y,y, g)] (:3z) (Vx eE ~)(3y u) ( :3y eE z)F(~, ( �o -Col) (vg)(w)[(w (VV) (Vu) [(Vx E9 ~)(3y)F(~, u) (:3y)F(x,y,y, v-3 (Ao-Col) 17)] V) ~--+ (3z)(W for Ao-formulas �o-formulas F(x, F(x, g) and F(x, F(x, y,y,~), if) , respectively. respectively. We We first first prove prove v) and for L). for Lim. � ( �o -Sep)L~ for AA EE Lim. !--(Ao-Sep) (198) (198) Let {a, {a, aall, . . ." , a, na}n } C_ LA and and aa := : = max{stg(a), max{stg(a), stg(az),..., stg(al), ' " , stg(an)} stg (an) } ++ 1. 1 . Define Define Let � k~ l ' an) } . F(x, aail,l. .". , " an)}. La l xx EE aa A/\ F(x, {x EE k~l bb ::= - {x (i) (i) Then we we obtain obtain for for tt EE T~: fa : Then , an) by by (Taut)and (Taut) and (Sent) (Sent) � t tI b, b , tt EE aa A/\ F(t, F(t, az,... a l , " . ,an) F-t (ii) (ii) b) � (Vx (Vx EE b)[x b) [x EE aa A/\ F(x, F(x, aaI,i , ... .. ,. , a,)] an)] from from (ii) (ii) by by (V (Vb) (iii) (iii) � tt fI a, a, ~F(t, -,F(t, ~), it), tt EE aa A/\ F(t, F(t, g) it) A/\ tt == tt for for tt EE Tstg(,) Istg(a) F- (iv) � tt fI a,-~F(t, a, -,F(t, ~), it), tt e€ bb A/\ tt == tt reformulation reformulation of of (iv) (iv) (v) (v) � t If a,-~F(t,~) a, -,F(t, it) VV tt EE bb }--t (vi) (vi) from from (188)and (188) and (189) (189) by by (Sent) (Sent) from (v) by from (v) by (E) (E) and and (V (V')') a) � E a)[F(x, a) [F(x, it) E b]b] from (vi) by ~- (Vx (Vx E ~) --+ -4 x xE from (vi) by (v (Va) � (:3z EE LA) [(Vx EE z) (x EE aa A/\ F(x, (F(x, it) b (3z L~)[(Vx z)(x F(x, it)) ~)) /\ A (Vx (Vx E E a) a)(F(x, ~) --+ --+ xx E E z)] z)] >' from and (vii) ). from (iii) (iii)and (vii) by by (/\' (A')) and and (:3 (3~). From �o -Sep) LL~ ). by From (viii) (viii) we we finally finally obtain obtain ((Ao-Sep) by inferences inferences (V>'). (Y~). We We show show � ). for I- (Inf (Inf')' ) LLx for w < A~ EE Lim. Lim. (iv) (vii) (viii) o D (199) Let Let aa EE Tw T~ and and ac~ := := stg(a) s t g ( a )+ + 11. . Then Then � ~ aa EE La L~ and and � ~-aa EE Ll~w by by (191). (191). This (:3y E Lw ) [y EE LL~] (:3z E Lw ) [a EE z] This entails entails � ~(3yEL~)[y and � ~-(3zEk~)[a z] for for all all tt EE Tw T~.. Hence Hence w ] and � Lk~ (Vx EE Lk~)(3z Since w w << AAwe we get get the the claim claim by by an an inference inference w i-~=00 /\A (Vx w ) [x EE z]z].. Since w ) (:3z EE LL~)[x (V (V')') . We We summarize summarize Lemma Lemma 3.4.6.3, 3.4.6.3, (194) (194),, (196), (196), (197) (197),, (198) (198) and and (199). (199). 3.4.6.7. Let 3.4.6.7. Lemma. Lemma. Let A~ be be aa limit limit ordinal ordinal and and A A be be one one of of the the axioms axioms (Ext) (Ext),, ' ) and tl ' ) , (�o-Sep) ' ) , (Union (Pair (Pair'), (Union'), (Ao-Sep) or or (Inf (Inf')and 7-l an an acceptable acceptable operator. operator. Then Then there there is is an an w().+n ) ALx nn << ww such that tl [{A}] � AL). such that 7/[{A}] Io(x+") Next and ",n EE Reg Next we we prove prove that that for for acceptable acceptable tl 7/and Reg there there isis an an nn << ww such such that that n �o-Col)".~. tlH [{",}] (200) I { 4 ] I� ~+ ((~o-CoD (200) '0 . . . , ,an} Let an } �C_T,. Let ",n EE Reg Reg and and {a, {a, aI, al,... T~.. By By Lemma Lemma 3.4.6.1 3.4.6.1 we we obtain obtain (:3y EE LL~)F(x tl7/[par(~, [par(it, a)a)U{n}] U { "'}] � (:3y EE LL~)F(x, ~'0 -,(Vx ~(Vx EEa) a)(3y ~),(Vx EEa) a)(3y (i) ,. )F(x, y,y,it),(Vx ,. )F(x, y,y,it)~).. (i) Set Theory and Second Order Order Number Theory 323 323 By (Ref)", (Ref)~ this this implies implies By 7-/[par(E, a) U U {{n}] ~ , -~(Vx a)(3y L,)F(x, y, y ag)), (Vx EE a) (3y EE L",)F(x, 1i [par(a, a) �}] � ). (~ Ee LL,)(W a)(3y z)F(~, y, y, a~). (3y Ee z)F(x, (3z ", ) (Vx Ee a) I0 (V) ~ By By two two clauses clauses (V) and and some some clauses clauses (A) this this entails entails (/\) n (Vg EE LL,)(Vu 74[{n}] u)(3y y, V) 0") 1i [{ � }] I~'0�+ (Vv (3y EE LL,)F(x ", ) (Vu EE LL,)(Vx ", )F(x, y, ", ) (Vx EE u) --+ (3z Ee LL,)(Vx u)(3y z)F(x, y, y, v). g). (3y EE z)F(x, -+ ", ) (Vx EE u) To deal deal with with the the foundation foundation scheme scheme (and (and axiom) axiom) we we prove prove aa lemma. lemma. To 3.4.6.8. Foundation Foundation Lemma. Lemma. 3.4.6.8. operator 1i 7i.. Then (ii) (ii) D (A} U U par(F(Lo)L par(F(Lo) L~) C_ 1i 7-/ for for an acceptable Let {>.} >. ) � L ·rk(F(a) >')+3 .(stg(a)+ 1 ) ,F 17-/[par(a)] _~F(a)L~ (3x EE LLx)[F(x) (Vy E E x)'F(y) x)-,F(y) LL~] i [par(a )] 1�Io2.rk(F(a)'~)+3.(str A) [F(x) LL~ (a) L\' (3x >.] >. 1\A (Vy for all a E T>. T~.. for We prove prove the the lemma lemma by by induction induction on on stg(a) stg(a) and and get get We L ·rk(F(b) >')+3 .(stg(b)+ 1 ) ,F(b) 1n[p~r(b)] L\' (3x 2.rk(f(b)'~)+S.(,tg(b)+l) _~F(b)Lx (3x Ee LA L~)[F(x) (Vy E E x),F x)-,F(y) (i) i [par(b)] 1� ) [F(x) LLx>. 1\A (Vy >.] (i) (y) LLx] Io for all all bb E E Ts Tstg(a). By the the structural structural rule rule (151) (151) this this entails entails for tg(a) . By L ·rk(F(b) >' ) +3.(stg(b)+ 1 ) bb If a,a,-~F(b) 12"rk(F(b)L~)+3"(stg(b)+l) ,F(WL~>' ,' 1i [par(a, b)] b)] 1� 7-l[par(a, ,o (3z L~)[F(~) (Vy Ee x) ~)~F(y)L~]. [F(x) L~>. 1\^ (Vy (3x Ee LA) 'F(y) L >. ]. this implies implies Using (/\) Using (A) this (ii) (ii) L ·rk(F(b) >' )+3 .(stg(a))+ 1 (Vz a)~F(z) (Vz EE a) 1i [par(a)] Io 1�2.rk(F(b)4,)+3.(,tr 'F(z)LL~>. ,' 7-/[par(a)] (iii) (iii) L~)[F(x) (Vy Ee x)-~F(y)L~]. x) 'F(y) L >. ]. [F(x) LL~>. 1\A (Vy (3x Ee LA) By Lemma 3.4.6.1 we have have By Lemma 3.4.6.1 we 12"rk(fCa)L~) 2r FaL 1i [par(a)] 1,0. k( ( ) >. ) F(a) 7-/[par(a)] L~,, -~F(a) A (Vy (iv) (Vy eE x)-~F(y)L~]. x) 'F(y) L>. ] . (iv) LA ) [F(x) LL~>. 1\ F(a) L>. (3x eE L~)[F(x) L>. , (3x 'F(a) L~, 0 L>. ) we Putting we obtain Putting aa := 2. 2 . rk(F(a) rk(F(a) L~) obtain o . (stg(a))+2 la+3'(stg(a))+2 ,F(a), F(a) F( L~, ~F(a), F(a) L~ A (Vz (Vz EE a)-~F(z) 1i [par(a)] ,;l 0 +3 "//[par(a)] L>. 1\ a)' z)L>., (3x L~ A (Vy EE x)-~F(y) x) 'F(y) L~] (3x EE L~)[F(x) 1\ (Vy L >. ] LA)[F(x) L>. (v) (v) from (iii)and from (iii) and (iv). (iv). Hence Hence o a +3.(stg( ))+3 ~F(a)L~, L>. ] 1i [par(a)] 1o 1\ (Vy (Vy EE x)-~F(y) x) 'F(y) L~] l~+3.(st,(,))+3 (3x EE L~)[F(x) LA) [F(x) LL~>. A 'F(a)L>. , (3x 7-/[par(a)] 0 � (V). by (V)" We get We get the the following following theorem theorem as as a a corollary corollary of of the the Foundation Foundation Lemma. Lemma. (vi) (vi) D [] 324 324 w. Pohlers Pohlers W. 3.4.6.9. Foundation Foundation Theorem. Theorem. Let Let F(x,Z) F(x, x) be be an an L:(E, C ( E, Ad) Ad) formula formula without without 3.4.6.9. Lim and and 7-l 1i an an acceptable acceptable operator operator with with AA EE 7-l. 1i . Then Then there there further free free variables, variables, A A EE Lira further is an an nn << co w such such that that is n +A+ (Vs ?t ~'~+~+n L,\ ) [(:3x eE L~)F(x,s x)--,F(y, X) L>. ]] . L,\ )F(x, x) L>. -+ (:3x EE L~)[F(x,s L,\ ) [F(x, x) L>. 1\ (Vy EE x)~F(y,s ("Ix eE L~)[(3x 1i I1�·'\ '0 To prove prove the the theorem theorem we we introduce introduce the the abbreviation abbreviation Found({x Found( {x eE L~ L II F(z, F(x, sx) L >. }) } ) :r: ¢:} To '\ (Vy eE x)--,F(y, x) L>.]. (:3x eE L,\ )F(x, x) L>. -+ (:3x eE L,\ ) [F(x, x) L>. ^1\ (Vy L>. ) <� co. Observe that that for for {a, { a, aa1,l , .. ...., ,aa, n} w . AA ++ kk for for � 7"~ T,x we we have have rk(F(a, rk(F(a, aaI,l , ..... ., a, an) Observe } C_ n ) [~) w. From From the the Foundation Foundation Lemma Lemma we we obtain obtain some kk << co. some Found({x 1i[par(a1 , . . . , an)] [w.~+~ I�M'\ Found ( {x eE k~[ L,\ I F(x, ii)L>. }) F(x, ~)Lx }) ?/[par(al,...,a~)] ,0 by an an inference inference (A)" Applying nn inferences inferences (A) (/\) we we finally finally obtain obtain the the claim. claim. by (/\) . Applying 0 D So are already already prepared to compute compute an upper bound bound for for [[KPwl[~cK. IIKPw I lw1CK . So far far we we are prepared to an upper KPw, such If P w f-~ F F then have finitely A 1 , .. .. .., ,AAn n of of KPw, such If K KPw then we we have finitely many many axioms axioms AI, that A1 Al -+ -. ". . -~ is valid valid in in first first order order logic logic with Hence that Ann -+ with identity. identity. Hence -+ A -+ FF is w A�o -+ K by II�· co~K . . --+ FLo for for f~ 0 ::=- Wf by Theorem 3.4.6.6. By 1io 7/0 '~+no --+ ."'" An�L,o ---> Theorem 3.4.6.6. By -+ FLfl -+ A ..*'~+nmo AlL, Lemma 3.4.6.7, (200) 3.4.6.9) we Lemma 3.4.6.7, (200) and and the the Foundation Foundation Theorem Theorem (Theorem (Theorem 3.4.6.9) we therethere2 w FL. by by cuts. cuts. Applying Applying the Predicative Elimination Lemma fore obtain 7to 1�· [n.2+~ the Predicative Elimination Lemma fore obtain 1io . n+ FLo .ft+n + n -l)1)((fl.2+w) n'2+w) cp�(n- F Lo L" . Assuming that F F is is aa � ~1Assuming that F 1n 1 ) + w) < formula and = ~n-1)(~.~. < e Cf~_l_ the Collapsing Collapsing Theorem Ml1 we formula and putting putting aa ::= rp� - (0 · 22 -[-co) we get get by by the Theorem +Q) (wo tP o Lo r F L" . Finally applying Theorem 3.4.2.2 we obtain (Theorem (Theorem 3.4.5.3) 3.4.5.3) 1i 7/~,+~ wo+Q I]rtPo (w +Q) F . Finally applying Theorem 3.4.2.2 we obtain + tP (wo Q) F F LL,. Therefore we we have have shown shown o . Therefore I]r o K we 3.4.6.10. := Wf 3.4.6.10. Theorem. Theorem. For For 0 f~ := co~K we have have IIKPw IIKP~,IIn _< 1/I r n(en+1 ) . l in � I fM1 (Lemma 7t0 [~o (Lemma 3.4.3.6) 3.4.3.6) we we then then obtain obtain 1io l+l 0 To To obtain obtain also also an an ordinal ordinal analysis analysis for for the the theories theories KPI KPI and and KPi K P i we we need need Let controlling operators for axioms (Ad 1) , (Ad2), (Ad3) and (Lim) . controlling operators for axioms (Adl), (Ad2), (Ad3) and (Lim). Let 1i 7t be under regular be an an acceptable acceptable operator operator which which is is closed closed under regular successors successors and and AA an an < A) (:3x; E A)[X; EE Reg ordinal satisfying ordinal satisfying (V� (V~ 1i[par(a) (a) -+ 7/[par(a) U {A}] {A}] � ~ Ad Ad(a) --+ w co E E aa 1\ A 1Tan(a) Tran(a) for for some some a a < < AA depending depending on on a. a. By By an an inference inference (/\) (A) we we obtain obtain from from (i) (i) n[{A}] 1i[P}] (i) Ad 1 )L>. (Vx E E LL~)[Ad(x) --+ w co E Ex x 1\ A 1Tan(x)] Tran(x)],, i.e., i.e., 1i[P}] 7/[{A}] � ~ ((Adl) L~.. (201) (201) �~ ("Ix ,\ )[Ad(x) -+ Set Set Theory Theory and and Second Second Order OrderNumber Number Theory Theory 325 325 Similarly L" EE LI' V L" = V LI' and (188) and Similarly we we obtain obtain f~-L~ LuVL~ = LI' LuV Lu EE L" L~ by by (191) (191)and (188)and .(L" EE LI' V L" V LI' a E bb V aa== Lu, bb r=1= L", L~,--(L~ LuV L~ = = LI' LuV Lu EE L,,), L~),aE b Vb bVE b E aa ffor o r aall ll f- aa~:=1= LI" p" #, Kn EE AAnMReg. Reg. As As in in the the previous previous case case this this implies implies ((202) 202 ) Now Pair ' ) , ((Union'), Union ' ) , ((A0-Separation) Llo-Separation) or Llo Now let let A A be be one one of of the the axioms axioms ((Pair'), or ((A0K Collection). For any any Kn EE AAMReg we get get by by Lemma Lemma 3.4.6.7 and and ((200) n[{~}] � ~ AL A L" Collection ) . For n Reg we 200) 1l[{K}] "k/[{A}] ~ (Ad2) L~. + . As for some some Q: a < < K~+. As in in the the previous previous cases cases this this implies implies for ((203) 203) "k/[{A}] ~ (Ad3) [~. For For bb EE 0. T~ we we have have Kn := stg(b) stg(b) ++ < < A. A. By By (188) (188) and and (191) (191)itit follows follows ' [Ad(y) /\A bb EE y] f~- Ad(L,,) Ad(L~) /\ A bbeE L" L~.. By By (V (V')) it it follows follows f~- (3y (By EE L.x) L~)[Ad(y) y] and and by by (1\ (A')' ) we we ] . Hence finally obtain obtain f~ (Vx (Vx EE L.x) L~)(3y In)lAd(y) Hence finally (3y EE L.x) [Ad(y) /\A xx EE yy]. N[{A}] i m ) /L~. �. 1l [P}] I;~ ->.+- n( L(Lim . co.X+n ((204) 204 ) Since f2 ~co satisfies (Vx (VxEf~co)(3~E~co)[~ Reg /\ A xx EE K] ~] we we obtain obtain from from Since E f2w ) (3K E f2w ) [K EE Reg w satisfies Lemma Lemma 3.4.6.2, Lemma Lemma 3.4.6.7, Lemma Lemma 3.4.6.3, the the Foundation Foundation Theorem Theorem 3.4.6.9 201 ) through 204 ) that and ((201) through ((204) that for for every every sentence sentence F F there there is is an an m < <w co such such that that and KPI ~ F ((205) 205 ) =~ "kt0 [n~.2+coFL.~ n~+m Cl K -sentence If 205 ) is �r CK If we we assume assume that that F F in in ((205) is aa E ~11 -sentence we we can can apply apply the the Predicative Predicative Elimination Lemma Elimination Lemma 3.4.3.6, the the Collapsing Collapsing Theorem Theorem 3.4.5.3 and and Theorem Theorem 3.4.2.2 to to obtain obtain COCK K ==~ (€nw + d . < 1Pn KPI ~f- FF and and FF EE E KPI Y]I�f 1 for some some aQ: < Cn(~'ftw_}_l). => ~p FF for So we have have shown shown the So we the following following theorem theorem 3.4.6.11. Theorem. 3.4.6.11. Theorem. IIIKPlll I KPl llna < :5 Cn(cn~+l). 1Pn (€nw + 1 ) ' The ordinal ordinal II not not only only satisfies satisfies (Vx (Vx EE f~co)(3~ f2w ) (3K EE f~co)[~ f2w ) [K EE Reg Reg A /\ xx E E n] K] but but also also The E Reg. Reg. So So we we obtain obtain by by (200), ( 200) , Lemma Lemma 3.4.6.2, Lemma Lemma 3.4.6.7, Lemma Lemma 3.4.6.3, the the II E (Theorem 3.4.6.9)and 3.4.6.9) and (201)through (201 ) through (204) (204) Foundation Theorem (Theorem Foundation Theorem I .2+W FL' L/ K P i ~IrFF =~ 1l 0 IIr2+coF => 7/0 KPi I+n I +n for some some nn << w. w. As As before before this this implies implies that that there there is is an an aQ: << Cn(r such that that ~p FF for 1Pn (€l+ l ) such for for ~l~-sentences E p -sentences which which are are provable provable in in KPi. KPi. So So we we have have 3.4.6.12. Theorem. 3.4.6.12. Theorem. IIKPi lln IIKPill n _< :5 Cn(r 1Pn (€l+l) . To obtain obtain also also an an analysis analysis of of the the theory theory KP1 KPlrr we we have have to to do do aa bit bit more. more. First First we we To show show ~+.+2k Lop+n (206) ( 206) (a)L%+Ll ( x) =~ => (3n p+n + 326 326 W. Pohlers Pohlers W. limit ordinal ordinal AA and and aa finite finite set set A(Z) b.(i') of of E-formulas �-formulas which which contain contain only only the the for aa limit for shown free free variables variables and and an an acceptable acceptable operator operator 7/which 1£ which isis closed closed under under ~� ~-~ 1-+ ~+. �+ . shown The proof proofisis by by induction induction on on kk and and runs runs essentially essentially as as that that of of Lemma Lemma 3.4.6.2. 3.4.6.2. But But The there isis the the additional additional (and (and critical) critical) case case that that the the main main formula formula of of the the last last inference inference there p. ...,Lim, (Vy)[-~Ad(y) (Vy) [...,Ad(y) V xiXi q~ is -~Lim. ...,L im. Then Then we we have have the the premise premise ~-~-~Lim, i y], y] , A(~) b.(i') which which by by is inversion yields yields ~-~Lim,-~Ad(y) p. ...,Lim, ...,A d(y) VV x,Xi ri y,y, A(Z) b.(i') for for aa free free variable variable yy not not occurring occurring inversion w such such that that b.(i') . By By induction induction hypothesis hypothesis there there isis an an no no << w in A(Z). in n�+nO+I +2ko � b] In"+"~176 � L nno+~+1 +m + l b.(a) 1£ [a, ...,Ad(b) VV a,ai ~d'F b,b, A(g)L""o n[,~, b] I~,,+,.o+~+ n�+no+ l +,l ---,Ad(b) So we we obtain obtain for all all gii EE 7"~, and all all bb EE Tn,+~. Tn�+I ' So Tn� and for 7/[d] ,n,+.o+,+ll n"+"~176 (Vz e Ln,+,)[~Ad(z) V a/it z], A(g)L%+"o+~ . ��+ 1£ [{ai}] ~'0I 2 _I (3z (3z eE Ln,+l)[Ad(z) ai eE z] and and obtain obtain By (188) (188) and and (191) (191) we we have have 7/[{a/}] By Ln�+ I ) [Ad(z) A1\ ai n�+nO+ l +2k Ln�+nO + 1 1£[ii] I.,.§ I~u+no+l+2k n[~] /,,(~)'~247247 n�+no+ l + l b.(ii) by cut. by cut. Next observe that (Union') and Next we we observe that the the axioms axioms (Pair'), (Pair'), (Union') and (b.o-Separation) (A0-Separation) are are dispensable. They can be be derived from (Ad (Ad3) with (Lim). dispensable. They can derived from 3) together together with (Lim). The The same same is is true true for for (Inf') (Inf').. The The scheme scheme of of b.o-Foundation Ao-Foundation can can be be formulated formulated as as (b.o-Found) (Ao-Found) ((Vu)[ Tran(u) 1\ A ((Vx u)[(Vy E e x)F(y) x)F(y) -t -+ F(x)] F(x)] -t -+ ((Vx u)F(x)] Vx Ee u)F(x)] Vx Ee u)[(Vy Vu)[ Tran(u) where nite set where F(x) F(x) is is aa b.o-formula. A0-formula. So So if if KPlr KPF � ~ b.(i') A(Z) for for aa fi finite set b.(i') A(Z) of of �-formulas E-formulas then there . . . , AAkk such then there are are finitely finitely many many instances instances of of axioms axioms Ab A1,..., such that that � Ak , b.(i') ~- ...,Lim, -~Lim, ...~, Ab A , , .. .. .., , ..., ~Ak, A(Z).. ,Ai is aa �El-formula. With With the the exception exception of of axiom axiom (Lim) (Lim) every every of of these these formulas formulas ...-~A/is For 1 -formula. For aa limit limit ordinal ordinal AA and and aa tuple tuple ii~ EE L Ls~ we get get by by (206) (206) J.L #< For For aa �-sentence E-sentence F F this this entails entails u [ �::� 1£0 Ilu~+~n~+lF KPlr E w) [7/0 KP1 r � ~ FF =} =~ (3m (3mEw) F ] nw LLn~ (208) (208) which which for for aa �p-sentence E~-sentence F F by by the the Collapsing Collapsing Theorem Theorem and and Theorem Theorem 3.4.2.2 3.4.2.2 implies implies .pn (nm2 'Ww ) F . Since limm I]r F . Since limm~ Cn(gt~.. WwW~)) = = 'l/JCa(~o,) we have have Ew 'l/Jn(O-;" n (Ow ) we 327 327 Set Theory and Second Order Number Theory 3.4.6.13. Theorem. IIgPl~[In_ KPI and and K KPi there isis no no difference difference Though there there isis aa tremendous tremendous gap gap between between KP1 Though P i there between KPff KPlr and and KPff. KPir . If If we we try try the the same same analysis analysis as as we we did did for for KPff KPlr the the between axiom of of Ao-Collection �o-Collection will will spoil spoil the the argument. argument. This, This, however, however, can can be be remedied remedied by by axiom augmenting the the logical logical calculus calculus by by the the following following non-logical non-logical rule. rule. augmenting ( �o-Collection Rule) Rule) (A0-Collection F A, �, (Vx (Vx eE a)(3y)F(x, a)(3y)F(x, y) y) ~=} 1 m+ ! zx, �, (3z)(W (3z)(Vx eE a)(3y a) (3Y eE z)F(x, y). y) . By [Ao] [�o] ~F A� we we denote denote that that A � isis provable provable in in the the extended extended calculus. calculus. We obtain for for By We obtain �(il) of of E2:- formulas formulas and and an an ordinal ordinal A .A finite set set A(zT) aa finite ~,+2,~ 1 A(g)L.~,+2m] (209) [A0] ~- ~Lim, A(g) =v (V# e A)(Vde Tn,)[n[d] la,+~m+ The proof proof parallels that of of (206). case is an application The parallels that (206). The The additional additional case is an application of of the the �o-Collection Rule. induction hypothesis then have A0-Collection Rule. By By the the induction hypothesis we we then have +2mo y). (3y eE La,+2mo)f(x, ai)(3y Ln,.+2mo )F(x, y). VX eE ai) 1�JJ".+2mo + 1 �(a) n"+2mO , ((Vx By an (Refn,.+2mo+ l ) and and Upward Persistency this this implies implies By an inference inference (Refa,+2~o+l) Upward Persistency n,.+2mO+2 >'+2mo+2 ) (3Y Ee z)F(x, [a] I,a,+2mo+~+ll 1£ 7/[g] A(d)a~+2~o +2,, (3z (3z eE LLa,+~o+2)(Vx z)F(x, yy)) . n,.+2m +2 )(Vx Ee aia,)(3y na€176 ,.+2mO+2 + 1 �(a) n[~] 1£ [a] � n 0 Now replacing (206) Now replacing (206) by by (209) (209) we we obtain obtain with with the the same same strategy strategy as as in in the the case case of of KPlr KPF ] E w) [1£0 I�: KPir KPff � ~ F F =} =v (3n (3nEw)[7io In"a.+l + ! FF] for for 2:p E~ sentences sentences F F.. So So we we have have 3.4.6.14. Theorem. [IKPi~lln_ Cn(f~). We We will will roughly roughly sketch sketch how how an an analysis analysis for for W-KPI W-KP1 and and W-KPi W - K P i can can be be ob obtained tained without without going going too too much much into into details. details. The The stumbling stumbling block block here here is is the the scheme scheme of of Mathematical Mathematical Induction. Induction. The The remedy remedy is is to to augment augment the the calculi calculi F ~and and [flo] [A0] F ~ - ,, respectively, respectively, by by an an ww- and and aa cut-rule. cut-rule. Call Call these these extended extended cal calculi w] � culi [w] [w] � ~ and and [flo, [A0,w] ~ , , respectively. respectively. Then Then if if W-KPI W-KP1 � ~-F F or or W-KPi w-gPi � ~ F F we ,AI, ' . . , ,Ak, ,MIl , ' . . , ,MIt, F we obtain obtain [(flo)] [(Ao)] F ~----,A1,...,~Ak,~MI1,...,~MIt, F for for finitely finitely many many instances instances Mlj Since MIi of of Mathematical Mathematical Induction. Induction. Since [(flo), [(A0) ' w] w] I�+n ~' 0 M1i MI, we we obtain obtain by by cut cut + cf. w ,Al l " " ,Ak, FF which [(A0),w] which by by the the usual usual cut-elimination cut-elimination for for w-logic w-logic ((cf. [(flo), w] I;~-~---~A1,...,-~Ak, Lemma w] �~ ,Al l ' . . , ,Ak, Lemma 2.1.2.9) 2.1.2.9) entails entails [(flo), [(A0),w] -~A1,..., ~Ak, F F for for some some 0: a < < co. Co. Now Now we we are are in in situation similar similar to to that that in in the the analyses analyses of of KPlr KP1 r and and KPir KPff.. While While (209) (209) modifies modifies aa situation directly directly to to [flo, w] FnoQ,Lim, fl(il) =} Q fl(a) Lo>. +2Q ], (Va E TnJ [1£ [a] 1�>'+2 >.+2Q + 1 (2 o) (210) 328 328 W. Pohlers we cannot cannot directly directly adapt adapt (206) (206) to to the the calculus calculus [w] [w] � ~ because because we we have have infinite infinite we derivations. We We have have to to refine refine the the argument argument and and prove prove derivations. [w] � ~-~Lim, A(~) '* =~ (VA (VA Ee Lim) Lim)(V~, T~)[A(~) c_ � E~" '* =~ [�(a)b~ S; [w] (21 1) (Vii, bb Ee TnJ . Lim, �(u) �� 1iw0>' +3e> [a, b] 1I'~+1 30 �(a)b] for aa finite finite set set of of �-formulas E-formulas �(u) A(~) by by induction induction on on a. a. The crucial crucial The for case is is again again that that the the main main formula formula of of the the last last inference inference is is .Lim. -~Lim. Let Let case such that that �(a)b A(~) b S; C_ � E "~.. From the the induction induction hypothesis hypothesis we we obtain obtain a~,b From , b such n>. +300 -~Ad(c) V ai r C,/k(~) b for all c E T~+ which by an inference 7/~,~+3~oo [a, [~, b, b, c] c] I[a~+a~o 1iw0>.+3e> '~++1 "++ 1 .Ad(c) V ai � c, �(a)b for all c E 7,.+ which by an inference n +300 + l ~(3Z L~+)[Ad(z) A ai E z], A(a) b By cut we (A) implies ~/o,n~+3~o[~, b] I]flX+3c~o+l �+1 (1\) implies 1iw0 . (:Jz E L,,+ ) [Ad(z) 1\ ai E Z] , �(a)b . By cut we >. +3e>o [a, b] "K+-I-I 2 + +n>. + +2 < wn>. n + + n~,x+3oo[~,b] A(~)b. wn>. n~+3~~ +w w,,~++n~+3~o+2 +300 + +30 we � 300 � (a)b . Since w obtain 1iw o>. +3e>o [a, b] I1a~+3~o+2 300 < W ~+3~ ',r" +1 obtain 1i 7/~,~+3~[~, b] 1I~+1 "~+3~ A(~) by the the Collapsing Collapsing Theorem. Theorem. obtain 30 � (a)b~ by W0>. +3e> [a, b] By the the now now familiar familiar technique technique we we obtain obtain from from (210) (210) By �� [ �: ] W - K P i ~� - F F '* =~ (:Ja ( 3 a eEcco) ~ 1io 7/~ IIn~n~+lFL"~] F Loe> W-KPi (212) (212) +l Collapsing Theorem for for �l-sentences ~l-sentences F F.. By By the the Collapsing Theorem and and Theorem Theorem 3.4.2.2 3.4.2.2 this this implies implies 3.4.6.15. Theorem. Theorem. 3.4.6.15. W-KPi ll n ::; _< 'ljJ Cn(~o). IIIIW-KPilln n (neo ) . From (211) (211) we we obtain obtain as as in in the the proof proof of of (208) (208) From [ W-KP1 ~ F f '* =~ (:Ja (3a E e co) Co) 1iw no,,~+~ W -KPI � ow +e> [~+~+t EL"] 213) ((213) n+nw +O) -= 'ljJ Since Cn(w 'ljJn (w~+n~+~) WO) < r'ljJn (nw . Co) co) for for �l-sentences F < Co co we for ~l-sentences F.. Since r n (nw . w ~) < for aa < we get from (213) by the Collapsing Theorem and and Theorem Theorem 3.4.2.2 get from (213) by the Collapsing Theorem 3.4.2.2 3.4.6.16. Theorem. 3.4.6.16. Theorem. o) . 'ljJn (nw ' cc0). [IW-KPlll II W-KPl ll ~n _< ::; Cn(fl~" The theories for for iterated iterated admissibles admissibles needs needs serious extra work The analysis analysis of of theories serious extra work which which has first first been been done done by by M. M. Rathjen. Rathjen. We there are are operator controlled has We have have to to show show that that there operator controlled ItAd( a, f). f) . This This is is prepared is provable provable derivations for for the the axioms derivations axioms ItAd(a, prepared by by aa Lemma Lemma which which is in K PF.. in KPlr 3.4.6.17. Lemma. Let aa and and uu be be admissible admissible sets sets such such that that aa EE uu and and Lemma. (KPI (KPn 3.4.6.17. r) Let a. Then Then (V~ (V� << a)(3f a) (:Jj eE a)ltAd(~, a) ltAd(�, f)f) implies implies (3g (:Jg eE u)ltAd(a, u) ltAd(a, g). g) . let aa eE a. let Proof. Since Since ItAd(~, ItAd(�, f)f) isis aa Ao-formula �o -formula we we get get from the hypothesis hypothesis by by Ao-Collection �o-Collection Proof. from the set bb eE aa such such that that (V� < a)(3f a) (:Jj eE b)ltAd(~, b) ItAd(�, f). f) . By By (130) ( 130) for for every every relative to to aa aa set relative (V~ < � << aa there there is is exactly exactly one one such such f~ b. By By A0-Separation �o-Separation relative relative to to aa we we j{ EE b. < := U~<~ get hh := h E a. Clearly h is a function. Let c be an E-least element get f~ E a. Clearly h is a function. Let c be an E-least element U{ o Since Ad(a) Ad(a) A 1\ aa eE uu the the set set cc exists exists by by A0�o Ad(x) A 1\ xx~�r nrng(h) of {{xx eEuu Il Ad(x) of g ( h ) }}. . Since Foundation and we we define Foundation and define 329 329 Set Set Theory Theory and and Second Second Order Order Number Number Theory Theory E Lim U {0} {O} iiff aa E g : = { hhhhUU{{(" e) } iiff aa==9 ' ,+ l+. l. ( 7 , c)} Then gg isis aa function function with with domain domain aa and and we we easily easily check check ItAd(a, ItAd( a, 9). g) . Then 0o Recall the the function function A~. .x� . T~ which enumerates enumerates the the admissible admissible ordinals. ordinals. We Recall We have have 7{ which g:= {w if a = O w fl,~ n 7" == Qa+l Ta n,,"+ 1 II if a = 0 for 00 << aa << w W for for for w W << aa << II for f o r a==I I From for aa limit From Lemma Lemma 3.4.6.17 3.4.6.17 and and (207) (207) we we get get for limit ordinal ordinal uv and and aa << uv aa kk << w such such that that w m ,Ad(LTJ , -~Ad(L~-o+,) a)(3f E L~)[ItAd(~ 1l0[{a}] I Ta+k+ LTa , ,(\f� E a)(3f LTJ [ltAd(�, f)] 1)], ,Ad(LTa+ 1), aa ~� L~,---,(V~ Ta+k + 1 ~Ad(L~) . g) ] ( 3g EE L~+,)[ItAd(a, L~ LTa+1 ) [ 1tAd( a, 9)1. LTa+! , (39 LTa r� L,~+,, For a :::: Ad(L Ad(LTao+,) Ad(LTa) , 1lo[{a}] +1 ) , I- a Ee LTa and For a < < II we we have have 7~0[{a}] ~/o[{a}] I,~.+~ 1l0[{a}] ~I:::: Ad(L,~) IT~+I I- L~ So we we get get by by some some cuts cuts LTa+!1.· So LTa EE L~+ ) ) Ta+k+ g) ] . ~. +,+,,�' -,(V~ L,~+l)[ItAd(a, ) [ 1tAd(a, g)]. a) ( 3f EE L,~)[ItAd(~, )], (3g EE LTa+1 1lo[{ a}] Ii,o+,,+,," LTJ [ ltAd(�, 1.f)], Ta+k+n ,(\f� Ee a)(3.f (214) Now Now we we show show by by induction induction on on a a < < II H( O + 1 ) (3f (215)) (215 a }] ,n,~+., ( 3 f EE LTa+1 )ltAd(a, :). "~/o[{a}] L~+,)ltAd(a, 1lo[{ 1) . (nla+w a+w nHw H'({+ 1 ) On, ( By the induction n'+~+4"(~+1) By the induction hypothesis hypothesis, 7/0[{a,~r aa �r On, (3f LTH1 )ltAd(�, f) 1) 3f EE L,,+,)/tAd(~, 1lo[{ a, O ] IIg16+,, n(+w, Fta+~+4a+3 for < a. a. This This implies implies 7/0[{a}] 1lo[{a}] 1In~+~ LTJ ltAd(�, 1f)) by �:::Ho+3 ((V~ for all all � ~c < < a) a ) ((3 3 ff E E Lr~)ltAd(~, by two two \f� < inferences 214) we inferences (V), ( V ) , an an inference inference (/\ ( A )) and and Upward Upward Persistency. Persistency. From From this this and and ((214) we obtain the claim by obtain the claim by a a cut. cut. Since �~ < implies �~ + 215) Since < v v implies + w :=:; _ v v for for v v E E Lim Lira we we obtain obtain from from ((215) w (3 f EE LnJ n~ u)(3f Lfl~)ltAd(~, 1lo[{ ltAd(�, 1f)) \f� << v) n o [ { v4 }] ] II�: ''" ++"v ((V~ 0 [3 (216) (216) im. The we applied applied for ordinal analysis KPI together together with for for v u E E LLira. The strategy strategy we for the the ordinal analysis of of KP1 with ((216) 216 ) now yields now yields v .2+W F L v a] Il �n''2+'' KPl" (a) o (217) n~+m F(g)L"~ (217) K P 1 , I~ - FF((ir) g ) :::} =~ 1lo[ 7/o[~1 �,,,, +m for limit ordinal for aa limit ordinal vu and and all all a ~E E Tn T ~v. ' By By the the meanwhile meanwhile familiar familiar argument argument this this entails entails 3.4.6.18. Theorem. Theorem. 3.4.6.18. (€nv + 1) KPl" lln :=:; IIIIKPI.IIa --<~1/ln CD(Ef~u+l) An An ordinal ordinal analysis analysis of of the the theory theory KPl� KPI~ for for limit limit ordinals ordinals vu is is obtained obtained in in in in aa similar similar way 206) to way as as that that of of KPlr KP1 r.. We We modify modify ((206) to Lim, ,(\f� < ) ltAd(�, 1f),) , � (31f)ltAd(~, < v) u)(3 A(~) =~ �~- ,-,Lim,-,(V~ (u) :::} ( a) LOa+ww]] ((w }] 1�:::H(O+k) �A(~)Lfla+ "-r,o) {o,}1 \fa << v)(W1 Ee Tr J [1l[a, {a s (218 (218)) 330 330 W. Pohlers for aa finite finite set set b.( A(g) of � ~-formulas and an an acceptable acceptable operator operator which which is is closed closed under under for il) of -formulas and A~.. O{ f ~ .. The The proof proof is is analogous analogous to to that that of of (206) (206) with with the the additional additional case case that that the the A� main formula formula of of the the last last inference inference is is "'{V� ~(V~ < < 1I) u)(3f)ltAd(~, Then we we have have the the main (3j) ltAd(�, jf).) . Then premise premise II ) (3 jf)ItAd(~, ) ltAd(�, jf),) , Ui < 1Iu)(g ui Ee On O n 1\ A Ui ui < < u 1\ A (\fj)-./tAd(Ui, (Vf)-~ltAd(ui, jf).) . � --~Lim,-~(V~ .Lim, -. (\f� < induction hyApplying inversion inversion we we obtain obtain for for a c~,, aaii < < IIu,, a ~E e Tr T~a and and ff E e Tr T~.+I by induction hyApplying a+1 by na+w +4(a+I+ko ) � a+w a ° ~ .. By -~ltAd(ai, A(~)L"~+ By an an mference inference (( A ) pothesis a} , ff]] IIn.+ [ a, {{c~}, pothesis 11. 7-/[~, -. /tAd( ai , ff),) , b. nn~+~+4(~+,+ko) a+w~ l l o na+w in~+~+4(~+ko+l)+l . . to. + +k )+ + ( � � 4 a IS Yyields Ields 11. (a ) L°a+w~ which th a}] I'a.+~ this "k/[~, L~.+l)ltAd(c~, A(g)L"-+ which to( 3f Ee LT -.-,(3f [a, {{c~}] a+ 1 ) ItAd( a, ff),) , b. na+w gether with with (215) (215) entails entails the the claim claim by by aa cut. cut. gether If KPI� KPlr, f~- F F for for aa �-sentence 2-sentence F F then then there there are are II 111 sentences AI A1,..., Ann - the the If ,...,A I sentences axioms different different from from Lim Lim and and the the iteration iteration axiom axiom -- such such that that by by (218) (218) we we obtain obtain axioms na+w +na --, A-Lna+~ LOa+w ,,..., f~a+~+ft,, ALnn~+o, FLn~+o, Loa+w . . .,A 11. nLoa+w ,, F n0[{~}] ~A~1 o [{ Q }] II,~ 00+"" . (�) L for some some a a < < u. some cuts cuts we we get get for II. Applying Applying some 1�:::'2 [ 1\) ] no+~.2FFL~ LOa +w ] 11. 0 [11] Ifl~+~ KPI~ ~ FF => =~ (3 (3aa < < IIu)) [n0[u] KPI� f- (219 (219)) for �-sentences E-sentences F. If we we assume assume that that w w < < II u and and IIu is is additively additively indecomposable indecomposable for F. If then a < < IIu implies c~+ + W w< < u. So we we obtain obtain from from (219) (219) implies a II. SO then a 3.4.6.19. 3.4.6.19. Theorem. Theorem. V .. II 'ljin (Ov) for additively _< Cn(ft~)for additively indecomposable indecomposable ordinals ordinals II[]KPI~IIn KPI� ll n ::; We can say something limit ordinals ordinals which additively decomposable. decomposable. We We can also also say something for for limit which are are additively We generalize (211) to (211 ) to generalize [w] -.(\f� < ((220) 220) [w] ~� -'Lim, -~Lim, ~(V~ < u)(3f)ltAd(~, => j )b.(il) =~ 1I)(3j) ltAd(�, f)A(~) 4a b. b] 1I�:; (v~, bb eE Tn. T~)[~X(~) c � r ~I< => ~ 1I. n~~176 "~+'" ~x(~) (\fa, ) [b.(a) ~b � (a) b~]]. wOv+4a [a, b] Therefore we obtain from from KPI� f- F for a ~-sentence �-sentence F a k < w Therefore we obtain KPI~ ~ co such such that Ln v+ L n ' o k 'lJ IS shows 7~,..+~ F This TL wOv+k II n~+k n-l-1 n 1 F ". Th shows -- + IIKPl',,lln < ~;-I-I " Cn(~ .~) for limit limit ordinals ordinals which which are are additively additively decomposable. decomposable. II have, have, however, however, never never checked checked for whether way meaningful. whether this this ordinal ordinal is is the the precise precise one one or or in in other other way meaningful. In In all all relevant relevant applications applications the the ordinal ordinal uII is is additively additively indecomposable. indecomposable. F for We also analysis of - K P I ~ . . If - K P I ~ ~f- F for aa �-sentence ~-sentence We also obtain obtain an an analysis of W W-KPlv If W W-KP1v such that that by by (220) (220) FF we - K P I an we obtain obtain similarly similarly as as in in the the case case of of W W-KPI an c~ a << c0 co such �:: L a EF L"o . Applying 7-/~o,~+.[{oL}] Applying Collapsing Collapsing and and Theorem Theorem 3.4.2.2 3.4.2.2 we we obtain obtain 1I.wOv+a [{ a}] IIf~+~ n+l 3.4.6.20. heorem. 3.4.6.20. TTheorem. For limit limit ordinals ordinals uII we we have have IIW-KPluI[~ IIW-KPlv ll n <::; Cn(f~v" 'ljin (Ov ' Co). co ) . For SetTheory Theoryand andSecond Second Order Number Theory Set Order Number Theory 331 331 Recall from from (169) and and (170) (170)that that'lhCx0is isthetheleast least fixed point of the function fixed point of the function O A~c. Recall R e . By (215)(169) we get therefore A� . 0( . By (215) we get therefore 7/o ~ (Vc~e Lr Lr f). 'r ("10'. (221)(221) tio � E L.p]o) (3J EeL.p]o) /tAd(O'., J). .pI 0 By the now familiar procedures we obtain from (221) By the now familiar procedures we obtain from (221) [_r A u t - K P 1 ~ F =~ 7t0 ir n - FL+,o (222) (222) I �F W - A u t - K P 1 ~ F =} =~ (30'. (3a << coe0)(Vb Lr [H¢ [HrnI.a[{O'.}] I�:;" ~ ) (Vb EEL.plo) W-Aut-KPI 'f~+l for E-sentences F. Therefore we obtain for E-sentences Therefore we obtain F. (223) (223) FLnFL,] ] (224)(224) 3.4.6.21. Theorem. 3.4.6.21. Theorem. (i)(i)IIAut-KPF Ila _< Ca(r II Aut-KPlr lin ::; 'l/Jn ('lhO) (ii) ~'o) (ii)[IW-Aut-KPF IIW-Aut-KPlrII~ 'l/Jn('l/JIO · 9co) l i n _< ::; r162 (i)(i)IIAut-KPlll a <::;r162 II Aut-KPllin 'l/Jn(c.p]O+l ) � (V~ (V� eE lLan,n)) ((33Jf Ee Lla,)ltAd((, whichtogether togetherwith with nn ) ltAd(�, Jf)) which From(215) (215)we weget getalso also 7i0 tio ~ From ourour previous work shows previous work shows 3.4.6.22. IIKPl* llan <~::; ~)~t(E~n.C1) 'l/Jn (cnn +l ) '" Theorem. [IKPl,[I 3.4.6.22.Theorem. Theseexamples examplesshould shouldsuffice sufficetotodemonstrate demonstrate the These the technique technique of ofordinal ordinalanalysis. analysis. alreadymentioned mentionedthat thatititisisnot not the the most most recent recent state WeWe already state of of the the art. art. On Onthe theone one hand analysesofofmuch muchstronger strongeraxiom axiom systems systems such such as reflection and even hand thethe analyses as IT Ha reflection and even 3 E1 -Separation- which - whichcorresponds correspondstotoIIl-comprehension IT� -comprehension -_ are El-Separation are known known today. today. The The collapsing functions needed there are based on a IT g -reflecting ordinal which collapsing functions needed there are based on a H~ ordinal which-- when when stick simplificationofofusing usingregular regularcardinals cardinals instead instead of we we stick to to thethe simplification of recursively recursivelyregular regular ones corresponds to the fi r st weakly compact cardinal. The cardinals ones - corresponds to the first weakly compact cardinal. The cardinals which which are are in this connectedto tocollapsing collapsingfunctions functions for for El-separation E I -separation are in this wayway connected are considerably considerably larger. other handwewealso alsoomitted omitteda awhole wholezoo zooof oftheories theories which larger. OnOn thethe other hand which are are between between a�-CA) (a�-CA) and + (Bi) , i.e., between and KPi. Examples eo KPl ( (A~-CA) and ( A ~ - C A ) + (Bi), i.e., between KPleo and KPi. Examples for for omitted systems (E-FOUND )which whichon on the the side side of of subsystems subsystems of suchsuch omitted systems areare WW - K-PKPi i + +(E-FOUND) of Number Theory corresponds autonomouslyiterated iteratedA~ Ll�comprehension comprehension ((Aut-A~) Aut-a� ) Number Theory corresponds to to autonomously whose ordinal is 'l/Jn ('l/JI (cI+l)) ). (and(and whose ordinal is r162 We arranged the theories in Table 1 in such a way that in every row the theories We arranged the theories in Table 1 in such a way that in every row the theories are embeddable from right.WeWehave haveshown shownthat thatthe theordinal ordinal inin the the column column are embeddable from leftleft to to right. IIAxllwCK is an upper bound for the "strongest" theory, i.e., the Set Theoretic theory. IIAxlIwcK isI an upper bound for the "strongest" theory, i.e., the Set Theoretic theory. To show that the bounds are precise it suffices to arithmetize the notation system To show that the bounds are precise it suffices to arithmetize the notation system (whose construction is indicated on page 308) and to give a well-ordering proof for (whose construction is indicated on page 308) and to give a well-ordering proof for within the theory in the leftmost column. All the shown the segment below I I Ax ll wCK the segment below IIAxll~cK Iwithin the theory in the leftmost column. All the shown 332 W. Pohlers Pohlers W. Theories for Inductive Definitions Restricted Comprehension Set Theoretic Theories NT22 and Choice in NT AxllwC1 K IIIIAxll~o~ lSt-order 2Dd-order 2nd-order Ist-order IDI ID1 ID~ ID� (Ill-CA) (II�CA ) -- KPw '!fin {cn+d Cn(~n+l) ID< ID ID�,.,)o ((ID~,)o (II I- C CA)o {II�A)o A)o (.6.�( A1- C CA)o KPlr KPff KPi r KPir KPfiF KP,ar (!1w ) Cn(n~) '!fin W-ID~t ID�", ID~ W-ID,.,t (II1-CA) �- CA ) (II W-KPI W-KPI '!fin{!1w . co) ", ID,., ID BID~ BID� + ((Bi) II�- C A ) + Bi) ((I-II-CA) KPI KPI '!fin {cn",+d ID< w'" ID<w~ BID~w~ BID� w'" I D ~ ... ID�", (III-CA<~) �- C A<w'" ) (II (A~-CR) - CR) (.6.� KPl::, K P I ~... { !1w'" ) '!fin Cn(~) ID<e ID<eoo B I D ~ 6 eoo BID� iD2eo ID�£o KPl�o KPI~o W-KPi W-KPi W-KP(3 W-KP/3 '!fin (!1,o) Cn(~,o) ID BID~.v BID� II�- CA II�- CA<eo) ((II1-CA<~o) (A~-CA) ( ~ - -A A CC) ) (.6.�- CA ) ,' (E� KPl� KPlr' , i JI '!fin{!1v)� Cn(nv)* {ID�)o (ID~)o II�- CA,,)o ((Illl-CAv)o KPl� KPIr, Cn(nv)** '!fin {!1v)H W-IDv W -ID..,tt ID~ ID� II�- CA,,) ((Illl-CAv) W-KPl" W-KPlv eo)tt tt 'Cn(nv" !fin { !1 v ' co) ID" IDv BID2v BID� ID 2 + (Bi) ID� + ( Bi) (II�- CA,,) ++ (Bi) (III'CAv) (Bi) KPl" KPlv Cn(en~+l)tt '!fin {cnv+d tt ID.<, ID-<. I BID2* BID2, ID 2. + ID2* + (Bi) ( Bi) KPl* {cno+d KPI* = = KPln KPI~ '!fin Cn(enn+l) {Aut-ID)o (Aut-ID)o Aut-lIDo ((Aut-Illl)o Aut-KPff Aut -KPlr r162 '!fin ('!fiIO) Aut-ID Aut-ID �) (Aut-II1) (Aut-II W-Aut-KP1 W-Aut-KPI r162 · co) co) '!fin{'!fiIO (Aut-Hll)+(Bi) (Aut-IID+{Bi) Aut-BID Aut-BID i ! i i (Bi) (A~-CA) (.6.�-C A) ++ (Bi) ' (E~-AC) + (Bi) + -A (Bi) (E� C) r'!fin 1 6 2{c'PI o+d Aut-KPI Aut-KPI i i KPi KPi KPf} KP(3 '!fin {cI+d Cn(el+l) K Fig. some impredicative Notice that that ~n ::= - w~ Fig. 1: 1: H~-ordinals n t -ordinals of of some impredicative theories. theories. Notice WfK tWe theory. For details cf. Buchholz et al. [1981] t We did did not define this this theory. [1981] �For vII == wP, wP , p PE Lim E Lira E Lira Lim ~For ttFor vII E ~For HFor additively indecomposable vII Set Theory and Second Order Number Theory 333 333 upper bounds bounds are are precise precise ones. ones. Unfortunately Unfortunately there there is is not not enough enough space space to to indicate indicate upper the well-ordering well-ordering proofs. proofs. Details Details for for the the well-ordering well-ordering proof proof for for ID" IDv are are in in Buchholz Buchholz the and Pohlers Pohlers [1978] [1978] and and Buchholz Buchholz et et al. al. [1981]. [1981]. More More well-orderings well-orderings proof proof are are carried carried and through in in Rathjen Rathjen [1988]. [1988]. through Notice that that the the remark remark in in the the beginning beginning of of Section Section 2.2.2 2.2.2 applies applies to to all all the the ordinal ordinal Notice analyses given given in in this this article. article. So So these these analyses analyses are are all all profound profound and and can can therefore therefore be be analyses extended to to rr II~ g -analyses. extended References References H.. G. ACZEL, ACZEL, H H.. SIMMONS, SIMMONS, AND AND S. S. 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[1975] Proof Theory, North-Holland, [1975] North-Holland, Amsterdam. Amsterdam. A. WEIERMANN A. WEIERMANN [1996] [1996] How How to to characterize characterize provably provably total total functions functions by by local local predicativity, predicativity, Journal of of Symbolic pp. 52-69. Logic, 61, 61, pp. 52-69. This Page Intentionally Left Blank CHAPTER V CHAPTER V GSdel's Functional Interpretation Godel '8 Function al (("Dialectica") "Di alectica" ) Interpret ation Jeremy Avigad Jeremy A vigad Department Department of Philosophy, Carnegie Carnegie Mellon University Pittsburgh, PA 15213 Solomon Feferman Feferman Solomon Departments of Mathematics Mathematics and Philosophy, Stanford University Stanford, CA 94305 9~305 Contents Contents Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1. Introduction 2. T h e Dialectica Dialectica interpretation i n t e r p r e t a t i o n of of arithmetic arithmetic .. .. .. . . . . . . . . . . . . . . . . . . 2. The 3. Consequences and and benefits benefits of of the the iinterpretation nterpretation . . . . . . . . . . . . . . . . . . 3. Consequences 4. of T, T, type t y p e structures, structures, and and normalizability normalizability . . . . . . . . . . . . . . . . . 4. Models Models of T h e interpretation i n t e r p r e t a t i o n of of fragments fragments of of arithmetic arithmetic . . . . . . . . . . . . . . . . . . . 5. The 6. T h e interpretation i n t e r p r e t a t i o n of of analysis analysis .. .. .. .. .. . . . . . . . . . . . .. . . . . . . . . . . . . . . . . 6. The 7. Conservation Conservation results results for for weak weak Konig's KSnig's lemma l e m m a .. .. . . . . . . . . . . . . . . . . . Non-constructive interpretations interpretations and and applications applications . . . . . . . . . . . . . . . . . 8. Non-constructive 9. The T h e interpretation i n t e r p r e t a t i o n of of theories theories of of ordinals ordinals .. . . . . . . . . . . . . . . . . . . . . . 10. Interpretations 10. I n t e r p r e t a t i o n s based based on on polymorphism polymorphism . . . . . . . . . . . . . . . . . . . . . . References References .. .. .. .. .. .. . . . . . . . . . . . . . .. .. .. . . .. .. . . . . . . . . . . . . . . . . . . . . HANDBOOK PROOF H A N D B O O K OF OF P R O O F THEORY THEORY Edited S. R. Edited by by S. R. Buss Buss © 1998 Elsevier B.V. All 9 1998 Elsevier Science Science B.V. All rights rights reserved reserved 338 341 351 356 362 365 371 377 386 393 393 400 338 338 J. A vigad and S. S. Feferman 1 1.. Introduction Introduction 1.1. 1.1. Functional F u n c t i o n a l interpretations interpretations In Gi:idel published journal Dialectica In 1958, 1958, Kurt Kurt GSdel published in in the the journal Dialectica an an interpretation interpretation of of intuitionistic theory of finite type, type, an intuitionistic arithmetic arithmetic in in aa quantifier-free quantifier-free theory of functionals functionals of of finite an ' s functional interpretation which interpretation which has has since since come come to to be be known known as as Gi:idel GSdel's functional or or Dialectica Dialectica interpretation. interpretation. When When combined combined with with Gi:idel's GSdel's double-negation double-negation interpretation, interpretation, which which reduces classical classical arithmetic arithmetic to to intuitionistic intuitionistic arithmetic, arithmetic, the the Dialectica Dialectica interpretation interpretation reduces (also (also referred referred to to below below as as the the D-interpretation) D-interpretation) yields yields a a reduction reduction of of the the classical classical theory well. This since been theory as as well. This approach approach has has since been extended extended and and adapted adapted to to other other theories, theories, ' s original but but the the pattern pattern usually usually follows follows Gi:idel GSdel's original example: example: •9 first, first, one one reduces reduces aa classical classical theory theory C C to to aa variant variant II based based on on intuitionistic intuitionistic logic; logic; •9 then er-free functional then one one reduces reduces the the theory theory I I to to a a quantifi quantifier-free functional theory theory F. F. Functional interpretations interpretations of of this this form form can can be be interesting interesting for for a a number number of of reasons. reasons. Functional 's To To begin begin with, with, the the work work can can be be seen seen as as a a contribution contribution to to aa modified modified form form of of Hilbert Hilbert's program, from a a foundational foundational point point of of view, view, the the consistency consistency of of C C is is thereby thereby program, since, since, from F. Subsequent reduced reduced to to the the consistency consistency of of F. Subsequent analyses analyses of of F F often often lead lead to to further further gains, yielding, for gains, yielding, for example, example, reductions reductions of of (prima (prima facie) facie) •9 infinitary finitary ones, infinitary systems systems to to finitary ones, •9 non-constructive non-constructive systems systems to to constructive constructive ones, ones, and and •9 impredicative impredicative systems systems to to predicative predicative ones. ones. Secondly, Secondly, functional functional interpretation interpretation provides provides a a way way of of extracting extracting (or (or "unwinding" "unwinding")) C. For example, as constructive constructive information information from from proofs proofs in in I I or or C. For example, as a a direct direct consequence consequence of interpretation one of the the interpretation one usually usually obtains obtains the the result result that that any any recursive recursive function function whose whose F. Via totality totality can can be be proven proven either either in in II or or in in C C is is represented represented by by aa term term of of F. Via an an additional interpretation I, this additional interpretation of of F F in in I, this characterization characterization is is in in fact fact usually usually shown shown to to be the terms terms of be exact. exact. It It often often turns turns out out that that the of F F represent represent a a natural natural class class of of functions, functions, such such as as the the primitive primitive recursive recursive or or polynomial-time polynomial-time computable computable functions. functions. In In other other cases, cases, the the theory theory F F embodies embodies independently independently interesting interesting computational computational constructs, constructs, such such as as bar-recursion bar-recursion or or polymorphism, polymorphism, which which are are discussed discussed in in this this chapter. chapter. Finally, Finally, functional functional interpretations interpretations often often provide provide aa useful useful stepping-stone stepping-stone to to other other goals. goals. For For example, example, the the analyses analyses of of Gi:idel's GSdel's functional functional calculus calculus T T due due to to Tait Tait and and Howard Howard provide provide an an alternative alternative means means to to the the ordinal ordinal analysis analysis of of classical classical arithmetic, arithmetic, and and non-constructive non-constructive interpretations interpretations due due to to the the second second author author yield yield consequences consequences for for various various subsystems subsystems of of second-order second-order arithmetic. arithmetic. Many Many of of these these results results can can also also be be ob obtained syntactic transformation, tained using using Herbrand-Gentzen Herbrand-Gentzen methods methods of of syntactic transformation, and and in in certain certain domains domains (for (for example, example, in in the the ordinal ordinal analysis analysis of of strong strong subsystems subsystems of of analysis analysis and and l Nonetheless, set-theory) these methods are set-theory) these latter latter methods are the the only only ones ones currently currently known. known. 1 Nonetheless, 11Whether Whether there there is is aa fundamental fundamental obstacle obstacle against against the the use use of of D-interpretations D-interpretations for for such such purposes purposes is is aa matter matter of of methodological methodological interest. interest. Godel's Gb'del 's Functional Interpretation 339 339 functional interpretations interpretations have have proven proven to to be be aa relatively relatively powerful powerful and and versatile versatile tool, tool, functional with distinct distinct advantages advantages that that will will be be illustrated illustrated below. below. with 1.2. Historical H i s t o r i c a l background background 1.2. Although the the Dialectica Dialectica interpretation interpretation was was not not published published until until 1958, 1958, G6del GSdel began began Although 's as to develop develop these these ideas ideas in in the the latter latter part part of of the the 1930 1930's as aa possible possible modification modification of of to ' s program Hilbert's program ((cf. and Parsons Parsons [[1994]), and the the D-interpretation D-interpretation itself itself was was cf. Sieg Sieg and 1994]) , and Hilbert arrived at at by by 1941 1941 and and presented presented in in aa lecture lecture to to the the Mathematics Mathematics and and Philosophy Philosophy arrived 's Nachlass Clubs Clubs at at Yale Yale University. University. Full Full notes notes for for that that lecture lecture are are found found in in G6del GSdel's Nachlass and have have been been made made available available in in Volume Volume III III of of his his Collected Collected Works Works as as G6del GSdel [[1941]. and 1941 J. The interpretation interpretation was was first first brought brought to to the the attention attention of of the the logic logic community community The at large large in in aa lecture lecture by by Georg Georg Kreisel Kreisel at at the the Summer Summer Institute Institute in in Symbolic Symbolic Logic Logic at held at at Cornell Cornell University University in in 1957 1957 ((cf. the notes notes Kreisel Kreisel [[1957]). The publication publication held cf. the 1957]) . The GSdel [[1958], in German, was for for an issue of of Dialectica Dialectica in in honor honor of of Paul Paul Bernays Bernays'' G6del 1958] , in German, was an issue 70th birthday. birthday. GSdel worked on on a a translation and expansion expansion of of that that article article for for 70th G6del worked translation and another issue issue in in honor honor of of Bernays Bernays a a decade decade later, but though though it it reached reached the the stage stage of of another later, but proof it never never appeared appeared in in print print until until it it was was retrieved retrieved from from the Nachlass for for proof sheets sheets it the Nachlass publication in in Volume Volume II II of his Collected Collected Works Works as as GSdel publication of his G6del [[1972]. 1972J . Subsequent work work on on functional functional interpretations was carried carried out out in in the the 1960s 1960s Subsequent interpretations was through through the the early early 1980s, 1980s, following following the the initial initial developments developments by by Kreisel Kreisel in in the the late late 1950s. After After aa lapse lapse in in the the latter latter part part of of the the 1980s, 1980s, there there has has been been aa resurgence resurgence of of 1950s. interest in in these these methods methods in in the the 1990s, 1990s, yielding yielding aa number number of of new new applications. applications. Some Some interest of the the different different general general directions directions of of work work may be indicated indicated roughly roughly as as follows follows ((more more of may be or less less along along the which should for or the lines lines of of Troelstra Troelstra [1990,pp. [ 1990,pp. 236-239], 236-239J , which should be be consulted consulted for publication information information concerning found in the references references to to this publication concerning items items not not found in the this chapter). chapter) . 1. The functionals functionals in in G6del's interpretation are defined by schemata for explicit 1. The GSdel's interpretation are defined by schemata for explicit definition natural extension primitive recursion recursion to are definition and and aa natural extension of of primitive to finite finite types, types, and and are therefore recursive functionals of finite finite type. have been been therefore called called primitive primitive recursive functionals of type. 22 There There have aa number number of of investigations class of of functionals, which have investigations of of this this class functionals, which have set-theoretic, set-theoretic, recursion-theoretic and term models. Prominent latter are recursion-theoretic and term models. Prominent in in the the study study of of the the latter are various methods methods of related assignments Among various of normalization normalization and and related assignments of of ordinals. ordinals. Among the contributors here one should should mention mention are are S. Diller, W. W. Tait Tait the contributors here that that one S. Hinata, Hinata, J. J. Diller, and and W. W. Howard. Howard. 2. 2. Next, Next, GSdel's G6del's interpretation interpretation has has been been adapted adapted and and extended extended both both to to stronger stronger and weaker weaker theories. theories. Here, Here, briefly, briefly, in in semi-chronological semi-chronological order, order, are are some some of of and the kinds kinds of of systems to which which the the D-interpretation D-interpretation has has been been extended, extended, either either the systems to directly in in the the case case of of intuitionistic intuitionistic systems, or indirectly with directly systems, or indirectly by by combination combination with (We also indicate some some the negative negative translation in the the case case of of classical the translation in classical systems. systems. (We also indicate of of the the main main contributors contributors to to each): each ) : 2This will Kleene [1959b] will be distinguished distinguished below from from the class class of functionals functionals introduced by Kleene [1959b] using a weaker weaker predicative predicative extension extension of primitive primitive recursion to finite types. 340 340 J. J. Avigad A vigad and and S. S. Feferman Fe.ferman Intuitionistic arithmetic arithmetic with with principles principles of of transfinite transfinite induction induction ((a) a) Intuitionistic ((G. G. Kreisel ). Kreisel). ((b) b) Impredicative "full" ) classical Impredicative (("full") classical analysis analysis formulated formulated with with function function variables variables ((G. G. Kreisel, H. Luckhardt Kreisel, C. C. Spector, Spector, W. W. Howard, Howard, H. Luckhardt).) . ((c) c ) Subsystems C. Parsons Subsystems of of classical classical arithmetic arithmetic ((C. Parsons).) . Impredicative systems systems of of classical classical analysis analysis formulated formulated with with set set variables variables ((d) d ) Impredicative (J.-Y. (J.-Y. Girard Girard).) . ((e) e) Intuitionistic Intuitionistic and ordinals (W. (W. Howard, S. Feferman and classical classical theories theories of of ordinals Howard, S. Feferman).) . ((f) f) Predicative classical analysis S. Feferman ). Predicative systems systems of of classical analysis (W. (W. Maass, Maass, S. Feferman). ((g) g) Classical ). Classical analysis analysis with with aa game game quantifier quantifier (W (W.. Friedrich Friedrich). ((h) h) Systems arithmetic ((S.A. S.A. Cook Systems of of feasible feasible arithmetic Cook and and A. A. Urquhart Urquhart).) . fixed point point theories theories (J. (J. Avigad Avigad). ((i) i ) Iterated Iterated arithmetical arithmetical fixed ). 3. the interpretations have been been applied applied towards towards aa number number of of inter inter3. Furthermore, Furthermore, the interpretations have esting proof proof theoretic ends. These esting theoretic ends. These applications applications include: include: a) The G. Kreisel ). The no-counterexample no-counterexample interpretation interpretation for for Peano Peano Arithmetic Arithmetic ((G. Kreisel). ((a) ((b) b) Closure intuitionistic systems A.S. TroelTroel Closure and and conservation conservation results results for for intuitionistic systems ((A.S. stra stra).) . ((c) c) Conservation classical systems characterization of Conservation results results for for classical systems and and characterization of the the prov provC. Parsons, S. Feferman, U. Kohlenbach ably ably recursive recursive functionals functionals ((C. Parsons, S. Feferman, U. Kohlenbach).) . 's original original interpretation been de 4. Finally, Finally, useful 4. useful variants variants of of Godel Ghdel's interpretation have have also also been developed, J. Shoenfield, J. Diller Diller and veloped, among among them them those those due due to to J. Shoenfield, J. and W. W. Nahm, Nahm, M. M. Beeson, Beeson, U. U. Kohlenbach. Kohlenbach. These These lists, lists, as as well well as as the the treatment treatment below, below, are are not not comprehensive. comprehensive. For For more more information [1990] and information we we refer refer the the reader reader to to the the surveys surveys Troelstra Troelstra [1990] and Feferman Feferman [1993], [1993], the the encyclopedic encyclopedic treatment treatment of of Troelstra Troelstra [1973], [1973], and and the the related related articles articles Feferman Feferman [1977] [1977] and and Troelstra Troelstra [1977] [1977].. 33 1.3. 1.3. An A n overview overview of of this chapter In this chapter In this chapter we we try try to to give give aa broad broad and and self-contained self-contained survey survey of of the the D Dinterpretation interpretation and and its its applications. applications. First First we we provide provide the the details details of of the the interpretation interpretation of arithmetic, explicitly of arithmetic, explicitly presenting presenting the the relevant relevant axioms axioms and and the the functional functional theory theory T T.. IInn section section 33 we we present present some some of of the the useful useful information information that that can can bbee gleaned gleaned from from the the interpretation, look at interpretation, and and take take aa broader broader look at the the general general form form of of the the interpretation interpretation in in order order to to understand understand better better how how it it might might be be adapted adapted to to other other contexts. contexts. The The presentation presentation of of aa functional functional theory theory raises raises the the issue issue of of what what its its models models look look T, that like. like. In In the the case case of of Godel's Ghdel's T, that issue issue is is addressed addressed in in section section 4. 4. In In section section 55 we we 33Unfortunately, Unfortunately, notation notation in literature is not uniform, uniform, and here here we have have struck struck some some compro compromises. primitive recursive mises. For example, example, though though the use of PRW PR" to denote denote Gi:idel's Ghdel's theory theory of the primitive recursive functionals of finite [1977,1990] is more finite type in Feferman Feferman [1977,1990] more descriptive, descriptive, here here we follow follow Gi:idel's Ghdel's original use of the name T T.. Codel GSdel'8's Functional Functional Interpretation Interpretation 341 341 show how how to to weaken weaken T T in in order order to to obtain obtain useful useful characterizations characterizations of of the the provably provably show total total recursive recursive functions functions of of certain certain fragments fragments of of arithmetic, arithmetic, namely namely IEJ I21 and and SJ S~.. In section section 66 we we go go in in the the opposite opposite direction direction and and consider consider aa strengthening strengthening of of T T In that suffices suffices to to interpret interpret full full second-order second-order arithmetic arithmetic (("analysis"). In the the following following that "analysis" ) . In three sections sections we we then then consider consider ways ways in in which which the the D-interpretation D-interpretation can can also also be be used used three to to obtain obtain information information regarding regarding aa number number of of interesting interesting subsystems subsystems of of analysis. analysis. Finally, Finally, in in the the last last section section we we show show how how functional functional theories theories based based on on polymor polymorphic phic types types arise arise in in aa natural natural way way from from aa functional functional interpretation interpretation of of full full analysis, analysis, formulated using using predicate predicate variables variables instead instead of of function function variables. variables. formulated The The authors authors are are very very much much indebted indebted to to Ulrich Ulrich Kohlenbach Kohlenbach for for numerous numerous com comments and and suggestions, suggestions, as as well well as as corrections corrections to to aa draft draft of of this this chapter. chapter. ments 2.. The The Dialectica Dialectica interpretation interpretation of of arithmetic arithmetic 2 2.1. Theories T h e o r i e s of of arithmetic arithmetic and and the the double-negation double-negation interpretation interpretation 2.1. The The first-order first-order theory theory Peano Peano arithmetic, arithmetic, or or PA PA,, has has already already been been discussed discussed in in Chapter Chapter II. II. Peano Peano arithmetic arithmetic has has its its intuitionistic intuitionistic analogue analogue in in aa theory theory known known as Heyting Heyting arithmetic, arithmetic, or or HA HA,, which which differs differs from from the the former former only only in in that that it it uses uses as intuitionistic axioms and rules as intuitionistic axioms and rules as the the underlying underlying predicate predicate logic. logic. For For concreteness, concreteness, cf. we take take the the following following list list of of axioms axioms and and rules, rules, which which is is that that used used by by Godel GSdel [1958J [1958] ((cf. we also Troelstra [1973,1977]): [1973,1977]): also Troelstra From cp, ~, cp ~ � -+ 'ljJ r conclude conclude 'ljJ r 11.. From 2. From cp ~ � --+ r'ljJ, 'ljJ r � --+ e0 conclude ~ � --+ e0 conclude cp 2. From 3. ~ VV~cp- -� + ~cp, p , pcp- +�~ A cp ~1\ cp 3 . cp 4. - + ~cpVVr 'ljJ 1 6, 2cp 1\ 'ljJ � cp 4. ~cp-� 5. ~Vr162162 5. cp V 'ljJ � 'ljJ V cp, cp 1\ 'ljJ � 'ljJ 1\ cp 6. F r o m ~cpr � 1 6 2'ljJ conclude e V cp � e V 'ljJ 6. From e, and and conversely � ('ljJ � e) conclude ~cp A 7. From ~cp --+ (r -+ 0) conclude conversely 7. From 1\ r'ljJ ~� 0, 88.. • -1 � e � r'ljJ conclude conclude ~cp --+ � Vx 'r/x r'ljJ, assuming assuming xx is is not not free in cp 9. From ~cp --+ free in 9. From 10. 10. Vx 'r/x ~cp -+ � ~[t/x], cp[t/x] , assuming assuming tt isis free free for for xx in in cp 11. 11. ~[t/x] cp[t/xJ ~� 3x :Jx ~p, cp, assuming assuming tt is is free free for for xx in in cp assuming xx isis not not free free in in r'ljJ 12. From ~cp ~� r'ljJ conclude conclude 3x :Jx ~cp --+ � r'ljJ , assuming 12. From Here ~[t/x] cp[t/xJ denotes denotes the the result result of of replacing replacing all all free occurrences of of the the variable variable xx by by tt Here free occurrences in the the formula formula ~. cp. ItIt isis common common in in intuitionistic intuitionistic systems systems to to define define negation negation by by in ...,A == AA- ~� •-L ~A where _L -1 is is an an identically identically false false statement, statement, or or "contradiction"; "contradiction" ; _L -1 may may be be taken taken to to be be where 1 . We We take take the the equality equality axioms axioms to to be be closed atomic atomic formula, formula, or or identified identified with with 00 -= 1. aa closed given given by by J. A A vigad vigad and and S. S. Feferman Feferman 342 342 1. x X -=- Xx 1. 2. xx = 2. = y y -+ -+ (cp[x/z] (~[x/z] -+ cp[y/z]) ~[y/z]),, where where cp ~ is is atomic. atomic. Finally, Finally, classical classical logic logic is is obtained obtained by by adding adding to to this this list list tertium tertium non non datur, datur, the the law law of of excluded excluded middle: middle: cp~ V --'cp. ~. Classical Classical predicate predicate logic logic can can be be reduced reduced in in aa simple simple way way to to intuitionistic intuitionistic predicate predicate logic or negative) independently) logic via via the the so-called so-called double-negation double-negation ((or negative) translation translation due due ((independently) to G6del and Gentzen. This to Gbdel and Gentzen. This is is defined defined as as follows: follows: N ~-- -'-'cp, ~(~, for cp ~ atomic atomic for 11.. cp(/9g = = cpN 1\ 'lj;N -,cpN 1\A -,'Ij; (v V v 'Ij;) r N= = -,( ~(~vN ~r N ) (cp -+ -+ (V --~ 'Ij;) r N= = cp ~N N _+ CN 'lj;N (cp ("Ix (v~ cp(X)) ~(~))~N = = "Ix v~ cp(x) ~(~)~N (3~ cp(x))N V(~))N = = -,Vx ~V~ -,cp(x) ~V(~)NN (::Ix 1\ r N 2. (~A 2. (cp 'Ij;) a3. 4. 4. 5. 66. The "double negation" appellation is The "double negation" appellation is due due not not only only to to clause clause 11,, but but also also the the fact fact that that -,(cpN V 'lj; N +-+ (~ r N +-+ ++ -, _~(~N c gN)) and and (3x ~)N ++ ~ 3 X cpN ~N are are provable provable intuitionistically. intuitionistically. -,-,::Ix (cp V 'Ij;) (::Ix cp) Clearly, point of Clearly, from from aa classical classical point of view view every every formula formula is is equivalent equivalent to to its its N Ninterpretation. Moreover, one the following interpretation. Moreover, one has has the following Suppose set of Suppose aa set of axioms axioms S S proves proves aa formula formula cp ~ using using classical classical logic. Then Then S SN Iv proves proves cp ~ gN using using intuitionistic intuitionistic logic. logic. logic. 2.1.1. 2.1.1. Theorem. Theorem. For For aa proof proof of of this this and and more more general general results, results, see see Troelstra Troelstra [1973] [1973] or or Troelstra Troelstra [1977, [1977, section 3.8]. For the case at hand, the preceding theorem provides section 3.8]. For the case at hand, the preceding theorem provides the the following following useful useful 2.1.2. 2.1.2. Corollary. Corollary. Suppose proves aa formula proves cpN Suppose PA PA proves formula cpo ~. Then Then HA HA proves ~N.. 2.1.3. 2.1.3. Proof. Proof. We We only only need need to to verify verify that that HA HA proves proves the the N-interpretation N-interpretation of of each each HA proves using aa double axiom rule of axiom and and rule of PA. PA. Since Since HA proves x x= = y y V -,(x -~(x = = y) y) ((using double induction induction on on x x and and y), y), the the N-interpretations N-interpretations of of the the quantifier-free quantifier-free axioms axioms of of PA PA follow follow from from their the N-interpretation the induction their HA HA counterparts. counterparts. Finally, Finally, the N-interpretation of of an an instance instance of of the induction 0 scheme scheme is is again again an an instance instance of of the the induction induction scheme. scheme, n 2.2. 2.2. The T h e primitive p r i m i t i v e recursive recursive functionals functionals of of finite finite type type The The Dialectica Dialectica interpretation interpretation reduces reduces HA HA to to aa theory theory T T which which axiomatizes axiomatizes aa class called the class of of functionals functionals that that G6del Gbdel called the "primitive "primitive recursive recursive functionals functionals of of finite finite ,, type. While T type. ''44 While T is is quantifier-free, quantifier-free, its its language language is is many-sorted, many-sorted, in in that that each each term term is is assigned short. The assigned aa type type symbol, symbol, or or type type for for short. The set set of of types types is is generated generated inductively inductively by by the following following rules: rules: the 4For 4For a variant of the D-interpretation D-interpretation that applies applies directly directly to PA, see Shoenfield Shoenfield [1967). [1967]. Gadel's GSdel 's Functional Interpretation 343 343 1. is a a type. type. 1. 00 is 2. If a a and and 'I T are are types types then then so so is is a a -+ --+ '1 T55 2. If The intended use is is that that objects of type type 00 are are considered to be be natural numbers The intended use objects of considered to natural numbers and objects objects of of type type a a -+ --} 'I T are are considered considered to to be be functions functions from from objects objects of of type type a a and to T. The to objects objects of of type type T. The latter latter may may be be interpreted interpreted as as constructive constructive functions functions in in some ed type. some sense sense or or other, other, or or set-theoretically set-theoretically as as all all functions functions of of the the specifi specified type. By By convention convention we we interpret interpret T 1 - - - } T2 ---Y . . . ----} Tn by associating parentheses right, i.e. by associating parentheses to to the the right, i.e. as as -+ arguments and Objects Objects of of a a type type (a (a -+ --+ 'I) 7-) -+ --+ P p have have function function arguments and are are usually usually called called type) of functionals. functionals. An An interpretation interpretation (M (M~u :: a a a a type) of such such a a typed typed language language is is called called a a functional functional type type structure structure or or simply simply a a type type structure. structure. One list the One might might also also wish wish to to include include on on the the preceding preceding list the following following additional additional closure closure condition: condition: 3. If a a and and 'I T are are types types then then so so is is a a x x T. T. 3. If Here a a x • 'I r denotes denotes the the set set of of ordered ordered pairs pairs of of objects objects ((s, t) with with ss an an object object of of type type Here s, t) a and and tt an an object object of of type type T. This closure closure condition is eliminable eliminable in in favor favor of of 22 by by a T. This condition is "currying," that is, the type the type type pp -+ (a (a -+ "currying," that is, interpreting interpreting the type (p (p x x a) a) -+ 'I T as as the --+ 'I) T) and the term-reading such choices and adopting adopting the term-reading conventions conventions described described below. below. (Always (Always in in such choices there conditions on es description there are are trade-offs: trade-offs: fewer fewer closure closure conditions on type type symbols symbols simplifi simplifies description of models, normalization terms, etc., of models, normalization of of terms, etc., but but more more closure closure conditions conditions provide provide added added flflexibility exibility and and naturalness naturalness of of formulation.) formulation.) To natural number To each each type type a a we we can can assign assign a a natural number lev(a) lev(a) as as its its type type level, level, by: by: 11.. lev(0) lev(O) = = 00 2. lev(a 2. lev(a -+ --+ 'I) 7-) = = max(lev(a) m a x ( l e v ( a )+ + 11,, lev(T» lev(T)).. nite level The be of The language language of of T T is is said said to to be of finite finite type type since since every every type type is is assigned assigned a a fi finite level by (n) are by this this convention. convention. The The pure pure types types (n) are defined defined by by 1. (0) = 1. (0) = 0 0 2. (n (n + 1) = (n) -+ 2. + l) = (n) ~ O. 0. Where Where the the context context determines determines that that we we are are dealing dealing with with type type symbols, symbols, we we drop drop the the n) = parentheses parentheses around around symbols symbols for for pure pure types. types. Then Then lev( lev(n) = n n for for each each n n < < w. We terms of We now now define define the the set set of of terms of T, T, as as well well as as the the relation relation tt :: 'I T (read (read "term "term tt has ) , inductively has type type 'I" T"), inductively via via the the following following rules: rules: T , Zz T~,, .. ..,. of 1. There are are infinitely infinitely many many variables variables x x T~,, y y~, of each each type type T. T. 1. There If s is a term of type a and t a term is of type a then t(s) 'I 2. 2. If S is a term of type a and t a term is of type a -+ -~ T then t(s) is is a a term term of of type type T. T. 3. 3. 0 0 is is a a constant constant of of type type O. 0. 5Godel a) for 5G5del [1958] [1958] used used (T, (%a) for our our a a -+ --+ T. r. There There are are many many alternative alternative notations notations in in use use in in the the literature such (a) 'I or T( a) or literature such as a s (a)T o r T(a) o r 'I" q.a,, etc.; etc.; caveat caveat emptor. 344 344 J. A vigad and S. S. Feferman Feferman J. 4. Sc Sc is is aa constant constant of of type type 00 -+ --~ O. 0. 4. 5. For For each each pair pair of of types types a a,, 7, T, Koo K~,r is aa constant constant of of type type a a -+ ~ 7T -+ --+ a. a. 5. ,T is 6. For For each each triple triple of of types types p, p, a, a, 7, T, Sp Sp,~,r is aa constant constant of of type type 6. ,oo,'T is (p -+ ao -+ 7) -+ (p (p -+ ao)) -+ (p (p -+ 7) . (p 7. For For each each type type a, R~ is is aa constant constant of of type type a a -+ --+ (0 (0 -+ --+ a a -+ --+ a a)) -+ --+ 00 -+ --+ a. a. 7. a, Roo The intended intended interpretation interpretation is is that that 00 denotes denotes the the constant constant zero, zero, Sc Sc(t) denotes the the The (t) denotes successor successor of of tt (which (which we we will will also also write write tl) t'),, and and t(s) t(s) denotes denotes the the result result of of applying applying the function(al) function(hi) tt to to the the argument argument s. The intuitive intuitive meanings meanings of of Koo,'T) K~,~, SSp,~,~, and Roo R~ the s. The p,oo,'T) and will become become clear clear when when we we present present their their defining defining equations equations below. below. will Where possible possible without without ambiguity, ambiguity, we we will will suppress suppress the the type type superscripts superscripts on on the the Where constants K, R and and on on variables, variables, and and write, write, for for example, example, x, x, y, y,z,Z, ..... We We will will K, S, S, R constants sometimes use use capital capital letters letters X, X, Y, Y, Z Z ... .. . to to denote denote variables variables of of functional functional type. type. To To sometimes improve readability, readability, if term of of type type p p -+ --+ ((a --+ 7 T), is aa term term of of type type p, p, improve if tt is is aa term a -+ ) , rr is and ss is is aa term term of of type type a, a, we we will will write write t(r, t(r, s) s) instead instead of of t(r) t(r)(s) (which we we in in turn turn and (s) (which interpret by by associating associating to to the the left, left, yielding yielding (t(r)) (t(r))(s)). Similarly t(rl' t(rl, Tr 2 ,, .. ... ., r, nrn) ) interpret (s) ) . Similarly denotes t(rl t(rl)(r2) (rn). 66 denotes ) (r2) .. .. .. (rn). At the the risk risk of of confusion, confusion, sequences sequences of of variables variables (possibly (possibly empty) empty) will be indicated indicated At will be by by the the same same font, font, e.g. ( x l ,, .. ... ., , xn) or X X = - (X ( X 1l ,, .. .. .., , Xn). X~). If If x x is is such such a a sequence sequence e.g. xx -= (x! xn) or then the the term term t(x) t(x) should should be be interpreted interpreted as as t(XI t ( x l ,,. .. .. . ,,xn) and the the prefix prefix 3x 3x should should then xn) and be interpreted interpreted as as the the quantifier quantifier string string 3XI 3x~ 3X 3x22 ..... . 3xn 3x~.. In In general, general, we we will will rely rely on on be context to to determine determine whether whether we we are are dealing dealing with with a a single single variable variable or or sequence sequence of of context such. If x x and and y y denote of variables, variables, then denotes their their concatenation, concatenation, denote sequences sequences of then x, x, yy denotes such. If as in in t(x, t(x, y) y) or or 3x, 3x, y. y. as In the the intended intended interpretation, interpretation, of of course, course, 00 and and Sc Sc satisfy satisfy In . • . Xl z' =1= # 00 and and Xl x' -= y' --+ X x = = y. yl -+ y. The constants in (5) (5) and and (6) (6) above are the the usual whose The constants K g and and S S in above are usual typed typed combinators combinators whose interpretation is is given given by interpretation by K(s, K ( s , tt)) ==s s ff oorr ss: a: aan and d t : r ,t : 7, and and S(r, s, t)t) = r(t) (s(t)) for S(r, s, = r(t)(s(t)) for rr: : (p (p --+ -+ aa -+ -+ T), 7) , ss: : pp --+ -+ a, a, and and tt: : p. p. The of equality at higher The meaning meaning of equality at higher types types is is aa delicate delicate matter matter which which will will be be taken taken up up 2.5 below. below. For For the the time time being being we we read read these these equations equations naively. naively. in section section 2.5 in If terms terms are are generated generated from the variables variables and and constants constants solely solely by by the the operation operation of of If from the application application then then one one obtains obtains combinatory combinatory completeness completeness as as usual, usual, i.e. i.e. we we can can associate associate 6In dropped altogether 6In the the literature literature on on functional functional interpretations, interpretations, parentheses parentheses are often often dropped altogether for for the the sake sake of of brevity, brevity, so so that that one one may may encounter encounter e.g. e.g. Xxyz Xxyz instead instead of of X(x,y,z). X(x, y, z) . When When this this isis the case, case, convention convention and and the the types types of of the the associated associated terms terms and and variables variables dictate dictate the the appropriate appropriate the reading. 345 345 Godel's G6del 's Functional Interpretation t AX.t with with each each term term t and and variable variable x x another another term term Ax.t whose whose free free variables variables are are all all those those of es the of t other other than than x, x, and and which which satisfi satisfies the equation equation t (AX.t)(S) t [s/x] . It closed xn , then It follows follows that that if if tt has has free free variables variables among among X x ll ,, .. ... . ,,xn, then (AX (Axl... Axn.t) n .t) isis aa closed l . . . AX = term term with with = Si Xi . . . , . for n. Alternatively, for all all si of of the the same same type type as as xi for for ii = 11,, .. .. . , , n. Alternatively, we we could could have have taken taken the the A operation operation as as a a basic basic term-forming term-forming operation, operation, where where the the terms terms thereby thereby formed formed have have the the defining defining equation equation above. above. Finally, we Finally, we have have the the equations equations for for the the recursors recursors R, which define form of of R, which define aa simple simple form primitive primitive recursion: recursion: A = == == R(J, R(f, g, g, O) O) = f f R(J, g, g, n') R(J, g, g, n)). R(f, n') = g(n, g(n, R(f, n)). That That is, is, R(J, R(f, g) g) is is a a function function h h of of type type 00 -+ a a,, with with defining defining equations equations h(O) h(0) = f f h(n') h(n') = g(n, g(n, h(n)). h(n)). There There is is no no need need to to mention mention parameters parameters to to h h explicitly, explicitly, since since these these can can be be absorbed absorbed g. We in in the the types types of of f f and and g. We note note that that this this kind kind of of higher-type higher-type iteration iteration is is clearly clearly anticipated anticipated in in Hilbert Hilbert [1926], [1926], and and even, even, to to some some extent, extent, in in Weyl Weyl [1918]. [1918]. The The atomic atomic formulas formulas of of T T consist consist of of assertions assertions of of equality equality between between terms terms of of the the same formulas are are obtained same type,7 type, 7 and and more more complex complex formulas obtained by by combining combining these these with with the the usual usual propositional propositional connectives. connectives. The The axioms axioms of of T T consist consist of of the the defining defining equations equations Sc, K, described above, substitution of of of 0, 0, Sc, K, S, S, and and R R described above, aa rule rule allowing allowing for for the the substitution of arbitrary terms for variables of type, equality equality axioms, axioms, the axioms of arbitrary terms for variables of the the same same type, the axioms of classical classical propositional propositional logic, logic, and and the the scheme scheme of of induction induction from ~(x') conclude t for in the for arbitrary arbitrary formulas formulas
346 346 J. J. A vigad and S. S. Feferman 2.3. 2.3. The The Dialectica Dialectica interpretation interpretation To each each formula formula cP qo in in the the language language of of arithmetic arithmetic we we associate associate its its Dialectica Dialectica (or (or To D-) interpretation interpretation cp ~oD D,, which which is is a a formula formula of of the the form form D-) 3X Vy Vy CP ~PD cp~pDD =__ 3x D where T. Here Here the where CP ~0D (quantifier-free) formula formula in in the the language language of of T. the free free variables variables D isis aa (quantifier-free) cP, together of of CP ~oD D consist consist of of those those free free in in ~, together with with the the sequences sequences of of variables variables (possibly (possibly empty) empty) x x and and y. However, the the free free variables variables of of cP qo are are generally generally suppressed suppressed and and we we y. However, of cP are exhibited, as for If one or more free variables also write write CP ~D(X, y) for ~D. If one or more free variables z of ~ are exhibited, as also z . D (X, y) CPD cp~(z), then we write for then we write ~OD(Z, y, z) for ~0D. y . z (z) , CPD (X, , ) CPD D and inductively as The The associations associations (( ))D and (( )D are defined defined inductively as follows, follows, where where )D are D D ::IX Vy Vy CP ~ DD and and 'lj; ~/)D = _. 3u ~U "I VV cp~D == 3x v 'lj;~/)D. D. 1. For 1. For cP ~ an an atomic atomic formula, formula, x x and and y y are are both both empty empty and and cp ~0D D = = CP ~ DD = --- cpo ~. D 2. ((rcp /\ ^ 'Ij;) r176= = 3x 3~,, U ~ Vy, vy, V~ (CP (r D /\ ^ 'lj;D) r . 2. 0 ~/\DCP)D ) VV ((zz ==I A1 C/\D'lj;D)) 3. ( z ==0 A )). . 3. ((cp~ vVr 'Ij;) D = 3z , xx,, uuVy, Vy, vv (((z 4. (Vz cp ~(z)) x Vz, Vz, y y CP ~ (DX(X(Z), (z), y y,, zz).) . 4. (Vz (z)) D~ == 33X 5. 3z,~ vy CP ~ (D~(X, , y ,y , zz).) . 5. (3z (3z cp~(z)) (z)) D~ == 3z , x Vy 6. -~ 'Ij;) r D= = 33u, r "Ix w , , v~ (CP ( ~D( ~(X, , Y(x, Y(~, v)) ~)) -+ 'lj;D(U(X), r ~)).. 6. ((~cp -+ v)) U, Y The special explanation The case case of of -+ --+ has has been been put put last last here here because because this this requires requires special explanation below. below. Since Since we we have have defined defined -' ~ pCP to to be be cP ~ -+ --+ 1.. A_,, from from 66 we we obtain obtain 7. -,cp) D == 3Y 7. ((-~)~ ~v "Ix w -' - ~CP(D~(X, , Y(x v(~)). )) . In clause identification of In clause 11 we we assume assume the the obvious obvious identification of the the symbols symbols + + and and x of of HA with addition and T. The definition of with the the terms terms that that represent represent addition and multiplication multiplication in in T. The definition of D for and of for atomic atomic ~, of ((~ r D and of (3z (3z cp ~(z)) needs no no comment. comment. The The definition definition cp~pD cP, of cp /\A 'Ij;) (z)) DD needs cp VV 'Ij;) of ((~ r D is is also also clear clear on on aa constructive constructive reading: reading: the the new new parameter parameter zz tells tells which which of definition disjunct disjunct is is being being established, established, according according as as to to whether whether zz = = 00 or or zz = - 11.. The The definition D to of (Vz (Vz cp ~(z)) obtained by by prefixing prefixing aa universal universal quantifier quantifier to to cp ~pD to obtain obtain of (Z)) DD isis obtained Vz 3x 3x Vy Vy CP ~PD(X, y, zz)) Vz D (X, y, and "skolemizing" the and then then "skolemizing" the existentially existentially quantified quantified variable. variable. cp -+ The behind the definition of G6del is The motivation motivation behind the definition of ((~o ~ 'Ij; r ) D given given by by Gbdel is as as follows: follows: from from a a witness witness x x to to the the hypothesis hypothesis cp ~ DD one o n e should should be be able able to to obtain obtain aa witness witness U u to to D , such the the conclusion conclusion 'lj; cO, such that that from from a a counterexample counterexample vv to to the the conclusion conclusion one one should should be be able able to to find find a a counterexample counterexample y y to to the the hypothesis. hypothesis. In In short, short, one one uses uses equivalences equivalences (i (i-iv) below to to bring bring quantifiers quantifiers to to the the front, front, and and then then skolemizes skolemizes the the existential existential -iv) below variables: variables: ++ (i) (3~ Vy vy CP ~ (D~(X, , yy)) -+ -~ 3u 3~ "I wv 'lj; r D (U, v)) ~)) (3x w (Vy (vy CP ~ (D~(X, , y) y) -+ -~ 3u ~ "I wv 'lj;D r (U, v)) ~)) (ii) ++ (ii) "Ix ++ (iii) (iii) w 3u 3u (V (vyy CP ~ (D~(X, , yy)) -+ -~ "I wv 'lj;D(U, Co(u, V)) ~)) "Ix (iv) ++ (iv) v~ 3u 3u "Iv v~ (V (vyy CPD ~ ( ~(X, , y) y) -+ -~ 'lj;D r (U, v)) ~)) "Ix (U , v)) ++ (v) (v) v~ 3u 3u "Iv v~ 3y ~y (CP ( ~ D( ~(X, , y) y) -+ -~ 'lj;D Co(u, ~)) "Ix v~ 3u, 3u, Y ~ 1 "Iv v~ (CPD(X, ( ~ ( ~ , Yi ~(~)) -+ 'lj;D r (U, v)) ~)) (vi) ++ (vi) (v)) -+ "Ix Y "Ix w , , v~ (CP ( ~D( ~(X, , Y(x, Y(~, ~)) -~ 'lj; r D (U(X), v)) ~)).. v)) -+ 33u, U, Y Godel's G~del's Functional Functional Interpretation 347 347 The definitions of (~ A r D , (9~ r D , and and (3z are all all justified justified from from a a The definitions of (cp /up) ( z )) DD are (cp VV 'IjJ) (3z cp~(z)) constructive definition of constructive as as well well as as classical classical point point of of view. view. The The definition of (' w 3y av cp(x, ~(~, y) y) --+ -~ 3Y 3Y ' (AC) (AC) accepted reading of hypothesis is accepted by by many many constructivists, constructivists, since since the the reading of the the hypothesis is that that one one has has aa constructive proof of provide aa means constructive proof of ' (' is special case principle called is a a special case of of aa principle called independence independence of of premise. premise. Though Though this this is is valid valid in classical classical logic, it is is not not generally generally accepted accepted constructively, constructively, since since the the constructive constructive in logic, it reading of of the the hypothesis (0 --+ -+ 3u 3u TJ) 7) is is that that we we have have aa constructive constructive means means of of turning turning reading hypothesis (() any with aa witness any proof proof of of the the premise premise ()0 into into aa proof proof of of TJ r/with witness for for the the existential existential quantifier quantifier applied to to r]. In general, the choice choice of of such such aa u u will will then then depend on the the proof proof of of (), 0, TJ. In general, the depend on applied Pi) tells while while (J (IP') tells us us that that u u can can be be chosen chosen independently independently of of any any proof proof of of that that premise. premise. ' s principle, Equivalence Equivalence (iv) (iv) can can be be justified justified by by a a generalization generalization of of Markov Markov's principle, namely namely (MP') (MP') -Ny -,Vy ()0 --+ --+ 3y 3y -.() -,0 in in which which ()0 is is assumed assumed to to be be quantifier-free. quantifier-free. (Assuming (Assuming that that the the law law of of excluded excluded middle middle holds holds for for 'ljJ CO, argue thus: thus: if if 'ljJ CO true then then (iv) (iv) is is justified, justified, and and if if 'ljJ CO false D isis true D , argue D isis false apply apply (MP') (MP').).) The The problem problem with with (MP') (MP') is is that that there there is is no no evident evident way way to to choose choose constructively constructively a a witness witness y y to to -.() -,0 from from a a proof proof that that ' (MP) (MP) w (-.-.3y ( ~ s v cp(x, ~(~, y) y) --+ -~ 3y 3v cp(x, ~(~, y)) v)) ' which which is is accepted accepted in in the the Russian Russian school school of of constructivity constructivity for for cp ~ quantifier-free. quantifier-free. While While the the reasoning reasoning leading leading to to the the form form of of the the D-interpretation D-interpretation is is not not fully fully constructive constructive it it can can still still be be used used as as aa tool tool in in constructive constructive metamathematics metamathematics and and to to ' ll see derive derive constructive constructive information. information. In In section section 3.1 3.1 we we'll see that that the the D-interpretation D-interpretation verifies verifies the the three three principles principles (AC), (A C), (IP') (IP'),, and and (MP') (MR') just just discussed, discussed, and and hence hence allows allows one one to to use use the the D-interpretation D-interpretation to to extract extract constructive constructive information information from from non-constructive non-constructive proofs. proofs. 2.4. 2.4. Verifying Verifying the t h e axioms a x i o m s of of arithmetic arithmetic ' s main Gi:idel Ghdel's main result result is is as as follows. follows. 348 348 S. Feferman J.J. A vigad and S. 2.4.1. Theorem. Theorem. is aa formula formula in in the the language language of of arithmetic, arithmetic, and and HA HA 2.4.1. uppose cP~ is SSuppose proves cpo ~p. Then Then there there is is aa sequence sequence of of terms terms tt such such that that T T proves CPD(t, ~D(t, y) y).. proves We express express this this by by saying saying that that HA HA is is D-interpreted D-interpreted in in T. T. Combining Combining Theorem Theorem 2.4.1 2.4.1 We with Corollary Corollary 2.1.2 2.1.2 we we obtain obtain with is aa.formula in the the language language of of arithmetic, arithmetic, such such that that uppose cP~ is formula in SSuppose PA cpo Then ). PA proves proves ~. Then there there is is aa sequence sequence of of terms terms tt such such that that T T proves proves (cpN)D(t, (~g)D(t, yy). 2.4.2. Corollary. Corollary. 2.4.2. In short, short, PA PA is is ND-interpreted ND-interpreted in in T. T. In One proves Theorem 2.4.1 by induction on the the length length of of the the proof proof in in HA HA.. One One One proves Theorem 2.4.1 by induction on only has has to to verify verify that that the the claim claim holds holds true true when when cP ~ is is an an axiom axiom of of HA, HA, and and that that it it is is only maintained under under rules rules of of inference. inference. maintained We begin begin by by considering considering the the axioms axioms of of rules rules of of intuitionistic intuitionistic logic, logic, listed listed in in We section section 2.1. 2.1. For For most most of of these these the the verification verification is is routine, routine, and and we we only only address address aa few few key examples. For For example, example, consider consider the the rule rule "from ~ -+ --+ 'l/J r and and cP ~ conclude conclude 'l/J." r key examples. "from cP Given a a term term aa such such that that T T proves proves CPD(a, ~PD(a,y) and and terms terms bb and and cc such such that that T T proves proves Given (pD(x,b(x, v)) -+ ~ 'l/JD( CD(C(X), V),, we we want want a a term such that T proves proves 'l/JD CD(d, v). By By CPD(x, b(x, v)) C( X) , v) term dd such that T (d, v). substituting b(a, b(a, v) for y y in in the hypothesis and and aa for for x x in in the second, we we see see v) for the first first hypothesis the second, substituting that taking works; so, so, in that taking d d= - c(a) in a a sense, sense, modus modus ponens ponens corresponds corresponds to to functional functional c( a) works; application. The The reader reader can can verify verify that, that, similarly, similarly, the the axiom axiom "from "from cP ~p -+ --+ 'l/J r and and application. r -+ --+ (J0 conclude conclude cP ~ -+ --+ (J" 0" corresponds corresponds to to the the composition composition of of functions. functions. 'l/J Handling the the axiom axiom cP ~ -+ --+ cP ~p1\ A cP ~ requires requires aa bit bit more more work. work. If If the the interpretation interpretation of of Handling the D (x, y) the hypothesis hypothesis is is given given by by :Ix 3x Vy Vy cP ~D(X, y),, the the interpretation interpretation of of the the conclusion conclusion is is According to clause for we need need to provide terms terms cl(x) According to the the clause for implication, implication, we to provide Cl (X) and and c2(x) C2 (X) for the for this this taking witness xx for for the the conclusion; conclusion; for taking aa witness the hypothesis hypothesis to to witnesses witnesses xl Xl and and Xx22 for purpose, we can take c~ Cl (x) (X) and and c2(x) C2 (X) to to be x. But But we also need purpose, we can simply simply take be x. we also need aa functional functional that will will take take yl Yl and witnessing the d(x, yt , Yy2) d(x, yl, and Y2 the failure failure of of CPD( ~D(X, A ~D(X, X , Yl) Yl) 1\ CPD(X, Y2) Y2 ) Y2 witnessing 2 ) that to aa value This functional functional must must effectively effectively value dd representing failure of of ~D(X, CPD (x, d) to representing the the failure d).. This determine which CPD (X , y~) and ~D(X, false. We determine which of of ~D(X, is false. We need need the the following following CPD(X , Y2) Y2 ) is Yl ) and 2.4.3. Lemma. 2.4.3. L emma. proves proves such that T If cP formula of t", such If ~ is is aa formula of arithmetic, arithmetic, there there is is aa term term tv that T t",(X, y) = 00 ~H ~D(X, y) = CPD(X , y). y) . t~(x, The lemma CPD ; the the key key instance instance occurs occurs when when ~D CPD lemma is is proved proved by by induction induction on on the the size size of of ~PD; The is simply an atomic formula sl Sl -= s2, 82 , for for which case the the required required tt can can be be obtained obtained is simply an atomic formula which case from primitive recursion. from an an application application of of primitive recursion. Another primitive recursion Another instance instance of of primitive recursion yields yields aa functional functional Cond Cond such such that that TT proves proves { u if w = O Cond(w, u, v) v) == ~ u ifw = 0. Cond (w, u, 0therWlse. ( vv otherwise. 349 349 Giidel's Ghdel 's Functional Functional Interpretation Interpretation Taking d d = - Cond(tcp(x, Cond(tv(x, Yl) yl),, Yl, yl, Y y2) above then then suffices suffices to to complete complete the the proof proof for for Taking 2 ) above 1\ cp. ~o --~ ~o A ~o. cP ---+ cP Aside from from the the two two instances instances just just described, described, the the recursors recursors R R are are not not otherwise otherwise Aside used used to to verify verify the the logical logical axioms. axioms. Their Their primary primary purpose purpose is is to to interpret interpret the the induction induction cp(u') conclude ." Inductively rule, "from "from ~a(0) and ~a(u) -+ ~(u') conclude cp(u) ~(u)." Inductively we we are are given given terms terms cp(O) and cp(u) ---+ rule, a, b, b, and a, y) a, and cc so so that that T T proves proves CPD(O, ~aD(0, a, y) and and v.(u, b(u,z, v.(u', c(u, y,). We want want a a term such that that cp ~aD(U, y).. Using the recursors recursors we we can can define using We term dd such D (u, d(u), d( u), y) Using the define dd using primitive primitive recursion, recursion, so so that that d(O) d(0) - aa d(u')') = c(u, c(u, d(u)). d(u)). d(u This This yields yields CPD (O, d(O) ~aD(O, d(O),, y) y) and and CPD(U, ~aD(U, d(u), d(u), b(u, b(u, d(u), d(u), y)) y)) ---+ --+ CPD(U ~D(U',', d(u'), d(u'), y). y). D (u, d( u) , y) The following following lemma lemma will will then then allow allow us us to to conclude conclude cP ~D(U, d(u), y),, as as desired. desired. The 2.4.4. 2.4.4. Lemma. Lemma. r 1j;(u, y) . From y) b(u, y) From 1j;(O, r y) and and 1j;(u, r b(u, y)) y) ) ---+ --+ 1j;(u', r y) one one can can prove, prove, in in T, T, , The idea idea is is to to work work backwards: backwards: if if "" � - ", denotes denotes truncated truncated (or (or "cut "cut off" off")) subtraction, subtraction, The note y) note that that 1j;(u, r y) follows follows from from 1j;(u � - 11,, b(u � - 11,, y)), y)), which which in in turn turn follows follows from from 1j;(u � 2, b(u 2, b(u r - 2, b(u � - 2, b(u � - 11,, y))), y))), and and so so on, on, until until the the first first argument argument is is equal equal to to O. 0. More More formally, formally, one one defines defines in in T (y, z)) T aa function function e(y, e(y, z) z) by by e(y, e(y, O) O) = = yy and and e(y, e(y, z') z') = = b(u b ( u -� z', z', ee(y, z)),, and and then then uses uses induction to induction to prove prove that that r1j;( w , e(y, e(y, uu -� w w))) ) holds holds for for every every w w less less than than or or equal equal to to u. u. For For details, details, see see Spector Spector [1962J [1962] or or Troelstra Troelstra [1973J [1973].. This 0, Sc, This leaves leaves only only the the quantifier-free quantifier-free axioms axioms regarding regarding 0, Sc, + +,, and and x in in HA HA,, which immediately from which follow follow immediately from their their counterparts counterparts in in T. T. Once Once more more we we emphasize emphasize that that the the recursors recursors of of T T are are only only essentially essentially needed needed for for speaking, it the the nonlogical nonlogical axioms. axioms. Roughly Roughly speaking, it is is the the combinatory combinatory completeness completeness of of T T that that allows allows us us to to verify verify the the axioms axioms of of intuitionistic intuitionistic predicate predicate logic, logic, whereas whereas the the recursors induction. This recursors are are the the functional functional analogue analogue of of induction. This observation observation allows allows one one to to generalize generalize the the D-interpretation D-interpretation to to other other theories, theories, as as discussed discussed in in section section 3.3 3.3 below. below. 350 350 J. A vigad and S. S. Feferman Fe.ferman J. 2.5. Equality E q u a l i t y at at higher h i g h e r types 2.5. We return return to to the the question question as as to to how how equality equality at at higher higher types types is is to to be be treated treated in in We T. There There are are two two basic basic choices, choices, the the intensional intensional formulation formulation taken taken in in G6del GSdel [1958] [1958] T. ' s formulation, and the the (weakly) (weakly) extensional extensional formulation formulation of of Spector Spector [1962]. [1962]. In In G6del GSdel's formulation, and we have have aa decidable decidable equality equality relation relation =(7 =~ at at each each type, type, i.e. i.e. all all formulas formulas ss = =~(7 tt with with we of type type a a are are taken taken to to be be atomic, atomic, and and the the law law of of excluded excluded middle middle is is accepted accepted for for s,s, tt of these. Suppressing Suppressing type type subscripts subscripts where where there there is is no no ambiguity, ambiguity, the the equality equality axioms axioms these. for T T in in this this version version are, are, as as usual, usual, for 1. 1. s8 -=- 8s = tt 1\ ^ cp[s/x] -+ - + cp[t/x] . 2. s = The axioms axioms for for K, K, SS and and R R at at each each type type may may be be read read as as they they stand. stand. The In Spector's Spector's formulation, formulation, the the atomic atomic formulas formulas are are equations equations between between terms terms of of In type 00 only, only, and and the the law law of of excluded excluded middle middle is is accepted accepted only only for for these. these. For For a a = = type (all -+ --+ ..... . -+ --+ an an -+ --+ 0) 0),, an an equation equation ss =(7 =~ tt between between terms terms of of type type a a is is regarded regarded as as an an (a abbreviation for for abbreviation 2. = t(Xb . . . , xn) S(Xb · . . , xn) = where the the Xi xi are are fresh fresh variables variables of of type type ai ai for for ii = = 11,, . . .., , n. In In particular, particular, the the axioms axioms where for K, K, SS and and R R are are to to be be read read as as such such abbreviations. abbreviations. Now Now in in this this version, the axioms axioms for version, the 1, 22 are are taken taken only only for for terms terms ss,, tt of of type type 0; denote these these by by 10 10 and and 20 20 respectively. respectively. 0; denote 1, For ss,, tt of of type type a a # r 00 we we must further adjoin adjoin a a rule rule For must further 2'. From S(Xl s(xl,... = t(Xb t ( x l , ..... ,. , xn) Xn) and and cp[s/x] ~[s/x] infer infer cp[t/x] ~[t/x].. xn) = 2'. From ' . . . ',xn) An G6del in An alternative alternative suggested suggested by by GSdel in aa footnote footnote to to his his revision revision [1972] [1972] of of [1958] [1958] and made made explicit explicit by by Troelstra Troelstra [1990] [1990] is is to to follow follow Spector Spector in in taking taking only only equations equations and and 220, at as basic assume as axioms, besides besides 10 at type type 00 as basic but but to to assume as axioms, 10 and just the the following following 0 , just consequences of 2' 2' for for K, K, S, S, and and R: R: consequences of s[K(u, s [ K ( u , v)/x] v)lz] = = s[u/x], s[S(u, v, w)/x] = = s[u(v, u(w))/x], ')/x] = v, ~))/~], w))/x], s[R(u, 0)/~] = = ~[~/~], s[R(u, v, w')/~] = s[v(w, ~[v(~, R(u, R(u, v, v, O)/x] s[u/x], s[R(u, v, w s[R(u, v, where six] s[x] isis aa term term of of type type 00 and and u, u, v, v, w w are are terms type. Note Note that that where terms of of appropriate appropriate type. 's versions. the is contained contained in GSdel's and and Spector Spector's the resulting resulting theory theory is in both both G6del's versions. Most of the results results for the functional to which Most of the for the functional interpretation interpretation are are insensitive insensitive to which of of these formulations of of T systems like like T T to be considered below)) is T ((and and of of other other systems to be considered below is these three three formulations taken. When it is necessary necessary to make a a distinction, distinction, as as for for example in the section taken. When it is to make example in the next next section when considering extensions of adjunction of of quantifiers, 4.1 when considering extensions of TT by by adjunction quantifiers, and and in in section section 4.1 when considering considering models models of T, we we shall shall use use WE-T WE- T to denote Spector's Spector 's version version and and when of T, to denote To to to denote denote the the version, version, just just described, described, elicited elicited by by Troelstra, Troelstra, while while reserving reserving TT To itself itself for for GSdel's G6del's version. version. However, However, one one further further system system must must be be considered considered along along with the the latter. latter. ItIt is is implicit implicit in in the the idea idea of of intensional equality at at higher higher types that with intensional equality types that we have an effective effective procedure procedure to to decide decide whether whether any two objects of the the same same type type we have an any two objects of are E(7 for for the the characteristic characteristic are identical. identical. This This is is made made explicit explicit by by adjoining adjoining aa constant constant E~ function function of of =~ =(7 at at each each type type a, a, with with axiom: axiom: Ea(x, ~ y. E(7(x, y) y) --= 00 e+ +-+ xx -=(7 y. Codel's GSdel 's Functional Functional Interpretation Interpretation 351 351 The system system T T with with these these additional additional constants constants and and axioms axioms is is then then denoted denoted 1I - TT.. The 3 and 3.. Consequences Consequences a n d benefits b e n e f i t s of o f the t h e interpretation interpretation 3.1. 3.1. Higher H i g h e r type t y p e arithmetic arithmetic Before Before turning turning to to some some of of the the immediate immediate consequences consequences of of the the Dialectica Dialectica inter interpretation, pretation, we we pause pause to to consider consider some some easy easy generalizations. generalizations. First First of of all, all, note note that that the D-translation applies equally equally well well if if the the source source language language is is also also typed. In other other the D-translation applies typed. In words, words, if if we we define define HAw HA W to to be be a a version version of of T T with with quantifiers quantifiers ranging ranging over over each each finite finite type, with with the the axioms axioms and and rules rules of of intuitionistic intuitionistic predicate predicate logic logic and and induction induction for for all all type, formulas formulas in in the the new new language, language, then then we we can can apply apply the the D-translation D-translation to to this this theory theory as as well. well. A A problem, problem, however, however, arises arises in in verifying verifying the the interpretation interpretation ofthe of the axiom axiom
or, or, equivalently, equivalently, w w which cannot in general be the special which cannot in general be proved proved in in HA# HA # .. It It is, is, however, however, provable provable for for the special case QF-A C) cpo For, case ((QF-A C) of of the the axiom axiom of of choice choice with with quantifier-free quantifier-free matrix matrix ~. For, in in that that case, case, w ..., ...,3y cpN (x, y) y) ++ "Ix w 3y 3y cpN (x, y) y) "Ix 8But note that WE-HAw~ as that Troelstra [1990,p. [1990,p. 351] 351] says says that ""WE-HA as an an intermediate possibility possibility [between [between I-HAW I-HA ~ and and HA'O'] HA~] is not very very attractive: the deduction theorem does does not hold hold for this theory." theory." Yet Yet another another alternative alternative for for dealing dealing with with the the problem problem of of verifying verifying the the axioms axioms cp ~o-t ~ cp ~oII A cp without without restriction restriction on on atomic atomic formulas, formulas, but but without without additional additional Eu E~ functionals, functionals, makes makes use use of of aa variant of the D-interpretation due to Diller and Nahm Nahm [1974], [1974], described described in Troelstra [1973,pp. 243-245]. 243-245]. We We shall shall not not go go into into that that variant variant in in this this chapter. chapter. Godel's Gbdel 's Functional Functional Interpretation 353 353 using (MP') is is provable provable in in HA# HA # using (MP').. Then Then from from (AC) (A C) in in that that system system we we infer infer N (X, Y(x) ) , which 3Y "Ix Vx cp (flY(x,Y(x)), intuitionistically its double negation. negation. The The which implies implies intuitionistically its double 3Y conclusion is is that that ((QF-AC) QF-A C) is is ND-interpreted ND-interpreted in in T. T. The The following following observation observation conclusion by Kreisel [1959,p. proper analogue by Kreisel [1959,p. 120J 120] then then gives gives the the proper analogue to to Theorem Theorem 3.1.1: 3.1.1: 3.1.4. 3.1.4. Theorem. Theorem. equivalent. equivalent. D are Over classical classical logic, logic, the the schemata schemata (( QF QF-A C) and and cp ~ t-t ++ cpN ~ND are Over -AC) 3.1.5. consider negative 3.1.5. Proof. P r o o f . In In the the forward forward direction, direction, one one need need only only consider negative formulas formulas cp, i.e. those which contain disjunction existence symbols. ~, i.e. those which do do not not contain disjunction or or existence symbols. For For such such formulas formulas D has the it is is easily easily proved proved by by induction on cp ~ that that cp ~9N N D has the form form 3X 3X Vy Yy 'ljJ(X(y) r (y),, y) y) where where it induction on 'ljJ er-free and y) r is is quantifi quantifier-free and where where cp 9~ t-t ++ 3X 3X Vy Yy 'ljJ(X(y) r (y),, y) y) t-t ~ Vy Yy 3x 3x 'ljJ(x, r y).. The The reverse reverse 0 direction is direction is immediate. immediate. [] Thus Thus PA# PA # can can equally equally well well be be thought thought of of as as PA PA w~ + (QF-AC). (QF-A C). When When analyzing analyzing classical classical or or intuitionistic intuitionistic theories theories of of firstfirst- or or second-order second-order arith arithmetic, it often often turns turns to to be be useful useful to to embed embed them them in in fragments fragments or or extensions extensions of of PA# PA # metic, it and HA# HA # respectively; cf. the the discussion in section section 3.3 3.3 below. below. and respectively; cf. discussion in 3.2. 3.2. Some Some consequences consequences of of the the D-interpretation D-interpretation Since D-interpretations of Since the the D-interpretations of HA HA and and HA# HA #,, as as well well as as the the ND-interpretations ND-interpretations of they can of PA PA and and PA# PA #,, are are purely purely syntactic, syntactic, they can be be formalized formalized in in a a weak weak theory theory of of arithmetic. This yields arithmetic. This yields the the following following theorem, theorem, which which is is of of foundational foundational importance. importance. Let Let S S be be any any of of the the theories theories HA HA,, HA# HA #,, PA, PA, or or PA# PA #.. Then Then aa weak weak base base theory theory proves proves 3.2.1. 3.2.1. Theorem. Theorem. Con( T) -+ Con(S) Con(T) Con(S).. Of interpretations yield yield far Of course, course, the the interpretations far more more information information than than just just the the relative relative l ' X2 , ' . . , xn) in consistency consistency of of the the theories theories involved. involved. Recall Recall that that aa formula formula (}(X O(xl,x2,...,xn) in the the language nes language of of arithmetic arithmetic is is �g A ~ if if all all its its quantifiers quantifiers are are bounded, bounded, in in which which case case it it defi defines aa primitive primitive recursive recursive relation relation on on the the natural natural numbers. numbers. The The characteristic characteristic function function of of language of T, such this relation this relation can can be be represented represented by by a a term term tt in in the the language of T, such that that PA PA # # or or HA# O. Though it is HA # proves proves (}(x) 0(~) t-t ~ t(x) t(~) = = 0. Though it is somewhat somewhat an an abuse abuse of of notation, notation, when when we we T proves say below that say below that ""T proves (}(x)" 0(Y~')" for for such such a a (}, 0, we we mean mean that that it it proves proves t(x) t(Z) = = O. 0. 3.2.2. 3.2.2. Theorem. Theorem. formula formula Let Let S S be be any any of of the the theories theories above, above, and and suppose suppose S S proves proves the the rrg II ~ "Ix 3y (}(x, Vx 3y O(x, y), y), there is where where (}0 is is �g A~o.. Then Then there is aa term term f f such such that that T T proves proves (}(x, f(x)). J. J. A A vigad and S. S. Feferman 354 354 3.2.3. Proof. P r o o f . By By embedding embedding HA HA and and PA PA in in HA HA # # and and PA PA # # respectively respectively and and prov prov3.2.3. ing the the equivalence just discussed, that case case ing equivalence just discussed, we we can can assume assume that that ()0 is is quantifier-free. quantifier-free. In In that the D-interpretation of of Vx Vx 3y 3y ()(x, O(x, y) is 3Y Vx ()(x, 0(x, Y(x)) Y(x)),, and and its its ND-interpretation ND-interpretation the D-interpretation y) is 3Y Vx is 3Y 3Y Vx Vx -,-,()(x, -~--0(x, Y(x) Y (x)). The conclusion conclusion then then follows follows from from Theorem Theorem 3.1.2 3.1.2 in in the the case case ) . The is D of HA HA # # ,, and and Theorem Theorem 3.1.3 3.1.3 in in the the case case of of PA PA # #.. [] of Now defined by Now suppose suppose h h is is aa recursive recursive function function whose whose graph graph is is defined by aa E� T ~ formula formula model. We ~(x, y) y) in in the the standard standard model. We say say that that the the theory theory S S proves proves h h to to be be total total cp(x, 3!y cp(x, if if it it proves proves Vx Vx 3[y ~(x, y) y).. In In section section 2.2 2.2 we we axiomatized axiomatized the the primitive primitive recursive recursive functionals finite type, functionals of of finite type, without without addressing addressing the the issue issue of of what what exactly exactly the the terms terms below, for denote. this discussion discussion to denote. Deferring Deferring this to section section 44 below, for now now let let us us assume assume that that at at least the closed type 11 terms least the closed type terms denote denote functions. functions. As As aa corollary corollary to to Theorem Theorem 3.2.2 3.2.2 we we have have 3.2.4. Every 3.2.4. Corollary. Corollary. Every provably provably total total recursive recursive function function of of HA, HA, HA HA # #,, PA, PA, or or PA PA # # is is denoted denoted by by aa term term of of T. T. In In fact, fact, in in section section 4.1 4.1 we we will will see see that that there there are are models models of of T T that that can can be be formalized formalized in in the the language language of of arithmetic, arithmetic, yielding yielding an an interpretation interpretation of of T T in in HA HA.. This This yields yields the the following interesting in following result, result, which which is is interesting in that that it it makes makes no no mention mention of of T T at at all: all: 3.2.5. 3.2.5. Corollary. Corollary. sentences. sentences. PA, PA, and and hence hence HA HA + + (MP) (MP),, is is conservative conservative over over HA HA for for rrg II ~ When When it it comes comes to to PA, PA, Theorem Theorem 3.2.2 3.2.2 is is sharp sharp in in the the following following sense. sense. Consider Consider the the rrg y, such II ~ sentence sentence "for "for every every x x there there exists exists aa y, such that that either either yy is is aa halting halting computation computation doesn't halt." halt." Though for for the the Turing Turing machine machine with with index index x, or Turing Turing machine machine x x doesn't Though x, or this this statement statement is is provable provable in in PA PA,, any any function function returning returning such such a a yy for for every every x x cannot cannot be be recursive recursive since since it it solves solves the the halting halting problem. problem. Later Later we'll we'll see see that, that, on on the the other other hand, the terms of hand, the functions functions represented represented by by terms of T T are are recursive, recursive, so so that that the the analogue analogue of of does not hold for Theorem Theorem 3.2.2 3.2.2 does not hold for rrg II ~ formulas. formulas. Nonetheless, one Nonetheless, one can can extract extract a a different different kind kind of of constructive constructive information information from from PA-proofs A (or (or PA given by PA-proofs of of complex complex formulas. formulas. Suppose Suppose P PA PA # #)) proves proves aa formula formula cp ~p given by Vxl 3yl Vx2 3y2 ... Vxn 3yn 0(xl, x 2 , . . . , xn, yl, y2,..., yn) where is quantifier-free. The N -interpretation of where ()0 is quantifier-free. The N-interpretation of this this formula formula intuitionistically intuitionistically implies implies --3xl Vyl 3x2 Vy2 ... 3x~ Vy~ ~0(xl, x 2 , . . . , xn, yl, y2,..., y~), whose whose D-interpretation D-interpretation is is the the same same as as that that of of n V VX1, X 2,, .. .· .·, ' X~ Y2,, . 9. 9. 9, Y Yn )(n 33yl, Yl , Y2 )(1 , )(2 -'-'() ()(b )(2 ( Yl ) , . . . , )(n(Yl , Y2 , · · . , Yn-l ) , Yl , Y2 , · · . , Yn) . . As As a a result result we we have have . . , . . . , . . . , y , ) . GSdel's Functional Interpretation Godel's 3.2.6. Corollary. Corollary. 3.2.6. 355 355 Suppose PA PA proves proves cp ~p as as above. above 9 Then Then there there are are terms terms Suppose tx = F~ (Xx, X2, . . . , X~) tn = Fn(X1, X2, . . . , Xn) such that that T T proves proves such o(x~, x~(t,), . . . , x,(t~, t~, . . . , t,_x), t~, t~, . . . , t,). The The observation observation that that the the functionals functionals Ft F 1,,F2, F 2 ,.. .. .., ,FFn n can can be be taken taken to to be be recur recur' s "no-counterexample sive in in their their arguments arguments is is known known as as Kreisel Kreisel's "no-counterexample interpreta interpretasive tion" (cf. (cf. Kreisel Kreisel [1951,1959]). [1951,1959]). To To make make sense sense of of the the name, name, think think of of the the func function" tions tions Xl X 1 ,, X 2 , ... .. ,. X, Xn n as as trying trying to to provide provide counterexamples counterexamples to to the the truth truth of of cp ~p,, by by X2, making (YI), . . . , Xn(Yt, making (}(XI 0(X1,, X2 X2(Yl),..., X n ( y l , ... .. ,. , Yn Yn-1), Y l ,, .. ... ., , Yn) Yn) false false for for any any given given values values of of - d , YI in which Y2,..., Y~;; in which case case FI F1,, F2, F 2 , .. .. .., , Fn F~ effectively effectively provide provide witnesses witnesses that that foil foil the the . . . , Yn YYl, t , Y2, purported counterexample. counterexample. purported 3.3. Benefits Benefits of of the t h e D-interpretation D-interpretation 3.3. In general, the In general, the D-interpretation D-interpretation is is aa powerful powerful tool tool when when applied applied to to the the reduction reduction of an an intuitionistic intuitionistic theory theory I I to to a a functional functional theory theory F. As we we have have seen, seen, the the very very F. As of form of of the the D-interpretation D-interpretation automatically automatically brings brings a a number number of of benefi benefits Sifice little form ts.9 Shice little more more than than the the combinatorial combinatorial completeness completeness of of F F is is necessary necessary to to interpret interpret the the logical logical axioms of of I I,, one one only only has worry about interpreting the axioms of of I I has to to worry about interpreting the non-logical non-logical axioms axioms accordingly.9 As As an bonus, II can can often often be be and tailor the the functionals functionals of of F and tailor F accordingly an added added bonus, embedded in a a higher-type which we we can can add add the the schemata (AC), embedded in higher-type analogue analogue IIW~ to to which schemata (AC), (IP'),, (MP') (MR') at no extra (A C), (IP') (IP'),, and and (MR') prove the (IP') at no extra cost. cost. Taken Taken together together (AC), (MP') prove the scheme cp ~ ~ fact that often useful useful in allows one one to pull facts scheme H ~D, cpD , aa fact that is is often in practice practice since since itit allows to pull facts about the to the being interpreted. interpreted. This observation about the D-interpretation D-interpretation back back to the theory theory being This observation will employed in in sections will be be employed sections 66 and and 77 below. below. If one is is trying analyze aa classical C by by reducing to I I via via a a doubleIf one trying to to analyze classical theory theory C reducing itit to double negation has, of of course, course, to is strong negation interpretation, interpretation, one one has, to ensure ensure that that II is strong enough enough to to prove doubly-negated axioms axioms of of C. Once again, prove the the doubly-negated C. Once again, though, though, the the choice choice of of aa DD interpretation for has some can often embedded in some advantages: advantages: C C can often also also be be embedded in aa interpretation for II has higher-type analogue analogue C c~w in in aa natural natural way, way, and and the the fact that Markov's Markov' s principle principle is is higher-type fact that verified verified in in the the interpretation interpretation guarantees guarantees that that one one ultimately ultimately obtains obtains Skolem Skolem terms terms for provable 17 for provable rrg~ sentences. sentences. This This in in turn turn yields yields aa characterization characterization of of C's C 's provably provably total total recursive recursive functions. functions. [1993]: These advantages advantages of of the the D-interpretation D-interpretation are are summed summed up up in in Feferman Feferman [1993]: These Applied of the the underlying Applied to to intuitionistic intuitionistic systems systems itit takes takes care care of underlying logic logic once once and verifies the and for for all, all, verifies the Axiom Axiom of of Choice Choice AC AC in in all all types, types, and and interprets interprets various various forms forms of of induction induction by by suitably suitably related related forms forms of of recursion. recursion. This This and S. S. Feferman Feferman J. AAvigad vigad and 356 356 then then leads leads for for such such systems systems to to aa perspicuous perspicuous mathematical mathematical charactericharacteri zation of of the the provably provably recursive recursive functions functions and and functionals. functionals. For For applicaapplica zation tion to to classical classical systems, systems, one one must must first first apply apply the the negative negative translation translation tion (again (again taken taken care care of of once once and and for for all). all). Since Since the the D-interpretation D-interpretation verifies verifies Markov ' s principle principle even even at at higher higher types types ... . . . at at least least the the provably recursive Markov's provably recursive functions and and functionals functionals are are preserved, preserved, as as well well as as QF-AC QF-AC in in all all types types functions and and induction induction schemata. schemata. The The main main disadvantage, disadvantage, though, though, comes comes with with the the analysis analysis of of other other statements statements whose whose negative negative translation translation may may lead lead to to aa complicated D-interpretation; special may have have to complicated D-interpretation; special tricks tricks may to be be employed employed to handle handle these. these. to A behavior with A further further distinguishing distinguishing feature feature of of the the D-interpretation D-interpretation is is its its nice nice behavior with respect to modus modus ponens. ponens. In In contrast contrast to to cut-elimination, cut-elimination, which global (and (and respect to which entails entails aa global computationally infeasible) infeasible) transformation transformation of of proofs, the D-interpretation extracts computationally proofs, the D-interpretation extracts and constructive information information through through aa purely purely local local procedure: procedure: when when proofs proofs of of ~cp and constructive --+ 'ljJ r are combined to to yield for the the antecedents antecedents of of cp -+ are combined yield aa proof proof of of r'ljJ, witnessing witnessing terms terms for this last last inference to yield term for the conclusion. conclusion. As As this inference are are combined combined to yield aa witnessing witnessing term for the of this this modularity, modularity, the the interpretation interpretation of theorem can can be be readily readily obtained aa result result of of aa theorem obtained from of the the lemmata in its proof. from the the interpretations interpretations of lemmata used used in its proof. The of applying to specific can The process process of applying the the D-interpretation D-interpretation to specific classical classical theorems theorems can sometimes be be used used to to obtain obtain appropriate appropriate constructivizations sometimes constructivizations thereof, thereof, or or to to uncover uncover additional implicit in in their proofs; cf., cf., for for additional numerical numerical information information that that is is implicit their classical classical proofs; or Kohlenbach Kohlenbach [1993]. [1993]. example, Bishop Bishop [1970] example, [1970] or 4. M Models T,, ttype 4. o d e l s oof f T y p e structures, s t r u c t u r e s , aand n d normalizability normalizability In In section section 2.2 2.2 we we presented presented the the set set of of terms terms of of the the theory theory T T without without aa discussion discussion of In this of what what these these terms terms denote. denote. In this section section we we exhibit exhibit several several kinds kinds of of functional functional and and term term models. models. 4.1. 4.1. Functional F u n c t i o n a l models models The model of The most most obvious obvious model of T T considered considered in in either either the the intensional intensional or or extensional extensional sense sense is is the the full full (set-theoretic) (set-theoretic) hierarchy hierarchy of of functionals functionals of of finite finite type, type, in in which which the the objects the natural natural numbers, objects of of type type 00 are are the numbers, and and each each type type (J a -+ --+ 7 T represents represents the the set set of of all type (Ja to those of 0, Sc, all functions functions from from the the objects objects of of type to those of type type 7 T.. The The denotations denotations of of 0, Sc, K, K, S, S, and and R R are are then then apparent, apparent, as as well well as as the the denotation denotation of of terms terms built built up up through through the the application application of of these these constants. constants. The The equality equality relation relation in in T T is is taken taken to to denote denote true true (extensional) (extensional) equality equality in in this this model. model. By By the the primitive primitive recursive recursive functionals functionals in in the the set-theoretic set-theoretic sense sense we we mean mean those those denoted denoted by by closed closed terms terms of of T. T. One One can can obtain obtain aa "smaller" "smaller" type type structure structure in in which which the the elements elements of of each each type type are are indices for indices for recursive recursive functions, functions, as as follows. follows. Let Let CPe ~ denote denote aa standard standard enumeration enumeration of of the the recursive recursive functions, functions, say, say, using using Kleene's Kleene's universal universal predicate. predicate. Define Define Mo M0 = - N, N, and and = yy)}. M~_~ = = {e Vx E e Mu M~ 3y 3y E e MT M~ ((CPe ~ ((x) x ) .}$= {e II \Ix n· MU-+T Gbdel 's Jilunctional Functional Interpretation Interpretation Godel's 357 357 One One can can associate associate recursive recursive indices indices to to constants constants of of T T in in a a natural natural way, way, and and interpret interpret equality indices. The model M equality as as equality equality of of indices. The resulting resulting model A/i of of II - TT (and (and of of I-HAW I-HA ~)) is is known as as the the hereditarily hereditarily recursive recursive operations operations (though (though "hereditarily "hereditarily total total recursive recursive known operations" might be be more more accurate) accurate) and and is is usually usually denoted by HRO HRO.. Equality Equality is is operations" might denoted by not not extensional extensional in in this this model, model, since since many many different different indices indices can can represent represent the the same same functional. functional. If If instead instead one one wants wants a a model model of of WEWE- T T (and (and of of WE-HAW WE-HA ~ or or even even E-HAW) E-HA~),, one one can can consider consider the the hereditarily hereditarily effective effective operations operations N Af,, usually usually denoted denoted by by HEO HEO.. For For this this model, model, sets sets Nu N~ and and the the equality equality relation relation =u =~ are are defined defined inductively inductively as as follows: relation for follows: set set No No = - N N and and = =00 the the usual usual equality equality relation for natural natural numbers, numbers, Nu......,. = {e I "Ix E Nu :Jy E N.,. ( ipe (x) .j..= y) 1\ w E e Nu, y y E e Nu (x =u Y y -+ ipe (x) =.,. ipe (y))}, "Ix and and =~-.r f = - Vz Vz E e Nu g~ ((~pe(z) = r,. ip ~l(z)). ee =u ..... .,. f J (z)). ipe (z) =. Both Both HEO HEO and and HRO HRO can can be be formalized formalized in in HA HA in in the the sense sense that that the the sets sets Mu M~ (resp. Nu) N~) and and the the equality equality relations relations =~ are defined defined by by formulas formulas in in the the language language of of (resp. =u are arithmetic, arithmetic, HA HA proves proves each each axiom axiom of of T T true true in in the the interpretation, interpretation, and and the the natural natural Notice, however, numbers numbers in in the the model model correspond correspond to to the the natural natural numbers numbers of of HA HA.. Notice, however, that the the complexity complexity of of the the formulas formulas defining defining Mu M~ and and Nu N~ grow grow with with the the level level of of (1 a,, so so that prove the the consistency that that HA HA (or (or PA) PA) cannot cannot prove consistency of of T T outright. outright. By By generalizing generalizing the the notion notion of of a a continuous continuous function function to to higher higher types, types, Kleene Kleene and and Troelstra [1973]); Kreisel independently models of Kreisel independently obtained obtained further further models of T T (cf. (cf. also also Troelstra [1973]); these these 6. will significance in in section will be be of of significance section 6. All of models described described in All of the the models in this this subsection subsection contain contain more more than than just just the the primitive primitive recursive the indices model extensions recursive functionals functionals (or (or the indices for for such) such),, and and hence hence model extensions of of T T as as well. well. 99 In In contrast, contrast, aa "minimal" "minimal" model model of of T, T, which which only only contains contains objects objects denoted denoted by by terms, terms, is is provided provided by by the the term term model, model, which which we we now now describe. describe. 4.2. 4.2. Normalization N o r m a l i z a t i o n and a n d the t h e term t e r m model model The S, and The defining defining equations equations for for the the typed typed combinators combinators K K,, S, and R R in in section section 2.2 2.2 describe describe aa symmetric symmetric equality equality relation. relation. From From aa computational computational point point of of view, view, it it is is often think of defining equations describing aa directed often more more useful useful to to think of these these defining equations as as describing directed relation, relation, in in which which terms terms on on the the left-hand left-hand sides sides of of the the equations equations are are "reduced" "reduced" to to more more basic basic ones example, the defining equations yield the ones on on the the right. right. For For example, the defining equations of of the the theory theory T T yield the following following reduction reduction rules: rules: 1. K(s, 1. K(s, t) t) I> t> Ss 2. S(r, s, t)t) I>~ r(t) (s(t)) 2. S(r, s, r(t)(s(t)) 3. 3. R(s, R(s,t,t, 0) 0) I> ~, Ss 9Some, 9Some, like like the the recursion recursion theoretic theoretic models, models, have have generalizations generalizations to to transfinite transfinite types; types; see, see, e.g. e.g. Beeson [1982]. [1982]. Category-theoretic Category-theoretic methods have also been been used used to construct models models of functional theories, though we will will not discuss discuss such such models models here. J. A vigad and S. S. Feterman Fe.ferman 358 358 4. R(s, R ( s , tt,, xx') ' ) ~t>t t(x ( x , ,RR(s, ( s , t ,t,x x)) )). . 4. If combinators, the If one one uses uses lambda lambda terms terms instead instead of of combinators, the first first two two clauses clauses can can be be replaced replaced (t) t> t[s/x] include product product types, types, reduction rule by the the reduction rule (>.x.s) ()~x.s)(t)~ t[s/x].. If If one one wants wants to to include by corresponding reduction rules pairing and projection operations operations can corresponding reduction rules for for the the pairing and projection can be be defined defined as well. as well. If that ss reduces If ss and and tt are are terms, terms, we we say say that reduces to to tt in in one one step, step, written written ss -t --+ tt,, if if tt can can be obtained by replacing some subterm u of s by a v such that u t> v . We say be obtained by replacing some subterm u of s by a v such that u ~ v. We say that that ss reduces obtained from nite sequence reduces to to tt if if tt can can be be obtained from ss by by a a fi finite sequence of of one-step one-step reductions. reductions. * is In In other other words, words, the the reducibility reducibility relation relation -t --+* is the the reflexive-transitive reflexive-transitive closure closure of of -t --+.. Such a a reducibility reducibility relation relation is is an an example example of of a a rewrite rewrite system system (cf. (cf. Dershowitz Dershowitz and and Such Jouannaud Jouannaud [1990]). [1990]). The The following following terminology terminology is is standard. standard. 4.2.1. 4.2.1. Definition. Definition. 1. If s, then 1. If ss is is a a term term and and u u is is a a subterm subterm of of s, then u u is is a a redex redex of of ss if if a a reduction reduction rule rule Uj i.e. can can be be applied applied to to u; i.e. there there is is some some vv such such that that U u t> ~, vv.. 2. If If ss has has a a redex, redex, then then ss iiss reducible. reducible. Otherwise, Otherwise, ss iiss irreducible, irreducible, or or iinn normal normal 2. form. form. 3. A term term is is normalizable normalizable if if it it can can be be reduced to one one in in normal normal form. form. A A system of 3. A reduced to system of reduction normalizible. reduction rules rules is is normalizing normalizing if if every every term term is is normalizible. 4. nite (one-step) 4. A A term term ss is is strongly strongly normalizable normalizable if if there there are are no no infi infinite (one-step) reduction reduction sequences sequences beginning beginning with with Ss;j that that is, is, every every such such sequence sequence eventually eventually leads leads to to aa term term in in normal normal form. form. A A system system of of reduction reduction rules rules is is strongly strongly normalizing normalizing if if every every term term is is strongly strongly normalizable. normalizable. 5. 5. A A system system of of reduction reduction rules rules is is confluent, confluent, or or has has the the Church-Rosser Church-Rosser property, property, if if * vv then * U * tt and whenever whenever ss -t --+* u and and ss -t --+* then there there is is a a term term tt such such that that u u -t --+* and * t. vv -t ---~* t. If If a a system system of of rules rules is is confluent confluent and and ss is is any any term, term, then then ss has has at at most most one one normal normal form. Furthermore, if form. Furthermore, if the the system system is is also also normalizing, normalizing, then then ss has has exactly exactly one one normal normal form. form. Identifying Identifying closed closed terms terms with with their their normal normal forms forms then then provides provides a a term term model model for for the the defining defining equations equations corresponding corresponding to to the the reduction reduction rules. rules. Under Under this this very very simple semantics, think of nothing more simple semantics, one one can can think of each each closed closed term term representing representing nothing more than than the the "program" "program" it it computes, computes, when when applied applied to to other other closed closed terms terms in in normal normal form. form. Strong Strong normalizability normalizability implies implies that that the the "programming "programming language" language" is is insensitive insensitive to to its its implementation, implementation, in in the the sense sense that that every every reduction reduction sequence sequence terminates terminates regardless regardless of of the the order order in in which which the the reductions reductions are are performed. performed. In all all the the functional functional theories theories we we consider consider in in this this chapter chapter (together (together with with the the In associated associated reduction reduction relations) relations),, the the only only closed closed irreducible irreducible terms terms of of type type 00 are are in in fact fact numerals. numerals. In In that that case, case, we we can can identify identify type type 00 objects objects of of the the corresponding corresponding term term model with model with natural natural numbers. numbers. Showing Showing that that the the reduction reduction relation relation associated associated with with a a functional uent and normalizing then functional theory theory is is confl confluent and normalizing then has has two two benefits: benefits: •9 it it implies implies that that the the functional functional theory theory is is consistent, consistent, that that is, is, it it cannot cannot prove prove 00 = - 11;j and and G6del 's Functional Interpretation Code/'s 359 359 • 9 it it justifies justifies the the intuition intuition that that the the functional functional theory theory describes describes "computable" "computable" entities. entities. 4.3. 4.3. Strong S t r o n g normalization n o r m a l i z a t i o n for for T T Given reduction rules Given the the reduction rules corresponding corresponding to to the the theory theory T T described described above, above, one one can combinatorial force can apply apply brute brute but but judicious judicious combinatorial force to to verify verify the the following following The The reduction reduction relation relation -t* --+* associated associated with with T T is is confluent. confluent. Fur Furthermore, term of thermore, any any closed closed irreducible irreducible term of type type 00 is is aa numeral. numeral. 4.3.1. 4.3.1. Lemma. Lemma. The this, due The "shortest "shortest known known proof" proof" of of this, due to to Tait Tait and and Martin-Lof, Martin-LSf, can can be be found found in in Hindley Hindley and and Seldin Seldin [1986,appendix [1986,appendix 1]. 1]. Assuming we we can can prove prove that that the relation -t* --+* is is also also normalizing, normalizing, the the resulting resulting Assuming the relation term term model model will will satisfy satisfy the the axioms axioms of of T: the uniqueness uniqueness of of normal normal forms forms implies implies T: the that that terms terms on on either either side side of of an an equality equality axiom axiom have have the the same same interpretation interpretation under under any any instantiation instantiation of of the the variables; variables; and and the the fact fact that that the the objects objects of of type type 00 in in the the term term model model are are numerals numerals reduces reduces induction induction in in T T to to induction induction in in the the metatheory. metatheory. Proving bound to Proving that that -t* --+* is is normalizing, normalizing, however, however, is is bound to be be tricky. tricky. Because Because it it implies Peano arithmetic, proof must somehow go beyond the implies the the consistency consistency of of Peano arithmetic, the the proof must somehow go beyond the capabilities theory. W. capabilities of of that that theory. W. Tait Tait developed developed an an elegant elegant and and flexible flexible technique technique for for proving proving normalization, normalization, using using appropriate appropriate "convertibility" "convertibility" predicates predicates (cf. (cf. Tait Tait [1967] [1967] and and Tait Tait [1971]). [1971]). 4.3.2. a, we 4.3.2. Definition. Definition. For For each each type type a, we define define the the set set of of reducible reducible terms terms of of type type a, a, denoted denoted by by RedO" Red~:: 11.. If 0, then If tt is is aa term term of of type type 0, then tt is is in in Redo Redo if if and and only only if if tt is is normalizing. normalizing. 2. If 2. If tt is is aa term term of of type type a a -t --+ T, then then tt is is in in Redo"-+ Red~_.rT if if and and only only if if whenever whenever ss in in RedO" t(s) is . Red~,, t(s) is in in Red Red~. T Writing Writing the the second second clause clause symbolically, symbolically, we we have have that that tt is is in in Redo" Red~_,~ if and and only only if if -+T if Vs (s EE RedO" Ys (s Red~ -t --+ t(s) t(s) E e Red Redr). T ). Notice Notice that that the the quantifier quantifier complexity complexity of of the the first-order first-order formula formula expressing expressing "t "t E E Red Redp" p" grows grows with with the the complexity complexity of of p. p. The The normalization normalization proof proof proceeds proceeds in in two two steps: steps: 11.. One terms, that One shows, shows, by by induction induction on on terms, that every every tt of of type type a a is is in in RedO" Red~.. 2. One induction on 2. One shows, shows, by by induction on the the type type a, a, that that every every tt in in RedO" Red~ is is normalizing. normalizing. The The only only aspect aspect of of the the proof proof that that cannot cannot be be carried carried out out in in aa weak weak base base theory theory is is 2, when the the verification verification of of clause clause 2, when tt is is the the recursor recursor RO" R~:: at at this this point point the the argument argument requires predicate RedO" requires induction induction on on aa formula formula involving involving the the predicate Red~.. As As aa result result we we have have J. AAvigad and and S. S. Feferman Feferman J. 360 360 is normalizing. normalizing. Moreover, Moreover, for for each each term term tt of of T, T, PA PA proves proves TT is that tt isis normalizable. normalizable. that 4.3.3. TTheorem. 4.3.3. heorem. This does does not not mean mean that that Peano Peano arithmetic arithmetic proves proves that that "for "for every every term term t,t , tt isis This the corresponding corresponding PA-proof FA-proof can can grow grow increasingly increasingly normalizable," since since for for various various tt the normalizable," complex. The The latter latter result, result, however, however, follows follows from from the the soundness soundness of of PA. PA . complex. Another Another approach approach to to proving proving normalization normalization involves involves assigning assigning ordinals ordinals (or (or notanota to terms terms in in such such aa way way that that each each one-step one-step reduction reduction leads leads tions representing representing ordinals) ordinals) to tions T, this this task task isis carried carried out out by by to aa decrease decrease in in the the associated associated ordinal. ordinal. For For terms terms of of T, to Howard [1970] [1970] using using notations notations below below the the ordinal ordinal e0. co . Via Via aa formal formal treatment treatment of of the the Howard term model, model, this this yields, yields, as as aa by-product, by-product, an an ordinal ordinal analysis: weak base base theory theory term analysis: over over aa weak the assertion assertion that that "there "there are are no no infinite infinite descending descending sequences sequences of of ordinal ordinal notations notations the co " implies implies the the consistency consistency of of PA. PA. beneath ~0" beneath 4.4. 4.4. IInfinitely n f i n i t e l y long long tterms erms Another term term model model for which is of special special interest interest was provided by Tait [1965]. [1965]. Another for T T which is of was provided by Tait This uses uses infinitely infinitely long terms to replace the produces aa system system This long terms to replace the recursors, recursors, and and thus thus produces of terms which which is is closer closer in the ordinary ordinary typed >.-calculus. The of terms in character character to to the typed A-calculus. The closure closure conditions are as as follows: conditions on on terms terms are follows: There . . . of There are are infinitely infinitely many many variables variables x", x ~, y", y~, z", z~,.., of each each type type T T.. 00 is a constant of type o. is a constant of type 0. Be Sc is is a a constant constant of of type type (0 (0 -+ --+ 0) 0).. If t( s) is If ss is is a a term term of of type type a a and and tt is is a a term term of of type type (a (a -+ --+ T T)) then then t(s) is a a term term of of type type T T.. 5. ). 5. If If tt is is aa term term of of type type T T then then >'XCT.t Ax~.t is is aa term term of of type type (a (a -+ --+ T T). (n = 0 , 1 , 2 , . . . ) is a sequence of terms of type T then 6. 6. If If tt,~ (tn} is aa term term of of n ) is n (n = 0, 1, 2 , . . . ) is a sequence of terms of type then (t type ). type (0 (0 -+ --+ T T). n (O) Then Write Write n n for for Bc Sc"(O). we translate translate each each term term tt of of T T into into aa term term tt + + of of this this . Then we + system system of of infinite infinite terms terms by by taking taking tt + = = tt for for tt aa variable, variable, 00 or or Be, Sc, 11.. 2. 2. 3. 3. 4. 4. K K ++ = = >'x>'y.x, AxAy.x, SS++ = = >.x>.y>.z.x(z, AxAyAz.x(z, y(z)) y(z)) and and (n, ttn), n) , R n+ ! = R ++ = - >.j>.g>.x.(tn} AfAgAx.(tn} where where ttoo = - ff and and ttn+l - gg(n, + to and and by by requiring requiring (-) (.)+ to preserve preserve application. application. Each Each term term tt of of the the infinite infinite system system is is assigned assigned an an ordinal ordinal It[t[I as as length length in in aa natural natural way, way, with with It[tlI = = 11 for for tt aa variable variable or or constant, constant, I>.x.tl [Ax.t] = = It[t[I + + 11,, It(s) [t(s)[1 : ls Is[i + + ItItII and, (tn} 1 == ssupn<~([t~[ UPn<w (ltn l + and, finally, finally, l[(tn)[ + 1) 1).. Note Note that that for for each each of of the the constants constants G C of of T, T, + I s:;<_Ww,, so + 1 << Ww.· 22.. IG [C+[ so for for each each term term tt of of T, T, ItIt+[ We We have have three three immediate immediate reduction reduction rules rules for for this this system system of of infinite infinite terms: terms: = 11.. (>.x.t[x]) (s) t>~,t[s/x] (Ax.t[xl)(s) t[~/z] 2.2. (t(tn> } {m ) t> t (m)~> tm m n Godel's G6del 's Functional Functional Interpretation Interpretation 361 361 3. ({tn} ((tn)(r))(s)~ (tn(s))(r), when rr is is not not aa numeral numeral and and (tn}{r) (t~)(r) is is not not of of type type O. O. {r)) (s) t> (tn(s)) (r) , when 3. * The relation relation -t -~* is is then then the the least least reflexive reflexive and and transitive transitive relation relation which which extends extends the the The t> relation relation and and preserves preserves application. application. As As before, before, aa term term tt is is said said to to be be in in normal normal form form * uu then whenever tt -t --+* then tt is is identical identical with with uu.. ifif whenever For For each each term term tt of of T T we we can can find find aa term term to t ~ in in normal normal form form such such that tt++ -t -+** to t ~ and and ItO It~I < co ~o.. that 4.4.1. 4.4.1. Theorem. Theorem. The idea idea of of Tait's Tait's proof proof of of Theorem Theorem 4.4. 4.4.11 is is very very much much the the same same as as that that for for the the The ' s classical cut-elimination theorem theorem for for the the extension extension of of Gentzen Gentzen's classical propositional propositional sequent sequent cut-elimination calculus calculus to to that that for for logic logic with with countably countably long long conjunctions conjunctions II II and and disjunctions disjunctions L E. Derivations Derivations in in PA PA are are translated translated into into derivations derivations in in this this calculus, calculus, by by first first translating translating formulas cp ~ into into propositional propositional formulas formulas cp+ ~+,, using using (Yx (Vx cp[x] ~[x])) ++ = = II I Inn<<w~CP++[[n/ n / xx] ] . . Then Then formulas each each derivation derivation dd from from PA P A is is translated translated into into an an infinite infinite propositional propositional derivation derivation dd++ 2. Each with finite finite cut-rank cut-rank and and IId+l w.. 2. Each derivation derivation whose whose cut-rank cut-rank is is :::; < m+ + 11 and and with d+ I < w ordinal length length is is :::; _< Q a is is effectively effectively reduced reduced to to aa derivation derivation of of the the same same end-formula end-formula ordinal whose whose cut-rank cut-rank is is :::; < m and and ordinal ordinal length length is is :::; _ wO<~.. So So for for each each derivation derivation dd of of T we we eventually eventually reduce to aa derivation derivation dO d ~ of of length length < co ~0.. Similarly, Similarly, we we can can T reduce dd++ to assign aa "cut-rank" "cut-rank" to to reducible infinite terms, terms, and and lower lower cut-complexity cut-complexity at at the the same same assign reducible infinite exponential bounds. See exponential cost cost of of increasing increasing ordinal ordinal bounds. See Tait Tait [1968], [1968], Schwichtenberg Schwichtenberg [1977], [1977], or in this this volume or Chapters Chapters III III IV IV in volume for for more more details details concerning concerning cut-elimination cut-elimination for for sequent for infinitary or Feferman sequent calculi calculi for infinitary languages, languages, and and Tait Tait [1965] [1965] or Feferman [1977] [1977] for for details details concerning for infinitary infinitary term concerning normalization normalization for term calculi. calculi. More More information information can can be be extracted extracted from from these these procedures procedures as as follows. follows. Schwicht Schwichtenberg detail how derivations generated generated enberg [1977,section [1977,section 4.2.2] 4.2.2] shows shows in in detail how the the infinitary infinitary derivations those in in PA, PA, as as described described above, above, may may be be coded coded by by indices indices for from for primitive primitive recursive recursive from those functions. For each derivation derivation d' d' in in the reductions from to ddO~, the the functions. For each the sequence sequence of of reductions from d d+ + to of d' d' both both determines determines the the structure tree and and contains bound on code of structure of of d' d' as as a a tree contains aa bound on code its cut-rank and on its length length (< « e0, co , in notation system its cut-rank «(< w w)) and on its in aa primitive primitive recursive recursive notation system for ~0). co ) . Exactly Exactly the the same same kind kind of of thing thing can can be be done done for for each each infinite infinite term term t't' in the for in the + to reduction sequence sequence from to tto~. This This allows allows one one to to define define an an effective effective valuation valuation reduction from tt+ function when the the initial initial tt isis aa closed closed term term of of type type 11 (i.e. (i.e. 00 ~-t 0), 0), and and that that function on on tto~ when leads leads one one to to 4.4.2. TTheorem. The functions functions of of type generated by by the the primitive primitive recursive recursive 4.4.2. heorem. The type 11 generated functionals may may be be defined defined by by schemata schemata of of effective effective transfinite transfinite recursion recursion on on ordinals ordinals functionals < co < ~o.· The schemata schemata referred referred to to define, define, for for any any given given ordinal ordinal aQ << ~0, co , aa function function F(x, The F(x, y) y) of numbers numbers x, x, yy by by F(O, F(O, y) y) == G(y) G(y) and and for for x 0, F(x, F(x, y) y) in in terms terms of of F(Hi(x, F(Hj(x, y), y), y) y) of x ~=1= O, where the the Hi Hi are are one one or or more more functions functions with with Hi(x, Hj(x, y) y) <~ <0< x. x. The The collection collection of of FF where definable using using such such schemata schemata coupled coupled with with the the usual usual schemata schemata for for primitive primitive recursive recursive definable definition isis denoted denoted REC( J. AAvigad S. Feferman Feferman J. vigad and S. 362 362 The provably recursive recursive functions functions of of PA PA (and (and hence hence also also of HA) The provably of HA) REC( 4.4.3. TTheorem. 4.4.3. heorem. A A similar similar characterization characterization can can be be obtained obtained of of the the functionals functionals which which are are asserted asserted to exist exist in in Kriesel's Kriesel's no-counterexample no-counterexample interpretation interpretation of of PA, PA, here here via via the the NDND to interpretation of PA in T (Corollary 3.2.6). interpretation of PA in T (Corollary 3.2.6). Returning to to Theorem Theorem 4.4.2, 4.4.2, for for its proof one makes use use of of the the following following simple simple Returning its proof one makes description of of the the normal normal terms terms t[xl,..., t[xt , . . . , xn] xnl of of type type 00 in in which which all all the the free variables description free variables Xi are are of of type type 0: 0: either either xi 1. tt-=0 o0r or 1. 2. tt == Sc(s) Sc(s) where where ss is is normal normal or or 2. 3. tt isis aa variable of type type 00 or or 3. variable of 4. tt == (tn)(s) (tn ) ( s) where where each each t~ tn and and ss is is normal normal of of type type 0, 0, but but ss is is not numeral. 4. not aa numeral. closed normal It follows the only terms of of type It follows that that the only closed closed terms type 00 are are numerals. numerals. The The closed normal terms terms of type type O. just those the form form )~x~ Axo .t[xl where of type of type 11 are are just those of of the where tt is is normal normal of 0. 5 . The of 5. T h e interpretation interpretation o f fragments f r a g m e n t s of o f arithmetic arithmetic 5.1. I571 lEJ and recursive functions functions 5.1. a n d the t h e primitive p r i m i t i v e recursive In the of PA PA and HA the the recursors Ru were were used interpret the the In the interpretation interpretation of and HA recursors R~ used to to interpret induction axioms, it should should not not be be surprising surprising that that weaker forms of induction axioms, and and it weaker forms of recursion recursion lEJ be be the the fragment fragment of can be be used used to to interpret forms of of induction. induction. Let can interpret weaker weaker forms Let IL'I of PA which induction is restricted to EI:r~ formulas. formulas. A A nice nice application application of of the PA in in which induction is restricted to the D-interpretation D-interpretation due due to to Parsons Parsons (cf. (cf. [1970,1972]) [1970,1972]) shows shows that that any any provably provably total total recursive recursive function function of of lEJ IEI is is in in fact fact primitive primitive recursive. recursive. The definition of "impredicative," in The definition of the the recursors recursors Ru R~ in in section section 2.2 2.2 for for (7 a =f:. r 00 is is "impredicative," in that g, n') n') at g, n) that the the evaluation evaluation of of Ru(f, R~(f, g, at aa given given argument argument x x presumes presumes that that Ru(f, a~(f, g, n) has z. A has already already been been defined defined for for arbitrary arbitrary arguments arguments z. A "predicative" "predicative" restriction restriction of of this scheme scheme is recursors It which have this is given given by by the the recursors R~ due due to to Kleene, Kleene, which have the the defining defining schemata schemata R u (f, g, g, 0, 0, b) l~a(f, b) = = f(b) f(b) R g, n', n', b) u(f, g, g, n, n, b), l~au(f, (f, g, b) = = g(n, g(n, R l~a(f, b), b). b). Note Note that that each each type type (7 a is is uniquely uniquely of of the the form form ((71 ( a l ,, .. ...., , (7k) ak) -+ 00 for for some some sequence sequence ((7 ( a 1l ,, .. ... ., , (7k) ak).. In In the the equations equations above, above, then, then, ff is is of of type type (7, a, 9g is is of of type type 00 -+ 00 -+ (7, a, (f, g, and and bb = = (br ( b ~1 ,, . . . , , b� b~k~)) is is aa sequence sequence of of variables variables chosen chosen so so that that Ru R~(f, g, n, n, b) b) is is of of type type 10 We 0.1~ We let let T T denote denote the the restriction restriction of of T T which which only only allows allows this this type type of of recursion. recursion. 11 11 0. • • • 1lO0 In In contrast contrast to to Godel's Ghdel's recursors, recursors, each each R Ra can in fact fact be be defined defined from from Ro ~ by by the the equation equation q can in R n, b Ro( R~q = V, Af, g, g,n,b I:to(f(b),Ak, g(k,l,b),n). I, b), n) . f (b), >'k, Il g(k, 1 In the 1llIn the versions versions To To and and 1I-T1' we we need need to to add, add, as as in in Parsons Parsons [1972], [1972], conditional conditional functionals functionals = Condp Condp : 9(O, (0, p, p, p) p) -+ --~ o. 0. These These functionals functionals have have defining defining equations equations Condp(w, Condp(w, u,v) u, v) = u u if if W w= = 00 and and Condp(w, Condp(w, u, u, v) v) = vv otherwise; otherwise; in in these these theories theories they they are are no no longer longer definable definable using using the the R 1~ recursors. recursors. = = 363 363 GSdel 's Functional Gadel's Functional Interpretation Interpretation 5.1.1. TTheorem. The closed closed level level i1 terms terms of denote primitive primitive recursive recursive funcfunc 5.1.1. heorem. The of TT denote tions. Moreover, Moreover, there there isis aa natural natural translation translation of of type type 00 terms terms tt of whose only only free free tions. of TT whose PRA of of PPRRA such that that if proves tt == s,s , then then PPRRA variables A , , such if TT proves A variables are are of of type type 00 toto terms terms ttPRA proves proves ttPPRRA A "=- - 8SPPRRA A .9 A 5.1.2. PProof. The proof proof of of the the first first assertion assertion is is by by adaptation adaptation of of Kleene Kleene [1959b, [1959b, 5.1.2. roof. The 12 The sections The idea class K sections 1.5-1.7] 1.5-1.7] to to the the present present framework. framework. 12 idea is is to to define define aa class KLL of of functionals F(b) F(b) of of lists lists of of arguments arguments bb of of arbitrary arbitrary finite values of functionals finite type type but but with with values of t~pe that each by aa term term in term tt in only, such such that each FF in in KKLL isis defined defined by in T, T , while while each each term in t;ype 00 only, represented by an FF in in KKLL either either directly directly (if (if itit is is of of type 0) or or by abstraction T by an type 0) by abstraction T isis represented on some some of of the the variables variables of of FF (otherwise). (otherwise). on In defining defining KKLL and and in in the the proof proof of of these these facts we modify modify the the conventions conventions of of In facts we section 2.2 2.2 as as follows. follows. We will use use aQ. to to denote denote (possibly (possibly empty) empty) sequences of types types section We will sequences of and by by (a (Q. --+ -+ 0) 0) we we mean mean 00 ifif kk == 00 and and (al -+ ak -+ 0) 0) otherwise. otherwise. ((0" h 1i ,, .. ... ., , ak), + . . .. . --+ O"k) , and O"k -+ (0"1 --+ to denote denote aa sequence of terms terms (ur and Au!!.. . F(u, . . . ) means We will use use uu!!..~ to We will sequence of ( u ~'' ,, .. ... ., , U�k), u~ k), and AuZ.F(u,...) means AUr' .F(uJ , . . . , Uk,...) Uk, . . . ) otherwise. So each T is F(( ... .. ). ) when F when nn- = 00 and and A u ~ . .. .. . AU�k AU~kk.F(Ul,..., otherwise. So each type type T is (Q. -+ Q., and and lev(T) = maxl:5:i:5:k (lev(O"i ) + 1). uniquely of the the form form (a uniquely of --+ 0) 0) for for some some a, lev(T) -maxl 2. F(n, 2. F(n, b) b) = = nn F(n, b) = n' nl 3. 3. F(n, b) = -+ 0) (possibly 0) (Q. --+ -+ 0), 0) , TT ~=1= O, 0, O"i 0) (aT~,, b) b) == aa(tl tk) where where 7T -= (a 4. F 4. F(a ( t l ,, ... .. ,. , tk) ai = = (P-i --+ 0) (possibly (!!.i and ttii -= Awf· Gi (Wi , a, a, b) b) for for ii -= 11,, .. ... ., , kk and Aw~ ~Gi(wi, 5. F(b) 5. F(b) = - G(H(b), G(H(b), b) b) where where the the first first argument argument of of G G is is of of type type 00 6. F(b) = G(b,,) where b" is a permutation of b by 1f 6. F(b) - G(b~) where b~ is a permutation of b by r 7. F(O, F(O, b) b) = = G(b), G(b), F(k F(k',1 , b) b) = = H(k, H(k, b, b, F(k, F(k, b)) b)) 7. Then Then the the following following facts facts are are established: established: 11.. The The functionals functionals F F of of KL K L are are closed closed under under substitution substitution of of one one or or more more terms terms tt of of type T T for for variables variables aa Tr of of F F where where T T= -- ((a_ 0), T T =1= -~ 00 and and tt = -- Au!!.. )~uZ.H(u,...). . H(u, . . . ) . Q. -+ 0), type A 2. For each each F(b) F(b) in in KL K L we we can can find find a a term term tt of of T T with with free free variables variables b, which 2. For b, which defines defines F. F. Q. -+ 3. If If tt is is a a term term of of T T with with free free variables variables b, b, and and tt is is of of type type ((a_ --+ 0) 0) (possibly (possibly 0) 0),, 3. then we can find a functional F( u, b) of K L such that u) = u, b) then we can find a functional F(u, b) of K L such that t( t(u) - F( F(u, b) for for all all u, u ~ bb. . 4. 4. If If F(b) F(b) has has all all its its variables variables bJ b l,, .. .. .., , bbtl of of type type 0, 0, then then F F is is primitive primitive recursive. recursive. Fact Fact 1I corresponds corresponds to to the the Full Full Substitution Substitution Theorem Theorem of of Kleene Kleene [1959b,section [1959b,section 1.6] 1.6].. It It is is proved proved by by induction induction on on the the maximum maximum m m of of the the levels levels T T of of the the variables variables aT a r being being substituted substituted for for and, and, for for given given m, my by by induction induction on on the the generation generation of of F F in in KL. K L . Facts Facts 22 and using 11 for and 33 are are straightforward, straightforward, using for the the application application step step in in 3. 3. Finally, Finally, fact fact 44 1' , with its additional functionals Ep. 12Here t2Here we do not consider the version version 1I-T, Ep. J. A vigad and S. Avigad S. Feferman 364 364 follows by by observation observation that that scheme scheme 44 of of the the definition definition of of K K LL can can never never be be applied applied in in follows the the generation generation of of functionals functionals all all of of whose whose arguments arguments are are of of type type O0.. This This shows shows that that the the type type 11 terms terms of of T T denote denote primitive primitive recursive recursive functions. functions. The The conservation conservation result result of of Theorem Theorem 5.1.1 5.1.1 is is obtained obtained by by using using the the argument argument above above schemata 1-7, then using using fact to to transform transform proofs proofs in in T T to to proofs proofs from from schemata 1-7, and and then fact 44 to to transform this this to proof in RA . transform to a a proof in P PRA. ((Remark. Remark. One think of normalization argument One can can think of the the proof proof of of fact fact 11 as as aa normalization argument together with for for aa system system of of terms terms generated generated from from schemata schemata 11-7 with aa scheme scheme of of full full -7 together 4. Another Another route substitution substitution in in place place of of scheme scheme 4. route to to the the proof proof of of Theorem Theorem 5.1.1 5.1.1 should should be possible possible by by normalization normalization of of the the terms terms of of T T,, but but that that would would apparently apparently require require be more more work. work. For For a a proof proof of of an an analogous analogous result result for for aa system system of of feasible feasible functionals functionals of of 0 finite Theorem 5.2.1 finite type, type, see see the the reference reference after after Theorem 5.2.1 below. below.)) [] A - - Let HA# and denote variants Let H'A# and PA ~ # # denote variants of of HA# HA # and and PA# PA # respectively, respectively, in in which which induction induction is is restricted restricted to to existential existential formulas. formulas. The The reader reader can can verify verify that that the the recursors recursors R are are sufficient sufficient to to interpret interpret induction induction for for these these formulas, formulas, yielding yielding 5.1.3. 5.1.3. Theorem. Theorem. JiA ~ # # is is D-interpreted D-interpreted in in T T,, and and FA ~A # # is is ND-interpreted ND-interpreted in in T T.. # proves Since Since FA P'A# proves that that any any �� E ~ formula formula is is equivalent equivalent to to an an existential existential one, one, IEJ IZI can in this can be be embedded embedded in this theory, theory, and and Theorems Theorems 5.1.1 5.1.1 and and 5.1.3 5.1.3 yield yield 5.1.4. IEJ 5.1.4. Theorem. Theorem. IY,1 is is conservative conservative over over PRA PRA for for rrg II ~ formulas, .formulas, in in the the sense sense that that if if IEJ IE~ proves proves "Ix Vx 3y 3y recurswe. recursive. Since Since IEJ IEI can can prove prove each each primitive primitive recursive recursive function function to to be be total, total, this this last last character characterization is ization is exact. exact. 5.2. 5.2. 81 S~ and the the polynomial-time polynomial-time computable computable functions functions When bounded arithmetic cf. Chapter ) , IEJ Buss' When it it comes comes to to bounded arithmetic ((cf. Chapter II II), I•1 is is analogous analogous to to Buss' 's theory theory theory 81 S~,, and and PRA PRA is is analogous analogous to to Cook Cook's theory PV, P V, which which axiomatizes axiomatizes the the polynomial-time computable functions polynomial-time computable functions as as characterized characterized in in section section 1.3.7 1.3.7 of of Chap Chapter II. In ter II. In Buss Buss [1986] [1986] it it is is shown shown that that 81 5'~ is is conservative conservative over over P P VV in in the the sense sense of of Theorem Theorem 5.1.4. 5.1.4. This This result result has has been been reobtained reobtained using using aa D-interpretation D-interpretation in in Cook and cf. also Cook and Urquhart Urquhart [1993], [1993], and and it it is is this this presentation presentation that that we we now now sketch sketch ((cf. also Feferman Feferman [1990]). [1990]). Cook Cook and and Urquhart Urquhart start start by by defining defining a a higher-type higher-type version version P PV Vw~ ofthe of the theory theory P PV V.. Aside Aside from from a a careful careful choice choice of of initial initial functions, functions, which which hinge hinge oonn the the fact fact that that one one is is supposed to think of representations, the supposed to think of the the natural natural numbers numbers in in terms terms of of their their binary binary representations, the computational computational strength strength of of P PV Vw~ comes comes from from recursors recursors Il, 1~, which which allow allow for for "higher-type "higher-type Codel's G6del 's Functional Interpretation 365 365 limited notation." These limited recursion recursion on on notation." These are are described described by by the the equations equations � f( 0 (f, g, R(f, g, h, h, 0, 0, 0 b) = = f(b) _ g, h, h, n', n', b) R(j, R(f, g, b) = = { g(n, (j, g, g(n, � fit(f, g, h, h, In/2J [n/2J,, b) b),, b) b) if if this this has has length length less less than than that that of of h(n, h(n, b) b) otherwise. h(n, b) b) otherwise. h(n, h, 0, 0, b) Here Here bb once once again again denotes denotes aa sequence sequence of of variables variables chosen chosen so so that t h a t / ~R((j, f , g, g, h, b) is is of of type 0, is a a type type 11 function function which provides aa bound bound on on the the growth growth of of the the function function type 0, hh is which provides being defined, defined, U2 [./2JJ is is among among the the initial initial functions functions of of PV P V wW,, and and additional additional initial initial being functions representing "length" subtraction and a conditional functions representing "length" subtraction and a conditional are are used used to to express express the the 3 The second equation. equation. 113 The following following theorem theorem is is analogous analogous to to Theorem Theorem 5.1.1. 5.1.1. second 5.2.1. The 5.2.1. Theorem. Theorem. The level level 11 terms terms of of PVW P V ~ denote denote polynomial-time polynomial-time computable computable functions. functions. Moreover, Moreover, there there is is aa natural natural translation translation of of type type 00 terms terms tt of of PVW P V ~ whose whose V of only type 00 to only free free variables variables are are of of type to terms terms tt P PV of P PV V,, so so that that if if PVW P V ~ proves proves tt = = s, s, V= then P PV V proves proves tt P PV = s S pv PV.. then Full Cook and 1993,pp. 140-146 Full details details are are provided provided in in Cook and Urquhart Urquhart [[1993,pp. 140-146].] . Finally, IPVw I P V W and and CPVw CPV ~ are are defined defined to to be be quantifier quantifier versions versions of of PVw P V ~ based based on on Finally, intuitionistic and and classical classical logic logic respectively, respectively, where where only only type type 00 equality equality is is allowed. allowed. In In intuitionistic these theories, induction induction is these theories, is allowed allowed for for "NP-predicates," "NP-predicates," that that is, is, formulas formulas of of the the form form the free < tt (r (r = = s) s) where where all all the free variables variables of of tt have have type type O0.. Since Since the the recursors recursors R R are are 33yy :::; sufficient Theorem 5. 1.3, sufficient to to interpret interpret this this form form of of induction, induction, we we have, have, in in analogy analogy to to Theorem 5.1.3, 5.2.2. IPVW 5.2.2. Theorem. Theorem. I P V ~ ++ (MP) (MP) is is D-interpreted D-interpreted in in PVw, P V W, and and CPVw CPV ~ is is NDNDinterpreted in in PVW P V W.. interpreted 81 can be be embedded in CPVw the latter can prove 5'~ can embedded in CPV W,, since since the latter theory theory can prove any any ~E� formula formula equivalent equivalent to to an an NP-predicate NP-predicate as as described described above. above. Hence Hence we we have have the the following following 5.2.3. PV 5.2.3. Theorem. T h e o r e m . 81 S~ is is conservative conservative over over P V for .for 'v'E� V ~ sentences, sentences, in in the the sense sense that that such that if if 81 S~ proves proves 'v'x Vx 3y 3y cp(x, ~(x, y) y) for for cp ~o aa E� ~ formula, formula, then then there there is is aa term term f f such that PV PV proves -definable function proves cp(x, ~(x, f(x) f (x)).) . Every Every E� ~-definable function of of 81 S~ is is polynomial-time polynomial-time computable. computable. By By Theorem Theorem 1.3.4.1 1.3.4.1 of of Chapter Chapter II II this this last last characterization characterization is is exact. exact. 6. T The of 6. h e interpretation interpretation o f analysis analysis 6.1. 6.1. Towards Towards the the interpretation interpretation of of stronger stronger theories theories At paper, G6del made the At the the end end of of his his 1958 1958 paper, GSdel made the following following suggestions suggestions regarding regarding the the functional functional interpretation interpretation of of stronger stronger theories. theories. l3In 1993,section 6] only the type 0 recursors are taken to be basic, 13In Cook Cook and Urquhart [[1993,section basic, with the others defined defined as in footnote 10. 366 366 J. A vigad and S. S. Feferman Fe]erman It It is is clear clear that, that, starting starting from from the the same same basic basic idea, idea, one one can can also also construct construct systems T, for systems that that are are much much stronger stronger than than T, for example example by by admitting admitting trans transfinite finite types types or or the the sort sort of of inference inference that that Brouwer Brouwer used used in in proving proving the the "fan "fan theorem." theorem." 10 we In In section section 10 we will will address address the the first first proposal, proposal, expanding expanding the the notion notion of of "type" "type" to to the the transfinite. transfinite. In In this this section section we we explore explore the the second second proposal. proposal. The The fan fan theorem theorem that that Godel refers is, from classical point Ghdel refers to to is, from aa classical point of of view, view, equivalent equivalent to to weak weak Konig's Khnig's lemma, lemma, to to be be dealt dealt with with in in section section 7. 7. Brouwer Brouwer was was able able to to prove prove the the fan fan theorem theorem using using a a principle of induction" which interpretation principle of "bar "bar induction" which he he felt felt was was justified justified by by aa constructive constructive interpretation of the the terms terms involved involved (cf. (cf. Beeson Beeson [1985], [1985], Troelstra and van van Dalen Dalen [1988]). [1988]). In In a a Troelstra and of posthumous paper, C. Spector [1962] generalization of posthumous paper, C. Spector [1962] used used aa generalization of this this principle principle to to justify justify aa computational he dubbed computational scheme scheme which which he dubbed "bar "bar recursion." recursion." With With this this scheme scheme Spector Spector was was able able to to provide provide a a functional functional interpretation interpretation of of full full second-order second-order arithmetic, arithmetic, that that 4 is, the the theory theory PAll PA 2 + + ((CA) defined in in section section 6.2 below. 114 is, CA) defined 6.2 below. While Spector Spector used induction to recursion, he While used bar bar induction to justify justify bar bar recursion, he apparently apparently intended intended to show show that, conversely, bar bar recursion recursion could could be be used used to to obtain obtain a a functional functional interpre interpreto that, conversely, tation induction. This tation of of bar bar induction. This task task was was in in fact fact carried carried out out by by Howard Howard [1968], [1968], and and is is the approach approach we we outline outline here. here. the 6.2. 6.2. Analysis A n a l y s i s and a n d higher h i g h e r type t y p e extensions extensions The second-order arithmetic The language language of of second-order arithmetic extends extends the the language language of of Peano Peano arithmetic arithmetic with with variables variables that that range range over over sets sets of of numbers, numbers, and and a a binary binary membership membership relation relation Y.. In In this this language language we we only only allow allow first-order first-order equality, equality, defining defining the the assertion assertion xx E Y Y = = Z Z to to mean mean Y "Ix Vx (x (x E ~ Y Y ++ ~ x x E ~ Z) Z).. The second-order arithmetic, arithmetic, PAll CA) , extends arithmetic with The theory theory of of second-order PA 2 + ((CA), extends Peano Peano arithmetic with induction induction for for formulas formulas in in the the expanded expanded language, language, and and the the comprehension comprehension scheme scheme ((CA) CA) 3Y aY "Ix Vx (x (x E e Y Y ++ ~ cp(x)) r for for arbitrary arbitrary formulas formulas cp ~.o (As (As usual usual we we denote denote the the scheme scheme in in which which cp ~ is is restricted restricted to - CA) .) The CA) is to formulas formulas in in r F by by (r (F-CA).) The theory theory PAll PA 2 + + ((CA) is often often called called "analysis" "analysis" due that, via coding of due to to the the observation observation that, via the the coding of real real numbers numbers and and continuous continuous functions functions as as sets sets of of natural natural numbers, numbers, one one can can develop develop a a good good theory theory of of the the continuum continuum from from these of Weyl these axioms. axioms. (In (In fact, fact, by by the the work work of Weyl [1918,1994]' [1918,1994], first-order first-order arithmetic arithmetic comprehension comprehension with with parameters parameters suffices suffices for for most most practical practical purposes. purposes.)) In order to CA) in In order to embed embed PAll PA ~ + + ((CA) in an an extension extension of of PA# PA #,, we we identify identify sets sets Y Y with with (t) f:. o. This their their characteristic characteristic functions functions Xy XY,, and and read read tt E Y Y as as Xy xy(t) r O. This provides provides aa 14 14 We should mention in this connection connection Kreisel Kreisel [1959], [1959], in which which it is shown shown that if second-order arithmetic proves cp, the witnessing witnessing functions can be taken to be "recursively proves a formula formula ~, "recursivelycontinuous," continuous," in a sense defined, defined, independently, independently, by Kreisel greisel lop. lop. cit.] cit.] and Kleene Kleene [1959a] [1959a] (cf. (cf. also Feferman Feferman [1993]). [1993]). For an interesting extension extension of bar recursion recursion to transfinite types, types, see Friedrich Friedrich [1985]. [1985]. Godel's GSdel 's Functional Interpretation 367 367 natural way way of of interpreting interpreting the the set set variables variables of of PA PA*2 + + ((CA) with function function variables variables natural GA) with of of type type l. 1. Recall Recall the the axiom axiom of of choice choice (AG) (AC) Vx w :3y ~v We will will denote denote the the scheme scheme in in which which the and yy are are restricted restricted to to be be of of type type We the variables variables xx and (j a and and T respectively respectively by by (AGuT) (A C~).. We We have have seen seen that that adding adding the the axiom axiom of of choice choice (AG) (A C) to HA HA ww results results in in a a theory theory that that is is no no stronger stronger than than Heyting Heyting arithmetic. arithmetic. In In contrast, contrast, to includes ((CA), GA) , since the addition addition of of even even (A (ACoo) to PA PA ww results results in in aa theory theory that that includes since one one the Goo ) to can can apply apply this this choice choice principle principle to to the the classically classically valid valid formula formula w :3y 3y ((y ((y = = 00 /\ ^ -' 6.3. The T h e principle p r i n c i p l e of of bar b a r induction induction 6.3. In induction used In the the following following statement statement of of the the principle principle of of bar bar induction used in in the the Spector Spector(eo, CI Howard interpretation, the Howard interpretation, the variable variable cc represents represents aa finite finite sequence sequence (Co, c l ,, . .. .., , Ck-l Ck-1)) of of the variable variable ff ranges ranges over objects objects of of type type (j a (suitably (suitably coded) coded),, while while the over infinite infinite sequences sequences of objects of (j, which (j. Given of objects of type type a, which is is to to say say that that f f is is aa functional functional of of type type 0 � ~ a. Given such such an initial segment an f f,, let let f(k) f(k) to to denote denote the the initial segment of of f f of of length length k, k, i.e. i.e. the the finite finite sequence sequence m U(O), ( f ( O ) ,f(l f ( 1)), ,. ..... ,, ff(k ( k - 1 ) 1)). ). - Finally, Finally, if if u u is is an an element element of of type type (j a,, c'u c^u denotes denotes the the sequence sequence obtained obtained by by appending appending o c c. . uu tto 6.3.1. 6.3.1. Principle P r i n c i p l e of of bar bar induction i n d u c t i o n at at type t y p e (j a.. Suppose Suppose
To To make make sense sense of of this this principle, principle, imagine imagine the the tree tree of of all all finite finite sequences sequences of of objects objects of of type type (j a.. Clauses Clauses (1 (1-3) imply that that every every path path through through the the tree tree passes passes through through aa -3) imply node cc where well. The nodes cc where node where 't/J r holds, holds, and and hence hence
J. A vigad and S. S. Feferman 368 368 The principle principle of of bar bar induction induction may may be be read read as as aa statement statement of of induction induction over over aa The well-founded relation. relation. Classically, Classically, it it follows follows from from an an appropriate appropriate axiom axiom of of dependent dependent well-founded choices. choices. Assuming Assuming Clause Clause (4) (4),, the the failure failure of of cp ~o to to hold hold at at the the empty empty sequence sequence would would allow allow one one to to construct construct an an infinite infinite sequence sequence CO, CI, C2~... Co, . Ck-l ) . Defining with initial segment with the the property property that that cp ~o does does not not hold hold at at any any initial segment ((Co,... C~-l). Defining the the function function f f by by f(k) def Ck f (k) = -'def Ck would would then then provide provide a a counterexample counterexample to to (1) (1) and and (3). (3). In In the the special special case case in in which which a a is is the the type type of of natural natural numbers, numbers, Principle Principle 6.3.1 6.3.1 is is Kleene gleene's's exposition exposition of of Brouwer Srouwer's's "bar "bar theorem" theorem" (cf. (cf. Kleene gleene and and Vesley Vesley [1965]). [1965]). The The generalization due to generalization to to higher higher types types is is due to Spector. Spector. To To express express this this principle principle in in the the language language of of HA# HA # ,, note note that that we we may may identify identify each each sequence ( C( x) C(x) = { au if if x x < < kk 0~ otherwise. otherwise. Cx c~ I u .) (Inductively ne for (Inductively one one can can defi define for each each a a aa constant constant "zero" "zero" functional functional denoted denoted by by a0~.) References References to to such such Cc can can then then be be taken taken as as shorthand shorthand for for references references to to such such pairs pairs C, C, k. k. For For this this representation representation we we will will use use cfi and and length(c) length(c) to to denote denote C C and and kk respectively. respectively. principle of bar induction induction for objects of Let (E1u) (BI~) denote denote the the principle of bar for sequences sequences of of objects of type type Let a, and (EI) denote principle for will see a, and let let (BI) denote the the same same principle for arbitrary arbitrary a. We We will see in in section section 6.5 6.5 that that the the theory theory HA# HA # + + (EI) (BI) is is in in fact fact strong strong enough enough to to interpret interpret PAW PA" + + (AGoo) (ACoo),, and and hence hence analysis. analysis. First, First, however, however, we we show show that that bar bar induction induction has has aa computational computational analogue analogue in in a a form form of of recursion, recursion, much much in in the the way way that that arithmetic arithmetic induction induction has has its its computational computational analogue analogue in in the the recursors recursors R R of of section section 2.2. 2.2. 6.4. 6.4. The T h e interpretation i n t e r p r e t a t i o n of of bar b a r induction i n d u c t i o n using using bar b a r recursion recursion For For each each a a and and T, the the principle principle of of bar bar recursion recursion uniformly uniformly associates associates with with given given functionals G, H functionals G, H,, and and Y Ya a new new functional functional F F which which maps maps finite finite sequences sequences Cc of of objects objects of T, according of type type a a to to objects objects of of type type T, according to to the the defining defining equation equation _ F (c) = F(c) - G(c) { H(AU.F(cAu), H(Au.F(du), c) c) G(c) if if Y(c) Y(a) < < length(c) length(c) if Y(C) Y(a) ;::: _> length(c) length(c).. if To To make make the the uniformity uniformity clear, clear, and and in in analogy analogy to to the the recursions recursions in in section section 2.2, 2.2, one one introduces introduces aa single single functional functional Bu B~rr for for each each a a and and T T and and replaces replaces F F above above by by Bur (G, H, Y) . B~.(G,H,Y). F(c) is From definition of From the the definition of F F we we see see that that if if Y(C) Y(~) < < length(c) length(c) then then F(c) is defined defined outright, and F(c) depends u. One outright, and otherwise otherwise F(c) depends on on the the values values F(cAu) F(du),, for for arbitrary arbitrary u. One should think think of the values being computed should of the values of of F F as as being computed by by recursion recursion along along a a well-founded well-founded Code/'s GSdel's Functional Interpretation 369 369 tree of of sequences sequences determined determined by by Y, in aa way way that that will will be be made made more more explicit explicit below. below. tree Y, in Spector Spector showed showed that that the the principle principle of of bar bar induction induction can can be be used used to to justify justify bar bar recursion, recursion, when the the argument argument Y Y is is taken taken to to be be aa continuous continuous functional functional (in (in the the Kleene-Kreisel Kleene-Kreisel when sense sense referred referred to to in in section section 4.1 4.1 and and footnote footnote 14) 14).. The induction comes The following following informal informal argument argument gives gives aa hint hint as as to to how how bar bar induction comes into play. play. Given Given the the functional functional F F described described above, above, we we would would like like to to see see that that the the into value denote property Y(c) value F(()) F(()) is is "defined." "defined." Let Let 'l/;(c) r denote the the property Y(~) < < length(c) length(c),, and and let let ~(c) denote denote the the property property that that F(c) F(c) is is defined. Clauses (3-4) (3-4) of of Principle Principle 6.3.1 6.3.1 are are cp(c) defined. Clauses clearly appropriate formalization clearly seen seen to to hold, hold, and and in in an an appropriate formalization clause clause (2) (2) can can be be justified justified intuitionistically intuitionistically as as well. well. Seeing Seeing that that the the well-foundedness well-foundedness condition condition (1) (1) holds holds requires requires verifying verifying that that for for k. Suppose every every function function f f,, there there is is aa finite finite initial initial segment segment cc = = f(k) f(k) so so that that Y(C) Y(~) < < k. Suppose n. The Y(J) Y ( f ) = n. The computation computation of of Y Y on on f f can can only only depend depend on on finitely finitely many many values values of among the list f(O), some m. In of f f,, contained contained among the list f(0), f(1), f ( 1 ) , .. ...., , f(m) f(m) for for some In particular, particular, if if kk = ( m , nn + 1) and then - max max(m, + 1) and cc = = f(k) f(k),, then Y(c) Y(~) = Y(J) Y(f) = = n n < < kk = = length(c) length(c),, as as desired. desired. The The following following lemma lemma shows shows that that conversely, conversely, bar bar recursion recursion can can be be used used to to justify justify (BR) to principle of bar bar induction. induction. We We use use HA# HA # + + (BR) to denote denote HA# HA # augmented augmented by by the the principle of bar bar recursion recursion for for arbitrary arbitrary types. types. HA# (BR) proves HA # + + (BR) proves the the principle principle of of bar bar induction, induction, (BI) (BI).. 6.4.1. Lemma. 6.4.1. L emma. While While the the proof proof requires requires some some effort, effort, the the underlying underlying idea idea is is straightforward. straightforward. By By Theorem Theorem 3.1.1, 3.1.1, HA# HA # proves proves (BI) (BI) equivalent equivalent to to its its D-interpretation, D-interpretation, which which asserts asserts the the existence existence of of various various functionals. functionals. One One shows shows how how to to define define these these functionals functionals explicitly, explicitly, using using bar bar recursion recursion in in a a key key instance, instance, and and employing employing a a trick trick due due to to Kreisel Kreisel to properties. We to verify verify that that these these functionals functionals have have the the desired desired properties. We refer refer the the reader reader to to Howard Howard [1968J [1968] for for details. details. On .2, HA# On the the other other hand, hand, by by Theorem Theorem 3.1 3.1.2, HA # + + (BR) (BR) is is clearly clearly D-interpreted D-interpreted in in T T+ + (BR) (BR).. Combining Combining this this observation observation with with the the previous previous lemma lemma yields yields 6.4.2. Theorem. 6.4.2. T heorem. HA# BI ) is ). HA # + + ((BI) is D-interpreted D-interpreted in in T T+ + (BR (BR). 6.5. 6.5. Interpreting I n t e r p r e t i n g PA PA"w + + (AGoo) (A Coo) We We have have seen seen that that full full second-order second-order arithmetic arithmetic can can be be embedded embedded in in the the theory theory 's interpretation PA ) . Spector PA"w + + (A (A Goo Coo). Spector's interpretation applies applies not not only only to to this this but but more more generally generally to to (A (A Gou) Co,~) and and an an even even stronger stronger axiom axiom scheme, scheme, (DGu), (DC,,), which which asserts asserts the the existence existence of of sequences formed dependent choices: sequences formed by by making making dependent choices: 3f \Ix w , a 3b cp(x, a, b) b) --t 3f W cp(x, f (x) , f(x + + 1)), 1)), \Ix, J. J. A Avigad vigad and and S. S. Feferman Feferman 370 370 where where x x is is of of type type 00 and and aa and and bb are are of of type type a. Let Let (DC) (DC) denote denote the the union union of of the the (DC,,) (DC~).. In In section section 6.3 6.3 we we pointed pointed out out that, that, classically, classically, (DC) (DC) can can be be used used to to justify justify bar induction. induction. The The following following theorem theorem represents represents aa kind kind of of converse, converse, and and shows shows the the bar full full strength strength of of Spector Spector's's interpretation. interpretation. 6.5.1. Theorem. Theorem. 6.5.1. PAwW+ + (DC) is N-interpreted N-interpreted in in HA# HA # + + (B/) (BI).. PA (DC) is By By Theorem Theorem 3.1.1 3.1.1 this this is is tantamount tantamount to to the the assertion assertion that that the the double-negation double-negation interpretation interpretation of of (DC) (DC) is is provable provable in in the the latter latter theory. theory. Howard Howard was was able able to to obtain obtain this this result result by by first first reducing reducing (DC) (DC) to to aa special special case case of of bar bar induction induction in in an an appropriate appropriate extension extension of of PA PA wW (which (which is is plausible plausible when when one one considers considers the the contrapositive contrapositive of of (DC) (DC)),), and and then then deriving deriving the the double double negation negation of of this this special special case case in in HA# HA # + (B/) (BI).. The The entire entire proof proof would would take take us us too too far far afield, afield, and and so so once once again again we we refer refer the the reader reader to to Howard Howard [1968] [1968] for for details. details. Together Together with with Theorem Theorem 6.4.2 6.4.2 this this yields yields 6.5.2. Corollary. Corollary. 6.5.2. PA wW+ (DC) (DC) is is ND-interpreted ND-interpreted in in T T+ + (BR) (BR).. PA Since PA wW+ + (A (A Coo) Coo) is is contained contained in in PA PAwW+ + (DC) (DC),, we we have have Since PA 6.5.3. The 6.5.3. Corollary. Corollary. The consistency consistency of of T T+ + (BR) (BR) implies implies the the consistency consistency of of PAll ( CA) . Moreover, ( CA) is PA ~ + +(CA). Moreover, every every provably provably total total recursive recursive function function of of PAll PA ~ + +(CA) is represented represented by by aa type type 1i term term of of T T+ + (BR) (BR).. We both aa functional We have have both functional and and aa term term model model for for T T+ + (BR) (BR).. The The former former is is given given by continuous functionals Kleene and Kreisel indicated indicated in section 4.1 by the the continuous functionals of of Kleene and Kreisel in section 4.1 and and with the continuous functionals. footnote footnote 14, 14, with the constants constants interpreted interpreted by by recursively recursively continuous functionals. The The latter latter is is established established by by work work of of Tait Tait [1971] [1971] and and independently independently Luckhardt Luckhardt [1973], [1973], which (BR) is which shows shows that that T T+ + (BR) is normalizing normalizing and and confluent. confluent. Both Both models models can can be be formalized CA) , thus proving formalized in in PAll PA2 + + ((CA), thus proving The provably total total recursive CA) are The provably recursive functions functions of of PAll PA ~ + + ((CA) are exactly exactly the ones represented bar-recursive terms. the ones represented by by bar-recursive terms. 6.5.4. Theorem. 6.5.4. T heorem. 6.6. 6.6. Evaluation E v a l u a t i o n of of Spector's interpretation Spector was induction to Spector was careful careful not not to to claim claim that that the the generalization generalization of of bar bar induction to higher higher types, types, which which he he used used to to justify justify bar bar recursion recursion for for continuous continuous functionals, functionals, should should be be accepted intuitionistic grounds. accepted on on intuitionistic grounds. In In fact, fact, he he offers offers the the following following caveat: caveat: The The author author believes believes that that the the bar bar theorem theorem is is itself itself questionable, questionable, and and that that until bar theorem until the the bar theorem can can be be given given aa suitable suitable foundation, foundation, the the question question of of whether whether bar bar induction induction is is intuitionistic intuitionistic is is premature. premature. The can be The question question of of whether whether bar bar recursion recursion can be justified justified on on constructive constructive grounds grounds was was taken in aa seminar the foundations led by in taken up up in seminar on on the foundations of of analysis analysis led by G. G. Kreisel Kreisel at at Stanford Stanford in 371 371 Gjjdel's GSdel 's Functional Interpretation the the summer summer of of 1963. 1963. The The seminar seminar's's conclusion, conclusion, summarized summarized by by Kreisel Kreisel in in an an ensuing ensuing report, report, was was that that ..... . the the answer answer is is negative negative by by aa wide wide margin, margin, since since not not even even bar bar recursion recursion of type type 22 can can be be proved proved consistent consistent [by [by constructively constructively accepted accepted principles]. principles]. of Despite disappointing assessment, reduction of Despite this this disappointing assessment, we we feel feel that that Spector Spector's's reduction of all all of of classical classical analysis analysis to to the the prima prima facie facie simple simple computational computational construct construct of of bar bar recursion recursion is, in is, in the the end, end, rather rather remarkable. remarkable. 7. Conservation 7. Conservation rresults e s u l t s for f o r weak w e a k Konig's K S n i g ' s lemma lemma 7.1. 7.1. The T h e theory t h e o r y WKLo WKLo In arithmetic known In this this section section we we study study aa subsystem subsystem of of second-order second-order arithmetic known as as WKLo WKLo.. WKLo interesting theory WKLo is is an an interesting theory because because it it is is just just strong strong enough enough to to prove, prove, among among other things, things, the the Heine-Borel theorem, and and the the completeness completeness and and compactness compactness of of other Heine-Borel theorem, first-order logic (cf. more surprising first-order logic (cf. Simpson Simpson [1987]). [1987]). These These facts facts make make it it all all the the more surprising that that from theory is weak. from aa proof-theoretic proof-theoretic standpoint standpoint the the theory is fairly fairly weak. Formally WKLo the fragment CA) in which induction Formally WKLo is is the fragment of of PA2 PA ~ + + ((CA) in which induction is is restricted restricted to �� E ~ formulas formulas (set (set parameters parameters are are allowed) allowed),, and and instead instead of of full full comprehension the comprehension the to only principles are comprehension scheme, only set set existence existence principles are given given by by aa recursive recursive comprehension scheme, (RCA) (RCA),, ' s lemma. which -definable sets, which asserts asserts the the existence existence of of ll� A~-definable sets, and and aa weak weak version version of of Konig KSnig's lemma. WKL) , asserts nite binary The The latter, latter, denoted denoted ((WKL), asserts that that every every infi infinite binary tree tree has has aa path. path. In WKL) , let In order order to to express express ((WKL), let {0, 1} k (resp. (resp. {0, 1} <~, {0,1} ~) denote denote the the set set of nite, infinite) binary sequences, of length-k length-k (resp. (resp. fi finite, infinite) binary sequences, and and fix fix aa reasonable reasonable encoding encoding of of fifinite nite binary binary sequences sequences as as natural natural numbers. numbers. If If ss and and tt are are sequences sequences so so coded, coded, let let length( s) denote length(s) denote the the length length of of ss and and let let the the primitive primitive recursive recursive predicate predicate tt � C_ ss assert assert that denote the that tt is is an an initial initial segment segment of of s. s. If If bb is is an an element element of of {0, 1} ~, let let bb denote the initial initial segment segment function function b( x) = b( ), b(I), . . . , b(x b(x) -def b(1),..., b(x - 1)). def ((b(0), Finally, Finally, define define the the predicates predicates {O, I}k {O, I} <w , {O, I}W ) {O, I}W , O x (b(x) BinFunc(b) = ----def VX (b(x) = - °0 V k/ b(x) b(x) = -- 1) 1) def V BinFunc(b) {a, l}W , <w A VtVt �c_ ss (t(t e I)) BinTree(J) (s E {a, BinTree(f) ----defV V8s E ff (8 {0, I} 1} <w /\ f)) which which asserts asserts that that bb is is an an element element of of {0, 1 }~, =def E E E which which asserts asserts that that f f is is aa binary binary tree, tree, that that is, is, aa set set of of binary binary sequences sequences closed closed under under initial initial segments, segments, and and {a, l}k Bounded(J, Bounded(f, k) k) = ~def VS E {0, 1} k ((ss ¢ r I) f) def Vs which which asserts asserts that that the the height height of of the the binary binary tree tree ff is is less less than than or or equal equal to to k. k. Weak Weak 's lemma Konig be now be written KSnig's lemma can can be now be written Vf Vf (BinTree(J) (BinTree(f) /\ A Vk Vk -,Bounded(J, ~Sounded(f, k) k) --T --+ 3b 3b (BinFunc(b) (SinFunc(b) /\ A Vk Vk b(k) b(k) E e I)) f)).. S. Feferman Feferman J. A vigad and S. 372 372 In In words, words, ifif ff isis aa binary binary tree tree with with branches branches at at every every level, level, then then there there is is aa path path bb through f. through f. Although the the predicate predicate Bounded Bounded is is primitive primitive recursive recursive (since (since there there are are only only Although finitely many many binary binary sequences sequences of of length length k), k), both both BinFunc BinFunc and and BinTree BinTree require require aa finitely universal quantifier. quantifier. To To avoid avoid these, these, itit turns turns out out to to be be useful to employ employ the the following following universal useful to bin by trick. by trick. Given Given aa function function b, b, define define bbbin bbin bin b -def _ =def { if b(x) b(x) = 00 / 0O if 11 otherwise, otherwise, denoting (Formally, bin denoting the the casting casting of of bb to to aa binary binary function. function. (Formally, bin is is aa functional functional of of type type (0 ftree SO (0 --+ -+ 0) 0) --+ -+ (0 (0 -+ -+ 0).) 0).) Similarly, Similarly, define define ftree so that that for for any any ss < � wA Ss EE ftree 1\ kit Vt C� ss (t(t eE f)), ree +.~ t-t (S (s EE {0, {O, 1} 1} <w J)), so that extraneous sequences. tree is is aa binary binary tree tree obtained obtained from from ff by by pruning pruning away away extraneous sequences. so that ftree The then prove The theory theory ~HA can can then prove in ) BinFunc(b bin)) 1\ A (BinFunc(b) --+ bb = - bbbbin) BinFunc(bbin (BinFunc(b) -+ and, similarly, and, similarly, BinWree(f tree) --+ f f = ftree).. ee ) A (BinTree(J) -+ = ree) 1\ (BinTree(f) BinTree(r -..w As aa result, result, when are working the language language of of H HA we can can just just as as well As when we we are working in in the ' ~ , , we well take take ((WKL) WKL) to to be be given given by by bin (k) Ee ree)). Vf tree, k) k) -+ ~ 3b 3b Vk (bbin(k) ftree)). Vk (b Vf (Vk (Vk ~Bounded(f -,Bounded(pree, comprehension follows from ((QF-A QF-A C), C ) , we take WKLo WKLo to Since Since A A ~� comprehension follows from we can can take to be be con contained tained in in -# PA + P'A# + (( WKL) WKL).. The The fact fact that that WKLo WKLo is is proof-theoretically proof-theoretically weak weak is is shown shown by by the the following following celebrated celebrated theorem H. Friedman. theorem of of H. Friedman. 7.1.1. 7.1.1. Theorem. Theorem. WKLo RA WKLo is is conservative conservative over over P PR A for for II� II ~ formulas. formulas. While Friedman original proof . 1 was While Friedman's' s original proof of of Theorem Theorem 7.1 7.1.1 was model-theoretic, model-theoretic, Kohlen Kohlenbach [1992] [1992] showed showed how how to to use use aa D-interpretation D-interpretation to to obtain obtain the the same same result. result. In In fact, fact, bach Kohlenbach Kohlenbach's's work work was was dedicated dedicated to to somewhat somewhat more more general general results; results; the the approach approach we we present present here here is is aa simplification simplification of of his his methods methods applied applied to to the the specific specific case case at at hand. hand. The The details details are are manageable manageable enough enough so so that that we we can can present present them them here here in in full. full. (The (The realization realization that that hereditary hereditary majorizability, majorizability, defined defined in in the the next next section, section, can can be be used used to obtain Theorem to obtain Theorem 7.1.1 7.1.1 is is due due to to Sieg Sieg [1985], [1985], who who used used it it with with Herbrand Herbrand methods. methods. 's strengthening For For other other proof-theoretic proof-theoretic approaches, approaches, as as well well as as Harrington Harrington's strengthening of of this this 1 5 theorem, theorem, see see Hajek H~jek [1993], [1993], Avigad Avigad [1996a].) [1996a].) 15 15 A number 15A number of other interesting applications applications of the technique technique of majorization in combination combination with functional interpretation have been been made made by Kohlenbach Kohlenbachin a series series of papers; cf. cf. Kohlenbach [1996a,1996b] [1996a,1996b] among among others. GSdel 's Functional Functional Interpretation Interpretation Cadel's 373 373 7.2. Hereditary H e r e d i t a r y majorizability majorizability 7.2. 's lemma If If one one puts puts the the discrete discrete topology topology on on the the set set {O, {0, I}, 1 }, weak weak Konig KSnig's lemma expresses expresses the compactness compactness of of {O, {0,1} under the the associated associated product product topology. topology. Recall Recall that that T I}W~ under the has models models in in which which functionals functionals F F of of type type (0 (0 -+ 0) 0) -+ --~ 00 are are computable, computable, and and hence hence has continuous, continuous, since, since, for for any any g, g, the the computation computation of of F(g) F(g) depends depends on on only only finitely finitely many many values of of g. g. As As aa result, result, the the compactness compactness of of {O, {0, I1} implies that that any any such such F F is is }W~ implies values necessarily bounded bounded when when restricted restricted to to functions functions in in this this space. space. necessarily The The notion notion of of hereditary hereditary majorizability, majorizability, due due to to Howard Howard [1973] [1973],, is is an an effective effective generalization of of this this observation. observation. generalization 7.2.1 7.2.1.. Definition. Definition. For For each each type type (1, a, we we define define aa relation relation aa ::;. <, bb for for terms terms aa and and bb of type type (1, a, as as follows. follows. of I f a(1- = 0 , 0, aa _::;. , b ibs jisujust s t a _ a< b::;. b. 1.1. If 2. If If (1 a = - ((T ~ p) p),, then then aa ::;. _ , bb if if and and only only if if T -+ 2. Vx, yy (x (x ::;. <_, Yy -+ a(x) a(x) ::;. <_, b(y)) b(y)) Vx, where the the variables variables x x and and yy are are of of type type T. where The relation relation aa ::;. <, bb is is read read "a "a is is hereditarily hereditarily majorized majorized by by b." b." The Notice that that Notice Vf ::;. Ax.l). Vf (BinFunc(J) ( B i n F u n c ( f )t-t ~ f f _<, Ax.1). It is is not not difficult difficult to to majorize majorize type type 11 functions: functions: It 7.2.2. 7.2.2. Proposition. Proposition. Given Given aa type type 1I term term ff,, define define f(y) j*(x) def max f* (x) = =de~ max y<x f (y).· Y: Then ff <, ::;. f*. j* . Then A Though we we cannot T that that for functional F there is another Though cannot prove prove in in T for every every functional F there is another G that that majorizes majorizes it, it, we we can can majorize majorize closed closed terms. terms. functional G functional 7.2.3. PProposition. For every closed closed term term FF in in the the language language of is another another 7.2.3. roposition. For every of TT there there is --w closed HA proves closed term term GG such such that that HA proves FF <, ::;. G. G. ...""'..4M 7.2.4. 7.2.4. Proof. Proof. Inductively, Inductively, for for each each term term F[Xl,... F[X b . . . ,xn] , xn] with with free free variables variables shown, shown, one G[[YyIl,, .. ...., , yn] Yn] such such that that TT proves proves one constructs constructs aa term term G Xl <_, ::;. Yl Xn <_, ::; . Yn Yn ~-+ FF <, ::;. G. G. YI A/\. .. .. . A/\ Xn Xl The main main case case isis for for FF -= R. R. Here Here we we can can take take G G defined defined by by The aG(J, ( f , g, f(b) g, 0,0, b)b) -= f(b) g, n,n, b), b) , b)). b)). n', b) G(J, g, n', b) -= max(G(f, max(G(J, g, g, n,n, b), b), g(n, g(n, GG(J, G ( f , g, ( f , g, See See Howard Howard [1973] [1973] for for further further details. details. D J. AAvigad vigad and and S. S. Feferman J. 374 374 Notice that that ifif FF isis aa closed closed term term of of type type (0 (0 --+ -t 0) 0) --+ -t 0, 0, FF <_, ::;. G, G, and and kk == G(Ax.1), G(Ax.l) , Notice then Definition Definition 7.2.1 7.2.1 and and the the remark remark immediately immediately following following imply imply that that for for every every bb in in then {O, I }W, we have that F(b) < k . {0,1} ~, we have that F(b) k. 7.4. The following following technical technical lemma lemma will will be b e needed needed in in section section 7.4. The 7.2.5. LLemma. 7.2.5. emma. If the the term term BB is is of of type type (0 (0 -~ -t O) 0) -+ -t (0 (0 ~-t 0), 0) , then then TT proves proves If A "If BinFunc(B(f)) BinFunc(B(J)) <--> f-+ B B <, ::;. A VAX.1. Vf fAx.1. 7.2.6. Proof. Proof. 7.2.6. B <_. ::;. AVAx.l is equivalent equivalent to to B fAx.1 is Vf, (J <_, ::;. g9 ---> -t B B(J) ::;. Ax.l), Ax.l), V f, g (f ( f ) <_, that is, is, that Vf, ::;. g9 -t BinFunc(B(J))). k/f, gg (J (f _<, --> BinFunc(B(f))). (I) (1) Taking from Proposition Proposition 7.2.2 7.2.2 we we see see that (1) implies implies "If BinFunc(B(J)) , Taking 9 g = = 1 f** from that (1) Vf BinFunc(B(f)), 0 and the the converse and converse is is trivial. trivial. [] - 7.3. Reducing to H-A# HA # ++ (WKL') ( WKL' ) 7.3. R e d u c i n g WKLo WKLo to Since WKLo WKL) , to Since WKLo is is contained contained in in fiA# P"A# + + ((WKL), to obtain obtain Theorem Theorem 7.1.1 7.1.1 it it is is suf sufficient, by ficient, by Theorems Theorems 5.1.3 5.1.3 and and 5.1.1, 5.1.1, to to show show that that the the latter latter theory theory is is conservative conservative sentences. Our over over JiA# H'A# for for rrg II ~ sentences. Our first first step step is is to to reduce reduce it it to to an an intuitionistic intuitionistic variant. variant. 7.3.1. 7.3.1. Lemma. Lemma. The ). The theory theory fiA ~## + + (( WKL WKL)) is is N-interpreted N-interpreted in in JiA ~# # + + (( WKL WKL). # is 7.3.2. 7.3.2. Proof. Proof. fiA P-A# is N-interpreted N-interpreted in in ii}/, H'A#, and and the the double-negation double-negation interpreta interpretation of WKL) , given tion of ((WKL), given by by "If Vf (Vk (Vk-~Bounded(ftree, k)-->-~-~3b Vk (b (bbin(k) ftree)), -,Bounded(ree , k) -t -,-,3b Vk hin (k) EE ree )), WKL) since follows follows from from ((WKL) since the the double-negation double-negation only only weakens weakens the the conclusion. conclusion. 0 C] We WKL) as We define define aa convenient convenient variant variant of of ((WKL) as follows: follows: ((WKL') WKL' ) 'v'f 3b 3b Vk Vk (-,Bounded(ree, (-~Bounded(f tree,k) k) -t --+ bbbin(k) ftree). hin (k) Ee ree). "If Proving Proving the the following following lemma lemma is is aa simple simple exercise exercise in in intuitionistic intuitionistic logic. logic. 7.3.3. 7.3.3. Lemma. Lemma. Over ). Over intuitionistic intuitionistic logic, logic, (( WKL WKL)) is is implied implied by by (( WKL' WKL'). In In fact, fact, the the two two principles principles are are equivalent equivalent over over JiA ~ # ,# , but but the the converse converse direction direction requires requires more more work work and and is is not not needed needed below. below. Godel's GSdel 's Functional Functional Interpretation Interpretation 375 375 - - # ++ ((WKL') # 7.4. Reducing R e d u c i n g HA H-'A# to HA H-'A# 7.4. WKL') to ' t prove # doesn Though WKL) , we WKL) doesn't Though HA H-A# doesn't prove ((WKL), we can can show show that that adding adding ((WKL) doesn't # allow H'A# to prove any new II ~ sentences. The main avenue to this result allow HA to prove any new rrg sentences. The main avenue to this result isis abstracted in in the the following following abstracted Lemma. 7.4.1. Lemma. # proves Suppose HA ~ffA# proves Suppose a, b,b,c) 3~ Vb, vb, c~ S( S(a, ~) -t ~ Vx w 3y 3y R(x, R(~, y) y). 3a (2) (2) (a, x) Then Then there there is is aa specific specific term term c8(a, x) (whose (whose other other free free variables variables are are among among those those of of R R # proves and S) S) such such that that HA ~ffA# proves and (a, x)) w 3a 3~ Vb vb S(a, s(~, b,b, ce(a, ~)) -t -+ Vx w 3y 3y R(x, R(~, y) y). Vx # to The proof proof is is straightforward, straightforward, using using the the scheme scheme cp ~ +-+ ++ cpD ~D of of JiA H-A# to convert convert (2) (2) The with "Vx" "Vx" deleted deleted to to its its D-translation, D-translation, applying applying Theorem Theorem 5.1.3 5.1.3 to to extract extract aa term term with c 8 of of T T,, and and then then manipulating manipulating quantifiers. quantifiers. The The upshot upshot is is that that to to eliminate eliminate the the # , it assumption assumption 3a 3a Vb, Yb, cc S(a, S(a, b, b, cc)) from from aa proof proof of of Vx Yx 3y 3y R(x, R(x, y) y) in in JiA H-A#, it suffices suffices to to show that that one one can can prove prove show 3a vb S(a, S(a,b,~(a,~)) 3a Vb b, c(a, x)) A for any any specific specific term term c8.. for We now now apply apply this this to to the the situation situation at at hand. hand. We 7.4.2. 7.4.2. Lemma. Lemma. 7.4.3. Proof. 7.4.3. P roof. proves proves HA # + WKL) is conservative over HA # for for rrg H-A# + (( WKL) is conservative over ~ffA# II~ sentences. sentences. Lemma 7.3.3 if JiA# By + ((WKL) WKL) By Lemma 7.3.3 and and the the deduction deduction theorem, theorem, if H-A#+ Vx 3y R(~, R(x, y), w 3y y), then proves then H'A# HA # proves Vf 3b 3b Vk Vf Vk (~Bounded(f ~y R(x, R(x, y).16 bhin(k) EE ftree) -+ VX ( -,Bounded (ftree, tree , k) k) --+ -t bbin(k) ree) _.~ Vx 3y y). 16 Applying C) to Applying (A (AC) to the the hypothesis, hypothesis, we we obtain obtain 3B f, kk (~Bounded(f tree, ee , k) k) ~-t B(f)bin(k) B(f)hin(k) eE ftree) ree) ._~ 3B VVf, ( -,Bounded (r -t Vx Vx 3y 3y R(x, R(x, y). y). Applying Lemma Lemma 7.4.1 7.4.1 with with kk in in place place of of c, c, we we are are reduced reduced to to showing showing that that HA HA## Applying proves proves 9B (3) ree ) (3) ( -,Bounded (ree , k(S,x)) k(B, x)) ~-t B(f)bin([~(B,x)) B(f)hin(k(B, x)) eE ftree) 3B Vf Vf (~Sounded(ftree, # - - # - # to use H'Ao HAo# or or I-H'A I-HA# instead instead of WEfffA WE-HA# , since the the deduction theorem fails 16Here we need to for the the latter latter theory; cf. 3.1. However, However, for for aa way around around this, this, see Kohlenbach [1992], [1992), p. p. 1246. for J. Avigad and S. Feferman Feferman 376 376 for for any any closed closed term term k k.. We now now bring in the the notion of hereditary hereditary majorizability to bound bound the the value value of of We bring in notion of majorizability to kk(B, (B, x) find aa term x).. By By Proposition Proposition 7.2.3, 7.2.3, we we can can find term k* k* that that hereditarily hereditarily majorizes majorizes kk.. Define Define k'(x) def k*().x).j.1, x). k'(x) = --def Since xx :::: HA # proves Since <.: x, from Lemma Lemma 7.2.5 7.2.5 we we see see that that H-A# proves * x, from (B, x) :::: (4) Vf Vf BinFunc(B(J)) SinFunc(S(f)) -+ ~ kk(B,x) <_: k'(x). k'(x). (4) - To B. Define f) as To verify verify (3), (3), we we only only need need construct construct an an appropriate appropriate B. Define B'(x, S'(x,f) as follows. Let maximum value than or equal to to k'(x) such such that that rree ftree has has follows. Let ll be be the the maximum value less less than or equal an element element of of length length l, l, and and let let s be be any any (e.g. (e.g. the the leftmost) leftmost) such such element. element. Let Let an Y) = B'(x def B'(x,, f)( f)(y) --def { 0Sy sy if i f y < so so that that B'(x, B'(x, f) f),, as as an an element element of of (0, 1}w~,, consists consists of of s followed followed by by aa string string of of zeros. zeros. {O, l} -# HA proves In In particular, particular, H'A# proves <_: k'(x) k'(x)A1\ ""Bounded(Jtree, ~Bounded(ftree,/)l) -+ ~ B B'(x, f)(l) (l) l :::: ' (x, f) e E ftree. ree. (5) (5) -# Finally, =def ).f HA proves Finally, let let B(x) B(x)=def Af.S'(x, f).. Then Then H'A# proves · B'(x, f) Vf Vf BinFunc(B(x, SinFunc(B(x, f)) f)) and and so, so, using using (4) (4),, k (B(x) , x) :::: <_: k'(x)). k'(x)). Vx ((k(B(x),x) By By (5) (5) we we then then have have V Bounded(Jtree, kk(B(x),x)) Vff (..., (~Bounded(ftree, ~ B(x, B(x,f)bin(k(B(x),x)) ftree).) . (B(x), x)) -+ f) bin ( k (B(x), x)) Ee ree SSoo B(x) S(x) witnesses witnesses the the existential existential quantifier quantifier iinn (3). (3). o Friedman . 1 now Friedman's' s Theorem Theorem 7.1 7.1.1 now follows follows from from Lemmas Lemmas 7.3.1 7.3.1 and and 7.4.2, 7.4.2, together together with with Theorems Theorems 5.1.3 5.1.3 and and 5.1.1. 5.1.1. Harrington Harrington's's theorem theorem states states that that WKLo is is conservative conservative over over its its subtheory subtheory RCAo (which WKL) ) for (which omits omits ((WKL)) for m II~ sentences. sentences. Since Since RCAo RCAo is is easily easily interpretable interpretable in in IEJ this, together IZI,, this, together with with Theorem Theorem 5.1.4, 5.1.4, yields yields another another proof proof of of Friedman Friedman's' s result. result. However, However, the the question question as as to to whether whether aa functional functional interpretation interpretation can can be be used used to to obtain obtain Harrington's strengthening open. Harrington's strengthening is is still still open. The be adapted The proof proof above above can can be adapted to to yield yield similar similar conservation conservation results results for for weaker weaker fragments "elementary analysis," whose provably fragments of of second-order second-order arithmetic, arithmetic, such such as as "elementary analysis," whose provably total recursive total recursive functions functions are are all all elementary elementary recursive recursive (cf. (cf. also also Kohlenbach Kohlenbach [1996b]). [1996b]). It It is is reasonable reasonable to to conjecture conjecture that that functional functional interpretations interpretations can can be be used used to to obtain obtain similar similar results results for for systems systems of of polynomial-time-computable polynomial-time-computable arithmetic, arithmetic, especially especially in in Gode/'s Functional Interpretation GSdel's Functional 377 377 view view of of the the conservation conservation result result of of Ferreira Ferreira [1988,1994] [1988,1994], which showed that that aa suitable suitable ' which showed version version of of ((WKL) is conservative conservative over over 81 S~ for for rrg II ~ formulas formulas (cf. (cf. also also Cantini Cantini [1996]). [1996]). WKL) is However, k) from However, the the functional functional B' B ~ and and the the relation relation Bounded(f, Bounded(f,k) from the the proof proof above above cannot be defined say, PVw, P V ~, so so new considerations seem seem to to be be necessary. (If cannot be defined in, in, say, new considerations necessary. (If the reader reader is is concerned concerned with with polynomial polynomial bounds bounds rather rather than than a a polynomial-time polynomial-timethe computable Skolem function, should consult Corollary 4.28 computable Skolem function, he he or or she she should consult Corollary 4.28 of of Kohlenbach Kohlenbach [1996bj [1996b].).) 8 interpretations and 8.. Non-constructive Non-constructive interpretations a n d applications applications 8.1. 8.1. Overview O v e r v i e w of of the t h e section; section; general general pattern. pattern. This This section section shows shows how how the the D-interpretation D-interpretation can can be be extended extended by by inclusion inclusion of of certain non-constructive functional functional operators operators in in order to obtain obtain conservation conservation results results certain non-constructive order to of various various finite finite type type systems systems contained contained in in PAw PA ~ + + (AC) (A C) over over related related second-order second-order of systems. QF-A C) is systems. As As we we have have noted noted in in section section 3.1 3.1,, ((QF-AC) is automatically automatically preserved preserved C) for under the under the ND-interpretation. ND-interpretation. This This can can be be extended extended to to (r-A (F-AC) for some some larger larger classes classes of of formulas formulas r F by by adjunction adjunction of of suitable suitable (necessarily (necessarily non-constructive) non-constructive) Skolem Skolem functionals functionals with with associated associated axioms axioms which which imply imply that that each each formula formula in in r F is is equivalent equivalent to to aa QFQF- (i.e. (i.e. quantifier-free) quantifier-free) formula. formula. The The paradigm paradigm case case is is given given by by adjunction adjunction of of the the non-constructive non-constructive minimum minimum operator /1-, which operator #, which allows allows us us to to reduce reduce every every arithmetical arithmetical formula formula to to QF QF form. form. + ((QF-AC) QF-A C) imply Then Then axioms axioms (/1-) (#) for for this this with with PAW PA~+ imply the the second-order second-order system system Ef -DC, and is shown (/1-) + QF-A C) Z/-DC, and it it is shown that that the the system system PAW PA ~ + + (#) + ((QF-A C) is is ND-interpreted ND-interpreted infinite terms in T T+ + (/1-) (#).. Normalization Normalization of of aa system system of of infinite terms for for the the latter, latter, just just as as in described in methods of described in section section 4.4 4.4 above above for for T T by by the the methods of Tait Tait [1965] [1965],, shows shows that that the the type type 11 functions functions thus thus generated generated lie lie in in the the hierarchy hierarchy of of hyperarithmetic hyperarithmetic functions functions Ho: formalization of Ha for for a c~ < < co e0.. Then Then formalization of this this model model can can be be carried carried out out in in a a theory theory < (lIPCA) of relative arithmetical comprehension iterated up to co , yielding co (II~ of relative arithmetical comprehension iterated up to e0, yielding aa number results, including number of of conservation conservation results, including aa well-known well-known one one due due to to Friedman Friedman [1970j. [1970]. The The arguments arguments for for this this are are detailed detailed in in sections sections 8.2 8.2 and and 8.3. 8.3. It It is is also also shown shown that that when when the the Bar Bar Rule Rule is is adjoined adjoined to to the the above above finite finite type type system, system, we we obtain obtain conservation conservation over over the system (lIPo the full full predicative predicative system (II~ CA) CA) < J. J. A A vigad vigad and and S. S. Feferman Feferman 378 378 8.2. 8.2. Finite F i n i t e type t y p e systems s y s t e m s with w i t h non-constructive non-constructive {l-operator, p-operator, and and related related second-order second-order systems systems 8.2.1. Subsystems 8.2.1. S u b s y s t e m s of of PAw PA ~ Besides the nite type includes the Besides the classical classical fi finite type theory theory PAw PA ~ which which includes the constants constants K, K, SS and and R induction for shall consider consider also R in in all all types types and and full full induction for all all formulas, formulas, we we shall also its its subsystems subsystems -w PA in in which which R R is is replaced replaced by by the predicative recursion recursion operator operator R R introduced in PA the predicative introduced in section section 5.1, 5.1, and and subsystems subsystems of of both, both, obtained obtained by by restricting restricting induction induction as as follows: follows: � (Res-Ind) (Res-Ind) o0 Ee X x /\ ^ Vx w (x E e X x -+ x' E e X) x ) -+ Vx w (x E e X). x). Here Here X X ranges ranges over over sets sets of of type type 11 (identified (identified with with characteristic characteristic functions functions as as in in CA) for section section 6.3) 6.3).. When When we we add add principles principles which which imply imply (r(F-CA) for various various classes classes of of formulas formulas r F,, we we can can infer infer from from (Kes-Ind) (I~es-Ind) all all substitution substitution instances instances for for X X by by formulas formulas
The The functional functional Eo E0 of of type type 22 with with values values 00 and and 11 only only is is defined defined by by (E0) Eo(J) E0(f) = - 00 t+ ~ 3x 3x (J(x) (f(x) = - 0) 0).. 2E (This functional Using this (This functional is is also also denoted denoted 2E in in the the recursion-theoretic recursion-theoretic literature.) literature.) Using this axiom, axiom, every every arithmetical arithmetical formula formula is is equivalent equivalent to to aa QF-formula. QF-formula. If If we we are are to to use use it it with an with an ND-interpretation, ND-interpretation, we we should should consider consider the the two two implications: implications: ff ((x) x) = = 00 -+ --+ Eo(J) E0(f) = = 00 and and Eo(J) E0(f) = = 00 -+ -+ 3x 3x (J(x) (f(x) = = 0). 0). The The ND-interpretation ND-interpretation preserves preserves the the first first of of these these but but requires requires for for the the second second aa O. Combining Combining this functional functional X X such such that that Eo(J) E0(f) = --- 00 -+ ~ f f ((X(J)) X(f)) = - O. this with with the the first first Skolem implication o. Such implication gives gives f(x) f(x) = = 00 -+ --~ f f ((X(J)) X(f)) = - 0. Such an an X X is is then then aa choice choice or or Skolem functional cation. We shall use functional for for type type 00 quantifi quantification. We shall use the the new new symbol symbol {l p for for such such aa functional functional and and take take as as its its axiom axiom f(x) = = 0 o -+ f({l(J)) = = o. o. Then 0, Then Eo E0 may may be be defined defined simply simply in in terms terms of of {l p by by Eo(J) E0(f) = = 00 if if f f (({l(J)) p(f)) = = O, 1. The otherwise otherwise Eo(J) E0(f) = = 1. The advantage advantage of of using using {l p in in place place of of Eo E0 is is that that its its axiom axiom is is preserved preserved directly directly under under the the ND-interpretation ND-interpretation without without requiring requiring any any supplementary supplementary functionals. the non-constructive functionals. Note Note that that the non-constructive minimum minimum operator operator in in its its usual usual sense, sense, defined defined as as the the least least x x such such that that f(x) f(x) = = 00 if if 3x 3x (J(x) (f(x) = = 0) 0) and and 00 otherwise, otherwise, is is also also definable in bounded minimum minimum operator. definable in terms terms of of {l p and and the the primitive primitive recursive recursive bounded operator. Godel's Gbdel 's Functional Interpretation 379 379 8.2.3. Second-order S e c o n d - o r d e r forms forms of of ((CA), (AC) and and (DC) (DC) 8.2.3. CA) , (AC) We shall shall consider consider various various restricted restricted forms forms of of these these principles principles for for the the usual usual arith arithWe metical (II� (H ~ and and � ~E�~) and and analytical analytical (II� (II~ and and ��) ~1) hierarchies. hierarchies. That That is, is, for for r F one one of of metical these classes classes of of formulas, formulas, we we shall consider the the schemata schemata these shall consider (r-CA) (rCA) (r-ACo~)) (r-ACoJ (F-DC~)) (r-DCJ BY Vx Vx (x (x Ee Y Y f-+ ~ cp(x)) ~(x)) 3Y Vx Vx 3f 3 f cp(x, 9~(x, j) f) ---+ ~ 3g Vx Vx cp(x, 9~(x,ggx) x) Vx, Vx, ff 3g 3g cp(x, 9~(x,f, f, g) g) ---+ --> Vf Vf 3g 3g (go (go = = ff /\ A Vx Vx cp(x, 9~(x,ggx, x , ggx+l)) x+l )) where in in each each case case cp ~ ranges ranges over over formulas formulas in in r F,, and and in in the the two two choice choice principles principles gg,x where is written written for for >.. ,ky.g(x, y).. In In addition, we shall shall consider the scheme scheme is y .g(x, y) addition, we consider the r - CA) (At-CA) (Ll Vx (cp(x) Vz (~(z) ++ f-+ ~ 3Y 3Y Vx Vz (x (x E e Y Y f-+ e cp(x)) ~(z)) -,~r1jJ(x)) ---+ where cp ~ and and 1jJ r again again range range over over formulas formulas in in r. F. In In the the context context of of second-order second-order where systems, we we omit omit the the subscripts subscripts from from the the (AC) (AC) and and (DC) (DC) principles. Also we we use use systems, principles. Also these these principles principles to to name name the the second-order second-order system system S S obtained obtained by by adjoining adjoining them them to to PA together full induction induction for for second-order second-order formulas. formulas. Then, Then, in in accordance accordance PA together with with full with 8.2.1, 8.2.1, we we use use Res-S Res-S to to name name the the same same system system with with restricted restricted induction. The induction. The with systems of of particular particular concern concern to to us us in in this this section section and and the the next next are: are: Ill-CA IIf- CA,, Lli-CA A ~- CA,, systems Z~-AC, while iinn section shall take Ef -A C , Z~-DC, Ef -DC, and and their their restricted restricted versions, versions, while section 8.4 8.4 we we shall take up corresponding systems level up up in in the analytic hierarchy. up the the corresponding systems one one level the analytic hierarchy. Note Note that that the system IIf-CA as the the system A CA based CA proves proves the the same same theorems theorems as system ACA based on on the the the system Illaxiom, i.e. CA) where where rF is the class class of arithmetical arithmetical comprehension comprehension axiom, i.e. (r(F-CA) is the of arithmetical arithmetical formulas. For, we we can can derive derive closure under complement complement from from (II~ and hence formulas. For, closure under (Ill-CA) and hence the the principle (~~ CA) CA),, and we obtain obtain (A in general. We note note and then then by by iterating iterating these these we (A CA) CA) in general. We principle (Elthis record in are established this for for the the record in the the following; following; the the further further relationships relationships below below are established just easily. just as as easily. Theorem. 88.2.4. .2.4. Theorem. 1. Ill -CA = ACA ACA 1. II~ 2. Ef -A C 2. II~ Ill -A C = X{-AC 3. II~ Ill -DC = = Z[-DC Ef -DC ~. 4 · II~ Ill-CA c� A~-CA Lli -CA c� X:-A Ef -A CC c_ � E]-DC Ef -DC 5. The The same results hold for the corresponding theories theories with with ((Res -Ind) . 5. same results hold.for the corresponding Res-Ind). Apropos of of this this last, last, itit is is aa familiar familiar result result that that Res-II~ Res-Ill-CA or or AACAo is aa conservative conservative Apropos CAo is extension extension of of PA; PA ; aa model-theoretic model-theoretic argument argument gives gives the the quickest quickest proof proof of of that. that. S. Feferman J. A vigad and S. 380 380 8.2.5. Transfinite T r a n s f i n i t e induction i n d u c t i o n below below co. c0. 8.2.5. In /3" 7 , .. .... to In the the following following we we shall shall use use variables variables 0:, c~,/~, to range range both both over over ordinals ordinals and and over terms terms in in aa natural natural recursive recursive ordinal ordinal representation representation system system for for ordinals ordinals up up to to co c0.. over We write write < arithmetic and and each each 0: a < < co Co.. Analysis Analysis of of his his argument argument actually shows more: more: arithmetic actually shows 8.2.6. Suppose 8.2.6. Theorem. Theorem. Suppose S S is is any any system system whose whose language language contains contains that that of of PA PA and and which which contains contains full full induction. induction. Then Then for for each each formula formula cp ~ of of S S and and each each 0: c~ < co Co we we S. can prove ( T1o) ( cp) in S. The argument argument for for this this proceeds proceeds by by showing showing that that transfinite transfinite induction induction up up to to WO w~ The with respect respect to to cp ~p can can be be reduced reduced to to transfinite transfinite induction induction up up to to 0: a with with respect respect to to with formula cpo ~* which which is built by by propositional propositional operations operations and and numerical numerical (first-order) (first-order) is built aa formula quantification from from ~. (For aa simple simple choice choice of of cpo ~* due due to to Schfitte, cf. Feferman Feferman [1977,p. [1977,p. cpo (For Schutte, cf. quantification 946].) 946].) 8.2.7. Transfinitely T r a n s f i n i t e l y iterated i t e r a t e d arithmetical a r i t h m e t i c a l comprehension comprehension 8.2.7. The The hyperarithmetic hyperarithmetic hierarchy hierarchy up up to to any any 0: a and and relative relative to to any any initial initial set set X X is is x 8.2.8. T Theorem. 8.2.8. heorem. (ilP-CA) The proof proof consists consists in in showing, showing, for for each each c~ 0: << Co, co , the the existence existence of of HX(~) HX (/3) for for The /3 < by the the scheme scheme of of transfinite transfinite induction induction up up to to c~. 0:. When When passing passing to to the the limit limit < a0: by makes use x ('y) exists uniquely determined determined by by /3 one one makes use of of the the fact fact that that ifif H HX(J) exists then then itit is is uniquely its set Hx(~) defined in H (/3) can can be be defined in A~ �} form form (cf. (cf. its recursive recursive defining defining conditions, conditions, so so the the set Feferman Feferman [1977,pp. [1977,pp. 944-947] 944-947] for for further further details). details) . Note that that what what is is essential essential for this proof proof is is availability availability of of transfinite transfinite induction induction for for Note for this �� formulas formulas in in the the system system ALl1-CA, which we we do do not not have have in in its its restricted restricted version. version. Our Our E~ ~-CA, which next step these second-order together with next step is is to to put put these second-order systems systems together with the the finite-type finite-type systems systems of two chains of 8.2.1. 8.2. 1. This This leads leads to to two chains of of interest: interest: x Godel's Ghdel 's Functional Interpretation 381 381 8.2.9. 8.2.9. Theorem. Theorem. (lI~l -CA)« o � C_ Lli A~-CA C_ Ej Z]-AC C_ Ej Z]-DC C_ PA PA wW+ + (p,) (#)++ ((QF-AC) QF-AC) 11.. (II -CA � -A C � -DC � 2. PA PA � c_ Res-Ill Res-II~ -CA � c_ Res-Lli Res-A~-CA c_ Res-Ej Res-Z/-A C -CA � -AC 2. Plt + (p,) c_ ResRes-P"A'+ (#) + + (( QF-AC). QF-A C). � What's added added here here to to the the information information from from above above are are the the final final inclusions. inclusions. In In 1, 1, What's QF-A C) from from \/x, Vx, I f 3g 3G :::lG \/x, Vx, If As As noted noted above, above, the the (p,) (#) axiom axiom is is preserved preserved under under the the ND-interpretation; ND-interpretation; in in combination combination with with the the Theorems Theorems 3.1.3 3.1.3 and and 3.1.4, 3.1.4, this this immediately immediately yields: yields: 8.3.1. 8.3.1. Theorem. Theorem. PA -A C) PA wW+ + (p,) (#) + + (QF ( QF-A C) is is ND-interpreted ND-interpreted in in T T+ + (p,) (#).. The The next next step step is is to to show show how how T T+ + (p,) (#) can can be be modeled modeled in in (II (H~l- CA) « 0 '. This This is is by by an an extension symbol, of infinite term model of extension to to p" #, as as a a type type 22 constant constant symbol, of Tait's Tait's infinite term model of T T from from section new arguments and so, section 4.4. 4.4. No No new arguments are are needed needed for for normalization normalization and so, just just as as before, before, every every term term of of T T+ + (p,) (#) translates translates into into a a (possibly) (possibly) infinitely infinitely long long term term which, which, in in turn, turn, It I < reduces term tt with reduces effectively effectively to to a a normal normal term with Itl < co ~0.. Only Only the the description description of of normal normal modified. For infinitely infinitely long long terms terms tt at at type type 00 needs needs to to be be modified. For this this purpose, purpose, one one examines examines 1, and such such tt which which may may contain contain free free variables variables x x of of type type 00 and/or and/or I f of of type type 1, and arrives arrives the following at the following possibilities possibilities (cf. (cf. the the end end of of 4.4): 4.4): either either at 1. 1. tt -= - 00o ror 2. tt = Sc(s) 2. Sc(s) where where ss is is normal normal or or 3. tt is 3. is a a variable variable of of type type 00 or or (s) where 4. tt = 4. = (tn) (t~)(s) where each each tn t~ is is normal normal and and 8s is is normal normal or or 5. 5. tt = = 1(8) f(s) where where I f is is a a type type 11 variable variable and and ss is is normal normal or or 6. 6. tt = = p,(AX.8) #(Ax.s) where where 8s is is normal normal or or 7. p,( (tn)) where 7. tt == tt((t~)) where each each tn t~ is is normal. normal. Every Every closed closed term term of of type type 11 in in T T+ + (p,) (#) then then reduces reduces effectively effectively to to a a normal normal term term normal of type 00 or (tn) where of the the form form AX.t Ax.t where where tt is is normal of type or of of the the form form (t~) where each each tn t~ is is of normal. What transfinite iteration normal. What these these represent represent as as functions functions is is the the effective effective transfinite iteration of of the the p,# operator, operator, or or equivalently equivalently the the jump jump (Eo (E0)) operator, operator, up up to to ordinals ordinals less less than than c~0. Thus o . Thus J. Avigad A vigad and S. S. Feferman 382 382 each each closed closed term term tt of of type type 1I in in T T+ + (1-£) (#) represents represents aa function function which which is is recursive recursive in in Ho. Ha for if tt is is aa term of type for some some a c~ < < ce0. More generally, generally, if term with with aa free free variable variable f f of type 11,, then then o . More tt represents represents a a function function recursive recursive uniformly uniformly in in H! H~ for for some some a c~ < < co e0.. Formalization Formalization of of this argument argument leads leads fifinally nally to this to the the following: following: PAw -A C) is PA W+ + ( (1-£) # ) ++ (QF (QF-AC) is aa conservative conservative extension extension of of (IIP-CA)<£ (II~ <,oo for for m II~ sentences. sentences. 8.3.2. 8.3.2. Theorem. Theorem. For, g) with arithmetical, we For, given given aa provable provable m II~ sentence sentence "If Vf 3g 3g cp(J, ~o(f,g) with cp ~o arithmetical, we use use t[f]) , where 8.3.1 obtain aa term 8.3.1 to to obtain term t[f] t[f] of of type type 11 such such that that T T+ + (1-£) (#) proves proves cp'(J, ~o'(f,t[f]), where ~o' is is quantifierquantifier- free free and and equivalent equivalent to to cp ~o under (#),, and and then apply the the preceding preceding cp' under (1-£) then apply modeling argument. modeling argument. 8.3.3. Corollary. sentences. sentences. Ef -DC is X]-DC is aa conservative conservative extension extension of of (IIP (II~ -CA) <£o for for m HI This originally to established it This result result is is due due originally to Friedman Friedman [1970], [1970], who who established it by by model modeltheoretic theoretic techniques. techniques. Subsequently Subsequently to to the the appearance appearance of of the the above above approach approach in in the the publications publications referred referred to to in in section section 8.1, 8.1, Feferman Feferman and and Sieg Sieg [1981] [1981] used used Herbrand Herbrandto the Gentzen Gentzen methods methods with with the the 1-£ # operator operator to the same same end. end. Now Now turning turning to to the the restricted restricted systems, systems, the the main main result result is is -w 1-£) + (QF 8.3.4. Res-PA -A C) is 8.3.4. Theorem. Theorem. Res-PA + + ((#)+ (QF-AC) is aa conservative conservative extension extension of of Res-IIp-CA for Res-II~ for II� II~ sentences, sentences, and and hence hence of of PA PA for for arithmetical arithmetical sentences. sentences. A W Here, use of Here, for for the the proof, proof, one one makes makes^use of the the ND-interpretation ND-interpretation of of our our higher higher type type system picture, since since system in in T T + + (1-£) (#).. The The recursor recursor Ii R can can then then be be eliminated eliminated from from the the picture, it be defined simply of 1-£. Terms it can can be defined arithmetically, arithmetically, hence hence by by means means simply of + + ,,. , , and and #. Terms of of the resulting then normalize without passing the resulting system system then normalize without passing to to infinite infinite terms, terms, and and so so the the functions functions represented represented by by closed closed terms terms of of type type 11 are are also also arithmetical. arithmetical. The The same same holds holds t [ f ] of type 1 uniformly in a type 1 variable f . for terms for terms t[f] of type 1 uniformly in a type 1 variable f. . Res-Ef -A C is aa conservative Res-Z]-A C is conservative extension extension of of Res-IIp Res-II~ -CA for for m II~ sentences, sentences, and and hence hence of of PA PA for (or arithmetical arithmetical sentences. sentences. 8.3.5. 8.3.5. Corollary. Corollary. This established by model-theoretic methods methods by Schlipf This was was first first established by model-theoretic by Barwise Barwise and and Schlipf Again, Feferman [1975] [1975] and and independently independently Friedman Friedman [1976] [1976].. Again, Feferman and and Sieg Sieg [1981] [1981] used used Herbrand-Gentzen -DC Herbrand-Gentzen methods methods with with 1-£ # to to obtain obtain the the same same result. result. Note Note that that Res-Ef Res-Y,~-DC is proves the is stronger stronger than than Res-IIpRes-II~ CA because because it it proves the existence existence of of H Hw. w. GSdel 's Functional Interpretation Codel's 383 383 8.3.6. Predicatively P r e d i c a t i v e l y provable p r o v a b l e ordinals ordinals 8.3.6. By formalizing formalizing work work of of Kleene, Kleene, the the systems systems (I1f(IIf-CA)<~ are another another form form of of By CA) < a are ramified analysis RAa RAn when when a c~ - w .. a c~.. The The proof-theoretic proof-theoretic ordinals17 ordinals 17 of of these these ramifi ed analysis were established established by by Schutte Schiitte [1960] [1960] (cf. (cf. also also Schutte Schiitte [1977]) [1977]) in in terms of the the Veblen Veblen were terms of hierarchy of of normal normal functions functions ~a of of ordinals, ordinals, defined defined by: by: hierarchy 11.. ~0(~) = w ~ 2. For For a c~ =1= # 0, 0, CPa ~ enumerates enumerates {~: ~ Z ( ~ ) = ~ for for all all/3 < a a}. {3 < 2. }. xed points Each ~ + 1 is is then then the the critical critical function function of of ~ , , i.e. i.e. it it enumerates enumerates the the set set of of fi fixed points Each {~ : ~,(~) = = O ~}.· In In particular, particular, ~1 (~c) = Q, etc. etc. The The diagonal diagonal function function A~c.~(0) is also also normal normal and and its its least least fixed fixed point point is is denoted denoted F0. There There is is aa natural natural notation notation is system for for ordinals ordinals up up to to F0 (Feferman (Feferman [1968b]), [1968b]), and and we we shall shall now now use use a, a,/3, 7 , - ... to to {3, I, system range over over that that system system as as well well as as over over ordinals ordinals in in the the usual usual sense. sense. It It follows follows from from range ' s work Schfitte's work referenced referenced above that Schutte above that CPa CPo(�) = w{ CPa+l {� : CPa(�) ro =w {� : cpp(�) = � CPa, CPl(�) = C{, roo A�.cp{(O) .. =w CPo For a not For a a = w .. a a ((~ not 0), 0), the the proof-theoretic pro@theoretic ordinal ordinal of of RAa RAn or, or, equivalently, of of (Ilf (II~ -CA) 8.3.7. 8.3.7. Theorem. Theorem. Now, the the so-called ordinals of of ramified ramified analysis analysis were were defined defined to to be be Now, so-called autonomous autonomous ordinals those generated generated by by the the following following "boot-strap" "boot-strap" procedure: procedure: those 1. is an an autonomous autonomous ordinal. ordinal. 1. 00 is 2. If If/3{3 is is an an autonomous autonomous ordinal ordinal and and a c~ is is a a provably provably recursive recursive ordinal ordinal of of RA RAzp 2. then a a is is an an autonomous autonomous ordinal. ordinal. then The The systems systems RAa RAn for for a c~ autonomous autonomous were were proposed proposed by by Kreisel Kreisel as as a a characterization characterization of predicative predicative analysis, analysis, and and so the autonomous autonomous ordinals are identified identified with with the of so the ordinals are the predpred icatively provable 8.3.7, itit was established independently Schutte icatively provable ordinals. ordinals. Using Using 8.3.7, was established independently by by Schfitte and Feferman Feferman in in the 1960s that least non-autonomous ordinal, hence and the mid mid 1960s that F0 is is the the least non-autonomous ordinal, hence least impredicative impredicative ordinal. ordinal. In In order order to to investigate the extent predicativity in in the investigate the extent of of predicativity the least ordinary, analysis, Feferman second-order ordinary, unramified, unramified, analysis, Feferman [1964] [1964] described described an an unramified unramified second-order system with whose proof-theoretic ro o This This was was system with finitely finitely many many schemata schemata whose proof-theoretic ordinal ordinal is is F0. strengthened later in Feferman Feferman [1979] [1979] to following: strengthened later in to the the following: ro 8.3.8. Theorem. The -DC + has proof-theoretic proof-theoretic ordinal ordinal 8.3.8. Theorem. The system system EJ ,F,]-DC + (Bar-Rule) (Bar-Rule) has it is over (II~ (ilf-CA) ro; The Bar-Rule, Bar-Rule, which which isis related related to to the the principle principle of of Bar Bar Induction Induction described described in in The section 6.3 6.3 above, allows one one to to infer infer the the scheme of transfinite transfinite induction induction on on aa recursive recursive section above, allows scheme of ordering when when its its well-foundedness well-foundedness has has been been established. established. The The idea idea of of the the proof proof of of ordering Theorem 8.3.8 8.3.8 sketched op. cit. cit. isis to to apply apply aa functional functional interpretation interpretation to to the the still still larger larger Theorem sketched op. (Jl) + (( QF-A PA~w ++ (#)+ QF-A C) C) ++ (Bar-Rule), (Bar-Rule) , carrying carrying itit into into TT ++ (( Jl# )) ++ (Bar-Rule), (Bar-Rule) , system PA system 1 7The proof-theoretic is defined in in several several ways, e.g. e.g. as as the the least least ordinal ordinal proof-theoretic ordinal ordinal of of aa system SS is XTThe aa which which is is not not the the order-type order-type of of aa provably provably recursive recursive well-ordering well-ordering of of the the system, system, or or as as the the least least (in aa standard standard notation notation system) system) for which which the the consistency consistency of of S can can be be proved proved by by transfinite transfinite aa (in induction are equivalent induction up up to to aa over aa weak base system. system. These are equivalent in practice. 384 384 J. Avigad A vigad and S. S. Feferman and and then then to to model model the the latter latter in in a a boot-strap boot-strap fashion fashion in in aa system system of of infinitely infinitely long long terms of of length length less less than than ro F0.o The The final final result result is is then then obtained obtained by by formalization formalization terms of the the type type 11 functions functions of of that that model as for for Theorem Theorem 8.3.2 8.3.2 above. above. The The details details of of of model as this are rather complicated, and and in in its its place place aa more more digestible digestible proof of Theorem Theorem this are rather complicated, proof of 8.3.8 obtained by 8.3.8 was was subsequently subsequently obtained by Feferman Feferman and and Jager Jiiger [1983] [1983] by by Herbrand-Gentzen Herbrand-Gentzen methods methods with with the the J.L # operator. operator. But But the the functional functional interpretation interpretation served served an an important important intermediate, heuristic role arriving at intermediate, partly partly heuristic role in in arriving at this this result. result. 8.4. 8.4. Functional F u n c t i o n a l interpretations interpretations using using the the Kleene Kleene basis operator operator non- ) well-foundedness as 8.4.1. 8.4.1. Testing Testing for for ((non-)well-foundedness as a a functional functional As coding the As in in sections sections 66 and and 7, 7, we we write write g(x) y(x) for for the the sequence sequence number number coding the sequence sequence (g(O) ( g ( 0 ), ,. .. ...,, g(x g ( x -- 1») 1)) when when 9 g is is of of type type 11.. Then Then aa type type 11 function function f f represents represents aa 0, where well-founded well-founded tree tree if if \l Vgg 3x 3x f(g(x» f(-~(x)) =1= :/: O, where the the tree tree consists consists of of all all sequence sequence numbers O. Kleene's predicates provides numbers 8s with with f(8) f(s) = - 0. Kleene's basis basis theorem theorem for for ~E� predicates provides aa functional functional J.Ll #1 which which chooses chooses a a descending descending branch branch in in this this tree tree if if it it is is not not well-founded, well-founded, recursive 1, given recursive in in the the Suslin Suslin quantifier quantifier El E1 with with values values 00 and and 1, given by by (Et) El E1 (J) (f) = 00 +-+ ~-~ 3g 3g \Ix Vx f(g(x» f(-~(x)) = O. O. Specifically, 0, J.L Specifically, when when E E1l (J) (f) = = 0, #1l is is the the left-most left-most descending descending branch branch in in this this tree, tree, (J) = ned recursively g(x) to with with the the successive successive values values of of J.Ll #l(f) = 9 g defi defined recursively by by taking taking g(x) to be be the the least least y y such such that that there there is is an an infinite infinite descending descending branch branch in in the the f-tree f-tree extending extending g(xt(y) es the -~(x)^(y).. Thus Thus J.Ll #t satisfi satisfies the axiom axiom O. \Ix l (J)(X» = Vx f(g(x» f(-g(x)) = = 00 -+ --~ \Ix Vx f(J.L f(ptl(f)(x)) - O. Of terms of Of course course then then E E1l is is definable definable in in terms of J.Ll #1.. This This axiom axiom is is preserved preserved under under the the N-interpretation, N-interpretation, and and in in the the presence presence of of the the axiom axiom (J.L) (#) is is equivalent equivalent to to aa quantifier-free quantifier-free statement,which the D-interpretation. statement,which is is further further preserved preserved under under the D-interpretation. So So this this leads leads us us to to consider the systems the axioms axioms for Then . consider the systems of of 8.2.1 8.2.1 augmented augmented by by the for both both J.L # and and J.L #t. Then l J - CA, we we obtain obtain conservation conservation results results for for these these over over second-order second-order systems systems involving involving II II]-CA, -DC . In EJ -AC Z~-A C and and EJ E~-DC. In addition, addition, we we have have to to consider consider the the following. following. 8.4.2. - CA) 8.4.2. Transfinitely Transfinitely iterated iterated (IIJ (II]-CA) The - CA) < ", is The explanation explanation of of the the systems systems (IIJ-CA)", (H]-CA)a and and (IIJ (H]-CA) 385 385 Gode/'s GSdel 's Functional Interpretation 2. Res-Ill Res-II]-CA C_ Res Res-A~-CA c_ Res-Ei Res-Z~-AC 2. -CA � -Ll�-CA � -A C P)/ C_ ResRes-~ ~ + + (#) + (P,l (#I)) + + ((QF-AC) (p,) + QF_AC) 1188 � In analogy analogy to to 8.3.2, 8.3.2, using using the the ND-interpretation ND-interpretation of of PAW PA w + + (p,) (#) + + (P,l (#i)) + + (( QF-A QF-A C) C) In in obtain: in T T+ + (p,) (#) + + (P,l (/.I)) we we can can also also obtain: 8.4.4. PAW ) + (( QF -A C) is 8.4.4. Theorem. Theorem. PAw + + (p,) (#)++ (P,l (#2)+ QF-AC) is aa conservative conservative extension extension of of (II/ -CA) < g O for (H]-CA)<~o for m II~ statements. statements. 88.4.5. .4.5. Corollary. Corollary. statements. statements. Ei -DC is -CA) Similarly -AC over s-II/ - CA . Similarly we we obtain obtain a a conservation conservation result result for for Res-IIi Res-H~-AC over Re Res-H{-CA. 1970]. For Corollary 8.4.5 8.4.5 is is again again aa result result first first obtained obtained by by Friedman Friedman [1970]. For more more details details of of Corollary the arguments arguments using using the the approach approach sketched sketched here here through through functional functional interpretations, interpretations, the see see Feferman Feferman [1977,section [1977,section 8]. 8]. [ 88.4.6. .4.6. Autonomously - CA Autonomously iterated iterated II/ II]-CA Just iterated ramified Just as as for for the the case case of of autonomously autonomously iterated ramified analysis analysis or or IIfII~ CA, we we may may explain what what are are the the autonomous autonomous ordinals ordinals for for iteration iteration of of (II/ (II{-CA). A much much larger larger explain - CA) . A recursive recursive ordinal ordinal notation notation system system is is needed needed to to determine determine these, these, using using the the so-called so-called subsequently simplified simplified through Bachmann ordinal functions Bachmann hierarchies hierarchies of of ordinal functions ((subsequently through work work of of Feferman, Aczel and Feferman, Aczel and Buchholz Buchholz - - ccf. f . Schutte Schiitte [1977,Chapter [1977,Chapter IX]). IX]). We We shall shall not not try try to to describe up the describe these these here, here, except except to to take take up the first first ordinal ordinal of of proof-theoretical proof-theoretical interest interest from that hierarchy, so-called Howard Howard ordinal, below. We from that hierarchy, the the so-called ordinal, in in section section 9.7 9.7 below. We content content ourselves analogue of ourselves instead instead with with the the statement statement of of an an analogue of Theorem Theorem 8.3.8 8.3.8 in in the the following following form. form. Ei -DC + Z~-DC + (Bar-Rule) (Bar-Rule) is is aa conservative conservative extension extension of of auautonomously / -CA for tonomously iterated iterated II 1111-CA for II� II 1 statements. statements. 88.4.7. .4.7. Theorem. Theorem. This This can can be be proved proved by by an an extension extension of of the the functional functional interpretation interpretation meth methods ods indicated indicated above, above, thus thus also also giving giving conservation conservation for for the the finite finite type type system system PAW QF-AC) + - CA. PA w + + (p,) (#)++ (P,l (#I)) + + ((QF-AC) + (Bar-Rule) (Bar-Rule) over over autonomously autonomously iterated iterated II/ II~-CA. Theorem Theorem 8.4.7 8.4.7 has has also also been been proved proved by by Herbrand-Gentzen Herbrand-Gentzen methods methods with with the the func functionals tionals p, # and and P,l #1 in in Feferman Feferman and and Jager J~iger [1983]. [1983]. 8.4.8. 8.4.8. Further F u r t h e r results results and and methodological methodological discussion discussion Friedman Friedman [1970] [1970] obtained obtained further further analogues analogues to to Corollaries Corollaries 8.3.5 8.3.5 and and 8.4.5, 8.4.5, of of the the following following form: form: 118Actually, 8 Actually, it is known known from from the KondO-Addison KondS-Addison theorem theorem that we can strengthen two of the inclusions in 11 by Llk-CA -AC = EJ -DC. inclusions A~-CA = EJ ,F,~-AC ,F,~-DC. = = 386 386 J. A vigad and S. S. Feterman Feferman Fornn � >_ 2, i 8.4.9. For 2, E�+rDC is -CA) <<6o 8.4.9. Theorem. Theorem. Y'~+I-DC is aa conservative conservative extension extension of of (II� (H~-CA) oo for for II! H~ statements. statements. This This can can be be improved improved to to conservation conservation results results for for finite finite type type extensions extensions of of E�+rDC Z~+I-DC by by means means of of additional additional axioms axioms for for suitable suitable Skolem Skolem functionals. functionals. However, However, those those functionals functionals do do not not occur occur naturally naturally in in practice, practice, unlike unlike the the non-constructive non-constructive minimum minimum operator and operator, so cial operator and the the Kleene Kleene basis basis operator, so such such results results would would be be somewhat somewhat artifi artificial to state. Friedman's own own proof proof of of this this theorem theorem was was by by model-theoretic methods, to state. Friedman's model-theoretic methods, and and later later Feferman Feferman and and Sieg Sieg [1981] [1981] gave gave aa proof proof by by Herbrand-Gentzen Herbrand-Gentzen methods. methods. Subsequently Subsequently Feferman Feferman and and Jager Jiiger [1983], [1983], using using the the same same methods, methods, obtained obtained the the related related analogues analogues of of Theorem Theorem 8.4.7: 8.4.7: For + For nn � > 2, 2, E�+rDC E~+I-DCI + (Bar-Rule) (Bar-Rule) is is aa conservative conservative extension extension of -CA for of autonomously autonomously iterated iterated II� II~-CA .for II! II~ statements. statements. 8.4.10. 8.4.10. Theorem. Theorem. 9. T The of 9. h e interpretation interpretation o f ttheories h e o r i e s of o f ordinals ordinals 9.1. 9.1. A A classical classical theory t h e o r y of of countable c o u n t a b l e tree t r e e ordinals ordinals We methods of We now now extend extend the the methods of the the previous previous section section to to aa finite finite type type theory theory OR! OR"i of of order to countable the J-L countable (tree) (tree) ordinals ordinals including including the # operator, operator, in in order to obtain obtain aa conservation conservation result over over a a classical classical theory theory IDl IDI of of one one arithmetical inductive definition. definition. 119 This result arithmetical inductive 9 This work work is is compared compared with with that that of of Howard Howard [1972] [1972] on on analogous analogous constructive constructive theories; theories; it it is is an an interesting interesting open open question question how how these these results results may may be be combined. combined. The The work work described described here here is is based based on on an an unpublished unpublished paper, paper, Feferman Feferman [1968a] [1968a].. ND-interpretation is applied, is The The system system OR! OR~,, to to which which the the ND-interpretation is to to be be applied, is aa variant variant the of of that that used used in in Feferman Feferman [1968a]. [1968a]. It It extends extends PAW PA w + + (J-L) (#) as as follows. follows. We We expand expand the type ground type, type, denoted denoted O. type structure structure by by an an additional additional ground ~. Then Then types types a, r, % .. ... . are are a -+ generated from ground types generated from the the ground types 00 and and 0 ~ by by closing closing under under ((a --+ r) T) as as before. before. We We have of each have infinitely infinitely many many variables variables of each type; type; we we use use the the letters letters a ~,/~, 7 , .. ... ., , �, ~, 7], 77,(r for for , {3, "I, variables O. These variables of of the the new new type type ~. These are are informally informally understood understood to to range range over over countable countable tree tree ordinals, ordinals, which which are are closed closed under under formation formation of of suprema suprema in in the the sense sense that that if if f f is is immediate of of type type 00 -+ ~ 0 ~ then then Sup(J) Sup(f) E E 0 ~ and and Sup(J) Sup(f) represents represents the the tree tree whose whose immediate number. Formally, subtrees subtrees are are given given by by the the sequence sequence of of values values f(x) f (x) for for x x a a natural natural number. Formally, those of PAww + the the constants constants of of OR! OR~ besides besides those of PA + (J-L (#)) are are as as follows: follows: 11.. OOn is a a constant constant of of type type 0 n is 2. Sup -+ 0) 2. Sup is is a a constant constant of of type type (0 (0-~ ~) -+ 0 l 3. SUp 3. Sup -1 is is a a constant constant of of type type 0 ~ -+ ~ (0 (0 -+ --+ 0) ~) 4. For a, Rn 4. For each each a, Rn,~ is a a constant constant of of type type (0, (~, (0 (0 -+ -~ a a)) -+ --+ a a),) , a, 0 ~ -+ -~ a. a. ,,, is 19Recent 19Recent work work of Burr and Hartung [n.d.] [n.d.] and Burr [1997] [1997] extends extends these these results results with an interpretation of KPw (essentially (essentially a set-theoretic set-theoretic analogue analogue of IDJ ID1 ) in a theory of primitive recursive set functionals functionals of finite finite type, instead instead of the ordinal ordinal functionals functionals of ORw OR~ •. Godel's GSdel 's Functional Interpretation 387 387 We shall omit subscript '' 17 We shall omit the the subscript a '' from from the the ordinal ordinal recursor recursor whenever whenever there there is is no no ambiguity. In In the the following following we we shall shall write write ax ax for for (Sup-l (Sup-l(a))(x) for x x of of type type 0; for (a)) (x) for 0; for ambiguity. a non-zero, non-zero, this this represents represents the the immediate immediate subtree subtree at at position Just as as we we do do not not position x. x. Just a assume assume extensionality extensionality for for functions, functions, we we do do not not assume assume extensionality extensionality for for ordinals, ordinals, i.e. {3x) that {3. Nevertheless i.e. it it does does not not follow follow from from "Ix Vx (ax (ax = =/3~) that a a = - ft. Nevertheless we we shall shall assume assume there one-one correspondence non-zero ordinals there is is aa one-one correspondence between between non-zero ordinals and and functions functions of of which which - l will they are are the the suprema, suprema, so so that that S Sup indeed be be the the inverse inverse of of the the Sup Sup operation. operation. they will indeed Up -1 This This is is not not necessary, necessary, but but simplifies simplifies some some points. points. The The logic logic taken taken for for OR"j OR"/ is is full full classical classical quantificational quantificational logic logic in in its its finite finite type type consist of: language. language. The The axioms axioms of of OR"j OR"/consist of: 1. AW~ + (p,) with with the the induction 1. The The axioms axioms of of P PA + (#) induction scheme scheme extended extended to to all all formulas formulas of the the language language of l p(J)) = I , for I of type (0 -+ 0) 2. Sup(J) 2. Sup(f) =1= 7~ On 0a 1\ AS Sup (Sup(f)) = f, for f of type (0 --+ ~t) Up --1 (S U l a)) = 3. a a =1= # On 0n -+ --+ Sup(Sup Sup(Sup -- l ((a)) = a a 3. 4. 4 . (On)x = = On 5. (a) Rn(J, 5. (a) R a ( f , aa, , On) 0a) = = aa (b) a 17, where (b) a =1= 7(=On On -+ -+ Rn(J, Ra(f, a, a) c~) -= I(a, f ( a , Ax.Rn(J, Ax.Rn(f, a, ax)) c~)) for for each each type type a, where the the 17, and variable variable a a is is of of type type a, and the the variable variable I f is is of of type type 0, ft, (0 (0 -+ --+ 17 a)) -+ --+ 17 a ~(0a) 1\ A Va Voz (a =1= r On 0n 1\ A "Ix Vx rp(ax) ~(ax) -+ --+ rp ~(a)) ~ Va rp(a) ~(a) 6. rp(On) ( a)) -+ 7. 7. (( QF-A QF-A C) C) may be The er-free subsystem The quantifi quantifier-free subsystem of of OR"j OR"/may be axiomatized axiomatized as as follows. follows. I'. 1'. T T + + (p,) (#),, with with induction induction rule rule extended extended to to all all quantifierquantifier- free free formulas formulas of of OR'l OR~ 2'.-5'.. The The same same as as 2-5 2-5 in in OR"j OR"/ 2'.-5' 6' induction on ordinals, i.e. x rp(ax) 6'.. The The rule rule of of induction on ordinals, i.e. from from rp(On) ~(0n) and and (a (or =1= # On 0n 1\ A "I Vx ~o(a~) -+ --+ ~(c~)) infer infer rp(a) ~(c~) for for all all quantifier-free quantifier-free rp. ~o. rp(a)) (This (This last last is is to to be be expressed expressed in in quantifier-free quantifier-free form form using using the the p, It operator.) operator.) We We denote denote this this system system by by Tn T9 + + (p,) (It).. Just Just as as for for Theorem Theorem 8.3.1 8.3.1 we we readily readily obtain: obtain: 9.1.1. 9.1.1. Theorem. Theorem. is ND-interpreted OR"j OR"/is ND-interpreted in in Tn T~ + + (p,) (It).. 9.2. classical systems 9.2. The T h e classical s y s t e m s IDJ IDI of of one one aarithmetical r i t h m e t i c a l inductive i n d u c t i v e definition definition We We remind remind the the reader reader briefly briefly of of these these relatively relatively familiar familiar systems. systems. Here Here B(P+, 0(P +, x) x) denotes denotes a a formula formula in in the the language language of of arithmetic arithmetic with with one one additional additional predicate predicate or or set set B. This symbol symbol P P that that has has only only positive positive occurrences occurrences in in 0. This determines determines a a monotonic monotonic x) } , which operator operator from from sets sets P P to to sets sets {x { x : : B(P, 0(P, x)}, which thus thus has has aa least least fixed fixed point. point. The The theory theory IDJ IDI (B) (0) associated associated with with B0 is is given given by by the the following following axioms axioms expressing expressing that that P P is is such rst-order language: such a a least least fixed fixed point, point, to to the the extent extent possible possible within within the the given given fi first-order language: 1. The 1. The axioms axioms of of PA PA with with induction induction expanded expanded to to include include all all formulas formulas containing containing the the symbol symbol P. P. 2. 2. B(P, O(P,x) x ) --+ ~ P(x) P(x) 3. x (B( / P, 3. "I Vx (O(r x) -+ --+ 'IjrJ (x)) -+ --+ "I Vxx (P(x) (P(x) -+ 'IjrJ (x) ) , for for each each formula formula V'. r 'Ij J x) J. J. A A vigad vigad and and S. S. Feferman Feyerman 388 388 Here O(r denotes the the result result of of substituting substituting any any occurrence occurrence of of the the form form P( P(t)t) Here 8( 1j;/P, x) denotes in . The in 80 by by 1j;(t/x) r The logic logic of of IDJ ID1 (8) (0) is, is, of of course, course, classical. classical. When When referring referring to to any any system this way, system described described in in this way, we we simply simply write write IDJ ID1.• We will use corresponding theories We will use IDj ID~ to to denote denote the the corresponding theories based based on on intuitionistic intuitionistic logic. Here, Here, however, however, we we need need to to specify specify how how the the positivity positivity requirement requirement on on 8O is is to to logic. be interpreted. interpreted. We We will will say say that that 8O is is weakly weakly positive positive if if it it is is positive positive in in the the classical classical be or strictly) sense, sense, and and strongly strongly ((or strictly) positive positive if if there there are are no no occurrences occurrences of of P P in in the the antecedent implication, where antecedent of of an an implication, where the the basic basic logical logical connectives connectives are are taken taken to to be be those those of of section section 2.1. 2.1. An An even even more more restrictive restrictive requirement requirement on on 8O is is that that it it be be an an accessibility 9.8 for accessibility inductive inductive definition, definition, as as described described in in section section 9.6; 9.6; cf. cf. section section 9.8 for a a discussion. discussion. IDJ 9.3. 9.3. Translation T r a n s l a t i o n of of IDI into into OR1 OR~ The (8) into The obvious obvious way way to to carry carry out out the the translation translation of of IDJ IDI(O) into OR1 OR"i is is to to define define the the approximations approximations to to the the least least fixed fixed point point from from below. below. For For this this purpose, purpose, we we need need form of of ordinal ordinal recursion recursion which which defines defines aa function function at at an an ordinal ordinal a a in in terms terms of of its its aa form (3. The ordinals is values smaller ordinals ordinals ~. values at at all all smaller The appropriate appropriate less-than less-than relation relation for for tree tree ordinals is introduced as finite sequences introduced as follows. follows. We We use use the the letter l e t t e r ''s' s ' to to range range over over numbers numbers of of finite sequences of numbers. The of natural natural numbers. The number number 00 is is chosen chosen to to code the empty empty sequence, sequence, and and the the code the A extension (X ) . extension of of a a sequence sequence ss by by one one new new term term x x is is denoted denoted Ss^(x). For ordinal c~, a, the s" is For an an ordinal the predecessor predecessor as c~, of of a c~ "down "down along along s" is defined defined recursively recursively by: by: 1. ao 1. OL0 = -- a OL 2. as 2. ~<~> = (as)x. (~,)~. '(x) = Then Then we we define define i= A0 Z1\=(3a=~ as) 3. (3 Z < 0 [] a a - = Sup(f) Sup(f). . Given a, the restriction of Given a a function function f f of of type type n ~ -+ --+ a, the restriction of f f to to a a for for a a i= #- On 0a is is AS, x. both as definable as definable as As, x. f(as' f(as~(~)), which we we also also write write both as AS As i= r O. 0. f(as) f ( a s ) and and as as (x» ) , which A(3 ((3) . Then A/~ < < a. c~. f f(/3). Then one one can can derive derive by by use use of of our our recursor recursor Rn Rn the the following following more more a, given general general form form of of recursion recursion with with values values in in any any type type a, given any any aa of of type type a a and and G G of of type n, type ~, (0 (0 -+ --+ a a)) -+ --+ a: a: «(<-Rec,) -Recf}) F(On) F(0n) = = a, a, and and for for a c~ i= r OOn, F(~) = = G(a, G(c~, A(3 A/~ < < a. c~. F((3)). F(/~)). n , F(a) GSdel 's Functional Functional Interpretation Interpretation Godel's 389 Correspondingly we we can can derive derive the the following following more more general general form form of of induction induction on on n: ~: Correspondingly « -Indn) f(o,-,) w (a. (o, =/: -r O0,-, vgt < < a. cp((3) cp(O n ) 1\A Va. n 1\A V(3 --+ --t w cp(a.) cp(a.)) ----+t Va. 9.3.3. Lemma. Lemma. Given arithmetical arithmetical 6(P, 8(P, x) x),, we we can can define define aa function function F F of of type type 9.3.3. Given n, x) = x) for f2, o0 --t ~ 00 in in OR1 OR~ satisfying: satisfying: F(a., F(a,x) = 00 ++ +-~ 6( 8({y 3/~ < a. ce F((3, F(~, y) y) = = O}, 0},x) for all all {y II 3(3 a.0~.. The proof proof of of this this depends depends essentially essentially on on the the use use of of J.L # to to eliminate eliminate all all the the quantifiers quantifiers The in 68 and and to to eliminate eliminate the the quantifier quantifier ''3/~ < a. ~',' , which which is is just just aa quantifier quantifier over over non-zero non-zero in 3(3 < sequence numbers, numbers, so so that that we we can can then then apply apply the the principle principle «(<-Reck) above. sequence -Recn) above. Now suppose suppose that that 6(P, 8(P, x) x) has has just just positive positive occurrences occurrences of of P. P. Using Using the the function function Now F from from the the preceding preceding lemma, lemma, defi define F ne so that that so P(x) ++ ~ 3a. 3a P(a., P(a, x), x), where where P(a., P(a, x) x) ++ ~ F(a., F(a, x) x ) == 0, O, P(x) (6) (6) P(a., x) ++ 6( {y II 3(3 P(a,x) ++ 0({y 3~ < a. a P((3, P(~,y)},x), for all all a.. a. y)}, x), for (7) (7) 9.3.4. Theorem. Theorem. 9.3.4. IDI (6) (8) in ORal.. IDJ in OR1 9.3.5. Proof. Proof. 9.3.5. The The predicate predicate P P thus thus defined defined provably provably satisfies satisfies the the axioms axioms of of First, to show show 6(P, O(P, x) x) --t -~ P(x) P(x),, we we must must show show First, to 6( {y II 3(3 x). y)}, x) --t 8({y 3/~ P((3, P(l~,y)},x) --+ 3a. 3a P(a., P(a,x). Using the the positivity of P P in in 68,, we bring the hypothesis of this implication to Using positivity of we may may bring the hypothesis of this implication to prenex normal prenex normal form form QlZl...Qnzn 3/~1 ...3/~m 80(X, Zl,..., Zn, ~1,..., t3m), where Qi is V or and 80 60 is is quantifier-free. quantifier-free. Then Then by by successive application of of parts where Qi is V or 3 3 and successive application parts 22 and and 3 of we obtain of Lemma Lemma 9.3.1 we obtain 3ce QlZl... Qnzn 3/~1 < oL ... 3/~rn < OL80(X, Z l , . . . , Zn, ~ 1 , . . . , ~m), which is is equivalent equivalent back back to which to 3a. 0({y 6( {y II 3/3 3(3 << aa. P(/~, 3a P((3, y)}, y)}, x). x). Hence (6) and P(a, x), i.e. P(x). Hence by by (6) and (7) (7) we we may may conclude conclude 3a P(x) . 3a. P(a., x) , i.e. Next, Next, to to show show Vx x) --+ --t Vx Vx (P(x) (P(x) --+ --t r'l/J (x)), Vx (8(r (6(1jJf P, x) --t r'l/J (x)) --+ we simply simply prove prove by by use use of the principle principle (<-Inda) « -Indn) that, that, under under the the hypothesis, hypothesis, we we have have we of the [] 0 V~ This proceeds proceeds as as usual. usual. Va. Vx Vx (P(a,x) (P(a., x) -~ --t r'l/J (x)) . This 390 390 J. A vigad and S. S. FeJerman Feferman 9.4. 9.4. Models M o d e l s of of Tn Ta + + (J-t) (#) We We can can build build models models of of this this system system which which parallel parallel those those of of T T in in section section 4.1. 4.1. First First of all, aa full, of all, full, extensional, extensional, set-theoretical set-theoretical model model (Ma) (M~) is is definable definable as as follows: follows: 11.. Mo M 0 ==NN 2. Mn 2. Mn is is the the smallest smallest set set X X which which contains contains 00 and and which which is is such such that that whenever whenever If " : N X. Y --* ~ X X,, then then (1, (1, j) f) E E X. 3. -H = 3. Ma M~_~ = {{Jf i rI 1 maps maps Ma M~ into into M. My,. }}.. Then Then we we define define - l ((l, j)) 4. On 0, Sup (f) = 4. On = = 0, Sup(f) = (1, (1, j) f) for for I f": N g --* --+ Mn Mn,, and and Sup Sup-l((1, f)) = = I f.. = I(x) . With each a E Mn is associated SO So for for a c~ = - Sup(f) Sup(f) we we have have a c~x = f(x). With each ~ E Mn is associated an an ordinal ordinal lal, I~I, in in the the usual usual set-theoretical set-theoretical sense, sense, by by 5. 101 5. IoI = = 00 and and ISup(f) ]Sup(f) II = = sup sup {11(x) {If(x)I1 + + 11 I1x x E E N} Y}.. al whenever Then 0, and definable by ordinal ax l < Then lIcz~I < lIc~I whenever a cz i= r 0, and thus thus Rn Rn is is definable by classical classical ordinal J-t) . recursion so recursion so as as to to satisfy satisfy axiom axiom 55 of of Tn Ta + + ((#). For intensional recursion-theoretic recursion-theoretic model model analogous RO we For an an intensional analogous to to H HRO we make make use use of of recursion which is recursion in in the the J-t # operator operator as as a a type type 22 functional, functional, which is a a special special case case of of the the Kleene Kleene [1959b] development development of of recursion recursion in in finite finite type type objects. objects. In In the the following following we we shall shall use use [1959b] f , gg,, hh,, . .. ,. . to to range range over over N N and and write write I(x f ( x ll,,. .. .. ,. x, nxn) ) for for Kleene's Kleene's {J}(J-t, { f } ( # , XXll,,. .. .. ,.x, ~xn) ) I, whenever whenever it it is is defined. defined. Now Now the the model model (Na) (N~) is is defined defined by by 11.. No N 0 ==NN 2. Nn 2. Nn = - the the smallest smallest set set X X � C_ N N such such that that 00 E E X X and and whenever whenever j f" : N N --* ~ X X then the pairing pairing function then (1, (1, I) f) E EX X (where (where the function (x, (x, y) y) is is assumed assumed to to be be from from Nl N2 to to N N - {O} { 0 } )). . 3. Na-t.,. = 3. = {{I f l wI ' x.r(f, a, a, g(x rr(f, (f, a, a, 0) a, and a, (1, O) = = a, and r(f, r(f, a, (1, gg)) _~ I(g, f(g, >')~x.r(f, g(x))). ))) . )) � Then a, a) Then induction induction on on lal ]czI shows shows that that r(f, r(f, a, ~) is is defined defined on on the the objects objects j, f, a a of of appro appropriate type, for priate type, for all all a c~ E E Nn Nn.. Finally, hereditarily extensional Finally, one one can can define define an an analogue analogue of of the the hereditarily extensional recursion recursionmodel by defining the theoretic H theoretic H RRE E model by first first defining the notion notion a ~ =n =n {3 /~ inductively, inductively, and and then then defining =a-t.,. defining =~_,~ in in terms terms of of =a =~ and and =.,. =~ as as for for H HRRE, E , with with the the objects objects at at each each type type being those being those which which preserve preserve the the defined defined equality equality relations relations at at each each type. type. Godel's Functional Interpretation GSdel's Functional 9.5. Interpreting 9.5. Interpreting OR1 OR~ in in 391 IDl IDI 0) , where is the OR1 in in ID1 We can can interpret interpret OR~ IDI ((0), where 0 O is the set set of of Church-Kleene Church-Kleene con conWe structive notations, by structive ordinal ordinal notations, by first first applying applying the the ND-interpretation ND-interpretation of of OR! OR~ in in its its quantifier-free RO model quantifier-free subtheory subtheory Tn Ta + + (J-L) (#) and and then then formalizing formalizing the the H HRO model of of the the latter latter in in ID1 IDI.• For For this, this, though, though, instead instead of of making making use use of of Kleene's Kleene's definition definition of of recursion recursion in in J-L # as special case recursion in finite type objects, one as aa special case of of recursion in finite type objects, one takes takes aa more more concrete concrete version version which possible from which is is possible from the the fact fact that that the the partial-recursive-in-J-L partial-recursive-in-# functions functions are are exactly exactly the II~ functions; those those can can be be enumerated enumerated by by uniformizing uniformizing the the II� II~ relations relations the II� partial partial functions; in aa standard standard enumeration. enumeration. Since Since 0 O is is aa complete complete II� 1-I~ predicate, predicate, this this generalized generalized in 0) and, recursion recursion theory theory can can be be formalized formalized in in IDl IDI ((0) and, further, further, the the definitions definitions of of the the N~ can can be be given given for for each each (7 a in in that that theory. theory. The The resulting resulting interpretation interpretation preserves preserves N" arithmetical statements, statements, and and is is such such that that with with each each closed term tt of of type type n ~ is is arithmetical closed term which ID 1 ( 0) proves associated number nntt for associated a a number for which 101(0) proves n ntt EE O O,, and and such such that that It ItlI � <_ In Intl. It t I . It follows follows that that the the provable provable ordinals ordinals of of OR1 OR~ are are the the same same as as those those of of ID1 ID1 (O) (O),, and and that that is ordinal. The described in is the the same same as as its its proof-theoretic proof-theoretic ordinal. The latter latter will will be be described in section section 9.8. 9.6. Functional Functional interpretation interpretation of of a a constructive c o n s t r u c t i v e theory t h e o r y of of countable countable tree tree 9.6. ordinals ordinals Howard introduced aa fifirst-order rst-order theory Howard [1972] [1972] introduced theory U U that that he he called called a a system system of of abstract abstract constructive constructive ordinals, ordinals, with with just just two two sorts sorts of of variables, variables, numbers numbers and and ordinals. ordinals. In QF-AC) In place place of of ((QF-A C) this this made made use use of of the the principle principle of of w-upper w-upper bounds bounds (Lemma (Lemma 9.3.1 above, nite type above, clause clause 3). Howard's Howard's system system can can be be translated translated directly directly into into a a fi finite type theory theory UW that we U ~ which which is is the the same same as as OR1 OR~,, except except that we omit omit the the J-L # operator, operator, and and base base it it on on intuitionistic intuitionistic logic logic in in place place of of classical classical logic. logic. Then Then UW U ~ has has aa D-interpretation D-interpretation in in the the system system Tn Ta of of section section 9.1, again again without without the the J-L # operator; operator; that that system system is is just just another another 's quantifier-free version version of of Howard Howard's quantifier-free finite-type finite-type theory theory V V of of abstract abstract constructive constructive (tree) op. cit. (tree) ordinals ordinals op. cit. Howard Howard gave gave a a term term model model of of V V which which is is an an extension extension of of Tait using infi nitely long long terms Tait's's term term model model of of T T using infinitely terms (cf. (cf. section section 4.4), 4.4), and and used used that that to to obtain obtain an an upper upper bound bound to to the the proof-theoretic proof-theoretic ordinal ordinal of of U U as as the the least least ordinal ordinal greater greater than n. The It I for than Itl for tt aa normal normal term term of of type type ~. The so-called so-called (Bachmann-}Howard (Bachmann-)Howard ordinal ordinal thus obtained will described in section. Howard thus obtained will be be described in the the next next section. Howard also also showed showed how how to to (B) for U. This translate translate intuitionistic intuitionistic ID; ID~ (0) for B0 of of a a certain certain special special form form into into his his system system U. This special special form form includes includes "accessibility" "accessibility" inductive inductive definitions, definitions, where where B(P, 0(P, x) x) is is of of the the form form nition -+ P(y)) Vy (y (y < J. A vigad and S. S. Feferman Fe?erman 392 392 9.7. The T h e Howard H o w a r d ordinal ordinal 9.7. The original description description of ordinal made The original of this this ordinal made use use of of an an extension extension of of the the Veblen Veblen hierarchy in aa form hierarchy of of ordinal ordinal functions functions in form due due to to Bachmann Bachmann [1950], [1950], which which gives gives sense sense suitable uncountable a. For to to CPa ~ for for suitable uncountable c~. For the the specific specific purposes purposes of of 9.6, 9.6, it it is is sufficient sufficient to to to the tell tell how how this this is is to to be be done done for for a c~ up up to the first first epsilon epsilon number number greater greater than than the the least least 0 Roughly speaking speaking one uncountable uncountable ordinal ordinal WI Wl,, namely namely C~1+1.2~ one first first assigns assigns to to each each WI + 1 ' 2 Roughly a fundamental term limit ordinal ordinal in in aa notation notation system ordinals up term a c~ for for aa limit system for for ordinals up to to Cc~WI1+1 a fundamental +1 sequence sequence of of order order type type :=:; _ WI wl in in a a reasonably reasonably canonical canonical way. way. If If the the cofinality cofinality type type of of terms of this sequence sequence is this is countable, countable, one one proceeds proceeds to to define define CPa 7~ in in terms of the the simultaneous simultaneous fixed fixed points points of of the the CP(3 ~Z for for the the terms terms (3 fl = - av ~ in in that that fundamental fundamental sequence sequence as as with with the the Veblen diagonalizes, Veblen hierarchy; hierarchy; if if its its fundamental fundamental sequence sequence is is of of length length WI Wl then then one one diagonalizes, (v) to i.e. i.e. takes takes CPa 7~(v) to be be CPa" 7 ~ ( (O) 0 ) . . The The Howard Howard ordinal ordinal is is then then defined defined to to be be CPa(O) 7~(0) for for 21 21 I . a l O/ = "-- c gW l [ l " W+ 9.S. Discussion 9.8. Discussion From a a foundational foundational standpoint, standpoint, it it is is desirable desirable to to have have aa reduction reduction of of classical classical From to to a a constructive constructive theory. theory. Although Although a a slight slight variant variant of of the the double-negation double-negation interpretation interpretation serves serves to to reduce reduce IDI to to its its formally formally intuitionistic intuitionistic counterpart counterpart ID~ (cf. (cf. Buchholz Buchholz et et al. al. [1981,p. [1981,p. 56]), 56]), the the latter latter theory theory is is not not evidently evidently constructive, constructive, in in the the sense sense that that there there is is no no direct direct constructive constructive justification justification of of axioms axioms 22 and and 33 of of section section 9.2 9.2 for for f)8 in in which which the the predicate predicate symbol symbol P P occurs occurs only only in in aa weakly weakly positive positive way. way. (An (An indirect indirect justification justification is is provided provided by by the the intuitionistic intuitionistic theory theory of of species species - i.e. i.e. the the formal formal counterpart counterpart of of second-order second-order analysis analysis - whose whose constructivity constructivity is, is, however, however, a a matter dispute; cf. discussion in matter of of dispute; cf. the the discussion in Feferman Feferman [1982b,pp. [1982b,pp. 77-78].) 77-78].) What What we we really really desire desire is is a a reduction reduction of of IDI to to an an intuitionistic intuitionistic theory theory of of accessibility accessibility inductive inductive definitions, whose definitions, whose very very form form provides provides a a clear clear picture picture of of how how the the corresponding corresponding sets sets are are generated generated from from the the "bottom "bottom up" up";; or, or, alternatively, alternatively, aa reduction reduction of of IDI to to the the constructive constructive theory theory U UW~ without without the the J.L # operator. operator. In In fact, fact, the the first first type type of of reduction reduction has has been been given given by by work work of of Pohlers Pohlers and and Buch Buchholz in holz in aa series series of of steps steps beginning beginning in in the the mid-1970s mid-1970s using using interesting interesting (prima-facie) (prima-facie) uncountably uncountably infinitary infinitary extensions extensions of of Gentzen-Schiitte Gentzen-Schiitte style style proof proof theory; theory; cf. cf. the the reports al. [1981] reports in in Buchholz Buchholz et et al. [1981].. Indeed, Indeed, they they have have succeeded succeeded in in determining determining the the proof-theoretic proof-theoretic ordinals ordinals of of classical classical theories theories of of iterated iterated inductive inductive definitions definitions ID,~ and and in reducing them in reducing them to to corresponding corresponding intuitionistic intuitionistic theories theories of of iterated iterated accessibility accessibility inductive be the the same. inductive definitions, definitions, thus thus showing showing the the proof-theoretic proof-theoretic ordinals ordinals to to be same. In In particular, in the case of their work work yields the Howard particular, in the special special case of IDa, the the result result of of their yields the Howard ordinal whether taken in its ordinal as as the the proof-theoretic proof-theoretic ordinal ordinal of of the the system system whether taken in its classical classical or or restricted restricted intuitionistic intuitionistic form. form. IDJ IDI IDJ IDj IDJ IDJ IDa IDJ , 20Here, ordinals are treated set-theoretically. 2~ set-theoretically. 21The Bachmann since been superseded Feferman-Aczel-Buchholzapproach Bachmann approach has since superseded by the Feferman-Aczel-Buchholz 21The described in Schiitte Schiitte [1977,Chapter [1977,Chapter IX]: IX]: the latter is simpler simpler in not requiring a prior assignment assignment of of fundamental fundamental sequences. sequences. 393 393 Godel's Ghdel's Functional Functional Interpretation We do do not, not, however, however, know know how how to to achieve achieve this this same same result result via via the the method method of of We functional functional interpretation, interpretation, nor nor do do we we have have a a direct direct reduction reduction of of IDI1 to to U U W~.. In In addition, interpretation to addition, no no one one has has yet yet extended extended the the method method of of functional functional interpretation to iterated iterated IDa, either either classical classical or or intuitionistic, intuitionistic, in in an an informative informative way way specific to those those systems. systems. 22 22 specific to It would would be be of of interest interest to to know know whether whether there there is is some some fundamental fundamental methodological methodological It obstacle obstacle for for doing doing so, so, or or if if it it is is simply simply for for lack lack of of aa new new idea idea - - or or simply simply lack lack of of trying trying hard hard enough. enough. ID IDo, 1 0 . Interpretations 10. Interpretations based based on on polymorphism pol ym orphis m 10.1 10.1.. Transfinite T r a n s f i n i t e types t y p e s and a n d polymorphism polymorphism In In this this section section we we address address strengthenings strengthenings of of T T that that provide provide mechanisms mechanisms for for defining "transfi nite" types. (n) from defining "transfinite" types. For For example, example, recall recall the the types types (n) from section section 2.2, defined 1) = defined by by (0) = - 0 and and (n (n + + 1) - (n) (n) -+ --+ O. 0. One One might might want want to to define define a a function function f f that, for that, for each each natural natural number number n, n, returns returns an an object object of of type type (n) (n).. Such Such a a function function f f is is an an element element of of the the product product type type II n) , I]~0 nEo ((n), and capabilities of and clearly clearly goes goes beyond beyond the the finite-type finite-type capabilities of T. T. The The function function f f is is also also "polymorphic" in that, for n, the n ) depends depends on "polymorphic" in the the sense sense that, for each each n, the type type of of f f ((n) on its its argument. argument. A polymorphism arises A down-to-earth down-to-earth example example of of polymorphism arises in in the the context context of of writing writing aa sorting algorithm. writing separate sorting algorithm. Instead Instead of of writing separate routines routines that that sort sort lists lists of of integers, integers, real real numbers, on, one numbers, strings, strings, and and so so on, one would would prefer prefer to to write write a a general general routine routine that, that, given given aa type type X X and and a a comparison comparison function function in in X X X x X X -+ --+ 00,, sorts sorts lists lists of of objects objects of of type type X X. . Assuming Assuming such such lists lists are are represented represented by by the the type type List(X), List(X), for for each each type type X X we we want want Sort(X) the type type Sort(X) to to have have the (X xx X (X X -+ --+ 0) 0) x x List(X) List(X) -+ --+ List(X) List(X).. The The type type of of the the function function Sort Sort itself itself can can then then be be written written IIx Hx ((X ((X x x X X -+ --+ 0) 0) x x List(X) List(X) -+ -+ List(X)) List(X)) where where the the product product IIx Hx now now ranges ranges over over (some (some collection collection of) of) types. types. In polymorphism. In In this this section section we we will will consider consider two two different different kinds kinds of of polymorphism. In the the first, first, the the variable variable X X in in the the preceding preceding example example is is allowed allowed to to range range over over all all types, types, include include the the type type of of Sort Sort itself. itself. This This scheme scheme is is known known as as impredicative impredicative polymorphism polymorphism and and was was discovered discovered independently independently by by J.-y' J.-Y. Girard, Girard, who who was was looking looking for for aa D-interpretation D-interpretation for J. Reynolds, for second-order second-order arithmetic arithmetic (cf. (cf. Girard Girard [1971]) [1971]),, and and J. Reynolds, who who was was exploring exploring recursive ordinal ordinal of analysis we can, in principle, treat IDa as a subsystem 22For Q o~ a provably recursive 22For of analysis and then apply Spector's Spector's interpretation (section (section 6) or Girard's interpretation (sec (section 10). descriptions of the associated 10). But it would would appear very difficult to extract meaningful descriptions proof-theoretic proof-theoretic ordinals ordinals and and reductions reductions to to the the corresponding corresponding intuitionistic intuitionistic systems systems via via those those interpretations. interpretations. At At any any rate rate this this does does not not look look at at all all like like aa practical practical possibility. possibility. J. Avigad and S. S. Feferman Feferman 394 type-theoretic type-theoretic constructs constructs from from aa computational computational point point of of view view (cf. (cf. Reynolds Reynolds [1974]) [1974]).. ' s main Given Given these these independent independent motivations, motivations, Girard Girard's main theorem theorem is is quite quite satisfying: satisfying: the the provably total total recursive recursive functions functions of of second-order second-order arithmetic arithmetic are are exactly exactly the the ones ones provably computable computable in in the the Girard-Reynolds Girard-Reynolds framework. framework. The (X) to The circularity circularity of of allowing allowing the the variable variable X X in in aa type type Il Hx T(X) to range range over over x T T(X) itself, might seem disconcerting. Predicative poly all types, including including Il IIx T(X) itself, might seem disconcerting. Predicative polyall types, x morphism benign in since in this framework morphism is is more more benign in that that regard, regard, since in this framework the the variable variable X X is is restricted to range range over over aa pre-set pre-set universe universe of of "smaller" "smaller" types. types. For For example, example, in in the the restricted to type type Il (X) Ilx~uo T(X) x E uo T the the variable variable X X takes takes values values in in the the fixed fixed universe universe Uo U0.. This This kind kind of of polymorphism polymorphism was developed developed by by Martin-LOf Martin-Lbf as as a a framework framework for for constructive constructive mathematics mathematics (cf. (cf. was Martin-Lbf [1973]) [1973]),, and and has has been implemented in in the the underlying underlying language of the Martin-Lof been implemented language of the Nuprl Chapter X al. [1986] ) . Once Nuprl proof-development proof-development system system (cf. (cf. Chapter X or or Constable Constable et et al. [1986]). Once again, again, there there is is aa result result that that nicely nicely characterizes characterizes the the axiomatic axiomatic strength strength of of this this kind kind of of polymorphism: polymorphism: the the provably provably total total recursive recursive functions functions of of predicative predicative analysis analysis (cf. (cf. section the ones can be defined using using nested nested universes universes of section 8.3.6) are are exactly exactly the ones that that can be defined of types. types. Here will focus Here we we will focus on on the the interpretative interpretative strength strength of of polymorphism. polymorphism. For For a a detailed detailed discussion of subject, we discussion of the the various various computational computational aspects aspects of of the the subject, we refer refer the the reader reader to to Mitchell Mitchell [1990] [1990] and and Gallier Gallier [1990]. [1990]. 10.2. 10.2. The T h e second-order s e c o n d - o r d e r polymorphic p o l y m o r p h i c lambda l a m b d a calculus, calculus, F F 's theory We F, aa polymorphic polymorphic extension We now now define define Girard Girard's theory F, extension of of T T strong strong enough enough to to interpret interpret second-order second-order arithmetic. arithmetic. This This allows allows for for terms terms which which can can be be applied applied to to (terms (terms representing) representing) types, types, and and abstraction abstraction across across types. types. It It is is simpler simpler here here to to take take abstraction basic operators rather than defined in abstraction as as basic operators rather than defined in terms terms of of combinators. combinators. The The types defined inductively, types of of F F are are defined inductively, as as follows: follows: 1. There infinitely many 1. There are are infinitely many type type variables variables X X,, Y Y,, Z, Z, ..... . is a a type. type. 22.. 00 is 3. If T and If a a and and T are are types, types, so so are are a a --+ --+ T and a a x • T. 4. 4. If If a a is is a a type type and and X X is is aa type type variable, variable, then then Il Hx and � ~ xx a a are are also also types. types. x aa and a denotes a polymorphic function that, for each type T, returns A A term term of of type type Il Hx a denotes a polymorphic function that, for each type ~', returns x T, tt), an an object object of of type type arT/Xl a[T/X].. A A term term of of type type � ~ xx a a denotes denotes a a pair pair ((% where T r is is a a ) , where type /X] . type and and tt is is an an object object of of type type arT a[T/X]. The The terms terms of of F F are are also also defined defined inductively, inductively, as as follows: follows: 1. For 1. For every every type type a a there there are are infinitely infinitely many many variables, variables, x x "~,, y" y~,, zz "~,, ..... . ) , and 2. 00,, Sc, Sc, the the recursors recursors R" R~,, pairing pairing operators operators ((.,- , ."/, and projection projection operators operators 11" 7r0, 7rl 2. 0 , 11"1 are are terms terms of of appropriate appropriate type. type. 3. a, then 3. If If tt is is a a term term of of type type T T and and x x is is aa variable variable of of type type a, then A )~x.t is a a term term of of type type X .t is a a --+ ---~T. 7-. 4. If ( s ) is If tt is is aa term term of of type type a a --+ --+ T and and ss is is aa term term of of type type a, a, then then tt(s) is of of type type T T.. Ghdel 's Functional Interpretation Interpretation Code/'s 395 395 5. If T, and the type 5. If tt is is a a term term of of type type T, and the type variable variable X X is is not not free free in in the the type type of of aa free free term of type Hx I1x T. variable variable of of t, t, then then AX.t A X . t is is a a term of type T. 6. If If tt is is a a term term of of type type I1x IIx T T and and a a is is a a type, type, then then t(a) t(a) is is a a term term of of type type T[a 7"[•/X]. /X] . T[a /X] , then 7. 7. If If a a is is a a type type and and tt is is a a term term of of type type T[a/X], then (a, (6, t) t) is is a a term term of of type type �x Ex T. T. If t[x t[x Tr]] is is a a term term of of type type a a and and the the type type variable variable X X is is only only free free in in the the type type of of 8. If then Xx T~,, then is is a a term term of of type type �x Ex T T -r -+ a. a. While While clauses clauses (1-4) (1-4) are are essentially essentially imported imported from from T, T, clauses clauses (5-8) provide provide the the new new polymorphic polymorphic capabilities. capabilities. The The choice choice of of lambda lambda terms terms instead instead of of combinators combinators in in clause (3) conforms clause (3) conforms with with the the majority majority of of the the literature literature on on this this subject. subject. The The term term AXt A X . t in in clause clause (6) denotes denotes a a function function that that associates associates to to every every type type a a an T[a /X] , with with defining defining equation an object object of of type type TIc/X], equation (AX.t)(a) = = t[a/X] t[a/X]. . (AXt)(a) The requirement requirement that that X X is is not not free free in in the the type type of of any any free free variable variable of of tt precludes precludes The terms terms like like AX.x X X,, AX.x in which which the the free free variable variable x x cannot cannot be be assigned assigned any any reasonable reasonable type. type. On On the the other other in hand, hand, it it does does allow allow constructions constructions such such as as the the polymorphic polymorphic identity identity function function AX.Ax x .x x and and the the Sort Sort function function defined defined in in section section 10.1. 10.1. Strictly Strictly speaking, speaking, the the application application operation (6) is (s) operation (AXt)(a) (AX.t)(a) in in clause clause (6) is different different from from the the application application operation operation (Ax.t) (Ax.t)(s) in in clause clause (4) (4),, though though we we will will use use the the same same notation. notation. T ).t in The (X, xxr).t The function function \7 V(X, in clause clause (8) (8) is is a a bit bit trickier: trickier: it it takes takes a a pair pair (a, (a, s) s),, for for which T[a /X] , and the value which ss is is of of type type TIc~X], and returns returns the value of of tt with with X X replaced replaced by by a a and and [a/Xl replaced xr this term defining equation x r[~/z] replaced by by s. s. In In other other words, words, this term has has the the defining equation x z ) . t.t) ) ( o(a, , s) = = t[a /X] [s/xr [a/ l ] . \7 (X, x) ((v(x, As As usual, usual, the the variable variable X X becomes becomes bound bound in in clause clause (6) (6),, and and both both variables variables X X and and x x become bound in become bound in clause clause (8) (8).. For technical reasons section also For technical reasons the the interpretation interpretation in in the the next next section also requires requires aa constant constant zero defining its behavior, though zero functional functional 017 0~ of of each each type. type. We We omit omit the the rules rules defining its behavior, though they they can can be be found found in in Girard Girard [1971]. [1971]. 10.3. 10.3. The The interpretation interpretation of of analysis analysis Whereas Whereas in in Spector's Spector's interpretation interpretation of of analysis analysis the the second-order second-order objects objects of of 's interpretation PAt CA) become with function in Girard PA 2 + + ((CA) become identified identified with function variables, variables, in Girard's interpretation it it is is more predicates. (Moreover, interpretation extends more natural natural to to treat treat them them as as predicates. (Moreover, the the interpretation extends to to aa 396 396 J. AAvigad vigad and and S. S. Feferman Fe]erman J. higher-type higher-type version version of of this this system, system, with with predicates predicates instead instead of of functions functions at at each each level.) level.) As As in in the the interpretation interpretation of of PA PA,, we we first first apply apply the the double-negation double-negation interpretation interpretation to to reduce PAl! PA ~ + + ((CCA) A ) to to its its intuitionistic intuitionistic variant, variant, HAl! HA ~ + + ((CCA A )).. Unlike Unlike the the case case of of reduce ' s interpretation, (A C ) iin n Spector Spector's interpretation, ((CA) follows directly directly from from ((CA)in HA ~.. (AC) CA) Ng follows CA) in HAl! The The D-interpretation D-interpretation we we are are about about to to define define translates translates formulas formulas cp ~oin in the the language language of second-order second-order arithmetic arithmetic to to formulas formulas cpD ~D of ofthe the form form of vV y) = 0 (8) (8) where ers where CP r is quantifier-free with with only only type type 00 equality. equality. Adding Adding dummy dummy quantifi quantifiers D is quantifier-free as as necessary necessary and and using using the the pairing pairing and and projection projection operations operations to to combine combine quanti quantified variables, variables, we we can can assume assume that that there there is is always always exactly exactly one one variable variable after after each each fied quantifier. quantifier. The rst-order formulas The first first new new step step needed needed in in defining defining cpD ~D beyond beyond fi first-order formulas is is to to find find aa suitable suitable translation translation for for formulas formulas of of the the form form tt E E Z. Z. Since Since we we are are aiming aiming to to interpret interpret the comprehension comprehension axioms, axioms, we we want want the the variable variable Z Z to to "range" "range" over over arbitrary arbitrary formulas formulas the ~o.. We We define define (t (t E E Z) to be be Z)DD to cp 3x x Vy YY f(x, f (x, y, y, t) t) = -- 0 (9) (9) 3xX where now now X X and and Y Y are are type type variables variables and and f f is is aa function function variable variable of of type type where X xxY Y x Ox~ 0O -+ . O. X Intuitively, (t (t E Z)D Z) D represents represents an an "arbitrary" "arbitrary" formula formula of of the the form form (8) (8).. Intuitively, The translation is The translation is extended extended to to the the logical logical connectives connectives and and first-order first-order quantifiers quantifiers as before. before. To To define define (3Z ~o(Z))) D, suppose cp(Z)D ~o(Z) D is is given given by by (3Z cp(Z) D , suppose as ,Yj F(x, 3x u~[X'YI ~[X'Y] = 0 [X,Yj VyT[X F(x , y, f) 3x f) = where X, X , Y, Y, and and ff are are the the variables variables we Z in in the preceding where we have have associated associated with with Z the preceding paragraph. The formula 3Z ~o(Z) translate to cp(Z) should should then then translate to paragraph. The formula 3Z 3X, ]I, f, x ~[x'Y] F(x, y, f) = O. Yj Vy [X,y] F(x, f) = VyT~[X'Y] 3X, Y, f, xu[X, But now now the the type type of depends on the existentially existentially quantified quantified types types X X and and Y, Y, which which But of yy depends on the with an an element element of of the the appropriate appropriate is problematic. problematic. We remedy this this by by replacing replacing yy with is We remedy product product type type to to obtain obtain 3X, f) = = O. O. Y), f) y(X, Y), IIx,y ~[x,Y] T[X,Yj F(x, u[X ,Yj Vy F(x, y(X, Vynx,v Y, f,f, xx~[X'Y] 3X, ]I, Finally, we we replace replace the the existentially existentially quantified quantified variables variables X, X , Y, Y , and and ff by by aa single single Finally, variable uu of of type type Ex �x EY � Y (X (X •x Y -+ 0). 0) . Then Then uu may may be be paired paired with with xx to to put put variable Y •x 00 --+ (:]Z D again (3Z rcp(Z))D again in in the the form form (8). (8) . Similar manipulations manipulations are are used used to to define define the the translation translation of of the the formula formula VZ VZ ~o(Z). cp( Z) . Similar When this this isis done, done, the the interpretation interpretation of of HA HAl!e is is verified verified just just as as in in section section 2.4. 2.4. The The When ( CA) are are also also easily easily dealt dealt with, with, as as aa consequence consequence of of the the comprehension axioms axioms (CA) comprehension 23 translation we we have have chosen chosen for for xx E Z. Z. Details Details can can be be found found in in Girard Girard [1971]. [1971] .23 translation This This yields yields and in the the literature, literature, FF is often used more specifically specifically to to denote the the set of of reduction 23There, and Codel's GSdel 's Functional Functional Interpretation 10.3.1. Theorem. Theorem. 10.3.1. 397 PA 2 + + ((CA) is ND ND-interpreted in F F.. PA2 CA) is - interpreted in 10.4. 10.4. Strong S t r o n g normalization n o r m a l i z a t i o n for for F F In doctoral dissertation dissertation Girard In his his doctoral Girard [1972] [1972] presented presented a a powerful powerful generalization generalization of of Tait cf. section section 4.3) Tait's's convertibility convertibility methods methods ((cf. 4.3),, and and used used it it to to show show that that F F is is strongly strongly normalizing. normalizing. Since the the normalization normalization of of F F implies implies the the consistency consistency of of second-order second-order arithmetic, arithmetic, Since the the full full argument argument cannot cannot be be formalized formalized in in that that theory. theory. To To appreciate appreciate the the difficulties difficulties that that arise arise in in trying trying to to adapt adapt the the argument argument of of section section 4.3, 4.3, consider consider the the problem problem of of 7. We extending extending the the reducibility reducibility predicate predicate defined defined there there to to a a term term tt of of type type Ilx Hx T. We would to say would like like to say that that tt is is reducible reducible if if for for every every type type a a the the term term t(a) t(a) is reducible. But But is reducible. aa is is arbitrary, arbitrary, and and could could very very well well be be the the type type Ilx Hx 7 T itself. itself. In In that that case, case, determining determining the t(a) will the reducibility reducibility of of t(a) will require require some some knowledge knowledge of of what what it it means means for for terms terms of of type type Ilx Hx 7 T to to be be reduciblereducible-- which which is is exactly exactly the the notion notion we we are are trying trying to to define. define. ne the Girard Girard's's clever clever dodge dodge is is to to defi define the notion notion of of a a "reducibility "reducibility candidate," candidate," which which is a a predicate predicate of of terms terms which which satisfies satisfies certain certain closure closure conditions. conditions. The The reducibility reducibility is predicate candidate, and, and, in predicate of of section section 4.3 4.3 is is an an example example of of a a reducibility reducibility candidate, in fact, fact, it it is just these closure conditions necessary induction is just these closure conditions that that allow allow one one to to carry carry out out the the necessary induction on terms. terms. Girard Girard then then declared declared tt E Ilx Hx 7 ~" to to be be reducible reducible if if for for every every reducibility reducibility on candidate C C and and every every type type a, t(a) is reducible reducible of of type type 7 T(a), where now now reducibility reducibility candidate a, t( a) is (a) , where for t( a) is ned in for t(a) is defi defined in terms terms of of C. C. Since Since the the definition definition of of reducibility reducibility for for terms terms of of type type Ilx 1-Ix 7 T involves involves prefixing prefixing aa second-order second-order quantifier quantifier to to a a formula formula involving involving the the definition definition of of reducibility reducibility for for the definition terms type 7T,, for terms of of type for arbitrary arbitrary polymorphic polymorphic types types the definition requires requires second-order second-order formulas formulas of of arbitrary arbitrary complexity. complexity. The The net net result result is is the the following following For F, PA2 CA) proves For each each term term tt of of F, PA ~ + + ((CA) proves that that tt is is strongly strongly normalizing. addition, PA2 CA) proves proves the normalizing. In In addition, PA 2 + ((CA) the confluence confluence of of F F.. 10.4.1. 10.4.1. Theorem. Theorem. Thus CA) via Thus F F can can b bee interpreted interpreted iinn PA2 PA ~ + § ((CA) via its its model model iinn the the normal normal terms. terms. This This yields yields the the following following 10.4.2. The CA) are 10.4.2. Theorem. Theorem. The provably provably total total recursive recursive functions functions of of PA2 PA2 + + ((CA) are exactly exactly the the ones ones that that are are represented represented by by terms terms of of F F.. The The method method of of reducibility reducibility candidates candidates extends extends to to stronger stronger functional functional theories, theories, including aa typed including typed extension extension of of F F also also introduced introduced in in Girard Girard's's dissertation, dissertation, and and CoCorules rules corresponding corresponding to to the the defining defining equations equations of of the the theory theory described described above; above; we we have have chosen chosen to to blur blur this this distinction. distinction. A A more more minimal minimal version version of of F F which which omits omits the the base base type type 00 and and the the sum sum type type - essentially A--+'V of essentially the the system system )~-~,v of Mitchell Mitchell [1990], [1990], Gallier Gallier [1990] [1990] -- is is discussed discussed in in Girard, Girard, Lafont Lafont and Taylor [1989]. The The theory theory Y of of Girard Girard [1971] [1971] is is essentially essentially aa logic-free logic-free version version of of our our theory theory F. More More precisely, precisely, if gic-free if F F proves proves aa quantifier-free quantifier-free formula formula cp ~ involving involving only only type type 00 equality, equality, then then Y proves proves the the lo logic-free translation of 'P ~p when free variables are instantiated by any closed closed terms. 398 S. Feferman J. A vigad and S. quand's "Calculus "Calculus of of Constructions." Constructions." In In fact, fact, by by identifying identifying natural natural deductions deductions with with quand's terms of of an an appropriate appropriate functional functional calculus, calculus, Girard Girard [1971] [1971] was was also also able able to to use use these these terms 's methods to to give give aa new new proof proof of of "Takeuti's "Takeuti's conjecture," conjecture," an an extension extension of of Gentzen Gentzen's methods Hauptsatz to to higher-order higher-order deductive deductive sytems; sytems; this this was was realized realized independently independently by by Hauptsatz Martin-Lhf [1971] [1971] and and Prawitz Prawitz [1971]. [1971]. See See Gallier Gallier [1990], Coquand Coquand [1990], [1990], Girard, Girard, Martin-Lof Lafont and and Taylor Taylor [1989] [1989] for for more more details. details. Lafont 10.5. Theories T h e o r i e s based b a s e d on on predicative p r e d i c a t i v e polymorphism polymorphism 10.5. A more more restrictive, restrictive, "predicative," "predicative," method method of of extending extending the the theory theory T T polymorphi polymorphiA cally is is represented represented by by aa sequence sequence oftheories of theories Pn Pn,, based based on on Martin-Lofs Martin-Lhf's theories theories MLn MLn.. cally The theory theory Po is is equivalent equivalent to to T, while the the theory theory Pn+I Pn+l adds adds n + + 11 "universes" "universes" of of The T, while types, Uo U0 through through Un U~,, each each of of which which will will itself itself be be aa type. type. types, Like the the theory theory F, the theories theories Pn P~ have have product product and and sum sum types, types, but but rather rather than than F, the Like taking products products over over types, types, one one takes takes products products over over terms terms of of aa given given type. type. That That is, is, taking one has has the the following following type type formation formation rules: rules: one 1. 0 is is a a type. type. 2. If If a a and and T[XO~]' ] are are types, types, then then so so are are II Hxe~ and �XEO' E x ~ T. xE O' T and Terms of of type type II II~e~ denote functions functions f f which which take take elements elements aa of of type type a a to to elements elements Terms x EO' T denote f (a) of of type type T[a/x] T[a/x],, and and terms of type type �XE E~e~ T denote denote pairs pairs (a, (a, b) b),, where where aa is is an an object object f(a) terms of O' is an constructions can be of of type type a a and and bb is an object object of of type type T[a/x] T[a/x].. The The --+ -+ and and x constructions can be seen special cases seen as as special cases of of the the II H and and � E constructions constructions respectively, respectively, in in which which the the type type T ~doesn't depend on the the variable variable xxO'~.. doesn't depend on One has has to to modify modify the the usual usual formation formation rules rules for for terms terms to to accommodate accommodate these these new new One dependent types. types. For For example, example, the the new new rule rule of of explicit explicit definition definition takes takes the the form form dependent t[xO'] term of of type type r[x~], T[XO'] , then then )~x.t Ax.t isis aa term term of of type If t[x If ~] is is aa term type II I-[xeaT. xE O'T. Similar rules take of those of T application, pairing, pairing, projection, Similar rules take the the place place of those of T regarding regarding application, projection, and the the recursors. recursors. and We for defining depend on We haven't haven't yet yet provided provided aa mechanism mechanism for defining types types that that depend on variables. variables. P,n polymorphic polymorphic is types can can be be represented by terms, What makes What makes the the systems systems P is that that types represented by terms, since elements since elements of type Ui are to denote types. For For example, example, PI P1 has has are themselves themselves taken taken to denote types. of type the rules: the following following rules: 1. Uo U0 is is a a type type 2. 0 (the type of the natural numbers)) is is an U0 (the type of the natural numbers an element element of of Uo and ExerT. �XEO' T. If aa and and T[X T[XO'] are elements elements of of U0, Uo , then then so so are are I-[xeaT IIxE O'T and 3. If 3. ~] are 4. If If tt is is aa term of type is aa type. term of type U0, Uo , then then tt is type. The last last clause clause places places the the theory theory PI PI in in sharp sharp distinction distinction to to T, T, where where there there is is no no The In P1, PI , one one can can define define aa term term tt of of type Uo , interplay terms and and types. types. In interplay between between terms type U0, conclude conclude that that tt is is aa type, type, and and then then define define another another term term ss of of type type t.t . For For example, example, the the (n) of of section section 10.1 can can be be defined defined by by aa simple simple instance instance of of primitive primitive recursion recursion types (n) types Uo , whereupon whereupon the the term term IIne0(n) IInEO(n) is with range range Uo, is an an element element of of U0 Uo and and hence In with hence aa type. type. In Pn has has nn universes universes U0,..., Uo, . . . , U~-I, Un-I , each each of of which which contains contains terms terms general, each each theory theory P~ general, denoting denoting all all "smaller" "smaller" universes. universes. Giidel's GSdel's Functional FunctionalInterpretation Interpretation 399 399 10.6. The interpretation interpretation of predicative theories theories of analysis 10.6. The theory theory m ID~1 is is aa weakening weakening of of the the theory theory IDI ID~ defi defined in section section 9.2. 9.2. Here, Here, The ned in rather than than assert assert that that P P is is aa least least fixed fixed point point of of the the positive positive arithmetic arithmetic operator operator rather given by by 0, one simply simply asserts asserts that that P P is is some fi fixed point; that that is, is, one one replaces replaces 22 and and given 0, one xed point; from section section 9.2 by the the single single axiom axiom 33 from 9.2 by A Vx (O(P, (O(P,x) ++P(x)) P(x)).. \Ix x) ++ A - In general, general, each each theo t h e o� r ~ I1Dn D , allows allows n nested nested induct inductive definitions, by by allowing allowing one one to to In �e definitions, use any any predicate predicate of of ID; IDi to to define define aa fixed-point fixed-point in in 1Di+I IDi+~.. Also Also we we set set use A nn <<ww It is is known known (cf. (cf:.Feferman [1982a], Friedman, McAloon and and Simpson [1982], Avigad It Feferman [1982a], Friedman, McAloon Simpson [1982] ' Avigad [1996b]) that that m ID<~ has strength strength ro F0,, and and hence hence proves proves the the same same arithmetic arithmetic formulas formulas [1996bJ) <w has as predicative predicative analysis analysis (cf. (cf. the the discussion discussion preceding preceding Theorem Theorem 8.3.8) 8.3.8) as as well well as as an an as important second-order theory known known as as A A TRo TRo.24 The following following theorem theorem is is due due to to . 24 The important second-order theory Avigad [n.d.] [n.d.].. Avigad Eachtheory theory mn IDn is is ND-interpreted ND-interpreted in in Pn p,.25 Each . 25 A 10.6.1.. Theorem. Theorem. 10.6.1 The The interpretation interpretation is is somewhat somewhat tortuous, tortuous, so so we we only only provide provide aa rough rough outline outline here. The The first first reduction reduction relies relies on on the the following following lemma, lemma, due due to to P. P. Aczel Aczel (cf. (cf. Feferman Feferman here. [1982a]). [1982aJ). mI is interpretable in Ef -A G generally, each IDI is interpretable in S[-A C.. More More generally, each theory theory fD,+l AC (~,). . mn+ I is interpretable interpretable in in SEf] --A G(mn) 10.6.2. Lemma. 10.6.2. Lemma. A Here the theory theory Ef-AG(mn) is aa second-order second-order version mn together together with Here the Z[-AC(~n) is version of of I'D, with a a choice scheme scheme for any arithmetic I:� formulas formulas in in the the expanded expanded language. language. Given Given any arithmetic (or (or choice for E~ even Y) in Y occurs even E~) I:D formula formula ~(x, rp(x, Y) in which which the the predicate predicate Y occurs positively, positively, one one can can obtain obtain fixed-point of of the corresponding set set operation. operation. (The proof aa E~ I:� formula formula r7jJ defining defining aa fixed-point the corresponding (The proof is similar to that that of lemma, while (Ef-AG) is is similar to of Godel's G5del's fixed-point fixed-point lemma, while the the scheme scheme (Z[-AC) is needed needed that that that formula formula r7jJ works advertised.) One One then then uses formulas 7jJ to show that works as as advertised.) uses these these formulas to show to to interpret interpret the the fixed-point fixed-point constants constants of of ID,+I. mn+ I . By Theorem Theorem 8.3.1, 8.3.1, one one can can interpret interpret S]-AC Ef -A G in in the the theory theory TT ++ (#), (/-L) , in in which which By formulas in in the the language language of of Peano Peano arithmetic arithmetic translate translate to to formulas formulas that that are are quantifierquantifier formulas free. We can instead instead interpret interpret Z[-AC Ef -AG in in P1 PI if if we we interpret interpret formulas formulas tt E ZZ by by free. We can equation (9) (9) of of section section 10.3, 10.3, where where now now the the type type variables variables range range over over the the universe universe U0 Uo equation instead instead of of all all types. types. Iterating Iterating this this idea idea leads leads to to Theorem Theorem 10.6.1. 10.6.1 . A 24The first functional interpretation interpretation of predicative ramified analysis analysis was was given by Maass [1976] [1976] via a specialization of Girard's Girard's interpretation interpretation described in section 10.3 10.3 above. 25The theories Pn of Avigad [n.d.] [n.d.] are logic-free logic-free subsystems subsystems of the theories Pn defined here. J. A A vigad and S. Feferman 400 400 P<w Pn) Pn, One [n.d.] ' Martin-Lof One can can show show (cf. (cf. Coquand Coquand~[n.d.], Martin-LSf [1973]) [1973]) that that terms terms of of P<~ (= (-- U (J P , ) are normalizing. Since each can define a recursion-theoretic model for are normalizing. Since each IDu can define a recursion-theoretic model for P~, we we obtain obtain mw A The The provably provably total total recursive recursive functions functions of of each each mn IDn are are exactly exAactly the n. The provably total the ones ones represented represented by by terms terms of of P P,. The provably total recursive recursive functions functions of of m ID<~, <w , and hence predicative analysis analysis and and ATRo A TRo,, are are exactly exactly the the ones ones represented represented by by terms terms and hence predicative of of P P<~. <w · 10.6.3. 10.6.3. Corollary. Corollary. 10.7. 10.7. Final F i n a l comments c o m m e n t s and a n d questions questions The functional functional interpretation interpretation of of aa classical classical theory theory provides provides aa nice nice interplay interplay The between logic and theoretical computer science. Given a a set set of of classical axioms, between logic and theoretical computer science. Given classical axioms, the corresponding corresponding computational schemata provide provide aa constructive constructive understanding understanding of of the computational schemata their strength, and and normalization normalization proofs proofs provide provide evidence evidence that that these these abstract axioms their strength, abstract axioms have have interesting interesting computational computational consequences. consequences. Conversely, Conversely, given given some some computational computational schemata, schemata, the the calibration calibration of of their their classical classical axiomatic axiomatic strength strength helps helps round round out out our our understanding understanding of of their their capabilities. capabilities. Martin-Lofhas Martin-LSf has described described extensions extensions of of the the theories theories MLn with with "elimination "elimination rules" rules" for the the universes, universes, yielding yielding the the same same proof-theoretic proof-theoretic strength strength as as the the impredicative impredicative for theories theories ID, (cf. (cf. Griffor Griffor and and Rathjen Rathjen [1994,Theorem [1994,Theorem 4.14]). 4.14]). Can Can such such elimination elimination rules rules be be used used to to give give the the theories theories 1Dn ID, a a direct direct functional functional interpretation? interpretation? The The interpretations interpretations of of Spector Spector and and Girard Girard show show that that bar-recursion bar-recursion and and impred impredicative polymorphism each icative polymorphism each exactly exactly capture capture the the provably provably total total recursive recursive functions functions of of second-order second-order arithmetic. arithmetic. Are Are there there natural natural characterizations characterizations for for interesting interesting frag fragments and extensions the latter the ones ments and extensions of of the latter theory, theory, other other than than the ones we we have have already already described described in in this this chapter? chapter? MLn IDn References References J. AVIGAD AVIGAD [n.d.] [n.d.] Predicative Predicativefunctionals and an interpretation of m<w ID <~.. To appear in the Annals Annals of Pure Applied Logic. and Applied Logic. 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Troelstra1 Troelstra 1 Language, Logic Logic and Computer Science, Science, University of Amsterdam Institute for Language, NL-1018 TV Amsterdam, The Netherlands NL-l018 Contents Contents 1. Numerical Numerical realizability realizability .. .. .. .. .. . . . . . . . . . . . . . .. .. . . . . . . . . . . . . . 2. A b s t r a c t realizability realizability and and function function realizability realizability . . . . . . . . . . . . . . . . . . 2. Abstract realizability .. .. .. . . . . . . . . . . . . . . . . . . . . . . . . . 3. Modified Modified realizability Derivation of of the the Fan Fan Rule Rule . . . . . . . . . . . . . . . . . . . . . . . . . 4. Derivation 5. Lifschitz Lifschitz realizability realizability .. .. .. . . . . . . . . . . . . . . . . . . . . . . . . . 6. Extensional Extensional realizability realizability .. .. . . . . . . . . . . . . . . . . . . . . . . . . . 6. 7. Realizability for for intuitionistic intuitionistic second-order second-order arithmetic a r i t h m e t i c .. . . . . . . . . . . . . . . 7. Realizability Realizability for for higher-order higher-order logic logic and and arithmetic arithmetic . . . . . . . . . . . . . . . . . 8. Realizability 9. u r t h e r work 9. FFurther work .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References .. .. . .. .. . . . . . . . . . . . . .. .. .. .. .. .. .. .. . .. . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 408 422 422 429 434 437 437 439 441 445 458 462 H ANDBOOK O F PPROOF ROOF T HEORY HANDBOOK OF THEORY Edited Edited by by S. S. R. R. Buss Buss 9 1998 Science B.V. rights reserved 1998 Elsevier Elsevier Science B.V. All All rights reserved © iS. Buss, U. and J. IS. Buss, U. Kohlenbach, Kohlenbach, H. H. Luckhardt, Luckhardt, J.R. J.R. Moschovakis Moschovakis and J. van van Oosten Oosten have have commented commented on drafts of Van Oosten for section on earlier earlier drafts of this this paper. paper. Van Oosten also also provided provided aa sketch sketch for section 88 which which has has been been used used in in composing composing the the final final version. version. 408 408 A.S. Troelstra 1. N Numerical 1. umerical rrealizability ealizability 1.1. IIntroduction 1.1. ntroduction There is is not not just just one one single single notion notion of of realizability, realizability, but but aa whole whole family family of of notions, notions, There which of of course resemble each other in in certain This section is devoted devoted to to which course resemble each other certain respects. respects. This section is S.C. fairly detailed detailed discussion discussion of of the the earliest and most most basic basic notion notion of of realizability, realizability, S.C. aa fairly earliest and Kleene's realizability realizability by numbers. In In later later sections sections we discuss more more briefly briefly variations variations Kleene's by numbers. we discuss of of the the basic basic notion. notion. We We do do not not aim aim at at an an exhaustive exhaustive description description of of all all possible possible proof-theoretic applications applications of of realizability, realizability, but but rather rather aim aim at illustrative proof-theoretic at presenting presenting illustrative examples. Most Most of of the the sections sections are are followed followed by by "Notes", "Notes" , containing containing suggestions suggestions for for examples. further reading, reading, some some historical historical comments, comments, etc. etc. The The historical historical comments comments concern concern further mainly the the period period after after 1972, 1972, since since the the history up till till 1972 1972 is is fairly fairly completely completely mainly history up documented in in Troelstra Troelstra [1973a]. [1973a] . documented Realizability by by numbers numbers was was introduced introduced by by Kleene Kleene [1945] [1945] as semantics for for Realizability as aa semantics intuitionistic arithmetic, by defining defining for for arithmetical arithmetical sentences sentences A A aa notion notion "the intuitionistic arithmetic, by "the number n realizes A" intended to to capture capture some some essential aspects of of the the intuitionistic number n realizes A",, intended essential aspects intuitionistic A. Here Here nn is not aa term term of of the the arithmetical arithmetical formalism, but an meaning of of A. meaning is not formalism, but an element element of of IN . The definition is is by by induction induction on the complexity A: the natural natural numbers numbers IN. the The definition on the complexity of of A: holds; •9 n n realizes = s holds; realizes tt = = s iff iff tt = • A /\ B iff A and and pPIn B; 9 nn realizes realizes A AB iff pon pon realizes realizes A i n realizes realizes B; • realizes A AV B iff pon == 00 and and PIn realizes A A or and PIn 9 nn realizes VB iff pon P i n realizes or pon pon = = 11 and p i n realizes realizes B; B; •9 n n realizes ff for for all all m m realizing realizing A, A, n.m defined and realizes A A -+ -~ B B iiff nom iiss defined and realizes realizes B B;; • A if realizes A 9n n realizes realizes ...., -,A if for for nnoo m m,, m m realizes A;; • 9n n realizes realizes 3y 3y A A iff iff P P iInn realizes realizes A[y/pon] A[y/~d-~].. •9 n A[y 1m] , for n realizes realizes Vy Vy A A iff iff n.m nom is is defined defined and and realizes realizes A[y/~], for all all m m.. Here Here P PlI and and Po P0 are are the the inverses inverses of of some some standard standard primitive primitive recursive recursive pairing pairing function function P coding IN IN, and m is p coding IN22 onto onto IN, and ~ is the the standard standard term term SmO sm0 (numeral) (numeral) in in the the language language of intuitionistic arithmetic corresponding to of intuitionistic arithmetic corresponding to m m ;; 9 is is partial partial recursive recursive function function application, application, i.e. i.e. n.m n ~ is is the the result result of of applying applying the the function function with with code code n n to to m m.. (Later (Later on on we we also also use use in, rh, ii, ~ , .. ... . for for numerals.) numerals.) The The definition definition may may be be extended extended to to formulas formulas with with free free variables variables by by stipulating stipulating that that n n realizes realizes A A if if n n realizes realizes the the universal universal closure closure of of A. A. Reading Reading "there "there is is aa number number realizing realizing A" A" as as "A "A is is constructively constructively true" true",, we we see see that that aa realizing realizing number number provides provides witnesses witnesses for for the the constructive constructive truth truth of of existential existential quantifiers quantifiers and and disjunctions, disjunctions, and and in in implications implications carries carries this this type type of of information information from from premise premise to to conclusion conclusion by by means means of of partial partial recursive recursive operators. operators. In In short, short, realizing realizing numbers numbers "hereditarily" "hereditarily" encode encode information information about about the the realization realization of of existential existential quantifiers quantifiers and and disjunctions. disjunctions. Realizability, Realizability, as as an an interpretation interpretation of of "constructively "constructively true" true" is is reminiscent reminiscent of of the the well-known well-known Brouwer-Heyting-Kolmogorov Brouwer-Heyting-Kolmogorov explanation explanation (BHK (BHK for for short) short) of of the the "p proves intuitionistic intuitionistic meaning meaning of of the the logical logical connectives. connectives. BHK BHK explains explains "p proves A" A" for for • 409 409 Realizability Realizability compound A A in in terms terms of of the the provability provability of of the the components components of of A. For prime prime formulas formulas compound A. For the notion notion of of proof proof is is supposed supposed to to be be given. given. Examples Examples of of the the clauses clauses of of BHK are: the BHK are: p proves proves A A -+ --~ B B iff iff P p is is a a construction construction transforming transforming any any proof proof cc of of A A into into a a •9 P proof p(c) p(c) of proof of B; B; •9 P iff Pp = PI) and Po proves p proves proves A A 1\ AB B iff = (Po, (p0,Pl) and P0 proves A, A, P plI proves proves B; B; •9 P PI) with } , and p proves proves A AV VB B iff iff P p= = (PO (p0,, pl) with Po p0 E E {O, {0, I1}, and P plI proves proves A A if if Po p0 -- 0, 0, PI pl proves O. proves B B if if Po P0 =1= ~ 0. Realizability BHK if Realizability corresponds corresponds to to BHK if (a) (a) we we concentrate concentrate on on (numerical) (numerical) information information concerning the ers and concerning the realizations realizations of of existential existential quantifi quantifiers and the the choices choices for for disjunctions, disjunctions, and the constructions and (b) (b) the constructions considered considered for for V, V, -+ ~ are are encoded encoded by by (partial) (partial) recursive recursive operations. operations. Realizability Realizability gives gives a a classically classically meaningful meaningful definition definition of of intuitionistic intuitionistic truth; truth; the the set of of realizable realizable statements statements is is closed closed under under deduction deduction and and must must be be consistent, consistent, since since set 1=0 cannot 1=0 cannot be be realizable. realizable. It It is is to to be be noted noted that that decidedly decidedly non-classical non-classical principles principles are are realizable, realizable, for for example example -Nx[3yTxxy V Vy.Txxy] -~Vx[3yTxxy Vy-~Txxy] 's T-predicate, is easily seen to to be be realizable. realizable. (T is Kleene Kleene's T-predicate, which which is is assumed assumed to to is easily seen (T is be available available in in our our language; language; Txyz Txyz is is primitive primitive recursive recursive in in x, x, y, y, zz and and expresses expresses be that algorithm with code xx applied yields aa computation that the the algorithm with code applied to to argument argument y yields computation with with code zz;; U is code zz code is aa primitive primitive recursive recursive function function extracting extracting from from a a computation computation code the z .) For realizable iff A, and the result result U Uz.) For .A -~A is is realizable iff no no number number realizes realizes A, and realizability realizability of of Vx[3yTxxy Vx[3yTxxy V Y Vy.Txxy] Vy-~Txxy] requires requires aa total total recursive recursive function function deciding deciding 3yTxxy, 3yTxxy, which which does does not not exist exist (more (more about about this this below) below).. In In this this way way realizability realizability shows shows how how in in constructive constructive mathematics mathematics principles principles may may be be incorporated incorporated which which cause cause it it to to diverge diverge from classical theory, included in from the the corresponding corresponding classical theory, instead instead of of just just being being included in the the classical classical theory. theory. Some notational habits adopted Some notational habits adopted in in this this paper paper are: are: dropping dropping of of distinguishing distinguishing sub suband and superscripts superscripts where where the the context context permits; permits; saving saving on on parentheses, parentheses, e.g. e.g. for for aa binary binary predicate x, yy we (this habit just predicate R R applied applied to to x, we often often write write Rxy Rxy instead instead of of R(x, R(x, y) y) (this habit has has just been demonstrated above) literal identity been demonstrated above).. The The symbol symbol == - is is used used for for literal identity of of expressions expressions modulo renaming renaming of bound variables. metamathematical consequence modulo of bound variables. * =~ is is used used as as metamathematical consequence particular A, relation, and relation, and in in particular .A, B B* ==~C C expresses expresses a a rule rule which which derives derives C C from from premises premises A, B. FV (A) is A, B. FV(A) is the the set set of of free free variables variables of of expression expression A. A. = 1.2. 1.2. Formalizing realizability in in HA HA In In order order to to exploit exploit realizability realizability proof-theoretically, proof-theoretically, we we have have to to formalize formalize it. it. Let Let us us first istic first-order arithmetic HA first discuss discuss its its formalization formalization in in ordinary ordinary intuition intuitionistic first-order arithmetic HA intuitionistic predicate (("Heyting's "Heyting' s Arithmetic" Arithmetic"),), based based on on intuitionistic predicate logic logic with with equality, equality, symbols for all primitive and and containing containing symbols for all primitive recursive recursive functions, functions, with with their their recursion recursion equations equations as as axioms. axioms. y, z, x, y, z , .. ... . are are numerical numerical variables, variables, S S is is successor. successor. We We use use the the notation notation fi ~ for for the the x, n o; such Po, PI term term s Sn0; such terms terms are are called called numerals. numerals. P0, Pl bind bind stronger stronger than than infix infix binary binary operations, (Pot) + R, Rt operations, i.e. i.e. Pot pot + + ss is is (pot) + s. s. For For primitive primitive recursive recursive predicates predicates R, Rtl... l . . . ttnn 410 410 A.S. A.S. Troelstra Troelstra may treated as prime formula since the symbol for may be be treated as a a prime formula since the formalism formalism contains contains aa symbol for the the characteristic characteristic function function X XR. R. Now Now we we are are ready ready for for aa formalized formalized definition definition of of "x "x realizes realizes A" A" in in HA. HA. rn A, xx ¢r FV(A) Definition. Definition. By By recursion recursion on on the the complexity complexity of of A A we we define define x xr___nnA, FV(A),, "x "x numerically numerically realizes realizes A" A" : 9 x rn (t = s) := (t = s) xx rn E) := ((p0xr___nnA) r___n_n(A (A /\ A B) A ((Pp lX l x rrn n SE), ), Pox rn A) /\ rn (A -+ E) rn A -+ B ) : =:= 'v'y(y Vy(yr___nnA ~ 3z(Txyz 3 z ( T x y z /\ A U Vzr___nnB)), xxr__nn(A z rn E)), r__nn'v'y Vy A A := 'v'y3z(Txyz V y 3 z ( T x y z /\ AU U zz rn r__nnA), A), xx rn xx rn r___nn3y 3y A A p l xx rn r._nnA[yfpoxJ. A[y/pox]. :"= = Pl Note rn A) c {x} U Note that that FV(x FV(xr___nnA) C {x} 0 FV(A) FV(A).. 0 [] Remarks. (i) We R e m a r k s . (i) We have have omitted omitted clauses clauses for for negation negation and and disjunction, disjunction, since since in in arithmetic 0, AAv /\ (x =Iarithmetic we we can can take take -,A -,A := A A -+ 11 = = O, V BE := 3x((x 3x((x = = 00 -+ -~ A) A)A(x ~ 00 -+ --+ E)) B)).. If If we we spell spell out out x x rn r___~n(A (A V VE B)) on on the the basis basis of of this this definition, definition, we we find find xr__nn(A V B ) " = (pox = 0 --+ (p0plx)0 r___~nA) A (pox # 0 --+ (plplx)0 r__~nB), (ii) (ii) The The definition definition of of realizability realizability permits permits slight slight variations, variations, e.g. e.g. for for the the first first clause clause we we might might have have taken taken n'(t = = s ) := : = (x = = tt /\ ^ tt = = s) . x rn'(t However, this variant -realizability is However, it it is is routine routine to to see see that that this variant rn' rj-realizability is equivalent equivalent to to rn r__nnrealizability realizability in in the the following following sense: sense: for for each each formula formula A A there there are are two two partial partial recursive recursive functions functions ¢J CA and 1/J CA such that that A and A such f- xx r___nn rn A A A -+ -+ ¢J CA rn__ZA A (X) rn' f~- x x rn'A rn A. rn___'A-+ --+ 1/JA(X) CA (x) r___nn A. (If (If in in the the future future we we shall shall call call two two versions versions of of aa realizability realizability notion notion equivalent, equivalent, it it will will always always be be in in this this or or aa similar similar sense.) sense.) Similarly, Similarly, if if we we treat treat V V as as aa primitive, primitive, the the clause clause rn (A for for x x r___nn (A V E) B) given given above above may may be be simplified simplified to to A) VV ((pox xxr___nn(A rn (A VV E) := ((pox B):= = 00 /\ A Pl plxX rn rnA) ~: 00 /\ A Pl plxX rn r___nnE), S), Pox =IPox = which which yields yields an an equivalent equivalent notion notion of of realizability. realizability. (iii) terms of partial recursive recursive function (iii) In In terms of partial function application application 9 and and the the definedness definedness predicate (t-l- means "t is predicate -l$ (t$ means "t is defined" defined"),) , we we can can write write more more succinctly: succinctly: 0 xx rn r._~n(A (A -+ ~ E) B) r___n_'v'y Vy n A A xx rn := := 'v'y(y V y ( y rn r__~nA A -+ --+ xo x . yY$-l- /\ A xoy x . y rn r___~nE), B), := 'v'y(xoY-lV y ( x . y $ /\ A xoy xoy rn r__~nE). B). := where where t-lt$ expresses expresses that that tt is is defined defined (cf. (cf. next next subsection). subsection). Of Of course, course, the the partial partial operation operation ~ and and the the definedness definedness predicate predicate -l$ are are not not part part of of the the language, language, but but expressions expressions containing containing them them may may be be treated treated as as abbreviations, abbreviations, using using the the following following equivalences: equivalences: 0 ttll = = tt22 +-+ e+ 3X(tl Bx(tl = = x x /\ A tt22 = = x), x), ttl~ = x x +-+ +-~ 3 3Y y zZU(tl u(tl = = Y y /\ A tt22 = = zz /\ A Tyzu T y z u /\ A Uu Uu = = x), x), l ot2 = t-lt$ +-+ 3z(t 3z(t = = z). z). 411 411 Realizability (t1, t2 , x, y, A(t), t tt, t2 )' (tl,t2 terms terms containing containing .o, x , y , zZ,, uu not not free free in in tl,t2). However, However, note note that that the the logical complexity complexity of of A(t), where where t is is an an expression expression containing containing ., o, depends depends on on logical the complexity complexity of of t[ (On (On the the other other hand, hand, t$ is is always always expressible expressible in in I:�-form.) E~ the For metamathematical metamathematical investigations investigations it it is is therefore therefore more more convenient convenient to to formalize formalize For realizability in in aa conservative conservative extension extension HA* HA* of of HA H A in in which which we we can can treat treat "~ as aa realizability "." as primitive. Treating Treating t~ = - t2 for for partially partially defined defined tl, t2 as as an an abbreviation in aa rigorous rigorous abbreviation in primitive. way is is possible, possible, but but involves involves aa good good deal deal of of lengthy lengthy inductions, inductions, as as demonstrated demonstrated in in way Kleene [1969] [1969].. Since Since ordinary ordinary logic logic deals deals with with total total functions functions only, only, we we first first need need to to Kleene extend extend our our logic logic to to the the (intuitionistic) (intuitionistic) logic logic of of partial partial terms terms LPT, LPT, or or intuitionistic intuitionistic + -logic, in E+-logic, in the the terminology terminology of of Troelstra Troelstra and and van van Dalen[1988,2.2.3 Dalen[1988,2.2.3].. LPT LPT first first E appeared in in Beeson Beeson [1981 [1981].. appeared t! t..j. t1 t2 t1, t2 ] ] 1.3. Intuitionistic I n t u i t i o n i s t i c predicate p r e d i c a t e logic logic with w i t h partial p a r t i a l terms t e r m s LPT LPT 1.3. Variables are are supposed supposed to to range range over over the the objects objects of of the the domain domain considered, considered, so so always always Variables denote; arbitrary arbitrary terms denote, so so we we need predicate E, expressing E, expressing denote; terms need need not not denote, need aa predicate definedness; Et reads reads "t denotes" denotes" or or "t is defined".. Instead Instead of of Et Et we we shall write t$, definedness; Et is defined" shall write in the the notation notation commonly commonly used used in in recursion recursion theory. theory. in If we we also also have have equality equality in in our our logic, and read read t = - ss as as "t and and ss are are both both defined defined If logic, and and equal" equal",, we we can can express express t$ as as t = = t. and The following following axiomatization axiomatization is is a a convenient convenient (but (but not not canonical) canonical) choice choice for for The arguments proceeding proceeding by by induction induction on the length length of of formal formal deductions: deductions: arguments on the L1 A -4 A, Ll L2 A, A ---+ 4 B * ~ B, L2 L3 A - 4 B, B --+ -4 C C * ~ A ---+ 4 C, C, L3 L4 A A B -4 A, A A B -4 B, L4 L5 A C ~* A -4 C, A -4 --+ B, A -4 C --+ B A C, L5 L6 A -4 A V V B, B -4 --+ A V V B, L6 L7 A -4 C, B --+ -4 C C * ~ A V L7 --+ C, V B -4 C, C, L8 A A B -4 C ~ C), --+ C * A -4 --+ (B -4 L8 --+ C), L9 A -4 (B ---+ 4 CC) ) =v 4 CC, , L9 * A A B ---+ ..l -4 --+ A, L10 _l_ LID --+ Vx A (x r� FV(B)), L ll B --+ -4 A =~ * B -4 FV ), L11 L12 Yx A A t$ -4 A[x/t], L12 t..j. --+ L13 L13 A[x/t] A t$ --+ -4 3x A, * 3x A -4 --+ S (x r� FV(B)) L14 A -4 --+ B =~ FV (B)) L14 where t$ := := t = = t. For For equality equality we we have have ( F function function symbol, symbol, R relation relation symbol symbol of of where the the language): language) : "t "t t..j. t t. t..j., t "t A --+ A, A, A B B, A A --+ B, B A /\ B --+ A, A /\ B --+ B, B, A --+ A B /\ A --+ A B, B A B, B A B --+ A A (B A /\ B A --+ (B A /\ B A, B A B Vx A (x (B) Vx A /\ A[x/t], A[x/t] /\ t..j. :Jx A, A B :Jx A B (x t..j. t t. (F R Vxyz(x == yy ^/\ yY == zZ --+ x == z),z) , {{wvyxy(x ( z == yy --+ yy == x), Vzyz( EQ EQ V~g(~ Fi..j. -4--+ FFi~ = Fg), Fi/), V~g(R~ Vifi(Ri A/\ ~i == ~7fi -~--+ Ry--') Ri/) Vifi(i == ~7fi A/\ F~$ Basic predicates and and functions functions of of the the language language are assumed to to be be strict: strict: Basic predicates are assumed ti..j. STR F ( t ~ , .. ... ., , t~)$ ( t ~ ,, ." . . ", t,) ti..j., RR(t1 tn) -4--+ t~$ STR F(t1, tn)..i- -4--+ t~$, Note reduces to ordinary first-order intuitionistic logic Note that that this this logic logic reduces to ordinary first-order intuitionistic logic if if all all functions functions are total, terms t. are total, i.e. i.e. V2(f25), Vi(fi..j.) , since since then then t$ t..j. for for all all terms t. = A.S. Troelstra 412 For For the the notion notion "equally "equally defined defined and and equal equal if if defined" defined" introduced introduced by by t � _~ s := : = (t~ V v s$) -~ t = = s, we can can prove prove the the replacement replacement schema schema for for arbitrary arbitrary formulas formulas A we t � ~_ ~ 1\ ^ A[~/t] ~ A[~/~] t s (tt st) -+ t s, A t s A[x/t] -+ A[x/s]. Conservativeness of of defined defined functions functions 11.4. .4. Conservativeness Relative to to the the logic logic of of partial partial terms, terms, the the following following conservative conservative extension extension result result is is Relative easily proved. proved. Let Let r F be be aa theory theory based based on on LPT, LPT, such such that that easily F ft- A(2, y) y) 1\ A A(2, zz)) --+ y = = z. z. r Then we we may may introduce introduce aa symbol symbol cP CA for partial function function with with axiom axiom Then for aa partial A A(i, -+ y A(i, AX(r A(i, A(2, y) y) ++ yy = cP CA(2). A (i). B = The conservativeness conservativeness of of this this addition addition can can be be proved proved in in aa straightforward straightforward syntactic syntactic The way; the the easiest easiest method, however, uses uses completeness completeness for for Kripke Kripke models, see Troelstra Troelstra way; method, however, models, see and van van Dalen Dalen [1988,2.7]. and Let r* F* consist consist of of r F and and all all substitution substitution instances instances of of the the axiom axiom schemata schemata w.r.t. w.r.t. Let the extended extended language, language, and and let let cP(r*) r be the the result result of of systematically systematically eliminating eliminating the the the be function symbol symbol cP CA from the the elements elements of of F, and assume assume cP(r*) r to be be provable provable from from function r, and to A from r, then "r* + ( cPA) is F, then the the conservative conservative extension extension result result still still holds holds in in the the form: form: "F* + Ax AX(r is conservative over over r" F".. conservative This ned below, This extended extended result result applies applies to to HA* HA* defi defined below, since since eliminating eliminating the the symbol symbol for partial partial recursive recursive function function application from instances instances of of induction for application from induction yields yields instances instances of in the language of HA. of induction induction in the language of HA. 1.5. Formalizing eelementary recursion theory 1.5. Formalizing l e m e n t a r y recursion theory in HA* HA* HA* is the conservative extension of HA, formulated in logic of of HA* is the conservative extension of HA, formulated in the the intuitionistic intuitionistic logic partial terms, with primitive binary binary partial partial operation operation .9of of partial partial recursive recursive function function partial terms, with aa primitive ) (association ( association to to the left ) . application. tl.t2*t3.., abbreviates abbreviates (... ( . . . ((tl.t2).t3)...) ( application, the left). Note that strictness strictness entails the application application for the Note that entails in in particular particular totl$ ~ t$ A1\ t~$ for operation. Of Of course course we to require primitive recursive functions; operation. we have have to require totality totality for for the the primitive recursive functions; In all all other other cases cases the the primitive recursive functions it suffices to demand it suffices to demand 05, Sx$. In primitive recursive functions =, characterizing characterizing them them inductively inductively in in terms terms of of functions functions satisfy equations equations with with --, satisfy introduced ( e.g. z ++ 00 = = x, x + + Syy --= S(x + + y) y) ).). By By induction induction one one can can then then introduced before before (e.g. prove FX for each each primitive primitive recursive recursive function function symbol symbol F. F. prove F X ll. .. .. . xn$ for HA* can can be be given given by by A smooth smooth formalization formalization of elementary recursion recursion theory theory in in HA* A of elementary using Kleene's Kleene 's index index method method in in combination combination with with the the theory theory of of elementary elementary inductive inductive using definitions in in arithmetic arithmetic Troelstra Troelstra and and van van Dalen Dalen [1988,3.6, 3.7]. 3.7]. In In particular particular we we definitions obtain the the smn-theorem, smn-theorem, the the recursion recursion theorem theorem (Kleene's ( Kleene 's fixed-point fixed-point theorem), theorem) , the the obtain Kleene normal normal form form theorem, theorem, etc. etc. Moreover, Moreover, by by the the normal normal form form theorem, theorem, every every Kleene partial partial recursive recursive function function is is definable definable by by aa term term of of the the language language of of HA*. HA * . t1.t2 .t3 xnt • • • (t1.t2 ).t3) t.t't -+ tt t't Ot, Sxt. x x, x S • • • S(x Realizability Realizability 413 413 N o t a t i o n . If If tt is is aa term term in in the the language language of of HA HA*, then Ax. Ax.tt is is aa canonically canonically chosen chosen Notation. *, then code number number for for tt as as aa partial partial recursive recursive function function of of x, x, uniformly uniformly in in the the other other free free code variables; by by the the smn-theorem smn-theorem we we may may therefore therefore assume assume Ax.t Ax.t to to be be primitive primitive recursive recursive variables; in F FV(t) {x}.. AXI A x l ... .. x. xn.t n . t abbreviates abbreviates AXI Axl (Ax (Ax2... (Axn.t)...). V(t) \\ {x} in n .t) . . . ) . 0[] 2 . . . (Ax We note note the the following following We Lemma. � -formulas of L e m m a . In In HA HA** the the � 2O_formulas of HA H A are are equivalent equivalent to to prime prime formulas formulas of of the the � -formula of form tt = = tt for for suitable suitable t, t, and and each each formula formula tt = = ss is is equivalent equivalent to to aa � ~O_formula of form HA. HA. Proof. P r o o f . Systematically Systematically using using the the equivalences equivalences mentioned mentioned above above transforms transforms any any formula tt = = ss of of HA* HA* into into aa � ~~� -formula of of HA H A .. Conversely, Conversely, let let aa � ~~� -formula formula be be given; given; by by the the normal normal form form results results of of recursion recursion theory, theory, we we can can write write this this in in the the form 33zT(~, (~),, z) z) for for aa numeral numeral n; ~; this this is is equivalent equivalent to to no(x} ~.(~) = = no(x} ~.(~).. 0 [] form zT(n , (x) We are are now now ready ready to to formalize formalize x x rn rn A A directly directly in in HA HA*. We *. 1.6. Formalizing F o r m a l i z i n g rn-realizability r__nn-realizability in in HA* HA* 1.6. Definition. Definition. x x rn r__nnA A is is defined defined by by induction induction on on the the complexity complexity of of A A,, x x � ~ FV(A) FV(A).. P prime, prime, P "= P /\ A X4. x$ for for P := P xx r___n rn P xx rn r___n(A (A /\ A B) B) "= pox rn r__~nA A /\ AP plx r__nnB, B, := pox Ix rn B)) " := Vy(y r__nn(A ( A --t --+ B = V y (y rn r___nnA A --t -+ Xoy x. y rn r__nnB) S ) /\ A x.j., x $, xx rn rn Vy Vy A A xx rn xx rrn ___nn3y 3y A A Vy(x.y rn r__nnA), A), :"= = Vy(xoy := Ix rn "= P plx r___nnA[y/pox]. A[y/pox]. We also define aa combination combination of of realizability realizability with A; the the clauses clauses are We also define with truth, truth, xx rnt r n t A; are the the same as as for clause for for implication excepted, which which now now reads: rn , the the clause implication excepted, reads: same for r__~n, 0 B) "= := Vy(y A --+ --t Xoy B) /\ x.j. /\ (A --+ --t B). B) . n t (A (A --t -+ B) Vy(y rnt rnt A x.y rnt r n t B) A x$ A (A [] xx rrnt Remarks. rn A A is is 3-free 3-free (i.e. (i.e. does does not not contain contain 33)) for Note that, that, by by our R e m a r k s . (i) for all all A. A. Note our (i ) tt r___n definition definition of of V V in in terms terms of of the the other other operators, operators, 3-free 3-free implies implies V-free. V-free. (ii) The The clauses clauses "Ax$" "/\ x.j." have have been been added added for the cases cases of of prime prime formulas formulas and and (ii) for the of the the following following lemma. lemma. implications, in in order order to to guarantee guarantee the the truth truth of of part part (i) implications, (i) of negations we we have have xx r___n-~A rn -,A ~f-+ Vy(-~y Vy( -,y r___~n rn A) A) A/\ x$, x.j., and and xx rrn f-+ (iii) For negations ___~n~-,-,A A ++ (iii) For -,-,3z(z r___~n rn -~A) Vy(~y Vy( -,y r___~n -,A) A/\ x$ X4. ~f-+ Vy-~Vz-~(z Vy Vz-,(z r___n rn A) A) A/\ x$ x.j. ++ rn A) A) A/\ x$. x.j.. f-+ ~-~3z(z -, The The following following lemmas lemmas are are easily easily proved proved by by induction induction on on A. A. Lemma. (Dejinedness of of realizing realizing terms; terms; Substitution Substitution Property) Property) For {rn, rrnt} L e m m a . (Definedness nt} For RR E {rn, (i) (i) ~I-- tt RRAA ~--t t$, t.j., (ii) U FV(t), (ii) (x (x RR A)[y/t] A) [y/t] -== xx I~ R (A[y/t]) (A[y/t]) (x (x r� FV(A) FV(A) U x) . FV(t) , yy ~::j x). By induction induction on on the the complexity complexity of of A. A. Let Let e.g. e.g. tr___nn3yA, t rn 3yA , then then pltr__~nA[y/pot], P I t rn A[y/Pot] , PProof. r o o f . By hence hence by by induction induction hypothesis hypothesis PitS, and so so by by strictness strictness t$. t.j.. O0 P It.j., and A.S. Troelstra A.S. 414 414 t rnt A A. Lemma. HA* FI- t rnt A ~-+ A. L e m m a . HA* A A similar similar lemma lemma holds holds for for all all combinations combinations of of realizability realizability with with truth truth (i.e. (i.e. realizabilrealizabil 1 in in their their mnemonic mnemonic code) code) we we shall shall encounter encounter in in the the sequel; sequel; we we shall shall not not ities with with 1; ities bother to to state state itit explicitly explicitly in in the the future. future. We We can can readily readily prove prove that that realizability realizability isis bother HA* : sound for sound for HA*" 1. 7. TTheorem. (Soundness theorem) theorem) 1.7. h e o r e m . (Soundness A rn A rnt A HA* FI- A =v => HA* HA * FI- ttr n A A A ttr n t A HA* for aa suitable suitable term term tt with with FV(t) FV(t) Cc FV(A). FV(A) . for Proof. The proof proof proceeds proceeds by by induction induction on on the the length length of of derivations; derivations; that that is is to to say, say, P r o o f . The we have have to to find find realizing realizing terms terms for for the the axioms, axioms, and and for for the the rules rules we we must must show show how how we to find find aa realizing realizing term term for for the the conclusion conclusion from from realizing realizing terms terms for for the the premises. premises. We to We check some some cases. cases. check L5. Assume B) , tt'~r_~n(A -+ -+ C), G) , and and x r_~nA; then then p(t.x, f . x ) r_~n(BAG) Lh. Assume tt r_nn(A -+ B), (BAC),, so Ax.p(t.x, t'ox) r___~n(A --+ -+ BB A C). G) . so L14. Assume L14. Assume tr___nn(A-+ --+ B) B),, x r¢ FV(B), FV(B) , and and let let yr___nn3xA, then then plyr._nnA[x/poy], -+ S). B) . hence t[x/poy].(ply)r___nB, so hence so Ay.t[x/poy].(ply)r___nn (3x A -~ Of the the non-logical requires attention. attention. Suppose Of non-logical axioms, axioms, only only induction induction requires Suppose rn (A rn (A x rn A; p(tox, t'ox) rn Ax.p(tox, t'ox) rn (A t rn (A x yrn3x A, Pl y rnA[x/poY]' t[x/POY]O(P1Y) rnB, Ay.t[x/POY].(P1Y) rn (3x A xx r___nn rn (A[y/O] (A[y/O] A Vy(A 'v'y(A --+ A[y/Sy])). A[y/Sy])). Then Then -+ poxrnA[y/O], rnAA -+ (Plx)oyozrnA[y/Sy]. pox rn A[y/O], zz r___~n ~ (p~x).y.z rn A[y/Sy]. So So let let tt be be such such that that tt.0 oO �~ p0x, pox, to(Sy) t . ( S y ) ~� (P1X)o (p,x).y.(t.y). YO(toy). The application of theorem, or The existence existence of of tt follows follows either either by by an an application of the the recursion recursion theorem, or is is immediate immediate if if closure closure under under recursion recursion has has been been built built directly directly into into the the definition definition of of recursive function. recursive function. It It is is now now easy easy to to prove prove by by induction induction that that tt realizes realizes induction induction for for A . O0 A call this A statement statement weaker weaker than than soundness soundness is is I~- A => =~ IF- 3x(x r___~nA); we we might might call this weak weak soundness. soundness. We We can can also also prove prove a a stronger stronger version version of of soundness: soundness: A. A 3x(x rn A); A 1.8. 1.8. Theorem. T h e o r e m . (Strong (Strong Soundness Soundness Theorem) Theorem) For For closed closed A HA HA** IF A => ~ HA HA** IFn ~ r_~nA A An ~ r n t A for for some some numeral numeral n ~.. A rn A rnt A A; Proof. P r o o f . Let Let HA* HA* IF A; from from the the soundness soundness theorem theorem we we find find aa term term tt such such that that tt r_nnA, hence hence t..\.. t$. t..\., � -formula of t$, i.e. i.e. tt = - tt is is equivalent equivalent to to aa E E~ of HA, H A , say say 3x(s = - 0) 0),, and and HA H A proves proves only true E only true E~� -formulas, from from which which we we see see that that tt = - n 5 must must be be provable provable in in HA* HA* for for some some numeral numeral n ft.. Similarly Similarly for for r n t . 0 o rn A, 3x(s rnt. 415 Realizability R e m a r k . If If one one formalizes formalizes the the proof proof of of the the soundness soundness theorem, theorem, it it is is easy easy to to 1.9. Remark. see that that there there are are primitive primitive recursive recursive functions functions '1/;, r 1 r such such that that see H A fk- Prf Prf(x, 7) -t --+ Prf Prf(r HA (x , rrAAI) ( 1 (x) , Sub ( r y rn A' , yy,r , 'I/; (x))) where "Prf" "Prf" is is the the formalized formalized proof-predicate proof-predicate of of HA* HA*,, rr~7 is the the godelnumber gbdelnumber of of where c is expression �, ~, and and Sub(rB 7, x, x, rrss ,7)) is is the the godelnumber gbdelnumber of of E[x/ B[x/s]. expression Sub(r E', s] . In fact, fact, the the whole whole implication implication is is provable provable even even in in primitive primitive recursive recursive arithmetic. arithmetic. In But the the statement statement expressing expressing aa formalized formalized version version of of the the strong strong completeness completeness But theorem: theorem: Prf(x, rrAT) --+ Prf(1(x), Prf(r rn AI) A 7) Prf(x, A ') -t rre(x) 'I/; (x) rn (A closed, closed, for for suitable suitable provably provably recursive recursive 1, r '1/;) r is is not not provable provable in in HA H A (see (see sec sec(A tion 1. 1.16). tion 16). 1.10. Lemma. L e m m a . (Self-realizing (Self-realizing formulas) formulas) For For 33-free formulas, canonical canonical realizers realizers 1.10. -free formulas, exist, that is to to say say for for each each 33-free A we we have have in in HA* HA* exist, that is -free A (i) fF 3x(x 3x(x rrnn A A)) -t -+ A, A, (i) (ii) tA with (ii) fFA A -t --+ ttArn A for for some some term term tA with FV(tA) FV(tA) c C FV(A) FV(A).. A rn A (iii) A A formula formula A A is is provably provably equivalent equivalent to to its its own own realizability, realizability, i.i.e. A ++ ~ 3x(x rn A)), A)), 3x( x rn (iii) e. A iff -free formula. iff A A is is provably provably equivalent equivalent to to an an existentially existentially quantified quantified 33-free formula. (iv) Realizability is idempotent, i.e. 3x(x rn 3y(y rn A)) ++ 3x(x rn A) ; in in fact, fact, even even (iv) Realizability is idempotent, i.e. :Jx(x rn 3y(y rn A)) ++ 3x(x rn A); 3x(x rn (A ++ 3y(y rn A))) holds. := Ax.tA (x rfr tAAB := P(tA' Proof. Proof. Take Take tts=s, 0, tA^s P(tA, ttB), AX.tA,, tA tA-~B Ax.tsB (X VxA "-B ) , ttVxA -+ B := Ax.t S=SI := 0, FV(ts)), and prove prove (i) and (ii) by simultaneous simultaneous induction on A. A. (iii) (iii) and and (iv) (iv) are are FV(t (i) and (ii) by induction on B ) ) , and immediate corollaries. 0 [] immediate corollaries. := := R e m a r k . An of practical practical usefulness is the any definable definable Remark. An observation observation of usefulness is the following. following. For For any predicate with with canonical realizers (i.e. (i.e. a a predicate predicate A definable by A definable by an an 3-free 3-free formula) formula) predicate canonical realizers . . . )) we obtain an equivalent realizability we obtain an equivalent realizability if if we we read read restricted restricted quantifiers quantifiers Vx(A(x) Vx(A(x) -t -+ ... and 3x(A(x) quantifiers VxEA, VXEA, 3xEA 3xEA over over aa new domain with with realizability and 3x(A(x) A as quantifiers new domain realizability /\ .. ... . )) as clauses copied from numerical quantification, i.e. i.e. clauses copied from numerical quantification, xx rr__~n r__~nB) A x$, E) /\ x.j.. , n VyEA.B VyEA.E :"= = VyEA(x~ VY EA(x.y rn rn B[x/pox] E[x/poX] A /\ A(p0x). A(pox) . xx rn r___~n33Yy eEA.E A . B "= plx PI X r___n In short, we may may simply forget about In short, we simply forget about the the canonical canonical realizers. realizers. := realizability 1.11. x i o m a t i z i n g pprovable r o v a b l e realizability 1.11. A Axiomatizing As we we have seen already already in in the the introduction, introduction, realizability realizability validates validates more more than than what what is is As have seen provable prove realizability provable in in HA; HA; in in fact, fact, we we can can formally formally prove realizability of of in in HA* HA* an an intuitionistic intuitionistic version version of of Church's Church's thesis: thesis: CTo Vx3y A(x, CT0 Vx3y A(x, y) y) --+ -t 3zVx(A(x, 3zVx{A(x, zox) z.x) A/\ z~ z.x.j..) . is certainly certainly not not provable provable in in HA, HA, since since itit is is in fact refutable refutable in classical arithmetic. arithmetic. CT00 is in fact in classical CT This version of well-known version This version of Church's Church's thesis thesis is is in in fact fact aa combination combination of of the the well-known version which which states states "Each "Each humanly humanly computable computable function function is is recursive" recursive" and and the the intuitionistic intuitionistic 416 A.S. Troelstra A.S. reading y) which which states states that that there there is reading of of \fx3yA(x, Vx3yA(x, y) is aa method method for for constructing, constructing, for for each each given method describes humanly computable given x, x, a a y y such such that that A(x, A(x, y) y).. Such Such a a method describes aa humanly computable function. function. We now now ask ask ourselves: ourselves: is is there there aa reasonably reasonably simple simple axiomatization axiomatization (by (by aa few few We axiom in HA? axiom schemata schemata say) say) of of the the formulas formulas provably provably realizable realizable in H A ? The The answer answer is is yes, yes, 0, the axiomatized by the provably provably realizable realizable formulas formulas can can be be axiomatized by aa generalization generalization of of CT CTo, namely namely "Extended "Extended Church Church's's Thesis Thesis":": ECTa \fx(Ax ECTo Vx(Ax -+ 3y 3y Bxy) B x y ) - -+ ~ 3z\fx(Ax 3zVx(Ax -+ zox-!z.x$ /\ A B(x, B ( x , zzox)) . x ) ) (A (A 3-free) 3-free).. Lemma. * -realizable. L e m m a . Each Each instance instance of of ECTa ECT0 is is HA HA*-realizable. Proof. P r o o f . Suppose Suppose uu rrn n \fx(Ax Vx(Ax -+ --+ 3yBxy) 3yBxy) rn 3yBxy) , and Then rn Ax -+ uoxov Then \fxv(v Vxv(vr___nnAx u~ and since since A A is is 3-free, 3-free, in in particular particular \fx(Ax oxot A rn ( u xot A ) rn a (Uoxot A )) ' Then Vx(Ax -+ uu.x.tA r n 33yBxy) y B x y ) , , so so \fx(Ax Vx(Ax -+ Pl pl(U.X.tA) r n BB(x, (x, P po(U.X.tA)). Then it it is is straightforward straightforward to to see see that that p ( Ax · Pa( UoxotA) , A xv. p ( O , Pl p(Ax.po(uox~ Axv.p(0, Pl (UOxotA))) (U~ )) realizes realizes the the conclusion. conclusion. 0 [] o condition "A "A is Remark. R e m a r k . The The condition is 3-free" 3-free" in in ECTa ECT0 cannot cannot be be dropped: dropped: applying applying -,3zTxxz, Bxy 3zTxxz) V unrestricted ECTa unrestricted ECT0 to to Ax Ax := 33zTxxz zTxxz V V-~3zTxxz, B x y := (y (y = = 00 /\ A 3zTxxz) (y = -- 11 /\ A -,3zTxxz) -~3zTxxz) yields yields a a contradiction. contradiction. In In fact, fact, this this example example can can be be used used to to (y show a ! fails a! is show that that even even unrestricted unrestricted ECT ECT0! fails (ECT (ECT0! is like like ECT ECT0a except except that that 3y 3y in in the the premise premise is is replaced replaced by by 3!y; 3!y; 3!y 3!y means means "there "there is is aa unique unique y y such such that" that").) . Theorem. T h e o r e m . (Characterization (Characterization Theorem Theorem for for rn-realizability) r__nn-realizability) A) for (i) (i) HA* HA* + + ECTa ECT0 fFA A +-+ ~ 3x(x 3x(x R rt A) for R rt E e {rn, {rn, rnt} r n t },, (ii) (ii) For For closed closed A, A, HA* HA* + + ECTa ECT0 fFA A {::} r HA* HA* fFn ~t rn r___nnA A for for some some numeral numeral n ft.. Proof. P r o o f . (i) (i) is is proved proved by by a a straightforward straightforward induction induction on on A. A. The The crucial crucial case case is is A C; then rn B) -+ rn C)) (by A == _-- B B -+ --+ C; then B B -+ ~ C C +-+ ~ (3x(x (3x(xr___~nB) ~ 3y(y 3y(yr__gnC)) (by the the induction induction hypothesis) hypothesis) +-+ ~ Vx(x Vx(x rn r___~nB B -+ --+ 3y(y 3y(y rn rn C)) C)) (by (by pure pure logic) logic) +-+ ~ 3zVx(x 3zVx(x rn r__~nB B -+ --+ zox zox rn r___nC) C) (by rn B (by ECTa ECT0,, since since x x r___n B is is 3-free) 3-free) == = 3z(z 3z(z rn r__~n(B (B -+ --+ C)) C)).. (ii) (ii).. The The direction direction � =v follows follows from from the the strong strong soundness soundness theorem theorem plus plus the the lemma; lemma; {= r is is an an immediate immediate consequence consequence of of (i). (i). 0 o Curiosity classically provably Curiosity prompts prompts us us to to ask ask which which formulas formulas are are classically provably realizable, realizable, i.e. i.e. provably provably realizable realizable in in first-order first-order Peano Peano Arithmetic Arithmetic PA, P A , which which is is just just HA H A with with classical classical logic. logic. The The answer answer is is contained contained in in the the following following Proposition. 3x(x rn P r o p o s i t i o n . PA P A fF 3x(x r___~nA) A) {::} r HA HA + + M M+ + ECTo ECT0 f~- -,-,A, -~-~A, 's principle: where where M M is is Markov Markov's principle: M M Vx(A V -,A) -,A) /\ A. ~' Vx(A A ~-,-,3x 3x A A -+ 3x 3x A. Realizability 417 417 Proof. rn A) let B P r o o f . Let Let PA P A I}- 3x(x 3x(x r__nn A),, and and let B be be aa negative negative formula formula (i.e. (i.e. aa formula formula in in the the - -Nx-,(x A, V, V, -+-fragment) --+-fragment) such such that that HA H A ++MM lF- x xr__~nA ++ B(x) B(x).. Then Then PA P A I~~Vx~(xr__~nA), rn A t-t rn A) , /\, and since since PA P A is is conservative conservative over over HA H A for for negative negative formulas formulas (in (in consequence consequence of of and Ghdel's's negative negative translation), translation), also also HA H A IF --~Vx-B, i.e. HA HA + + M M IF -,-,3x(x ~3x(xr__nnA), ,Vx-,B, i.e. rn A) , Godel simpler. 0 and and thus thus it it follows follows that that HA HA + + M M+ + ECTo ECT0 IF -,-,A. ~-~A. The The converse converse is is simpler, o Extensions of of HA* HA* 11.12. .12. Extensions For For suitable suitable sets sets r F of of extra extra axioms, axioms, we we may may replace replace HA HA** in in the the soundness soundness and and r. Weak characterization characterization theorem theorem by by HA HA** + + F. Weak soundness soundness and and the the characterization characterization theorem theorem require require for for all all A AE Er F HA* HA* + + r F IF 3x(x 3x(x rrn ___n_A) A). n . (1) r Soundness requires Soundness requires for for all all A AEF HA* (2) HA* + + r F I~- tt rn r_nnA A for for some some term term t, t, and and strong strong soundness soundness requires requires (2) and and in in addition: addition" HA HA** + + r F proves proves only only true true ~~ ��-formulas. Examples Examples (a) For (a) For r F any any set set of of 3-free 3-free formulas formulas soundness soundness and and the the characterization characterization theorem theorem soundness holds. extend. extend. If If HA HA** + + r F proves proves only only true true �� ~~ -formulas, strong strong soundness holds. The The next next two examples permit permit characterization two examples characterization and and strong strong soundness. soundness. (b) Let Let -< -~ be be a a primitive primitive recursive recursive well-ordering well-ordering of of iN, provably total total and and linear linear IN, provably (b) in * ; for in HA HA*; for r F we we take take all all instances instances of of transfinite transfinite induction induction over over -< -~:: TI( -<) Vy(Vx-.y.B, x) Vx(A(Ay.B, x) -+ ~ B[y/x]) B[y/x]) ~-+ Vx Vx E Tr.B[y/x] Tr.B[y/x] 418 418 A.S. Troelstra Troelstra A.S. B Tr. for the language language extended with the for all all B in in the extended with the new new primitive primitive predicate predicate Tr. Then Then we we can extend extend rn-realizability can rn-realizability simply simply by by putting putting x ran (t (t E ~ Tr) Tr) := := tt E ~ Tr. Tr. xrn A(Tr, x) x Tr Let Let us us check check that that the the soundness soundness theorem theorem extends. extends. A(Tr, x) is is equivalent equivalent to to an an 3-free 3-free formula, so so its its realizability realizability implies implies its its truth, and x E Tr follows. follows. As As to to the the schema, schema, formula, truth, and assume assume --+ B[y/x]), or or uu rn Vx(A(Ay.B, x) -+ ur_nnVx[(x = 0 -+ B(O)) 1\ A (pox -- 11A 1\ VyB(plxoy) -+ Bx)]. u So So rn \t'X (A(Ay.B, x) B[y/x]), rn\t'x[(x = B(O)) (Pox = \t'yB(Pl xoy) Bx)]. po (uoO)o(O, 0) O)r__~n B(O), Po(UoO)o(O, rnB(O), rn (pox PPll (uox)ov (u.x).v rnB(x) B(x) if if PoX pox = = 11 and and vv r__n_n (pox = = 11 1\ A \t'yB(PIxoy). VyB(plxoy). Assume Assume \t'y(eo(P Vy(eo(p~xoy)r__~nB(plx.y)), pox = = 11.. Then Then l xoy) rnB(P lxOY)), pox = p(O, p(O, Ay.eO(P Ay.e.(pxx.y))r__nn (p0x = = 11 1\ A \t'yB(PIXO VyB(pxx.y)). Y)). vv = l xOY)) rn (Pox Therefore Therefore if oX = if P pox = 11 and and \t'y(eO(PIxO Vy(e.(plx.y) r__~nS(plx.y)) Y) rnB(PIxoy)) then then PI Pl (uox)o(O, (u.x).(0, Ay.eO(PIxO Ay.e.(plx.y)) r__nnB(x). B(x). Y)) rn Now Now we we construct construct by by the the recursion recursion theorem theorem an an ee such such that that if x = 0, po(UOO)OO P0 (u*0)*0 if x = 0, o = eox e.x � "-~ P Pl(U*x)*p(0, Ay.e.(plx.y)) if if P pox = 11,, l (UOX)Op(O, Ay.eO(PIxOY)) undefined undefined otherwise. otherwise. We induction on We then then prove prove by by induction on Tr Tr that that 'I Vxx E Tr(eox Tr(e.x rn r__~nB(x)). B(x)). This This is is straightforward. straightforward. rn { X This This example example is is capable capable of of considerable considerable generalization, generalization, namely namely to to arithmetic arithmetic enriched enriched with with constants constants for for predicates predicates introduced introduced by by iterated iterated inductive inductive definitions definitions of of higher higher 1981,IV, section level; level; see see e.g. e.g. Buchholz Buchholz et et al. al. [[1981,IV, section 6]. The mentioned also The examples examples just just mentioned also permit permit extension extension of of rnt-realizability. rnt-realizability. We end the the section some applications and rnt-realizability. We end section with with some applications of of r___~n-and rnt-realizability. 6]. rn- 1.13. 1.13. Proposition. P r o p o s i t i o n . (Consistency (Consistency and and inconsistency inconsistency results) results) (i) HA* + HA* (and (i) HA* + ECTo ECT0 is is consistent consistent relative relative to to HA* (and hence hence also also relative relative to to PA P A )). . (ii) (ii) -,Yx(A V Y - , A ) , -,(Vx-~-,S -+ --+ -,-,YxS) are are consistent consistent with with HA* HA* for for certain certain arithmetical A, B. arithmetical (iii) (iii) The The schema schema "Independence "Independence of of Premise" Premise" IP IP (-,A -+ ~ 3zS) -+ --+ 3z(-,A -+ ~ B) is is not not derivable derivable in in HA* HA* + + CTo CT0 + + M; M; in in fact, fact, HA* HA* + + IP IP + + CTo CT0 + + M M fF 11 = 0. -Nx(A -,A), -,(\t'x-,-,B A, B. -,-,\t'xB) (-,A 3zB) 3z( -,A B) = O. Proof. i ) Immediate P r o o f . ((i) Immediate from from the the characterization characterization theorem. theorem. ii ) is is a a corollary corollary of of the the realizability realizability of of CTo CT0:: take take A == - 3yTxxy, B == = 3yTxxy V ((ii) -,3yTxxy . iii) By By M, M, -,-,3yTxxy -+ ~ 3zTxxz; apply apply IP IP to to obtain obtain Vx3z(-,-,3yTxxy -+ --+ ((iii) Txxz), , then then by by CTo CT0 there there is is a a total total recursive recursive F such such that that -,-,3yTxxy -+ ~ T(x, x, Fx), and and this this would would make make 3yTxxy recursive recursive in in x. 0 O We We next next give give an an example example of of a a conservative conservative extension extension result. result. -,3yTxxy. Txxz) -,-,3yTxxy 3zTxxz; F 3yTxxy x. A 3yTxxy, B 3yTxxy \t'x3z(-,-,3yTxxy -,-,3yTxxy T(x, x, Fx), 419 419 Realizability A A, B Lemma. (rn) we L e m m a . For For A A E CC CC(rn) we have have f[- 3x(xrnA) 3x(x r n A ) --+ -+ A. A. Proof. Proof. B Byy induction induction on on the the structure structure of of A. A. Consider Consider the the case case A A - B B --+ --+ C; C; then B. Assume then B B is is 3-free, 3-free, so so there there is is aa tB tB such such that that fF B B --+ ~ tB tB rn r___~nB. Assume B B and and induction hypothesis xxr__nn(B rn (B --+ xotB..1. /\A xotB rn C hence --+ C), C), then then x.tB$ x.tBr__nnC, hence by by the the induction hypothesis C C;; therefore rn (S (B --+--+ C)) --+ (B (B --+--+ C). therefore (x (xr___n C))--+ C). 0 o Definition. CC CC(rn) r___~n-ConservativeClass Class)) is is the the class class of of formulas formulas A 11.14. .14. Definition. ( rn ) ((the the rn-Conservative such such that that whenever whenever B --+ --+ C is is aa subformula subformula of of A, then then B is is 3-free. 3-free. 0 [] B C E == , The lemma in combination with The lemma in combination with the the characterization characterization theorem theorem yields yields o is conservative over ) ( w.r.t. formulas in (rn ) : Proposition. CT is conservative over HA* P r o p o s i t i o n . HA* HA* + + E ECT0 HA* w.r.t, formulas in CC CC(.r___n)" (HA* + CTo n rn) = (rn) . (HA* + E ECT0) N CC CC(rn) = HA* HA* n N CC CC(r___~n). The proposition follows The following following proposition follows from from rnt-realizability. rnt-realizability. (Derived rules) In (i) For For sentences sentences fFA Av YB =v fF A A or or F- B B (Disjunction (Disjunction property property D DP), (i) B => P) , (ii) sentences f-F 3xA some numeral (ii) For For sentences 3xA => =~ fF A[x/nl A[x/~] for ]or some numeral n ~ (Explicit (Explicit Definability Definability for for Numbers Numbers EDN), EDN), (iii) (iii) Extended Extended Church' Church'ss Rule: Rule: for for 3-free 3-free A A ECR f-F Vx(A zox)). ECR Vx(A --+ -~ 3yBxy) 3yBxy) => =~ fF 3zVx(A 3zVx(A --+ ~ zox z.x$..1. /\ A B(x, S(x,z.x)). 1.15. 1.15. Proposition. Proposition. (Derived rules) In HA* HA* f- ( actually, ((i) i ) and ii) are Proof. i) follows Proof. ((i) from ((ii) and ((ii) are equivalent equivalent for for systems systems follows from ii) (actually, containing minimum of 1975]) . As ii) , let containing aa minimum of arithmetic, arithmetic, see see Friedman Friedman [[1975]). As to to ((ii), let F 3xA, then m, then by by the the strong strong soundness soundness for for rnt-realizability rnt-realizability fFm rh rnt r n t 3xA for for some some numeral numeral rh, so m rnt p mJ and hence f-~- A[x/p0rh]. /poml . so fF plrh r n t A[x/p0rh], and hence ((iii) iii) Assume then for Assume F Vx(A --~ 3yBxy), then for aa suitable suitable t fF t r n t Mx(A --+ 3yBxy), i.e. i.e. F VxVz(z rnt r n t A --+ Pl (t~176 rnt r n t B(x, po(tox~ Since Since tA rnt r n t A, F Vx(A --+ ~ Pl (toXotm) rnt r n t S(x, Po(t~ and So and therefore therefore F Vx(A -~ B(z, Po(t~176 So we we can can take take z = - Ax.po(tozotA). 0 0 f- 3xA, 3xA PI A[x/ o , A[x f- Vx(A --+ 3yBxy), t t rntVx(A --+ 3yBxy), f- VxVz(z A --+ PI (toxoz) B(x, Po (toxoz) ). tA A, f- VX(A PI (toXotA) B(x, Po(toxotA))), f- Vx(A --+ B(x, Po(toxotA)). z Ax.Po(toxotA). be formalized 1.16. 1.16. Remark. R e m a r k . The The DP DP cannot cannot be formalized in in any any consistent consistent extension extension of of HA HA 's argument itself Myhill [[1973a], 1973aJ , Friedman 1977a]) . We the result itself ((Myhill Friedman [[1977a]). We sketch sketch Myhill Myhill's argument ((the result of of Friedman Friedman is is even even stronger stronger).) . Assume Assume that that there there is is aa provably provably recursive recursive function function f satisfying satisfying r F Prf Prf(x, (x, rrA V S ~) ~ ((fx == 0 A Pr Pr(rAT)) V ((fx = = 1 A Pr Pr(rB~))). r So So f = = {p} (p},, and and ~- Vx3yT~xy. Let Let F enumerate enumerate all all primitive primitive recursive recursive functions, functions, i.e. i.e. ~n.F(i, n) is is the the i-th i-th primitive primitive recursive recursive function. function. Put Put f fA V B"') --+ ((Ix 0 /\ ( A"')) V ((Ix 1 /\ f f- Vx3yTnxy. F )..n .F(i, n) D(n) := poF(n, poF(n, n)) r1 0, O, D(n) := B"'))). 420 420 A.S. 1'roelstra Troelstra A.S. n then r k- Vn(Dn Vn(Dn V -,Dn) , D n ) (i.e. (i.e. Prf(k, rVn(Dn rVn(Dn V V -,D - D n )) "'7)) for for aa specific specific k), k), then Prf(k, from which which we we can can find find aa particular particular primitive primitive recursive recursive )..)~n.F(rh, n) such such from n .F(m, n) that r }- Prf(F(m, Prf(F(rh, fi), ~), rDfi r D ~ V ,D --D~7). Then Dm Orb -+ p.F(m, #.F(rh, m) rh) i= ~: 00 -+ -~ that fi "' ) . Then Prf(F(rh, m) rh),rDrh - O r b 7) /\ A Pr(,Dm"') pr(r~Drh7),, hence hence ,Dm --Orb follows, follows, since since HA* is is Prf(F(m, , rDm VV ,Dm"') consistent. If If we we start start assuming assuming ,Dm, --Drh, we we similarly similarly obtain obtain aa contradiction. contradiction. consistent. From this this we we see see that that DP DP cannot cannot be be proved proved in in HA* HA* itself; itself; for for if if DP DP were were provable provable From in HA HA*, then aa function function f S as as above above would would be be given given by by * , then in f(x) := p0(the s.t.(x does does not not prove prove aa closed closed disjunction disjunction and and Yy = - 0) 0) := P o (the least least Yy s.t.(x f(x) or (for (for some some closed closed rrA Prf(x, rrA B 7) /\ AP P0Y = 00 /\ A Prf( Prf(ply, A"')) A V B "'7,, Prf(x, or A V B"') oY = P1Y , rrA7)) or (for (for some some closed closed rrA S "'7,, Prf(x, Prf(x, rrA B 7) /\ A PlY PlY = = 11 /\ A Prf( Prf(ply, B"')) ) . or AVB A V B"') P1Y , rrB~))). This in in turn turn implies implies that that the the strong strong soundness soundness theorem theorem is is not not formalizable formalizable in in H HA*, This A* , since strong strong soundness soundness for for rn-realizability r___~n-realizability immediately immediately implies implies EDN EDN for for HA HA** + + since ECT0.· ECTo otes 11.17. . 17. N Notes Slash relations. Already Already in in Kleene [1945], aa modification of numerical numerical realizability realizability Kleene [1945], modification of Slash relations. was considered, considered, namely namely r F k-realizability; let us us use use "R" "P," as as aa short short designation designation for for this this was r-realizability; let kind of realizability. realizability. The The clauses clauses for for V, V, /\ A and and for for prime prime formulas formulas are are as as for for ordinary ordinary kind of realizability; the clauses clauses for for V, --+ become: V, 3, -+ become: realizability; the •9 n B) iff 0, Pln A, or 0, Pln B and n RR((A A V V B) iff pon p0n = = 0, p l n RRAA and and r Fr ~- A, or pon p0n i= -fi 0, pin R P.B and r Fr F- B B;; nR 1~3xA 3xA iff iff pln pin R P. A[x/pon] A[x/~-~] and and r Fr t- A[x/pon] A[x/p---~];; •9 n m, if A, then ned and •9 n B) iff nR P. (A (A -+ --+ B) iff for for all all m, if m m RR A A and and r F r k A, then n.m nom is is defi defined and nomR n.m RBB.. Kleene [1952,Example [1952,Example 22 on on page page 510] 510] used used this this notion notion to obtain aa version Church 's Kleene to obtain version of of Church's thesis. Later Kleene [1962] observed that that by by dropping dropping the the realizability realizability part and thesis. Later Kleene [1962] observed part and retaining only only the part, one one obtained obtained an an inductively inductively defined defined property retaining the provability provability part, property of of formulas quite simple proofs of of (generalizations (generalizations of) formulas which which could could be be used used to to obtain obtain quite simple proofs of) the the disjunctionand existence existence properties properties for for logic logic and and arithmetic. arithmetic. For reference, let disjunction- and For easy easy reference, let us A")) for putting riA (("F 'T slashes slashes A" for arithmetic, arithmetic, treating treating V,-, v" as as defined, defined, and and putting us define define FIA FfII r F- A A as as short short for for "FIA "flA and and Fr Fr A": A" : iff r krP prime sentences F P iff F P for for prime sentences P, P, flP iff PIA flA and and riB, fiB, B) iff rf I (A (A /\ A B) -+ B)iff B) iff F[Ffl r A F (A -+ A => f I (A '* rIB, fiB, r A[x/fz] A[x/fi] for for some some numeral fi, iff rl3xA F 3xA iff FfIIknumeral ~, rflVxA VxA iff for all numerals ~. iff rlA[x/n] flA[x/fi] for all numerals fi. formula A A is is defined defined as as FIB fiB for for some some universal universal closure closure B B of of A. A. (For riA for FIA for aa formula (For predicate have to V, _1_ ..1 have to be be added.) added.) predicate logic, logic, clauses clauses for for V, is sometimes sometimes called called aa realizability, realizability, but think itit better better to to reserve reserve the term "1" but we we think the term " I " is realizability for notions explicitly. Since " I " in in realizability for notions where where realizing realizing objects objects appear appear explicitly. Since the the "1" "riA" has has nothing nothing to to do do with with division, division, the the term term "divides" "divides" for "I" is is also also not not advisable. advisable. "FIA" for "1" Therefore we we call call the the notions notions derived from, or or similar similar to to Kleene's Kleene's FIA riA simply simply slash Therefore derived from, relations or or slashes. relations slashes. slash 421 421 Realizability In In one one respect respect FIA flA is is not not well well behaved; behaved; itit is is not not closed closed under under deduction, deduction, since since itit may flA, but but not not rl(A f I (A vV A). A) . Aczel Aczel [1968] [ 1968] gave gave aa simple simple modification modification may happen happen that that FIA, which for V, v, 3:l are are which overcomes overcomes this this defect: defect: the the deducibility deducibility requirements requirements in in the the clauses clauses for dropped, dropped, and and for for implication implication and and universal universal quantification quantification we we require require instead instead and Pr Ff- A A --+ � B, B, riB) and f I (A -~ � B) B) iff iff (PIA (flA ~:::} FIB) F[(A fI'v'xAx iff 'v'xAx and and rlA[x/n] rIA[x/n] for for all n. rlVxAx iff rr ~f- VxAx all n. Now Now F[A flA =~ A holds holds for for all all A, A , and and the the modified modified slash slash yields yields the the same same applications applications :::} Fr Ff- A as the the original original one. one. In In fact, fact, one one easily easily proves proves by by formula formula induction induction that that FfIIF A A in in the the as sense of of Kleene Kleene iff iff F[A flA in in the the sense sense of of Aczel. Aczel. The The Aczel Aczel slash slash also also has has an an appealing appealing sense model-theoretic interpretation; interpretation; see see e.g. e.g. Troelstra Troelstra and and van Dalen [1988,13.7]. [ 1988,13.7] . model-theoretic van Dalen It is is also also worth worth noting noting that that CIC GIG is is both both necessary and su]flcient sufficient for for the the validity validity It necessary and G --+ � A[x/~] A[x/n] for for some some ~" n" (Kleene ( Kleene [1962], [ 1962] of A, F C G --+ � 3xA :lxA ~:::} F C of the the rule rule "For "For all all A, Troelstra [1973a,3.1.8]). [ 1973a,3.1.8]) . Troelstra Slash operators operators in in many many variants been widely widely used used for for obtaining obtaining metamathmetamath Slash variants have have been ematical based on on intuitionistic intuitionistic logic. logic. ematical results results for for formalisms formalisms based The slash as to sentences the use of partial The slash as defined defined above above applies applies to sentences only, only, but but the use of partial reflection principles in combination combination with formalized versions of the the slash relation, reflection principles in with formalized versions of slash relation, restricted to formulas of of bounded bounded complexity, free numerical restricted to formulas complexity, may may be be used used to to deal deal with with free numerical variables, see Troelstra 1973a,3.1.16] . variables, see Troelstra [[1973a,3.1.16]. Suitable slash slash relations relations for systems beyond beyond arithmetic may be be defined defined by by concon Suitable for systems arithmetic may sidering conservative "witnessing constants" constants" for for existential sidering conservative extensions extensions with with extra extra "witnessing existential statements. The explicit explicit definability definability property for numbers then be be statements. The property for numbers EDN EDN can can then proved by proving proving soundness soundness of of slash slash for for the system (a ( a typical proved by the extended extended system typical example example is Moschovakis [ 1981 ] ) . is Moschovakis [1981]). 1973] describes Friedman Friedman [[1973] describes the the extension extension of of the the Kleene Kleene slash slash ttoo higher-order higher-order logic. logic. 1982] it In In Scedrov Scedrov and and Scott Scott [[1982] it is is shown shown that that this this extension extension of of the the slash slash is is in in fact fact see equivalent the free due to Freyd ((see equivalent to to aa categorical categorical construction construction on on the free topos topos due to P. P. Freyd Lambek and Scott [[1986]). 1986]) . Lambek and Scott Friedman 1983] use Friedman and and Scedrov Scedrov [[1983] use slash slash relations relations and and numerical numerical realizability realizability combined q-realizability, see combined with with truth truth ((q-realizability, see below) below) to to obtain obtain the the explicit explicit set-existence set-existence explicit defin ability property intuitionistic second-order second-order arith property property ((explicit definability property for for sets sets)) for for intuitionistic arithcf. 7.1 intuitionistic set metic metic HAS H A S ((cf. 7.1)) and and intuitionistic set theory theory plus plus countable countable choice choice or or relativized relativized dependent dependent choice. choice. Friedman 1986] use Friedman and and Scedrov Scedrov [[1986] use aa slash slash relation relation to to establish establish aa very very interesting interesting result: result: there there is is aa particular particular number-theoretic number-theoretic property property A(n) A(n) such such that that if if HA H A proves proves transfinite recursive binary transfinite induction induction for for aa primitive primitive recursive binary relation relation -< -~ w.r.t. w.r.t. A, A, then then -< -~ is is well-founded The corresponding well-founded with with ordinal ordinal less less than than fO e0.. ((The corresponding result result is is false false for for PA, PA, cf. 1953] . ) If cf. Kreisel Kreisel [[1953].) If transfinite transfinite induction induction is is proved proved for for -< -~ w.r.t. w.r.t. A A for for the the theory theory HA H A ++ obtained obtained by by adding adding transfinite transfinite induction induction for for all all recursive-wellorderings, recursive-wellorderings, then then -< -~ is is well-founded. well-founded. Of Of the the many many papers papers discussing discussing or or making making use use of of slash slash relations relations we we further further mention: mention: [ 1975,1976b,1977a] , Beeson 1984] , Dragalin [ 1980,1988] , fried Beeson Beeson[1975,1976b,1977a], Seeson and and Scedrov Scedrov [[1984], Dragalin[1980,1988], Friedman 1977a] , Krol 1977] , Moschovakis 1967] , Myhill [1973b,1975] , Robinson [ 1965] . man [[1977a], Krol'' [[1977], Moschovakis [[1967], Myhill[1973b,1975], Robinson[1965]. f- f- m f- ' A.S. Troelstra Troelstra 422 q-realizability. in soundness theorems for q-realizability. Since Since in soundness theorems for formalized formalized realizability realizability we we prove prove deducibility instead instead of of just truth, one one can can replace replace deducibility deducibility in in the the definition definition of of deducibility just truth, n--realizability this realizability. The clauses FF--realizability by by truth; truth; let let us us use use "q" "q" for for this realizability. The clauses for for 3, 3, -+ then then become: become" x qq (A -~+ B) B ) : = Vy(yqA A A -+ ---4x . y qqBB) ) A x$, m x (A x q 3yA x�3yA ::== \fy(y q A I\. A x.y I\. xt, plx �-A[y/pox] I\. A[Y/Pox]. := plx q A[y/pox] A A[y/pox]. Such Such aa q-variant q-variant was was used used in in Kleene Kleene [1969] [1969] to to obtain obtain derived derived rules rules for for intuitionistic intuitionistic analysis-with analysis with function function variables. variables, q-realizability q-realizability is is also also not not closed closed under under deducibility deducibility (think of HA; then (think of an an instance instance A of of CT CT00 unprovable unprovable in in HA; then A is is q-realizable, q-realizable, but but A V VA is is not). not). Grayson Grayson [1981a] [1981a] observed observed that that an an Aczel-style Aczel-style modification modification could could be be used used instead q-realizability; this instead of of q-realizability; this corresponds corresponds to to our our rnt-realizability. rnt-realizability. Friedman Friedman and and 1984 use -realizability and Scedrov Scedrov [1984] use rn r__~n-realizability and q-realizability q-realizability to to obtain obtain consistency consistency with with Church Church's's thesis, thesis, the the disjunction disjunction property property -and and the the numerical numerical existence existence property property for for set on intuitionistic the existence set theories theories based based on intuitionistic logic, logic, with with axioms axioms asserting asserting the existence of of very very large cardinals, thereby demonstrating large cardinals, thereby demonstrating that that the the metamathematical metamathematical properties properties just just mentioned, mentioned, often often regarded regarded as as aa test test for for the the constructive constructive character character of of aa system, system, are are not not affected affected by by assumptions assumptions concerning concerning large large cardinals. cardinals. A [ A A A ] Shanin's S h a n i n ' s algorithm. a l g o r i t h m . In In aa number number of of papers papers Shanin Shanin presented presented aa systematic systematic way way of of making making the the constructive constructive meaning meaning of of arithmetical arithmetical formulas formulas explicit. explicit. His His method method is is logically equivalent equivalent to to rn-realizability, rn-realizability, as as shown shown by by Kleene Kleene [1960] [1960].. On On the the one one hand hand logically Shanin's Shanin's algorithm algorithm is is more more complicated complicated than than realizability, realizability, on on the the other other hand hand it it has has the being the the advantage advantage of of being the identity identity on on 3-free 3-free formulas. formulas. 2. Abstract 2. A b s t r a c t realizability r e a l i z a b i l i t y and a n d function f u n c t i o n realizability realizabillty 2.1. in the 2.1. After After the the leisurely leisurely introduction introduction to to numerical numerical realizability realizability in the preceding preceding section, we we now now turn turn to to variations variations and and generalizations. generalizations. In In order order to to distinguish distinguish easily easily section, the the various various concepts concepts of of realizability, realizability, we we shall shall use use aa certain certain mnemonic mnemonic code: code: !:r_ signifies signifies !! n__signifies signifies f f__ signifies signifies !!! m signifies signifies 1 t__ signifies signifies 1 ! signifies signifies � e_ signifies signifies "realizability" "realizability",, "numerical" numerical or or "by "by numbers" numbers",, "by "by functions" functions",, "modified" "modified",, "combined with "combined with truth" truth",, "Lifschitz "Lifschitz variant variant of' of",, "extensional" extensional .. "realizability by Thus Thus "rft" " r f t " refers refers to to "realizability by functions functions combined combined with with truth" truth" etc. etc. Strictly Strictly speaking, speaking, the the !: 1" is is redundant redundant in in many many of of these these mnemonic mnemonic codes. codes. A A simple simple generalization generalization of of numerical numerical realizability realizability is is realizability realizability with with aa different different set abstractly, the set of of realizing realizing objects objects and/or and/or different different application application operator; operator; abstractly, the realizing realizing objects objects with with application application have have to to form form aa combinatory combinatory algebra. algebra. We We shall shall first first sketch sketch an abstract version namely realizability an abstract version of of numerical numerical realizability, realizability, namely realizability in in aa combina combinatory consider the interesting special tory algebra algebra with with induction, induction, then then consider the interesting special case case of of function function Realizability Realizability 423 realizability. realizability. (The theory 2.2. 2.2. Definition. D e f i n i t i o n . (The theory APP) A P P ) The The language language is is single-sorted, single-sorted, based based on on LPT. LPT. The only only non-logical non-logical predicate predicate is is N N (natural (natural numbers) numbers).. There There is is an an application application The operation operation .9and and constants constants o0 (zero) (zero),, S S (successor) (successor),, P P (predecessor) (predecessor),, o, PPlI (pairing P,, P P0, (pairing with with inverses) inverses),, P k, (combinators),, d d (numerical by cases). cases). (numerical definition definition by k, ss (combinators) (We (We have have used used the the same same symbols symbols for for pairing pairing and and inverses inverses as as in in the the case case of of HA H A **,, even corresponds to even if if there there is is a a slight slight difference difference in in syntax: syntax: p(t, p(t, t') t') in in HA* HA* corresponds to (pt)t' (pt)t' in in APP A P P . ).) For For tl.t tlot22 we we simply simply write write (tl (tit2), and we we use use association association to to the the left, left, i.e. i.e. t2 ) ' and . . . ((t tIt t i t 22 ... .. t. ~tn is is short short for for ((... ( ( t Il tt22 )t ) t 33)) .. ... t. ntn) ). . Axioms Axioms for for the the constants: constants: NO, Nx NO, N x --+ -+ N(Sx) N ( S x ) , , Nx N x --+ -~ N(Px), N(Px), P(St) � ~_ t, t, PO PO = = 0, O, 00 =1= ~ St, St, P(St) kx kx$, kty � ~_ t, t, sxy..!., sxy$, stt't stt't"" � ~_ tt tt"(t't"), ..!. , kty " (t't" ) , ox..!., P IX..!., P pxy. pxy$,.!., P p0x$, plx$, p 0o(ptx) ( p t x ) == tt,, P p lI((pPxxt) t ) == t, t, Nu Nu A A NNv v --+ -+ ((uu =1= ~ vv --+ d x y u v = x) A d x y u u = y. y. x) A dxyuu = dxyuv --+ = Observe Observe that that by by the the general general LPT-axioms LPT-axioms we we have have tt'..! tt'$. --+ --+ t..! t$. A A t'..!.. t'$. Finally Finally we we have have induction: induction: N(A --+ A[x/Sx]) --+ A[x/O] A[x/O] A A \/x Vx E e N(A ~ A[x/Sx]) --+ \/x Vx E e N. N. A D [:] The A-abstraction defined induction on The combinators combinators k, k, ss permit permit us us to to have have A-abstraction defined by by induction on the the construction construction of of terms: terms: Ax.t k t for for tt a a constant constant or or variable variable =/:. ~ x, x, AX.t ::=- kt AX.X Ax.x := := skk, skk, AX.tt' := S(AX.t) (AX.t'). Ax.tt' := s(Ax.t)(Ax.t'). For For this this definition definition FV(AX.t) FV(Ax.t) == -= FV(t) FV(t) \\ {x}, {x}, t'$ --+ --+ (AX.t)t' (Ax.t)t' � ~_ t[x/t'] t[x/t'] if if t' t' is is free free for for x x in in t, t, t'..!. AX.t. Ax.t$.!. for for all all t. t. It It is is not not generally generally true true thae that 2 [y/t'] , (t[y/t']) � if x x rt r FV(t'), FV(t'), y y =/:. ~ x x then then AX. Ax.(t[y/t']) ~_ (AX.t) (Ax.t)[y/t'], if (1) (consider (consider e.g. e.g. tt == - yy,, t' t' == - kk) kk) but but we we ddoo have, have, for for x x rt r FV(t') FV(t'), , yy rt r FV(t FV(t"), ~ x x " ) , yy =/:. [x/t"] . [y/t'])t" � (2) t" t"$..!. --+ -~ ((AX.t) ((Ax.t)[y/t'])t" ~_ t[x/t"][y/t'] t[x/t"][y/t'] == - t[y/t'] tly/t'][x/t"]. Property definition of Property (1) (1) can can be be guaranteed guaranteed by by an an alternative alternative definition of abstraction: abstraction: Xx.x A ~x.x := := skk, skk, Xx.t := := kt A'x.t kt if if x x rt r FV(t) FV(t),, Xx.tt' := := s(Xx.t) (Xx.t') if xx Ee FV(tt'), A'x.tt' s(A'x.t)(A'x.t')if FV(tt'), A 2This was overlooked in the proofs in 'Troelstra Troelstra and van Dalen [1988,section [1988,section 9.3], 9.3], but is easily remedied by the use of (2). Strahm [1993] [1993] recently showed that this problem might also be overcome by considering considering instead of APP APP based on combinators, a theory based on lambda-abstraction lambda-abstraction as a primitive (without a �-rule ~-rule of the form t = Ss =} =~ AX.t Ax.t = - AX.S Ax.s !) and an explicit substitution operator as part of the language. A.S. A.S. Troelstra Troelstra 424 424 t. but then we lose the property property that that AX.t..j.. )~x.t$ for for all A recursor recursor and and a a minimum minimum operator operator but then we lose the all t. A may point operator may be be defined defined with with help help of of aa fixed fixed point operator (see (see e.g. e.g. Troelstra Troelstra and and van van which permits APP Dalen Dalen [1988,9.3]) [1988,9.3]) which permits us us to to define define in in A P P all all partial partial recursive recursive functions. functions. It follows follows that that HA H A can can be be embedded embedded into into APP A P P in in aa natural natural and and straightforward straightforward It way. way. R e m a r k . Partial Partial combinatory combinatory algebras algebras are are structures structures (X, ~ k, s),, k k i~ ss,, satisfying satisfying Remark. k, s) the structures we the relevant relevant axioms axioms above; above; in in such such structures we can can always always define define terms terms forming forming IN, and aa copy copy of of IN, and appropriate appropriate S, S, P, P, P p,, PO p0,, PI pl,, and and we we might might simply simply have have postulated postulated induction for models it induction for this this particular particular copy copy of of IN IN.. However, However, in in describing describing models it is is more more convenient convenient not not to to be be tied tied to to a a specific specific representation representation of of IN IN relative relative to to the the combinators. combinators. Also, we Also, we want want to to leave leave open open the the possibility possibility that that the the interpretation interpretation of of N N is is non nonstandard. standard. 0, 2.3. 2.3. The T h e model m o d e l of of the t h e partial p a r t i a l recursive operations operations PRO PRO The basic basic combinatory combinatory algebra algebra is is The Axy.x, Axyz.xozo(yoz)); (IN, (IN,., Axy.x, Axyz.x.z.(y.z)); 0, where where ~ is is partial partial recursive recursive function function application application for for IN IN.. 0, 0, S, S, P P get get their their usual usual interpretation interpretation (more (more precisely, precisely, we we choose choose codes codes Ax.Sx, Ax.Sx, Ax.Px etc.; etc.; for for d d take take l J. Auvxy[u ·. sglx sglx - Yll + + v(l":'lx v(1-1x - Yl)]. Auvxy[u A* we In In H HA* we can can prove prove PRO PRO to to bbee a a model model of of APP A P P , , iinn the the sense sense that that APP A P P I- }A =~ HA* HA* I-F- [A]pRo. Here Here and and in in the the sequel sequel we we use use "interpretation "interpretation brackets" brackets":: given given indicate the some some model model M A/I,, we we use use I r i s , [A]~ to to indicate the interpretation interpretation of of term term t, formula formula A in in the the model model M M . . Thus Thus "[A]~ holds" holds" means means the the same same as as M M F ~ A. A. 0 y [A]PRO. - A ::::} A Ax.Px y) [t]M, [A]M "[A]M (Abstract (Abstract realizability) realizability) x!:A x r_A in in APP A P P is is defined defined by by t, 2.4. 2.4. Definition. Definition. ::= xx !:r_P P - P P /\ Ax x$..j.. for for P P prime, prime, (A /\ A B) B) := := ((p0xr_A) A ((plxr_B), PI X !:B), xxr_!: (A Pox !: A) /\ xx!: r_(A (A -+ B) B) := := Vy(y r_A -+ ~ X x.y !: r B) B) /\ A x..j. x$,., \iy(y!:A oy x!:\iyA := x r_Vy A := \iy(xoy!:A), Vy(x.y r_A), xx !:r_33yyA A := PI plxx !: r_A[y/p0x]. 0 := A[Y/PoxJ. 0 !: \iyEN.A becomes literally \iyz(z!: Remark. Remark. x xr_VyEN.A becomes literally Vyz(zr_ (y (y E E N) N) -+ ~ (xoyoz) (xoyoz) r.A)). r_A)). It It is is easy easy to special clause to see see that that realizability realizability with with a a special clause for for the the relativized relativized quantifier quantifier x!:'\iyEN.A := \iyEN(xoy!:' xrZVyEN.A := VyCN(x.yrZ A) A) is is in in fact fact equivalent. equivalent. * , i.e. The The 3-free 3-free formulas formulas play play the the same same role role in in APP A P P as as they they do do in in HA HA*, i.e. 3-free 3-free formulas with their their truth, formulas have have canonical canonical realizing realizing terms, terms, their their realizability realizability coincides coincides with truth, and and equivalence equivalence of of realizability realizability with with truth truth for for aa formula formula A means means that that A is is equivalent equivalent to B 3-free; schema characterizing to aa formula formula 3xB, 3xB, B 3-free; the the schema characterizing !:-realizability r_-realizability is is an an Extended A Axiom Axiom of of Choice Choice EAC EAC 3yBxy) 3z\ix(Ax zox..j.. A zox)) (A Extended \ix(Ax Vx(Ax -+ --+ 3ySxy) -+ ~ 3zVx(Ax -+ --+ z.x$ /\ A B(x, S ( x , z . x ) ) (A is is 3-free) 3-free) 425 425 Realizability Realizability etc. etc. We may may specialize specialize !:-realizability r_-realizability to to PRO-!:-realizability PRO-r_-realizability by by interpreting interpreting APP APP We in PRO. PRO. It It is is then then not not difficult difficult to to show show that that the the resulting resulting realizability realizability of of HA H A (as (as in embedded embedded in in the the obvious obvious way way into into APP) A P P ) becomes becomes equivalent equivalent to to rn-realizability; r__nn-realizability; cf. cf. Renardel Renardel de de Lavalette Lavalette [1984]. [1984]. 2.5. 2.5. Partial Partial continuous continuous function function application application Another model of the model Another important important model of APP A P P is is the model peo PCO of of functions functions with with partial partial continuous application. Before continuous application. Before we we can can discuss discuss this, this, we we need need some some preliminaries. preliminaries. Notations. N o t a t i o n s . From From primitive primitive recursive recursive p, p, Po, P0, PI Pl we we can can construct construct primitive primitive recursive recursive n-tuples of natural numbers, numbers, with encodings p pnn of of n-tuples of natural with primitive primitive recursive recursive inverses inverses p� p~ encodings finite sequences sequences of numbers to (0 ::::; (0 < ii < < n). n). We We may may also also assume assume finite of natural natural numbers to be be coded coded nx - l ; . . ,,nx-1; onto IN; we . . . ,,nx-1) nx -l ) for nite sequence onto IN; we write write (no, (n0,... for (the (the code code of) of) the the fi finite sequence no, n0,... is the the (code (code of of the) the) empty empty sequence sequence (which (which may may be be assumed assumed to to be be equal equal to to 00;; see see (( )> is below) below).. If m m is is (a code of) of) a a sequence, sequence, lth(m) lth(m) is is its its length; is aa primitive primitive recursive recursive If (a code length; *9 is concatenation concatenation function function for for codes codes of of sequences. sequences. We We abbreviate abbreviate · nn ~�mm ."= = )m) 3 n ' ( n .*nn' '=m , , - 3n'(n " - ( n - ~(n m A�n ~m m )1\ , n =I- m) , nn - ~-<m m := .i; := (x) . 9= The The primitive primitive recursive recursive inverse inverse function function AXY.{X)y A x y . ( x ) y of of sequence sequence encoding encoding satisfies satisfies m m) y = m) y = m = = (no, ( n o , .. .. .., , n nx-l> =~ ((m)y = n nyy for for y y< < x, ((m)y = 00 for for y > x. x. x -l) => For reasons reasons of of technical technical convenience convenience we we assume assume monotonicity monotonicity in in the the arguments arguments for for For encodings of pairs, n-tuples n-tuples and encodings of pairs, and finite finite sequences: sequences: n' -+ -+ p(n, p(n, m) m) < < p(n', p(n', m), m), m m < < m' m' -+ --+ p(n, p(n, m) m) < < p(n, p(n, m') m'),, nn << n' and and similarly similarly for for p-tuples; p-tuples; nn ::::; lth(n) = lth(m) 1\ m.. _ n n *9m; m; lth(n) = lth(m) A Vx((n) V x ( ( n ) ~ x ::::; <_ (m) ( m ) ~x)) -+ --+ n n ::::; < m For For example, example, for for p p we we may may take take p(n, p(n, m) m) = = !l ((n n + + m) m ) ( n(n + + m m + + 11)) + + m. m. These These monotonicity conditions in n-tuples (c~1,..., c~n) of monotonicity conditions in fact fact enforce enforce ((}) = - 0. Encoding Encoding of of n-tuples of sequences sequences is is obtained obtained by by (ao, . .. .. , oln) ":=- AX. (Olo, A x . ppn n ( o(aIX, l l x , . . . , olnx). For use For initial initial segments segments of of functions functions we we use := ((>, ) , ~(x+l)'-+ 1) : = (~O,...,olx>. 0 ~0"= Q x, O. . . . , anx). (aO, . . . , ax). . , an) a (x aO � (aI, . . . , an) Definition. Definition. Elementary E l e m e n t a r y Analysis A n a l y s i s EL EL is is aa conservative conservative extension extension of of HA H A obtained obtained by by adding adding to to HA H A variables variables (c~, f3, ~, 1', ~/, 0, 5, f.e)) and and quantifiers quantifiers for for (total) (total) functions functions from from IN IN to infinite sequences to IN IN (i.e. (i.e. infinite sequences of of natural natural numbers) numbers).. There There is is A-abstraction A-abstraction for for explicit explicit (t aa numerical numerical term, definition recursion-operator Rec definition of of functions, functions, and and aa recursion-operator Rec such such that that (t term, p (t , t' )) = Cp(t, r aa function function term; term; r (t , t' t')) ::= t')) Rec(t, Rec(t, = Rec(t, r = t, t, Rec(t, Rec(t, r = r Rec(t, r = Induction Induction iiss extended extended to to all all formulas formulas iinn the the new new language. language. (a, ¢ ¢ ¢)(O) ¢ ¢)(Sx) ¢ (x, ¢)(x)). 426 426 A.S. Troelstra A.S. The functions functions of of EL EL are are assumed assumed to to be be closed closed under under "recursive "recursive in", in" , which which isis The A: expressed by by including including aa weak weak choice choice axiom axiom for for quantifier-free quantifier-free A: expressed QF-AC QF-AC Vn3mA(n, m) m) --+ -t B~VnA(n, 3aVnA(n, an) an) Vn3mA(n, 0 [] In EL EL we we introduce introduce abbreviations abbreviations for partial continuous continuous application application Definition. In Definition. for partial 0), ~(~) = 33y(a(j3y) y ( ~ ( ~ y ) == x9 ++ 11 ^/\ vVyl.n.a(x *9n)(~) n)(fJ) == ~/x) 0, or or equivalently equivalently "I ""Ix) A/\ (~0 Vx3y((~(~ Vx3y(a(x *9~y) j3y) == ?x /\ Vy' - is aa conservative conservative extension extension of of EL EL based based on on the the logic logic of of partial partial Definition. EL* EL * is Definition. terms, ~ . c ~ ( ~ ) have have been >.afJ.alfJ and and )>.afJ.a(fJ) been added added as as primitive primitive operations. operations. terms, to to which which "~.(~1~ Numerical lambda-abstraction satisfies: Numerical lambda-abstraction satisfies: ~ ^/\ ((>.x.t).J. ~ . t ) ~ . -+ ~.t)~ = t[~/~], ((>.x.t) ~ . t ) ~.J.. ++ ++ Vx(t.J. w(t~). -t ((>.x.t)s = t[x/s], s.J.. .) . For function function application we require require For application we ¢t /\ r¢.J... Ct~.J.. ++ ~ r¢.J.. A (The implication implication from hold since is supposed to imply imply totality totality (The from left left to to right right must must hold since ¢.J.. r is supposed to For Rec Rec we we have have of the function function denoted denoted by of the by r¢.) For Rec(t, r¢).J.. ++ t.J.. /\ Rec(t, ++ t$ A r¢.J... [] o 2.6. 2.6. The T h e model m o d e l of of the t h e partial partial continuous continuous operations operations PCO PCO This IN; the This model model has has as as domain domain all all the the total total functions functions from from IN IN to to IN; the application application is is II * . Some defined defined above. above. The The elementary elementary theory theory of of the the model model can can be be formalized formalized in in EL EL*. Some work PCO is model of work is is needed needed to to show show that that PCO is actually actually aa model of APP. APP. The class Definition. Definition. ((The class of of neighbourhood neighbourhood functions) functions) aa E e K* K* := aO c~0 = = 00 /\ A Vnm(an rum(an > > 00 -t -+ an c~n = = a(n a(n *9m)) m)) /\ A VfJ3x(a(j3x) V~3x(c~(/?x) > > 0) 0).. o O Crucial is Crucial is the the following following Lemma. L e m m a . To To each each function function term term ¢, r and and each each numerical numerical term term tt of of EL* EL*,, we we can can construct construct function function terms terms ", ~r E E K* K*,, ~tt E E K* K* respectively, respectively, such such that that (i) (i) ",la ~1~ � ~- ¢; r (ii) (t la).J.. iff t.J.. ; (iii) (iii) t.J. t$. -t -+ ( (r t la)O = = t; t; (iv) C FV(t) \\ {a} ) (iv) FV( FV((I)t) C FV(t) {a},, FV(",) FV((I)r t their their free free variables. variables. C C FV(¢) FV(r \\ {a}, {c~}, (~t, (~ primitive primitive recursive recursive in in t , ", 427 427 Realizability Realizability () ( ) are proved proved by by simultaneous simultaneous induction induction on on the the construction construction of of nunu PProof. r o o f . (i)-(iv) i - iv are merical merical and and function function terms. terms. The The reason reason that that we we need need aa function function term term with with partial partial · ( . ) ) isis to represent numerical term term (instead (instead of of application application .(.)) continuous application application ]I to continuous represent aa numerical that may contain contain function-terms function-terms as as subterms, subterms, which which all all have have to to be be that aa numerical numerical term term t may defined strictness condition defined by by the the strictness condition of of the the logic logic of of partial partial terms; terms; this this is is aa II~ rr -condition and cannot cannot be be expressed expressed by by definedness definedness of of aa numerical term. and numerical term. 4>0 == 0, 0, (I)t0 t O = 0. O. We consider aa few few typical cases. In In all all cases put ~r We consider typical cases. cases we we put 1. t -== x. x. Take Take ~t(z Similarly for for t -== 0. O. t ('z ** ~n) == xx ++ 1.1. Similarly Case 1. a. Take Take Case 2. 2. r -== a. . * aa + 1 if < ) = f a z + 1 ifz < n, ( ~ ( s 9 ~n) = ( 00 otherwise. otherwise. == r162 Take Take Case 3. t =_ /\ z+ + 11 A *9 ~n) = (u)y + + 1 if B u < n V y < l t h ( u ) ( ~ r 4>(u) = z ++ 1) (I)r162 ~r 1) A /\ ~r 4>(x *9 &n) > 00 A/\ (I)r 4>("') (~ ", (x *9 &n) > 0, 0, (x *9 &n) = 0o otherwise. otherwise. == r 'ljJ1(. Take Take Case 4. r -z+ + 1 iif Bue( *9 &n) = (u)y + *9 u) = z + "'le(x 0, r162 *9~n) = r162163 + 1) A ",(x r162 *9c)n) > 00 /\ A e(x ~(s *9 ~n) > 0, ",(x 1) /\ otherwise . 0o otherwise. 0 The other cases are to the the reader. reader. [] The other cases are left left to It is is now in EL pca is model of APP do not confuse It now easy easy to to prove prove in EL** that that PCO is aa model of A P P (do not confuse the A-abstraction EL* with with the the defined A-operator in in APP) For example, example, an an in EL* the A-abstraction in defined A-operator A P P ) . . For interpretation of of Is] [s] is is found found as follows. If If r is term of of EL * , let let us us write interpretation as follows. is a a function function term EL*, write Aa.¢J for for the the ~r 4> given by the the lemma. lemma. Then Aa.r given by Then we we put put t g = Case t an) t Case ¢J { z z n, '" z n Case t ¢J('ljJ) . {z iE lu", (Y an) = (u)y an) > an) > =z an) = Case 4. ¢J {z Elu n y u Y an) = (u)y an) = u) = z an) > an) > ( ¢J [s]pco := AaA,3ky .(al 'Y ) I CB I 'Y) · Clearly la) l,3 = total, hence Clearly ([s] ([s][a)[~ = A A')'(al')')l(/3l'~' since all all terms terms 4> ~r are are total, hence defined. defined. 'Y (al 'Y ) I (,3I 'Y)),, since Moreo M o r e o vver e r (([s] l a) I,3) I 'Y � = (al 'Y) I (,3I 'Y ) · We spell out definition of realizability for application which We spell out a a definition of realizability for this this application which is is not not literally literally what -realizability in what one one obtains obtains by by interpreting interpreting r. r_-realizability in pca, PCO, but but equivalent equivalent to to it: it: 2.7. 2.7. Definition. D e f i n i t i o n . (Realizability by functions) With With each each formula formula A A of of EL* EL* we we rf A (a rf.r FV(A) associate associate a ar__tf A (a FV(A))) as as follows: follows: aa r_f rf P "= P /\ Aa aS+ (P prime prime),, (Realizability by functions) P ::== P (P ) ::== := rf A) aar__f rf (A (A /\ A B) B) "= (poa (p0ar___f A) /\ i ((plar_ff B), Pl a rf B), aa rf +, r__f(A (A --+ -e B) B) := \f,3(,3 V/9(fl rf r___fA A --+ -e al,3 al/9 rf r___fB) B) /\ Aa aS, aa rf r___f\fx Vx A A "= \fx(aIAn.x V x ( a l A n . x rf r___fA), A), := aa rf r___f\f,3 Vfl A A ": \f,3(al,3 vfl(alZ rf rAA), A), rf aa rrf :lx A a !BxA "= Pl p ~ a r ! A A[xj(poa)O] [x/(poa)O], , aar___f rf :l,3 a rf A[,3jpoa]. :"= 313 A A plarAA[19/poa]. = Pl rft-realizability rft-realizability is is defined defined by by modifying modifying rf-realizability rf-realizability as as before. before. 0 El Now Now the the theory theory runs runs to to aa large large extent extent parallel parallel to to numerical numerical realizability. realizability. The The role role ECTo is taken over by the following schema of of of ECT0 is taken over by the following schema of Generalized Continuity: a(A --+ la) ) (A la+ /\A B(a, GC a ( A --+ -~ :I,3B(a, B ~ B ( a , ,3)) ~)) --+ ~ :I'Y\f BBVa(A --+ 'Y -),[a$ B ( a , 'Y ^/[a)) (A :I-free B-free) GC \fV a(A Generalized Continuity: ) 428 428 A. S. Troelstra A.S. Troelstra where where 3-free 3-free in in EL* EL* is is defined defined as as before; before; in in EL, EL, 3-free 3-free formulas formulas correspond correspond to to the the class of of formulas formulas constructed constructed from from tt = = ss,, 3x 3x(t s),, 3a 3c~(t s) by by means means of of -+, --+,/\, A, V. V. class (t == s) (t == s) 2.8. Proposition. P r o p o s i t i o n . (Examples (Examples of of applications) applications) For For EL EL** we we have have 2.8. (i) (i) I-I- Va(A V~(A -+ 3(3B(a, 3/~B(~, (3)) fl)) � =~ I-I- 3,Va(A 37V~(A -+ ,Ia.!. ~ 1 ~ /\ ^ B(a, B(~, ,Ia)) ~1~)) for almost almost negative negative AA (Generalized ). (Generalized Continuity Continuity Rule Rule GCR GCR). (ii} For For 3aAa 3c~Ac~ closed, closed, I-F 3a 3c~Ac~ =v there there exists exists some some nf~ such such that that I-I- A( A({~})A (ii) Aa � {n}) /\ Vm(nom.!.) e. ifif I-F 3a A(a) , there Vm(~om$),, i.i.e. 3c~A(c~), there is is aa total total recursive recursive function function ff such such that that A(I). I-~- A U) . (iiO CC C C ((rf r f )) n r7 EL* = = CC CC(rf) r7 ((EL* + GC GC), where the the conservative conservative class class CC CC ((rf) (rf ) n EL* + ) , where rf ) (iii) for rf rf -realizability -realizability is is defined defined in in complete complete analogy analogy to to CC CC ((rn). rn) . for Proof. ((Of The strong strong soundness soundness theorem theorem yields yields in in this this case case aa particular particular Proof. Of ((ii).) ii) . ) The function Aa, hence function term term ¢J r such such that that I-F ¢J C rft r f t 3a 3ceAc~, hence IF PI P l ¢J Crft r f t A[a/po¢J] A[c~/p0r , and and thus thus I-F A[ce/p0r A[a/po¢J] /\A PP0r o¢J'!'; PP0(r o (¢J) is is aa closed closed function function term term in in the the language language of of EL EL which which may be be written written as as {n} (fi,}.. ((5} is short short for for/~x.(~ox).) may ({ n} is AX. (nox) . ) D[] E x a m p l e s of of extensions extensions 2.9. Examples Bar Induction Induction for for Decidable predicates is is an an induction principle Bar Decidable predicates induction principle Vc~3x P(ax) P(&x) /\ A Vn(Pn Vn(Pn -+ --~ Qn)/\ Qn)A Va3x Vn(Pn V -,Pn) ,Pn) /\A Vn(Pn vn(vm(Q(,~ * 9(m») (m)) -+ ~ Qn) -+ ~ Q( ) Vn(Vm(Q(n An principle is BI! with P(ax) /\ Vn(Pn An equivalent equivalent principle is BI! with Va3x Vc~3xP(&x)A Vn(Pn V - ,Pn) , P n ) replaced replaced by by Va3!x ax) . BID BIo implies the Fan Vc~3!x P( P(hx). implies the Fan theorem theorem for for Decidable Decidable predicates predicates FANo Va�(33x ,An) -+ 3zVa�(33x�z FAND Vc~<_~3xA(ax) A(hx) /\ A Vn(An Vn(An V ~An) 3zVc~<_/~3x<_zA(ax) A(&x) where c~ a< (3 := Vx(ax � (3x) . Since rf-realizability validates continuity where fl "= Vx(c~x < ~x). Since r___f-realizability validates continuity principles, principles, � BID BIo in fact the stronger in fact the stronger FAN V~3x 3zVc~ Kleene [1965b] [1965b] contains contains the the first first formalization formalization of of r__f-realizability. rf-realizability. In In this this paper paper Kleene Kleene shows a.o. that that formulas formulas such such that that all all their their subformulas subformulas in in the the scope scope of of an an Kleene shows a.o. ( provable classically). classically) . universal function function quantifier quantifier are are 3-free 3-free are are true true iff iff r___f-realizable rf -realizable (provable universal Kleene [1969] [1969] aa thorough thorough formalized formalized treatment treatment of of function function realizability realizability is is given, given, In Kleene In (version of) of) rft-realizability. rft-realizability. Another Another notion notion of of realizability realizability by by functions functions of aa (version also of also is is found found in in Moschovakis Moschovakis [1993,1994]. [1993,1994] . Vesley [1996] [1996] considers considers aa notion notion of of function-realizability function-realizability which which combines combines the the idea idea Vesley of r__f-realizability rf-realizability with with the the topological topological model model of of elementary elementary intuitionistic intuitionistic analysis analysis in in of Moschovakis [1973] (itself (itself an an adaptation adaptation of of aa model model due due to to Scott Scott [1968]). [1968]). Moschovakis [1973] Realizability Realizability 429 429 Barendregt [ 1973] used used an an abstract abstract version version of of realizability realizability to to show show consistency consistency of of Barendregt [1973] an axiom axiom of of choice choice with with combinatory combinatory logic. logic. Staples[1973,1974] Staples [ 1973,1974] used used realizability realizability with with an combinators for for higher-order higher-order logic logic and and set set theory. theory. Abstract Abstract realizability realizability for for theories theories combinators was introduced introduced by by Feferman[1975,1979]. Feferman [ 1975,1979] . including AAPP including P P was Of the the researches researches using using abstract abstract versions versions of of realizability realizability we we further further mention mention Of Beeson [ 1977b,1979b,1980,1985] , Renardel Renardel de de Lavalette[1984,1990]. Lavalette [ 1984,1990] . Beeson[1977b,1979b,1980,1985], 3. M Modified 3. o d i f i e d rrealizability ealizability In the the case case of of numerical and function function realizability, realizability, we we started with the the concrete concrete In numerical and started with and ended ended with with the the abstract abstract version. version. and For For modified modified realizability realizability on on the the other other hand, hand, itit is is advantageous advantageous to to start start with with the abstract setting, and and afterwards afterwards to to specialize specialize to to more more concrete concrete versions. versions. The The the abstract setting, abstract setting setting of of modified modified realizability realizability is is not not aa type-free type-free theory theory such such as as A APP, abstract PP, sketched above, above, but but a a system system HAw H A w of of intuitionistic intuitionistic finite-type finite-type arithmetic. arithmetic. sketched 3.1. 3.1. Description Description of of intuitionistic intuitionistic finite-type finite-type arithmetic arithmetic HAw HA w The set set of type symbols symbols T is generated generated by by the the clauses E T type of the The of finite finite type T is clauses 00 E T ((type of the natural l' E T then (a formation of natural numbers numbers);) ; if if a, ~T then (a •x 1' T)) E T T ((formation of product product types types)) and and formation of . . . for (a --+ --+ 1' T)) E E T T ((formation of function function types) types).. We We use use a, a, aa', ' , ... .. ,. , 1', T, 1" T ',, .. .. .., , p, p, p', pt,.., for arbitrary arbitrary type type symbols. symbols. As 00) , nn + As an an alternative alternative for for (a (a --+ -+ 1' T)) we we write write (aT) (aT);; 11 is is short short for for ((00), + 11 for for (nO) in type type symbols (nO).. Outer Outer parentheses parentheses in symbols are are usually usually omitted. omitted. Further Further saving saving on on parentheses parentheses is is obtained obtained by by the the convention convention of of association association to to the the right, right, i.e. i.e. aOaIa2a3 a0ala2a3 . . . (( abbreviates (al (a2a3))) ; al al x• 2(2) an) . abbreviates (ao (ao(al(a2a3))); al x• a2 x• . . . X • an abbreviates abbreviates ((-.. ((al 1 5 x (3) . . . X Xan). The The language language of of of of intuitionistic intuitionistic finite-type finite-type arithmetic arithmetic HAw H A w is is aa many-sorted many-sorted language language with with variables variables (XU (x ~,, yU, y~, zU z ~,, .. ... .) of of all all types; types; for for each each a a ET T there there is is aa primitive primitive equality , and there are some constants listed below, and an application operation equality = =~, and there are some constants listed below, and an application operation u A PP u from . . . In App~,~ --+ l'T and and a a to to T. T. For For arbitrary arbitrary terms terms we we use use t, t, t', t', til, t " , .. .. .., , s, s, s', s ~,S", s", ..... In 'r from aa --+ u If tt EE aa --+ order order to to indicate indicate that that tt is is aa term term of of type type a a we we write write tt E Ea a or or tft.. If --+ 1' T,, t' t ~E E a, a, then PPu, (t , t') E T. For PPu ( t , t') then A App~,~(t,t') For A App~,~(t, t') we we simply simply write write (tt') (tt') or or even even tt' tt';; we we save save on on r ,r . . . (( tl t2 )ta) . . . tn) parentheses parentheses by by association association to to the the left: left: tl t l ... .. t. ntn is is short short for for ((... ((tlt2)t3)'" tn).. As As constants constants we we have have for for all all a, a, 1', T, pp E E T: T: o0 EE 00 ((zero), zero) , S EE 00 successor) , 00 ((successor), u,r E aT(a ) , ((pairing) pairing) pp~'~ aT(a xx 1'7), p�, ) 1', ((unpairing) unpairing) p~'~r EE (a (a xx T)a, pr, p~'~r E (a xx 1'T)T, ,u,r E (paT) kk u~'~ (pa)pr ((combinators), combinators) , s p'~'~ (paT)(pa)pT ,r E aTa, sp rU recursor . r ~ E a(aOa)Oa ((recursor). Here Here again again we we use use the the same same symbols symbols (k, (k, s, s, P,P0, Pl) for for operations operations closely closely analogous analogous to to the the operations operations denoted denoted by by the the same same symbols symbols in in APP A P P . . We We shall shall drop drop type type sub suben tting" en superscripts superscripts wherever wherever it it is is safe safe to to do do so; so; types types are are always always assumed assumed to to be be "fi "fitting" i.e. ifif tt' ). ((i.e. tt' is is written written then then tt EE aT, aT, t't' EE aa for for suitable suitable a, a, 1'T). ) S ) P,Po, PI) A.S. A.S. Troelstra Troelstra 430 430 The logical logical basis of HAw H A W is is many-sorted many-sorted intuitionistic intuitionistic predicate predicate logic logic with with The basis of equality; equality; the the constants constants satisfy satisfy the the following following equations: equations: y, p( ( PXY) = Po (pxy) = x, PI po(pxy) = x, Pt(pxy) = y, p(pox)(plx) = X, x, PO X) (PIX) = kxy = = x, xyz = = xz(yz), x z ( y z ) , rxyO rxyO = = x, rxy(Sz) = = y(rxyz)z. y(rxyz)z. x, ssxyz x, rxy(Sz) kxy Finally, --+ xx = y, and Finally, we we have have 00 i= :/: Sx, S x , Sx Sx = - Sy Sy ~ - y, and full full induction. induction. (Actually, (Actually, define aa predecessor Sx = - Sy S y --+ --~ x x = - y y is is redundant, redundant, since since we we can can define predecessor function function P P such such Sx that P(St) that P(St) = = x.) x.) There There is is defined defined A-abstraction, A-abstraction, as as in in for for APP A P P ; ; we we can can use use the the second second recipe recipe x. HA mentioned mentioned in in 2.2, 2.2, with with h.t Ax.t = - kt k t for for all all tt not not containing containing x. H A is is embedded embedded in in H A Win in the the obvious obvious way. way. HAw 3.2. T h e systems s y s t e m s I-HA I-HA w W,, E-HA E-HA W W 3.2. The I-HAw I - H A W,, iintensional n t e n s i o n a l finite-type finite-type arithmetic arithmetic is is a a strengthening strengthening of of HAw H A W obtained by obtained by including all aa EE TT,, satisfying including an an equality equality functional functional e" ea E E aa aaOO for for all satisfying ex"y" 1, ex"y" ex~y ~ � <_1, ex~y o = - 0 O +-+ ++ x" x~ = = y", ya, so so equality equality is is decidable decidable at at all all types. types. On the the other other hand, e x t e n s i o n a l finite-type finite-type arithmetic arithmetic E-HA E - H A WW is is obtained obtained from from On hand, extensional H A Wby by adding adding extensionality extensionality axioms for all a: HAw axioms for all types types a: "T +-+ "T "T \lx" Vx ~ (y" ( y ~Txx =T --~ Zz ~ xX) ) ~ y y ~ ="T --a~ Zz ~ . . This permits us This permits us to to define define equality equality of of type type a a in in terms terms of of =0 =0,, via via yy ="T := \lx"(yx =or Zz := Vx ~(yx =T = r zx) z x ) ,, Z. YY ="XT ==• Zz := := PoY PoY = =a" PoZ poz /\ A PlY PlY =T =~ PI plz. Therefore Therefore we we may may assume assume E-HAw E - H A W to to be be formulated formulated in in a a language language which which contains contains only only =0 - o as as primitive primitive equality, equality, so so that that prime prime formulas formulas are are always always decidable. decidable. 3.3. 3.3. Models M o d e l s of of HAw HA ~ A A model model of of HAw H A W is is given given by by a a type type structure structure (M", (M~, "'" ,,~)aET, with M" M~ a a set, set, "'" ,,~ an an ) ,,ET , with equivalence equivalence relation relation on on M" Ma,, plus plus suitable suitable interpretations interpretations of of ApP" App~,~ and the the various various ,T and constants. constants. (i) FTS, the (i) FTS, the Full Full Type Type Structure. Structure. Take Take lN IN for for Mo M0,, for for M"T M ~ take take the the set set of of all all functions functions from from M" M~ to to MT M~,l for for M"XT M~x~ take take M" M~ x x MT M~;; this this is is the the full full type type structure; structure; "'" ~ at at each each type type is is set-theoretic set-theoretic equality, equality, and and it it is is obvious obvious how how to to interpret interpret App App and and the the constants. constants. (ii) HRO, (ii) HRO, the the Hereditarily Hereditarily Recursive R e c u r s i v e Operations. Operations. Put Put HROo := lN, HRO0 := IN, HRO"XT oZ EE HRO" HRO~x~ := := {z {z :: P p0z HRO~ /\ AP p iIZ z E E HROT}, HRO~}, HRO"T H R O ~ := := {z {z :: \Ix Vx E E HRO,, HRO~ (zox (zox E E HROT)}. HRO~) }. App partial recursive App is is interpreted interpreted as as partial recursive application application (i.e. (i.e. as as 0), .), =" =~ as as equality equality between between numbers (as (as elements elements of numbers of HRO" HRO~),) , := 0, 0, [S] := Ax.Sx, [k] := Axy.x, [s] := := Axyz.xz(yz) [0) [0] := [ s l := Ikl := A yz. z(yz),, := Axy.p(x, [p] := hx.pox, Ax.pox, [[Pl] PI] ::== AX,PIX, Po] := iP] := Axy.p(x, y) y),, [[Po] AX.plx, = Axy .sg l x [r] e] ::= [r] := := aa suitable suitable code code for for aa recursor, recursor, [[e] Axy.sglx - YI. Yl. Realizability Realizability 431 431 The existence existence of of aa suitable suitable code code for for aa recursor recursor either either follows follows directly directly from from the the The definition of of recursive recursive function, function, or or by by an an application application of of the the recursion recursion theorem theorem yielding yielding definition solution rr to to r.(x, ro(x, y,y, O) 0) ~_ -::,- O, 0, r~ ro(x, y,y, Sz) Sz) ~_ -::,- y.(r~ yo(ro(x, y,y, z), z), z), z) , as as in in Troelstra Troelstra and and van van aa solution Dalen [1988,3.7.5]. [1988,3.7.5] . The The result result isis aa model model of of II-HAw Dalen - H A w.. (iii) HEO, HEO, the the model model of of the the Hereditarily Hereditarily Effective Effective Operations. Operations. We define aa partial partial (iii) We define by equivalence relation relation ,"'(J between natural natural numbers numbers for for each each aa EE Tr by equivalence ~ between y xx ~0 "'0 Y where where y, : = xx == y, := X "'(J x r Y := ( Pox "'(J PO y) 1\ (P IX "'r PlY)' "'r x.z' 'izz' (z ,,~ "'(J z'z' --+ -+ xxoz "'r y.z' yoz' A1\ xxoz : = Vzz'(z xx ,"'(Jr , ~ yY := . z ,,~ . z ,,~ xoz' A1\ y.z y.z ,,~ "'r y.z') Yoz') HEO(J ::= z. zz E HEO~ - zz ~"'(J z. s , p, For the rest, rest, the the definition definition of of interpretations interpretations of of 0, 0, S, S, k, k, s, proceeds as as For the PI , rr proceeds Po, Pl, P , P0, as ~"'(J and we we obtain obtain aa model model of of EE-HAw before, we we interpret interpret -=(J before, ~ as , , and - H A ~.. (J mr A, A, for for formulas HAw 3.4. 3.4. Definition. D e f i n i t i o n . (Modified (Modified realizability) realizability) We We define define xxr m__rr formulas of of H A ~,, by induction induction on the complexity complexity of of A as follows. follows. The The type type a is determined determined by by on the A as a of of xx is by the structure structure of of A the A.. XO mr ((tt == s ) :: == ((tt == s), mr B, xx mr B)) := A A1\ Plx m__rr(A ( A 1\ AB := pox po x mr m__rrA plxm__rrB, xx mr :='iy(ymr A -+ m__~r(A ( A -+ --~ B) B):=Vy(ym__~rA --+ xy xy mr m__~rB), B ), xx mr m__rr'ix Vx A A xxm__rr mr :Jz 3z A A := := 'iz(xz Vz(xz mr m__rrA) A),, mr A[z/pox]. := PIX := plxm__rr A[z/pox]. We -realizability. All We also also consider consider mrt-realizability, mrt-realizability, which which is is similar similar to to rnt rnt-realizability. All clauses clauses are implication clause are the the same same as as for for mr, m__~r,the the implication clause excepted, excepted, which which now now reads reads xx mmrt mrt A mrr tt B) r t (A (A -+ -+ B) B) := := 'iy(y Vy(ymrt A -+ --+ xy xym B ) A1\ (A (A -+ --+ B). B). 0 [] Remark. nition (cf. R e m a r k . In In the the usual usual defi definition (cf. Troelstra Troelstra [1973a,3.4.2]), [1973a,3.4.2]), one one realizes realizes with with sequences A. The sequences of of terms terms r, t', of of length length and and types types depending depending on on the the structure structure of of A. The attractive attractive feature feature of of this this definition definition is is that that :J-free 3-free formulas formulas are are literally literally self-realizing: self-realizing: for for :J-free 3-free A, A, tmr t*m__rrA A := : - A, A, so so tis t'is empty. empty. For For our our definition definition above, above, the the choice choice of of type type 00 for for the the realizing realizing objects objects of of prime prime formulas formulas is is somewhat somewhat arbitrary; arbitrary; aa more more canonical canonical choice choice might might have have been been obtained obtained by by (conservatively) (conservatively) adding adding aa singleton singleton type type to to HAW H A ~ and and letting letting the the single single element element of of this this type type realize realize tt = = ss iff iff true. true. A A concrete concrete version version of of mr-realizability mr-realizability is is obtained obtained by by interpreting interpreting HAW H A ~ in in aa model model M A4;; this this yields yields M-mr-realizability. ~4-m__rr-realizability. The The difference difference between between Kleene's Kleene's realizability realizability and and mr-realizability m___rr-realizability becomes becomes clear clear by by comparing comparing rn-realizability r___nn-realizability and and HRO-mr-modified HRO-m__~r-modified realizability realizability of of statements statements of of the the form form 'iy-Nz-,Txyz Vy-~Vz-~Txyz -+ --+ B B For For rn, r___~n,this this requires requires aa realizer realizer tt which which must must be be applicable applicable to to the the canonical canonical realizer realizer Ay.O Ay.0 of of 'iy-,'iz-,Txyz Vy-~Vz-~Txyz if if this this is is true. true. On On the the other other hand, hand, in in the the case case of of HRO HRO,'iz-,Txyz isis true mr-realizability m_xr-realizability toAy.O toAy.O must must be be defined, defined, whether whether 'iyVy-~Vz-~Txyz true or or not. not. In In 432 Troelstra A.S. Troelstra other words, words, in in modified modified realizability, realizability, realizing realizing objects objects for for implications implications have have aa larger larger other domain of of definition definition than than what what is is required required by by "pure" "pure" realizability. realizability. domain Soundness now now takes takes the the form form Soundness 3.5. T h e o r e m . (Soundness) (Soundness) 3.5. Theorem. H A WfFA A =} ~ HAw H A WfF ttm__~r A ;\ A tt m rt A A for for some some term term tt with with FV(t) FV(t) c C FV(A). FV(A). HAw mr A mrt P r o o f . By By aa straightforward straightforward induction induction on on the the length length of of derivations. derivations. D. 0. Proof. As noted above, for 3-free formulas there are canonical realizers, and truth truth and and As noted above, for :I-free formulas there are canonical realizers, and realizability coincide coincide for for :I-free 3-free formulas. formulas. Therefore Therefore the the :I-free 3-free formulas formulas of of HAw HA W realizability play the the same same role role w.r.t. w.r.t, mr-realizability mr-realizability as as the the :I-free 3-free formulas formulas of of HA* HA* w.r.t. w.r.t. play rn-realizability. rn -realizability. For an an axiomatization axiomatization we we need need the the following following For 3.6. Lemma. L e m m a . For For each each instance instance F F of of one one of of the the following following schemata schemata 3.6. -free) , B) --+ (A --+ --+ :Ix"" 3x~B) --4 :ly""(A 3y~(A --+ --4 B) B) (y (y rf. r FV(A) FV(A),, A A :I3-free), (A TVX"" A(x, zx), T A(x, y) AC Vx~3yrA(x, y) --+ -~ :lZ,," 3z~rVx~A(x, zx), AC VX"":ly there is is aa term term tt such such that that fF tt mr m__~F, F, r with with FV(t) FV (t) c C FV(A) FV (A).. there IPef IPef 3.7. Theorem. of modified realizability) 3.7. T h e o r e m . (Axiomatization (Axiomatizationofmodifiedrealizability) H A W+ + AC AC + + IPef IPef fFA A f-+ +4 :l3x(x m__~A) A) r HAW x(x mr and for H E { H A W, I H A W, E H A W } and for H {HAW, I-HAw, E-HAW} H + + AC AC + + IPef IPef fFA A {:} r H H f~- tt mr m__rrA A H for some with FV(t) FV(t) C C FV(A) \{x}. for some tt with FV(A) \{x}. R e m a r k . In T with with decidable prime formulas the schema schema Remark. In aa theory theory T decidable prime formulas IPef IP ef iiss implied implied by by the IP To see 3-free formulas formulas are see this, this, note note that that in in TT :I-free are logically logically equivalent equivalent IP defined defined in in 11.13. .13. To to negated negated formulas, formulas, since since -....,~..B.., B of-+ B to induction on of B). B (by (by induction on the the construction construction of B). On On the IP is is m_xr-realizable A W,, so so the holds with the other other hand, hand, IP mr-realizable in in H HAw the preceding preceding theorem theorem also also holds with IP replacing replacing IPef, for H equal to - H A Wor - H A W.. IPef , for H equal to II-HAW or EE-HAw IP 3.8. TTheorem. (Applications of modified realizability) realizability) Let H EE { H A W, I - H A W, 3.8. h e o r e m . (Applications of modified Let H E-HAW}, and let let H' H' be be H H 4± IPef IPef 4± AC. AC . Then Then E-HAw } , and (i) H' is is consistent. consistent. (i) H' H' Ff- AA Vv BB =~ =} H' H' Ff- AA or or H' H' Ff- BB (for (for AA yV BB closed) closed) (Disjunction (Disjunction Property Property (ii) H' (ii) DR). DP). (iii) ~] for (iii) H' H' Ff- 3x~A :Ix"" A =~ =} nH'' Ff- A[x/t A[x/t""] for aa suitable suitable term term t,t, FV(t) FV(t) Cc FV(A) FV(A) \\ {x} {x} (Explicit (Explicit Definability Definability El)). ED). (Rule of ~hoi~ choice ACR). (iv) nH'' ~f- W~v~A(~, VX"":lyT A(x, y) y) ~=} nH'' ef- 3z~W~A(~, AC R). (/,) zx) (Rul~ :lZ,,"T VX"" A(x, z~) H'' Ff- (A (A -4 --+ 3x~B) :Ix"" B) =v =} H' H' Ff- 3x~(A :lx""(A --4 --+ B) B) where where AA isis 3-free :I-free (IPRef-rule). (lPRef -rule). (v) H (v) 433 433 Realizability 3.9. Concrete C o n c r e t e forms f o r m s of of modified m o d i f i e d realizability realizability 3.9. The proof-theoretic proof-theoretic applications applications of of mr-realizability m__~r-realizability obtained obtained by by specifying specifying aa model model The for HAw H A ~ have have in in fact fact two two "levels "levels of of freedom" freedom"": (a) (a) the the choice choice of of aa model model M M , , defi definable for nable in aa language language .c s say, say, and and (b) (b) the the theory theory formulated formulated in in .c s which which is is available available for for proving proving in facts about about M M , , i.e. i.e. the the metatheory metatheory for for M M.. facts By M definable nable iinn .c By "".M definable iinn .c s " we we do do not not mean mean that that M M is is globally globally defi definable s , but but only that that locally, locally, for for each each A A of of HAw H A ~,, we we can can express express [A]M [ A ] ~ by by aa formula formula of of .c. s Thus Thus only choosing HRO for for M M is is the the first first level level of of freedom, freedom, and and choosing choosing some some theory theory r F in in the the choosing HRO language of of HA HA** for for proving proving facts facts about about HRO HRO is is the the second second level level of of freedom. freedom. language An interesting interesting example example of of this this occurs occurs in in connection connection with with two two models models of of HAW HA ~ An which are are similar similar to to HRO HRO and and HEO HEO respectively, respectively, but but based based on on partial partial continuous continuous which function application application 1] instead instead of of partial partial recursive recursive application application 9 function The Intensional Intensional Continuous Continuous Functionals Functionals ICF ICF are are an an analogue analogue of of HRO; HRO; we we give give The the intuitively intuitively simplest simplest defi definition (which does does not not mean mean the the technically technically slickest) slickest) of of the nition (which the types: types: the ICFo ICF0 := "- IN, IN, ICF00 := "- IN IN � ~ IN, IN, ICFoo ,8 E ICFu (a(,8).j.) } ((aa =IICF~o := "= {a {a : 9\f VfleICF~(a(fl)$)} # 0), 0), ICFuo ICF0, := "- {a {a : 9\fx(An.a((x) Vx()m.a((x},* n) n) E e ICFu)} ICF,)} ((aa =I# 0), 0), ICFou ,8 E ICFu (a l ,8 ES ICFT)} a, T ICFu ICF~rT := "= {a {a : 9\f YflSICF~(a]fl ICFr)} ((a, ~- =I# 0). 0). Application is is then then defined defined in in the the obvious obvious way: way: Appu App~,0(a, fl) :"a(fl),, Appo App0,~(a, n) Application = a(,8) ,u (a, n) ,o (a, ,8) P U, T (a , ,8 := (n) *9 m) 9= Am.a( Am.a({n} m),, Ap App~,~(a, fl)) := "- al,8, a[fl, etc. etc. Equality Equality at at type type a a is is interpreted interpreted by by equality of of numbers numbers (for (for a a = - 0) or functions functions (for (for a a =I~ 0). 0). 0) or equality The The Extensional Extensional Continuous Continuous Functionals Functionals ECF ECF are are related related to to ICF ICF in in the the same same way as related to defines hereditary equivalence relation based based way as HEO HEO is is related to HRO" HRO: one one defi nes aa hereditary equivalence relation ECF coincides coincides with with Kleene's Kleene 's countable countable functionals Kreisel's on ]1 instead on instead of of oo.. ECF functionals or or Kreisel's continuous functionals. continuous functionals. Both and ECF ECF are locally defi definable in the the language language of EL*, soundness Both ICF ICF and are locally nable in of EL * , and and for for soundness of and ECF-mr ECF-m___rrrelative relative to EL* nothing nothing more more is of ICF-m__~r ICF-mr and to EL* is needed. needed. But But additional additional axioms added added to may result in different different properties properties of of the the models, and hence axioms to EL* EL * may result in models, and hence of M-mr-realizability. A/t-m__~r-realizability. Two Two mutually incompatible additional additional axioms axioms we we can can add add to of mutually incompatible to EL* are are FAND version of of Church Church's's Thesis EL* FANo and and aa version Thesis CT Va3xVy(ax x.y). CT \fa3x\fy(ax = xoy). CT states that the the function EL** range the total total recursive CT states that function variables variables in in EL range over over the recursive functions; functions; the incompatibility of CT with FAND FANo follows follows from well-known example of the incompatibility of CT with from Kleene's Kleene's well-known example of recursive tree tree well-founded well-founded w.r.t, w.r.t. all all total total recursive recursive functions functions but but not not aa primitive primitive recursive w.r.t, w.r.t. all all functions, functions, since since the the depth depth of of the the tree tree isis unbounded unbounded (cf. (cf. Troelstra Troelstra and and van van Dalen Dalen [1988,4.7.6]). [1988,4.7.6]). Assuming FAND, FANo , we we can can show show that that ICF ICF and and ECF ECF contain contain aa Fan Fan Functional Functional r162 ¢>uc Assuming satisfying of Uniform satisfying the the axiom axiom for for aa Modulus Modulus of Uniform Continuity Continuity MUC V ~ V ~a�"(\f,8� < ~ V f l < ~ ( ~"(( r( {i (¢>uc z"() == ~(r ~ == zz,8). fl). j3(¢>uc z"() ~� zza \fzz22V\f"(\f If we we add add MUC MUC to HAW we can can m_xr-interpret mr-interpret FAND. FANo. If, If, on the other other hand, hand, we we If to H A ~,, we on the EL* ++ CT CT as as our our metatheory metatheory for for ICF-m__~r, ICF-mr, we we can can realize realize aa statement statement positively positively use EL* use 0 • 434 434 A.S. 1'roelstra Troelstra contradicting MUC. MUC. See See Troelstra Troelstra [1973a,2.6.4, [1973a,2.6.4, 2.6.6, 2.6.6, 3.4.16, 3.4.16, 3.4.19]. 3.4.19]. contradicting As an an example example of of an an application application of of aa concrete concrete version version of of mr-realizability m__Er-realizabilitywe we can can As show e.g. e.g. the the consistency consistency of of HAW HA ~ + + IP IPeff + + AC AC + + WC-N WC-N + + FANn FAND + + EXT EXT1,0, show 1 ,o , e where WC-N WC-N is is the the schema schema Va3n Va3n A(a, A(a, n) n) -+ -4 Va3n, Va3n, mVf3(iim mVt~(~m = = (3m ~m -+ -4 A(f3, A(~, n)) n)),, where and EXT EXT1,0 is Vaf3z2(a Vaflz2(a = =/~f3 -+ -4 Z2a z2ce = = z2f3) zZ/3).. (Use (Use ICF-mr-realizability ICF-m__Er-realizabilitywith with EL* EL* + + and 1,o is FAND as as metatheory.) metatheory.) FANn N o t a t i o n . Henceforth Henceforth we we write write mrn, mrn, mrf mrf for for HRO-mr HRO-m__Erand and ICF-mr-realizability ICF-m__Er-realizability Notation. respectively. D. [:]. respectively. 3.10. Notes Notes 3.10. Modified realizability realizability was was first first formulated formulated by by Kreisel Kreisel [1962b]; [1962b]; aa concrete concrete version version Modified equivalent to to our our ICF-mr-realizability ICF-mr-realizability was was used used in in Kleene and Vesley Vesley [1965] [1965].. equivalent Kleene and Cook and Harnik [1992] Cook and Urquhart Urquhart [1993] [1993] and and Harnik [1992] apply apply mrt-realizability mrt-realizability to to bounded bounded arithmetic and related systems, systems, improving improving on on earlier earlier results results obtained obtained by by Buss Buss [1986] [1986] arithmetic and related by means means of of numerical numerical realizability. by realizability. Vesley used modified Vesley [1970] [1970] used modified realizability realizability to to obtain obtain consistency consistency of of intuitionistic intuitionistic analysis with with aa restricted restricted form form of of IP IP (Vesley's (Vesley's principle) principle).. Moschovakis Moschovakis [1971] [1971] used used analysis realizability interpretation interpretation to obtain consistency consistency of of aa weak weak version version of of aa modified modified realizability to obtain Church's's thesis thesis with Kleene's's system system for for intuitionistic intuitionistic analysis analysis (i.e EL with with bar bar Church with Kleene (i.e EL induction induction and and GC GC for for the the case case A A == _= 00 = = 0), 0), together together with with Vesley's Vesley's Principle. Principle. The The weak Church's thesis may be as: "each "each numerical not not weak version version of of Church's thesis may be stated stated as: numerical function function is is not not recursive". observed that that the the modified realizability recursive" . In In Troelstra Troelstra [1973a,3.4.15] [1973a,3.4.15] itit is is observed modified realizability of in the of Moschovakis Moschovakis [1971] [1971] is is essentially essentially abstract abstract modified modified realizability realizability interpreted interpreted in the recursive elements ICF, and and that that the the consistency consistency type structure structure consisting consisting of of the the recursive type elements of of ICF, proof covers covers in fact full full IP IPeff (Troelstra [1973a,3.4.18]). proof in fact e (Troelstra [1973a,3.4.18]). Some examples of of papers using or realizability are Some further further examples papers using or discussing discussing modified modified realizability are Dragalin[1968], van Oosten [1990], Scedrov and Dragalin[1968] , Diller[1980], Diller[1980], Grayson[1981b,1982], Grayson[1981b,1982], van Oosten [1990], Scedrov and Vesley [1983]. also 9.8 on Berger Berger and [1995], Berger, Berger, Schwicht Vesley [1983]. See See also 9.8 on and Schwichtenberg Schwichtenberg [1995], Schwichtenberg and Seisenberger [1997] enberg and Seisenberger [1997].. 4. D 4. erivation oof f tthe h e Fan Fan R ule Derivation Rule This section section is is devoted devoted to to an an "indirect "indirect application" application" of of modified modified realizability: realizability: itit is is This shown from mrt.-realizability, may shown how how closure closure under under the the rule rule of of choice choice ACR, ACR, obtained obtained from mrt-realizability, may be combined combined with with the the (intrinsically (intrinsically interesting) interesting) notion notion of of "majorizable "majorizable functional" functional" be to Fan Rule. to obtain obtain closure closure under under the the Fan Rule. We can relative to We can define define the the so-called so-called majorizable majorizable functionals functionals relative to any any finite-type finite-type structure. They They are are introduced introduced via via aa relation relation of of majorization, majorization, defined defined as as follows. follows. structure. 435 435 Realizability Realizability t* ut, to, t E a, a: t* ot := t* ;::: t, t* maj uxrt := Pot* maju Pot P1t* majr PIt, t* rut := Vy*y(y* majr y t*y* maj u ty, tOy). t E Maj := 3t* ut ("t Lemma. t* maj maj tt =v => t* t* maj maj t*. t* . L e m m a . t* Proof. Induction on on the the type type of of t.t. P r o o f . Induction + EE 0aOa by 4.2. define tt+ 4.2. Definition. Definition. For For each each tt EE 0a Oa we we define by induction induction on on the the structure structure of a. a. of + z, t(Sz)} t+ O == to, t+ (Sz) == max{t t(Sz)} for for aa == 0, 0, t+O tO, t+(Sz) max{t+z, := An.[Ay((An.tny)+n)] for a == a1a tt+ + := An.[Ay((An.tny)+n)] for a -a, a2, 2, + n)((An,P1(tn)) +n) for for aa - al a1 xx a2. a2 . = An.p((An.po(tn)) tt+ += )m.p((An.po(tn))+n)((An.pl(tn))+n) Lemma. F++ maj L e m m a . II/Vn f V n ~O (Fnmaj maj Gn), Gn), then then F maj G+, G +, G. G. Proof. a. Let Let Fn, P r o o f . We We use use induction induction on on a. Fn, Gn Gn E E a. a. Case (i) O. Almost Almost immediate. immediate. Case (i) a a =_ O. Case (ii) (ii) aa === ala2. a1a2 ' The yields Case The assumption assumption yields s* maj Fns* maj Fns, Gns Gns s* maj ss =~ Fns* maj Fns, for all nn EE IN. IN. By By the the induction for all induction hypothesis hypothesis we we have have + , (An.Fns), +, (An.Gns), (An.Fns*) (~n.Fns*) + + maj maj (An.Fns) ()m.Fns) +, (~n.Fns), (An.Gns) (~n.Gns) +, (~n.Gns), (1) Now nition of Now by by defi definition of F+, F +, G+ G + and and beta-conversion: beta-conversion: (An.Fns*) ( )m.Fns* ) + +kk = - F F+ +ks ks** + ks + k == FF+ks (An.Fns) (An.Fns)+k + ks (An.Gns)+k ()m.Gns)+k =G = G+ks If If n n ;::: _ m, m, we we obtain obtain from from (1) (1) +ns, FF+ms, +ns** maj +ms, Gms. +ms, Fms, +ns** maj FF+ns maj F F+ns, Fms, F F+ns maj G G+ms, Gms. + and F+n maj m, Fm, Since m, itit follows and from from this this F+n maj F F+m, Fm, G+m, G+m, Gm. Gm. Since n n ;::: _ m, follows that that FF ++ maj maj G G+ +,, G. G. Case Case (iii) (iii) a a - a1 al x x a a2. We are are given given Vn(Fnmaj Vn(Fn maj Gn), Gn), so so 2 . We Vn(pi(Fn) (i EE {0, {O, I}). Vn(pi(Fn) maj maj pi(Gn)) p,(Gn)) (i 1}). SO So we we have have Vn((An.pi(Fn))nmaj Vn((An.pi(Fn))n maj (An.Pi(Gn))n) (An.pi(Gn))n) and induction hypothesis and hence hence by by the the induction hypothesis (An,Pi(Fn)) ( ) m . p , ( f n)) + + maj maj (An,Pi(Gn)) (~n.pi(Gn)) ++,, An )m.pi(Gn). ,Pi(Gn). From m, ii EE {O, From this this we we obtain obtain for for n n ;::: > m, {0,1I}} +m, (An,Pi(Gn)) +n maj +m, (An.Pi(Gn))m, (An,Pi(Fn)) (~n.pi(Fn))+n maj (An,Pi(Fn)) ( ) m . p , ( f n))+m, (~n.pi(Gn))+m, (~n.p,(Gn))m, + n) maj +m), Pi(G +m), pi(Gm) Pi(F pi(F+n) maj Pi(F pi(F+m), p,(G+m), p,(Gm) and and therefore therefore hence hence F F+ + maj maj G G+ +,, G. G. 4.1. 4.1. Definition. Definition. t* maj maj ~t, for for t*, t E a, is is defined defined by by induction induction on on a: maj t*maj0t "=t*_t, /\ Pl t* majr Pit, t* maj ~xrt "= pot* maj~ pot A t* maj maj ~ t "= Vy*y(y* maj~ y -+ ---* t'y* maj~ ty, t'y). Furthermore we we put put Furthermore maj ~t ("t is is majorizable"). majorizable" ). t E Maj "- 3t* maj == == => == 436 436 A.S. Troelstra 4.3. Let all all free 1; then then there 4.3. Proposition. P r o p o s i t i o n . Let free variables variables in in tt E T ~" be be of of type type 00 or or 1; there is is aa term t* E T with t* maj term t* with FV(t*) FV(t*) C C FV(t) FV(t),, such such that that HAw H A WfF t* maj t*, t*, t. t. Proof. P r o o f . For For each each constant constant or or variable variable of of type type 00 or or 11 of of HAw H A W (c (cT~ say say)) we we show show that that there c. there is is aa c* c* E T with with c* c* maj maj~T c. maj 0, 0, S S maj maj S S are are immediate; immediate; ((a) a) 00 maj o for yyll define +; maj x x~ ; for define y* y* by by recursion recursion as as yy+; ((b) b ) Xx O~maj ((c) c) k k maj maj k, k, ss maj maj s, s, p p maj maj p, p, po P0 maj maj po P0,, Pl Pl maj maj Pl Pl;; ((d) d ) If rOts = If rr is is the the recursor recursor with with rOts = tt etc., etc., take take rr** := rr + +.. 4.4. Theorem. 4.4. T h e o r e m . (Fan (Fan Rule) Rule) Let Let A A be be a a formula formula of of HAW H A W containing containing only only variables variables of of types 0 0 or or 1i free, free, then then HAw H A WfF \fa�,B3n Vc~_ =~ HAw H A WfF 3m 3 m \fa�,B3n�m Y a < ~ 3 n < m A(a, A(a, n) n),, types where ,B := \fm(an � ,Bn) . where a ~ � P r o o f . Let Let HAw H A WfF \f Va___~ A(c~, Fa) F a ) for for a a suitable suitable term term F F E (1)0. (1)0. F F is is majorizable, majorizable, Proof. a�,B A(a, so F* maj F*, F so there there is is an an F* F* such such that that F* maj F*, F which which means means in in particular particular that that \fa,B(,B Vc~/~(/~ 2:: _ F*# + + 2:: >_ Fc~) and hence hence HAw H A WF A(a, n). n). 0 [] Fa) and f- \fVc~_ Remarks. Switching from R e m a r k s . Switching from aa recursor recursor of of type type 0"(0"00")00" a(aOa)Oa to to aa recursor recursor of of type type O(aOa)aa is is purely matter of of technical convenience; these these recursors are inter inter0(0"00")0"0" purely aa matter technical convenience; recursors are definable. definable. 1992] the In In Kohlenbach Kohlenbach [[1992] the following following generalization generalization is is established established for for E-HAw E-HAW:: V~VXf- \fa\fx�psa3Y� a\fx�psa3yT A(a, x, y) T taA(a, x, y) for } , ss E Ip, closed, and for some some term term t, t, where where T E {O, (0, 1, 1, 22}, lp, ss closed, and where where ___~ is defined defined by by �u is induction on the the type induction on type structure structure by by X x O~ <_o yo ::=- x x � <_y, y, X x UT ~ � < U~T yy UT ~ := : - \fz Vz U~ (xy (xy � <~T yz) yz),, �o Yo XT � UXT := pox ___~ := pox �u PoY ___~x~ P0y /\ AP plx __~ PlY. Xx U~x~ iX � T PlY' U X T yy~X~ As As observed observed above, above, the the proof proof of of closure closure under under the the Fan Fan Rule Rule given given above above depends depends on on realizability realizability only only to to the the extent extent that that we we have have used used modified modified realizability realizability to to obtain obtain closure closure under under the the fan fan rule. rule. For For other other systems systems other other interpretations, interpretations, such such as as the the 1992] . Dialectica interpretation, Dialectica interpretation, yield yield closure closure under under the the rule rule of of choice; choice; cf. cf. Kohlenbach Kohlenbach [[1992]. 4.5. 4.5. Notes Notes The notions of introduced by The notions of majorization majorization and and majorizable majorizable functional functional were were introduced by Howard Howard called strong [[1973]. 1973] . The [ 1986,1989] ' called The present present version version is is aa modification modification due due to to Bezem Bezem[1986,1989], strong majorization him; we majorization by by him; we have have added added a a clause clause for for product product types. types. Kohlenbach 1990] introduced Kohlenbach [[1990] introduced aa version version of of Bezem's Bezem's definition definition with with a a special special clause clause the presence for for types types of of the the form form 0"0; a0; however, however, in in the presence of of product product types types we we found found it it more more 's definition. convenient convenient to to stick stick to to Bezem Bezem's definition. The 1992] . For The proof proof of of the the Fan Fan Rule Rule presented presented here here is is due due to to Kohlenbach Kohlenbach [[1992]. For 1977c] , Beeson 1985] , Troelstra and van other other proofs, proofs, see see e.g. e.g. Troelstra Troelstra [[1977c], Seeson [[1985], Troelstra and van Dalen Dalen [[1988,9.7.23]. 1988,9.7.23] . Realizability Realizability 437 437 5. Lifschitz L i f s c h i t z realizability realizability 5. 's This type type of of realizability realizability was was invented invented by by Lifschitz Lifschitz [1979] [1979] to to show show that that Church Church's This Thesis with with Uniqueness Uniqueness Thesis CT0! y) � -~ 3zVx(zox.J.. 3zVx(z.x$ /\ A A(x, zox» z.x)) CTo! Vx3!y A(x, y) does not not imply imply CT CT00 in in HA HA*. The idea idea to to achieve achieve this, this, is is to to use use as as realizer realizer for for an an does * . The existential formula formula not not aa single single instantiation instantiation for for the the quantifier, quantifier, but but aa finite finite inhabited inhabited existential set of of possible possible instantiations, instantiations, such such that that in in general general there there is is no no recursive recursive procedure procedure set for selecting selecting elements elements of of such such inhabited inhabited sets, sets, although although for for singletons singletons there there is is such such aa for procedure. The The sets sets we we use use are are given given by by procedure. }. V~ {y : 9yy � < PiX plx /\ A Vn-, Vn-,T(pox, y, n) n)}. V T (pox, y, x :"-= {y If we we know know that that V V,x is is aa singleton, singleton, say say {y {y : 900 = - O}, 0}, we we can can find find yy recursively recursively in in X x as as If follows: we we start start computing computing P poxoz for all values of of zz � _ PiX plx;; as as soon soon as as we we have have found found all values follows: O xoz for terminating computations computations for for PiX plx arguments, arguments, we we know know that that the the remaining remaining argument argument terminating <_ PiX plx is is the the required required y y.. � 5.1. Definition. Definition. The The clauses clauses for for rln-realizability rln-realizability are are identical identical to to the the clauses clauses for for 5.1. rn-realizability, except except for for the the existential existential quantifier: quantifier: rn-realizability, Inh(Vx) Vy E 6 V Vx(plyrlnA[y/poy]) x rrllnn33yyAA := Inh(V x (Pi y rln A[yjpoY]) x ) /\A Vy where "Inh "Inh(W)" means that that 3z(z 3z(z E 6 W W).) . 0 [] where ( W ) " means In this this form form the the notion appears as as a a modification of numerical numerical realizability. realizability. There There In notion appears modification of 's analogue is is also also a a Lifschitz Lifschitz's analogue of of function function realizability. realizability. In In that that case case the the sets sets of of realizers realizers for the the existential existential quantifiers quantifiers take take the the form for form n(po a(:yn) = =b Va V~ ::= {7 :: , 7 � _< Pi plaa /\ A V Vn(poa(~n) = O)}. 0)}. The are not not finite, compact. There is no general method method for for finding an element element The Va V~ are finite, but but compact. There is no general finding an ' s which in inhabited is continuous there is V~'s which are are Va which which is continuous in in a, a, but but there is aa method method for for Va in inhabited V~ singletons. There is is no no interesting interesting "abstract" version of of Lifschitz realizability. singletons. There "abstract" version Lifschitz realizability. 5.2. Definition. Definition. rlf-realizability defined as r!-realizability, for the 5.2. rlf-realizability is is defi ned as rf-realizability, except except for the clauses for for the the existential existential quantifi quantifiers, which become: become" clauses ers, which A [.B/Po']) , Inh(Va) /\ V'), V, 6E V~(pl"), Va (Pl/ rrlf a rrlf l f 3flA l f A[fl/poT]), 3,BA := Inh(V~)A a a rlf 3xA Inh(Va) A/\ V, Va (pl/ rlf A[xj(po')O]). arlf 3zA := V7 6E Va(p17rlf A[x/(poT)O]). 0 [] := Inh(V~) 5.3. SSummary of rresults 5.3. u m m a r y of e s u l t s for for rrln-realizability ln-realizability Definition. Definition. In In HA* HA* the the bounded bounded Er:.g~-formulas (BE~ ( Br:.g -formulas) are are formulas formulas of of the the form 3x 438 438 A.S. Troelstra 7roelstra A.S. same modulo modulo logical logical equivalence. equivalence. To To see see this, this, observe observe that that (a) (a) aa II~ II�-formula in in HHA same A ' can be be written written as as ~s -,s == ss inin HA*, HA· , and and (b) (b) 3x 3x << t~(s b( s == s') S ) in in HA* HA· isis equivalent equivalent to to aa can formula of ofthe the form form 3y(t 3y(t == y)y) A/\Vz(t Vz(t == zz ~-+ 3x g CB~ ~ ~-~A --+ A (for A in B2~ Van Oosten Oosten showed showed that that in in fact fact CBE equivalent to to the the following following principle principle Van C B�g~ isis equivalent Vnm(Pn V Qm) Qm) --+ -+ (VnPn (VnPn vV VmQm) VmQm) (P, (P, Q Q primitive primitive recursive), recursive), Vnm(Pn where nn r� FV(Q), FV (Q) , m m r� FV(P). FV(P) . Soundness Soundness now now holds holds w.r.t. w.r.t. HA', HA', i.e. i.e. for for all all where sentences A A sentences :::} H HA n rrIn A HA A => H A ' ' I-f- A A ' ' I-f- ~, ln A The following following properties properties of Vn are for aa suitable suitable numeral numeral n crucial in in the the proof for ~.. The of the the Vn are crucial proof of the soundness theorem: of the soundness theorem: (i) for for some total recursive recursive f0, fo , Vxy(y Vxy(y EE V! i.e. indices indices of of singleton (i) some total Vfo(x ~ yY = x) x),, i.e. singleton o(x)) t+ vt 'S may be found found recursively recursively in their (unique) (unique) elements. Vt's may be in their elements. (ii) x, total (ii) There There is is aa partial partial recursive recursive h fl such such that that for for any any operation operation with with code code x, total on on Vy Vy,, the the image image of of Vy Vy under under x x is is V" Vfl(x,y (x,y)) .. (iii) (iii) There There is is aa total total recursive recursive h f2 such such that that Vh Vf~(x) = U{V. U{V~ :: zz E EV V~}. x} . (x) = (iv) Inh (Vx )/\ VyEV (iv) There There is is aa partial partial recursive recursive h f3 such such that that HA' H A ' fk Vx( Vx(Inh(V~)A VyEV~(y r l n A) A) x (y rIn -+ h (x) rIn A) . With With respect respect to to the the class class of of self-realizing self-realizing formulas, formulas, we we note note an an interesting interesting deviation deviation from from the the notions notions of of realizability realizability considered considered hitherto: hitherto: these these are are not not just just the the 3-free 3-free formulas, formulas, but but the the wider wider class class of of B� B~~-negative formulas. formulas. Now Now we we can can axiomatize axiomatize rIn-realizability rln-realizability relative relative to to HA' H A ' by by means means of of the the following following scheme scheme for for B� BS~-negative A: A: g g ECTL (Vzex ) /\ VUEV Vx(Ax Vx(Ax -+ -+ 3yBxy) 3yBxy) -+ -~ 3zVx(Ax 3zVx(Ax -+ -+ ZeX,!z.x$ /\ A Inh Inh(Vz.x)A VuEV~.~ Bxu).. zex Bxu) An An interesting interesting special special case case of of ECTL ECTL is is ECTL ECTL!! which which can can be be formulated formulated as as g-negative) , Vx(Ax-+3!y ZeX)) (A Vx(Ax-+3!y Bxy) S x y ) --+ + 3zVx(Ax 3zVx(Ax -+ -~ zex.!-/\B(x, z.xSAB(x,z.x)) (A B� BE~ with with the the help help of of the the following following Lemma. L e m m a . There There is is aa partial partial recursive recursive ff55 such such that that z ) -+ H A '' Ff- Vz(3xVy(x Vz(3xVy(x = = yy t+ .v->.Yy EE V Vz) ~ f5(Z) fh(z) EE v.). ~). HA 439 439 Realizability Realizability 5.4. 5.4. (i) (i) (ii) (ii) (iii) (iii) Proposition. P r o p o s i t i o n . (Applications) (Applications) HA ECTL is consistent; H A '' + + ECTL is consistent; HA* + HA* + ECTL! ECTL! If V CTo CT0;; HA fortiori under H A '' is is closed closed under under the the rule rule ECRL ECRL and and aafortiori under the the rule rule ECRL! ECRL! (as (as ECTL and ECTL! but with Since HA ECTL and ECTL! but with main main -t --+ replaced replaced by by => =~ etc). etc). Since H A '' also also satisfies satisfies DP and and EDN, more strongly the the rules rules DP EDN, we we can can formulate formulate the the rule rule ECRL ECRL!! even even more strongly as: for as: .for A A in in B2:� B 2 ~, f- 'v'x(Ax W(A~ -t -~ :3!yBxy) 3!yB~y) => ~ f-e 'v'x(Ax W(Az -t -~ n.x.J.. n.z4 A ^ B(x, B(z, n.x) n.~) /or aa suitable suitable numeral numeral n. ~. for 5.5. 5.5. Summary S u m m a r y of of results results for for rlfrlf- and and rlft-realizability rlft-realizability * , namely The The basis basis theory theory is is now now an an extension extension of of EL EL*, namely EL EL'' == -_- EL EL** + + MQF MQF + -+- KLQF KLQF,, where MQF is Markov's principle principle for Lemma where MQF is Markov's for quantifier-free quantifier-free formulas, formulas, and and Konig's Kb'nig's Lemma for /or quantifier-free quantifier-]ree formulas formulas is is the the schema schema for for quantifier-free quantifier-free A. A. Vx3n(lth(n) An) 'v'x:3n (lth(n) -- xx A nn<(~ A An) $o: KLQF KLQF m)) -t A'v'nm(A(n ^Wm(A(~,* m ~ An) A ~ )-t ~ :3,B 3Z_<~W A(Z~) C Bn) $o:'v'n A (n$o: ((n)y $ (n<_~ "= 'v'y( 3~___r . . . ) := :33c~(c~ _ r A A...)). The analogue analogue in in . . ) ) . The 0:(0: $ EL '~of of the the B2:g-negative B~~ formulas are are the the B2:�-negative B~21-negative formulas formulas (a (a B2:�-formula B~21-formula is is aa EL formulas -negative formula formula of of the the form form :3o:$ 3c~___r s - tt (a (a Bounded Bounded 2:� ~ formula) formula);; the the class class of of B2:� B~-negative formulas obtained from B2:�-formulas and formulas is is obtained from formulas formulas B~21-formulas and prime prime formulas formulas by by means means of of -t .-+,, A A,, 'v'). V). D a As special case As aa typical typical result result one one obtains obtains that that GC! GC! (i.e.the (i.e.the special case of of GC GC with with uniqueness uniqueness for WC-N. for the the existential existential quantifier) quantifier) does does not not imply imply GC, GC, not not even even the the special special case case of of WC-N. = _ := --' = := . Remark. KLQF for R e m a r k . KLQF for the the language language of of EL EL** follows follows from from KLQF KLQF for for EL EL by by observing observing that KLQF in that KL KL with with A AE E 2:� T ~ is is derivable derivable from from KLQF in EL. EL. 5.6. Notes Notes Khakhanyan Khakhanyan [1980bj [1980b] defined defined Lifschitz' Lifschitz' realizability realizability for for certain certain set set theories theories and and uses uses it it to CTo from Other relevant to obtain obtain independence independence of of CT0 from CTo CT0!! for for these these theories. theories. Other relevant papers papers are are van van Oosten[1990,1991a,1991bj Oosten[1990,1991a,1991b]. . As As to to van van Oosten Oosten [1991bj, [1991b], see see 8.31. 8.31. It possible to to combine It is is possible combine Lifschitz Lifschitz realizability realizability with with modified modified realizability realizability for for HRO HRO van van Oosten Oosten [1991aj. [1991a]. It It is is not not known known whether whether for for some some or or all all results results perhaps perhaps weaker weaker theories theories than than HA HA',' , EL will suffice. EL'' will suffice. 6 Extensional realizability realizability 6.. Extensional It It is is also also possible possible to to combine combine the the idea idea of of realizability realizability with with extensionality, extensionality, by by defining defining not not just just aa notion notion of of the the form form "x "x realizes realizes A" A",, but but aa relation relation between between realizing realizing 440 440 A.S. Troelstra objects: "x "x and and yy equally equally realize realize A" A".. The The definition definition below below has has been been written written out out for for objects: HA' HA* and and partial partial recursive recursive application, application, but but also also makes makes sense sense in in the the abstract abstract setting setting of APP, A P P , if if we we read read everywhere everywhere re r_oefor for rne m e .. of 6.1. Definition. D e f i n i t i o n . We We define define "x "x = = x' x' rne rne A" A" (x, ( x , xx'' f/. r F FV(A), ~ y), y), by by induction induction 6.1. V(A) , xx =1= on the the complexity complexity of of A: A" on = x' x' rne rne P P (x = = x' x' /\ AP P /\ A x..! x$. /\ A x'..!.) x'$) (P (P prime prime), xx = := (x ), := ( = ox x = x' rne (A A B) := (pox = pox' rne A) = PIX' p , x ' rne rne B), B), x = x' rne (A /\ B) Pox' rne A) /\A ((p,x P PlX = = y' rne A -+ x = x' rne (A ~ B) := x$ A x'$ A Vyy'(y = y' rne A x = x' rne (A -+ B) : = x..!. /\ x'..!. /\ Vyy'(y x~ = - xoy' xoy ~rne rne B B /\ A x' xl~oy = -- x' x~oy rne B B /\ A Xoy x~ = x' x~~ rne B), B), xoy oy'~rne oy rne = x' x' rne rne Vy Vy A A := Vy(xoy Vy(x~ = x'oy x'.y rne rne A), A), xx = xx = := ((pox = x' x' rne rne 3y 3y A A = P pox') (plx = = P plx' rne A A[y/pox]), ox') /\A (PlX [Y /Pox]), IX' rne Pox = and we we put put and n e AA := := x x ==x rxnrne e A . A. xx rrne As always, rnet-realizability is obtained by by adding adding "/\ "A (A (A -+ --+ B)" B)" in in the the implication implication As always, rnet-realizability is obtained clause. 0 [] clause. R e m a r k s . Note Note that that x x r___~nA does not not in in general general imply = x x rne rne Aj A; for for if if x x = = rn A does imply xx = Remarks. xx rne rne [Vy(t [Vy(t = -- 0) 0) -+ --+ Vz(s Vz(s = - 0)] 0)],, then then x x must must yield yield the the same same value value when when applied applied to to extensionally equal equal realizers realizers z, z, z' z' for for Vy(t = 0) 0);j on on the the other other hand, hand, for for an an x x such such that that Vy(t = extensionally [Vy(t = = 0) 0) -+ --+ Vz(s - 0)] no such such restriction restriction applies. applies. rn [Vy(t Vz(s = O)J no xx r___nn The The definition definition may may also also be be formulated formulated as as aa simultaneous simultaneous inductive inductive definition definition of of "x extensionally extensionally realizes A" and "x and are equivalent equivalent realizers realizers for "x realizes A" and "x and yy are for A", A" , but but this this is is more cumbersome. cumbersome. more It It is is straightforward straightforward to to prove prove soundness. soundness. The 3-free play the same role role as r___~n-realizability. On On the The 3-free formulas formulas play the same as in in rn-realizability. the other other hand, hand, no simple simple axiomatization formulas is is known. no axiomatization of of the the provably provably me-realizable rne-realizable formulas known. For proofs van Oosten [1990J . For proofs of of the the following following facts facts we we refer refer to to van Oosten [1990]. 6.2. The The difference difference between between ordinary realizability and realizability is 6.2. ordinary realizability and extensional extensional realizability is is not rne-realizable: demonstrated by the fact fact that the following following instance of ECT demonstrated by the that the instance of ECT00 is not rne-realizable: Vz[Vx3y(-~-~3uTzxu T z x y ) --+ A (-~-~3uTzxu T(z, x, vox))] Vz[Vx3y(...,...,3uTzxu --+ -+ Tzxy) -+ 3vVx(v.x 3vVx(vox /\ (..., ...,3uTzxu --+ -+ T(z,x, vox))J On it is is not On the the other other hand hand it not hard hard to to verify verify that that the the following following "Weak "Weak Extended Extended Church's Church 's Thesis" isis provably provably me-realizable: rne-realizable: Thesis" 6.3. r o p o s i t i o n . In A we 6.3. PProposition. In HHA we can can rne-realize: rne -realize: WECTo WECT0 -+ 3y 3y BBxy) -+ -~-~3zYx(A ...,...,3zVx(A --+ -+ zzox,,!, /\ BB((x, zox)) YVX(A x ( A -+ x y ) --+ .x$ A x , z.x)) for 3-free 3-free A. A. for A nice nice application application of of rnet.-realizability rnet-realizability is the following following refinement refinement of ECR. A is the of ECR. Realizability Realizability 441 441 6.4. Proposition. P r o p o s i t i o n . Assume Assume.for 3-free B B that that in in HA* HA* 6.4. for 3-free ~-Vz(Vx3y Vz(Vx3y Bzxy Bzxy --+ ~ 3u 3u Czu) Czu) fthen for for some some n then ' (Vx(vox = Vz(~.z$. /\ A Vv, Vv, vv'(VX(VoX = v'ox v'~ /\ A B(z, B(z, x, x, vox)) vox)) --+ --~ noZoV ~~ = = nozov ~oz.v'' /\ A C(z, C(z, nozov)). 5oz~ f-F Vz(noz.J. 6.5. Notes Notes 6.5. Extensional realizability realizability appears appears for for the the first first time time explicitly explicitly in in some some unpublished unpublished Extensional notes by by Grayson Grayson [[1981c], and implicitly implicitly in in Pitts Pitts [[1981]. notes 1981c] , and 1981 ] . Renardel ] and [ 1979b,1985] use Renardel de de Lavalette Lavalette [1984 [1984] and Beeson Beeson[1979b,1985] use an an abstract abstract version version of extensional extensional realizability realizability in in combination combination with with forcing, forcing, to to prove prove that that MLo ML0 (the (the of arithmetical fragment fragment of of the the extensional extensional version version of of Martin-Lof's Martin-Lbf's type type theory) theory) is is arithmetical conservative over over HA. HA. MLo ML0 includes includes E-HAw E - H A W+ § AC AC as as aa subtheory. subtheory. See See also also Eggerz Eggerz conservative 1978j 33.. The proofs proofs by by Renardel aenardel and and Beeson Beeson extend extend earlier earlier work work of of Goodman Goodman [[1978] 1987] . The [[1987]. There is is aa close close similarity similarity between between "x "x = = x' x ~rne rne A" A" and and "x "x = -- x' x ~E A" A" in in the the type typeThere 1982,1984] ' so theories theories of of Martin-Lof Martin-Lbf [[1982,1984], so it it is is not not surprising surprising that that an an interpretation interpretation akin akin 's extensional to extensional extensional realizability realizability can can be be used used to to model model (parts (parts of) of) Martin-Lof Martin-Lbf's extensional to type theories, theories, cf. cf. Beeson Beeson [[1982]. See also also 9.3. 1982]. See 9.3. type rne rne -realizability -realizability for for the the language language of of arithmetic arithmetic does does not not lend lend itself itself to to aa straight straightforward axiomatization in forward axiomatization in the the same same manner manner as as ECT ECT00 might might be be said said to to axiomatize axiomatize HA. But 1993] showed axiomatization is rn-realizability rn-realizability relative relative to to HA. But van van Oosten Oosten [[1993] showed that that axiomatization is possible possible in in a a suitably suitably chosen chosen conservative conservative extension extension of of HA H A plus plus Markov's Markov's principle. principle. The of The same same paper paper discusses discusses also also rne-realizability rne-realizability for for higher-order higher-order logic logic in in the the form form of certain certain toposes. toposes. Realizability for intuitionistic second-order 7. Realizability for intuitionistic second-order arithmetic arithmetic 7.1. The T h e system system H AS 7.1. HAS order) isis aa two-sorted two-sorted extension extension of of H HA with (Heyting Arithmetic Arithmetic of of Second HAS H A S (Heyting A with Second order) HA is extended extended with with quantifiers P(IN) , the the powerset powerset of of IN. IN . So So the the language of H quantifiers over over P(]N), language of A is X, Y, Z, and and corresponding corresponding (second-order) (second-order) quantifiers VX, 3Y; 3Y; atomic atomic set variables variables X, set Y, Z, quantifiers VX, or Xt (also written t E formulas are are now now of of the the form form tt -= ss or X t (also written t E X) X ) for for individual individual terms terms formulas t,t, ss and and set set variable variable X. X. Instead of of formally formally introducing introducing set-terms set-terms Ax.B >"x.B (B ( B any any formula) formula) we we can can formulate formulate Instead the V as the axiom axiom for for second-order second-order ~/as VX.A VX.A --+ --+ A[X/)~x.B] A[Xj>..x .B] A[Xj >..x .B] isis obtained obtained from from A A by by replacing replacing every every occurrence occurrence of of XXt by B[x/t]. B[xjt] . where A[X/)~x.B] where t by Alternatively, Alternatively, we we restrict restrict the the V2-axiom ' VX.A VX.A --+ --+ A[X/Y] A[XjY] 3We do do not not know know whether whether the the treatment treatment in in Beeson Beeson [1979b] [1979b] isis really really equivalent equivalent to to the the one one in in 3We Beeson [1985]. [1985]. 442 442 A.S. Troelstra A.S. while adding adding the the axiom axiom schema schema of of full full comprehension comprehension while CA A) (X (X gf/. FV(A)). FV(A)) . CA 33X'v'x(Xx X V x ( X x ~+-t A) Moreover, we we require require sets sets to to respect respect equality equality Moreover, = yY --+ -+ Xy). Xy). V'v'Xxy(Xx X x y ( X x AI\ zx = HAS' is is related related to to H HAS in the the same same way way as as HA* HA' to to H HA. In particular, particular, for for the the set set HAS* A S in A . In t.J.. variables we we have have strictness: strictness: XXtt ~-+ t$. variables 7.2. R Realizability for HAS* HAS' 7.2. e a l i z a b i l i t y for It is is quite quite easy easy to to extend extend r___~n-realizability rn-realizability from from HA' to HAS* HAS' by "brute force"; force" ; we we It HA* to by "brute X aa new new set set variable variable X*, X ' , representing representing the the "realizability "realizability assign to to each each set set variable variable X assign HA' : predicate" and and then then add add the the following following clauses clauses to to r___n_n-realizability rn-realizability for for HA*" predicate" x rn Xt := X*(x,t) X'(x, t) (x$ (x.J. is is automatic automatic by by strictness) strictness) xr___nXt 'v'X A A := := VX* 'v'X'(x rn A), A) , xx rrn ___nnVX (x rn xx r___nn rn 3X 3X' (x (x rn rn A). A) . 3X A A := := 3X* Here Y(t, t') for set variable Y abbreviates abbreviates Y(p(t, Y(p(t , f)). t') ) . ((Nothing Nothing prevents us from Here Y(t, t') for any any set variable Y prevents us from taking X* X' == but in in discussions this is is sometimes sometimes inconvenient inconvenient and and confusing.) confusing. ) taking - X X, , but discussions this 7.3. Remark. second-order context, context, .1, are definable definable in of --+ -+ 7.3. R e m a r k . In In aa second-order _l_, 3 3 and and 1\ A are in terms terms of and 'v', in particular and V, in particular o := 'v'ZO('v'Y(A 3Y.A 3Y.A := VZ ~(VY (A -+ --+ ZO) Z ~ -+ --+ z Z ~, A := 'v'ZO A 1\ AB B := VZ~(((A -+ --+ (B (B -+ -+ Z)) Z)) -+ Z) Z),, o .1 J_ := 'v'Z VZ~.Z, where Strictly speaking, where ZO Z ~ ranges ranges over over propositions. propositions. ((Strictly speaking, we we do do not not have have variables variables over over propositions, propositions, only only over over sets, sets, but but the the addition addition of of proposition proposition variables variables is is conservative, conservative, since X)A(XO) for since one one may may render render (Q (Q ZO)A(ZO) Z~ ~ as as (Q (QX)A(XO) for Q Q E e {'v', {V, 3}.) 3}.) Using Using this this 3, the definition definition of of 3, the clause clause for for realizing realizing 3X.A 3X.A is is in in fact fact redundant, redundant, and and we we obtain obtain an an equivalent equivalent notion notion of of realizability. realizability. A A virtually virtually immediate immediate consequence consequence of of soundness soundness ' s thesis for for rn-realizability rn-realizability for for HAS H A S is is the the consistency consistency of of HAS H A S with with Church Church's thesis and and the the so-called so-called Uniformity Uniformity principle principle 'v'X3y UP UP VX3y A(X, A(X, y) y) -+ 3y'v'X 3yVX A(X, A(X, y). y). 7.4. Proposition. P r o p o s i t i o n . HAS' HAS* + + ECTo ECT0 + + UP UP + +M M is is consistent. consistent. Here Here ECTo ECT0 is is formulated formulated as as for for HA' HA*,, except except that that A A is is restricted restricted to to 3-free =l-free formulas formulas of HA' , while B is arbitrary. of HA*, while B is arbitrary. 7.5. 7.5. rnt-realizability r n t - r e a l i z a b i l i t y for for HAS' HAS* Extension Extension of of rnt-realizability rnt-realizability to to HAS' HAS* is is similar similar to to the the extension extension of of rn-realizability, rn-realizability, but but we we have have to to be be slightly slightly more more careful: careful: we we want want to to keep keep track track of of realizability realizability and and truth, truth, so so we we want want to to associate associate with with an an arbitrary arbitrary set set X X an an arbitrary arbitrary Y Y together together with with its its realizability realizability set set Z. Z. It It is is convenient convenient to to encode encode Y Y and and Z Z into into aa single single set set X' X*;; we we put put X * t := : = {{n n :: X'(2n) X*(2n)}, X * r:= : = {{ nn :: X' X *(2n ( 2 n+ + l I) )}} x't } , x,r Realizability Realizability 443 443 representing the the two two components components of of truth truth and and realizability realizability respectively. respectively. The The new new representing clauses in in the the definition definition of of rnt-realizability rnt-realizability now now become become clauses xxrrnt n t XXt t :-:= X*t(t)AX*r(x,t), X*t (t) t\ X*' (x , t) , := VX* 'v'X*(x A), 'v'X AA := xx rrnt n t VX (x rrnt n t A), 3X AA := := 3X*(x 3X* (x rrnt A) , xx rrnt n t 3X n t A), xey rrnt (A --+ -+ B) B) := := Vy(y 'v'y(y rrnt A --+ (A --+ -+ B)*, B)*, xx rrnt n t (A nt A n t B) B) At\ (A -+ x.y y*t (t). (t) . ItIt isis readily where C* C* isis obtained obtained from from CC by by replacing replacing all all occurrences occurrences of of Yt Yt by by y,t readily where verified that that for for all all A A with with second-order second-order variables variables contained contained in in {X1, {Xl , )(2, X2 , ... Xn} Xn} verified A* x�t]tl ~H A* In / XX;t, ; t , . .. .. ,. , Xn A[Xb . . . , XXn/ I- A[X1,..., I- x xrrnt A ~-+ A* A* ~nt A and we we find find that that soundness soundness holds. holds. An An interesting interesting corollary corollary is is and • • • 7.6. Proposition. HAS* is is closed the Uniformity Uniformity Rule Rule 7.6. P r o p o s i t i o n . HAS* closed under under the 'v'X3y A(X, A(X, y) y) ~{:} ~I- 3yVX 3y'v'X A(X, A(X, y) y) UR ~I- VX3y UR and ED N. and and satisfies satisfies D DPP and EDN. Second-order 7.7. 7.7. S e c o n d - o r d e r eextensions x t e n s i o n s of of other o t h e r types t y p e s of of realizability realizability The preceding preceding two something of to The two examples examples reveal reveal something of a a pattern pattern for for the the extension extension to second-order still clearer the second-order languages. languages. The The pattern pattern will will become become still clearer when when we we study study the extension section, but extension to to higher-order higher-order logic logic in in the the next next section, but let let us us already already now now indicate indicate what what has has to to be be done done to to extend extend extensional extensional and and modified modified realizability. realizability. In In the the case case of of extensional extensional realizability, realizability, set set variables variables should should get get assigned assigned variables variables ranging A partial ranging over over partial partial equivalence equivalence relations relations over over IN IN.. ((A partial equivalence equivalence relation relation ) . Second-order satisfies satisfies symmetry symmetry and and transitivity, transitivity, but but not not necessarily necessarily reflexivity reflexivity). Second-order quantification quantification is is treated treated in in the the "uniform" "uniform" way, way, just just as as for for ordinary ordinary realizability. realizability. In there is In the the case case of of modified modified realizability, realizability, there is no no immediate immediate generalization generalization of of the the abstract abstract version version for for HA H A WW,, but but we we can can generalize generalize HRO-mr-realizability; HRO-mr-realizability; we we shall shall "modified realizability abbreviate abbreviate this this as as mrn-realizability mrn-realizability (("modified realizability for for numbers" numbers" )).. In realizers, but In this this case case we we need need to to assign assign to to each each formula formula not not only only aa set set of of realizers, but also also aa set set of of "potential "potential realizers" realizers",, which which determine determine the the domain domain of of definition definition in in the the In the case case of of implication. implication. ((In the case case of of HRO-mrn-realizability HRO-mrn-realizability restricted restricted to to HA H A wW,, the the sets sets of of potential potential realizers realizers are are always always of of the the form form HROu. HRO~.)) In In particular, particular, we we must must assign representing the assign to to set set variable variable X X two two variables variables xr X r ((representing the realizing realizing numbers numbers)) and and representing the ) . We ne for Xd X d ((representing the set set of of potential potential realizers realizers). We then then defi define for each each formula formula A A the the set the predicates predicates xx mrn turn A A ("x ("x HRO-modified HRO-modified realizes realizes A" A")) and and Ad A d ((the set of of potential potential realizers ) . Some realizers). Some typical typical clauses clauses for for the the potential potential realizers: realizers: (t (t = = S)d s) d := := lN IN,, (A (A -+ -+ B) d := := {x {x :: 'v' VyeAd(x.y Bd)}, (VX.A) d := := 'v'XdAd VXdA d,, and and for for the the realizability realizability B)d } , ('v'XA)d Y EAd(Xey Ee Bd) X t := := xx EE Xd X d t\ A xr(x, Xr(x,t), (A -+ ~ B) B ) : := = xx EE (A (A -+ ~ B)d B) d t\ A 'v'y(y Vy(ymrnA xx m mrrnnXt t) , xzmrn mrn (A mrn A -+ x.ymrnB), := 'v'xdxr(x VXdXr(xmrnA). xey mrn B) , xxmrnVX.A mrn 'v'X.A := mrn A) . The The reader reader will will have have no no difficulty difficulty in in supplying supplying the the remaining remaining ones, ones, keeping keeping in in mind mind that that this this is is to to be be an an extension extension of of HRO-mr-realizability. HRO-mr-realizability. However, However, in in verifying verifying soundness, soundness, it it turns turns out out that that there there is is on on important important extra extra property property required required of of the the Ad Ad:: there there should should always always be be aa fixed fixed number number in in the the sets sets of of potential potential realizers, realizers, so so Troelstra A.S. Troelstra 444 that operations operations defined defined over over the the Ad A d must must be be defined defined at at least least somewhere. somewhere. If If we we that let the the variables variables Xd X d range range over over inhabited inhabited sets sets containing containing 0, 0, and and if if we we choose choose our our let g5delnumbering of of partial partial recursive recursive operations operations in in such such aa way way that that p(O, p(0, 0) 0) = = 00 and and godelnumbering Ax.0 = = 00,, it it follows follows that that 00 E Ad A d for for all all A. A. Ax.O 7.8. Realizability R e a l i z a b i l i t y as as a a truth-value t r u t h - v a l u e semantics semantics 7.8. It is is instructive instructive to to rewrite rewrite rn-realizability rn-realizability for for HA' HA* in in the the form form of of aa valuation valuation in in aa set set It of truth-values. truth-values. Let Let X, X, Y Y E P(lN) P(iN);; we we define define of Definition. Definition. X A1\YY := : = {{p(x, p ( x , yy) ) :: x x eEXX A y1\e YY E } , Y}, X X -t --+ Y Y := := {z {z :: Vx Vx Ee X(z.x X ( z . x Ee Y)}, Y)}, X X vVYY := : = {{p(O, p ( 0 , xx) ) :: x x eEXX} } u {Up ({p(Sz, S z , y) y) :: zz eEi NlN, , Y y eEYY}, }, X X ++ ++ Y Y := := (X (X -t --+ Y) Y) 1\ h (Y (Y -t --+ X). X). X We associate associate to to each each formula formula A A of of HA* set [A] [A] of of realizing realizing numbers: numbers: We HA' aa set [t=s] :={x:t=s}, [t = s] := {x : t = s} , = [AJ [A [A 1\ AB B]] ::= [A l 1\ A [B], [BI, [A [A -t -+ B] S] := := [A] [A] -t ~ [B] IS],, IVxA] := VX(zox E e [A])}, IA])}, := {z {z :: Vx(z.x [VxA] [3xA] := {p(y, {p(y, z) z) :: zz Ee [A[x/yJ]}. [A[x/y]]}. := [:3xA] The The defined defined set set contains contains the the free free variables variables of of A A as as parameters. parameters. Furthermore Furthermore we we can can put, in in keeping keeping with with our our definition of disjunction, disjunction, put, definition of [AV S] := [A] V [B] IS].. 0 [3 [A V B] := [A] The inhabited elements The elements elements of of P(lN) P(iN) act act as as truth-values; truth-values; all all inhabited elements represent represent "truth" "truth" in the the sense in sense of of realizability. realizability. If we we now want to HAS*,, we we should should put put If now want to extend extend this this to to HAS' [Xt X*(x,t)}, := {{xx : : X'(x, t)}, [Xt]I := [VX.A] ni A[A(X)). [VX.AI := "= N (X)]. x X and now [A] contains contains for any X X free in A A aa parameter parameter X' may do do without without and now [A] for any free in X*.. (We (We may an explicit explicit definition for the the cases for A, since these these are definable in in aa second-order an definition for cases for are definable second-order 1\, 3:3 since setting, cf. 7.3.) setting, cf. 7.3.) Note that that numerical and set set quantifiers treated in completely different way. Note numerical and quantifiers are are treated in aa completely different way. This be remedied remedied in in this this case case in in aa more more or or less we associate with This can can be less ad ad hoc hoe manner: manner: we associate with each domain domain D D aa set-valued set-valued function function ED ED on on the the elements, elements, giving giving their their "extent". "extent" . In In each the case of the case of HAS* HAS' we we take take E~(n) p ((JN ~ ))(X) ( X ) : =:= {0}, EJN (n) := := {n}, {n}, EE'P {O}, and and define define for for domains domains D D [A(x)]) D.A(x)] := := (Ex' ~-t [A(x)]) [Vx eE D.A(x)] [\Ix n ' N(E x' Xt where x' x' isis the the parameter parameter in in [A(x)] [A (x)] corresponding corresponding to to xx (i.e. (i.e. xx === x' x' for numerical x, x, where for numerical = X' is aa set set variable variable X). X). (To (To see see that that the the resulting resulting notion notion of realizability xx'~ = X* if if xx'~ is of realizability is equivalent equivalent in in the the sense sense of of 1.2, 1.2, take take for the r¢> and and r'ljJ: r¢>vxA(Y ) := := AX.r Ax.ifJA (x) (Y.x) , is for the Realizability 445 445 = = AX. ((Y) r AX-r r .A ((Y) := AZ. )~Z.r not free free in in ¢>A CA((Y)), r . A(Y (Y)) ::-Y) := Y )), 'l/JvX ¢>A ((y) Y ) ((Zz not 'l/JvxA Y ) ::= 7/JA(x) (YOX) , ¢>vX CA(y.0).) Such an ad hoc solution to enforce uniformity of definition will not be Such an ad hoc solution to enforce uniformity of definition will not be O) .) (YO 7/JA satisfactory satisfactory in in the the case case of of higher-order higher-order logic, logic, to to be be discussed discussed in in the the next next section. section. 7.9. Notes Notes 7.9. Troelstra [[1973b] extended mrn-realizability, mrn-realizability, and and Friedman Friedman [[1977a] extended q qTroelstra 1973b] extended 1977a] extended realizability to to HAS; H A S ; here here we we have have recast recast Friedman Friedman's' s defi definition as rnt-realizabilit rnt-realizability. Y. nition as realizability The idea idea of of realizability realizability as as aa truth-value truth-value semantics semantics occurred occurred to to several several researchers researchers The independently, shortly shortly before before 1980. 1980. The The first first documented documented reference reference to to "realizability "realizability independently, treated as as aa truth-value truth-value semantics" semantics" II could could find find is is Dragalin Dragalin [[1979], cf. also also Dragalin Dragalin treated 1979] , cf. Other authors authors credit credit W. W. Powell, Powell, or or D.S. D.S. Scott Scott with with the the idea. idea. [[1988]. 1988] . Other 8. Realizability Realizability for higher-order 8. higher-order logic and and arithmetic arithmetic 8.1. Formulation F o r m u l a t i o n of of HAH HAH 8.1. Higher-order logic logic is is based based on on a a many-sorted many-sorted language language with with aa collection collection of of sorts sorts Higher-order ' , . . . for or types; types; we we use a,a~,...,T, for arbitrary arbitrary types. types. There are are variables variables or use a, a', . . . , T, TTI,... There U , yU , zU , . . . ) for a. Relation (x~,y~,z~,...) for each each type, type, and and an an equality equality symbol symbol = =~u for for each each a. Relation (X symbols and and function function symbols symbols may may take take arguments arguments of of different different types. types. For For quantifiers quantifiers symbols ranging over over objects objects of of type type a a we we sometimes sometimes write write VXEa, VxEa, :lxEa 3xEa instead instead of of Vxu~,, :lx 3x u~.. ranging For intuitionistic and For intuitionistic and classical classical higher-order higher-order logic logic there there are are certain certain type-forming type-forming operations operations generating generating new new types types with with appropriate appropriate axioms axioms connecting connecting the the types. types. D e f i n i t i o n . (Axioms (Axioms and language for for higher-order In a a many-sorted many-sorted language Definition. and language higher-order logic) logic) In language for P, -+, � , i.e. i.e. for higher-order higher-order logic, logic, the the collection collection of of types types is is closed closed under under •x , P, (i) with with each is aa power type P(a); (i) there is type P (a ) ; each type type aa there (ii) with with each pair of of types types a, there is is aa product product type and aa function type (ii) each pair a, T there type a •x T and function type (~ -+ T T.. a� One often includes includes a a type P(a) may be -+ w. One often type w W of of truth-values; truth-values; then then P (a) may be identified identified with with aa � w. y) There is is aa binary binary relation E u with type a, P(a) ; instead of EE~(x, There relation E~ with arguments arguments of of type a, P(a); instead of u (x, y) we write sometimes y(x) (predicate applied to to argument argument).) . and sometimes (predicate applied we write xx E~ E u Yy and For types aa � there is application operation operation App~,r ApP U,T such For types -+ T, aa there is an an application such that that for for tt E E t') is a term of type Usually we write tt' for App ( t, t') . a�T , t' E a, a, ApP ( t, a---~T, t' E App~,r(t, t') is a term of type T. Usually we write tt ~ for App(t, t'). T U, O'~T T such For each each pair pair a, a, T there there are are functional functional constants constants p~,~, pU,T, Pg'~, such that that pp takes takes p�, For p�,T , Pl a, T and and yields yields aa value value of of type type aa xx T, T , P0, Po, Pl PI take take arguments arguments of of arguments of of type type a, arguments type and yield yield values values of of type type aa and and T respectively. respectively. The The pairing pairing axioms axioms are are type aa xx T and assumed: assumed: PAIR (i == 0,1) 0, 1 ) PAIR VXoxl(pi(p(xo, XI) == x~) VXOXI (Pi (P(XO, xl) Xi) (i sSURJ uRJ w x) == x) . VxUX T (p(POX, pPIX) For For power-types power-types we we require require replacement replacement REPL vxP(a)Vxaya(x XA � yY EE X), VXP(u) VxU yU (x EE X X), REPL 1\ xx == yY --+ A.S. A.S. Troelstra 446 as well as as well as extensionality extensionality and and comprehension: comprehension: P( P( ) ) U U U y EXT vxP(a)YP(a)(Vx a (x X ++ ~ x x E E Y) Y ) -+ --+ X Z = = Y)), Y)), EXT (VX (x EE X VX u P( ) VxU (x CA 3xP(~)Vz~ (x E EX X ++ ~ A(x) A(x)).). 3X CA For For function function types types the the corresponding corresponding requirements requirements are are -H' ZU" -4T EXTF Vy EXTF Vy U'~-~z ~ (VX (Vx U~ (yx (yx = = zx) zx) -+ --+ y y = = z) z) U-4T VxU A(x, zx) U 3!yT A(x, y) CAF Vx"3!y~A(x, y) -+ -+ 3z 3z"-"Vx'~A(x, zx).. VX CAF If (a ) is If the the type type w is is present present and and P P(a) is identified identified with with a a -+ --+ w w,, EXT EXT and and CA CA become become special cases cases of of EXTF EXTF and and CAF, and REPL REPL follows follows from from the the fact fact that that functions functions special CAF, and respect respect equality. equality. 8.2. arithmetic (("Heyting "Heyting Arith 8.2. Definition. Definition. HAH, H A H , intuitionistic intuitionistic higher-order higher-order arithmetic Arithmetic metic of of Higher Higher order" order")) is is a a specialization specialization of of higher-order higher-order logic logic based based on on aa single single basic or N) basic type type a0 ((or N) for for the the natural natural numbers; numbers; types types are are closed closed under under power-type power-type and and function-type function-type formation. formation. On On the the basis basis type type a0 an an injective injective function function S S :: a0 -+ -~ a0 is is given, given, with with axioms axioms Sx = = S S yy -+ ~ x x = = y, y, a0 i= :/= Sx. Defining Sx Sx. Defining xx Ee IN := := vVX(a x(0 E e X z 1\ ^ vVy(Xy y ( x y -+ X(Sy)) z ( s y ) ) -+ x9 E e X) x) we we add add an an axiom axiom stating stating that that all all elements elements of of type type a0 are are in in IN IN:: VX Vx~O (x E E IN) IN).. As As a a result, the induction axiom result, the induction axiom becomes becomes valid. valid. 0 [::1 Remarks. i ) E-HAw R e m a r k s . ((i) E - H A ~ is is aa fragment fragment of of HAH H A H based based on on type type a0 and and function-type function-type formation formation only. only. ((ii) ii) It special element It is is well well known, known, that that if if we we consider consider in in HAH H A H any any set set X X with with aa special element Xo x0 E EX X and and aa function function f f :: X X -+ --+ X X,, then then there there is is aa unique unique function function F F :: IN IN -+ --+ X X such particular, if such that that Fa FO = = Xo Xo,, F(Sx) F(Sx) = = f(Fx) f ( F x ) . . In In particular, if f f is is injective, injective, then then the the image image j[X] isomorphic to f[X] U U {xo} {x0} is is isomorphic to the the type type N. N. 8.3. 8.3. Numerical N u m e r i c a l realizability r e a l i z a b i l i t y for for many-sorted m a n y - s o r t e d logic Since logic, and HAH Since our our versions versions of of intuitionistic intuitionistic higher-order higher-order logic, and the the system system H A H are are based logic, we based on on intuitionistic intuitionistic many-sorted many-sorted predicate predicate logic, we first first discuss discuss realizability realizability for for many-sorted many-sorted logic. logic. Our Our definition definition of of realizability realizability will will be be motivated motivated by by the the truth-functional truth-functional reformulation reformulation of of realizability realizability for for HAS H A S in in 7.8. 7.8. We We start start with with realizability realizability for for many-sorted many-sorted logic logic without without function function symbols. symbols. Below Below n == = P(IN) 7'(IN),, n ~** is is the the collection collection of of all all inhabited inhabited subsets subsets of of IN. IN. We We first first introduce introduce n-sets, will serve Ft-sets, which which will serve to to interpret interpret the the types types with with their their equalities. equalities. 8.4. n-set X 8.4. Definition. Definition. An An f~-set X == - (X, ( X , ==x) x ) is is a a set set X X together together with with aa map map =x: =x: writing tt =x X 22 -+ ---4 n f~ such such that that the the following following is is true true ((writing = x t' t' for for =x(t, = x ( t , t') t')):) : X n nx,y(x =x y y -+ --+ yy =x = x xx)) E En f~*, °, X ,y (x =x Ax n y *. 1\ =x -+ Y z x Nx,y,z(x = x y A y = x z --+ x = x z) z) E En Ft*. y =x , x, =x 447 447 Realizability Realizability Here /\, A,-~ on the the left left have have to to be be understood understood as as defined defined for for elements elements of of 0, ~, as as in in 7.8. 7.8. --+ on Here We write write E x t for for t =x t.5 We The m-product of of two two O-sets ~t-sets X A' == - (X, (X, "' ,,~) and Y y == - (Y, (Y, ",I) ,,~') is is the the O-set ~t-set X ,1' xx Y y == The ) and (X xx Y, ]I, "'' ~")) where where (X ') : = y) ~" ~ /\ (y (y ", ~', yy,). ') . y) yy'):= . . Xx Xn A product product of of n factors factors X Xl1> ,.. .. .. ,, Xn A'n iiss defined defined as as (Xl (X1 xx .... Xn-1) A'n. A n. - l) Xx X We use use calligraphic calligraphic capitals capitals X A',, Y, y , ... .. . for for O-sets. m-sets. 0 [] We Ext t =x t.5 O-product ' (x, "," (x', n (x '" x') Examples. Examples. 0 ~ itself itself may may be be viewed viewed as as an an O-set ~-set (0, (~, t+) ~ ) where where X X t+ ~ Y Y is is defined defined as in in 7.8. 7.8. Another Another example example is is N Af := (IN, (IN, = ~ ) , where where n = ~ m := {n} n N {m} m} as {n 9n - m m }} . . =IN), {n : n = n =IN := {n} { 8.5. D e f i n i t i o n . Let Let X A' == - (X, (X, "' ~)) be be an an O-set ~2-set and and F F :: X 8.5. Definition. X -+) ~t aa map. map. We We put put 0 := n (Fx Ex), a e p l ((FF)) := := n ~ (Fx (Fx /\ Ax x '" ~ Yy --+ Fy) Fy).. Repl An O-predicate ~-predicate on on X A' is is an an F F :: X X --+ 0 ~ such such that that Strict Strict(F) and Repl Repl(F) are An ( F ) and ( F ) are belong to ) . An -relation on inhabited inhabited ((belong to 0* ~*). An 0 m-relation on X X l1,, .. .. .., , Xn A'n is is an an O-predicate gt-predicate on on Xl A'I x Xn If (X (X xx Y, the product product of of the the gt-sets O-sets (X, If Y,~)) is is the (X, =x) (]1, =y), = y ) , and F :: X = ) and and (Y, and F X xx Strict Strict(F) ~ Ex), ( F ) := A (Fx --+ xEX xEX --+ x,yEX x,yEX .9 ... • Y Y X X , ~ n . . -+~ ~, 0, --+ "' x we we define define =y := n (F(x, F(x, Total ( F ) := := N(E n(Ex UF( Total(F) UF(x,, y)). An ~-function O-function from from A' X to to yY isis an an FF : X X xx YY --+ ~0 such such that that Strict(F), Strict ( F ) , aRepl An e p l ((FF) ), , Fun ( F ) Fun(F) z) y) A /\ F(x, z) z) ~--+ yy =y z) := N (F(x, y) x~y~z x ,y,z x x --+ y y : --+ are inhabited. inhabited. The The definition definition of of ~-function O-function for for more more than than one one Fun ( F ) , Total(F) Total ( F ) are Fun(F), argument argument isis reduced reduced to to this this case case via via products products of of m-sets. O-sets. []0 interpretation [ ] with equality equality =~ =u anan ~-set O-set ~a] [a] -== (9,(<1, [=~]); [=u]); for for [=~](x,x) [=u](x, x) we we (i) to to each each type type aa with (i) Eux, and and we we shall shall permit permit ourselves ourselves in in the the sequel sequel aa slight slight abuse abuse of of write E~x, also write also language, [a] also also for for the the underlying underlying set set 9. <1. language, using using [a] (ii) constants cc of of type type aa an an element element [c] [c] ofof ~a~, (ii) toto constants [a], to each each n-cry n-ary relation relation symbol symbol R, R, taking taking arguments arguments of of sorts sorts aaI, an respecrespec(iii) l , . .. .. ,. , an (iii ) to tively, an an m-relation O-relation IR] tively, [R] onon J[aa il]'l , ..... ,. ' ~an]. [an l N.B. ItIt isis important important to to observe observe that that for for practical practical purposes purposes the the definition definition of of EEx for N.B. x for an E x +-~ (X, ~) ) may may be be liberalized, liberalized, itit suffices suffices that that ~nx ((Ex 0* . an m-set O-set (X, t+ xx ,,~ '" x)x) eE gt*. 8.6. DDefinition. An interpretation I ] of of aa many-sorted many-sorted relational relational language language assigns assigns 8.6. e f i n i t i o n . An 0 F'] "' x 448 448 A.S. Troelstm Troelstra Remark. in the the definition R e m a r k . If If in definition above above we we take take for for n f~ aa complete complete Heyting Heyting algebra, algebra, and and replace /\, and replace n ['1 in in the the conditions conditions above above by by the the meet meet operator operator A, and take take n* [2* := := {T} {T},, T n, we T the the top top element element of of f~, we obtain obtain precisely precisely the the interpretation interpretation of of many-sorted many-sorted intuitionistic n-sets, as described in intuitionistic logic logic in in f~-sets, as described in Fourman Fourman and and Scott Scott [1979] [1979] or or Troelstra Troelstra and and van van Dalen Dalen [1988]. [1988]. There There Et Et measures measures the the "degree "degree of of existence" existence" of of t. t. 8.7. 8.7. Definition. Definition. Let Let X A' == - (X, (X, ,.~)) be be an an n-set, f~-set, and and let let F F :: X X -+ --+ n, f~, then then V'XEXF(x) := n VxEXF(x) := A (Exx (Exx -+ -~ Fx) Fx),, :JXEXF(x) SxEXF(x) := "= U U (Exx (Exx 1\ A Fx). Fx). 0 0 tv xxEX EX xEX xEX N.B. indicate elements N.B. "V'x "Vx E E X X ... .. ". ", , ":Jx "2x E E X X ... .. ". " indicate elements of of n, f~, but but "V'x "Vx E EX X . .. .. ". ", , ":Jx "Sx E E X . .. .. ". " refer refer to to ordinary ordinary quantification. quantification. X 8.8. 8.8. Definition. Definition. The The interpretation interpretation of of formulas formulas of of aa many-sorted many-sorted relational relational lan language guage may may now now be be given given modulo modulo assignments assignments p p for for the the variables. variables. Let Let p p be be an an assignment a, for a. For assignment of of elements elements of of [a] [a] to to the the variables variables of of type type a, for all all a. For constants constants cc the [c]p is p(x) , and the interpretation interpretation [clp is supposed supposed to to be be given; given; for for variables variables [x]p [x]p := p(x), and for for prime prime formulas formulas := [= [t := [R] ([tdp, · . . , [tnJp) it = =~" t']p t']~ := ~=~l([t]~, F]~), [R(tl [R(t~,, ..... . ,, tn)]p t~)]~ := [Rl(It~]~,..., [t,]~),, ,, ]([t]p, [t']p), and and for for compound compound formulas formulas according according to to 7.8, 7.8, i.e. i.e. [A [m 1\ A B]p B]p := := [A]p [A]p 1\ A [B]p, [B]p, [A -+ --+ B]p B]p := [A]p -+ --+ [B]p, [B]p, [A := [A]p [--,A]p := [-~A]p := [A]p [A]p -+ --+ 0, [V'xEa.A]p [Vxea.A]p := := (V'dE (Vde[a])[A]d=/~, [amA]p[x/dJ ' [:JxEa.A]p := (:JdE[a]) [A]p[x/dJ ' [3xea.A]p := (SdeH)IA]p[~/~q, where [x/d] is (y) = pp(y) (y) for (x) = where p p[x/d] is the the assignment assignment given given by by p[x/d] p[x/d](y) for yy ;f:. ~ xx,, p[x/d] p[x/d](x) = d. d. Instead Instead of of using using assignments, assignments, we we may may also also use use aa language language enriched enriched with with constants constants as as names names for for each each n-set f~-set used used for for the the interpretation interpretation of of the the types, types, and and define define the the interpretation interpretation only only for for sentences. sentences. A A sentence sentence A A is is said said to to be be valid valid if if [A] [A] E E n* f~*.. 0 [:] N.B. shall sometimes N.B. In In the the sequel sequel we we shall sometimes used used "mixed" "mixed" expressions: expressions: for for an an n-set f~-set := UX X := n X == ~ (X, (X, ,.~) [Vx E E XA(x)] XA(x)] := N=ex(Ex ~ [A(x)]) [A(x)]),, [[Bx XA(x)] := U , eExx((Ex E x 1\ A ) [V'x :Jx EE XA(x)] XEx (Ex -+ [A(x)]) [A(x)]).. 0, tv 8.9. sound for 8.9. Proposition. P r o p o s i t i o n . Intuitionistic Intuitionistic many-sorted many-sorted predicate predicate logic logic is is sound for realizabil realizabil- ity. ity. Proof. The proof proof is P r o o f . The is routine. routine. The The definition definition of of an an interpretation interpretation says says that that for for the the n-relation f~-relation [R] [R] interpreting interpreting relation relation R R of of the the language language the the following following hold: hold: (1) Ux = y] -+ [y = x]) E n°, n x x,y ,Y 2) ((21 n n ([x = y] y] 1\ ^ [y ly = = zJ -+ + [x = = z]) E e n* a*,, x,y,z X~y~Z = 449 449 Realizability Realizability (3) ~'~ ([R](Xl,... ,x,) --+ EXl A . . . A Exn) e ~*, XXl l l ·~...~Xn · ·,Xn [n](g)) Ee n' n ([R] (x) /\^ [x = YJIyl -+ [R](Y)) x,y where where of of course course [x [s = = YJI y~ abbreviates abbreviates [XI ~x1 = YI] /\ . , . /\ [Xn YYn]; similarly we we may may n] ; similarly abbreviate EXI EXl /\A .. .... /\A E En Es abbreviate n asas Ex. (1) and and (2) (2) guarantee guarantee the the validity validity of of symmetry symmetry and and transitivity transitivity of of equality, equality, and and (1) (3) and and (4) (4) the the validity validity of of strictness strictness and and replacement replacement for for R. R. Reflexivity Reflexivity translates translates (3) into the the trivial trivial nx(Ex Nx(Ex -+ -~ [x Ix = x]) x]) EE n' ~*,, so so does does not not need need an an extra extra condition. condition. 0 [] into (4) (4) = i,g ---- Y l ] A ... A ~X n = "- = The definition definition of of the the interpretation interpretation of of aa language language with with function function symbols symbols is is reduced reduced The to the the case case of of relational relational languages, languages, by by regarding regarding functions functions as as special special relations relations ((a a to partial function function is is aa relation relation which which is is functional: functional: Vxyz(R(x, VZyz(R(Z, y) y) A R(x, R(s z) --+ y = = z) z),, partial and aa ((total) function is is aa relation relation which which is is functional functional and and total, total, i.e. i.e. satisfies satisfies and total) function /\ -+ Y V~3yR(~, y) y)). VX3yR(x, ). 8.10. Definition. Definition. (Interpretation (Interpretation of of function function symbols) symbols) To To each each function function symbol symbol 8.10. F ' a l x . .. .. x. X an --+ a we we assign assign an an n-function ~-function ~F] from from [al] x . . . . X x ~an] to to ~a]. In In full the the conditions conditions read: read: full A ( E Z -+ Uy[F](Z, ~*, (x, y)) y)) EE n', n F : (TI (Tn -+ (T [F] [(Td . . [(Tn] [(T]. (Ex -+ Uy [F] n' . [F](x, z) z) -+ /\ IF](Z, [y = z]) y) A z]) eE ~*. N ~ IY ([F] (x, y) n ([F](Z, x,y,z s Ii = ~.,y,z 0 For partial functions functions the may be be omitted. For partial the first first condition condition may omitted. [] As As to to the the reason reason for for using using relations relations to to interpret interpret functions, functions, see see 8.13. 8.13. 8.11. formulas for 8.11. Definition. Definition. (Interpretation (Interpretation of of formulas for languages languages with with function function symbols) symbols) We now now assume assume aa language language with with relation relation symbols symbols and and symbols symbols for for total total functions. functions. We for compound compound terms terms = t2 t2 and and R t l . . . t n for We have have to to say say how how to to interpret interpret tl -We This isis done done recursively: recursively: tl --= t2t2 for for arbitrary arbitrary t l , t 2t2 isis interpreted interpreted as as t l , . . . , t n . This = x is is interpreted interpreted aas 3 x ( t 1 -= - X A tt22 == X ) ; F t l . . . tn = s 3 X l . . . x n ( (t t 1l == x 1 A . . . A t n == x n = x); the given by by ~F](pxl,...,pxn, px). the value value of of F X l . . . x n == x isis given A Fxl...xn = RRt t ll. .. .. . tn isis interpreted interpreted as as 3Z(t'= 3x(£ = Zx A R(Z)). R(x)). []0 tl Rtl . . . tn t l , . . . , tn . tl tJ, 3xI . . . Xn XI /\ . . ' /\ tn Xn 3x(tl x /\ x); Ftl . . . tn x FXI , . . Xn x [F](PXI, " " pxn , px). /\ FXI " , Xn x); tn /\ 8.12. TTheorem. The interpretation interpretation above above isis sound soundfor for many-sorted many-sorted logic. logic. "8.12. h e o r e m . The xA -+ A[x/t] Proof. Almost Almost entirely entirely routine. routine. To To see see that that e.g. e.g. all all instances instances of of VxA V --+ A[x/t] are are Proof. valid, one one should should note note that that the the "unwinding" "unwinding" of oftl == t2t2 and and Rtl Rtl ... mentioned above above valid, . . tn mentioned isis precisely precisely what what one one does does inin showing showing syntactically syntactically that that the the addition addition of of symbols symbols for for definable functions functions with with the the appropriate appropriate axiom axiom isis conservative; conservative; the the standard standard proof, proof, definable e.g. e.g. Kleene Kleene [1952], shows shows that that the the "unwinding" "unwinding" translation translation of of VxA V ~ A[x/t] isis inin fact fact derivable derivableinin the the relational relational part part ofofthe the language. language. tl [1952], . tn xA -+ A[x/t] 450 450 A.S. Troelstra Troelstra In our case this means that the the soundness In our case this means that soundness reduces reduces to to the the soundness soundness for for a a relational relational language language with with an an extra extra relation relation symbol symbol RF RE for for each each function function symbol symbol F F in in the the original original language. o 0.. language. 8.13. 8.13. Remark. R e m a r k . The The reason reason that that we we have have not not imposed imposed the the stronger stronger requirement requirement . . X• O"n that that aa function function symbol symbol F F :: 0" al1 xx ...an -+ 0" a is is to to be be interpreted interpreted by by a a function function ~F] :: [0" [al] ~an] -+ --4 [0"] [a] lies lies in the fact fact that that this this sometimes sometimes not not sufficiently sufficiently in the [F] 1 ] xx . .. .. . Xx [O"n] general: interpretation of general: the the interpretation of Yx:3!yR(x, Vx3!yR(x, y) y) says says that that n ~x(Ex --4 U U ~y [R] R ] ((x, x , y)) y)) E e n' ft*,, x (Ex -+ z]) Ee n' and (x, zz)) -+ and n ~x,u,z([R](x, y) 1\ A [R] ~R](x, --4 [y [y = - z~) ~*,, but but there there is is no no guarantee guarantee that that we we x ,y,z([R] (x, y) that n n' . Cf. the similar can can find find aa function function f f such such that ~ [xRlR] ] ( x(x, , fx) fx) E eft*. Cf. the similar situation situation for for the the interpretations interpretations where where n ft is is aa complete complete Heyting Heyting algebra; algebra; only only for for sheaves sheaves the the situation situation simplifies; see simplifies; see Troelstra Troelstra and and van van Dalen Dalen [1988,1.3.16, [1988,1.3.16, chapter chapter 14]. 14]. Example. E x a m p l e . Let Let f f be be a a primitive primitive recursive recursive function function with with function function symbol symbol F F in in the the [=IN]) as language language of of HA. H A . Over Over the the domain domain N Af == - (IN, (IN, ~=~]) as defined defined above above we we can can introduce introduce the the interpretation interpretation of of F F by by the the relation relation RF(n, := En En 1\A[m [m = RE(n, m) m):= = fn] fn] we we add add En En to to guarantee guarantee the the strictness strictness of of RF) RE),, so so RF RE := :-- {p(n, {p(n, fn) fn) :: nn E E IN}. IN}. It It is that this this yields yields aa realizability to) the the one one is now now routine routine to to see see that realizability for for HA H A (equivalent (equivalent to) defined defined before. before. 8.14. modelling of described above 8.14. Remark. R e m a r k . The The modelling of many-sorted many-sorted logic logic described above is is an an inter interpretation n-sets, and pretation in in a a certain certain category category Efr, Eft, with with as as objects objects the the f~-sets, and as as morphisms morphisms (equivalence (equivalence classes classes of) of) n-functions. f~-functions. More More precisely, precisely, the the morphisms morphisms from from X 2( to to Y y are are given given by by n-relations f~-relations on on X X xY 3) such such that that (cf. (cf. definition definition 8.5) 8.5) ', Strict(F), Strict(F), Repl(F), Repl(F), Fun(F) Fun(F),, Total(F) T o t a l ( F ) eE n ~*, modulo defined as modulo an an equivalence equivalence � ,,~ defined as 'xy)) Eeft*. n' . (Fxy +-t := [Yxy(Fxy F'xy)] = F � ,~ F' F' "[Vxy(Fxy +-t ~ F'xy)] - N ( (Ex E x 1\ AE Eyy -+ --4 (Fxy e+ F F'xy)) F xx~y,y Composition morphisms F Composition of of morphisms F : 9X X -+ ~ yY,, G G : 9Y 3) -+ --4 Z Z is is given given by by the the relational relational product: G product: G oo F F is is the the relation relation on on X A' x Z Z given given by by (G 0o F) F)(x, = :3y ~y E e Y(F(x, y(F(x, y) y) 1\ ^ G(y, C(y, zz)). (G (x, zz )) ::= )) . The morphism idx The identity identity morphism idx : 9X A' -+ --4 X ,u is is simply simply (the (the equivalence equivalence class class of) of) idx(x, idx(x, y) y) := :-Xx ==x x y .y · Let be the mappings. There Let Sets Sets be the category category of of sets sets and and set-theoretic set-theoretic mappings. There are are functors functors � which may A : 9Sets Sets -+ --4 Eff Eft and and r F : 9Efr Eft -+ -4 Sets Sets which may be be described described as as follows. follows. �, (X, "') A, the the "constant-objects "constant-objects functor functor",' , maps maps aa set set X X to to the the n-set ft-set (X, ,,~) with with x x '" ,,~ Y, then {n Ee IN := {n IN : 9x x= = y}, y}, and and if if f f : 9X X -+ --+ Y, then �f A f is is represented represented by by the the n-relation ft-relation yy := Rf , Rf ::== (( II xx '" ~ y) y). . relation xx � "') = I�, where F(X, ,,~) = {x {x": Ex Ex inhabited} inhabited}/___, where � ~_ is is the the equivalence equivalence relation ~_ x' x' := "= r(X, naturally (x (x '" ,,~ x' x ~inhabited) inhabited).. r F is is usually usually called called the the global-sections global-sections functor, functor, since since it it is is naturally (X, ,,~) "') the (X, ,,~); ); isomorphic isomorphic tot tot the the functor functor which which assigns assigns to to (X, the set set of of morphisms morphisms T T -+ -4 (X, T {* }},, =,) T is is the the terminal terminal object object (({, =,) with with ((,* =, = , *,)) = = IN. IN. n "' Realizability 451 451 Ll A preserves preserves finite finite limits limits and and is is full full and and faithful; faithful; r F also also preserves preserves finite finite limits, limits, Ll. N and and r F is is left-adjoint left-adjoint to to A. Af is is a a natural-numbers natural-numbers object object in in Eff, Eft, and and Eff Eft is is in in fact fact with proofs the facts aa topos (see 8.17). the theory topos (see 8.17). For For the theory of of Eff, Eft, with proofs of of the facts mentioned mentioned above, above, see see Hyland Hyland [1982]; [1982]; the the general general theory theory of of realizability realizability toposes toposes is is treated treated in in Hyland, Hyland, Johnstone Johnstone and and Pitts Pitts [1980] [1980].. Other Other sources sources of of information information on on Eff Elf are are Robinson Robinson and and Rosolini Rosolini [1990] [1990] and and Hyland, Hyland, Robinson Robinson and and Rosolini Rosolini [1990] [1990].. The by Eff The categorical categorical view view provided provided by Eft suggests suggests the the following following 8.15. ), Y 8.15. Definition. D e f i n i t i o n . The The n-sets fl-sets X X' == - (X, (X, ,,~), Y == - (Y, (Y, ",I) ,,J) are are said said to to be be isomorphic isomorphic if there there are are n-functions gt-functions F F :: X X x x Y -+ fl, G :: Y Y x • X X -+ -+ n gt such such that that G G oo F F � ~ idx idx,, if Y -+ n, G F oo G G � ~ idy idy,, where where � ~ is is defined defined as as above, above, i.e., i.e., H H � ~ H' U' := [Vxy(Uxy f-+ ++H'xy)] U'xy)].. 0 [] F := [Vxy(Hxy It isomorphic sets It is is easy easy to to see see that that quantification quantification over over isomorphic sets yields yields equivalent equivalent results, results, in in the the following following sense: sense: "' Lemma. L e m m a . Let Let X A' and and Y Y be be isomorphic isomorphic via via F, F, G. G. Then Then (G(y, x) VxEX.A(x) VxeX.A(x) = = VyEY.A(Gy) Vyey.A(Gy) = = VyEYVxEX. VyeyVxeX.(G(y, x) -+ --+ A(x)), A(x)), and and similarly similarly for for existential existential quantification. quantification. The The proof proof is is routine, routine, relying relying on on the the soundness soundness of of logic. logic. Important Important special special cases cases are (a) (a) any any n-set fl-set X A' == - (X, (Z, "') ,,~) is is isomorphic isomorphic to to (X', (X', "' "~rf X' X' x X') where X' X' := := {x {x :: X') where are X and and E Exx E E n' ~t*}},, and and (b) the following following situation: situation: let let X A' == - (X, be an an n-set, gt-set, xx Ec X (b) the (X, ,,~) "') be and and let let X' X' c c X X such such that that ' (X '" Vx Vx E e X(E X ( E xx E e n' ~* -+ ~ :lx 3x'' E e X X'(x ,,~ x x')' ) E e n'). ~*). 8. 16. Products, 8.16. P r o d u c t s , powersets p o w e r s e t s and a n d exponentials exponentials Definition. interpretation of [a1] x [a D e f i n i t i o n . The The interpretation of aa product product [a1 [al x a a2] is ~al] ~a2l. The functions functions 2] is 2 ] . The ptT~T, ~T ",T , p�,T, are unpairing on p P0O'~T, p PlrO',T are simply simply represented represented by by the the pairing pairing and and unpairing on the the relevant relevant n-sets. ~-sets. 0 C:l In logic, the In order order to to interpret interpret higher-order higher-order logic, the interpretation interpretation of of type type a a and and types types interpretation of interpretation of P(a) and a a -+ --+ T relative relative to to the the interpretation of a, a, T, and and the the interpretation of P( a ) and the operator the the relation relation E E~" as as well well as as the operator App App must must be be such such that that extensionality extensionality and and comprehension are comprehension are valid. valid. Definition. X, P(X) D e f i n i t i o n . Let Let X X' == = (X, (X, ,,~)) be be an an n-set; f~-set; the the n-powerset ~-powerset of of X', P(X), , is is an an n-set gt-set X -+ -+ n: (X (X -+ -+ n, fl, �) ~) where where for for F, F, G G of of X fl" E(F) E(F) := := Strict(F) Strict(F) 1\ A Repl(F), Repl(F), E(G) 1\ E(F) F --~ G := E(F) A E(G) An ~ xx((F F xx f-+ ~ Gx) Gx),, 1\ F � G := EF. xx EExx F := F x F := Fx 1\ A EF. ",I) be be n-sets, then the -functionset X (X x• Y Let Let X A' == -___(X, (X, ~),) , Y y == - (Y (]I,, ,,J) ~-sets, then the n f~-functionset X -+ -+ Y y is is (X Y -+ -+ n, ~t, �) ~) such such that that for for F, F, G G E EX X x • Y Y -+ -+ n fl := E(F) E(F) := Str(F) Str(F) 1\ A Repl(F) Repl(F) 1\ A Fun(F) Fun(F) 1\ A Total(F) Total(F) F := n (Fxy f-+ F � ~ G G := ~x,y(Ex AE Eyy -+ --+ (Fxy ~ Gxy)) Gxy)) 1\ A EF E F 1\ AE EG, G, X,y ( Ex 1\ ApP x, y) := E F 1\A Fxy. Appx,y(F, y) := EF Fxy. x,y (F, x, "' "' 452 452 A.S. Troelstra Troelstra A.S. N.B. Here Here we we have have availed availed ourselves ourselves of of the the freedom freedom to to define define E(F) E ( F ) so so as as to to be be N.B. equivalent only only to to F F � ~ F F in in the the realizability realizability sense, sense, not not literally literally identical. identical. D [] equivalent R e m a r k . As As noted noted above, above, 0 R itself itself may may be be viewed viewed as as an an O-set R-set (R, ,,-.). It is is then then Remark. (0, ",. ) . It not isomorphic to not hard hard to to see see that that the the O-powerset R-powerset of of aa O-set f~-set X X is is in in fact fact isomorphic to the the ~-functionset X X � -~ O f~.. O-functionset Definition. IInn an an interpretation interpretation of of intuitionistic intuitionistic higher-order higher-order logic logic powertypes powertypes and and Definition. exponentials are interpreted interpreted such such that that [P( IP(a)] is the the P[ P~a] and such such that that [0' ~a � -+ 7 T]] is is exponentials are O'] and O' )] is [0'] [aI � --+ (7) iT].. [E,,] [E~] :=E :=E [H,,),, [App" [App~,~] := App App[~],[~]. D ,T ) := [,,),[T) ' D R e m a r k . We We can can easily easily introduce introduce subtypes subtypes of of aa given given type, type, as as follows. follows. Let Let X A' == Remark. (X, "') ~) be be an an O-set. R-set. Intuitively Intuitively an an O-predicate f~-predicate over over X X determines determines aa subset subset of of X. A'. We We (X, may again again make make this this into into an f~-set: may an O-set: Definition. Let Let X A' == - (X, (X, "') ~) be be an an O-set, f~-set, and and let let F F :: X X � --+ 0 R be be an an O-predicatej ~-predicate; Definition. then the the O-subset R-subset of of X X determined determined by by F F is is (X', ,,~') with with X' X' := {x (x :: Fx Fx E e O* R*}, (X', ",I) }, then x , ", , JIyY: =: = x ~xy .'" y . D X N.B. An An equivalent equivalent definition would have have been been obtained obtained by by taking taking X' X' = - X X , , and and N.B. definition would xx '" y. The defined ,,~ y y ::= - Fx F x /\ A Fy F y /\ Ax x '" ,,, y. The resulting resulting O-set f~-set is is isomorphic isomorphic to to the the one one defined above. above. 8 .17. Proposition. and comprehension valid. 8.17. P r o p o s i t i o n . Extensionality Extensionality and comprehension are are valid. The proof is routine. routine. The proof is Remarks. (i) In In categorical the preceding the category category R e m a r k s . (i) categorical terms, terms, the preceding facts facts mean mean that that the Eff products and and exponentials exponentials (i is cartesian cartesian closed) Eft has has products (i.e. closed),, and and moreover moreover has has a a . e. is classifying truth-value object, object, namely (0, f-t and hence is aa topos. topos. classifying truth-value namely (gt, ~ ) ), , and hence is The fact that the natural numbers are unique modulo in higher-order The fact that the natural numbers are unique modulo isomorphism isomorphism in higher-order logic (8.2) (8.2) corresponds corresponds in in categorical terms to number logic categorical terms to the the uniqueness uniqueness of of the the natural natural number object in a a topos. object in topos. (ii) Obviously, the notions notions needed realizability interpretation interpretation of of H AH HAH (ii) Obviously, the needed for for the the realizability to types according can be be formalized formalized in in H HAH If we we assign level g(a) can A H itself. itself. If assign aa level types a0' according £(0') to 1, l(r), max( to ( a ) ++ 1, a x (£l ((O'a )) ++ 1, a x r ) == 1 , g.(a-+r) 0, g(Pa) to ~(0) £ (7) , g£((O'X7) £(PO' ) == g£(0') £(0'�7) == m £(0) == ~?(w) £(w) == 0, (i.e. all all variables variables then the the interpretation interpretation of of aa formula formula of of level level _< ::; nn (i.e. max(g(a), max(£ ( 0'), t~(r)), £ ( 7) ) , then are n) isis definable definable by by aa formula formula of of level level _< ::; n. n. are of of level level _< ::; n) next aim aim will will bbee to to show show that that for for H HAS the resulting resulting notion notion of of realizability realizability is is Our A S the Our next in fact fact equivalent equivalent to to realizability realizability as as defined in 7.2. 7.2. in defined in 8.18. Definition. Definition. An An f~-set O-set X X -== (X, "') is is called called canonically canonically uniform uniform ifif ["lxex 8.18. (X, ,,,) x Ex nXEX E is inhabited. inhabited. X X isis uniform uniform ifif itit isis isomorphic isomorphic to to aa canonically canonically uniform uniform set. set. is Realizability Realizability 453 453 For uniform n-sets X (X, rv) interpretation of universal and existential quantifiers quantifiers may may be be simplified simplified to to existential \ixEX Vz~ X Fx Fz := := n Fz, 3xEX 3z~V Fx Fz := := U Fz; xNEX Fx, xUEX Fx; more more precisely, precisely, (nx(Ex (N~(Ex � ~ Fx) Fx) ~ nx N~ Fx) Fx) Ee n* ~*,, (Ux(Ex (U~(Ex 1\ A Fx) Fx) ~ Ux U~ Fx) Fx) E E n* ~*.. It suffices suffices to to prove prove this this for for canonically canonically uniform uniform ~-sets, and then then it it is is easy: easy: It n-sets, and let \ix(m.n EE Fx), let n n E nx Nx Ex, Ex, and and let let m m E e nx(Ex Nx(Ex � --+ Fx), Fx), then then Vx(mon Fx), i.e. i.e. m.n m~ EE nx Nx Fx, Fx, 8.19. Lemma. L e m m a . For uniform ~-sets A' == - (X,,,~) interpretation of universal and 8.19. xEX PROOF. PROOF. xEX f-+ f-+ etc. D [] etc. X (X, rv) X canonically separated x = y. (x rv y) X canonically proto-effective x = y. Ex Ey X separated [proto-effective] X X [canonically] effective X 88.20. .20. Definition. D e f i n i t i o n . Let Let A' == _-- (X, ~) be be an an ~-set. is canonically separated if if n-set. A' is (x ~ y) inhabited inhabited � =~ x = y. X is is canonically proto-e~ective if if Ex n fl Ey inhabited inhabited � =v x -- y. A' is is separated [proto-effective] if if A' is is isomorphic isomorphic to to aa canonically canonically separated separated [canoni [canonically proto-effective] proto-effective] ~-set. is [canonically] effective if if A' is is [canonically] [canonically] separated separated cally n-set. A' is and [canonically] effective. D • and [canonically] effective. 8.21. Proposition. P r o p o s i t i o n . Let Let X X == - (X, (X, rv) ,,,) be be aa uni uniform and Y y - (Y, (Y, rvl) ,,J) aa proto-effective proto-effective 8.21. form and ~-set. Then Then the the uniformity uniformity principle principle n-set. UP(X, Y) y) VxEX 3yEY 3yE:); A(x, A(x, y) y) � -~ 3yEY\iXEX 3 y E Y V x E X A(x, A(x, y) y) UP(X, \ixEX valid. isis valid. == P r o o f . Without loss of generality we we may may assume assume yy to be canonically canonically proto protoProof. Without loss of generality to be effective. Let so n E Uyey(Ey 1\ then effective. Let n E N~ex3y E y A ( x , y ) , so A A(x,y)), then D i.e. n E UyeY Ey 1\ V x E X 3 y e Y ( p o n E Ey), i.e. A N~ex A(x, y). [] The is not not needed needed in follows but the logical The following following proposition proposition is in what what follows but describes describes the logical significance significance of of separatedness: separatedness: n E n Ex 3y E YA(x, y), n E U Ey (Ey A(x, y)), \ixEX3yEY(pon E Ey), X n E UyEY Ey nXEX A(x, y).y An n-set X (X, rv) is separated iff \ix, yEX(...,...,xrvy 8.22. Proposition. 8.22. P r o p o s i t i o n . An ~-set X -== (X,,,~) is separated ij~ Vx, yEX(-~-~x,,~y --+ � xrvy) is valid. x,,~y) is valid. X xrvy). \ix, yEX(...,...,xrvy xrvy) x, y E X: {x' EE XX :9x'x' ~rv x inhabited}, inhabited}, [x] := [x] := (x' [x] [x] ~� [y] [y] :=:= (n{n EE ININ 9xx ,,~rv yy inhabited}, inhabited}, x' } := { [x] : x9 eE xX}. X' Then A X'~' - (X', (X', ~) � ) is canonically separated separated and and isomorphic isomorphic to X via via the the ~-relations n-relations Then is canonically to A' FF on on X X xx XX and and G G on on X X xx X', X', defined defined by by F([x], y) y) -== G(x, G(x, [y]):= [y]) := (n{n EE ININ :9yy ~rv xX inhabited}. F([x], inhabited}. Proof. It is to see canonically separated separated ~-set n-set A' satisfies P r o o f . It is easy easy to see that that aa canonically satisfies Conversely, assume assume Vx, yEA'(-~-~x,,~y --+ � x,,~y) to to be be Conversely, valid. We define for x, y E X" valid. We define for \ix, yEX(...,...,xrvy � x~y). Vx, yEX(-~-~x~y -~ := == : X 454 454 A.S. Troelstra A.S. We have have to to show show that that F, F, GG are are strict, strict, total, total, functional functional and and that that their their composition composition isis We the identity. identity. This This isis mostly mostly routine. routine. For For example, example, to to see see that that FF isis functional, functional, observe observe the such that that that our our hypothesis hypothesis gives gives the the existence existence of of an an nn such that (1) 't/m, m' ml(m Ex A/\ m' ml EE Ey Ey A/\ (3m" (3mll(mll x"'Y ) ~-+ n.p(m, nop(m, m') m/) EE (x,,,,y) (x"'y»).. (1) Vm, (m EE Ex (m" EE x,,,,y) The functionality functionality of of FF amounts amounts to to validity validity of of V[x], 't/[x], y,y, y'(F([x], yl(F([x], y) y) A/\ F([x], F([x], y') yl) -+ -+ y y '" The l. .e. Y I) , i.e. yl), (2) (2) nn nn [x]EX' y,y'ex y,y' EX [4ex' (E[x] A/\ Ey Ey A/\ Eyl {n EE IN IN :93m(m 3m(m EE xx ,.~ '" y)}A y)}/\ (E[x] Ey' A/\ {n yl) eE ~*. D*. 3m(m eE x9 ~'" y')} yl)} -~ -+ yy ~'" y') {{nElN ~ e ~ :: 3m(m and xx ~'" y' inhabited implies implies yy ,.~ '" yl inhabited, (2) (2) readily readily follows follows from from Since xx ~'" yy and Since y' inhabited, yl inhabited (1). []0 (1). The following following proposition, proposition, with with aa proof proof due due to to van van Oosten, Oosten, justifies justifies the the terminolterminol The ogy "uniform" . ogy "uniform". 8.23. An D -set XX -== (X, (X, ~) "') is uniform iff satisfies UP (X,N). 8.23. PProposition. r o p o s i t i o n . An f~-set is uniform iff X X satisfies UP (X,Af ). Proof. direction is is aa consequence 8.21. The The other P r o o f . One One direction consequence of of proposition proposition 8.21. other direction direction is is proved as follows. Given A', X, consider consider the the f~-set D-set Y Y -== (Y, (Y, -~) �) defined defined by proved as follows. Given by Y Y := := {p(x, {p(x, n) n) :: n n E E Exx}, Exx}, (x, {n :: n~ Ee ~x",y (~, n) ~) � _~ (y, (y, m) m ) : =:= {~ ~ y /\ ^ ~n = = m}. m} We We shall shall write write (y, (y, n) n) for for p(y, p(y, n) n) in in what what follows. follows. There There is is an an D-function f~-function G G :: Y Y -+ --4 X A" given given by by a((~, ~), Xl) ~ ' ) ::= = (x (~ '" ~ X ~ l)' ) ^/\ {n}. {~}. G « x, n), G G is is surjective surjective as as an an D-function, f~-function, i.e. i.e. 't/yEY3xEX.G(y, x), VyEy3xEX.G(y,x), i.e. i.e. n n (E (Ey(y,n) --} U (Exx (Exx /\ AG G((y,n),x)) ~*. y (y, n) -+ « y, n), x » EE D*. (y,n)EY (v,n)~Y xEX x~x Let Let H H : 9Y y -+ --4 N Af be be the the surjective surjective D-function f~-function Then Then H « x, n), H((x, n), m) m) := := {n {n :: nn = = m}. m}. 't/xEX V x e X 33YEY y e Y 3nEN(G(y, 3ne.Kf (G(y, x) x) /\ A H(y, H(y, n» n)) is is valid. valid. If If we we assume assume UP(X, UP (X, N), Af), it it follows follows that that 3nEN't/ xEX 3yEY(G (y, x) 3nEAf VxEX 3yEY(G(y, x) /\ A H(y, H(y, nn)) » isis valid, valid, which which means means that that for for some some nn EE IN IN (1) n N((Ex -+ U U (x '" Xl» xEX xEX x'EX,nEEx' x~ E X , n E E x ~ I) , Z isis inhabited. inhabited. Let Let now now Z Z := := (Z, (Z, ", ,,/), Z := := {xEX { x E X :: nEEx}, nEEx}, ",I ~' the the restriction restriction of of '" ned by to to Z. Z. Clearly Clearly F F defi defined by F(x, F(x, Xl) x') := := (x (x '" ~ Xl) x') is is an an injection injection of of Z Z into into Xj X; and and the the formula formula (1) (1) states states that that this this injection injection is is also also aa surjection, surjection, hence hence Z Z and and X A' are are isomorphic. isomorphic. 0 [] Realizability Realizability 455 455 8.24. PProposition. The ~2-powerset n-powerset of of aa separated separated ~n--set X == (X, (X, ~) rv) isis uniform. uniform. 8.24. r o p o s i t i o n . The set X PProof. r o o f . Let Let X X be be canonically canonically separated, separated, and and let let yY :: X X --+ -+ gt n be be an an element element of of Ak. Pok realizes realizes Repl(y); Repl(Y); nn EE Str(y) Str(Y) means means the ~2-powerset n-powerset P(X) P(X) of of X, X , then then Ak.p0k the -+ EEx) n eE fn' lXE ~ xX( ~(x eE yY --+ ~).. By By restricting restricting attention attention to to "normal" "normal" Y Y we we can can construct construct uniform uniform realizers realizers for for if Let us us call call yY nnormal Eyy. . Let E o r m a l if E EEx. m EE Y(x) Y(x) =~ � pPlm m tm E x. For Am, m EE Str(y), Str(Y) , and and so so always always For normal normal yy,, A m . Pl plm p(Ak. Pok, Am.plm) Am. Pl m) eE EE(Y) p(Ak.p0k, (y).. To show ~-isomorphism n-isomorphism of of PP((X) X) with with the the subset subset of of normal normal elements, elements, observe observe that that ifif To show we arbitrary Y ( y ) ((x) x):= ( z ) A 1\ m x } , , we we map map arbitrary Y to to ~(Y) (Y) with with ~(Y) := {p(n, {p(n, mm)) :: nn eE yY(x) m eE EEx} we have that that yY =P(X) =P(X) ~(Y) (Y) isis inhabited: inhabited: have n (x E Y t-t x E (Y)) E n° . xx EEX X 0 If nn Ee Str(y), Str(Y) , kk eE (x (x eE Y), Y) , then then nn.k Ex, (Y) (x) , etc. etc. El If . k eE Z z , and and p(k, p(k, nn.k) . k ) eE O(:P)(x), 8.25. PProposition. For HHAS the realizability as defined above isis equivalent to r___nn rn 8.25. r o p o s i t i o n . For A S the realizability as defined above equivalent to as as defined defined in in 7.2. 7.2. Proof. P r o o f . This This result result is is now now obtainable obtainable as as aa corollary corollary to to the the preceding preceding propositions. propositions. As As to to the the second-order second-order quantifiers, quantifiers, we we may may restrict restrict attention attention to to the the normal normal elements elements N. Let of of the the n-powerset ~-powerset of of Af. Let ) , y) /(X) (I)'(X) := {p(p(x, {p(p(x, V y), y)": p(x, p(x, y) y) E e X}, X}, = {p(x, O"(X) ::= {p(x, y) y)": p(p(x, p(p(x, V), y), y) y) E e X}. X}. "(X) ~' corresponds corresponds as as operation operation on on binary binary relations relations to to 9 above, above, " ~" is is its its inverse. inverse. ' Let Let rn r__~nbe be defined defined as as for for HAS-formulas HAS-formulas as as in in 7.2 7.2 relative relative to to an an assignment assignment second-order variables; rn'~be notion as X r-+ ~ X* X* for for second-order variables; and and let let rn___ be the the realizability realizability notion as defined defined X in section, relative in this this section, relative to to an an assignment assignment X X r-+ ~ XO X ~ of of normal normal binary binary relations relations to to the the second-order rn' Xt := t) ; xxr__nn'A rn' A := second-order variables variables (i.e. (i.e. x xr___nn'Xt := x x E e [Xt] [Xt] := := XO(x, X~ "= x x E e [A]). [A]). Then Then for for all all formulas formulas A A of of HAS H A S there there are are cP CA, such that that A , 'l/JCA A such r � ] ,, r__~nA(X1 A ( X t ,, .. ...., , Xn) X~) -+ -4 'l/J r A rn' r__n_n'A A (Xl ( X 1,, .. .. .., , Xn) X ~ ) [X [ Xf,[ , .. .. .., , X X n�o//'X O , X 1;, ,, .. ... . ,, 'X xz rn cP , . . . , "X� ] , /"X . . . , Xn)[X , . . . , X -+ rn A(Xl ' rn A(X1 , . . . , Xn) xx r___~nA ( X t , . . . , X n ) -~ CAA r___~nA(X1,... , X n ) [ X ~;, . . . , X ~�/ O " X [ ,f . . . , O"X~,], where where X X 1l ,, .. .. .., , Xn X~ is is aa complete complete list list of of the the second-order second-order variables variables free free iinn A. A. 0 El 8.26. -sets X 8.26. Lemma. L e m m a . For For n ~)-sets X == - (X, ( X , rv) ,,~),, Y y == - (Y, (I, rv/) ,,/),, Y y separated, separated, the the elements elements of of be represented the the n ~ --functionset functionset X X -+ -~ Y y may m a y be represented by b y /functions unctions f f : 9X X -+ --+ Y Y.. 456 A.S. Troelstra Troelstra P r o o f . Let Let F F :: X X xy :)) -+ --+ n gt be an arbitrary arbitrary element of the the n-exponent, gt-exponent, for for which which Proof. be an element of E F is is inhabited. Then for for certain certain k, k~ EF inhabited. Then k, k' kk Ee n Fxy) , M~ex(Ex -~ UyEY U:YFxy), XE x (Ex -+ k' (Fxy Fxy' 1\ k' E 6n Nxex,y,y, ey(Fxy A Fxy' -+ -+ Y y ", ~ 'I V y'). ') . / XEX,y,y Ey By By the the second second statement, statement, it it readily readily follows follows in in combination combination with with separatedness, separatedness, that that Fxy inhabited, inhabited, Fxy' inhabited => Fxy Fxy ~ inhabited ~ y y= - V', y~, and and from from the the first first statement statement that that for for all all x x with with Ex Ex inhabited, inhabited, there there is is a a y y such such that that Fxy is is inhabited; inhabited; so so let let I f be be the the function function defined defined for for x x with with Ex Ex inhabited, inhabited, such such that that Fxy F(x, Ix) fx) is is inhabited. inhabited. 0 D F(x, 8.27. Lemma. set (X, 8.27. L e m m a . For For separated separated [effective] [effective] y y == - (Y, (]I, ",I) N') the the n-Iunction ~-function set (X, "') ,.~) -+ -+ (Y, ",I) is separated [effective]. Proof. the preceding there is P r o o f . By By the preceding lemma lemma we we may may represent represent ((there is aa little little checking checking to to do, do, but but we we leave leave this this to to the the reader reader)) the the elements elements of of the the exponent exponent by by the the isomorphic isomorphic (X-+Y, �) ~) with with (X-+Y, (f � (f ~ g) g) inhabited inhabited => =~ I f = = g, g, If � ~ 1 f := := {m {m :: Vy Vy E e YVn YVn E e Ey(mon Ey(m.n E e E(fy))}, E(fy))}, oorr combined combined into into aa single single definition: definition: Ey(mon Ee E(fy))}. := {m {m :: ff = If � ..~ 9g := = 9 g 1\ A Vy Vy E e YVn YVn E e Ey(m.n E(fy))}. The The separatedness separatedness has has been been built built into into the the definition; definition; as as to to proto-effectiveness, proto-effectiveness, suppose suppose EI Ef n M Eg Eg inhabited, inhabited, then then for for some some m m Vy Vy E e YVn YYn E e Ey(mon Ey(m.n E e E(fy) E ( f y ) nn E(gy)); E(gy)); by by the the proto-effectivity proto-effectivity of of Y Y it it follows follows that that Vy Yy E E Y(fy Y(fy = = gy) gy),, i.e. i.e. f f = = g. g. 0 [3 Remark. R e m a r k . The The fact fact that that X X -+ -+ Y y hold for logical reasons ( cf. 8.22): hold for logical reasons (cf. 8.22): If ==g .g. is is separated separated for for separated separated Y y is is also also easily easily seen seen to to " Vx(fx = Vx(fx = -,-,fI = = 9g +-+ ~ " -,-,Vx(fx = gx) gx) -+ -+ Vx(fx = gx) gx) +-+ 8.28. -sets generated 8.28. Proposition. P r o p o s i t i o n . The The structure structure of of functional functional w w-sets generated from from N Af is is iso isomorphic to HEO as morphic to HEO as defined defined in in 3.3. 3.3. Proof. P r o o f . Induction Induction on on the the type type structure. structure. 0 [3 The The following following is is immediate: immediate: 8.29. 8.29. Proposition. P r o p o s i t i o n . For For all all function function types types u a generated generated from from type type 00 in in HAH, H A H , the the realizability realizability interpretation interpretation validates validates aa uniformity uniformity principle: principle: [ vxPE~ x) ~ 3:vxPE~ x). P P VX Ol3x" A(X, x) -+ 3x"VX [OlA(X, x). Realizability 457 457 8.30. 8.30. Generalization G e n e r a l i z a t i o n to to other o t h e r kinds kinds of of realizability preceding section indicated how generalizations of In the the preceding section we we have have already already indicated how the the generalizations of In realizabilities realizabilities to to second-order second-order logic logic follow follow aa pattern. pattern. If If we we combine combine this this with with the the "truth-value semantics" semantics" idea idea introduced introduced in in the the preceding preceding section section and and used used extensively extensively "truth-value above, consider other above, we we are are led led to to consider other choices choices for for n f~ and and n· 9t*.. E x a m p l e s . (a) (a) If If we we want want to to generalize generalize rnt-realizability, rnt-realizability, we we take take Examples. ~"~rnt :__ X c c lN,p C {O}}, {0}}, IN, p c := {{(X,p) (X, p) :: X nrnt ~'~rnt, := :__ {{ (X, (X, p) p) E e n 9t :: X X inhabited, inhabited, 00 E e pl. p}. nrnt• rnt : The crucial crucial operations operations we we have have to to defi define are Nnt A rnt,, -+ _.+rnt, Nrnt: The ne are rnt , n (X,p) A rat (Y, q) q) := (X, p) N := nt (Y, (X,p) _.+rnt (y, q) q ) ::= = rnt (Y, (X, p) -+ Nrnt (Zy, By) :-:= n��Y yEY ( Zy , Py ) ((X I\ A Y, {0 :: O0 E e p p I\ A O0 E e q}), q}), ((X Y, {o ((X -+ --+ Y), {O {0 :: 00 E e P p -+ -+ 0 E e q}), q}), ((X Zy n NyEY By). ((NyEY )· nyEy Zy, yEy Py N.B. We We do do not not really really need need to to define define an an operation A rnt,, since since in in aa second-order second-order N.B. operation Nnt context we we can can define define (7.3) (7.3) operation A'' in in terms terms of of the the other other operations, operations, producing producing context operation 1\ objects (X, (X, p) p) 1\ A'' (Y, (Y, q) q) isomorphic isomorphic to to (X, ( X , pp)) ANnt rat (Y, (Y, q) q).. objects (b) ed realizability (b) For For modifi modified realizability we we can can put put ~'~mrn :__ X , YY) ) nmrn := {{ ((X, flmrn, :: = = { {(X, Y) nmrn• (X, Y) Z c C Y Y c C IN, IN, O0 E e Y}, Y}, :: X eE n 9t :: X X inhabited}. inhabited}. mrn is (-~mrn is defined defined component-wise, component-wise, and and for for -+ ._+mrn we take take mrn we n (Z, ..._~mrn (X 't,, yt) :__ ((X ((X ~-+ Y) Y)["ln (X' (X t -+ .--.+Y'), yt), X' X t -+ ~ Y')). yt)). mrn (X (X, Y) Y) -+ Y') := (In (In order order to to guarantee guarantee that that 00 always always occurs occurs in in the the second second component component we we must must choose choose our our godelnumbering gSdelnumbering of of the the partial partial recursive recursive functions functions such such that that Ax.O Ax.O = = 0.) 0.) (c) (c) For For Lifschitz Lifsehitz realizability realizability another another idea idea is is needed, needed, aa reformulation reformulation of of the the original nition which original defi definition which makes makes Lifschitz Lifschitz realizability realizability fit fit the the general general pattern pattern van van Oosten Oosten [1991b]. [1991b]. 8.31. 8.31. Notes Notes The definition of higher-order logic The proper proper definition of realizability realizability for for higher-order logic emerged emerged from from the the study study of (Hyland [[1982], Hyland, Johnstone 1982] ' Hyland, 1980] , Pitts 1981 ] , of special special toposes toposes (Hyland Johnstone and and Pitts Pitts [[1980], Pitts [[1981], [ 1981a,1981b,1981c] ) . Aczel [ 1980] described described aa less common Grayson Grayson[1981a,1981b,1981c]). Aczel[1980] less far-reaching far-reaching common generalization of realizability semantics. generalization of Heyting-valued Heyting-valued and and realizability semantics. The The higher-order higher-order 1991 b] . By ofthis extension In-realizability is extension of of rrln-realizability is due due to to van van Oosten Oosten [[1991b]. By means means of this extension extension he he shows shows that that the the following following principle principle RP RP (Richman's (Richman's Principle) Principle) v x dd((V v yyd d ( (X x C C Y Y VX X n NY Y = = 0 -+ --+ 3nVx(x 3nVx(x E e X X -+ --+ x x = = n)) n)) VX (where (where V VXd X d,, 3yd 3 Y d are are quantifiers quantifiers ranging ranging over over decidable decidable subsets subsets of of IN) ]hi) is is false false in in the the "Lifschitz "Lifschitz topos" topos" and and true true in in Eff, Eft, that that is is to to say say false false in in the the higher-order higher-order extension extension of of Lifschitz Lifschitz realizability realizability and and true true in in the the realizability realizability interpretation interpretation for for higher-order higher-order logic os Lif logic described described in in the the preceding preceding section. section. More More information information on on the the Lifschitz Lifschitz top topos Lif is 1996] . is given given in in van van Oosten Oosten [[1996]. 458 458 A.S. Troelstra Troelstra 9. Further Further w ork 9. work 9.1. Realizability R e a l i z a b i l i t y for for set-theory set-theory 9.1. It It is is also also possible possible to to define define rn-realizability, rn-realizability, or or the the abstract abstract version version !:-realizability r_-realizability for for the language The definition the fact the language of of set set theory. theory. The definition is is straightforward straightforward except except for for the fact that that we build in we have have to to build in extensionality. extensionality. The problem becomes clear if we we try try to to extend extend the the definition definition of of rn-realizability rn-realizability The problem becomes clear if given in in 7.2 to intuitionistic intuitionistic third-order third-order arithmetic arithmetic HAS H A S 33 ((variables X 2, yy2, 2 , .. ... .)) in in given 7.2 to variables X2, which which we we can can also also quantify quantify over over PP(IN) 7~P(IN),, and and with with full full impredicative impredicative comprehension comprehension and and extensionality extensionality EXT VX2(Y l1 E EX X 22 1\ Ay Y~l = = Zl Z 1 -+ -~ Z Z~ X 2)) EXT l EE X2 V'X2(y E E yl "= where := Vz(z V'z(z E X where X X l~ = = Y~ X ++ ~-~ Zz E Y) Y).. If If we we take take as as clauses clauses x rrn n XX2(yl) 2 ( y ~) "= 1 , x) , x X*2(Y*I,x), xrnVX2.A(X 2) := V'X*2(X VX*2(xrnA(X2), etc., we we discover discover that that there there is is no no X*2(y* rn A(X2) , etc., rn V'X2.A(X2) problem problem in in proving proving soundness soundness except except for for the the axiom axiom EXT; EXT; this this imposes imposes aa restriction restriction on the the sets sets over over which which the the "starred "starred variables" variables" X*2 X .2 should should range. range. on Some 1985] , solve Some authors, authors, e.g. e.g. Beeson Beeson [[1985], solve the the problem problem in in the the case case of of set set theory theory by by first first giving giving aa realizability realizability interpretation interpretation for for aa set set theory theory without without the the extensionality extensionality axiom, combined combined with axiom, with an an interpretation interpretation of of the the theory theory with with extensionality extensionality into into set 1984b] build set theory theory without without extensionality. extensionality. Others Others such such as as McCarty McCarty [[1984b] build the the extensionality extensionality into into the the definition definition of of realizability. realizability. The defining realizability 1971 ] . Other The earliest earliest paper paper defining realizability for for set set theory theory is is Tharp Tharp [[1971]. Other 1974] ' Friedman papers using papers using realizability realizability for for set set theory theory are: are: Staples Staples [[1974], Friedman and and Sce Scedrov [ 1983,1984] , McCarty [ 1984b,1986] , Beeson 1985] , and drov[1983,1984], McCarty[1984b,1986], Beeson [[1985], and the the series series of of papers papers by by Khakhanyan. Khakhanyan. . 9.2. 9.2. Comparison C o m p a r i s o n with w i t h functional f u n c t i o n a l interpretations Another type modified) Another type of of interpretation interpretation which which is is in in certain certain respects respects analogous analogous to to ((modified) realizability, Dialectica inter realizability, but but in in other other respects respects quite quite different, different, is is the the so-called so-called Dialectica interpretation devised Gi:idel [[1958]. 1958] . There pretation devised by by Ghdel There is is also also aa modification modification due due to to Diller Diller and and 1974] . As modified realizability Nahm [[1974]. Nahm As we we have have seen, seen, modified realizability associates associates to to formul8.S formulas A A of of HA H A ww (XU~ aa new 3-free 3-free formulas formulas of of the the form form Amr(xU) Amr(X~) (x new variable variable not not free free in in A), A), expressing expressing "XU "x ~ modified-realizes modified-realizes A" A".. The The DialecticaDialectica- and and Diller-Nahm Diller-Nahm interpretation interpretation on on the the other other hand AD (xU, ADN(XU, hand associate associate with with A A formulas formulas V'yT VyrAD(x a, yT) yr) and and V'yT VyrADN(X ~, yT) y~) respectively, respectively, (1, a, T depending on the logical structure of A alone, AD , ADN quantifier-free; we may read depending on the logical structure of A alone, AD, ADN quantifier-free; we may read and "XU AD (xU, yT) , V'yT ADN(XU, yT) V'yT Vy~AD(x~,yr), VyrADg(X~,y r) as as "XU "x ~ D-interprets D-interprets A" A"and "x ~ DN-interprets DN-interprets A" A" respectively. respectively. For interpretation, the For a a soundness soundness proof proof for for the the Dialectica Dialectica interpretation, the prime prime formulas formulas of of the the theory theory considered considered have have to to be be decidable decidable with with aa decision decision function function of of the the appropriate appropriate theories with type; type; for for the the Diller-Nahm Diller-Nahm interpretation interpretation this this is is not not necessary. necessary. For For theories with decidable prime formulas e.g. I-HAW) interpretation is decidable prime formulas ((e.g. I - H A w) the the Diller-Nahm Diller-Nahm interpretation is equivalent equivalent to to the the Dialectica Dialectica interpretation. interpretation. For For background background information information the the reader reader may may consult consult 1958] in 1990] , and the Gi:idel [[1958] the commentary commentary to to Ghdel in Gi:idel Ghdel [[1990], and the the relevant relevant chapter chapter elsewhere elsewhere in volume. in this this volume. Realizability Realizability 459 459 Stein has has constructed constructed aa whole whole sequence sequence of of interpretations interpretations intermediate intermediate between between Stein the modified realizability; the DN-interpretation DN-interpretation and and modified realizability; see see the the papers papers by by Stein, Stein, and and Diller Diller [[1979]. 1979] . 9.3. 9.3. Formulas-as-types Formulas-as-types realizability realizability In paradigm, formulas representing propositions In the the formulas-as-types formulas-as-types paradigm, formulas ((representing propositions)) are are regarded regarded identified with as determined by as determined by ((identified with)) the the set set of of their their proofs. proofs. The The idea idea is is illustrated illustrated by by taking and writing taking aa natural natural deduction deduction formulation formulation of of intuitionistic intuitionistic predicate predicate logic, logic, and writing the the deductions deductions as as terms terms in in aa typed typed lambda-calculus. lambda-calculus. Normalization Normalization of of the the deductions deductions suggests suggests equations equations between between the the terms terms of of such such in particular "t proves A" for aa calculus calculus ((in particular beta-conversion beta-conversion)) and and "t proves A" for compound compound A A then then behaves behaves like like an an abstract abstract realizability realizability notion. notion. Of Of particular particular interest interest is is the the realizability realizability obtained obtained by by stripping stripping the the proof-terms proof-terms of of their their types. types. With With combinators combinators instead instead of of [ 1973,1974] . Tait lambda-abstraction, already used lambda-abstraction, such such aa realizability realizability is is already used in in Staples Staples[1973,1974]. Tait [[1975] 1975] uses uses this this concept elegant version version of proof of the normalization concept for for an an elegant of Girard Girard's's proof of the normalization theorem theorem for for second-order second-order intuitionistic intuitionistic logic. logic. For For another another version version of of the the proof proof see see Girard, Lafont Lafont and and Taylor Taylor [[1988]. Girard, 1988] . Mints 1989] , Barbanera Barbanera and Martini [[1994] 1994] study Mints [[1989], and Martini study completeness completeness questions questions for for such cally, one such realizabilities. realizabilities. More More specifi specifically, one is is interested interested in in completeness completeness results results of of the the S, I-~ss A following following type: type: for for all all formulas formulas A A of of aa certain certain formal formal system system S, A iff iff I-T ~T 3x(xrA) where intuitionistic or ) , and where T T is is aa suitable suitable formal formal system system ((intuitionistic or classical classical logic logic), and rr the the abstract abstract version version of of realizability realizability studied. studied. "Formulas-as-types" "Formulas-as-types" has has also also been been aa leading leading idea idea in in the the formulation formulation of of vari vari[ 1982,1984] , permitting ous theories of ous typed typed theories, theories, such such as as the the theories of Martin-Lof Martin-LSf[1982,1984], permitting to to absorb implication is absorb logical logical operations operations into into type-forming type-forming operations operations ((implication is subsumed subsumed under cation under under function-type function-type formation, formation, universal universal quantifi quantification under formation formation products products of of ' s type dependent types, etc. dependent types, etc.).) . In In the the proof-theoretic proof-theoretic investigations investigations of of Martin-Lof Martin-LSf's type [ 1989,1991,1992] realizability role. theories theories by by de de Swaen Swaen[1989,1991,1992] realizability plays plays an an important important role. 9.4. 9.4. Completeness C o m p l e t e n e s s questions questions for for realizabilities realizabilities Rose 1953] gave intuitionistically provable Rose [[1953] gave an an example example of of aa classically classically valid, valid, but but not not intuitionistically provable formula formula of of propositional propositional logic, logic, such such that that all all its its arithmetical arithmetical substitution substitution examples examples classically) realizable; Kleene [[1965b], 1965b] , who are are ((classically) realizable; this this result result was was improved improved by by Kleene who showed showed in the that rf- and that the the example example also also worked worked for for r_J_fand mrf-realizability mrf-realizability ((in the latter latter case case the the substitution substitution instances instances were were provably provably realizable realizable even even intuitionistically intuitionistically).) . See See also also hand, Gavrilenko 1981 ] proved Tsejtin 1968] . On Tsejtin [[1968]. On the the other other hand, Gavrilenko [[1981] proved that that the the principle principle "Every "Every formula formula for for which which all all arithmetical arithmetical substitution substitution instances instances are are realizable, realizable, is is provable but not provable in in intuitionistic intuitionistic propositional propositional logic" logic" is is consistent consistent with with HAS H A S ((but not with with HAS + + M M). ). HAS Kleene also Kleene also showed showed that that the the class class of of formulas formulas of of predicate predicatelogic logic which which are are realizable realizable for rn, ) . Similar under under substitution substitution is is not not recursive recursive ((for r__nn,rf r_f,, mrn turn). Similar questions questions have have been been studied Plisko. A studied at at length length in in aa series series of of papers papers by by Plisko. A typical typical result result of of this this kind kind is is 460 460 A.S. A.S. 1'roelstra Troelstra the following. Let Let n T~ [AnJ [dtT~] be be the the class class of of all all formulas formulas A(PI A(P1,..., P,)) of of predicate predicate the following. , . . . , Pn logic . . . , P;) logic such such that that all all arithmetical arithmetical substitution substitution instances instances (i.e. (i.e. formulas formulas A(Pt, A(P~,..., P~) with with Pt P~*,... ,P,~ arithmetical) arithmetical) are are rn-realizable r___~n-realizable[such [such that that VX VXl... A(X1,..., , . . . ,P; l . . . Xn A(X l , . . . , Xn) is is rn-realizable r__nn-realizableas as defined defined for for HASJ. HAS]. n; n Plisko [1983J Plisko [1983] showed showed that that An A R is is aa complete complete m-set, II]-set, and and that that An AR c c 7~; R is is also also not arithmetical as not arithmetical as shown shown in in Plisko Plisko [1977J. [1977]. Van method originally Jongh, gave Van Oosten[1991c], Oosten[1991c], adapting adapting aa method originally due due to to de de Jongh, gave aa semantical earlier established D. Leivant: semantical proof proof of of aa result result earlier established by by proof-theoretic proof-theoretic means means by by D. Leivant: if all all arithmetical instances of of aa formula formula of of predicate predicate logic logic are are provable provable if arithmetical substitution substitution instances in HA, the intuitionistic predicate "maximality of in HA, the formula formula itself itself is is aa theorem theorem of of intuitionistic predicate logic( logic("maximality of intuitionistic method uses which rn-realizability intuitionistic arithmetic" arithmetic").). The The method uses aa realizability realizability in in which r___~n-realizability and Beth-semantics are are combined. combined. His proof also also yields yields the following completeness completeness and Beth-semantics His proof the following result result for for realizability. realizability. Let Let HA H A ++ be be an an extension extension of of HA H A obtained obtained by by adding adding to to the the language primitive primitive constants constants .9(application) (application),, k, k, s (combinators), (combinators), with with axioms axioms saying saying language k, s) partial combinatory combinatory algebra. that ., k, that (IN, (IN,., s) is is aa partial algebra. Define Define !:-realizability r__-realizability for for HA H A ++ relative relative to to this this combinatory combinatory algebra. algebra. Then Then aa predicate predicate formula formula A A is is provable provable in in intuitionistic intuitionistic predicate predicate logic logic iff iff all all arithmetical arithmetical instances instances of of A A are are provably provably realizable realizable in HA H A ++ .. in A different different sort sort of of completeness completeness result result has has been been obtained by Liiuchli L/iuchli [1970J. [1970]. He He A obtained by defined defined aa modified modified realizability realizability for for predicate predicate logic logic with with aa set-theoretic set-theoretic hierarchy hierarchy as as models the finite-type models for for the finite-type functionals. functionals. All All formula formula of of predicate predicate logic logic is is realizable realizable by by an an element element of of this this hierarchy hierarchy iff iff it it is is classically classically provable; provable; but but if if we we require require that that the domains, we the realizing realizing functionals functionals are are invariant invariant under under permutations permutations of of the the basic basic domains, we obtain Inspection shows obtain precisely precisely the the intuitionistically intuitionistically provable provable formulas. formulas. Inspection shows that that the the 's construction "modified" "modified" aspect aspect of of Liiuchli L/iuchli's construction is is not not really really relevant. relevant. A A modern modern recasting recasting 's result, of linking it of Liiuchli L/iuchli's result, linking it with with the the category-theoretic category-theoretic interpretation interpretation of of logic, logic, was was ' Donnell [1996J given and Makkai Makkai [1992J. given by by Harnik Harnik and [1992]. See See also also Lipton Lipton and and O O'Donnell [1996].. 9.5. intuitionistic arithmetic 9.5. Realizability Realizability for for subsystems s u b s y s t e m s of of intuitionistic arithmetic Damnjanovic Damnjanovic [1994J [1994] considers considers realizability realizability for for primitive primitive recursive recursive arithmetic, arithmetic, using using primitive primitive recursive recursive functions functions instead instead of of partial partial recursive recursive functions. functions. In In particular particular the the clauses clauses for for --+ --4 and and V V require require modification. modification. This This is is achieved achieved using using levels, levels, which which are are provided provided by by the the Grzegorczyk Grzegorczyk hierarchy hierarchy for for the the primitive primitive recursive recursive functions. functions. In [1995J the idea is is applied In Damnjanovic Damnjanovic [1995] the same same idea applied to to obtain obtain aa realizability realizability for for HA, HA, using using so-called so-called < co-recursive c0-recursive functions functions instead instead of of general general recursive recursive functions. functions. (The (The < < co-recursive c0-recursive functions functions are are precisely precisely the the functions functions provably provably recursive recursive in in HA H A and and PA.) This PA.) This permits permits reproving reproving aa number number of of metamathematical metamathematical results results for for HA H A by by realizability to elementary and finite realizability methods. methods. The The technique technique also also applies applies to elementary analysis analysis and finitetype type arithmetic. arithmetic. Wehmeier Wehmeier [1996J [1996] uses uses rnr__p_n-and and rnt-realizability rnt-realizability in in aa study study of of intuitionistic intuitionistic arithmetic Vn3mA( n, m arithmetic with with � ~l-induction, I~1. Wehmeier shows shows that that whenever whenever fF- Vn3mA(n, m)) l ' Wehmeier l -induction, I� then t(n) ) . Ferreira then there there is is aa primitive primitive recursive recursive t(x) such such that that VnA(n, VnA(n,t(n)). Ferreira and and Marques Marques [1996J [1996] use use aa version version of of rnt-realizability rnt-realizability for for aa language language of of arithmetic arithmetic extended extended Realizability 461 461 with infinite infinite disjunctions disjunctions in in aa study study of of the the intuitionistic intuitionistic counterpart counterpart IS~ IS� of of Buss's Buss's with system S~ S� (Buss (Buss [1986]), [1986]), and and of of I~x n:: l just just mentioned. The authors authors obtain obtain aa new new system mentioned. The proof of of the the fact, fact, established established in in Cook Cook and and Urquhart Urquhart [1993] [1993] by by mrt-realizability, mrt-realizability, that that proof whenever whenever ~f- Vn3mA(n, 'v'n3mA(n, m)in m) in IS~, IS� , then then ~f- VnA(n, 'v'nA(n, t(n)) t(n)) where where t(x)is t(x) is polynomial-time polynomial-time computable in in x. x. They They also also sketch sketch another proof of of Wehmeier's Wehmeier ' s result. result. computable another proof 9.6. Combining realizability w with logic 9.6. C o m b i n i n g realizability i t h classical classical logic Lifschitz[1982,1985] considered considered an an extension extension of of classical classical arithmetic arithmetic with with an an additional additional Lifschitz[1982,1985] predicate predicate K K (x), (x) , "x "x isis computable". computable" . The The result result is is aa combination combination of of classical classical arithmetic arithmetic and category Eff Eft we we can can obtain obtain something something and realizability. realizability. ItIt is is to to be be noted noted that that in in the the category similar by by considering considering side side by by side side N" N and and A]N. �lN. similar 9.7. M Medvedev's calculus of of finite finite pproblems 9.7. e d v e d e v ' s calculus roblems The calculus of finite finite problems reminiscent The calculus of problems as as formulated formulated by by Medvedev, Medvedev, is is somewhat somewhat reminiscent of, actually diverges realizability. See the papers papers by of, but but actually diverges rather rather far far from from recursive recursive realizability. See the by Medvedev, and by by Maksimova, [1979] . Medvedev, and Maksimova, Shekhtman Shekhtman and and Skvortsov Skvortsov [1979]. 9.8. Applications to C Computer Science 9.8. A p p l i c a t i o n s to o m p u t e r Science For some examples, [1990], Streicher Streicher [1991] (realizability modeling modeling of For some examples, see see Scedrov Scedrov [1990], [1991] (realizability of the theory theory of Smith [1993] [1993] (slash relations for type theory) and the the the of constructions) constructions),, Smith (slash relations for type theory) and papers Tatsuta (program (program synthesis synthesis by Within the papers by by Tatsuta by "realizability-cum-truth" "realizability-cum-truth").) . Within the effective topos one can find models for for strong e.g. effective topos one can find models strong polymorphic polymorphic type type theories; theories; see see e.g. Hyland Hyland [1988] [1988].. classical proofs In Berger Berger and and Schwichtenberg Schwichtenberg [1995] [1995] program-extraction program-extraction from from classical proofs of of In statements 'v'n3mA(n, statements Vn3mA(n, m) m),, A A quantifier-free, quantifier-free, is is studied. studied. Two Two methods methods are are compared; compared; one method method is is based based on on normalization normalization of of classical classical proofs proofs formalized formalized in in aa calculus calculus of of one natural natural deduction, deduction, the the other other method method uses uses modified modified realizability. realizability. The The two two methods methods (n)) (i yield yield terms terms tl(x), tl (x), tt2(x) respectively such such that that 'v'nA(n, VnA(n, titi(n)) (i = = 1, 1, 2) 2) and and for for all all 2 (X) respectively numerals n, tl numerals ~, tl (n) (~) = = tt2(~). An ingredient ingredient in in the the second second method method is is the the combination combination 2 (n) . An of of the the Godel Gb'del-Gentzen translation (Troelstra (Troelstra and and van van Dalen Dalen [1988,2.3]) [1988,2.3]) with with the the - Gentzen translation Friedman-Dragalin Friedman-Dragalin A-translation A-translation (Troelstra (Troelstra and and van van Dalen Dalen [1988,3.5]); [1988,3.5]); by by this this combination combination aa classical classical proof proof of of 'v'n3m Vn3m A(n, m) m),, A quantifier-free, quantifier-free, can can be be transformed transformed into into an an intuitionistic intuitionistic proof proof of of the the same same statement. statement. Interesting Interesting applications applications of of the the second second method method are are given given in in Berger, Berger, Schwichtenberg Schwichtenberg and and Seisenberger Seisenberger [1997]. [1997]. One One of of the the examples examples concerns concerns the the following following special special case case of of ' Higman s lemma: Higman's lemma: If If X x l1,, yyll are are two two number-theoretic number-theoretic functions, functions, we we can can find find n n<m m such such that that l l m and l n :S 1m . xxlnn :S< xxlm and yyln < yylm. This This fact fact is is easily easily proved proved classically. classically. Applying Applying the the program program extraction extraction via via modified modified realizability, realizability, one one obtains obtains an an algorithm algorithm which which is is in in suitable suitable cases cases quadratically quadratically faster faster nition of than than "brute-force" "brute-force" search. search. The The syntactic syntactic defi definition of modified modified realizability realizability and and 462 462 A.S. A.S. 1roelstra Troelstra of the the Friedman-Dragalin Friedman-Dragalin translation translation are are used used in in a a variant variant which keeping the of which helps helps keeping the complexity program (term) complexity of of the the extracted extracted program (term) down. down. Abbreviations A b b r e v i a t i o n s in in the t h e references references AML AML = = Annals Annals of of Mathematical Mathematical Logic. Logic. AMS AMS Transl. Transl. = = American American Mathematical Mathematical Society Society Translations, Translations, Series Series 2. 2. APAL APAL = = Annals Annals of of Pure Pure and and Applied Applied Logic. Logic. Archiv Archiv = = Archiv Archiv fUr fiir mathematische mathematische Logik Logik und und Grundlagenforschung. Grundlagenforschung. Doklady Doklady = = Doklady Doklady Akademii Akademii Nauk Nauk SSSR. SSSR. Izv.Akad.Nauk Izv.Akad.Nauk = = Izvestiya Izvestiya Akademii Akademii Nauk Nauk SSSR. SSSR. Seriya Seriya Matematicheskaya. Matematicheskaya. JSL JSL = = The The Journal Journal of of Symbolic Symbolic Logic. Logic. LMPS LMPS = = Logic, Logic, Methodology Methodology and and Philosophy Philosophy of of Science. Science. Math. Math. Izv. Izv. = = Mathematics Mathematics of of the the USSR, USSR, Izvestiya. Izvestiya. SM SM = = Soviet Soviet Mathematics. Mathematics. Doklady. Doklady. ZLGM ZLGM = = Zeitschrijt Zeitschrifl fUr fiir Logik Logik und und Grundlagen Grundlagen der der Mathematik. Mathematik. Zapiski Zapiski = = Zapiski Zapiski Nauchnykh Nauchnykh Seminarov Seminarov Leningradskogo Leningradskogo Otdeleniya Otdeleniya Matematicheskogo Matematicheskogo Instituta A. Steklova Instituta imeni imeni V. A. Steklova Akademii Akademii Nauk Nauk SSSR SSSR (LOMI). (LOMI). References References P. H. H. G. G. ACZEL P. 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[1980] Modified Modified realization realization and and the the formulae-as-types formulae-as-types notion, notion, in: in: To H.B. Curry: Essays on [1980] Combinatory Logic, Logic, Lambda Lambda Calculus and Fonnalism, Formalism, J. J. P. P. Seldin Seldin and and J. J. R. R. Hindley, Hindley, eds., Academic eds., Academic Press, Press, New New York, York, pp. pp. 491-501. 491-501. DILLER AND J. DILLER J. AND W. W. NAHM NAHM 4] Eine [1974] Eine Variante Variante zur zur Dialectica-Interpretation Dialectica-Interpretation der der Heyting-Arithmetik Heyting-Arithmetik endlicher endlicher Typen, Typen, [197 Archiv, 16, pp. Archly, 16, pp. 49-66. 49-66. A. G. G. DRAGALIN A. DRAGALIN [1968] The The computability computability of of primitive primitive recursive recursive terms terms of of finite finite type, type, and and primitive primitive recursive recursive [1968] realization (Russian), (Russian), Zapiski, 8, pp. 32-45. 32-45. Translation Translation Seminars in Mathematics. 8, pp. realization V.A.Steklov Mathematical Mathematical Institute Institute Leningrad 8(1970), 8(1970), pp. pp. 13-18. 13-18. This This volume volume appeared appeared V.A.Steklov as: A.O. as: A.O. Slisenko Slisenko (ed.), (ed.), Studies in Constructive Mathematics and Mathematical Logic. II. Consultants Consultants Bureau, Bureau, New New York, York, London. London. Part II. [1969] Transfinite Transfinite completions completions of of constructive constructive arithmetical arithmetical calculus calculus (Russian), (Russian), Doklady, 189, 189, [1969] pp. pp. 458-460. 458-460. Translation Translation SM SM 10, 10, pp. pp. 1417-1420. 1417-1420. [1979] An An algebraic algebraic approach approach to to intuitionistic intuitionistic models models of of the the realizability realizability type type (Russian), (Russian), [1979] in: Issledovaniya Issledovaniya po Neklassicheskim Logikam Logikam i Teorii Mnozhestv (Investigations (Investigations on in: Non-Classical Logics Logics and Set Set Theory), A. A. I. I. Mikhajlov Mikhajlov et et al., al., eds., eds., Nauka, Nauka, Moskva, Moskva, pp. pp. 83-201. 83-201. [1980] [1980] New New forms forms of of realizability realizability and and Markov's Markov's rule rule (Russian), (Russian), Doklady, 251, 251, pp. pp. 534-537. 534-537. Translation Translation SM SM 21, 21, pp. pp. 461-464. 461-464. [1988] Mathematical Mathematical Intuitionism, Intuitionism, American American Mathematical Mathematical Society, Society, Providence, Providence, Rhode Rhode Island. Island. [1988] Translation Translation of of the the Russian Russian original original from from 1979. 1979. PP.. EGGERZ EGGERZ [1987] [1987] Realisierbarkeitskalkiile Realisierbarkeitskalkiile MLo MLo und vergleichbare Theorien im Verhiiltnis VerhSltnis zur zur Heyting Heytingthesis, Ludwig-Maximilians-Universitat, Arithmetik, Arithmetik, PhD PhD thesis, Ludwig-Maximilians-Universit~it, Miinchen. Mfinchen. S. FEFERMAN S. FEFERMAN [1975] [1975] A A language language and and axioms axioms for for explicit explicit mathematics, mathematics, in: in: Algebra and Logic, Logic, J. J. N. N. Crossley, Crossley, ed., Springer 87-139. ed., Springer Verlag, Verlag, Berlin, Berlin, Heidelberg, Heidelberg, New New York, York, pp. pp. 87-139. [1979] [1979] Constructive Constructive theories theories of of functions functions and and classes, classes, in: in: Logic Logic Colloquium Colloquium '78, '78, M. M. Boffa, Boffa, D. van Dalen, and K. McAloon, D. van Dalen, and K. McAloon, eds., eds., North-Holland, North-Holland, Amsterdam, Amsterdam, pp. pp. 159-224. 159-224. F. A. MARQUES F. FERREIRA FERREIRA AND AND A. MARQUES [1996] [1996] Extracting algorithms algorithms from intuitionistic intuitionistic proofs, Tech. Tech. Rep. Rep. Pre-publicac;oes Pr~-publica~Ses de de Matematica 26/96, Universidade Matem~tica 26/96, Universidade de de Lisboa. Lisboa. AND D S. SSCOTT COTT M M.. P P.. FOURMAN FOURMAN AND D.. S. [1979] [1979] Sheaves Sheaves and and logic, logic, in: in: Applications of of Sheaves, M. M. P. P. Fourman, Fourman, C. C. J. J. Mulvey, Mulvey, and and D. D. S. S. Scott, eds., Springer Scott, eds., Springer Verlag, Verlag, Berlin, Berlin, Heidelberg, Heidelberg, New New York, York, pp. pp. 302-401. 302-401. H. H. M M.. FRIEDMAN FRIEDMAN [1973] [1973] Some Some applications applications of of Kleene's Kleene's methods methods for for intuitionistic intuitionistic systems, systems, in: in: Mathias and Rogers Rogers {1973J, [1973], pp. pp. 113-170. 113-170. [1975] [1975] The The disjunction disjunction property property implies implies the the numerical numerical existence existence property, property, Proceedings Proceedings of of the National Academy Academy of of Sciences of of the United States of of America, 72, 72, pp. pp. 2877-2878. 2877-2878. [1977a] 42, pp. [1977a] On On the the derivability derivability of of instantiation instantiation properties, properties, JSL, 42, pp. 506-514. 506-514. [1977b] [1977b] Set Set theoretic theoretic foundations foundations of of constructive constructive analysis, analysis, Annals of of Mathematics, Series 2, 105, 105, pp. pp. 1-28. 1-28. Realizability Realizability 465 465 H. M. M. FRIEDMAN FRIEDMANAND AND A. A. SCEDROV SCEDROV H. [1983] Set Set existence existence property property for for intuitionistic intuitionistic theories theories with with dependent dependent choice, choice, APAL, APAL, 25, 25, [1983] pp. pp. 129-140. 129-140. Corrigendum Corrigendum in in APAL APAL 26 26 (1984), (1984), p.101. p.101. [1984] Large Large sets sets in in intuitionistic intuitionistic set set theory, theory, APAL, APAL, 27, 27, pp. pp. 1-24. 1-24. [1984] [1986] Intuitionistically Intuitionistically provable provable recursive recursive well-orderings, well-orderings, APAL, APAL, 30, 30, pp. pp. 165-171. 165-171. [1986] GALLIER J. H. GALLIER [1993] Proving Proving properties properties of of typed typed lambda-terms lambda-terms using using realizability, realizability, covers, covers, and and sheaves, sheaves, [1993] Tech. Tech. Rep. Rep. MS-CIS-93-91, MS-CIS-93-91, Computer Computer and and Information Information Science Science Department, Department, School School of of Engineering Engineering and and Applied Applied Science, Science, University University of of Pennsylvania, Pennsylvania, Philadelphia. Philadelphia. GAVRILENKO Yu. V. GAVRILENKO [1981] Recursive Recursive realizability realizability from from the the intuitionistic intuitionistic point point of of view view (Russian), (Russian), Doklady, Doklady, 256, 256, [1981] pp. 18-22. 18-22. Translation Translation SM SM 23, 23, pp. pp. 9-14. 9-14. pp. GIRARD, Y. LAFONT, LAFONT, AND AND P. TAYLOR TAYLOR J. Y. GIRARD, [1988] Proofs Proo]sand Types, Types, Cambridge University Press, Press, Cambridge U.K. [1988] GODEL K. GODEL [1958] Uber 0ber eine eine bisher bisher noch noch nicht nicht beniitzte beniitzte Erweiterung Erweiterung des des finiten finiten Standpunktes, Standpunktes, Dialectica, Dialectica, [1958] 12, pp. 280-287. [1990] Collected CollectedWorks, Works, Volume Volume 2, Oxford Oxford University University Press, Press, Oxford. Oxford. [1990] N. D. GOODMAN GOODMAN [1978] Relativized Relativized realizability realizability in in intuitionistic intuitionistic arithmetic arithmetic of of all all finite finite types, types, JSL, 43, 43, pp. pp. 23-44. 23-44. [1978] GRAYSON R. J. GRAYSON [1981a] [1981a] Derived Derived rules rules obtained obtained by by aa model-theoretic model-theoretic approach approach to to realisability. realisability. Handwritten Handwritten notes from from Miinster Miinster University. University. notes Modified realisability toposes. Handwritten [1981b] [1981b] Modified realisability toposes. Handwritten notes notes from from Miinster Mfinster University. University. Handwritten notes [1981c] [1981c] Note Note on on extensional extensional realizability. realizability. Handwritten notes from from Miinster Miinster University. University. Appendix to Handwritten notes [1982] [1982] Appendix to modified modified realisability realisability toposes. toposes. Handwritten notes from from Miinster Miinster University. University. V.. HARNIK HARNIK V [1992] Provably Provably total total functions functions of arithmetic, JSL, 57, 466-477. [1992] of intuitionistic intuitionistic bounded bounded arithmetic, 57, pp. pp. 466-477. V. MAKKAI HARNIK AND AND M. M. MAKKAI V. HARNIK 's abstract categorical proof 57, pp. pp. 200[1992] Lambek's Lambek's categorical proof theory theory and and Liiuchli L/iuchli's abstract realizability, realizability, JSL, 57, 200[1992] 230. A. HEYTING A. HEYTING [1959] [1959] ed., ed., Constructivity Constructivity in Mathematics, North-Holland, North-Holland, Amsterdam. Amsterdam. W. HOWARD W. A. A. HOWARD Appendix: Hereditarily majorizable functionals functionals of of finite finite type, type, in: in: 'Iroelstm [1973aJ, [1973] Hereditarily majorizable Troelstra [1973@ [1973] Appendix: Springer Springer Verlag, Verlag, Berlin, Berlin, Heidelberg, Heidelberg, New New York, York, pp. pp. 454-461. 454-461. J. J. M. M . E. E. HYLAND HYLAND The effective effective topos, topos, in: in: The L.E.J. L.E.J. Brouwer Brouwer Centenary Centenary Symposium, Symposium, A. A. S. S. Troelstra Troelstra and and [1982] [1982] The D. Dalen, eds., D. van van Dalen, eds., North-Holland, North-Holland, Amsterdam, Amsterdam, pp. pp. 165-216. 165-216. APAL, 40, 40, pp. pp. 135-165. 135-165. A small small complete complete category, category, APAL, [1988] [1988] A J. J. M. M. E. E. HYLAND, HYLAND, P. P. T. T. JOHNSTONE, JOHNSTONE, AND A. A. M. M. PITTS P ITTS Tripos theory, theory, Mathematical Mathematical Proceedings Proceedings of of the Cambridge Philosophical Society, 88, 88, [1980] [1980] Tripos pp. pp. 205-232. 205-232. J. J. M. M. E. E. HYLAND HYLAND AND AND C.-H. C.-H. L. 1. ONG ONG Modified realizability realizability toposes toposes and and strong strong normalization normalization proofs, proofs, in: in: Typed Typed Lambda [1993] [1993] Modified M. Bezem Bezem and and J.J. F. F. Groote, Groote, eds., eds., Springer Springer Verlag, Verlag, Berlin, Berlin, Calculi and and Applications, M. Heidelberg, Heidelberg, New New York, York, pp. pp. 179-194. 179-194. 466 466 A.S. Troelstra A.S. M.. E. E. HYLAND, HYLAND, E E.. ROBINSON, ROBINSON, AND AND G. ROSOLINI ROSOLINI M [1990] The The discrete discrete objects objects in in the the effective effective topos, topos, Proceedings Proceedings of of the London Mathematical [1990] 60, pp. pp. 1-36. 1-36. Society, 60, A.. JANKOV J ANKOV V. A [1963] Realizable Realizable formulas formulas of of propositional propositional logic logic (Russian), (Russian), Doklady, 151, 151, pp. pp. 1035-1037. 1035-1037. [1963] Translation SM SM 4(, 4(, pp. pp. 1146-1148. 1146-1148. Translation J.. J D.. H. DE JONGH JONGH D H. J. DE [1968] Investigations on the intuitionistic intuitionistic propositional calculus, calculus, PhD PhD thesis, thesis, University University of of [1968] Wisconsin, Madison. Madison. Wisconsin, [1970] A A characterization characterization of of the the intuitionistic intuitionistic propositional propositional calculus, calculus, in: in: Kino, Myhill and [1970] [1970], pp. pp. 211-217. 211-217. Vesley [1970J, [1980] Formulas Formulas of of one one propositional propositional variable variable in in intuitionistic intuitionistic arithmetic, arithmetic, in: in: Troe/stra Troelstra and van [1980] [1980], pp. pp. 51-64. 51-64. Dalen [1980J, A. JOYAL JOYAL AND AND I. I. MOERDIJK MOERDIJK A. [1995] Algebraic Set Set Theory, London London Mathematical Mathematical Society Society Lecture Lecture Series Series 220, 220, Cambridge Cambridge [1995] University Press, Cambridge, U.K. U.K. University Press, Cambridge, KABAKOV F. A. A. KABAKOV [1963] On On the the derivability derivability of of some some realizable realizable formulae formulae of of the the sentential sentential calculus calculus (Russian), (Russian), [1963] 9, pp. pp. 97-104. 97-104. ZMLG, 9, [1970a] Intuitionistic deducibility of some realizable formulae of propositional logic (Russian), 192, pp. pp. 269-271. 269-271. Translation Translation SM SM 11, pp. 612-614. 612-614. Doklady, 192, 11, pp. [1970b] On On modelling modelling of of pseudo-boolean pseudo-boolean algebras algebras by by realizability realizability (Russian), (Russian), Doklady, 192, 192, [1970b] pp. 16-18. 16-18. Translation Translation SM SM 11, pp. 562-564. 562-564. 11, pp. pp. V. KH. KHAKHANYAN KHAKHANYAN [1979] Consistency Consistency of of the the intuitionistic intuitionistic set set theory theory with with Brouwer's Brouwer's principle, principle, Matematika, 5. 5. [1979] Not seen. seen. Found Found in in another another bibliography bibliography and and included included for for completeness. completeness. Not [1980a] Comparative Comparative strength strength of of variants variants of of Church's Church's thesis thesis at at the the level level of of set set theory theory (Russian), (Russian), [1980a] Doklady, 252, pp. 1070-1074. 1070-1074. Translation SM 21, 21, pp. pp. 894-898. 252, pp. Translation SM 894-898. of intuitionistic intuitionistic set set theory Church's principle principle and the uniformiza [1980b] The consistency consistency of theory with with Church's and the uniformiza[1980b] The tion principle (Russian), Universiteta. Seriya Seriya 1. L Matematika, tion principle (Russian), Vestnik Moskovskogo Universiteta. Matematika, Mekhanika, 3-7. Translation Translation Moscow University Mathematics 1980/5, pp. pp. 3-7. Mathematics Bulletin Bulletin Mekhanika, 1980/5, 35/5, 35/5, pp. pp. 3-7. 3-7. [1980c] The consistency of intuitionistic theory with formal mathematical analysis (Russian), (Russian), [1980c] The consistency of intuitionistic set set theory with formal mathematical analysis Doklady, 253, 253, pp. pp. 48-52. SM 22, 22, pp. pp. 46-50. 46-50. Doklady, 48-52. Translation Translation SM some intuitionistic intuitionistic and constructive principles principles with theory, [1981] [1981] The The consistency consistency of of some and constructive with aa set set theory, Studia Studia Logica, 40, 40, pp. pp. 237-248. 237-248. [1983] Set theory theory and and Church's Church's thesis in: Issledovaniya Issledovaniya po Neklascheskims Neklascheskims Logikam Logikam thesis (Russian), (Russian), in: [1983] Set A. I.I. (Studies in Nonclassical logics and Formal Systems), A. Formal'nym Sistemam Sistemam (Studies Nonclassical logics i Formal'nym Mikhajlov, ed., pp. 198-208. 198-208. Mikhajlov, ed., Nauka, Nauka, Moskva, Moskva, pp. of the Church's thesis (Russian), MatemMatem [1988] Nonderivability of the uniformization uniformization principle principle from from Church's thesis (Russian), [1988] Nonderivability aticheskie Zametki, Zametki, 43, 43, pp. pp. 685-691,703. 685-691,703. Translation Translation Mathematical Mathematical Notes Notes 43, 43, pp. pp. aticheskie 394-398. 394-398. A. A. KINO, J.J . MYHILL, MYHILL, AND AND R. R. E. E. VESLEY [1970] eds., Intuitionism Intuitionism and and Proof Proof Theory, Theory, North-Holland, North-Holland, Amsterdam. Amsterdam. [1970] eds., M. M . M. M . KIPNIS The constructive constructive classification classification of of arithmetic arithmetic predicates predicates and and the the semantic semantic bases bases of of [1968] [1968] The arithmetic (Russian), (Russian), Zapiski, Zapiski, 8, 8, pp. pp. 53-65. 53-65. Translation Translation Seminars Seminars in in Mathematics. Mathematics. arithmetic V.A.Steklov volume appeared 8(1970), pp. pp. 22-27. 22-27. This This volume appeared V.A.Steklov Mathematical Mathematical Institute Institute Leningrad Leningrad 8(1970), Studies in Constructive Constructive Mathematics Mathematics and Mathematical Mathematical Logic. as: A.O. A.a. Slisenko Slisenko (ed.), (ed.), Studies as: Part II. Consultants Consultants Bureau, Bureau, New New York, York, London. London. Part II. Realizability Realizability 467 467 [1971] On On the the realizations realizations of of predicate predicate formulas formulas (Russian) (Russian) (English (English summary), summary), Zapiski, Zapiski, 20, 20, [1971] pp. pp. 40-48. 40-48. Translation Translation Journal of of Soviet Mathematics Mathematics 1(1973), 1(1973), pp. pp. 22-27. 22-27. KLEENE S. C. C. KLEENE S. [1945] On On the the interpretation interpretation of of intuitionistic intuitionistic number number theory, theory, JSL, 10, 10, pp. pp. 109-124. 109-124. [1945] [1952] Introduction Introduction to metamathematics, metamathematics, North-Holland, North-Holland, Amsterdam. Amsterdam. Co-publisher: Co-publisher: Wolters Wolters[1952] Noordhoffj 8th revised revised ed.1980. Noordhoff; 8th ed.1980. [1957] [1957] Realizability, Realizability, in: in: Summaries Summaries of Talks Talks presented at the Summer Summer Institute Institute for Symbolic for Defense Defense Analyses, Analyses, Communications Communications Research Research Division, Princeton, Logic, Institute Institute for Division, Princeton, pp. 100-104. 100-104. Also Also in in Heyting [1959J, [1959], pp. pp. 285-289. 285-289. Errata Errata in in Kleene and Vesley [1965J, [1965], pp. page 192. page 192. 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P SKVORTSOV axiomatization of logic of problems [1979] The The impossibility impossibility of of a a finite finite axiomatization of Medvedev's Medvedev's logic of finitary finitary problems [1979] (Russian), 245, pp. Translation SM SM 20, 20, pp. pp. 1051-1054. 1051-1054. Translation pp. 394-398. 394-398. (Russian), Doklady, 245, P. MARTIN-LI~F P . MARTIN-LoF [1982] Constructive mathematics and and computer computer programming, programming, in: in: LMPS Cohen, Constructive mathematics LMPS 6, L. J. Cohen, [1982] Los, H. H. Pfeiffer, Pfeiffer, and K.-P. Podewski, eds., North-Holland, pp. 153-175. 153-175. J. Lo~, and K.-P. Podewski, eds., North-Holland, Amsterdam, Amsterdam, pp. [1984] Intuitionistic Intuitionistic type theory. Notes by Giovanni Sambin of of a series of of lectures given in [1984] Bibliopolis, Napoli. Napoli. Padua, June 1980, 1980, Bibliopolis, A. R. R. D. D . MATHIAS MATHIAS AND AND H. A. H. 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Report version of of the the author's PhD thesis, University 1983. thesis, Oxford Oxford University 1983. Subcountability under under realizability, realizability, The The Notre Journal of L09ic, 27, 27, [1986] Notre Dame Journal of Formal Logic, [1986] Subcountability pp. pp. 210-220. 210-220. Markov's principle, principle, isols isols and Dedekind finite finite sets, sets, JSL, 53, 53, pp. pp. 1042-1069. 1042-1069. [1988] Markov's [1988] and Dedekind [1991] Polymorphism Dame Journal of L09ic, 32, Polymorphism and apartness, The [1991] and apartness, The Notre Dame of Formal Logic, 32, pp. pp. 513-532. 513-532. Realizability Realizability 469 469 Yu. T. T. MEDVEDEV Yu. MEDVEDEV [1962] Finite Finite problems problems (Russian), (Russian), Doklady, Doklady, 142, 142, pp. pp. 1015-1018. 1015-1018. Translation Translation SM3, SM3, pp. pp. 227-230. 227-230. [1962] [1963] Interpretation Interpretation of of logical logical formulae formulae by by means means of of finite finite problems problems and and its its relation relation to to the the [1963] 148, pp. realizability realizability theory theory (Russian), (Russian), Doklady, Doklady, 148, pp. 771-774. 771-774. Translation Translation SM SM 4, 4, pp. pp. 180-183. 180-183. [1966] Interpretation Interpretation of of logical logical formulae formulae by by means means of of finite finite problems problems (Russian), (Russian), Doklady, Doklady, 169, 169, [1966] pp. pp. 20-23. 20-23. Translation Translation SM SM 7, 7, pp. pp. 857-860. 857-860. [1969] A A method method for for proving proving the the unsolvability unsolvability of of algorithmic algorithmic problems problems (Russian), (Russian), Doklady, [1969] 10, pp. 185, pp. pp. 1232-1235. 1232-1235. Translation Translation SM SM 10, pp. 495-498. 495-498. 185, [1972] Locally Locally finitary finitary algorithmic algorithmic problems problems (Russian), (Russian), Doklady, Doklady, 203, 203, pp. pp. 285-288. 285-288. Errata Errata [1972] 204, pp. pp. 1286. pp. 382-386. Ibidem 204, 1286. Translation Translation SM SM 13, 13, pp. 382-386. G. C. [1973] An An interpretation interpretation of of intuitionistic intuitionistic number number theory, theory, in: in: LMPS LMPS 4, P. P. Suppes, Suppes, G. C. Moisil, Moisil, [1973] and and A. A. Joja, Joja, eds., eds., North-Holland, North-Holland, Amsterdam, Amsterdam, pp. pp. 129-136. 129-136. G. E. E. MINTS G. MINTS [1989] The The completeness completeness of of provable provable realizability, realizability, The Notre Dame Journal of of Formal Logic, Logic, [1989] 30, 420-441. 30, pp. pp. 420-441. J. R. J. R. 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[1981] A A disjunctive disjunctive decomposition decomposition theorem theorem for for classical classical theories, theories, in: in: Constructive Mathemat Mathemat[1981] ed., Springer Berlin, Heidelberg, ics, F. F. Richman, Richman, ed., Springer Verlag, Verlag, Berlin, Heidelberg, New New York, York, pp. pp. 250-259. 250-259. [1993] [1993] An An intuitionistic intuitionistic theory theory of of lawlike, lawlike, choice choice and and lawless lawless sequences, sequences, in: in: Logic Logic Colloquium gO, Springer Berlin, Heidelberg, pp. 191-209. ''90, Springer Verlag, Verlag, Berlin, Heidelberg, New New York, York, pp. 191-209. (Lecture (Lecture Notes Notes in in Logic Logic 2). 2). [1994] More More about about relatively relatively lawless lawless sequences, sequences, JSL, 59, 59, pp. pp. 813-829. 813-829. [1994] 81, pp. the intuitionistic [1996] [1996] A A classical classical view view of of the intuitionistic continuum, continuum, APAL, 81, pp. 9-24. 9-24. J. MYHILL J. 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Amsterdam. [1991b] [1991b] Extension Extension of of Lifschitz' Lifschitz' realizability realizability to to higher higher order order arithmetic, arithmetic, and and aa solution solution to to aa 56, pp. 964-973. problem problem of of F. F. Richman, Richman, JSL, 56, pp. 964-973. rchive for Mathematical Logic, pp. [1991c] [1991c] A A semantical semantical proof proof of of De De Jongh's Jongh's theorem, theorem, A Archive pp. 105-114. 105-114. [1993] [1993] Extensional realizability, realizability, Tech. Tech. Rep. Rep. ML-93-18, ML-93-18, Institute Institute for for Logic, Logic, Language Language and and Computation, University Submitted to Computation, University of of Amsterdam. Amsterdam. Submitted to APAL. [1994] [1994] Axiomatizing Axiomatizing higher-order higher-order Kleene Kleene realizability, realizability, APAL, 70, 70, pp. pp. 87-111. 87-111. [1996] Two Two remarks remarks on on the the Lifschitz Lifschitz realizability realizability topos, topos, JSL, 61, 61, pp. pp. 70-79. 70-79. [1996] 470 470 A . S. Troelstra A.S. w. W. PHOA PHOA [1989] [1989] Relative Relative computability computability in in the the effective effective topos, topos, Mathematical Proceedings Proceedings of of the Cambridge Philosophical Society, 106, pp. 419-420. 106, pp. 419-420. A M. PITTS A.. M. [1981] [1981] The Theory of of Toposes, Toposes, PhD PhD thesis, thesis, University University of of Cambridge. Cambridge. V V.. E. E. PLISKO PLISKO [1973] On On realizable formulae (Russian), (Russian), Doklady, Doklady, 212, 212, pp. pp. 553-556. 553-556. Translation realizable predicate predicate formulae Translation SM [1973] 14, pp. pp. 1420-1424. 1420-1424. 14, [1974a] [1974a] A A certain certain formal formal system system that that is is connected connected with with realizability realizability (Russian), (Russian), in: in: Teoriya Teoriya [1974b] [1974b] [1974c] [1974c] [1976] [1976] [1977] [1977] [1978] [1978] [1983] [1983] [1992] [1992] Algorifmov Algorifinov i Matematicheskaya Logika: Logika: Sbornik Statej (Theory of Algorithms and Mathematical Logic. Logic. Collection of of articles dedicated dedicated to Andrej Andrejevich Markov), B. Kushner eds., Vychislitel'nyj B. Kushner and and N. N. M. M. Nagornyi, Nagornyi, eds., Vychislitel'nyj Tsentr Tsentr Akademii Akademii Nauk Nauk SSSR, SSSR, pp. 148-158,215. pp. 148-158,215. On interpretations interpretations of of predicate predicate formulae formulae that that are are connected connected with with constructive constructive logic logic On (Russian), in: 93 Vsesoyuznaya (Russian), in: Vsesoyuznaya Konferentsiya po Matematicheskoj Logike. Logike. Tezitsy Dok Doklady i Soobshcheniya (9rd Union Conference on Mathematical Logic), (3rd AllAll-Union Logic), pp. pp. 170-172. 170-172. Recursive realizability realizability and and constructive constructive predicate predicate logic logic (Russian), (Russian), Doklady, Doklady, 214, pp. 520520214, pp. Recursive 15, pp. 523. Translation 523. Translation SM SM 15, pp. 193-197. 193-197. Some Some variants variants of of the the notion notion of of realizability realizability for for predicate predicate formulas formulas (Russian), (Russian), Doklady, 226, pp. 226, pp. 61-64. 61-64. Translation Translation SM SM 17, 17, pp. pp. 59-63. 59-63. The The nonarithmeticity nonarithmeticity of of the the class class of of realizable realizable predicate predicate formulas formulas (Russian), (Russian), Izv. Akad. Nauk., 41, 41, pp. pp. 483-502. 483-502. Translation Translation Math. Izv. 11, 11, pp. pp. 453-471. 453-471. Some Some variants variants of of the the notion notion of of realizability realizability for for predicate predicate formulas formulas (Russian), (Russian), Izv. Akad. Nauk., 42, 42, pp. pp. 637-653. 637-653. Translation Translation Math. Izv. 12, 12, pp. pp. 588-604. 588-604. Absolute Absolute realizability realizability of of predicate predicate formulas formulas (Russian), (Russian), Izv. Akad. Nauk., 47, 47, pp. pp. 315-334. 315-334. 22, pp. pp. 291-308. Translation Translation Math. Izv. 22, 291-308. On On the the concept concept of of relatively relatively uniform uniform realizability realizability of of propositional propositional formulas formulas (Russian), (Russian), Vestnik Moskovskogo Moskovskogo Universiteta Universiteta Seriya 1. L Mathematika, Mekhanika, pp. pp. 77-79. 77-79. Translated Translated in in Moscow Moscow University Mathematics Bulletin 47 47 (1992), (1992), 41-42. 41-42. G. R. R. RENARDEL G. RENARDEL DE DE LAVALETTE LAVALETTE [1984] Theories with type-free type-Iree application application and extended bar induction, PhD PhD thesis, thesis, Universiteit Universiteit [1984] van van Amsterdam. Amsterdam. 50, pp. [1990] [1990] Extended Extended bar bar induction induction in in applicative applicative theories, theories, APAL, 50, pp. 139-189. 139-189. G. ROSOLINI E E.. ROBINSON ROBINSON AND AND G. ROSOLINI [1990] [1990] Colimit Colimit completions completions and and the the effective effective topos, topos, JSL, 55, 55, pp. pp. 678-699. 678-699. T. T T. T.. ROBINSON ROBINSON [1965] [1965] Interpretations Interpretations of of Kleene's Kleene's metamathematical metamathematical predicate predicate r F II A in in intuitionistic intuitionistic arith arith30, pp. pp. 140-154. metic, JSL, 30, metic, 140-154. G. F. F. ROSE G. ROSE [1953] [1953] Propositional Propositional calculus calculus and and realizibility, realizibility, Transactions Transactions of of the American Mathematical Society, 75, 75, pp. pp. 1-19. 1-19. G. G. ROSOLINI ROSOLINI [1990] [1990] About About modest modest sets, sets, International Journal of Foundations Foundations of Computer Science, 1, 1, pp. pp. 341-353. 341-353. B B.. SCARPELLINI SCARPELLINI [1970] A A model model for for intuitionistic intuitionistic analysis, analysis, Commentarii Mathematici Helvetici, 45, 45, pp. pp. 440-471. 440-471. [1970] [1977] [1977] A A new new realizability realizability notion notion for for intuitionistic intuitionistic analysis, analysis, ZLGM, 23, 23, pp. pp. 137-167. 137-167. CEDROV A A.. SSCEDROV [1984] Differential Differential equations equations in in constructive constructive analysis analysis and and in in the the recursive recursive realizability realizability topos, topos, [1984] Journal of of Pure and Applied Algebra, Algebra, 33, 33, pp. pp. 69-80. 69-80. Realizability Realizability 471 471 [1986] On On the the impossibility impossibility of of explicit explicit upper upper bounds bounds on on lengths lengths of of some some provably provably finite finite [1986] algorithms in in computable computable analysis, analysis, APAL, 32, 32, pp. pp. 291-297. 291-297. algorithms [1990] Recursive Recursive realizability realizability semantics semantics for for Calculus Calculus of of Constructions. Constructions. Preliminary Preliminary report, report, [1990] in: Logical Logical Foundations of of Functional Programming, Programming, G. G. Huet, Huet, ed., ed., Addison-Wesley Addison-Wesley in: Publishing Company, pp. 419-430. A. SCEDROV AND AND P. SCOTT A. SCEDROV P. J. SCOTT [1982] A A note note on on the the Friedman Friedman slash slash and and Freyd Freyd covers, covers, in: in: Troelstra Troelstra and van Dalen [1980J, [198o], [1982] pp. 443-452. A.. SCEDROV SCEDROV AND AND R. R. E. E. VESLEY VESLEY A [1983] On On aa weakening weakening of of Markov's Markov's principle, principle, Archiv, Archiv, 23, 23, pp. pp. 153-160. 153-160. [1983] D. S. SCOTT SCOTT D. [1968] Extending Extending the the topological topological interpretation interpretation to to intuitionistic intuitionistic analysis, analysis, Compositio Compositio Mathe Mathe[1968] matica, 20, 20, pp. pp. 194-210. 194-210. matica, SCOWCROFT PP.. SCOWCROFT [1993] The The disjunction disjunction and and numerical numerical existence existence properties properties for for intuitionistic intuitionistic analysis, analysis, in: in: L0 Logical [1993] 9ical J. N. N. Crossley, Crossley, J. B. Remmel, R. R. A. A. Shore, Shore, and and M. M. E. E. Sweedler, Sweedler, eds., eds., Methods, J. B. Remmel, Birkhaiiser Boston, Boston, Inc., Inc., Boston Boston MA, MA, pp. pp. 747-781. 747-781. Birkhaiiser SHANIN N. A. SHANIN [1958a] On On the the constructive constructive interpretation interpretation of of mathematical mathematical judgements judgements (Russian), (Russian), Trudy [1958a] Steklova. Akademiya Akademiya Nauk Ordena Lenina Matematicheskogo Instituta imeni V. A. Steklova. 52, pp. pp. 226-311. 226-311. Translation Translation AMS Transl. Transl. 23, 108-189. SSSR, 52, 23, pp. pp. 108-189. [1958b] Uber Uber einen einen Algorithmus Algorithmus zur zur konstruktiven konstruktiven Dechiffrierung Dechiffrierung mathematischer mathematischer Urteile Urteile [1958b] (Russian) (German (German summary), summary), ZLGM, 4, 4, pp. pp. 293-303. 293-303. (Russian) [1964] Concerning Concerning the the constructive constructive interpretation interpretation of of auxiliary auxiliary formulas formulas II (Russian), (Russian), Trudy ~udy [1964] A.. Steklova. Steklova. Akademiya Akademiya Nauk Ordena Lenina Matematicheskogo Instituta imeni V. A pp. 348-379. 348-379. Translation Translation AMS AMS Transl. Transl. 99, 233-275. SSSR, 72, 72, pp. 99, pp. pp. 233-275. J.. J M. SMITH M. SMITH [1993] An interpretation interpretation of slash in type theory, theory, in: in: Logical Huet of Kleene's Kleene's slash in type Logical Environments, G. G. Huet [1993] An and G. eds., pp. and G. Plotkin, Plotkin, eds., pp. 189-197. 189-197. J. STAPLES J. 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VESLEY [1970] [1970] A A palatable palatable substitute substitute for for Kripke's Kripke's schema, schema, in: in: Kino, Myhill and Vesley [1970j, [1970], pp. pp. 197-207. 197-207. [1996] Realizing Realizing Brouwer's Brouwer's sequences, sequences, APAL, APAL, 81, 81, pp. pp. 25-74. 25-74. [1996] A. VORONKOV A. VORONKOV 71, Department [1991] N-realizability: one more constructive semantics, Tech. Tech. Rep. Rep. 71, Department of of [1991] N-realizability: Mathematics, Mathematics, Monash Monash University, University, Australia. Australia. [1992] On in: Computer Science Logic. [1992] On completeness completeness of of program program synthesis synthesis systems, systems, in: Logic. 5th E. Borger, G. Jager, H. Kleine M. M. M. Richter, workshop, CSL '91, '91, E. BSrger, G. J~iger, H. Kleine Buning, Brining, and and M. Richter, eds., eds., Springer Springer Verlag, Verlag, Berlin, Berlin, Heidelberg, Heidelberg, New New York, York, pp. pp. 411-418. 411-418. K. F K. F.. WEHMEIER WEHMEIER -induction. Preprint. [1996] Fragments Fragments of of HA HA based based on on El ~l-induction. Preprint. Part Part of of the the author's author's dissertation dissertation [1996] project project at at he he University University of of Munster, Mfinster, Westphalia, Westphalia, Germany. Germany. This Page Intentionally Left Blank CHAPTER VII The Logic Logic of Provability Provability The Giorgi Japaridze Department Department of Computer and Information Science, Science, University University of o.f Pennsylvania Pennsylvania 1910~-6389, USA Philadelphia, Pennsylvania 19104-6389, Jongh Dick de J ongh for Logic, Logic, Language Language and Computation, Computation, University University of Amsterdam Institute for NL-1018 TV Amsterdam, The Netherlands This chapter chapter is is dedicated dedicated to to the the memory memory of of George George Boolos. Boolos. From From the the This start of of the the subject until his his death death on on 27 May May 1996 he he was was the the prime prime inspirer inspirer start subject until of the the work work in in the the logic logic of of provability. provability. of Contents Contents Solovay's theorems theorems 1. Introduction, Introduction, Solovay's .......................... 2. Modal 2. Modal logic logic preliminaries preliminaries .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Proof of of Solovay's Solovay's theorems theorems .. . .. . ... . . . . . . . . . . . . . . . . . . . . . . . 3. Proof Fixed point point theorems theorems .. . .. . ... . .. . .. . ... . . . . . . . . . . . . . . . . . . . 4. Fixed 5. Propositional theories theories and and Magari-algebras Magari-algebras .. . . . . . . . . . . . . . . . . . . . . 5. Propositional The extent extent of of Solovay's Solovay's theorems theorems .. . .. . ... . .. . .. . . . . . . . . . . . . . . . . 6. The 7. Classification Classification of of provability provability logics logics .. . .. . ... . .. . .. . . . . . . . . . . . . . . . . Bimodal and and polymodal polymodal provability provability logics logics . . . . . . . . . . . . . . . . . . . . . 8. Bimodal 9. Rosser 9. Rosser orderings orderings .. . .. . ... . .. . .. . ... . . . . . . . . . . . . . . . . . . . . . . 10. Logic 10. Logic of of proofs proofs .. .. .. .. .. . . . . . . . . . . . .. .. .. .. .. . . . . . . . . . . . . . . . . . . . . 11. Notions Notions of of interpretability interpretability .. . .. . ... . .. . .. . ... . .. . .. . ... . . . . . . . . . . Interpretability and and partial partial conservativity. conservativity . . . . . . . . . . . . . . . . . . . . . . 12. Interpretability 13. Axiomatization, modal completeness Axiomatization, semantics, semantics, modal completeness of of IILM L M .. . . . . . . . . . . . . . 14. Arithmetic 14. Arithmetic completeness completeness of of ILM I L M .. . .. . ... . .. . .. . ... . .. . .. . . . . . . . . . Tolerance logic logic and and other other interpretability interpretability logics logics . . . . . . . . . . . . . . . . . . 15. Tolerance 16. Predicate Predicate provability provability logics logics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17. Acknowledgements Acknowledgements .. . .. . ... . .. . .. . ... . .. . .. . ... . . . . . . . . . . . . . . References References .. .. .. .. .. . . . . . . . . . . . . .. .. .. .. . . . . . . . . . . . . . . . . . . . . . . . . . HANDBOOK H A N D B O O K OF OF PPROOF R O O F THEORY THEORY Edited Edited by by S. S. R. R. Buss Buss Elsevier Science Science B.V. B.V. All All rights rights reserved reserved © 1998 1998 Elsevier 476 477 481 484 484 486 488 491 496 497 500 503 514 521 528 531 541 541 476 476 G. Japaridze Japaridze and D. de Jongh G. 1.. Introduction, Introduction, S o l o v a y ' s theorems theorems 1 Solovay's ' s incompleteness ' s undecidability Ghdel's incompleteness theorems theorems and and Church Church's undecidability theorem theorem for for arith arithG6del metic showed showed that that reasonably reasonably strong strong formal formal systems systems cannot cannot be be complete complete and and metic decidable, and and cannot cannot prove prove their their own own consistency. consistency. Even Even at at the the time time though though these these decidable, negative theorems theorems were were accompanied accompanied by by positive positive results. results. Firstly, Firstly, formal formal systems systems fare fare negative better and this can be better in in reasoning reasoning in in restricted restricted areas, areas, and this reasoning reasoning can be formalized formalized in in the the theories themselves. themselves. In In Hilbert Hilbert and and Bernays Bernays [[1939] one fi finds the formalization formalization of of 1939] one nds the theories the completeness completeness theorem theorem for for the the predicate predicate calculus, calculus, i.e. i.e.,, reasoning reasoning in in the the predicate predicate the calculus is is adequately adequately described described in in strong strong enough enough theories. theories. A A fortiori, fortiori, this this is is so so for for calculus the propositional calculus calculus in in which which reasoning reasoning is is even even ((provably) decidable. Secondly, Secondly, the propositional provably) decidable. there is is aa positive positive component component in in the incompleteness theorems theorems themselves. themselves. The The there the incompleteness formalized version version of of the the second second incompleteness incompleteness theorem, theorem, i.e., i.e., if if it it is is provable provable in in PA PA formalized that PA PA is is consistent, consistent, then then PA PA is is inconsistent, inconsistent, is is provable provable in in P PA itself. The The area area that A itself. here called called the the logic logic of of provability provability arose arose in in the the seventies seventies when when two two developments developments here took place almost almost simultaneously. simultaneously. The The two two facets facets mentioned above were, were, one one took place mentioned above might say, say, integrated integrated by by showing showing that reasoning about about the the formalized formalized might that propositional propositional reasoning provability predicate predicate is is decidable decidable and and can can be be adequately adequately described described in in arithmetic arithmetic itself itself provability And in in the the same same period period the the de de Jongh-Sambin Jongh-Sambin fixed fixed point point theorem theorem Solovay [[1976]). 1976]) . And ((Solovay Sambin [[1976], Smoryfiski [[1978,1985])was proved for for modal-logical modal-logical systems systems with with see Sambin 1976] , Smorynski 1978,1985]) was proved ((see the provability provability interpretation interpretation in in mind. mind. Since Since that that time time the the main main achievements achievements have have the been to that similar similar results results mostly mostly fail fail for for predicate predicate logic, logic, to to recognize recognize reasoning reasoning been to show show that about about more more complex complex notions notions like like interpretability interpretability where where arithmetic arithmetic can can be be shown shown to to 's results reason reason adequately, adequately, and and also also to to strengthen strengthen Solovay Solovay's results directly. directly. Extensive Extensive 1993b] and 1985] , aa overviews Boolos [[19935] overviews on on the the subject subject can can be be found found in in Boolos and Smorynski Smoryfiski [[1985], short in Boolos short history history in Boolos and and Sambin Sambin [1991]. [ 1991 ] . Let us proceed proceed somewhat farther in in formulating theorems, and and call call Let us somewhat farther formulating Solovay's Solovay's theorems, an arithmetic of the the language language of of modal section 2) an arithmetic realization realization of modal logic logic (see (see section 2) into into the the ( l.: l -sound and extending n:;l, sometimes language of of the the arithmetic arithmetic theory theory TT (El-sound language and extending IE1, sometimes 1.6.0 mapping '* that that commutes commutes with with the the propositional propositional connectives and such such that that IAo)) aa mapping connectives and (DA)' = PrT(rA*~) PrT r A" ) ((where where PrT PrT is is the the formalized formalized provability provability predicate for T, T, i.e., i.e., (OA)*= predicate for ProofT (x, y) y) where where ProofT ProofT is the formalized formalized proof proof predicate predicate of of it the form form 3y it is is of of the is the 3y PrOOfT(x, T). If we want stress the on T T we we write for (A)'. (A)*. More If we want to to stress the dependency dependency on write (A)~, (A)T for More T). standard the term term "interpretation" "interpretation" for for "realization" "realization" but that conflicts standard is is the but that conflicts somewhat somewhat with our our terminology terminology with with regard regard to to interpretability. interpretability. The "realization" is is used used with The term term "realization" by by Boolos Boolos [1993b]. [ 1993b] . 1.1. heorem. (Solovay's 1.1. TTheorem. ( Solovay's first first arithmetic arithmetic completeness completeness theorem) theorem) The The modal modal formula AA is is provable provable in in TT under under all all arithmetic arithmetic realizations realizations iff A isis provable provable in in formula iff A the the modal (see sections sections 2, 2, 3). 3) . modal logic logic L (see second arithmetic completeness theorem) theorem) The The modal modal 1.2. h e o r e m . (Solovay's arithmetic completeness 1.2. T Theorem. ( Solovay's second formula AA isis true true under under all all arithmetic arithmetic realizations realizations iff iff AA is is provable in the the modal modal logic logic formula provable in S S (see (see sections sections 2, 2, 3). 3) . The Logic Logic of Provability 477 477 This chapter chapter is is to to be be thought thought of of as as divided divided into into three three parts: parts: the the first first part part consists consists of of This sections to propositional sections 2-10 2-10 and and is is devoted devoted to propositional provability provability logic, logic, i.e., i.e., the the propositional propositional logic of of the the provability provability predicate predicate and and its its direct direct extensions, extensions, the the second second part part consists consists logic of 1-15 and of sections sections 111-15 and treats treats interpretability interpretability logic logic and and related related areas, areas, the the last last part part consists consists of of section section 16 16 and and discusses discusses predicate predicate provability provability logic. logic. 2 2.. Modal Modal logic preliminaries preliminaries The language language of of the the modal modal propositional propositional calculus calculus consists consists of of aa set set of of propositional propositional The variables, connectives connectives V v,, /\ A,, -+ --+,, H ~, , ' -~,, T, _L and and aa unary unary operator operator o [].. Furthermore, Furthermore, variables, T, ..1 o <} is is an an abbreviation abbreviation of of -~ 0 [] ' -~.. The The modal modal logic logic K K is is axiomatized axiomatized by by the the schemes schemes 11 and and 2: 2: ' 11.. All All propositional propositional tautologies tautologies in in the the modal modal language, language, 2. (A -+ -+ ((OA OA -+ 2. 0 O(A --+ B) B)--+ --+ OB) []B),, together with with the the rules of modus modus ponens and necessitation, i.e., AI A/[]A. The modal together rules of ponens and necessitation, i.e., OA. The modal logic logic L L is is axiomatized axiomatized by by adding adding the the scheme scheme 3: 3: ( oA -+ 3. 3. o O([]A ~ A) A) -+ -+ OA, []A, to K K and and keeping keeping the the rules rules of of modus modus ponens ponens and and necessitation. necessitation. The The system system is is often often to GL, e.g. e.g. in 1993b] , and 1985] . It called called GL, in Boolos Boolos [[1993b], and is is named named PrL P r L in in Smorynski Smoryfiski [[1985]. It is is an an exercise oA -+ oA is exercise to to show show that that []A --+ o [][]A is derivable derivable in in L L,, which which makes makes L L an an extension extension oA -+ o of oA . of K4, K4, the the system system axiomatized axiomatized by by the the axioms axioms of of K K together together with with [:]A--+ OEIA. Extensions Extensions of of K K such such as as K4 K4 and and L L that that are are closed closed under under necessitation necessitation are are called called normal normal modal modal logics. logics. ,...,A We ,... ,A We will will write write AI A1,..., Ann rK bg B B for: for: B B is is derivable derivable in in K K from from AI A1,..., Ann without without use use of of necessitation, necessitation, or or more more precisely, precisely, B B is is derivable derivable by by modus modus ponens ponens from from theorems theorems K4, L. L. To ,...,A of K K plus plus AI A1,..., An. Similarly for for K4, To this this notation notation the the deduction deduction theorem theorem of n . Similarly obviously applies: A BFrKKCC iff -+ C. We obviously applies: AI" A 1 , . '. ." , AA~n ,, B iff AI" A 1 , . .' ." , AA~nFrzK B B--+C. We will will write write c::J WIA for /\ oA. results codified codified in proposition are for A A/x []A. The The results in the the next next proposition are readily readily proved. proved. 2.1. 2.1. Proposition. Proposition. ((a) KB . . . ,oA a) IIff Al A , ,, ..... ., , A Ann rFK B,, then then OAl, DA,,... ,OAnFKOB (also for for K4, K4, L), L), n r K O B (also ((b) b) iiff OAI , . . . , OA . . . , OA OA1,..., DAnFK4 B,, then then OAl, [:]AI,..., I:IAnFK4DB (also for for L), L), n r K40B (also n r K4 B ((c) c ) if . . . , OA oB rL B, then . . . , OA if oAl, DA1,..., [:]An, then oAl, DA1,..., EIAnFLI:3B, n , DBFLB, n r L OB, ((d) d) rFKO(A/X K O (A /\ B) A /\ B) H ++o [:]A /X OB, [:]B, ((e) e) rFK40(A K4 0 (A ~H B) -+ o (C(A) H B)-~ [](C(A) ~ C(B) C(B)),) , ((f) f) rF zK4, D(A 0 (A H ) -+ (([3C(A) oC(A) H ++B B)-+ ++ OC(B) E]C(B)),) , K4 c::J ) -+ (C(A) (A H g) rFz4 E](A ++B B)--+ (C(A) H ++C(B)) C(B)),, ((g) (h (h)) if if rK,K4,L FK,K4,L A AH ~ B B,, then then rK,K4,L FK,K4,L C(A) C(A) H ++C(B C(B),) , (i) rK ~-K 0 V•..1 -+ ..1 , 478 478 G. Japaridze and D. de Jongh (j) O) h F r. Op (}p -+ --4 O (} (p (p l\ A D--,p) []~p) and, and, hence, hence, rLOp F r. (}p ++ +-~ O (} (p (p l\ AD [] -~p) ,p) and and (p 1\ ^D D---,p) p) V v O <>(p (p 1\ ^ D --'p) . r L pp -+ (p The The modal modal logic logic S S is is defined defined by: by: F s A if and only if []B1 -+ B 1 , . . . , [::lBk --4 Bk FL A for some B ~ , . . . , Bk. The The logic logic S S is is not not closed closed under under necessitation, necessitation, and and is is therefore therefore aa nonnormal nonnormal modal modal logic. logic. 2.2. 2.2. Definition. Definition. ((a) a) A A Kripke-frame gripke-frame for for K g is is aa pair pair (W, (W, R) R) with with W W aa nonempty nonempty set set of of so-called so-called worlds worlds or nodes and binary relation, so-called accessibility or nodes and R R aa binary relation, the the so-called accessibility relation. relation. ((b) b) A A Kripke-frame gripke-frame for for K4 K 4 is is a a Kripke-frame Kripke-frame (W, (W, R) R) with with R R aa transitive transitive relation. relation. ((c) c) A A Kripke-frame gripke-frame for for L L is is a a Kripke-frame Kripke-frame (W, (W, R) R) with with R R aa transitive transitive relation relation such Of course, nite transitive such that that the the converse converse of of R R is is well-founded. well-founded. ((Of course, aa fi finite transitive frame frame is is conversely conversely well-founded well-founded iff iff it it is is irreflexive. irreflexive.)) d) A is aa node ((d) A root root of of a a Kripke-frame Kripke-frame is node w w such such that that w wR R w' w' for for all all w' w' =1= -~ w w in in the the frame. frame. ((In In the R.) The the case case of of K K put put the the transitive transitive closure closure of of R R here here in in place place of of R.) The depth depth also height) node w maximal m height) of of a a node w in in aa conversely conversely well-ordered well-ordered frame frame is is the the maximal m for for ((also R W ' " R w . The height of the model is the which there exists which there exists a a sequence sequence w w= =W woR w l . . . R win. The height of the model is the l m O maximum maximum of of the the height height of of its its nodes. nodes. e) A A Kripke-model gripke-model for for K g (K4, (K4, L L)) is is aa tuple tuple (W, (W, R, R, If-) IF) with with (W, (W, R) R) aa Kripke-frame Kripke-frame ((e) for with aa forcing If- between between worlds worlds and for K K (K4, (K4, L L)) together together with forcing relation relation IF and propositional propositional If- isis extended variables. variables. The The relation relation IF extended to to a a relation relation between between worlds worlds and and all all formulas formulas by by the the stipulations stipulations w w lrA, w lflF --, ~A A iff iff w P~A, w A and w If-A IFA 1\ AB B iff iff w w IfIFA and w w If-B, IFB, and and similarly similarly for for the the other other connectives, connectives, If- DA iff wR w', w' If- A. w lF[]A iff for for all all w' w' such such that that wRw', w'IFA. w If KF ned as, A for If K K = = (W, (W, R, R, If-) IF),, then then K ~ A A is is defi defined as, w w IfIFA for each each w w Ee W, W, and and we we say say that that A A is is valid valid in in M. M. It It is is easy easy to to check check that that Kripke-models Kripke-models are are sound sound in in the the sense sense that that each each A A derivable Kripke-model for derivable in in K K (K4, (K4, L L)) is is valid valid in in each each Kripke-model for K K (K4, (K4, L L).) . In In fact, fact, the the Kripke-models K4 resp. L Kripke-models for for K 4 ((resp. L)) are are exactly exactly the the ones ones that that validate validate the the formulas formulas derivable, respectively K4 derivable, respectively in in K 4 and and L L.. One One says says that that K4 K4 and and L L characterize characterize these these classes For the modal logic, logic, see Chellas [1980] classes of of models. models. ((For the main main concepts concepts of of modal see e.g., e.g., Chellas [1980],, Hughes Hughes and and Cresswell Cresswell [1984] [1984].).) Something Something stronger stronger is is true: true: in in K K,, K4 K 4 and and L L one one can can derive derive all all the the formulas formulas that that are are valid valid in in their their respective respective model model classes classes (modal (modal completeness) The standard standard method modal logic logic for proving completeness completeness).. The method in in modal for proving completeness is is to to construct necessary countermodels taking maximal maximal consistent consistent sets construct the the necessary countermodels by by taking sets of of the the logic logic as as the the worlds worlds of of the the model model and and providing providing this this set set of of worlds worlds with with an an appropriate appropriate accessibility R. This This method accessibility relation relation R. method cannot cannot be be applied applied to to L L,, since since the the logic logic is is The The Logic Logicof o] Provability Provability .479 .479 not not compact: compact: there there exist exist infinite infinite syntactically syntactically consistent consistent sets sets of of formulas formulas that that are are semantically semantically incoherent. incoherent. We We will will apply apply to to all all three three logics logics aa method method in in which which one one restricts maximal maximal consistent consistent sets sets to to aa finite finite so-called so-called adequate adequate set set of of formulas. formulas. One One restricts obtains finite finite countermodels countermodels by by this this method method and and hence, hence, immediately, immediately, decidability decidability of of obtains the logics. logics. the 2.3. Definition. Definition. If If A A is is not not aa negation, negation, then then rvA ,,~A is is ,A, --A, otherwise, otherwise, if if A A is is ,B, --B, then rvA ,,~A is is B. B. then An adequate adequate set set of of formulas formulas is is aa set set <1>9 with with the the properties: properties: An (i) <1>9 is is closed closed under under subformulas, subformulas, (i) (ii) if if B B Ee <1>, ~, then then rvB ,,~B Ee <1> (I).. (ii) It is is obvious obvious that that each each formula formula is is contained contained in in aa fi finite adequate set. set. It nite adequate 2.4. Theorem. T h e o r e m . (Modal (Modal completeness completeness of of K, K, K4, K4, L.) L.) 2.4. If A A is is not not derivable derivable in in K K (K4, ( g 4 , L), L), then then there there is is aa frame frame for for K g (K4, ( g 4 , L) L) on on which which A A If is not not valid. valid. is Proof. (K) KA A. We Proof. (K) Suppose Suppose J' ]ZK A.. Let Let <1>9 be be aa finite finite adequate adequate set set containing containing A. We consider consider the set set W W of of all all maximal maximal K-consistent K-consistent subsets subsets of of <1> (I).. We We define define for for w, w, w' w' E6 W, W, the wR wR w' w' {::: ~ :::} for for all all DD O D eE w, w, D D Ee w'. w'. -B Furthermore, Furthermore, we we define define w w II-p IFp iff iff p p Ee W. w. It It now now follows follows that that for for each each B B Ee <1> (I),, w w IIIFB iff B. For propositional letters iff B B E6 w w,, by by induction induction on on the the length length of of B. For propositional letters this this is is so so by by definition the case the connectives that definition and and the case of of the connectives is is standard, standard, so so let let us us consider consider the the case case that B is [3C. DC. B is ==? DC E w. W . Then, all w' such that wR w' , C W' . By induction -:: Assume Assume [::]Ce Then, for for all w' such that wRw', C E6 w'. By induction hypothesis, w' w' IFC all such such w'. II-C for for all w' . So, SO, w w IFC]C. II- DC. hypothesis, {= Assume [::ICr DC1 w the set set {DlOD {DI D D E6 w} W} U {rvC} . We We will show this this set ~ : : Assume w.. Consider Consider the U {,,~C}. will show set to be be K-consistent which means, by the the conditions conditions on on adequate adequate sets, sets, that that aa maximal maximal to K-consistent which means, by K-consistent superset superset w' w' of exists inside inside ~. <1> . By By induction w' WC, and K-consistent of it it exists induction hypothesis, hypothesis, w' ]~C, and since w', this since wR wRw', this implies implies that that w w ~W OC. DC. w} U U {..~C} {rvC} is indeed K-consistent, K-consistent, suppose suppose that that itit is is show that that {{DI To 0 1 DD 0 0 6E w} is indeed To show Dk FKC rKC for for some some ClD1,..., D Dl , . . . , [:]D~ DDk 6E w. w . Then Then C]D~,..., DDl, . . . , [:]Dk DDk FKC:]C rKDC not, i.e., i.e., D Dl1,, .. .. .., , Dk not, immediately follows, follows, but but that that would would make make w w inconsistent, inconsistent, contrary contrary to to what what was was immediately assumed. assumed. (K4) Suppose Suppose lZK4 J'K4 A. A. Proceed Proceed just just as as in in the the case case of of K, K, except except that that now: now: (K4) wR w' w' ~{::::::} for for all all C]D DD eE w, w, both both [:]D DD eE w' w' and and D D eE w'. w' . wR The argument argument isis as as for for K; K; only only the the case case B B-==D CDC (=~) (=}) needs needs special special attention. attention. The D C 1 w . This This time, time, consider consider the the set set {DD, { DD, D][:]D6w}U{~C}. D I DD E W} U {rvC} . The The only only Let Let OCCw. additional fact fact needed needed in in the the argument argument to to show show that that this this set set isis K4-consistent K4-consistent isis that that additional DDi D Di FK4DDD/. rK4D DDi . Suppose ]zL J' L A. A. Again Again proceed proceed as as before, before, except except that that now: now: (L) Suppose (L) 480 480 G. Japaridze and D. de Jongh w R ww'~ {::: ~ :::} for for all allD D eEww, , both bothD D eEww'~ and andD wR DD DD DeEww'~and, and, for C f/.r w for some some DC DC Ee W' w ~,, D DC w.. The DC ((r¢= ) merits The argument argument is is as as for for K4; K4; again, again, only only the the case case B B -== DC merits some some D D, D I DD E W} U { D C, rvC}}.. Note special attention. time, consider special attention. This This time, consider {{DD, DIDDew}U{[:]C,,.~C Note that that DC will the the inclusion inclusion of of DC will insure insure that that w' w ~ really really is is aa successor successor of of w w.. The The argument argument that that this rests on that, if LD ( DC -+ this set set is is consistent consistent now now rests on the the fact fact that, if DDl, D D ~ , .. .. .., , DDk DDk IFT. O(DC --+ C) C),, -I then then D D DI D ~", . .' .", DDk n D k IFLL DC. DC. -t 2.5. Definition. D e f i n i t i o n . If If 9 is is an an adequate adequate set set offormulas, of formulas, then then the the rooted Kripke-model 2.5. rooted Kripke-model M -sound if M, w DB -+ DB in M is is O-sound if in in the the root root w w of of M, w IIIF[:]B --+ B B for for each each DB in ; O; M M is is A-sound A-sound if M M is is -sound O-sound for for the the smallest smallest adequate adequate set set 9 containing containing A. A. if If K K is is a a Kripke-model Kripke-model with with root root w w,, then the derived derived model model K' K ~of of K K is is constructed constructed then the If by by adding adding a a new new root root w' w ~ below below w w and and giving giving w' w ~exactly exactly the the same same forcing forcing relation relation as as w for for the the atoms. atoms. w If If K K is is aa -sound O-sound Kripke-model Kripke-model with with root root w w,, then then w' w ~forces forces in in K' K~ exactly exactly the the same same formulas formulas from from 9 as as w w in in K K .. Proof. be aa rooted with root root w P r o o f . Let Let K K be rooted -sound O-sound Kripke-model Kripke-model with w.. We We prove prove by by induction the length that w' II- A. nition for induction on on the length of of A A that w ~llIF- A A iff iff w w IF A. This This is is so so by by defi definition for the the l- DA, then DA, atomic atomic formulas formulas and and otherwise otherwise obvious obvious except except for for the the D D.. If If w' w ~lIFDA, then w w IIIFDA, A, but since since w wR R w' w ~.. If If w w IIIF DA, DA, then then not not only only for for all all w" w" such such that that w wR R w", w", w" w" IIIFA, but also, also, by K, w II- A. But then, for R w"",, w" II- A, by the the -soundness O-soundness of of K, wlFA. But then, for all all w" w" such such that that w' w~Rw w"IFA, -I i.e., i.e., w' w ~llIF- DA. DA. -t 2.6. 2.6. Lemma. Lemma. 2.7. Theorem. 2.7. T h e o r e m . (Modal (Modal completeness completeness of of S) S) I-sA F sA iff iff A A is is forced forced in in the the root root of of all all A -sound L -models. A-sound L-models. Proof. :=:} A. If Proof. ----~ :: Assume Assume K K is is A-sound A-sound and and root root w wW P~A. If we we assume assume to to get get a a contradiction that contradiction that IF ss A, A, then then A A is is provable provable in in L L from from kk applications applications of of the the reflection reflection k+ l ) obtained scheme: O BI -+ model K obtained from scheme: DB1 --+ BI B 1", . .' .", DBk DBk -+ -+ Bk Bk.. Consider Consider the the model K ((~+1) from K K by by taking the derived with K. K. Each time that that one one takes takes the taking k k+ + 11 times times the derived model model starting starting with Each time the derived model, one derived model, one or or more more of of the the OBi [:]Bi may may change change from from being being forced forced to to not not being being principle, that forced forced (never (never the the other other way way around around).) . This This implies, implies, by by the the pigeon pigeon hole hole principle, that one model K (0 � one of of the the times times that that one one has has taken taken the the derived derived model K (m (m)) (0 ~<m m� ~k k+ + 11)) the the forcing forcing value value of of all all the the OBi c:]Bi remain remain the the same. same. It It is is easy easy to to check check that, that, in in the the root root of of that Bi -+ lemma 2.6, that model model K K (m (m)) for for that that m m,, O [:]Bi --+ Bi Bi is is forced forced for for all all ii � ~ An An elegant elegant formulation formulation of of the the semantics semantics of of S S in in terms terms of of infinite infinite models, models, so-called so-called tail tail models models is is given given in in Visser Visser [1984]. [1984]. The The Logic Logic of of Provability Provability 481 481 The The well-known well-known normal normal modal modal system system S4 S4 that that is is obtained obtained by by adding adding the the scheme scheme -+ A [::]A-+ A to to K4 K4 plays plays aa role role in in section section 10. 10. It It can can be be shown shown that that S4 $4 is is modally modally DA complete with with respect respect to to the the (finite) (finite) reflexive, reflexive, transitive transitive Kripke-models. Kripke-models. complete 3. Proof P r o o f of of Solovay's Solovay's theorems theorems 3. We We rely rely mostly mostly on on Buss's Buss's Chapter Chapter II II of of this this Handbook. Handbook. One One can can find find there there an an intensional arithmetization arithmetization of of metamathematics metamathematics worked worked out, out, the the (Hilbert-Bernays) (Hilbert-Bernays)intensional 's Lhb derivability derivability conditions conditions are are given given and and proofs proofs ofthe of the diagonalization diagonalization lemma, lemma, Godel Ghdel's Lob 's theorem theorems and and Lob Lhb's theorem are are presented. presented. An An additional additional fact fact that that we we need need is is some some theorems formalization of of the the recursion recursion theorem. theorem. formalization To To repeat repeat the the statement statement of of Solovay's Solovay's first first arithmetic arithmetic completeness completeness theorem theorem (theorem 11.1), for �1 El-sound r.e. theories theories T T containing containing lEI I E I :: -sound r.e. (theorem . 1 ) , for iff T T fF- A A** for for all all arithmetic realizations **.. f-F- Lt.AA iff arithmetic realizations Proof . 1 . => P r o o f of of theorem t h e o r e m 11.1. =~ :: These These are are just just the the Hilbert-Bernays-Lob Hilbert-Bernays-Lhb conditions conditions and and Lhb's theorem theorem (see (see Chapter Chapter II II of of this this Handbook) Handbook).. Lob's {::: do is Pk) , then 1 , . . . ,,ak ak ~= :: What What we we have have to todo is show show that, that, if if }LL FL A(P1 A(pl .. .. ... . ,,Pk), then there there exist exist a ~1,... such * , where realization generated such that that TT}L F A A*, where ** denotes denotes the the realization generated by by mapping mapping P1 P l ,, .. ... ,. P, Pk k to A. Then, Then, by nite L-model to a1> a l , ... .. ,. a, ak k . .11 Suppose Suppose }LL FL A. by theorem theorem 2.4, 2.4, there there is is a a fi finite L-model -) in A is {W, R, R, 11IF-) in which which A is not not valid. valid. We We may may assume assume that that W W = = {I, { 1 , .. .. .., , l}, l}, 11 is is the the (W, root, and and 11 � ~ AA.. We We defi define new frame frame (W', R'):: root, ne aa new (W', R') w ' ==w W u { oU} {O} , , W' R' == R {(O, w) II W E wW} n' R uU {(o, }. . Observe that is aa finite Observe that {W', (W', R') R') is finite L-frame. L-frame. We are : w -+ W' (with (with embed this frame into T by means of of aa function We are going going to to embed this frame into T by means function h h :w -+ W' the nonnegative for each W E9W', which assert nonnegative integers) integers) and and sentences sentences Limw Lim~,, for each w W', which assert that that w the the limit limit of of h. h. This This function function will will be be defined defined in in such such aa way that aa basic basic lemma lemma 3.2 3.2 w way that W isis the holds about about the the statements statements that that TT can can prove prove about about the the sentences sentences Lim~. Limw . These These holds statements are are tailored tailored to to prove the next next lemma lemma 3.3 3.3 that that expresses that provability provability statements prove the expresses that in TT behaves behaves for the relevant relevant formulas formulas on on the the Kripke-model Kripke-model in in the the same same way as the the in for the way as D . This This will will allow us to to conclude conclude the the proof. proof. operator [:]. modal allow us modal operator 3.2. emma. 3.2. LLemma. V {Limr I r E W'} , proves that that hh has has aa limit limit in in W W'' , , i.e., i. e. , TT Ff- V (a) TT proves (a) f- -1 (Lim~ (b) (b) IfIf wW ~-=f:. u, U, then then TT F(Limw h1\ Limu), Limu), ..., {Limr I r 9 W'}, R' u, u , then then TT ++ Lim~ Limw proves proves that that TT F}L ~..., Limu, Limu , (c) (c) IfIf wW R' then TT ++ Lim~ Limw proves proves that that TT Ff- ~ ..., Lim~, Limu , and not not wW R' R' u,u, then (d) (d) IIff wW ~-=f:. 0° and 1 We will will use use italic italic capital capital letters letters for for modal-logical modal-logical formulas formulas and and Greek Greek letters letters for for arithmetic arithmetic 1We sentences and and formulas, formulas, except except that that we we will will use use Roman Roman letters letters for for descriptive descriptive names names like like "Proof". "Proof" . sentences 482 482 G. Japaridze and D. de Jongh e) Lim0 Limo is is true, true, ((e) For each each ii Ee W', W', Limi Limi is is consistent with T. T. f) For consistent with ((f) We We now now define define aa realization realization '* by by setting setting for for each each propositional propositional letter letter P Pi, i, p*= W{Lim~ IweW, wlF-pi}. This This pi p~. will will then then function function as as the the above-mentioned above-mentioned O!i hi.. 3.3. 3.3. Lemma. L e m m a . For For any any w w E~ W W and and any any L-formula L-formula B, B, - B, then ((a) a) if if w w lfI~-B, then T T + + Limw Lim~ fF B' B*,, b) if B, then then T if w wW P~B, T + + Limw Lim~ fF, ~ B' B*.. ((b) Proof. induction on atomic, then a) is Proof. By By induction on the the complexity complexity of of B. B. If If B B is is atomic, then clause clause ((a) is evident, and b) is ( b) . The evident, and clause clause ((b) is also also clear clear in in view view of of lemma lemma 3.2 3.2(b). The cases cases when when B B is is aa Boolean combination combination are So, only DC will Boolean are straightforward. straightforward. So, only the the case case that that B B is is [:]C will have have to be considered. to be considered. a) Assume DC. Then, If-C. By Assume that that w w lfl~-[:]C. Then, for for each each W/ w ' eE W W with with w w RRww' ' , , w' w'l~-C. By ((a) induction hypothesis, hypothesis, for for each such w' w',, T T+ + Limw' Lim~, fF C' C*,, and and this this fact fact is is then then provable provable induction each such in T. T. On On the hand, by by lemma 3.2(a) in T T itself itself)) and and ((c), T+ + Limw Limw the other other hand, lemma 3.2 ( a) ((proved proved in c) , T in ' proves R' w } . Together { Limwl II w proves that that T T fF V W{Lim~, wR'w'}. Together this this implies implies that that T T proves proves that that T DC)" . T fF- C" C*,, i.e., i.e., T T fF- ((C]C)*. b ) Assume DC. Then, W with w', w Assume that that w wW P~c]C. Then, for for some some w' w ' eE W with W w R' R'w', w '' ~WCC. . ((b) ,C" , i.e., -+ ,Limw' . By By hypothesis, T By induction induction hypothesis, T+ + Limw' Lim~, f~-~C*, i.e., T T fF C" C*--+-~Lim~,. By the the sec second ( c) implies (DC)' ond HBL-condition, HBL-condition, T T f~-(C IV)* -+ --+ Pr PrT(-~Lim~). But lemma lemma 3.2 3.2(c) implies that that T ( ,Limw ) . But --I ,(DC) " . T ,PrT ( ,Limw ) ' i.e., T+ + Limw Limw fF--~PrT(-~Lim~), i.e., T T+ + Limw Lim~ fF-~([:]C)*. -~ w IF-B" Observe If- B" f-+ Observe by by the the way way that that lemma lemma 3.3 3.3 expresses expresses that that T T+ + Limw Limw fF ""w ~ B' B*.. Assuming lemma 3.2 we can now complete the proof of theorem Assuming lemma 3.2 we can now complete the proof of theorem 1.1. By By the the construction construction of of the the Kripke-model, Kripke-model, 1 1f-,A. IF--~A. By By lemma lemma 3.3, 3.3, T T+ + Lim Liml! fF ,A" -~A*.. Since, Since, --I ( f) , T by by lemma lemma 3.2 3.2(f), T+ + Lim! Liml is is consistent, consistent, T TY lz A' A*.. -~ Our remaining duties ne the Our remaining duties are are to to defi define the function function h h and and to to prove prove lemma lemma 3.2. 3.2. The The recursion recursion theorem theorem enables enables us us to to define define this this function function simultaneously simultaneously with with the the sentences sentences for each Limw Limw ((for each w w eE W') W~),, which, which, as as we we have have mentioned mentioned already, already, assert assert that that w w is is the the limit limit of of h. h. 3.4. Solovay function 3.4. Definition. Definition. ((Solovay function h) h) We (O) = We define define hh(0) = O. 0. If If x x is is the the code code of of aa proof proof in in T T of of ,Limw -~Limw for for some some w w with with h(x) h(x) R R w, w, then then h(x h(x + + 1) 1) = =w w.. Otherwise, = h(x) Otherwise, h(x h(x + + 1) 1)= h(x).. It It is is not not hard hard to to see see that that h h is is primitive primitive recursive. recursive. f) , we Proof case below, below, except e) and P r o o f of of lemma l e m m a 3.2. 3.2. In In each each case except in in ((e) and ((f), we reason reason in in T. T. 483 483 The Logic of Provability By induction induction on on the For end end nodes nodes w w ((i.e., the ones ones with with no no R Ra) By the nodes. nodes. For i.e., the ((a) successors h(y) = successors),) , it it can can be be proved proved that that T T IF- "Ix Vx (h(x) (h(x) = =w ~ -+ --+ Vy Vy � >t x x h(y) - w) ~) by by induction induction on -+ Limw on x x,, and and hence hence T T IF ::Ix 3x h(x) h ( x ) -= w ffJ--+ Limw.. Next, Next, it it is is easily easily seen seen that, that, if if for for all all ' w successors h(x) = Limw" II w' } , then successors w' of of a a node node w w,, T T IF- ::Ix 3x h(x) = w' ~' -+ -+ V V {{Lim~,, T ' -= w" w" V v w' w' R R w" w"}, then T Limwl II w WR ( x) = T IF ::Ix 3x h h(x) = w ~ -+ --+ V V {{Lim~, w= = w' w' V vw R W/}. w'}. Therefore, Therefore, this this will will hold hold for for w w= = 0, 0, which a) . which implies implies ((a). cannot have have two two different different limits w and and u u.. ((b) b) Clearly Clearly hh cannot limits w Assume w w is is the the limit limit of o f hh and and w wR Let n n be be such such that that for for all all x x /� > nn, , c) Assume R'~u. u . Let ((c) h(x) Limu . Deny h(x) = - w w.. We We need need to to show show that that T T� )z _~Limu. Deny this. this. Then, Then, since since every every provable provable formula has has arbitrarily arbitrarily long there is is x x/> n such such that that x x codes proof of of ...,Limu --Limu;; �n codes aa proof formula long proofs, proofs, there h(x + 11)) -= u, which, which, as but then, according definition 3.4, but then, according to to definition 3.4, we we must must have have h(x as u u =I~ w w by irreflexivity irreflexivity of of R') R'),, is is a a contradiction. contradiction. ((by since d ) Assume =I- 0, 0, w w is Assume w w-7(: is the the limit limit of of h h and and not not WR' wR' u. If If u u= = w w,, then then ((since ((d) w 0) there h(x + w =I~= 0) there exists exists an an x x such such that that h(x + 11)) ==ww and and hh(x) ( x ) r=I- w w.. Then Then x x codes codes aa =I- w proof .., Limw is proof of of ..., -, Limw Lim~ and and .--Lim,o is provable. provable. Next Next suppose suppose u u-7(: w.. Let Let us us fix fix a a number number with h h (( zz )) -=ww . . Since Since h h is is primitive primitive recursive, recursive, T T proves proves that that h h (( zz )) -= w w.. Now Now zz with since uu is the limit (z) = number xx argue argue in in T T + + Limu Lim~:: since is the limit of of h h and and h h(z) =w w =I~ u u,, there there is is a a number with � zz such h(x + 1) 1) == u. This with x x/> such that that h(x) h(x) =I~ u u and and h(x This contradicts contradicts the the fact fact that that not not (w = = )h(z)R' )h(z)R' u, Thus, Thus, T T + Lim~ is inconsistent, i.e., i.e., T T IF -1Lim~. (w is inconsistent, Limu . + Limu e ) By a) , as sound, one true. Since Since for By ((a), as T T is is sound, one of of the the Limw Limw for for w w Ee W' W' is is true. for no no w w ((e) do d) means implies in do we we have have wR'w, wRiT, ((d) means that that each each Limw Lim~,, except except Limo Lim0,, implies in T T its its own own T-disprovability Consequently, Lim0 Limo is T-disprovability and and therefore therefore is is false. false. Consequently, is true. true. e) , ((c) -j-I c ) and By ((e), and the the soundness soundness of of T. T. ((f) f) By ..., ..., 's second To To repeat repeat the the statement statement of of Solovay Solovay's second arithmetic arithmetic completeness completeness theorem theorem 1.2): ((theorem theorem 1. 2) : I-[- ss A A iff iff IN IN F ~ A' A* for for all all arithmetic arithmetic realizations realizations '*.. Proof Since the DA -+ principles and P r o o f of of theorem t h e o r e m 1.2. 1.2. Since the [:]A --+ A A are are reflection reflection principles and these these are are true sound theory, soundness part true for for a a sound theory, the the soundness part is is clear. clear. So, So, assume assume � jz ss A. A. Modal Modal completeness see definition completeness of of S S then then provides provides us us with with an an A-sound A-sound ((see definition 2.5 2.5)) model model in in which which A A is is not not forced forced in in the the root. root. We We can can repeat repeat the the procedure procedure of of the the proof proof of of the the which we first first completeness completeness theorem, theorem, but but now now directly directly to to the the model model itself itself ((which we assume assume to to have root 0) lemmas 3.2 But have a a root 0) without without adding adding a a new new root, root, and and again again prove prove lemmas 3.2 and and 3.3. 3.3. But this time lemma 3.3 this time we we have have forcing forcing also also for for 00 and and we we can can improve improve lemma 3.3 to to apply apply it it also also to to w w= - 0 ,0, at at least least for for subformulas subformulas of of A. A. b ) -part of The The proof proof of of the the ((b)-part of that that lemma lemma can can be be copied. copied. With With respect respect to to the the Then, for restricting again again to to the the D-case, [:]-case, assume assume that that 00 IIIF- DC. [:]C. Then, for each each w w Ee W W ((a)-part, a) -part, restricting with 0, w the A-soundness A-soundness of with w w =I:/: 0, w II-C. I}-C. But But now, now, by by the of the the model, model, C C is is also also forced forced in in the the root O. By induction hypothesis, ( a) root 0. By the the induction hypothesis, for for all all w w Ee W W,, T T + + Limw Lim~ It- C' C*.. By By lemma lemma 3.2 3.2(a) DC) ' and then then T T IF C' C*,, so, so, T T I[- (([:]C)* and hence hence T T IF Limo Lim0 -+ -+ ((C]C)*. DC) ' . Applying Applying the the strengthened strengthened version version of of lemma lemma 3.3 3.3 to to w w= = 00 and and A, A, we we obtain obtain -j T lemma 3.2 ). T IF Limo Lim0 -+ --+ ..., -1 A' A*,, which which suffices, suffices, since since Limo Lim0 is is true true ((lemma 3.2). -~ 484 484 G. Japaridze and D. de Jongh 4. Fixed 4. F i x e d point p o i n t theorems theorems For the the provability provability logic logic L L aa fixed fixed point point theorem theorem can can be be proved. proved. One One can can view view For 's diagonalization Godel lemma as stating that theories the GSdel's diagonalization lemma as stating that in in arithmetic arithmetic theories the formula formula -,Op ~ [::lp ' s proof has has aa fixed fixed point: point: the the Godel GSdel sentence. sentence. Godel GSdel's proof of of his his second second incompleteness incompleteness theorem effectively consisted of theorem effectively consisted of the the fact fact that that the the sentence sentence expressing expressing consistency, consistency, the the arithmetic 0 1.. is provably equivalent point. Actually arithmetic realization realization of of ~[::lJ_, is provably equivalent to to this this fixed fixed point. Actually this fact fact is is provable provable from from the the principles principles codified codified in in the the provability provability logic logic L, L, which which this means L. This means then then that that it it can can actually actually be be presented presented as as aa fact fact about about L. This leads leads to to aa rather rather general xed point general fi fixed point theorem, theorem, which which splits splits into into aa uniqueness uniqueness and and an an existence existence part. part. It It concerns formulas A A with with aa distinguished variable p p that that only only occurs occurs distinguished propositional propositional variable concerns formulas boxed A, i.e., in A DB of boxed in in A, i.e., each each occurrence occurrence of of p p in A is is part part of of aa subformula subformula [::IB of A. A. -, , q 44.1. . 1 . Theorem. Uniqueness of T h e o r e m . ((Uniqueness of fixed fixed points points)) If If p p occurs occurs only only boxed boxed in in A(p) A(p) and and q L c:J ((p t-t not occur does not occur at at all all in in A(p) A(p),, then then iF-L E]((p ~ A(p)) A(p)) 1\ ^ (q t-t +-~A(q)) A(q)) -+ ~ (p (p t-t ++ q). does (q q). If L B t-t If p p occurs occurs only only boxed boxed in in A(p) A(p),, and and both both iF-LB +-~A(B) A(B) and and t-L C C t-t ++ A(C) A(C),, then then iF-L B t-t ~ C. C. i-L LB 4.2. 4.2. Corollary. Corollary. 4.3. Theorem. Existence of xed points 4.3. T h e o r e m . ((Existence of fi fixed points)) If If p p occurs occurs only only boxed boxed in in A(p) A(p),, then then there there exists containing pp and otherwise containing exists aa formula formula B, B, not not containing and otherwise containing only only variables variables of of A(p) A(p),, such such that that iF-L LB B t-t ~ A(B) A (B).. After original proofs see Sambin After the the original proofs by by de de Jongh Jongh and and Sambin Sambin ((see Sambin [1976] [1976],, Smorynski Smoryfiski r st proof [1978,1985] [1978,1985], and, and, for for the the fi first proof of of uniqueness, uniqueness, Bernardi Bernardi [1976]) [1976]) many many other, other, ' different, different, proofs proofs have have been been given given for for the the fixed fixed point point theorems, theorems, syntactical syntactical as as well well as as semantical ones, the latter e.g., semantical ones, the latter e.g., in in Gleit Gleit and and Goldfarb Goldfarb [1990j [1990].. It It is is also also worthwhile worthwhile which can to 4.3 follows theorem 4.1 to remark remark that that theorem theorem 4.3 follows from from theorem 4.1 ((which can be be seen seen as as aa kind kind of of ' s definability theorem that holds for implicit nability theorem implicit defi definability theorem)) by by way way of of Beth Beth's definability theorem that holds for L. L. The The latter latter can can be be proved proved from from interpolation interpolation in in the the usual usual manner. manner. Interpolation Interpolation can can 's consistency be standard manner manner via be proved proved semantically semantically in in the the standard via aa kind kind of of Robinson Robinson's consistency see Smoryfiski Smorynski [1978]), syntactically in lemma lemma ((see [197S]), and and syntactically in the the standard standard manner manner by by cut cutSambin and elimination calculus formulation elimination in in aa sequent sequent calculus formulation ofL ofL ((Sambin and Valentini Valentini [1982,1983]). [1982,1983]). In In an an important important sense sense the the meaning meaning of of the the fixed fixed point point theorem theorem is is negative, negative, namely namely in in the the sense sense that, that, if if in in arithmetic arithmetic one one attempts attempts to to obtain obtain formulas formulas with with essentially essentially new properties by new properties by diagonalization, diagonalization, one one will will not not get get them them by by using using instantiations instantiations of of purely except once Godel sentence, purely propositional propositional modal modal formulas formulas ((except once of of course: course: the the GSdel sentence, or or the ) . That the sentence sentence Lob LSb used used to to prove prove his his theorem theorem). That is is one one reason reason that that interesting interesting fixed see section fixed points points often often use use Rosser-orderings Rosser-orderings ((see section 9). 9). 5 theories 5.. Propositional Propositional t h e o r i e s and a n d Magari-algebras Magari-algebras A usually in A propositional propositional theory theory is is aa set set of of modal modal formulas formulas ((usually in aa finite finite number number of of closed under under modus but propositional propositional variables variables)) which which is is closed modus ponens ponens and and necessitation, necessitation, but Logic of of Provability The Logic 485 485 not necessarily necessarily under under substitution. substitution. not We We say say that that such such aa theory theory is is faithfully interpretable in in PA, PA, if if there there is is aa realization realization such that that T = - {A { A II P A fF A*} A*}.. (This (This is isan adaptation of of definition definition 111.1 to the the modal modal ** such PA an adaptation 1 . 1 to propositional language.) language.) Each Each sentence sentence a a of of PA P A generates generates aa propositional propositional theory theory propositional which is is faithfully faithfully interpretable interpretable in in PA, PA, namely namely Tha Th~ = = {A(p) {A(p) II PA P A fF A*( A*(ra'~)}. Of a "' ) } . Of which course, this this theory theory is is closed closed under under L-derivability: L-derivability: it it is is an an L-propositional theory. course, A question question much much wider wider than than the the one one discussed discussed in in the the previous previous sections sections is, is, which which A L-propositional theories theories are are faithfully faithfully interpretable interpretable in in PA P A and and other other theories. This L-propositional theories. This question was was essentially essentially solved solved by by Shavrukov Shavrukov [1993b] [1993b]:: question 5.1. e. L-propositional theory TT is faithfully interpretable in PA 5.1. Theorem. T h e o r e m . An r. r.e. PA iff TT is consistent and and satisfies the e., oA the strong strong disjunction disjunction property property (i. (i.e., DA E~ T T implies A E~ T, T, and oA DA V V OB DB E~ T T implies OA DA E~ T T or OB DB E~ T) T).. A Note that that faithfully faithfully interpretable interpretable theories theories in in aa finite finite number number of of propositional propositional Note variables are are necessarily necessarily r.e. r.e. The The theorem theorem was was given given aa more more compact compact proof proof and and at at the the variables same time time generalized generalized to to all all r.e. r.e. theories theories extending extending IAo + EXP EXP by by Zambella [1994].. lao + Zambella [1994] same If one one applies applies the the theorem theorem to to the the minimal minimal L-propositional L-propositional theory, theory, an an earlier earlier proved proved If 's theorem strengthening of of Solovay Solovay's theorem (Artemov (Artiimov [1980] [1980],, Avron [1984], Boolos Boolos [1982] [1982], strengthening Avron [1984] ' ' Montagna [1979], [1979], Visser Visser [1980]) [1980])rolls out. Montagna rolls out. 5.2. (Uniform 5.2. Corollary. Corollary. (Uniform arithmetic arithmetic completeness completeness theorem) theorem) There exists a sequence of arithmetic sentences So, ao, aI , . . . such that, for any n and modal for of arithmetic O/1,... formula AA(p0,... (p , . . . ,,pn), LA iff, under the arithmetic realization ** induced by setting Pn ) , fF LA p� = a~ n, , A* p~) = ao, c~0,..., A* is provable in PA. PA. P · · · , p* o= o Sets of of modal modal formulas formulas that that are are the the true true sentences sentences under under some some realization realization are Sets are closed under under modus modus ponens, ponens, but but not necessarily under such sets sets are are closed not necessarily under necessitation; necessitation; such generally not not propositional above sense. sense. Let Let us us call call aa set set T of modal T of modal generally propositional theories theories in in the the above formulas realistic if if there exists a a realization such that that A A** is true, for every A A E~ T. there exists realization ** such is true, for every T. formulas Moreover, say that that T well-specified if, if, whenever A eE T and B B is of Moreover, we we say T is is well-specified whenever A T and is aa subformula subformula of A, we also have T or [1997] result that that generalizes generalizes .B Ee T. T. Strannegard A, we also have B B Ee T or -~B Strannegs [1997] proves proves aa result both arithmetic completeness completeness theorem. We give give aa both theorem theorem 5.1 5.1 and and Solovay's Solovay 's second second arithmetic theorem. We weak but easy to version of of it. weak but easy to state state version it. Let TT be a well-specified well-specified r.e. r.e. set set of of modal formulas. Then TT is modal formulas. realistic T is with S. S. realistic iff iffT is consistent consistent with 5.3. 5.3. Theorem. Theorem. An even more more general general point point of of view view than than propositional propositional theories theories is look at at the the An even is to to look Boolean algebras algebras of of arithmetic arithmetic theories theories with with one one additional additional operator operator representing representing Boolean formalized provability provability and, and, more more specifically, specifically, at at the the ones ones generated generated by by aa sequence sequence of of formalized sentences in in the the algebras of arithmetic. arithmetic. The The algebras can be be axiomatized sentences algebras of algebras can axiomatized equationally equationally and and are are called called Magari-algebras Magan-algebras (after (after the the originator originator R. R. Magari) Magari) or or diagonalizable diagonalizable algebras. Of Of course, course, theorem theorem 5.1 5.1 can can be be restated in terms terms of of Magari-algebras. Magari-algebras. algebras. restated in Shavrukov beautiful and and essential essential additional additional results results concerning concerning the the Shavrukov proved proved two two beautiful 486 486 G. Japaridze Japaridze and D. de Jongh Magari-algebras of of formal formal theories theories that cannot naturally naturally be be formulated formulated in in terms terms Magari-algebras that cannot of theories. of propositional propositional theories. 5.4. Theorem. PA 5.4. T h e o r e m . (Shavrukov (Shavrukov [1993a]) [1993a]) The Magari algebras algebras of o/P A and ZF ZF are not isomorphic, and, in fact not elementarily equivalent (Shavrukov ). (Shavrukov [1997] [1997]). F proves The The proof proof only only uses uses the the fact fact that that Z ZF proves the the uniform uniform 2: El-reflection principle 1 -refiection principle A. AA corollary for for P PA. corollary of of the the theorem theorem is is that that there there is is aa formula formula of of the the second second order order propositional propositional calculus calculus that that is is valid valid in in the the interpretation interpretation with with respect respect to to PA, PA, but but not not in the the one one with with respect respect to to ZF. Beklemishev [1996b] [1996b] gives gives aa different different kind kind of of example example ZF. Beklemishev in of theories PA ID.o + EXP. of such such aa formula formula for for the the two two theories PA and and IA0 EXP. 5.5. the Magari algebra 5.5. Theorem. T h e o r e m . (Shavrukov (Shavrukov [1997]) [1997]) The first order theory of o/the algebra of o/ PA PA is undecidable. Japaridze Japaridze [1993] [1993] contains contains some some moderately moderately positive positive results results on on the the decidability decidability of of certain certain fragments fragments of of (a (a special special version version of) of) this this theory. theory. 6. h e extent e x t e n t of o f Solovay's S o l o v a y ' s theorems theorems 6. T The 's theorems An An important important feature feature of of Solovay Solovay's theorems is is their their remarkable remarkable stability: stability: aa wide wide class class of of arithmetic arithmetic theories theories and and their their provability provability predicates predicates enjoys enjoys one one and and the the same same L. Roughly, provability provability logic logic L. Roughly, there there are are three three conditions conditions sufficient sufficient for for the the validity validity of of 's results: Solovay the theory theory has be ((a) a) sufficiently (b) recursively recursively enu Solovay's results: the has to to be sufficiently strong, strong, (b) enu' s derivability merable (a predicate satisfying derivability conditions merable (a provability provability predicate satisfying Lob Lhb's conditions is is naturally naturally constructed from constructed from aa recursive recursive enumeration enumeration of of the the set set of of its its axioms) axioms),, and and (c) (c) sound. sound. Let Let us us see see what what happens happens if if we we try try to to do do without without these these conditions. conditions. The The situation situation is is fully fully investigated investigated only only W.Lt. w.r.t, the the soundness soundness condition. condition. Consider Le. theory Consider an an arbitrary arbitrary arithmetic arithmetic r.e. theory T T containing containing PA PA and and aa 2: 211 provability provability defined as predicate predicate Pr PrT(x) for T. T. Iterated consistency assertions for for T T are are defined as follows: follows: T (x) for Con~ := T; Con~+l(T):= Con(T + Conn(T)), n (T) is ' ) . In where, rp) stands where, as as usual, usual, Con(T Con(T + + ~) stands for for -, ~ Pr PrT(r-~ ~7). In other other words, words, Con Conn(T) is T ( rp realization of modal formula (up to to provable provable equivalence) equivalence) the the unique unique arithmetic arithmetic realization of the the modal formula (up n (T) is n+ ! (T) is n .1. We [:]n_l_. We say say that that T T is is of of height n if if Con Con n (T) is true true and and Con Conn+l(T) is false false in in the the -, O standard model. If standard model. If no no such such n exists, exists, we we say say that that T T has has infinite height. In In aa sense, sense, theories theories of of finite finite height height are are close close to to being being inconsistent inconsistent and and therefore therefore can considered as can be be considered as aa pathology. pathology. The The inconsistent inconsistent theory theory is is the the only only one one of of height height O. All All 2: infinite height, 0. El-SOund theories have have infinite height, but but there there exist exist 2: El-Unsound theories 1 -sound theories 1 -unsound theories n of infinite height. of infinite height. The The theory theory T T+ + ~ Con Con ~(T) (T) is is of of height height n, if if T T has has infinite infinite height. height. Moreover, Moreover, for for each each consistent consistent but but 2: El-Unsound theory T T and and each each n > 0, 0, one one can can 1 -unsound theory construct aa provability construct provability predicate predicate for for T T such such that that T is is precisely precisely of of height height n with with respect respect to to this this predicate predicate (Beklemishev (Beklemishev [1989a]). [1989a]). -, -, The Logic of o.f Provability 487 487 Let us us call call the provability of T T the the set set of of all all modal formulas A A such such that that provability logic of modal formulas Let T f-F (A)r, arithmetic realizations realizations •* with (A)~,, for for all all arithmetic with respect respect to to Pr PrT. The truth provability T . The the set set of of all all A A such such that that (A)r (A)~, is is true true in in the the standard model, for for all all logic of TT isis the standard model, realizations realizations *. • . 6.1. T h e o r e m . (Visser (Visser [1981]) [1981]) For an r. r.e. PA, the 6.1. Theorem. e. arithmetic theory TT containing containing PA, provability logic ofT of T coincides with 1. L, L, if T 1. T has infinite height, n 1- f-F LA}, 2. {A 1I O 2. {A D~-lLA}, if T T is of of height 00 ::;; <~n n< < 00 co.. n 1- is 's construction, Proof. Proof. B Byy Solovay Solovay's construction, using using the the fact fact that that the the formula formula o D~_k is valid valid in in -1 Kripke-models n, and Kripke-models of of height height < < n, and only only in in such such models. models. -t ' s second Generalization interesting. To it, Generalization of of Solovay Solovay's second theorem theorem is is more more interesting. To formulate formulate it, we introduce aa convenient notation. For modal formulas we first first introduce convenient notation. For aa set set of of modal formulas X X,, let let LX LX denote denote the the closure closure under under modus modus ponens ponens and and substitution substitution of of the the set set X X together together with with ' s logic { o A -+ all all theorems theorems of of L. L. In In this this notation, notation, Solovay Solovay's logic S S is is the the same same as as L L{DA ~ A}. A}. The following following two two logics logics have have been been introduced introduced by by respectively respectively Artemov Art~mov [1980J [1980] and and The Japaridze [1986,1988bJ Japaridze [1986,1988b] with with different different provability provability interpretations interpretations in in mind mind (see (see the the next next section) section):: .:= L{-~Dn_klnelN} L{ --, o n 1- 1 n E IN} (OA VV DB)} D D .:= L{-~ D_k, C](C:]AV v DB) DB) -+ ~ ([:]A DB)} L{ --, 01 - , o(oA A A - - Obviously, Obviously, A Ac c D D c c S S.. The The following following theorem theorem gives gives an an exhaustive exhaustive description description of of all all truth provability truth provability logics. logics. 6.2. 6.2. Theorem. T h e o r e m . (Beklemishev (Beklemishev [1989a]) [1989a]) For an r.e. arithmetic theory T T containing PA, PA, the truth truth provability logic for .for T T coincides with 1. SS iff T 1. T is sound, but not sound, 2. 2. D D iff iff T T is � El-sound l -sound but 3. A but is not �El-sound, A iff T T has infinite height but l -sound, n n l + 1-, --, 4. L{o 1-} iff T 4. L{C:ln+l_l_, ~ o D~_k} T is of of height 00 ::;; <~n n< < 00 co.. It It is is known known that, that, at at least least iinn some some natural natural cases, cases, the the other other two two sufficient sufficient conditions conditions can can also also be be considerably considerably weakened. weakened. Boolos Boolos [1979J [1979] shows shows that that the the non-r.e. non-r.e, predicate predicate of of w-provability (dual (dual to to w-consistency) w-consistency) over over Peano Peano arithmetic arithmetic has has precisely precisely the the same same provability logic as arithmetic itself, L. The holds for provability logic as Peano Peano arithmetic itself, i.e., i.e., L. The same same holds for the the natural natural "to be provable in PA together with all formalization predicate formalization of of the the �n En+l-complete predicate P A together l-complete + true TIn -sentences" (Smorynski Boolos [1993b]) showed that IIn-sentences" (Smoryfiski [1985]). [1985]). Solovay Solovay (see (see Boolos [1993b])showed that logic of predicate of analysis together L L is is also also the the logic of the the TI�-complete H~-complete predicate of provability provability in in analysis together ' s theorems with with the the w-rule. w-rule. However, However, no no results results to to the the effect effect that that Solovay Solovay's theorems hold hold for for broad hand, Solovay broad classes classes of of non-r.e. non-r.e, predicates predicates are are known. known. On On the the other other hand, Solovay found found an an 488 488 G. Japaridze Japaridze and D. D. de de Jongh Jongh G. axiomatization of of the the provability provability logic logic of of the the predicate predicate "to "to be be valid valid in in all all transitive transitive axiomatization models of of ZF" ZF",, which which happens happens to to be be aa proper proper extension extension of of L L (see (see Boolos Boolos [1993b]). [19935]). models 's construction If one one is is somewhat somewhat careful, careful, Solovay Solovay's construction can can be be adapted adapted to to show show If 's theorems that Solovay Solovay's theorems hold hold for for all all (r.e., (r.e., sound) sound) extensions extensions of of Ida Iz~o + + EXP EXP (de (de that Jongh, Jumelet Jumelet and and Montagna Montagna [1991]). [1991]). The The two two theorems theorems formulated formulated earlier earlier in in this this Jongh, section can can be be similarly similarly generalized. generalized. However, However, the the most most intriguing intriguing question, question, whether whether section ' s theorems ' s Si Solovay's theorems hold hold for for essentially essentially weaker weaker theories, theories, such such as as Buss Buss's S~ or or even even Solovay remains open. open. This This problem problem was was thoroughly thoroughly investigated investigated by by Berarducci Berarducci and and S$2, 2 , remains Verbrugge [1993] [1993],, where where the the authors, authors, in in aa modification modification of of the the Solovay Solovay construction, construction, Verbrugge only succeeded succeeded in in embedding embedding particular particular kinds kinds of of Kripke-models Kripke-models into into bounded bounded arith arithonly 's construction, metic. The The main main technical technical difficulty difficulty lies lies in in the the fact fact that that Solovay Solovay's construction, in in metic. its known known variations, variations, presupposes presupposes (at (at least, least, sentential) sentential) provable provable :lIIt-completeness 3II~-completeness of of its the theory in question. question. This This property to fail fail for for bounded bounded arithmetic arithmetic under under the theory in property happens happens to some reasonable reasonable complexity-theoretic complexity-theoretic assumption assumption (Verbrugge (Verbrugge [1993a]). [1993a]). As As far far as as we we some know, it is not not excluded excluded that that the answer to to the the question question what what is is the the provability provability logic logic know, it is the answer of Si S~ also also depends depends on on difficult difficult open open problems problems in in complexity complexity theory. theory. of 's ' Solovay'ss theorem theorem does does not not hold in its its immediately form for for Heyting Heyting's Solovay hold in immediately transposed transposed form arithmetic H A (the (the intuitionistic intuitionistic pendant of PA). PA). The The logic logic of of the the provability provability arithmetic HA pendant of predicate of of HA H A with with regard regard to to HA H A certainly certainly contains contains additional beyond predicate additional principles principles beyond the obvious obvious intuitionistic intuitionistic version version of of L: the intuitionistic intuitionistic propositional IPC the L: the propositional calculus calculus IPC plus ( O A -+ A) plus O D(DA--+ A ) ~-+ oA. DA. The The situation situation has has been been discussed discussed by by Visser Visser in in several several papers (Visser al. [1995]) unknown what logic is, papers (Visser [1985,1994] [1985,1994], Visser et al. [1995]).. It It is is unknown what the the real real logic is, ' Visser et for all all we we know know it it may may even even be be complete complete IIg II ~. In In any any case case it it contains contains the the additional additional for principles: principles: 9 D----DA--+ D DA • 9 [o: ](-,-,OA ( ~ D A -~ -+ O A -+ DA) oA) -+ o DoA 2) 9 D(A D(A O (A VV B) B) --+ -+ o (A VV DB) OB) (Leivant's (Leivant 's Principle Principle2) • But this is not an list. It is well well possible possible that that the the logic logic of the binary But this is not an exhaustive exhaustive list. It is of the binary operator of E -preservativity over over H HA is better than the the logic logic of operator of ~-preservativity A is better behaved behaved than of provability provability on its own: own: PA+A PA+B, E 1 -sentence from which on its P A + A E-preserves ~-preserves P A + B , if if from from each each ~l-sentence from which A follows follows in in PA, also PA-derivable. PA-derivable. In In classical classical systems is A PA, B B is is also systems E-preservativity ~-preservativity is definable in in terms terms of 12 to to 14 14 for for that concept) and and vice vice definable of II1-conservativity IIl-conservativity (see (see sections sections 12 that concept) versa, versa, but but constructively constructively this this is is the the proper proper version version to to study study (see (see also also Visser Visser [1997]). [1997]). 7. Classification off provability Classification o provability logics logics One of of the the important important methodological methodological consequences consequences of of Ghdel's G6del's second second incompleteincomplete One ness theorem theorem is is the the fact fact that, that, in in general, general, itit is is necessary necessary to to distinguish distinguish between between aa ness theory TT under under study, study, and and aa metatheory metatheory U in in which which one one reasons reasons about about the the properties properties theory 2added as as one of the "stellingen" "stellingen" (theses) (theses) to Leivant's Leivant's Ph.D. thesis, Amsterdam, 1979 1979 The Logic o~ of Provability Provability The 489 489 of T. Perhaps, Perhaps, the the most most natural natural choice choice of of U U is is the the full full true true arithmetic arithmetic TA, TA, the the of T. set of of all all formulas formulas valid valid in in the the standard standard model, model, yet yet this this isis not not the the only only possibility. possibility. set Other meaningful meaningful choices choices could could be be TT itself, itself, or or the the reader's reader' s favorite favorite minimal minimal fragment fragment Other of arithmetic, e.g., e.g., I]E1. I�l ' The The separate separate role role of of the the metatheory metatheory isis emphasized emphasized in in the the of arithmetic, definition of of provability provability logic logic of of aa theory theory TT relative relative toto aa metatheory metatheory U that that was was definition suggested independently independently by by Art~mov Artemov [1980] [1980] and and Visser Visser [1981]. [1981]. suggested I�o ++ EXP, EXP , TT r.e. r.e. and and U not not Let TT and and U be be arithmetic arithmetic theories theories extending extending IAo Let necessarily necessarily r.e. r.e. The The provability provability logic logic of of TT relative relative to, to, or or simply simply at, at, U isis the the set set of of all all modal modal formulas formulas ~cp such such that that U }-I- (~)~, (CP)r, for for all all arithmetic arithmetic realizations realizations *• (denoted (denoted PRLT(U) ), Intuitively, Intuitively, PRLT(U) PRLT (U) expresses expresses those those principles principles of of provability provability in in TT that that PRLT(U)). be verified verified my my means means of of U. U. As As aa set set of of modal modal formulas, formulas, PR[.T(U) PRLT(U) contains contains LL can can be and and isis closed closed under under modus modus ponens ponens and and substitution, substitution, i.e., i.e., is is aa (not (not necessarily necessarily normal) normal) modal logic logic extending L. modal extending L. Solovay' s theorems theorems can can be be restated restated as as saying saying that, that, ifif TT is is aa sound sound theory, theory, then then Solovay's S. A A modal modal logic is called arithmetically complete, ifif PRLT(T) = = L L and and PRLT(TA) -= S. logic is called arithmetically has the the form form PRI_T(U), PRLT(U) , for for some some TT and and U problem of of obtaining it it has U.. The The problem obtaining aa reasonable reasonable general characterization characterization of arithmetically complete modal logics logics has has become become known known general of arithmetically complete modal was one one of the early motivating problems for the the as 'the classification problem',' , and and was of the early motivating problems for as 'the classification problem Moscow school of of provability founded by Moscow school provability logic logic founded by Artemov. Art~mov. The solution solution to problem is is the outcome of work of of several auThe to this this problem the joint joint outcome of the the work several au Artemov [1985b], Japaridze [1986,1988b], thors Artemov [1980], Visser Visser [1981,1984], [1981 ,1984] ' Art~mov thors Art~mov [1980], [1985b], Japaridze [1986,1988b], Beklemishev [1989a]. [1989a]. Art~mov Artemov [1980] so-called uniform Beklemishev [1980] (applying (applying the the so-called uniform version version of of 's theorem, Solovay corollary 5.2) 5.2) showed showed that that all the form form LX for any Solovay's theorem, corollary all logics logics of of the LX,, for any set set of variable-free variable-free modal modal formulas, formulas, are are arithmetically arithmetically complete. complete. In In Artemov X of X Art~mov [1985b], [1985b], he by the the following families he showed showed that that such such extensions extensions are are exhausted exhausted by following two two specific specific families of logics: of logics: Lo L~ := := L{Fn L{Fn lI n n ~E}a} , , L; = L{ Fn} , L~ ::= L{V,r ~Fn}, Vn¢!J --, +! 1.. -+ o n 1... Some where , /3 <;:; nite, and where a c~,/3 c_ IN IN,, /3 /3 is is cofi cofinite, and Fn Fn denotes denotes the the formula formula on Uln+l_L--~ D~_L. Some of the the provability provability logics logics introduced introduced above above have have this this form: form: L L = : L0 L~,, A A = = LIN L~, , of o n+ ! 1.. , ~ on L{On+l• Dn•1.. } = = LLIN\(~}. L{ 1N\{n} ' The The families hmilies La L~ and and L L~; are are ordered ordered by by inclusion inclusion precisely precisely as as their their indices, indices, and and La included in a. Note Note that L~ is is included in L: L~ for for cofinite cofinite c~. that the the logics logics L; L~ are are not not contained contained in in S, S, and and therefore therefore correspond correspond to to unsound unsound metatheories metatheories U if if the the theory theory T is is sound. sound. Visser Visser [1984] [1984] showed showed that that these these are are the the only only arithmetically arithmetically complete complete logics logics not not contained contained in in S. S. Artemov Art~mov [1985b] [1985b] improved improved this this by by actually actually reducing reducing the the classification classification problem problem to to the the interval interval between between A A and and S. S. Any Any arithmetically arithmetically complete complete logic logic ee from from this this interval interval generates generates aa family family of of different different arithmetically arithmetically complete complete logics logics of of the the form form e~ n N L; L~,, for for cofinite cofinite /3, fl, and and Artemov Art~mov showed showed that that such such logics, logics, together together with with the the families families La L~ and and L; L~,, exhaust exhaust all all arithmetically arithmetically complete complete ones. ones. Japaridze Japaridze [1986,1988b] [1986,1988b] found found aa new new provability provability logic logic within within the the interesting interesting inter interval establishing that val by by establishing that PRLpA(PA PR[-pA(PA + + w-Con(PA)) w-Con(PA)) = =D D,, where where w-Con(PA) w-Con(PA) denotes denotes --, 490 490 G. Japaridze Japaridze and and D. D. de de Jongh Jongh G. the the formalized formalized w-consistency w-consistency ofPA. of PA. The The final final step step was was made made by by Beklemishev Beklemishev [1989a] [1989a],, who who showed showed that that D D is is the the only only arithmetically arithmetically complete complete modal modal logic logic within within the the interval interval between between A A and and S, S, thus thus completing completing the the classification. classification. This This result result was was also also crucial crucial for the the proof proof of of theorem theorem 6.2 6.2 of of the the previous previous section. section. We We denote denote S{3 S~ := : - SS nn Lf; L~,, for DZ := :- D n n Lf; L~ and and formulate formulate the the resulting resulting theorem. theorem. D{3 7.1. Theorem. T h e o r e m . (Classification (Classification Theorem, Theorem, Beklemishev Beklemishev [1989a]) [1989a]) The arithmetically 7.1. complete modal logics are exhausted by the four families: LLa, L-~,, S{3, SZ, D{3, DZ, for a , Lf; cofinite. a, (3~ �c_N, (3fl cojinite. From From aa purely purely modal-logical modal-logical point point of of view, view, the the meaning meaning of of the the classification classification theorem is is that that only only very very few few extensions extensions of of L are are arithmetically arithmetically complete. complete. The The word word theorem 'few' must must not not be be understood understood here here in in terms terms of of cardinality, cardinality, because because the the family family L L~a 'few' has already already the the cardinality cardinality of of the the continuum, continuum, but but rather rather less less formally. formally. E.g., E.g., there there has is aa continuum continuum of of different different modal modal logics logics containing containing A A (Artemov (Artiimov [1985b]), [1985b]), but but only only is four of of them them are are arithmetically observations hold hold for for other other natural natural four arithmetically complete. complete. Similar Similar observations intervals in in the the lattice lattice of of extensions extensions of of L L.. intervals All arithmetically complete logics logics have have nice nice axiomatizations, axiomatizations, and and are are generally generally All arithmetically complete well-understood, normal. An well-understood, although although most most of of them them are are not not normal. An adequate adequate Kripke-type Kripke-type semantics arithmetically complete logics: for semantics is is known known for for all all arithmetically complete logics: for L L,a and and Lf; L~ it it can can be be formulated the height models for formulated in in terms terms of of the height of of the the tree-like tree-like models for L; the the so-called so-called tail Boolos [1981] models for for S S were were suggested suggested independently independently by by Boolos [1981] and and Visser Visser [1984]; [1984]; aa produced by Beklemishev [1989b] similar kind similar kind of of semantics semantics for for D D was was produced by Beklemishev [1989b].. A A corollary corollary is is that that all all logics logics of of the the families families S{3 S~,, D{3 DZ,, and and Lf; L~ are are decidable, decidable, and and aa logic logic of of the the form form Laa is is decidable, decidable, iff iff its its index index a a is is aa decidable decidable subset subset of of IN IN,, i.e., i.e., iff iff it it has has aa decidable decidable L axiomatization. axiomatization. The fact fact that that arithmetically scarce tells tells us us that that inference inference The arithmetically complete complete logics logics are are scarce considerably strengthens modal-logical 'by strengthens the the usual usual modal-logical 'by arithmetic arithmetic interpretation' interpretation ' considerably consequence relation. relation. In In fact, the classifi cation theorem can be be understood understood as as aa consequence fact, the classification theorem can classification of of modally modally expressible expressible arithmetic classification arithmetic schemata. schemata. Familiar Familiar examples examples of of such such A schemata are: are: the the local reflection reflection principle principle for for T, T, that that is is the the schema schema PrT(rAT) Prr( A"') ~-+ A schemata for all all arithmetic arithmetic sentences sentences A, A, which which is is expressed by the the modal modal formula formula [:]p--+ Op -+ p; p; for expressed by IN } ; the the local w times times iterated iterated consistency consistency of of T, T, which which is is expressed expressed by by {~ {..., [::]n2_ o n .1. I1 nn eE IN}; w which can can be be expressed by the the axioms axioms of of D, D, etc. etc. In In general, general, �l -reflection principle, principle, which El-reflection expressed by modally expressible expressible over theory T, T, ifif itit is is deductively deductively equivalent to the the schema is is modally aa schema over aa theory equivalent to set of of all all arithmetic arithmetic realizations realizations with with respect respect to to TT of of aa family family of of modal modal formulas. formulas. set The classification classification theorem theorem gives gives us us aa complete complete description description of of all all modally modally expressible expressible The arithmetic schemata: schemata: they they precisely precisely correspond correspond to to axiomatizations axiomatizations of of arithmetically arithmetically arithmetic complete complete modal modal logics. logics. all such such schemata schemata are are built built up up from instances of of the the It is is very very surprising surprising that that all It from instances reflection principle, principle, sometimes sometimes aa little little twisted twisted by by axioms axioms of of L~ Lf; type. type. This This can can local local reflection be be considered considered as as aa theoretical theoretical justification justification of of the the 'empirical' 'empirical' rule rule that that in in the the study study of provability provability all all reasonable reasonable metatheories metatheories happen happen to to be be equivalent equivalent to to some some version version of of of the the reflection reflection principle. principle. The Logic of Provability 491 491 We We round round up up the the discussion discussion of of the the classification classification theorem theorem by by giving giving some some examples examples for natural pairs metatheory) of for natural pairs (theory, (theory, metatheory) of fragments fragments of of arithmetic. arithmetic. PRLII:1 (PA) PRLIzI(PA) PRLn:1 PRLml(I�n) (I~3,,) PRL PRLIAo+EXP(PRA I..:l.o + EXP (P RA)) PRLn::1 (I�l + Con(PA)) PRLI~(I]E1 +Con(PA)) PRLpRA (I�l) PRLpRA(I~I) S, S, D, D, = - D, D, = A, A, = L. L. = for for n >> 11,, All All such such results results follow follow easily easily from from the the classification classification theorem theorem and and the the usual usual proof prooftheoretic information principles for theoretic information about about the the provability provability of of reflection reflection principles for the the theories theories in in question. I�l + question. E.g., E.g., IE1 + Con(PA) Con(PA) obviously obviously contains contains w-times-iterated w-times-iterated consistency consistency for for I�I being aa finite IE1,, but, but, being finite Il IIl-axiomatized extension of of I�l IE1,' cannot cannot contain contain the the local local 1 -axiomatized extension 's theorem) L: I�l (by (I�l + (PA) ) El-reflection principle for for IE1 (by Lob Lhb's theorem).. Hence, Hence, PRLn:1 PRLI~I (IE1 + Con Con(PA)) 1 -reflection principle contains A A but but does does not not contain contain D. The classification classification theorem theorem implies in this this D. The implies that, that, in contains case, A. case, it it must must be be A. 8. B i m o d a l and a n d polymodal p o l y m o d a l provability p r o v a b i l i t y logics logics 8. Bimodal 's The The fact fact that that all all reasonable reasonable theories theories have have one one and and the the same s a m e- Lob Lhb's - - provability provability logic is, in sense, aa drawback: logic is, in aa sense, drawback: it it means means that that the the provability provability logic logic of of aa theory theory cannot distinguish between cannot distinguish between most most of of the the interesting interesting properties properties of of theories, theories, such such as as ' s theorem e.g., finite axiomatizability, e.g., finite axiomatizability, reflexivity, reflexivity, etc. etc. In In fact, fact, by by Visser Visser's theorem 6.1, 6.1, the the only only recognizable recognizable characteristic characteristic of of aa theory theory is is its its height, height, and and the the situation situation does does not not become logics. become much much better better even even if if one one considers considers truth truth provability provability logics. One One obvious obvious way way to to increase increase the the expressive expressive power power of of the the modal modal language language is is to consider provability to consider provability operators operators in in several several different different theories theories simultaneously, simultaneously, which which naturally leads modal naturally leads to to bi- and and polymodal provability logic. It It turns turns out out that that the the modal description of joint behaviour or more description of the the joint behaviour of of two two or more provability provability operators operators is, is, in in general, general, aa considerably than the unimodal provability considerably more more difficult difficult task task than the calculation calculation of of unimodal provability logics. logics. There that can There is is no no single single system system that can justifiably justifiably be be called called the bimodal bimodal provability provability logic rather, we particular systems logic - - rather, we know know particular systems for for different different natural natural pairs pairs of of provability provability operators, operators, and and none none of of those those systems systems occupies occupies any any privileged privileged place place among among the the others. others. Moreover, Moreover, the the numerous numerous isolated isolated results results accumulated accumulated in in this this area, area, so so far, far, give give us us no no clue clue as as to to aa possible possible general general classification classification of of bimodal bimodal provability provability logics logics for for pairs pairs of of sound r.e. remains one open problems sound r.e. theories. theories. This This problem problem remains one of of the the most most challenging challenging open problems in the state in provability provability logic. logic. A A short short survey survey of of the state of of our our knowledge knowledge in in this this field field is is given given below. below. The The language language £ s ( 0 , 6) of of bimodal bimodal provability provability logic logic is is obtained obtained from from that that of of propositional propositional calculus calculus by by adding adding two two unary unary modal modal operators operators 0 K] and and 6 A.. Let Let (T, (T, U) be be aa pair pair of of arithmetic arithmetic r.e. r.e. theories, theories, taken taken together together with with some some fixed fixed canonical canonical L:l E1 provability provability predicates predicates Pr PrTT and and Pru Pru.. An An arithmetic realization (')T (')~,uU with respect to to arithmetic (T, (T, U) is is aa mapping mapping of of modal modal formulas formulas to arithmetic sentences sentences tha thatt commutes commutes with with 492 492 G. Japaridze and D. de Jongh Boolean connectives connectives and and translates translates 0 [] as as provability provability in in T and and/~ as that in U: Boolean 6 as that in ([:]A)~.,u = PrT(r(A)~.,uT), (AA)~.,u = Pru Pru((F((A)~,u-~). (6A);',u A );',u ., ). The provability logic for (T, U) collection of U),, denoted denoted PRLT,u, PRLT,u, is is the the collection of all all £ L (( n0,, 6) A)formulas formulas A A such such that that T fF (A)y (A)~,~ and U fF (A)y (A)~,,u for every every arithmetic arithmetic realization realization **.. ,U and ,U ', for In general, general, as as in in the the unimodal case, one one can can consider consider bimodal bimodal provability provability logics logics In unimodal case, for (T, U) relative relative to to an an arbitrary (where PRLT,u PRLT,v corresponds corresponds to to arbitrary metatheory metatheory V (where for VV ==TTN Un )U). . Not Not too too much much can can aa priori priori be be said said about about PRLT,u PRLT,v,, for for arbitrary arbitrary T and and U. Clearly, Clearly, PRLT,u under modus modus ponens, PRLT,u is is closed closed under ponens, substitution substitution and and the the 0[]- and and 6-necessitation A-necessitation rules. bimodal system rules. Moreover, Moreover, PRLT PRLT,,u has has to to be be aa (normal) (normal) extension extension of of the the bimodal system es, CS, given given by by the the axioms axioms and and rules rules of of L formulated formulated separately separately for for 0 [] and and 6, A, and and by by the the obvious obvious mixed mixed principles: principles: oA 6o0A, oA --+ ~/X A, AA --+ ~ 06A. o/XA. 6A 's theorem By By Solovay Solovay's theorem we we know know that, that, whenever whenever both both T and and U have have infinite infinite height, height, £(0) of the in the the fragment fragment of of PRLT,u PRLT,v in the language language/2([]) of 0 [7 alone, alone, as as well well as as the the one one in in the the language 6, actually language of of A, actually coincides coincides with with L. L. Using Using the the uniform uniform version version of of Solovay's Solovay's theorem, [1985] showed theorem, Smorynski Smoryfiski [1985] showed that that es CS is is the the minimal minimal bimodal bimodal provability provability logic, i.e., with PRLT,u certain pair finite extensions extensions T, U of logic, i.e., it it coincides coincides with PRLT,u for for a a certain pair of of finite of Peano arithmetic. there is Peano arithmetic. Beklemishev Beklemishev [1992] [1992] showed showed that that there is even even aa pair pair of of provability provability predicates corresponding bimodal predicates for for Peano Peano arithmetic arithmetic itself itself for for which which the the corresponding bimodal provability provability predicates can logic with es. logic coincides coincides with CS. Such Such predicates can be be called called independent in in the the sense sense that that they little about other as possible in should be they 'know 'know'' as as little about each each other as is is possible in principle. principle. It It should be noted noted 's example, however that, neither however that, neither the the theories theories in in Smorynski Smoryfiski's example, nor nor the the independent independent provability provability predicates predicates are are natural n a t u r a l- they they are are constructed constructed by by aa tricky tricky diagonalization. diagonalization. Thus, es, which Thus, we we are are in in the the interesting interesting situation situation that that the the bimodal bimodal logic logic CS, which structurally structurally occupies occupies aa privileged privileged place place among among the the provability provability logics, logics, does does not not correspond correspond to to any any known natural pair pair of known of theories. theories. Deeper Deeper structural structural information information on on bimodal bimodal provability provability logics logics is is provided provided by by the the Classification Theorem Classification Theorem 7.1 7.1 for for arithmetically arithmetically complete complete modal modal logics. logics. With With every every bimodal logic £t~ we bimodal logic we can can associate associate its its type: ((e) £) 0~ ::= = {A {A E9£(O) L(O) 1I £e f~ 6A}. AA}. . ) 0 surjectively An An easy easy analysis analysis then then shows shows that that ((.)0 surjectively maps maps normal normal extensions extensions of of es CS onto onto L. Under Under the assumption of the lattice of unimodal logics containing L. the lattice of the the unimodal logics containing the assumption of � ~l-soundness l -soundness of of T n N U we we obviously obviously have: have: PRLT(U) PRLT(U) = = (PRLT,u (PRLT, u)) 0~. The classification theorem unimodal logic) The classification theorem not not only only shows shows that that not not every every type type (of (of unimodal logic) is is materialized materialized as as that that of of aa bimodal bimodal provability provability logic, logic, but but also also gives gives us us aa complete complete description of all all such such possible possible types. types. description of The Logic of Provability 493 493 Besides particular bimodal Besides the the general general observations observations above, above, aa number number of of particular bimodal provabil provability known. These ity logics logics for for natural natural pairs pairs of of theories theories are are known. These logics logics cover cover most most of of the the examples that come mind, but, examples of of pairs pairs of of arithmetic arithmetic theories theories that come to to mind, but, unfortunately, unfortunately, are are far being an all bimodal far from from being an exhaustive exhaustive list list of of all bimodal provability provability logics. logics. The 1986] , and The best best known known system system is is the the logic logic PRLpA,ZF PRLpA,zF discovered discovered by by Carlson Carlson [[1986], and with aa different 1987] . This independently ((with independently different interpretation interpretation in in mind mind)) by by Montagna Montagna [[1987]. This logic CS by logic can can be be axiomatized axiomatized over over CS by the the principle principle of of essential reflexivity A(DA--+ 6 ( OA -+ A). It is is the the only bimodal bimodal provability provability logic of type type S S and and aa maximal maximal one one among among the the It logic of bimodal bimodal logics logics for for pairs pairs of of sound sound theories. theories. In In other other words, words, PRLT,u PRLT,u = - PRLpA,ZF PRLpA,ZF, , whenever the the theories theories T, T, U U are are sound sound and and U U contains contains the the local local reflection reflection principle principle whenever for for T. T. Furthermore, Furthermore, we we know know two two natural natural bimodal bimodal provability provability logics logics of of type type D, D, intro introduced by by Beklemishev The first first one one corresponds corresponds to to pairs pairs of of theories theories (T, U) duced Beklemishev [[1996a]. 1996a] . The such -reflection principle such that that U U is is aa finite finite extension extension of of T T that that proves proves the the local local �l El-reflection principle for for T. Typical l Ao + l�m, l�n T. Typical examples examples are are the the pairs pairs ((IAo + EXP, EXP, L:lo IAo + + SUPEXP) SUPEXP),, ((IEm, I E , )), for ' for > m ;;?! >/1, The logic logic can can be be axiomatized axiomatized over over CS CS by by the the mono monotonicity axiom nn > 1 , etc. etc. The tonicity axiom D A -+ --+ 6A AA and and the the schema schema oA 6(05 /x(Ds -+ + 5) s),, where (possibly empty DB and where 5 S is is an an arbitrary arbitrary (possibly empty)) disjunction disjunction of of formulas formulas of of the the form form [-qB and 6B.3 AB.a The exive (see The second second one one corresponds corresponds to to III-essentially II~-essentially refl reflexive (see definition definition 12.3 12.3)) exten extensions theories of bounded arithmetic arithmetic complexity PRA) sions of of theories of bounded complexity such such as as e.g., e.g., (lAo (IAo + + EXP, EXP, P RA),, (IN., I� R (I�n, �tl) for 1 , where I�k but IN.+1) for n n ;;?! >/1, where I� IN~� is is defined defined like like I~k but with with the the induction induction for rule. The for �k-formulas 2k-formulas formulated formulated as as aa rule. The corresponding corresponding provability provability logic logic can can be be axiomatized CSM -essential reflexivity schema axiomatized over over C S M by by the the III IIl-essential schema 6A -+ ( 0 (A -+ -+ 5), AA ~ 6 A(O(A --+ 5) S)--+ S), where before. where 5 S is is as as before. We Beklemishev [[1994]). 1994]) . The We also also know know two two natural natural provability provability logics logics of of type type A A ((Beklemishev The first first system system corresponds corresponds to to pairs pairs of of theories theories (T, (T, U) U) such such that that U is is an an extension extension of of T T by -sentences and consistency of T, such by finitely finitely many many III IIl-sentences and proves proves w-times-iterated w-times-iterated consistency of T, such as as etc. This This logic logic can ( ZF )) , ((IE1, I�b I�l ( I�2 )) ' etc. e.g., pairs (PA, (PA, PA e.g., the the pairs PA + + Con Con(ZF)), IE1 + + Con Con(IE2)), can be be axiomatized axiomatized over over CSM C S M by by the the principle principle (P) (P) 6A AA -+ --+ 0 [::1(6.1 (A_I_ V v A), A), valid valid for for all all III-axiomatizable 1-Ii-axiomatizable extensions extensions of of theories, theories, together together with with the the schema schema A-~ nn_L, n/> 1. 3In the following, monotonicity axiom will following, CS CS together together with the monotonicity will be denoted CSM. CSM. 494 494 C. G. Japaridze and D. de Jongh The second The second system system corresponds corresponds to to reflexive reflexive II IIl-axiomatizable extensions of of theories, theories, I -axiomatizable extensions n (PA) I[ nn/>� I}), such as as e.g., e.g., (PA, (PA, PA PA + + {Con {Conn(PA) 1}), (I�I, (IIE1, I]E1 + {Con(IEn) {Con(IIEn) I[ n � >i I1}). It I�l + } ) . It such CSM (P) by can can be be axiomatized axiomatized over over C S M plus plus (P) by the the reflexivity axiom axiom 6A A A --7 ~ 60A. AOA. Finally, Finally, we we know know by by Beklemishev Beklemishev [1996a] [1996a] aa natural natural system system of of type type L that that corresponds corresponds to to finite finite extensions extensions oftheories of theories of of the the form form (T, (T, T T+ + A) A),, where where both both T T+ + cp ~o and and T T+ + -~ cp ~o are conservative conservative over over T T with with respect respect to to Boolean of El-sentences. ~l-sentences. are Boolean combinations combinations of � 11,, and Examples (PRA, I�l) I�n) ' for Examples of of such such pairs pairs are are (PRA, I]E1),' (I��, (IIE~,I~En), for n n/> and others. others. The The logic CSM logic is is axiomatized axiomatized over over C S M by by the the 8(EI)-conservativity B(P~l)-conservativity schema schema --, 6B --7 AB ~ DB, [::]B, where where B B denotes denotes an an arbitrary arbitrary Boolean Boolean combination combination of of formulas formulas of of the the form form DC [::]C and and 6C. AC. The six six bimodal bimodal logics logics described described above above essentially essentially exhaust exhaust all all nontrivial nontrivial cases cases The for which natural natural provability logics have been characterized. for which provability logics have explicitly explicitly been characterized. It It is is worth worth mentioning that all mentioning that all these these systems systems are are decidable, decidable, and and aa suitable suitable Kripke-style Kripke-style semantics semantics is them. Smoryfiski Smorynski [1985] is known known for for each each of of them. [1985] contains contains an an extensive extensive treatment treatment of of three arithmetic PRLpA,zF including including proofs proofs of of three arithmetic completeness completeness theorems theorems due due to to Carlson. Carlson. PRLpA,ZF These [1997] These theorems theorems are are extended extended by by Strannegiird Strannegs [1997] to to the the setting setting of of r.e. r.e. sets sets of of bimodal Visser [1995] presents aa beautiful bimodal formulas formulas (as (as discussed discussed in in section section 5). 5). Visser [1995] presents beautiful approach bimodal provability approach to to Kripke Kripke semantics semantics for for bimodal provability logics. logics. Beklemishev Beklemishev [1994, [1994, 1996a] 1996a] gives gives aa detailed detailed survey survey of of the the current current state state of of the the field. field. Apart Apart from from describing describing the the joint joint behaviour behaviour of of two two 'usual 'usual'' provability provability predicates, predicates, each each of of them them being being separately separately well well enough enough understood, understood, bimodal bimodal logic logic has has been been successfully some nonstandard, successfully used used for for the the analysis analysis of of some nonstandard, not not necessarily necessarily r.e., r.e., concepts concepts of provability. provability. The The systems systems emerging emerging from from such such an an analysis analysis often often have have not not so so much much in in of common CS, although common with with CS, although different different 'bimodal 'bimodal analyses analyses'' do do share share common common technical technical ideas. ideas. As 1986, Japaridze As early early as as 1986, Japaridze [1986,1988b] [1986,1988b] characterized characterized the the bimodal bimodal logic logic of of prov prov-provability (dual Later his ability ability and and w oa-provability (dual to to w-consistency) w-consistency) in in Peano Peano arithmetic. arithmetic. Later his study study was Boolos [1993b,1993a], was simplified simplified and and further further advanced advanced by by Ignatiev Ignatiev [1993a] [1993a] and and Soolos [1993b,1993a], who, who, among among other other things, things, showed showed that that the the same same system system corresponds corresponds to to some some other, other, so-called so-called strong, concepts concepts of of provability provability (taken (taken jointly jointly with with the the usual usual one). one). Other Other examples examples of of strong strong provability provability predicates predicates are are the the En+ En+l-complete from all l -complete provability from -sentences, for true arithmetic arithmetic IIn IIn-sentences, for n � i> 11,, and and the the m-complete II~-complete provability provability under the w-rule in analysis. Japaridze bimodal logic Japaridze's' s bimodal logic can can be be axiomatized axiomatized by by the the axioms axioms and and rules rules of of L, formulated 6, the formulated separately separately for for D El and and for for A, the monotonicity monotonicity principle principle DA [:]A --7 --+ 6A, AA, and and an I -completeness principle principle an additional additional II HI-completeness OA ~A --7 --+ 60A, A~A, The Logic Logic of of Provability Provability 495 495 which reflects reflects in in so so far far as as that that is is possible possible that t h a t //::, k is is strong strong enough enough to to prove prove all all true true which IIl-sentences (if 0 [] is is the the usual usual r.e. r.e. provability provability predicate predicate and a n d //::, k aa strong strong provability provability Il l -sentences (if 's logic predicate). Japaridze Japaridze's logic is is decidable decidable and and has has aa reasonable reasonable Kripke Kripke semantics. semantics. An An predicate). extensive treatment treatment of of Japaridze's Japaridze's logic logic is is given given in in Boolos Boolos [1993b] [1993b].. extensive Bimodal analysis analysis of of other other unusual unusual provability provability concepts concepts has has been been undertaken undertaken Bimodal by Visser Visser [1989,1995] [1989,1995] and and Shavrukov Shavrukov [1991 [1991,1994]. Using the the work work of of Guaspari Guaspari and and by ,1994] . Using Solovay [1979] [1979],, Shavrukov Shavrukov [1991] [1991] found found aa complete complete axiomatization of the the bimodal bimodal Solovay axiomatization of logic of of the the usual usual and and Rosser's provability predicate for for Peano Peano arithmetic arithmetic (see (see also also logic section 9). It It is is worth worth noting that Rosser's Rosser's provability provability predicate, predicate, although although numerating numerating section 9). noting that (externally) the the same same theory theory as as the the usual usual one, one, has has aa very very different different modal modal behaviour; behaviour; (externally) 's e.g., Rosser Rosser consistency consistency of of PA PA is is aa provable provable fact, fact, but but on on the the other other hand, hand, Rosser Rosser's e.g., provability predicate predicate is is not not provably provably closed closed under under modus modus ponens. Shavrukov [1994] [1994] provability ponens. Shavrukov characterizes the the logic logic of of the the so-called so-called Feferman provability predicate. predicate. This This work work characterizes was preceded by Visser Visser [1989,1995] [1989,1995], where where the the concept concept of of provability in PA PA from was preceded by and some some other other unusual unusual provability provability concepts concepts 'nonstandardly finitely many' axioms' and were bimodally bimodally characterized. These systems systems were were motivated motivated by by their connections were characterized. These their connections with interpretability interpretability logic, logic, but but another another motivation motivation originates originates with with Jeroslow Jeroslow and and with Putnam who who studied studied the the Rosser Rosser and and Feferman Feferman style style systems systems as as 'experimental 'experimental'' Putnam systems: their their self-correcting self-correcting behaviour behaviour is is supposed supposed to to be be closer closer to to the the way way humans humans systems: reason. Studying Studying ordinary ordinary provability provability and and self-correcting self-correcting provability provability can can provide provide aa reason. good heuristic for appreciating appreciating the the differences differences between between both both kinds kinds of of systems. systems. good heuristic for A fi final example of of such such an an analysis analysis of of an an unusual unusual proof proof predicate predicate by by the the A nal example ' s analysis development of of aa bimodal logic was was Lindstrom LindstrSm [19941 [1994]'s analysis of of Parikh provability, development bimodal logic i.e.,, the the proof proof predicate predicate that that allows allows OAf []A/AA as as aa rule rule of of inference. inference. i.e. Additional early early results in bimodal logic, e.g., e.g., aa bimodal bimodal analysis analysis of of the so-called Additional results in bimodal logic, the so-called Mostowski operator, operator, can [1985]. Mostowski can be be found found in in Smoryfiski Smorynski [1985]. Many results in bimodal provability logic logic can can be be generalized generalized to to polymodal logic. Many results in bimodal provability Such generalization is is particularly particularly natural natural in in the modal-logical study Such aa generalization the modal-logical study of of progressions progressions of topic in in proof proof theory theory that the work work of of Turing of theories, theories, aa topic that goes goes as as far far back back as as the Turing [1939] [1939].. From the of view, view, however, however, such such a a generalization, all known known From the modal-logical modal-logical point point of generalization, in in all cases, lead to to any any essentially essentially new Roughly, the cases, does does not not lead new phenomena. phenomena. Roughly, the resulting resulting systems fragments; therefore therefore we systems happen happen to to be be direct direct sums sums of of their their bimodal bimodal fragments; we shall shall not not go go into into the the details. details. bimodal logic logic (modalities, (modalities, inin Polymodal analogues analogues are are known for Japaridze's Japaridze 's bimodal Polymodal known for dexed n, correspond correspond to to the the operators operators to be provable from from all true dexed by by natural natural numbers numbers n, IIn-sentences), provability logics Iln -sentences), and and for for natural natural provability logics due due to to Carlson Carlson and and Beklemishev. Beklemishev. Here, theories of Here, the the modal modal operators operators correspond correspond to to the the theories of the the original original Turing-Feferman Turing-Feferman progressions progressions of of transfinitely transfinitely iterated iterated reflection reflection principles, principles, and and thus, thus, are are indexed indexed by by ordinals say, the ordinals for for some some constructive constructive system system of of ordinal ordinal notation, notation, say, the natural natural one one up up EO . Iterating Iterating full full reflection reflection leads leads to to the the polymodal polymodal analogue analogue of of PR[-pA,ZF, PRLpA,ZF , and and to ~0. to transfinitely transfinitely iterated iterated consistency consistency leads leads to to aa natural natural polymodal polymodal analogue analogue of of A-type A-type provability provability logics logics (Beklemishev (Beklemishev [1991,1994]). [1991,1994]). 496 496 G. Japaridze Japaridze and D. de Jongh 9 9.. Rosser R o s s e r orderings orderings To To discuss discuss Rosser Rosser sentences sentences and and more more generally generally the the so-called so-called Rosser Rosser provability provability predicate in in aa modal modal context, context, Guaspari Guaspari and and Solovay Solovay [1979] [1979] enriched enriched the the modal modal predicate oA and oA -< DB and language language by by adding, adding, for for each each DA and DB, E]B, the the formulas formulas CA -< E]B and oA OA � ~ DB, OB, with with as as their their arithmetic arithmetic realizations realizations the the �l-sentences El-sentences ''A "A** is is provable provable by by aa proof proof that is is smaller smaller than than any any proof proof of of B*" B*",, and and "A* "A* is is provable provable by by aa proof proof that that is is that so-called witness comparison formulas). smaller smaller than than or or equal equal to to any any proof proof of of B*" B*" ((so-called They They axiomatized axiomatized modal modal logics logics RR - and and R R = = RR - ++ the the rule rule oAIA, C]A/A, and and gave gave an an arithmetic completeness arithmetic completeness result result for for R. R. In In this this arithmetic arithmetic completeness completeness result result they they did did have to to allow allow arbitrary arbitrary standard provability provability predicates predicates in in the the arithmetic arithmetic realizations realizations have however, however, i.e., i.e., arbitrary arbitrary provability provability predicates predicates satisfying satisfying the the three three Lob Lhb conditions. conditions. Shavrukov see also Shavrukov [1991] [1991] ((see also the the end end of of section section 8) 8) showed showed that that this this restriction restriction can can be be -,A ((the oA -< the dropped dropped when when one one restricts restricts the the contexts contexts for for the the new new operator operator to to [:]A -< o C]-~A RA), and short: O Rosser provability predicate, for for short: vIRA), and de de Jongh Jongh and and Montagna Montagna [1991] [1991] showed showed that, that, allowing allowing formulas formulas with with free free variables variables as as arithmetic arithmetic substitutions substitutions leads leads to to Rthe arithmetically complete system. system. Guaspari Guaspari and R - as as the arithmetically complete and Solovay Solovay [1979] [1979] also also showed showed that standard provability predicates all sentences a that for for some some standard provability predicates all Rosser sentences (i.e., (i.e., sentences c~ "' ) -< such that that P PA a ++ ~ (PrpA (PrpA ( ( r --, a a-l) .< PrpA PrpA ( (r a a-l)) are equivalent, equivalent, and and that that for for some some such A f-F a .., ) ) are other other standard standard provability provability predicates predicates this this is is not not the the case. case. This This leaves leaves open open the the question question whether whether aa reasonable reasonable notion notion of of usual proof proof predicate predicate can can be be defined defined for for which the question "Is the the Rosser which the question "Is Rosser sentence sentence unique?" unique?" does does have have aa definite definite answer. answer. Hence also, Hence also, uniqueness uniqueness of of fixed fixed points points is is not not provable provable in in R. R. Finally, Finally, they they showed showed that that Simpler proofs also also the the existence existence part part of of the the fixed fixed point point theorem theorem fails fails for for R. R. Simpler proofs for for the the completeness completeness theorems theorems were were given given in in de de Jongh Jongh [1987] [1987] and and Voorbraak Voorbraak [1988] [1988].. There connections between up. There are are connections between this this work work in in provability provability logic logic and and speed speed up. 's First, de First, de Jongh Jongh and and Montagna Montagna [1988,1989] [1988,1989] gave gave aa new new simpler simpler proof proof of of Parikh Parikh [1971] [1971]'s theorem that, for theorem that, for any any provably provably recursive recursive function function 9g there there is is aa sentence sentence a a provable provable proof in A such a "' ) by in P PA such that that PA PA proves proves PrpA Prpn ( (raT) by aa much much shorter shorter proof in the the sense sense of of 9g in g(a) < b) than The The Logic Logicof of Provability Provability 497 497 results somewhat somewhat less less clear clear than than one one might might wish. wish. results Montagna Montagna [1992] [1992] applied applied the the results results on on provable provable fixed fixed points points in in aa study study of of rules, i.e., i.e., rules rules like like PrT( PrT(rc~-~)/c~ that can can be be considered considered as as metamathematical rules, a "')/a that ed these realizations realizations of of modal-logical modal-logical rules rules (in (in case: case: OA/A). [:]A/A). He He classifi classified these rules rules into into two types: types: rules rules giving giving only only polynomial polynomial speed speed up up in in proofs proofs in in arithmetic, arithmetic, and and rules rules two giving aa superexponential superexponential speed speed up. up. In In Hajek, H~jek, Montagna Montagna and and Pudlak Pudlak [1993] [1993] it it was was giving shown shown that that the the rule rule OA/A [:]A/A is is maximally maximally powerful powerful among among these these metamathematical metamathematical rules in in the the sense sense that that the the use use of of any any of of them them can can be be polynomially polynomially simulated simulated by by rules OA/A. Moreover, Moreover, in in that that paper paper natural natural examples examples of of statements statements of of which which the the proof proof is superexponentially superexponentially shortened shortened by by the the above above rule rule are are given. given. is 10. proofs 1 0 . Logic of proofs A provability provability reading reading of of the the modality modality 0 E:I as as "is (informally) provable" was was an an A intended intended semantics semantics for for the the classical classical system system S4 $4 of of propositional propositional modal modal logic logic (see (see ' s paper end of of section section 2) 2) since since Godel Ghdel's paper (Godel (Ghdel [1933]). [1933]). However, However, as as we we have have seen, seen, end the straightforward straightforward interpretation interpretation of of OF ElF as as Pr(r-F -~) leads leads to to the the logics logics L L and and Pre F"') the which are are incompatible incompatible with with $4. The reflexivity reflexivity principle principle of O F -+ ~ F F fails fails in in L, L, SS which S4. The and the the necessitation necessitation rule rule fails fails in in S. S. Nevertheless, Nevertheless, an an interesting interesting interpretation interpretation and of the the S4-modality S4-modality as as formal formal provability provability is is possible. One can can have have the the reflexivity reflexivity of possible. One principle as the necessitation rule if principle as well well as as the necessitation rule if one one incorporates incorporates into into the the modal modal language language machinery machinery to to keep keep all all proofs proofs "real" "real",, i.e., i.e., given given by by actual actual natural natural numbers numbers and and not not quantifying succeeded in quantifying over over them them as as in in the the provability provability predicate. predicate. Artemov Artiimov succeeded in doing doing this this by by replacing replacing the the quantifiers quantifiers by by a a kind kind of of Skolem Skolem functions functions in in his his logic logic of of proofs proofs L P (Artiimov LP (Artemov [1994,1995]). [1994,1995]). The language language of of LP contains besides besides the The L P contains the usual usual Boolean Boolean constants, constants, connec connecproof variables Xo,... XO, ,,xn,..., xn, . . . , proof tives and sentence tives and sentence variables, variables, proof proof axiom constants symbols: monadic monadic !!,, and and •x , and and finally finally the the aao0,, . .. .., , aan, n , .. ... ,. , function function symbols: and binary binary + + and modal operator symbol modal operator symbol [I[ ] (() ). . Terms and and formulas formulas are are defined defined in in the the natural natural way: way: proof proof variables variables and and axiom axiom Terms constants are are terms; terms; sentence sentence variables variables and and Boolean Boolean constants constants are are formulas; formulas; whenever whenever constants are again again terms, terms, Boolean Boolean connectives connectives behave behave and tt are are terms terms !t, !t, (s(s ++ t), t), (s(s xx t)t) are ss and conventionally, and and for for tt aa term term and and FF aa formula, formula, It] [t] FF isis aa formula. formula. We We will will write write conventionally, or even even st instead instead of of (s (s •x t)t) and and skip skip parentheses parentheses when when convenient. convenient. A A term term s . . tt or is ground ground ifif isis does does not not contain contain variables. variables. The The system system LPAs IPAS' has has as as its its axioms axioms all all is formulas formulas of of the the of of the the forms forms below, below, and and as as its its only only rule rule modus modus ponens: ponens: A0. AO. The The tautologies tautologies in in the the language language of of LP, LP, [t] F--+ F -+ FF AI. It] A1. "reflexivity" "reflexivity" [ts] G) G) ( [s] FF --+ A2. It] [t] (F (F --+ "application" A2. -+ G)--+ G) -+ (Is] -+ Its] "application" [t] FF -+ [!t] [!t] It] [t] FF A3. It] A3. "proof checker" checker" "proof A4. [s]] [s] FF ~-+ Is[s ++ t]t] F, F , It] [t] FF- ~-+ Is[s ++ t]!t] FF A4. "choice" "choice" AS. AA finite finite set set of of formulas formulas of of the the form form [c]A, [c] A, where where cc isis an an axiom axiom constant, constant, AS. "axiom specification" specification" and AA isis an an axiom axiom A0-A4 AO-A4 and "axiom . . • . 498 498 G. Japaridze and D. de Jongh 'S of The system system LP L P is is the the generic generic name name for for the the !PAS LPAs'S of the the various various axiom axiom specifications specifications The AS. AS. The The intended intended understanding understanding of of LP L P is is as as aa logic logic of of operations operations on on proofs, proofs, where where It] F F stands stands for for tit '2 is a code for a proof of F". F'. For For the the usual usual G6del Ghdel proof proof predicate predicate [t] Proof(x, y) in in PA P A there there are are provably provably recursive recursive functions functions from from codes codes of of proofs proofs to to codes codes Proof(x, of proofs proofs corresponding corresponding to to x and and !!:: x stands stands for for an an operation operation on on proof proof sequences sequences of which realizes realizes the the modus ponens rule rule in in arithmetic, arithmetic, and and !! is is aa proof proof checker checker operation operation which as it it appears appears in in the the proof proof of of the the second second G6del Ghdel Incompleteness Incompleteness theorem. theorem. The The usual usual as proof predicate predicate has has aa natural natural nondeterministic nondeterministic version version PROOF(x, y) here here called called proof standard nondeterministic proof predicate: "x is a code of a derivation containing ~'.. The The predicate predicate PROOF already already has has all all three three operations operations a formula with a code 11' of the the LP-Ianguage: LP-language: the the operation operation s + + t is is in in its its case case just just the the concatenation concatenation of of the the of (nondeterministic) proofs proofs ss and and t. (nondeterministic) The The system system LP L P reminds reminds one one of of propositional propositional dynamic dynamic logic logic (see (see e.g., e.g., Harel Harel [1984]), [1984]), but is is really really quite quite different different in in character, character, since since the the modalities modalities [t] It] (-) (.) do do not not satisfy satisfy the the but property [t] (p --t t] Pp --t [t] q) property It] --4 q) q) --t ~ ([ (It] --4 It] q) in in LP. LP. This This makes makes the the logic logic LP L P nonnormal nonnormal and not not a a polymodal polymodal logic logic in in the the sense sense of of section section 8. Nevertheless, the the entire entire variety variety and 8. Nevertheless, of labeled labeled modalities modalities in in LP L P can can simulate simulate 84. $4. For For example, example, the the necessitation necessitation rule rule of F~ DF [] F of of normal normal modal modal logics logics has has its its constructive constructive counterpart counterpart in in LP: if LP LP r F F, then Fj LP: if F, then LP r ~- It] for some some ground ground term term t. In In general, general, let let FO F ~ be be the the result result of of substituting O LP [t] FF for substituting 0 for all all occurrences of It] in F, and XO X ~= - {{ F F~ } for for any any set set X X of of LP LP-formulas. for occurrences of [t] in F, and O I F E~ XX} -formulas. It 84: (LP)O It is is easy easy to to see see that that LP L P is is sound sound with with respect respect to to $4: (LP)~ � c_ 84. $4. The The converse converse inclusion S4 (Lp) o turns out to to be be valid valid as as well: well: by by an an LP-realization LP-realization r = = r(AS') r(AS) 84 �c_ (LP)O turns out inclusion of aa modal modal formula formula F F we we mean mean of 1. LP-terms to all occurrences 1. An An assignment assignment of of LP-terms to all occurrences of of 0 [] in in F, F, 2. aa choice choice of AS. 2. of an an axiom axiom specification specification AS. Under FT F r we F under under the negative Under we denote denote the the image image of of F the realization realization r. r. Positive Positive and and negative occurrences sequent are are defined A occurrences of of modality modality in in aa formula formula and and aa sequent defined in in the the usual usual way. way. A realization negative occurrences occurrences of of 0 [] are by proof is normal normal ifif all all negative are realized realized by proof variables. variables. realization r is then 10.1. TTheorem. (Artemov [1995]) [1995]) If If rS4F, 10.1. heorem. (Art~mov F s 4 F , then A S and some normal normal realization r = = r(AS). r(AS') . specification AS r LPAsFT for some axiom axiom FT.pAs Fr .for The proof of the provides an algorithm which, for aa given given cutfree cutfree The proof of the theorem theorem provides an algorithm which, for derivation in 84, assigns LP-terms LP-terms to to all all appearances appearances of the modality r. derivation T 7" in $4, assigns of the modality in in 7". Let us agree use aa new new function symbol LZ any arithmetic formula Let us agree to to use function symbol tz The Logic Logic of of Provability Provability 499 499 these e.g., the these distinctions, distinctions, since since some some operations operations on on proofs, proofs, e.g., the proof proof checker checker !!,, really really depend depend on on the the name name of of the the argument, argument, not not only only on on its its value. value. -formula Prf(x, y) such cp, if A A proof proof predicate predicate is is aa provably provably .0'l Al-formula Prf(x,y) such that, that, for for all all ~, if "' ) holds. P A fk ~, then, for for some some n n Ee W, w, Prf(n, Prf(n, rr cp ~n) holds. A A proof proof predicate predicate Prf(x, Prf(x, y) is here here cp, then, PA y) is called normal if called normal if 11.. For k, the For every every proof proof k, the set set of of corresponding corresponding theorems theorems is is finite finite and and the the function function T(k) {i l] Prf(k, Prf(k, i)} T(k) = the the code code of of the the set set {l l)} is is provably provably total total recursive, recursive, 2. 2. For For any any finite finite set set X X of of codes codes of of theorems theorems of of PA P A there there exists exists aa natural natural number number nn such such that that X X � c_ T(n) T(n). . For V), For each each normal normal proof proof predicate predicate Prf Prf there there are are provably provably recursive recursive terms terms m(x, m(x,y), a(x, V), c(x) a(x,y), c(x) such such that that for for all all closed closed recursive recursive terms terms s, s,tt and and for for all all arithmetic arithmetic cp, 'lj; formulas formulas ~, r the the following following formulas formulas are are valid: valid: "' ) -+ Prf(s, rr cp ~a -+ ~ 'lj;"') Cn) /\ A Prf(t, Prf(t, rr cp ~7) _+ P Prf(m(s, t), rr r'lj; "' ) Prf(s, rf(m(s, t), Prf(s, rr~a ~ P Prf(a(s,t),r~7), Prf(s, cp "'n)) -+ rf(a(s, t), r cp "' ), Prf(t, --+ Prf P r f (a(s ( a ( s,, t) t ) ,, rr~cp "'n)) Prf(t, rr~o cp "'n)) -+ Prf(t, rr cp"') ~7) -+ ~ Prf(c( prf(c(rtn), ~n)n). "' ) ' ) . Prf (t , rr cp t "' ), rrprf(t, Prf(t, As nondeterministic Godel As we we have have noted noted above, above, the the nondeterministic Ghdel proof proof predicate predicate PROOF PROOF is is aa normal normal proof proof predicate. predicate. Let cation. An Let AS AS be be an an axiom axiom specifi specification. An arithmetic arithmetic AS-realization AS-realization ** of of the the LP-Ianguage LP-language has has the the following following parameters: parameters: AS, AS, aa normal normal proof proof predicate predicate Prf, Prf, an sentence letters letters by sentences of an evaluation evaluation of of the the sentence by sentences of arithmetic, arithmetic, and and an an eval evaluation of proof letters constants by closed recursive recursive terms. We uation of proof letters and and axiom axiom constants by closed terms. We .1* == put put T* T* == - (0 ( 0 ==00) ) and and ._L* - (0 ( 0 ==11) ) , , ** commutes commutes with with Boolean Boolean connectives, connectives, O O (t.· s)* s)* == - m(t* m(t*,, Ss*), (t + + s)* s)* == - a(t* a(t*,, Ss*), (!t)* == = cc(rt*n), -- Prf(t* Prf(t*,, rr FF* , "'). n). ) , (!t)* ) , (t [t] F)* == (t ( t* "' ) , ((It]IF)* We assume assume also also that that P PA G* for for all all G G eE AS'. AS. We A f-k G* Under Under any any AS-interpretation AS-interpretation ** an an LP-term LP-term tt becomes becomes aa closed closed recursive recursive term term t* t* (i.e., (i.e., aa recursive recursive name name of of a a natural natural number), number), and and an an LP-formula LP-formula F F becomes becomes an an arith arithmetic metic sentence sentence F* F*.. Also Also note note that that the the reflexivity reflexivity principle principle is is there, there, since since [t]F -~ F F [t] F -+ is under any interpretation **.. Indeed, Indeed, let is provable provable in in PA PAL under any interpretation let n n be be the the value value of of t* t*.. If If Prf(~, rr F* F* "'n)) is is true, true, then then PA P A fk F* F*,, thus P A fk Prf(n, Prf(n, rr F* F* "'n)) -+ __+F* F*.. If If Prf(n, Prf(~, rr F* F* "'n)) thus PA Prf(n, is false, false, then then PA P A fb --Prf(~, F* "'n), and again again PA P A fb Prf(~, n) -+ _+ F* F*.. Prf(n, rr FF*. "') is --,Prf(n, rr F* ) , and 10.2. Theorem. (Artemov arithmetic completeness LP AS F, 10.2. T heorem. (Artiimov [1995] [1995],, arithmetic completeness of of LP) LP) If If 'r F-Lpas F, then then PA P A f~- F* F* and and hence hence IN IN F D F* F*,, for for any any AS-interpretation AS-interpretation **.. Combining Combining theorems theorems 10.1 10.1 and and 10.2 10.2 provides provides arithmetic arithmetic completeness completeness of of 84: $4: 10.3. Theorem. 10.3. Theorem. specification specification AS. AS. If F, then some axiom If kf-Ss44 F, then PA P A fk Fr F r for for some some realization realization l'r and and some axiom 500 500 G. Japaridze and D. de Jongh G. 's translation By istic propositional into 84, By G6del GSdel's translation of of intuition intuitionistic propositional logic logic into $4, which which provides provides faithful embedding of intuitionistic intuitionistic propositional propositional logic logic in in to to 84 $4 (G6del (GSdel [1933] [1933],McKaa faithful embedding of ,McK automatically includes insey and and Tarski Tarski [1948]), [1948]), this this automatically includes an an arithmetic arithmetic completeness completeness insey result for for intuitionistic intuitionistic logic logic as as well. If one one considers considers this this in in the the light light of of the the Curry Curryresult well. If Howard term interpretation intuitionistic natural deduction (see Howard term interpretation of of intuitionistic natural deduction (see e.g., e.g., Troelstra Troelstra and and Schwichtenberg Schwichtenberg [1996]) [1996]),, then then one one notes notes that that many many more more terms terms are are used used in in the the LP-interpretation. LP-interpretation. It It seems seems worthwhile worthwhile to to search search in in this this light light for for aa naturally naturally restricted restricted subsystem subsystem of of LP. LP. The version of The logic logic LP L P is is a a version of 84 $4 presented presented in in aa more more rich rich operational operational language, language, with no no information information being being lost, since 84 $4 is is the the exact exact term-forgetting term-forgetting projection projection of of LP. LP. with lost, since A A transliteration transliteration of of an an 84-theorem S4-theorem into into LP-Ianguage LP-language may may result result in in an an exponential exponential growth of of its its length, because the the 84-derivations S4-derivations are are included included in in the the LP-formulas LP-formulas as as growth length, because proof terms. terms. However, However, this this increase increase looks much less less dramatic dramatic if if we we calculate calculate the the proof looks much complexity length of proof of complexity of of the the input input 84-theorem S4-theorem F F in in an an 'honest 'honest'' way way as as the the length of a a proof of F in in 84: $4: the the proof terms appearing appearing in in the the realization realization algorithm algorithm have have a a size linear of of proof terms size linear F the length the proof, the total the length of of the proof, so, so, the total length length of of an an LP-realization LP-realization of of an an 84-theorem S4-theorem F F is is bounded bounded by by the the quadratic quadratic function function of of the the length length of of a a given given 84-derivation S4-derivation of of F. F. 1 1 . Notions 11. Notions of of interpretability interpretability In 1-15) we In the the part part on on interpretability interpretability and and its its logics logics (sections (sections 111-15) we are are going going to to investigate investigate a a family family of of concepts concepts like like interpretability interpretability and and partial partial conservativity, conservativity, which, which, in in aa sense, sense, are are generalizations generalizations of of the the notion notion of of provability provability and and for for which which we we use use the common rst two the common name name "interpretability" "interpretability".. In In the the fi first two sections sections we we will will explain explain these these concepts concepts and and relate relate them them to to each each other. other. In In the the third third section section we we develop develop an an extension extension of of provability provability logic logic to to so-called so-called interpretability interpretability logic logic with with these these concepts concepts in mind. In will prove in mind. In the the fourth fourth section section we we will prove arithmetic arithmetic completeness completeness of of the the best bestknown known interpretability interpretability logic logic ILM I L M with with regard regard to to interpretability interpretability in in as as well well as as 111H1conservativity conservativity over over PA. PA. In In the the fifth fifth section section we we give give aa brief brief survey survey of of the the logics logics induced induced by by some some other other concepts concepts from from the the above above family. family. The The concepts concepts discussed discussed in in the the first first two two sections sections are are defined defined in in terms terms of of the the comparison included in comparison of of the the deductive deductive strengths strengths of of theories theories like like "one "one theory theory is is included in another" another" or or "one "one theory theory is is consistent consistent with with another" another".. To To compare compare the the strengths strengths of of two theories are two theories, theories, these these theories are not not necessarily necessarily to to be be written written in in the the same same language, language, it it is "interpretation" ) of language of is enough enough to to organize organize a a translation translation (("interpretation") of the the language of one one theory theory into into the the language language of of the the other other and and just just consider consider the the translated translated variant variant of of the the first first theory. While introducing introducing the theory. While the notions notions we we will will even even assume assume that that different different theories theories always coincide graphically. always have have different different languages, languages, even even if if the the two two languages languages coincide graphically. For simplicity we within the For simplicity we restrict restrict our our considerations considerations to to theories theories formalized formalized within the classical first classical first order order logic logic with with identity; identity; we we suppose suppose that that the the languages languages of of the the theories theories we we consider consider contain contain finite finite or or in in the the worst worst case case countable countable sets sets of of predicate predicate constants constants and and do do not not contain contain functional functional or or individual individual constants. constants. For For aa language language K K,, FmK Fmg denotes denotes the the set set of of formulas formulas of of K K and and St StgK the the set set of of sentences, sentences, i.e., i.e., closed closed formulas formulas of of Logic of of Provability Provability The Logic 501 501 K. IfIf DD isis aa nonempty nonempty set, set, St StD � denotes denotes the the set set of of sentences sentences of of K K with with parameters parameters K. D. More More precisely, precisely, the the elements elements of of St StD � are are pairs pairs (~, (rp, fJ)), , where where ~rp eE FFmK and ff isis in D. in m g and in D; D; we we usually usually write write ~rp((al an ) instead instead of of valuation of of the the free variables of of ~rp in aa valuation free variables a l ,, ... .. ,. , an) , xn ), fJ) if al al == ff((xl) an == f(xn). f (xn ) . (( ~rp(Xl ( X l ,, .". . . ,Xn), ) , , if x l ) , , .. ...., , an By By aa theory theory we we mean mean aa pair pair TT == (A, (A, K K)) , , where where K K isis aa language language and and A A c_ � StK. StK . The set set A A contains contains the the extra-logical extra-logical axioms axioms of of T; T; provability provability in in TT means means derivability derivability The from A in in the the classical classical predicate predicate logic logic with with identity. identity. Thus, Thus, here here we we do do not not identify identify aa from A theory with with the the set set of of its its theorems, theorems, but but rather rather with with the the set set of of its its nonlogical nonlogical axioms axioms theory (in we do particular, we we suppose suppose that that IE1 I:El is is finitely finitely axiomatized). axiomatized ) . However However we do say say that that (in particular, aa theory (A, K) isis aa subtheory subtheory of of aa theory theory TT'' == (A',K') (A', K') and and write write TT c_ � T', T', ifif theory TT== (A,K) � K' K' and and the the set set of of theorems theorems of of TT is is aa subset subset of of that that of of T'; T' ; ifif at at the the same same time time A A isis K K c_ finite, then then TT is is said said to to be be aa finite finite subtheory subtheory of of T'. T'. IfIf K K -= K', K' , we we denote denote the the theory theory finite, (A (A U U A', A', K) K) by by TT ++ T'; T' ; ifif M M c_ � Stg StK and and ~rp eE Stg, StK , we we may may also also use use TT ++ M M and and TT ++ rp (A t2 U M, M, K) K) and and (A (A t2 U {~}, {rp}, K K)) , , respectively. respectively. denote the the theories theories (A to to denote Let us be too first order for order model: model: for Let us not not be too lazy lazy to to define define the the well-known well-known notion notion of of first K-model is M = = (D, (D, G), where D D is nonempty set K, aa K-model is aa pair pair M G) , where is aa nonempty set ((of of aa language language K, D-"individuals") function that each n-place "individuals ") called called the the domain domain and and G G is is aa function that assigns assigns to to each n-place Dpredicate of K K an an n-cry n-ary relation on D, D, such such that is the predicate constant constant P P of relation Gp Gp on that G= G= is the identity identity relation. The The truth of ~rp eE St~ St � in in M, M, in in symbols symbols M M ~1= rp is defined defined in in the relation. truth of ~,, is the standard standard way: an atom (al , . . . , an) an ) is holds, truth commutes is true true in in M M iff iff Gp(ab . . . , an) an ) holds, way: an atom P P(a~,..., Gp(al,..., truth commutes with the connectives and and M ~ Vx(p(x) iff for for all all a with the Boolean Boolean connectives M I= Vxr.p(x) iff a ~E D, D, M M ~I= ~(a). r.p (a) . The The StK I M 1= M is is defined defined as And M M is said to to theory TM of aa K-model K -model M theory TM of as ({~ ~ ~rp}, } , gK) ) . . And is said ( { rp EeStglM be aa model if M M ~1= rp for each each rp A (of (of course the latter latter also also be model of of aa theory theory (A, K) K), , if ~ for ~ eE A course the for any closed theorem theorem rp implies that M ~1= ~r.p for implies that M any closed ~ of of T). T). Let Let K K and and K' K' be be languages. languages. We We may may suppose suppose that that the the set set of of individual individual variables variables of K K is is a a subset subset of of that of K' K' and and that that there there are are infinitely infinitely many many variables variables of of K' K' of that of K. Then Then aa relative into K' pair (f, (x) ) , not belonging belonging to to K. relative translation translation from from K K into K' is is aa pair (t~,a a(x)), not where: where: function which which •9 ft~ isis aa function assigns assigns to to each each n-place n-place predicate predicate constant constant P P of of K K a a bounded variables belong to formula formula p P ll((vvb l , .' ... , , vn) vn) of of K' K' whose whose bounded variables do do not not belong to K K and and whose alphabetical list whose free free variables variables are are the the first first n n variables variables of of the the alphabetical list of of the the variables variables of of K' K',, . • (x) is called the 9a a(x) is a a formula formula of of K' K' ((called the relativizing relativizing formula) formula) with with precisely precisely x x free free whose whose bounded bounded variables variables do do not not belong belong to to K. K. Henceforth (ly) " and Henceforth we we usually usually omit omit the the word word "relative "relative(ly)" and we we call call translations translations from from the the language language of of PA P A into into the the same same language language arithmetic arithmetic translations. translations. Now, Now, for for each each formula -translation of the following formula rp ~ Ee FmK F m g we we define define trp, t~, the the tt-translation of rp ~ into into K' K',, by by the following induction induction on on the the complexity complexity of of rp ~:: •9 t (x ( x ==yy)) is i s xx ==yy, , 9 for any other atom P(xl,..., xn), tP(xl,..., xn) is Pl(Xl,..., xn), G. G. Japaridze and D. de Jongh 502 502 • with Boolean ta -+ (3 , etc., 9 tt commutes commutes with Boolean connectives: connectives: t(a t(a -+ --+ (3) fl) = = ta --+ ttfl, etc., (x) IIA ta) 9 t(Vx t(Vx a) a) is is Vx(a(x) Vx(a(x) -+ ta) ta),, and and thus thus t(3x t(3x a) a) is is 3x 3x (a (a(x) ta).. • If If T T and and T' T' are are theories theories in in the the languages languages K K and and K' K' and and tt is is aa translation translation from from K K into into K' K',, we we define define the the theories theories { t
tt -- ll((T') ({
notion of The notion of translation translation is is a a formal formal analog analog of of that that of of model: model: aa translation translation The - (~, a(x))) from from K K into into K' K' in in fact fact defines defines aa K-model K-model in in the the language language K', K ~, where where a(x) a(x) (e, a(x) t= role of G; as plays the the role role of of D D and and ee the the role of G; as soon soon as as we we have have aa K'-model K'-model M' M' = -- (D', (D', G') G') plays such I M'~F 0, aa unique unique K-model such that that {a {a Ee D' D~]M ~ a(a)} a(a)} =1= ~ 0, K-model M M = = (D, (D, G) G) arises arises by by tak takn i ~F ingD {aeE D' D~ II M' M ~F ~ a(a)} a(a)} and and Gp Gp = { ((al' a l , ... .. ,. a' an) n ) Ee D Dn I] M' ~ pl(al P t ( a l ,, ... . ,. a, an)} n ) } for for ing D== {a each n-place n-place predicate predicate letter letter P P of of K; we call call this this model model the the K K-model induced by each K; we -model induced (t, M') . (t,M'). Suppose K K and and K' K' are are languages languages and and M M == (D, (D, G G)) and and M' U ' == (D', (D', G') G') are are K KSuppose and K'-models, K~-models, respectively. respectively. Then Then an an interpretation interpretation of of M M in in M' M ~ is is aa translation translation and tt from from K K into into K' K' such such that that for for all all
The Logic Logic of of Provability Provability The 503 503 Definition. 11.1. Definition. • 9T T is is interpretable interpretable in in T' T ~,, if if there there exists exists t, t, called called an an interpretation interpretation of of T T in in T' T ~such such that t(T) t(T) � a_T', that • 9 T' T' is is cointerpretable cointerpretable in in T, if there there is is t, t, called called aa cointerpretation cointerpretation of of T' T' in in T, T, such such T, if that C1 t -1 (T') � a_ T, that • 9T T is is faithfully faithfully interpretable interpretable in in T' T ~,, if if there there is is t, t, called called aa faithful faithful interpretation interpretation of of T in in T' T',, which is both an interpretation interpretation of of T T in in T' T' and and aa cointerpretation cointerpretation of of T' T' T which is both an in T, in • 9T T is is weakly weakly interpretable interpretable in in T' T ~if if there there exists exists t, t, called called aa weak weak interpretation interpretation of of T T in T' T ~such such that that T' T ~+ t(T) t(T) is is consistent consistent ((which is also also equivalent equivalent to to the the assertion assertion in which is that T T + C1 t-l(T is consistent consistent).) . that (T')') is The binary binary relation of weak weak interpretability interpretability has has a a natural natural many-place many-place generalizaThe relation of generaliza tion. Observe Observe that that T T is is weakly weakly interpretable interpretable in in T' T ~ if if and and only only if if T T is is interpretable interpretable tion. in some some consistent consistent extension extension of of T' T' which which has has the the same same language language as as T' T ~.. Instead Instead of of in pairs we can can consider consider arbitrary arbitrary nonempty of theories theories and and say say that that pairs we nonempty finite finite sequences sequences of such a a sequence sequence T1 T 1,, .. .. .., , T T~ is ((linearly) tolerant, if if there there are are consistent consistent extensions extensions such linearly) tolerant, n is T1+ "tl is interpretable in T + ,, .. ... ., , T: T + of of these these theories theories such such that that for for each each 11 � ~ G. G. Japaridze Japaridze and and D. D. de de Jongh Jongh 504 504 which contain contain the the axioms axioms of of PA. PA. Let call such such theories theories superarithmetic superarithmetic theories theories which Let us us call (again we we consider consider the the variant variant of of PA P A without without functional functional symbols; symbols; however, however, below below (again we we speak, speak, without without any any confusion, confusion, about about terms terms for for primitive primitive recursive recursive functions functions in in superarithmetic superarithmetic theories). theories). As As usual, usual, the the theorems theorems proved proved below below for for this this special special class class of of theories theories are are of of aa much much more more general general character; character; actually actually they they hold hold for for all all reasonable reasonable so-called so-called (locally) (locally) essentially essentially reflexive reflexive theories theories (see (see definition definition 12.3) 12.3).. The The main main theorems theorems that that we we are are going going to to prove prove in in this this section section establish establish that that for for such such theories theories interpretability nothing but interpretability and and cointerpretability cointerpretability are are nothing but IIJ II1-- and and LJ-conservativity, 21-conservativity, respectively (theorems 13) ; weak respectively (theorems 12.7 12.7 and and 12. 12.13); weak interpretability interpretability corresponds corresponds to to what what we we call call IIJ-consistency 111-consistency (theorem (theorem 12.8), 12.8), and and faithful faithful interpretability interpretability of of T T in in S S takes takes place exactly we have have interpretability interpretability of of T T in in S S and and cointerpretability cointerpretability of of S S in in place exactly when when we T (theorem (theorem 12.14). 12.14). For For finitely finitely axiomatizable axiomatizable theories theories the the situation situation is is considerably considerably T different. different. We We will will make make some some remarks remarks and and give give references references on on this this at at the the end end of of this this section. section. 12.1. w, a(xJ 12.1. Definition. Definition. Let Let R R be be an an n-ary n-cry relation relation on on co, c~(xl,...,x~) an arithmetic arithmetic ' . . . , xn) an formula, formula, and and T T aa superarithmetic superarithmetic theory. theory. We We say say that: that: • 9 a c~ defines defines R, R, • if , . . . , kn if for for all all kJ k~,..., k~ Ee w co,, we we have have R(kJ, R ( k l , . .. .. ,. , kn) k~) standard model of standard model of arithmetic, arithmetic, � ~ IN the IN 1= ~ a(kJ c~(kl,..., kn),, IN IN the ' . . . , kn) 9 a c~ numerates numerates R R in in T, T, • if if for for all all kJ k l ,, .. ... ., , kn kn E~ W, w, R(kJ, R ( k l , .. .. .. ,, kn) kn) ==> ==, T T fF- a(kJ a ( k l ,' .. ... ., , kn) k~),, 9a a binumerates binumerates R R in in T, T, if if a a numerates numerates R R and and -,a -,a numerates numerates the the complement complement of of R R in in T. T. We We need need some some more more terminology terminology and and notation. notation. The The formula formula class class LJ! El! is is the the set set of of arithmetic which have explicit LJ arithmetic formulas formulas which have an an explicit E1 form, form, i.e., i.e., 3x 3x 'P ~o for for some some primitive primitive recursive lId denotes recursive formula formula 'P ~o.. Similarly Similarly for for IIJ HI!.! . Simply Simply LJ E1 (resp. (resp. 111) denotes the the class class of of formulas which are I�l-equivalent to some LJ!(resp. IIJ !-) formula. formulas which are IEl-equivalent to some E1 !- (resp. IIl!-) formula. It It is is known known (see (see Smorynski Smoryfiski [1977]) [1977]) that that the the predicate predicate "x "x codes codes aa true true LJ! El!sentence" can be formalized by a LJ !-formula, which we will denote by True(x) sentence" can be formalized by a E1 !-formula, which we will denote by True(x).. This This formula proves that) formula is is such such that that (I�l (IE1 proves that) for I�l f-F- 'P each LJ!-sentence f o r each ~1 ! - s e n t e n c e 'P, ~o, IIE1 qo +-t +-~ True( True(rqpT). 'P..,). Next, Next, we we denote denote by by Regwit(y, Regwit(y, x) x) the the very very primitive primitive recursive recursive formula formula for for which which True(x) True(x) == - 3y 3 y RRegwit(y, e g w i t ( y , xx) ) "' ) is and and say say that that kk is is aa regular regular witness witness of of aa LJ!-sentence 21 !-sentence 'P ~o,, iff iff Regwit(k, Regwit(k, rr 'P ~o~) is true. true. And And kk is is said said to to be be aa regular regular counterwitness counterwitness of of aa IIJ!-sentence 111!-sentence Vz Vz 'P ~o,, iff iff kk is is aa regular regular witness of of 3z 3z -, 'P ~o.. witness 505 505 The The Logic Logic of of Provability Provability k
12.2. 12.2. Notation. Notation. For For any any arithmetic arithmetic formula formula We this notation notation to to the more common We prefer prefer this the more common dot dot notation, notation, because because it it avoids avoids the the need need to to specify specify the the free free variables. variables. 12.3. Definition. A T, the 12.3. Definition. A theory theory T, the language language of of which which contains contains that that of of PA, PA, is is essentially reflexive, said to to be be locally locally essentially reflexive, if if for for any any sentence sentence
It It is is known known that that superarithmetic superarithmetic theories theories are are globally globally essentially essentially reflexive. reflexive. At At the the same finite(ly axiomatized) same time time no no consistent consistent finite(ly axiomatized) theory theory which which satisfies satisfies the the conditions conditions ' s second of of G6del Gbdel's second incompleteness incompleteness theorem theorem can can even even be be locally locally essentially essentially reflexive, reflexive, for for otherwise otherwise such such aa theory theory would would prove prove its its own own consistency. consistency. In In fact, fact, it it is is shown shown in Visser Visser [1990J [1990] that that essential essential reflexivity reflexivity is is equivalent equivalent to to full full induction. induction. (The (The idea idea in of of the the proof, proof, by by the the way, way, is is already already present present in in Kreisel Kreisel and and Levy Ldvy [1968J [1968].).) That That local local essential essential reflexivity reflexivity is is much much weaker weaker than than global global essential essential reflexivity reflexivity follows follows from from the the following T, T following observation. observation. For For any any reasonable reasonable theory theory T, T plus plus local local reflection reflection for for T T is is easily local essential easily seen seen to to satisfy satisfy local essential reflexivity. reflexivity. However, However, by by aa result result from from Feferman Feferman [1962], T plus local local reflection reflection for for T T is is contained contained in in T T plus plus all all true true Il 1-II-sentences, [1962] l -sentences, ' T plus which which for for weaker weaker theories theories certainly certainly does does not not entail entail full full induction. induction. It It turns turns out out that that for for our our results results we we just just need need local local essential essential reflexivity. reflexivity. This This is is the the reason reason that, that, in in the the G. G. Japaridze Japaridze and and D. de de Jongh Jongh 506 506 following with "essential its local following we we will will with "essential reflexivity" reflexivity",, perhaps perhaps nonstandardly, nonstandardly, refer refer to to its local version. version. 12.4. Let K)) and K') be 12.4. Definition. Definition. Let T T := (A, (A,K and T' T ' := (A', (A',K') be theories theories and and suppose suppose that r F� c_ FmK Fmg n N FmK" Fmg,. Then Then that • -conservative over implies 9T T is is r F-conservative over T', T t, if if for for any any cp ~ Ee r Fn N StK StK,, we we have have that that T T Ik- cp ~ implies T' I- cp T'F ~,, • -consistent with 9T T is is r F-consistent with T', T', if if for for any any cp ~ Ee r Fn N StK StK,, we we have have that that T T Ik- cp ~ implies implies T ' j.l ]z -~ cp ~;; in in other other words, words, if if T T is is r F-conservative over some some consistent consistent extension extension T' -conservative over of of T' T' in in the the same same language. language. ..., Note !- and Note that that for for sufficiently sufficiently strong strong theories theories the the notions notions of of �l El!and �l-conservativity El-conservativity !- and (as (as well well as as IIl 111!and II IIl-conservativity) are equivalent. equivalent. 1 -conservativity) are (PA a (x) ) is (PA 1-:) k:) Suppose Suppose tt -= (£, (i,a(x)} is aa translation translation from ffom aa language language K K into (x ) --* into aa language language K' K' and and cp ~ Ee StK StK.. Then Then PA P A I I- Pr( P r (r~cp '~)) --* -~ Pr(3x Pr(~3x aa(x) -~ ttcp~ ') ). . 12.5. 12.5. Lemma. Lemma. Proof. Proof. Argue Argue in in PA. PA. Suppose Suppose P P is is a a proof proof of of cp ~ in in PC, P C , and and let let Xl> x l , .' ...., , Xn xn be be all all P. Let (xn ) . By variables variables occurring occurring freely freely in in P. Let then then .6. A= = a(Xl) a ( x l ) A . .. .. . A a a(Xn). By induction induction P, one pc .6. tcp and on on the the length length of of P, one can can easily easily verify verify that that It-pc A ---* + t~ and hence hence (as (as cp ~ is is closed) PC closed) P C Ik- 3.6. 3A --* ~ tcp, t~, where where 3.6. 3A is is the the existential existential closure closure of of .6. A.. On On the the other other hand, hand, -1 (x) --* 3.6.. Consequently, pc 3x (x) --* Ipc 3x k-pc 3x a a(x) --+ 3A. Consequently, Ik-pc 3x a a(x) -~ tcp. t~. -~ (PA (x ) defining (PA 1I-:):) For For any any formula formula a a(x) defining aa set set of of arithmetic arithmetic sentences, sentences, there there is is an an arithmetic arithmetic translation translation tt such such that that for for any any sentence sentence cp, ~, (a) PA PA + + Con" Con~ It- Pr,,( P r , ( r ~cp7')) --* __+tcp, t~, (a) (b) PA PA + + Con" Con, + + Compl Compl,o Ik- P r , (( r ~cp7')) H ++ tcp. t~. (b) Pr,, 12.6. Lemma. 12.6. L emma. It It is is easier easier to to explain explain the the idea idea of of the the proof proof of of this this lemma lemma than than to to give give aa strict strict ' s completeness proof. proof. G6del Gbdel's completeness theorem theorem for for the the classical classical predicate predicate calculus calculus (with (with 's identity) model. An identity) says says that that every every consistent consistent theory theory has has aa model. An analysis analysis of of Henkin Henkin's proof arithmeticly definable proof of of this this theorem theorem shows shows how how to to construct construct for for aa consistent consistent arithmeticly definable (say, (say, superarithmetic) superarithmetic) theory theory T T= = (A, (A, K) K} a a model model M M = = (D, (D, G) G),, where where both both D D and and each ned; the each relation relation G Gpp are are arithmetically arithmetically defi defined; the whole whole proof proof can can be be formalized formalized in in PA (Hilbert and P A (Hilbert and Bernays Sernays [1939]) [1939]).. As As we we noted noted above, above, to to define define aa K-model K-model in in some some language the language language of of PA) language (in (in our our case case in in the PA) means means to to give give a a translation translation from from K K into into this this language; language; for for each each concrete concrete sentence sentence cp ~,, PA P A plus plus the the assumption assumption that that T T is soon as T, cp is consistent consistent then then proves proves that that as as soon as cp ~ is is aa theorem theorem of of T, ~ is is true true in in M, M, and and the the clause clause (a) (a) of of the the lemma lemma expresses expresses just just this this fact; fact; as as for for clause clause (b), (b), it it is is an an immediate consequence consequence of of (a), (a), for for PA PA + + Com Compl~ -~ Pr Pr~( ~7) ~ Pr,, pr~(r_~ ~pT) and ' ) --* ( cp ' ) and pi" 1-k- ..., immediate o (r cp PA tcp . P A lt-- h t~ cp ~ H ~ ...-,, t~. ..., The The Logic Logic of Provability 507 507 12.7. [1971 ,1972]) (PA 12.7. Theorem. T h e o r e m . (Orey (Orey [1961] [19611, Hajek [1971,1972]) (PA f~ :) :) For For superarithmetic superarithmetic ' Hs T and S the theories theories T and S the following following are are equivalent: equivalent: T S, (i) (i) T is is interpretable interpretable in in S, (ii) (ii) for for all all m, m, S S fF-ConT. COnTSm, \.m , (iii) (iii) T T is is TI Hi-conservative over S. S. l -conservative over Proof. (i) (i)=~(ii): Suppose tt == (R, is is an an interpretation interpretation of of T T in in S. S. Let Let
-, (ii)=~(i): Let T(X) bbee aa primitive primitive recursive recursive formula formula defining defining the the set set of of axioms axioms (ii) :::} (i) : Let of T, and a(x) c~(x) the the formula formula T(X) T(X)AA Con ConT,x. as soon as condition condition (ii) (ii) holds, holds, Then, as soon as of T, and T.\.x . Then, a(x) binumerates c~(x) binumerates the the set set of of axioms axioms of of T T in in S S and, and, thus, thus, Pra(x) Prs(x) numerates numerates the the set set of of theorems of of T T in in S. According to to lemma lemma 12.6(a) 12.6(a),, there there is is aa translation translation tt such such that that S. According theorems PA + + Cona Cons fF- Pra P r s ((r ~ Taking nothing but Taking into into account account that that weak weak interpretability interpretability of of T T in in S S is is nothing but inter interpretability S, we pretability of of T T in in some some consistent consistent extension extension of of S, we get: get: 12.8. 12.8. Corollary. Corollary. equivalent: equivalent: theories T (P A f-F- ::)) For (PA For superarithmetic superarithmetic theories T and and S S the the following following are are T (i) (i) T is is weakly weakly interpretable interpretable in in S, S, (ii) ConT.\.m , (ii) for for all all m, m, S S J.L lz --1C0nT,m, (iii) -consistent with (iii) T T is is TIl IIl-consistent with S. S. -, Our Our next next goal goal is is to to find find aa similar similar characterization characterization for for cointerpretability. cointerpretability. We We need need some some preparatory preparatory lemmas. lemmas. 12.9. PA f-F-:):) Suppose 12.9. Lemma. L e m m a . (Guaspari (Guaspari [1979]) [1979]) ((PA Suppose S S is is aa superarithmetic superarithmetic theory theory and set of and r F is is aa recursively recursively enumerable enumerable set of natural natural numbers. numbers. Then Then there there is is aa � El-formula l -formula ,), ( x) such such that: 7(x) that: (i) (i) ,),(x) 7(x) numerates numerates r F in in S; S; -conservative over (ii) (ii) for for any any kk Ee W, w, if if kk rtr r, F, then then S S + --,,7')'(k) ( k ) is is �l El-conservative over S. S. 508 508 G. Japaridze Japaridze and and D. D. de de Jongh Jongh G. Proof. -formula which nes r. Proof. Let Let B(x) O(x) be be aa I;I El-formula which defi defines F. Let Let I;I!(X) Y]I!(27) be be aa primitive primitive recursive recursive formula expressing expressing that that xx is is (the (the code code of) of) aa I;I!-sentence El!-sentence and and let let -.; --~ be be aa term term for for formula the primitive primitive recursive recursive function function which which assigns assigns to to each each pair pair mI ml,, m m22 of of numbers, numbers, as as the soon as as they they code code some some formulas formulas (cl1 and and (e2, the code of the formula cl --+ e2. Finally, soon ( . Finally, , the code of the formula (1 + 2 2 let let a(x) a(x) be be aa primitive primitive recursive recursive formula formula defining defining the the set set of of axioms axioms of of 5. S. Applying Applying self-reference, we we can can construct construct aa I;I!-formula El!-formula ,(x) 7(x) such such that that self-reference, (1) PA P A f- ,(x) Regwit (y , [B( [0(x)]) Vz, tt � <_yy x ) ] ) AA'v'Z, (1) 7(x) H :33y Y ([Regwit(y, / (x) ] -'; (El !(!(z)/~ Prf~(t, [-~,7(x)] 2+ z) z) ---+ (I;I + Z) A Pr f,, (t , [, r' , [,(x)]))) r(Regwit(r, z) AA'v' Vr' Regwit (r', [7(x)])))).. r' �~ )/ The The formula formula ,(x) 7(x) expresses expresses that that there there is is aa regular regular witness witness yy of of B(x) 0(5:) and and any any I;I! Y]I!:::;y ' ,(x) sentence sentence AA with with 5 S fk- 6' ~-_ Then Then (3) S5 fk- ,(n) 7(~) H ++ !\ A{3r(Regwit(r, A { :3r( Regwit (r, r)~in) r A; 'l) A V ~ Argue in in S. 5. Suppose Suppose -,7,(n) Then, by by (3), (3), there there is (1 ~> rrA~) A 'l ) Then S5 k-e Then and S5 proves proves that that ifif 0(~) B(n) has has aa regular regular witness, witness, the the latter latter isis larger larger than than e.e. Argue Argue and in S, has aa regular regular 5, and and suppose suppose 7(~). ,(n) . Then, Then, by by (1) (1) and and the the above above remark, remark, 0(~) in B(n) has witness Then, again again by by (1),)~ (1), A has has aa and the the smallest smallest such such witness witness isis larger larger than than e.e . Then, witness and regular witness witness (smaller (smaller than than any any regular regular witness witness of of "),(~)), ,(n) ) , so so AA isis the the case. case. Thus Thus regular ,(n) k-f- AA and, and, since since we we have have assumed assumed that that S5 ++ -~ , 7(~) ,(n) k-f- ~, A, we we have have S5 kf- ~. A. This This S5 ++ 7(~) proves -q -j proves the the desired desired conservativity. conservativity. 12.10. LLemma. (Lindstrom [1984]) [1984]) (PA f- : ) For For superarithmetic superarithmetic theories theories TT and and 12.10. e m m a . (Lindstrbm (PA k-:) S, 5, there there isis aa formula/3(x) formula f3(x) such such that: that: (i) r ~n), Pri3 ( A 'l), then, then, for forsome some m, m, S5 k-f- PrT+m(ran), PrT.l.m ( A 'l ) , (i) for for all all ~,>., ififS5 ~-f- Prz( (ii) (ii) /3(x) f3(x) binumerates binumerates the the set set of ofaxioms axioms of ofTT inin S.5. 509 509 The Logic of Provability Provability Proof. Proof. Let Let X X be be the the set set of of all all the the sentences sentences f.e such such that that S S IF- Prytm PrT$m((r-ef. "'n)) for for some some !-formula 7(x) ,(x) such m. By By lemma lemma 12.9, 12.9, there there is is aa EI El!-formula such that that for for all all sentences sentences A,~,, (4) (4) if A )~E~ X, X , then then S S I~ ,(A"'), 7(r-i~n), if (5) (5) if ,(A"') if A f/-r X, then then S S + + ..., -,7(r-A ") is El-conservative El-conservative over over S. Let T(X) and and O"(x) a(x) be be primitive recursive formulas formulas defining defining the the sets sets of of axioms axioms of of T T Let primitive recursive and Applying self-reference, ,8(x) by and S, respectively. respectively. Applying self-reference, we we define define/3(x) by 13(x) =_ TT(X) h Vy, Vy, Zz � <~x(Prf x(Prf~(y, -+ ,(z)). 7(z)). (X) 1\ (x) == ,8 ". (y, [[PrB(z)] Pr;J (z)])) -+ To prove prove (i) (i),, suppose suppose S S l-mPr;J(A"') F-mPrB(can).. Clearly Clearly S S lF- rA"' c)~n � ~<m rh (unless (unless we we have have some some To pathological Gi:idel GSdel numbering) and thus thus S S+ + ..., -, ,( 7( c A A..,n)) I~_Vx(,8(x) Vx(/3(x) -+ --+ T TSrh(x)), where ..j,.m(x) ) , where pathological numbering) and denotes T(X) 1\ AXx <~ Hence S S ++ ..., ~ ,( 7(r,~A"') n) It- Pr;J( Prz(r-,~A"') n) -+ -+ Pr'T. Pr,,,~(r-)~ n) and, m. Hence j.m ( A"') and, T~-$rh(x) ..j,. m(x) denotes � r-n. since S SF-mPrz(rAn) and T is is primitive primitive recursive, recursive, since l- mPr;J (A "' ) and (6) (6) S+ + ..., ~ ,( 7(r-An) i-- Pr PrT,m (r-An). A"') . S A"') IT.j.m ( Suppose AA~/X. by (5) and (6) (as Pr PrT,m(r,~ n) E~ El E,!), S ~I- Pr PrT,m(r-ln), and Suppose f/- X . Then, Then, by (5) and (6) (as !)' S T.j.m (A "' ) , and T.j.m (A"') (i) is is proved. proved. (i) ..., T(X) and ..., ,8(x) . If If If x is is not not the the code code of of an an axiom axiom of of T, T, then then S S 1t---1T(2) and thus thus S S IF---113(2). If x x is aa code code an an axiom axiom of of T, then S S It- TT(2), and to to show show that that s I~- ,8(x) fl(2) it it is is enough enough to to T, then (X), and is . (y, [Pr;J x) , S (z)]) -+ show y, Zz (( � show that that for for all all y, ~<x), S IF- Prf" Prf~(~2, [Prz(2)]) --+ ,(z) 7(2).. This an S -proof of (z) . And And if I- y Pr;J (z) , This is is obvious obvious if if yy is is not not the the code code of of an S-proof of Pr;J Prz(2). if S SF-y Prz(2), --1 then, whence, by then, by by (i) (i),, Zz codes codes an an element element of of X X, , whence, by (4) (4),, ,(z) 7(2).. Hence Hence (ii) (ii).. In 0, and In the the sequel sequel we we will will use use the the following following convention: convention: Ai h i is is AA if if i = = 0, and is is ..., ~A A if if i ==1 1. . 12.11. 12.11. Lemma. L e m m a . (Scott (Scott [1962]) [1962]) (PA (PA IF- ::)) Suppose Suppose S S is a superarithmetic superarithmetic theory theory and and (x) is a Ell -formula. There is then a formula «(x) such that for all (arithmetically L,(x) E1 !-formula. There then formula 4(x) such that for (arithmetically v 9 defined) functions g, h :: W defined)functions w -+ --+ {O, {0, 1}, 1}, if if the the set set S9 Sg = =S S + {{v(~)g(n) w }} is consistent, consistent, v ( n ) ( n) II nn Ee w n ( then so is the set Sg,h = 59 + { « (n) h ) I n E w} . Proof. defining the Proof. Let Let O"(x) a(x) be be aa primitive primitive recursive recursive formula formula defining the set set of of axioms axioms of of S S.. We We define define the the formula formula a(x) a(x) by: by: ( ) (y)] 1\A--..., True([ [v (y)] 1\A True([ (x) Vv 3y a(x) 3y � <<.x x ( ((x x = -Iv(y)] True(In(y)])) (x = = [[-, True([u(y)]))). v (y)])) . ..., vu(y)] v (y)])) Vv (x O" -+ {O, 1} be v( n) is true, and Explanation: Explanation: Let Let tt:: W w--+ {0, 1} be such such that that t ( n) n ) ==0° iff iff u(fi) is true, and let let n) I n Ee W} S+ code S + == S S + + {v(n)t( {u(~)t(~)I w}.. Then Then the the formula formula a(x) a(x) expresses expresses that that x x is is the the code + . It of an an axiom axiom of of S S +. It is is easy easy to to see see that that of (7) (7) for -+ {O, a(x) binumerates set of for each function function 9g:: W w --+ {0, 1}, 1}, a(x) binumerates the the set of axioms axioms n (x, y) S9 Sg = - S S + {v(n)9( {,(~)9(n)) Il nneEww}} in S9 Sg.' Consequently, Consequently, Prf,, Prf~(x, y) binumerates binumerates . . " in the the relation relation "... is an S9-proof Sg-proof of of ...." in S9 Sg.' (t• • • 510 510 G. Japaridze and D. de Jongh G. (for, roughly roughly speaking, speaking, Sg S9 thinks thinks that that SS+ (for, + == SS9) a).' b e aa primitive primitive recursive recursive formula formula expressing expressing that that ss is Let Let Seq(s,/) Seq(s, l ) be i s the the code code of aa {0,1}-valued {a, 1 }-valued sequence sequence of of length length /l (i.e., (i.e., of of aa function" function: M--+ M --+ {0,1}, where of {a, I } , where M , . . . ,. /. -. 1, }l - l } ifif /l>>0 O, , and M- {=0 {O, and M=q} M = 0 ifif /l==0 )O). . Let Let Conj(s,u) Conj (s, u) be be aa term term for for u) , of of which which ss codes the primitive primitive recursive recursive function function that that assigns assigns to to each each pair pair (s, codes the (s, u), and uu codes codes finite (possibly (possibly empty) empty) {0, 1 }-valued sequence sequence ff== ((1(0) aa finite f ( 0 ) ,, ... .. ,. f, (f(m m ) )) ) and {a, 1}-valued formula A .\ which which contains contains exactly exactly one one free free variable, variable, the the code code of of the the conjunction conjunction aa formula A(0) f(~0) A f(m).. Now, construct aa formula ... A .\(0)/( Now, applying applying self-reference, self-reference, we we construct formula ((x) ((x) .\(m)f(m) /\ ... /\ A(rh) such that that such (8) (8) -.; [((x)])--+ Conj ( s , rc ) --~ [( (x) ] ) --+ /\ Prfo (y, Conj(s,r-(n) f- ((x) ( (x) ++ ++ VyVs(Seq(s,x) 'v'y 'v's (Seq(s, x) APrfa(y, PPA A b- 3t < ((x) ] )) ) . Prfo (t, Conj(s', [. ((x)]))). Conj ( s', r-(n) < y3s'(Seq(s',x)A y 3s' (Seq(s', x) /\ Prfa(t, r C) 4 -.; [--~ is proved proved by by some the type type The formula formula ((x) ((x) asserts asserts that, that, ifif ((x) The ((2) is some extension extension of of SS+ + of of the S+ + ++ +((0) or absence absence of of-,. ), ± ( (O) n/\ ..... . n/\ +± ((x ((x -- 1) (where (where +± means means the the presence presence or ) , then then S in some some extension of SS++ of of the the same type. there is is aa shorter of --((2) . ((x) in there shorter proof proof of extension of same type. Assume that the set S9 = S + {1I(n)9( I n E w} is consistent. Let Uo Assume that the set Sg = S {u(~)g(")]n e w} is consistent. Let U0 = = {S9} {Sg} and and n) UI+ 1 == {R {R ++ r((I) , R +~([)IReU~}. U~+l To prove it suffices to show by + • ((I) I R E UI } . To prove the the lemma lemma it suffices to show by R eE UI consistent. The The only element S9 is consistent consistent induction on on l1 that that each each R induction Ut is is consistent. only element Sg of of Uo U0 is by Suppose there by our our assumption. assumption. Suppose there is is aa theory theory in in UI Ut+l which is is inconsistent. inconsistent. Then Then + ! which there there is is aa theory theory in in UI Ut which which proves proves ((I) ((1) or or • -1 ((I) (([).. Let Let then then kk be be the the smallest smallest number number such that, for E UI and f- k ((I) ii.. More such that, for some some R R~UI and ii eE{{a, 0 , 1I}}, , we we have have R RF-k~([) More precisely, precisely, kk is is }-valued sequence the the smallest smallest number number such such that, that, for for some some {a, {0, 11}-valued sequence f f of of length length l1 and and theory some f- k {((n)/(n) I ° � n < l} --+ ((I) i , and some ii eE{{a, 0 , 1I}}, , we we have have S9 Sg[-kA{((n)I(n)lO<.n 1\ S9 + + 1\ {((n)/(n) II °0 � .< n < < lit.} . Below Below we we employ, employ, without without explicit explicit mention, mention, proposition proposition (7) (7),, the the primitive primitive }-valued recursiveness ) , Conj ( . , .).),, and recursiveness of of Seq(., Seq(., ..), Conj(., and the the fact fact that that the the number number of of {a, {0, l1}-valued sequences sequences of of length length ll is is finite. finite. Ca Case 1" ii -= 0. O. Then Then s e 1: (9) (9) Sg f-F Seq( Seq(r-f7, /-) /\ A Prfo Prfa(k, Conj ( (r-fn, n) -.; --~ [[((/)1). ( ( I) ] ) . 1' , rr-~ 1' , I) S9 C) Ue , Conj }-valued sequence By By our our choice choice of of k, k, there there is is no no number number tt < < kk and and no no {a, {0, 11}-valued sequence f' f' of of - k 1\ length {((n)f'(n) II °0 � length l1 such such that that S9f Sgi-k A{((~)f'(") ~ )])). - . 3t <> kk 'v's Vs 3t 3t << yy 3s 3s'(Seq(s', Prf,(t, Conj(s', F(7)) -.; 4 [[--((/-)])). . ( (1) ] ) ) . (11) The Logic of Provability 511 511 By } -valued sequence By our our choice choice of of k, k, for for each each yy � ~ kk and and any any {O, {0, 11}-valued sequence f' f' of of length length l, Sg { ( ( flY' (n) Ii °0 � Y- y 1\ ( T) , whence SgKy A{~(~)s'(~) ~ (Lindstrom Suppose TT and SS are super (Lindstrhm [1984]) [1984]) (PA (PA Ik-:)) superS, and S I- Con a . arithmetic theories, a(x) a(x) binumerates the set of axioms of T T in S, Sk-Cons. There iiss then an interpretation interpretation t of T T in S S such that, for any sentence A, => S S Ik Pra Pro(rAT). (A') . S l-k tA =? Proof. Assume Assume the the conditions the lemma lemma and Proof. conditions of of the and let let us us fix fix an an enumeration enumeration {Wn}nEw {~n},e~, of of all all arithmetic arithmetic sentences. sentences. Consider Consider the the following following recursive recursive definition, definition, where where X X is is 12.12. Lemma. 12.12. L emma. any any subset subset of of co: w:{ / wn, ~n, if if (13) n = (a) (a) T T kI-AI\{m { ~ m I ml <mn<} -n} + ~---+ , , own, r or (b) TY- I\{m l m < n} ---+ , Wn and n E X, , Wn, 9 ,, otherwise. otherwise. 1I m < n} n} isis identified (If 0, 1\ with 06 = (If n = - O, A {{m r identified with = 0.) 0.) Let (x) be Let �~(x) be the the formula formula given given by by lemma lemma 12.1 12.111 for for v(x) tJ(x) = = Pro(x) Pr~(x).. Next Next,, let let X(x, X(x, y) y) be be aa formalization formalization of of the the result result of of converting converting (13) (13) into into an an explicit explicit definition definition in in the the usual (x) and (x) to . . " and . . . E9X" usual way way using using Pro Pr,(x) and �~(x) to represent represent the the predicates predicates "T Ik- ...." and ""... X",, respectively, 1' respectively, and and let let f3(x) #(x) be be :3y 3y X(x, X(x, y) y).. Obviously, Obviously, S S Ik- Con/1 Con# and and S S Ik- Compl/ Compl#, whence by lemma 12.6(b) there is that for sentence A, whence by lemma 12.6(b),, there is aa translation translation t such such that for each each sentence S Ik- tA +-+ ++ Pr/1 Pr#((r A AT); clearly we we also also have have PA PA Ik- Pr/ Pr#( ++ f3( #(r A An) and thus thus 1 (r AAT) ' ) +-+ ' ) and S ' ) ; clearly (14) (14) S s I-tA e t~ +-+ +~ f3 # ((~A') ). . Suppose the proof Suppose now now S SK P r , (( r aAn') ). . To To complete complete the proof we we must must show show that that S S YK tAo tA. Let Let Y- Pro --+ {{0, be such such that = 11 and and gg:co : w ---+ that gg(rAT) ( A') = O , 11}} be ! (15) (15) S S+ +Y Yg9 is is consistent, consistent, where Pro (ii) g (n) I nn E9w}. where Yg Yg = = {{Pr~(~)g(")I co}. Next Next we we define define � ~" as as follows: follows: ~, = � = wn, 9 ~, if if (a) (a) (b) (b) , Wn, 9 ,, otherwise. otherwise. Pro Pr~((r A1\{:n { ~ ' [ ml <mn }<-n} - + ~--+ Wn') 7) E9Yg, or or Pr m < n} ---+ n ') � Y9 and Pr~( A {{~-:n 1[m --+-~ ~n~)~Yg and a (r 1\ 'W Pr~( A {{ r :n I[ m < n} !\ A wn 9 , ---+ --+ AAT) r yg, Pr ') �Yg, a (r 1\ {O, 11}} be Let Let h h ::co -+ {0, be such such that that w ---+ Wn ---+ A') ro ( 1\{ h ( n ) ==0O iff iff P pr~(r A { ~ ":n ll m< 512 512 G. Japaridze and D. de Jongh � (n) h(n ) 1 n Er w} lemma 12.11 the choice choice of and Yg,h = = Yg Y9 U U {{~(~)h(n)I w}.. Then, Then, by by lemma 12.11 and and the of �, sc, and let let Yg,h (15) implies implies that that (15) (16) S + + Yg,h is is consistent. consistent. (16) By induction induction on on n we we can can easily easily check check that that S + Y Yg,h X( r <J?� ~" "7, n ~), whence g,h f-~-x( ) , whence By + Yg,h Yg,h fF-fJ(<J?�'). ~(r~.7). (17) S + We now now show show by by induction induction on on n that for every every n, We that for (18) Pro Pr~((r /\ A {{<J?:" o - IIm --+ AAT)r (18) m < n} --t ') � Yg. Observe Observe that that {{ef 1I Pro Pr~((f') re 7) Ee Yg} is is closed closed under under logical logical deduction. deduction. If If n = O0,, (18) (18) holds g' Suppose k. holds by by our our choice choice of of Y Yd. Suppose (18) (18) holds holds for for n = k. Case 1:1" <J?� ~ = = Wk' ~k. Then Then either either (a) (a) Pro( Pr~( r /\{<J?:" A { ~ - li m < k} --t --+ Wk') ~k 7) Ee Yg Yg,, or (b) (b) Pr~( A {{<J?:" ~ - 1i m < kk + + 1} -+ A AT)r The subcase subcase (b) (b) just just means means that that (18) (18) holds holds I} --t Pr o(r /\ ' ) � Yg • The for n = - k+ + 11,, and and the the subcase subcase (a) (a) together together with with the the induction hypothesis also also implies implies induction hypothesis for (18) (18) for for n = = k + l1.. Case 2:2: <J?� ~ = = -, -~ Wk. ~k. Then Then Pro Pr~((r /\{<J?:" A { ~ - Ii m < k} A A Wk ~k --t --+ A') AT) Ee Yg, for for otherwise otherwise Pro k} --t Pr~((r /\ A {{<J?:" ~ - I m1 m < k} -+ -, ~ kW7k)') ( / �YYg g and and so so <J?� ~ = = Wk. ~k. But then (18) (18) for for n ==k k + 11 But then easily follows the induction This proves easily follows from from the induction hypothesis. hypothesis. This proves (18). (18). Finally, it follows A. Hence, Finally, in in view view of of (18), (18), it follows that that for for some some k, <J?� ~ = - -, -~A. Hence, by by (17), S + + Yg,h Yg,h f-I-/~(r clearly the the latter latter implies implies S + Yg,h Yg,h f-}- Pr,a( Pr#( r-~ and, fJ( -' A~7); ' ) ; clearly -' AA7) ') and, (17), (as S fk-Con#) + Yg,h Yg,h f~--~Pr#(rA7), whence, by by (16), (16), S ¥ K Pr,a( Pr#(rA7), and by by (14) (14),, (as Con,a) S + Pr,a( A') , whence, A'), and --I S ¥K tA. The The proof proof of lemma 12.12 of lemma 12.12 is is complete. complete. -t -, : 12.13. Theorem. (Japaridze (J aparidze [1993]) 12.13. Theorem. [1993]) (PA (PA fk- :)) For superarithmetic theories T T and S the following are equivalent: (i) (i) S S is cointerpretable in T, T, m( (ii) (ii) for for all A and m, m, if S S fF-PrT.lPrTsm (r A') AT),, then T T fk- A, (iii) -conservative over T. (iii) S S is is I:l El-conservative over T. => (ii) : Suppose 8) is T, and Proof. (i) (i)=>(ii): Suppose t == (7, (T,~) is aa cointerpretation cointerpretation of of S in in T, and also also Proof. m (r AAT), ') , i.e., f- Pr( Pr(r /\ the conjunction PrT~m( i.e., S SkA T T Stm m --t --+ A') A7),, where where /\ A T T Stm m is is the conjunction of of SSk-f- PrT.j. all TJm. Then, lemma 12.5, Sf- Pr(3x all axioms axioms of of T~m. Then, by by lemma 12.5, SkPr(r3x 8(x) ~(x) --t --+ t(/\ t ( A Ttm Tim --t --+ A)') A)~),, which which implies, 3x 8(x) --t implies, since since S is is essentially essentially reflexive, reflexive, S fk-3x5(x) --+ t(/\ t(A T T Stm m --t --+ A). Therefore, Therefore, SS f-k- t ((3x(x whence (as T) 3 x ( x = x) x) --t --+ (/\ (A T T Stm m --t --+ A)) A)),, whence (as t is is aa cointerpretation cointerpretation of of S in in T) 3x(x = -t (/\ TT f-~- 3x(z = x) x) ---+ (A T T Stm m --t --+ A) A),, and and T fk- A. (ii) => (i): Let from lemma (ii)=>(i): Let us us fix fix the the formula formula fJ # from lemma 12.10. 12.10. By By clause clause (ii) (ii) of of that that S. Then, Then, by lemma, lemma,/3fJ binumerates binumerates the the set set of of axioms axioms of of T T in in S. by lemma lemma 12.11, 12.11, there there is is aa translation such that that for sentences A, A, translation t such for all all sentences (19) if if S + Con,a Con# f-t>., ~-tA, then then S + Con,a Con# fk- Pr,a Pr#(rAT). (19) ( A'). Logic of Provability The Logic 513 513 We claim claim that that if if condition condition (ii) (ii) of of theorem theorem 12.13 12.13 holds, holds, then then t is is aa cointerpretation cointerpretation of of We - 0. S in in T. T. Indeed, Indeed, suppose suppose S S fFtA.. . Then, Then, by by (19), (19), S S + Conf3 Con~ f-~-Prf3( Pr~( rAT); on the the other other A "' ) ; on S - Prf3( - Prf3 hand, we we clearly clearly have have S S+ + .., -~Conf3 Con~ f~Pr~( r A"' AT). Consequently, S S fFPr~((r A"') AT).. Then, Then, hand, ) . Consequently, by lemma lemma 12.10(i) 12.10(i),, S S fF-PrT-l-m( PrTsm(rA~) for some some m, which which together together with with the the condition condition by A"') for (ii) of of our our theorem, theorem, implies implies that that T T ft-- >.A.. (ii) (ii)=~(iii): Assume (ii) (ii).. Suppose Suppose >.A iiss aa I:l-sentence E~-sentence and and S S fb- AA.. Let Let m bbee such such (ii) => (iii): Assume - PrT-l-m( that T..j.. T$ m contains contains Robinson Robinson's's arithmetic. arithmetic. Then Then S S fFPrT~m(rA~), whence, by by (ii), (ii), that A"') , whence, TF-A. T f- A. (iii)=~(ii): Suppose S S is is I:l-conservative Ex-conservative over over T T and and S SkPrT%m(rl-~). Since (iii) => (ii) : Suppose f- PrT-l-m ( >. "' ) . Since PrT~m (r >...,) A-l) is is aa I:l-sentence, E~-sentence, it it follows follows that that T T fF-PrT.j. PrT~m (r A"') 17) and and T, T, being being essentially essentially m( PrT-l -m( reflexive, proves proves AA.. -l--t reflexive, 12.14. Theorem. Theorem. (Lindstr6m [1984 [1984])(PA~ A superarithmetic superarithmetic theory T T is (Lindstrom 12.14. ] ) (PA f- ::)) A faithfully interpretable in a superarithmetic superarithmetic theory S S iff T T is III II1-conservative -conservative over SS faithfully T. and SS is El-conservative I:I -conservative over T. Proof. In In view view of of theorems theorems 12.7 12.7 and and 12. 12.13, the direction direction (=» (=~) is is straightforward. straightforward. To To Proof. 13, the is I:l-conservative prove ({=) (r , suppose suppose T T is Hi-conservative over over S S and and S S is El-conservative over over T T.. prove is III-conservative Then by by theorems theorems 12.7 12.7 and and 12.13, 12.13, we we have: have: Then (20) f- Cont (20) for for all all m, S S tContsm, -l-m , - A. (21) for for all all m m and and A, A, if if S S fF- Prt-l-m Prtsm(rAT), T fFA. ( A"') , then then T (21) Let Let f3 3 be be the the formula formula from from lemma lemma 12.10, 12.10, and and let let a(x) a(x) be be the the formula formula f3(x) fl(x) /\ A Conf3-l-x Conz~x.' Then (arguing as of theorem (20) implies a(x) (x ) binubinu Then (arguing as in in the the proof proof of theorem 12.7(ii)=~(i)), 12.7(ii) => (i)) , (20) implies that that a merates of axioms of T F- Cons. merates the the set set of axioms of T in in SS and and SS fCona . Consequently, Consequently, by by lemma lemma 12.10, 12.10, there is an an interpretation interpretation tt of of T in S S such such that that there is T in (22) t>. ~=> SS kf- Pro(rAT). (22) for for all all A, >., SS Ff- tA Pra ( >."') . To show that tt iiss also also aa cointerpretation cointerpretation of T, suppose S fThen, by t>.. Then, To show that of S S iinn T, suppose S ~-tA. by (22), P A ~f-PPra r a ( r(>'''') A -~) ~-+ Pro(rAT). (22) , S~Pr~(rA-~). S f- Pra (A"') . It It is is obvious obvious that that PA Prf3 (>'''' ) . Then, Then, by by -l lemma 12.10(i), for some whence, by by (21), (21), T T fI-- A. -t lemma 12. 10(i) , SS Ff- PrT~m(rA-1) PrT.j.m( A"') for some m, whence, A. 12.15. i n i t e l y aaxiomatized x i o m a t i z e d theories 12.15. FFinitely theories In In the the case case of of finitely finitely axiomatized axiomatized theories theories the the interpretability interpretability relations relations have have other other interesting characterizations. characterizations. E.g., E.g., aa theorem theorem due due to to Harvey Harvey Friedman (improved interesting Friedman (improved establishes that that for for finitely finitely axiomatized axiomatized sequential sequential theories theories TT and and by Visser Visser [1990]) by [ 1990]) establishes S, S, TT is is interpretable interpretable in in SS if if and and only only if if the the weak weak theory theory IAo 1.6.0 ++ EXP EXP proves proves that that the the consistency of of SS (with (with respect respect to to cutfree cutfree proofs) proofs) implies implies the the consistency consistency of of T T (with (with consistency respect to respect to cutfree cutfree proofs). proofs). 514 514 G. G. Japaridze Japaridze and D. de Jongh 12.16. 12.16. Feasible Feasible interpretability interpretability Visser introduced notion of Visser introduced the the notion of feasible feasible interpretability. interpretability. A A theory theory T T is is feasibly the language interpretable in in aa theory theory T' T' iff iff there there is is aa translation translation tt from from the language of of T T into into the x, if the language language of of T' T' and and aa polynomial polynomial function function P(x) such such that that for for any any A A and and x, if - xA, then T f- s P(x) tA o In similar manner T fFxA, then T' T'F<_ P(x)tA. In a a similar manner we we can can define define the the notion notion of of feasible feasible TIl-conservativity: polynomial P(x) Ilx-conservativity: T T is is feasibly feasibly TIl Hi-conservative over S S iff iff there there is is aa polynomial -conservative over such that for -sentence A, A, if f-xA, then then S f- S P(x)A . Verbrugge such that for any any x x and and TIl Hi-sentence if T TFxA, SF_ The The idea idea of of interpretability interpretability logics logics arose arose in in Visser Visser [1990] [1990] in in which which they they were were al already The modal modal completeness ready developed developed to to aa large large extent. extent. The completeness with with respect respect to to the the Kripke-semantics Kripke-semantics due due to to Veltman Veltman was, was, for for the the most most important important systems, systems, proved proved in in de de Jongh Jongh and and Veltman Veltman [1990] [1990].. Realizing Realizing that that one one cannot cannot cover cover the the concept concept as as well well as provability, since interpretability has nitary character, as provability, since interpretability has aa more more infi infinitary character, one one has has to to choose turns out choose primitives primitives of of course, course, and, and, somewhat somewhat surprisingly, surprisingly, it it turns out that that choosing choosing aa binary binary connective connective is is much much more more rewarding rewarding than than choosing choosing aa unary unary connective. connective. The The arithmetic t> B arithmetic realization realization of of A At> B in in aa theory theory T T will will be be that that T T plus plus the the realization realization of of plus the the realization (T plus plus A interprets T plus B), B is is interpretable interpretable in in T T plus realization of of A A (T T plus B), B or, or, alternatively alternatively (and, (and, as as we we have have seen, seen, iinn the the case case ooff PA PA equivalently), equivalently), that that T T plus plus -conservative over A. The the the realization realization of of B B is is TIl Hi-conservative over T T plus plus the the interpretation interpretation of of A. The unary unary pendant pendant "T "T interprets interprets T T plus plus A" A" is is much much less less expressive expressive and and was was studied studied in in de de Rijke 1997]. Rijke [1992] [1992].. For For aa recent recent complete complete overview, overview, see see Visser Visser [[1997]. We rst introduce We fi first introduce aa basic basic interpretability interpretability logic logic IL: IL: it it contains, contains, besides besides the the usual usual axiom ( OA -+ oA for the provability ponens axiom O [:]([:]A --+ A) A) -+ --+ OA for the provability logic logic L L and and its its rules, rules, modus modus ponens and and necessitation, necessitation, the the axioms: axioms: (A -+ t> B) (1) (1) O [-q(A --+ B) B)--+ (At> B),, -+ (A t> C) , (2) t> B) t> C) -+ (A (2) (A (At> B ) ^/\ (B (Bt>C)--+ (At>C), t> C) , (3) t> C ) /\^ ((Bt>C) B t> C ) -+ (3) (A (At>C) --+((gA V vB Bt>C), (4) (A t> B) (4) (At> B ) +-+ (((}A --+ O 0 BB) ) ,, OA -+ A t> A. (5) O (5) (}At> A. With respect treated as With respect to to priority priority of of parentheses parentheses t> t> is is treated as -+ --+.. Furthermore, Furthermore, in in this this section, section, we we will will consider consider the the extension extension ILM ILM = = IL IL + + M M of of IL IL where where M M is is the the axiom axiom (A t> B) /\ [:]Ct> o C t> B/x B 1\ oC) will write write hL (At> B) -+ --+ (A (A/~ [:]C).. We We will bIL and and hLM FIL M for for derivability derivability in in IL IL and ILM, but and ILM, but sometimes sometimes we we may may leave leave off off the the subscript. subscript. As As will will be be proved proved further further on, on, The Logic of of Provability 515 515 the the logic the logic logic ILM I L M is is the logic of of II1-conservativity IIl-conservativity of of PA, PA, and and therefore therefore also, also, as as shown shown in section, its in the the previous previous section, its interpretability interpretability logic. logic. We We will will not not treat treat here here the the logic logic ILP ILP (A [> B) which [> B) which arises arises by by extending extending IL IL by by the the scheme scheme (A (At> B) -+ --4 0 D(At> B) that that axiomatizes axiomatizes the interpretability logic of of the the most most common common finitely finitely axiomatizable axiomatizable theories theories (Visser (Visser the interpretability logic [1990] using aa modal [1990],, using modal completeness completeness result result of of de de Jongh Jongh and and Veltman Veltman [1990]). [1990]). 13.1. 13.1. Lemma. Lemma. (a) hL Fir.[3~ --4 (A (At> B),, (a) O-, A A -+ [> B) A [> A, (b) (b) f-1L Fir. A Av vO (~At> A, (c) -, A . (c) hL FIr. A A t[> >A A Ai\ o [3-~A. Proof. The The parts parts (a) (a) and and (b) (b) are are easy. easy. For For part part (c) (c) use use lemma lemma 22.1(j) to obtain obtain Proof. . 1 (j) to f-Ft.L A (A i\A O-,A) i\ o-,A) A -+ --4 (A [3~A) V v (}(A [3~A).. Then Then use use the the necessitation necessitation rule, rule, axiom axiom (1), (1), part part O (A A -l (b) and axiom axiom (2) (2).. -q (b) and 13.2. 13.2. Corollary. Corollary. B, A and AA [>t>B (a) formulas A A [> t> B, A/xi\ 0-, n-~ A A [> t> B B and B/xi\ 0-, [3-, B B are lL-equivalent. IL-equivalent. (a) The formulas (b) (b) The formulas formulas A A [> t> ..1 2- and 0 [3--,A A are IL-equivalent. Proof. (a) lemma 13.1 (c) and its converse, which is Proof. (a) By By lemma 13.1(c) and its converse, which is derivable derivable from from axiom axiom (1), (1), and transitivity transitivity of of [> t> (axiom (axiom (2)). (2)). and (b) The The direction direction from from right right to to left left follows follows from from lemma lemma 13. 13.1(a). The other other (b) 1 (a) . The (i) and direction direction is is obtained obtained by by using using axiom axiom (4) (4) with with ..1 2_ for for B B,, lemma lemma 2.1 2.1(i) and transitivity transitivity -l-t of of [> t>.. -, An IL-frame 13.3. 13.3. Definition. Definition. An IL-ffame (also (also Veltman-frame) Veltman-ffame) is is an an L-frame L-frame (W, (W, R) with, with, for for each each w Ee W, W, an an additional additional relation relation Sw S~,, which which has has the the following following properties: properties: {w' Ee W W IIw wR (i) (i) Sw S~ is is aa relation relation on on wt wJ" = = {w' R w'}, w'}, (ii) (ii) Sw S~ iiss reflexive reflexive and and transitive, transitive, (iii) if w', w$ and and w'R w'R w", then then w' w'S~w". w', w" Ee wt Sww". (iii) if We write SS for We may may write for {Swlw {S~lw E~ W W }}. . 13.4. W, R, S) combined 13.4. Definition. Definition. An An IL-model IL-model is given given by by an an IL-frame IL-frame ((W, combined with with forcing relation relation IIIF with with the the clauses: clauses: aa forcing uu llVv(uR I- A ) , I~-oA rnA {=} ~ Vv(uR v =} =~ v iIFA), u IIlFAt> - A [> B Vv( I-A =} vSuw and I- B)) . B {=} ~ V v (uu R R vv and and v IIFA =~ 3w( 3w(vSuw and w IIFB)). 13.5. 13.5. Definition. Definition. 1. If then we S) and A for 1. If F F is is aa frame, frame, then we write write F F F ~ A A iff iff F F = = (W, (W, R, S) and w IIIFA for every every W and and every every IF on F. F. w Ee W II- on K. 2. ff F 2. IIff K K: iiss aa class class ooff frames, frames, we we write write K K: F ~ A A iiff F F ~ A A for for each each F F Ee/C. 3. KM the class the class 3. /CM,, the class of of ILM-frames, ILM-frames, is is the class of of IL-frames IL-frames satisfying satisfying (iv) (iv) if u Sw v R z, then then u R z. 4. 4. A Ann ILM-model ILM-model iiss an an IL-model IL-model oonn an an ILM-frame. ILM-frame. 516 516 G. Japaridze Japaridze and and D. D. de de Jongh Jongh G. The scheme scheme M M characterizes characterizes (see (see section section 2) 2) the the class class of of frames frames KM K:M;; that that is is the the content content The of part part (b) (b) of of the the next next soundness soundness lemma. lemma. of 13.6. Lemma. Lemma. For all all IL-frames IL-frames F, F, 13.6. For (a) For For each each A, A, if if hL F IL A, A, then then F F F ~ A. A. (a) M. (b) F F F ~ ILM I L M iff iff F F Ee K ~M. (b) M F~ A. (C) For For each each A, A, if if hLMA, FILMA, then then K )~M A. (c) As before, before, in in the the case case of of L, L, we we work work inside inside aa so-called so-called adequate adequate set. set. It It is is convenient convenient to to As use the the fact fact that that 0 [] is is definable definable in in IL IL in in terms terms of of t> t> :: DA []A is is IL-equivalent IL-equivalent to to -, -1A E>.1 _k use A t> . 2 (b)) . This (corollary (corollary 13 13.2(b)). This means means that that we we can, can, in in constructing constructing countermodels, countermodels, restrict restrict our attention attention to to formulas formulas that that do do not not contain contain D [3.. The The entire entire following following discussion discussion will will our be based based on on the the presumption presumption the the formulas formulas discussed discussed do do not not contain contain D [3.. be The ned symbol. The other other side side of of the the coin coin is is that that this this will will allow allow us us to to use use 0 [] as as aa defi defined symbol. The A will The most most convenient convenient way way to to this this turns turns out out to to be be the the following: following: 0 ~A will be be an an abbreviation of of -, -~ (A (AC>_I_) and DA []A will will then then abbreviate abbreviate the the formula formula rvOrvA ~~A (i.e., abbreviation t> .1) and (i.e., rvA ,,~A t> E>.1) _l_).. We We need need to to adapt adapt the the concept concept of of adequate adequate set set to to the the new new situation. situation. es the 13.7. 13.7. Definition. D e f i n i t i o n . An An adequate adequate set set offormulas of formulas is is aa set set (I) that that satisfi satisfies the following following conditions: conditions: 1. (I) is is closed closed under under taking taking subformulas, subformulas, 1. 2. if if A AE e , (I), then ,,~A Ee , (I), 2. then rvA _l_t> E>._L (I), 1 Ee , 33.. .1 -formula in At> 4. A E>B B Ee (I) if if A A is is an an antecedent antecedent or or succedent succedent of of some some t> E>-formula in (I),, and and so so 4. is B B.. is is is an set, then then A B ~E (~ iff iff both both 0 are in 13.8. Lemma. If (~ 13.8. Lemma. I.f an adequate adequate set, A t> E>B ~ AA and and ~0BB are in in case contains contains no no doubly doubly negated iff both and DrvB are (~ (and (and in case (~ negated formulas) formulas) iff both DrvA [3,,~A and [],,~B are in (~.. in is obvious that each each formula formula is is contained contained in in aa finite finite adequate adequate set. In proving proving It is It obvious that set. In completeness we we can can of of course course restrict restrict our our attention attention to to formulas formulas without without double double completeness negations, and and will will therefore therefore be be able able to to use use adequate sets with with formulas formulas without without negations, adequate sets double negations, negations, so so that that we we can can apply the last last part of lemma lemma 13.8. 13.8. We will write write part of double apply the We will ILS remarks apply apply to LM. ILS if if our our remarks to both both IL IL and and IILM. Let Fr and and A .0. be be maximal maximal ILS-consistent ILS-consistent subsets subsets of of some some 13.9. D Definition. 13.9. efinition. Let Then Fr << A .0. ~¢=:::} for for each each rnA DA eE F, r, []A, DA, AA eE A, finite adequate adequate O. . Then and, for for some some finite b., and, r, []A DA eE A. b.. In In this this case case we we say say that that A .0. isis aa successor successor of of Fr (see (see the the proof proof for for LL DA r� F, EIA of 2.4). of theorem theorem 2.4). 13.10. D Definition. Let Fr and and A .0. be be maximal maximal ILS-consistent ILS-consistent subsets subsets of of some some 13.10. efinition. Let given adequate adequate O. . Then Then A.0. isis aa C-critical C -critical successor successor of of Fr (F (r << cc A) iff given b.) iff (i) b., (i) Fr << A, (ii) ~> C (ii) ,,,A, rvA, C:],,,A DrvA eE A.0. for for each each AA such such that that AAt> C eE F. r. The Logic Logic of of Provability Provability The 13.11. Lemma. Lemma. 13.11. 517 517 If r P< a n d Abo<< O ,e, then t h e n Pr< Suppose r F is is maximal m a x i m a l ILS I L S --consistent c o n s i s t e n t in in (~ and and , - -(B ( B r> DC C)) Ee Pr.. Suppose Then T h e n there there exists exists aa C C --critical c r i t i c a l successor successor bo A of of rP,, maximal m a x i m a l ILS I L S --consistent c o n s i s t e n t in in , ~ , such such that B B E~ bo. A. that Proof. Let Let r, F, <1> (I), B and and C C satisfy satisfy the the conditions conditions of of the the lemma. lemma. Take Take bo A to to be be aa Proof. ,B 13.12. 13.12. Lemma. Lemma. maximal ILS-consistent ILS-consistent extension extension of of maximal { D , ODI DDID D E~ r} F} U tO{{O D~,,A, ,,~A[AE>C F} U tO{B, {B, OrvB}. D,.~B}. {D, OD rv A, rvAIA r> C Ee r} Note Note first first that that the the adequacy adequacy of of ff~ ensures ensures that that all all the the formulas formulas in in bo A are are indeed indeed available, -critical successor available, and and second second that that such such aa bo A,, if if it it exists, exists, is is aa C C-critical successor of of r. P. (It (It is C and, is aa successor, successor, because because OrvB D,.~B IL-implies IL-implies B B r> t>C and, hence, hence, cannot cannot be be aa member member of of r.) F.) To To prove prove that that such such aa bo A exists exists it it is is sufficient sufficient to to prove prove that that the the above above set set is is IL-consistent. Suppose Suppose not. not. Then Then there there exist exist AI A ~, ,,. .' ." , Am Am and and DI D 1 ", . .'. ", Dk Dk with with IL-consistent. DI D1," . '. ". , Dk, Dk, OD [-ID1, nDk, , --, AI A1,, '. ". . , -n Am, Am, O, c:l--,AI A~,, .. .. .. ,, o rn--, A m , B, B , O, [::]--,B B r..l F_L , Am, ) , .. .. .. ,, ODk, or equivalently equivalently or (AI vv .. .. . . vV Am) D 1,, .. .. .., , Dk, Dk, ODI, v1D1,..., [::lDkr FB BA ^ O [:1-, B -+ -+ AI A~ v v ..... . v v Am Am v v O O(A1 Am).. DI . . . , ODk ,B Applying what what we we know know of of L L gives gives Applying (AI v ..... . v Am vv O Am)). E:]D1,..., rqDkFr O D(B/~ D - BB- +-+ A Al l V v ..... . v yAm O(AlV yAm)). ODI, . . . , ODk (B A O, Axiom implies Axiom (1) then implies (1) then (AI vv ... ... v Am). v Am vv O ODk B^A O ODI, O D 1 , .. .. .., , D D ~r FB [:]-IBt>A1 v ..... . yAm (}(A~ vAin). , B r> AI v From lemmas and and axiom axiom (2) it follows that (2) it follows then then that From lemmas . . . yAm. v Am. D D 1 , .. ... ., , nODk D k FrBBt >r>AAI 1 vv ... ODI, (3) that that Given that that AAlt t >r>CC, . ,mAm we also also have, have, by by using using axiom axiom (3) Given , . .. . ., A t > Cr>eC F , E r , we r Fr A1 Al vV ... Am r> C. So, So, finally, finally, we we obtain obtain Fr Fr B Btr> C which which contradicts contradicts the the . , . vV Amt>C. F >C consistency -~ --I consistency of of F. r. 13.13. LLemma. Let BBEr>> CC ~E F. r . Then, Then, ifif there there exists exists an an EE-critical successor Abo of of 13.13. e m m a . Let - c r i t i c a l successor r with with BB ~E Abo, , there there also also exists exists an an EE- -critical successor Abo'' of of Pr with with C, C, Do~,C F c r i t i c a l successor C ~E Abo' '. . Proof. Suppose Suppose B, B, C, C, E, E, Fr and and A bo satisfy satisfy the the assumptions assumptions of of the the lemma lemma and and there there Proof. is no no such such A'. bo' . Then Then there there would would be be D ODI, ODn eE F, Fk r> EE eE Fr r, and and F1D FI r> EE,, . .. .. ,. , FkD is D 1 , .. ... ., , [::lDn such such that that Dn , DOD DI1 ," . .'. ," Dn, DIt ,,... .. ,. , I::]D~, ODn , -~ , Fk, Fk, [:]--1 O , F~, FI , .. .. .. ,, C]-~ o , Fk, Fk, C, C, D-~C O,C Fr I..l , EFIl ,, ... .. , --1 D · , and, and, therefore, therefore, D ~ n, , D D 1 ,, ... ·. ,· , K I Dn~rF C C ^A D C-~ ~ V DI1 ,, ..... ,. D, D ODI OD O~,C -+F FI . . . vF~ V Fk vv (}(F~ . . . yFk), v Fk), v ... O (FI vv ... E. Since Since BB and and EE are are respectively respectively an an antecedent antecedent and and which as as before before implies implies Fr Fr BB Dr> E. which , the the adequacy adequacy conditions conditions imply imply then then that that this this succedent of of some some D-formula r> -formula in in (I), aa succedent can be be strengthened strengthened to to BBDr> EE eE P. r . As As Abo isis supposed supposed to to be be an an E-critical E-critical successor successor of of can F, --] r, this this implies implies ,.~B rvB ~E Abo and and we we have have arrived arrived at at aa contradiction. contradiction. --I 518 518 G. Japaridze daparidze and D. de Jongh G. 13.14. Completeness and and decidability then there 13.14. Theorem. T h e o r e m . ((Completeness decidability of of IL IL)) IfYIfJZ ILA, ILA, then there is is a a -model K finite finite IL IL-model K such such that that K K .JC ~ A. A. P r o o f . Take Take some some finite finite adequate adequate set set 9 containing containing A, A, and and let let r F b bee aa maximal maximal Proof. IL-consistent subset of (I) containing containing .-vA. ~ A . The The intuitive intuitive idea of the construction of of idea of the construction IL-consistent subset of the model the set successors of the model is is to to divide divide the set of of successors of each each constructed constructed world world w, starting w, starting with r with F,, into into different different parts, parts, each each part part containing containing the the E-critical E-critical successors successors w w for for some some !>-succedent E in in the the adequate adequate set. For occurrences occurrences of of the the same same maximal maximal consistent consistent r> -succedent E set. For set in in different different parts we use use distinct The Sw Sw are are defined defined to to be be the the universal universal set parts we distinct copies. copies. The relation inside the E-critical but to relation inside each each part part consisting consisting of of the E-critical successors successors for for some some E E,, but to be be such no other such as as to to make make no other connections connections between between worlds. worlds. Then Then lemmas lemmas 13.12 13.12 and and 13.13 13.13 give give the the theorem theorem rather rather straightforwardly. straightforwardly. With With some some care care this this program program can can be be executed, executed, but but we we take take a a slightly slightly more more complicated complicated road road that that points points the the way way to to the the completeness completeness proof proof for for ILM I L M where where the the straightforward straightforward manner manner does does not not work. work. with l:. consistent Set the smallest Set Wr Wr to to be be the smallest set set of of pairs pairs (l:., (A, T) T) with A aa maximal maximal consistent subset of of 9 and and T T a a finite finite sequence sequence of of formulas from 9 that that satisfy satisfy the following formulas from the following subset requirements: requirements: ((i)i ) rr <, ~, the the length length of of which which does does not not exceed exceed ((ii) ii) TT iiss aa finite formulas from the depth depth of minus the the depth So, e.g. with the the of r F minus depth of of l:. A.. ((So, e.g. r F is is only only paired paired off off with the empty sequence. sequence.)) empty It that Wr l:., if successor of then l:.' It is is clear clear that Wr is is finite, finite, since, since, for for any any A, if l:.' A ~ is is aa successor of l:. A,, then A~ has l:.. We (w)o < (w')o and has fewer fewer successors successors than than A. We define define R R on on Wr Wr by by w wR R w' w' iff iff (w)0 < (w')0 and required properties properties check (I) and (w)l c_ (w'h (w')l.. The The required check out out easily. easily. Let Let u u Swv S~v apply apply if if (I) and (w h c::;: (II) hold ((writing writing *9 for (II) hold for concatenation concatenation):) : (I) U, (I) u, vv Ee wt, wt, (II) either C) H C ) H' either (wh (w)~ = = (uh (u), c::;: c_ (vh (v),,, or or (uh (U)l = = (wh (w), *9 ((C) 9T and and (vh (v), = = (wh (w), *9 ((C) 9T' for in the the latter latter case -critical successor successor of for some some C, C, T, T, T T'~,, and and if if in case (u)o (u)0 is is a a C C-critical of (w)o, (w)0, then then so so is is (v)o. (V)o. Let i) - ( iii) Let us us check check that that under under this this definition definition the the Sw S~ will will have have the the properties properties ((i)-(iii) required by required by definition definition 13.3: 13.3: (i) (i) That That Sw S~ is is a a relation relation on on wt wJ" is is instantaneous. instantaneous. ((ii) ii ) Reflexivity Reflexivity and and transistivity transistivity of of Sw S~ are are also also easy easy to to check. check. ((iii) iii ) If w'R w", then immediate. For If w', w', w" w" eE wt w$ and and w ' R w", then (I) (I) is is immediate. For (II (II)) it it suffices suffices to to recall recall that successors successors of successors are lemma 13.11). that of C-critical C-critical successors are C-critical C-critical ((lemma 13.11). Finally, ff pp Ee (w)o. Finally, we we define define w w If-p IFp iiff (w)0. We We will will now now prove prove that, that, for for each each B B Ee (I) and and w Wr If- B iff the length length of w Ee W r,, w w lFB iff B B eE (w)o, (w)0, by by induction induction on on the of B B.. Of Of course, course, the the connectives connectives are are trivial, trivial, so so it it suffices suffices to to prove prove that that B B r> >C c Ee (w)o (~)o � * = , Vu(w w(~ R Ru ~ 1\ ^B B Ee (u)o (~)o -t -~ 3v(u 3v(~ Sw S~ v~ 1\ AC C Ee (v)o)). (v)0)). � : Suppose r Suppose B B r> t> C C f/-r (w)o. (w)0. Then Then ~ (B (B r> E>C) C) Ee (w)o. (W)o. We We have have ttoo show show that, that, for for some some uu with with w with (w)o wR R u, u, B B Ee (U)o (u)o and and Vv Vv (u (u SwV Swv -t --+ .-v ~ C C Ee (V)o) (v)0).. Let Let l:. A with (w)0 < 519 519 The Logic Logic of of Provability given by by lemma lemma 13. 13.12, and take take u u to to be be (bo, {A, (wh (w)l*{CI}. It is is clear clear that that u u fulfills fulfills the the given 12, and * (C) ) . It requirements. requirements. ==;. C Ee (w)o : . : : Suppose Suppose B B [> t>C (w)o.. Consider Consider any any u u such such that that B B Ee (u)o (u)0 and and w R R uu,, and and first assume assume (uh (u)l = = (w) ( w )lI*, (E) { E } *, TT and and (u)o (u)0 is is an an E-critical E-critical successor successor of of (w) (W)o. By o . By first lemma 13.13 13.13 we we can can find find an an E-critical E-critical successor successor bo' A' of of (w)o (w)0 with with C C Ee bo' A'.. It It is is clear clear lemma * (E) )} is that that vv = = (bo {A',' , (wh (W)l,{E} is a a member member of of Wr Wr and and fulfills fulfills all all the the requirements requirements to to make make uU~wV. Swv. If (uh (u)l = = (w ( wh ) I*, {( E E }) ,*TT but but (u)o (u)0 is is not not an an E-critical E-critical successor successor of of (w)o (w)0,, then then we we find find If aa successor successor bo A'' of of (w)0 with C C Ee N A' by by using using axiom axiom (4) (4) instead instead of of lemma lemma 13.13. 13.13. Again Again (w)o with it is is clear clear that that vv = = (bo', {A', (w (w)l,{E}} is a a member member of of Wr Wr and and fulfills fulfills all all the the requirements requirements h * (E ) ) is it to make make u u Swv Swv.. The The final final case case is is that that (uh (u)l = = (W) (w)l.l . In In that that case case also also we we apply apply axiom axiom to -l (4) to to obtain obtain bo' A' with with C C E~ bo' A' and and take take vv = = ({A', (w)~}. -~ (4) bo', (W) l) ' 13.15. Theorem. T h e o r e m . (Completeness (Completeness and and decidability decidability of of IILM) L M ) If If•J.L ILMA, ILMA, then then there there 13.15. is aa finite finite ILM-model K such such that that K K .Ii J~ A. A. ILM -model K is The main main problem problem iinn the the proof proof of of this this theorem theorem is is the the following. following. To To apply apply the the The characteristic axiom axiom (A (A [> ~>B) B) -+ --+ (A (A/~ 9 [> t> B B/~A P C ) we we seem seem to to be be forced forced to to add add the the characteristic A DC DC) succedent succedent of of this this formula formula to to the the adequate adequate set set whenever whenever we we have have the the antecedent. antecedent. A A straightforward definition definition of of adequate set for for the case of of IILM L M would therefore lead lead straightforward adequate set the case would therefore adequate sets to to be be always always infinite, infinite, which is of of course unacceptable. After After some some adequate sets which is course unacceptable. searching we are are lead to the following defi definition. searching we lead to the following nition. An 13.16. Definition. Definition. An ILM-adequate ILM-adequate set set (I) is is an an adequate adequate set set that that satisfies satisfies the the 13.16. additional condition: condition: additional if B B t> t> C' C' such such that if [> C, C, []D D D Ee ~, , then then there there is is in in (I) aa formula formula B' B' [> that B' is C' to to C/~ C A DD. DD. is ILM-equivalent ILM-equivalent to to B/~ B A []D DD and and C' B' Even though we require require only only equivalents equivalents to present in course no longer Even though we to be be present in (I) it it is is of of course no longer evident of formulas contained in in a a finite I L M - a d e q u a t e set, evident that that each each finite finite set set of formulas is is contained finite ILM-adequate set, since each each newly newly constructed B/~A [::]D gives rise rise to to a a new new [:]-formula: B/~A ODC> _L.. since constructed B DD gives D-formula: B DD [> ..l But we will will show show that To make we But we that this this iiss nevertheless nevertheless true. true. To make iitt easier easier on on ourselves ourselves we assume that that in in our formula A assume our formula of t>-formulas the A all all antecedents antecedents and and succedents succedents of [> -formulas have have the form B AD B, except ..l . In In view 13.2 ( a) this this is is not an essential form B A [:] ,,~ B, except for for _1_. view of of corollary corollary 13.2(a) not an essential The restriction restriction is is not not really really necessary, necessary, see Berarducci [[1990].) 1990] .) restriction. restriction. ((The see Berarducci rv Each in an an ILM-adequate set (~ Each formula formula AA is is contained contained in ILM-adequate set that that contains only only aa finite finite number number of contains 9 ILM - equivalence classes. classes. 13.17. emma. 13.17. LLemma. be the the set set PProof. r o o f . Let Let (I) be be the the smallest smallest IL-adequate IL-adequate set set containing containing A. A. Let Let W9 be of We obtain by of antecedents antecedents and and succedents succedents of of t>-formulas [> -formulas in in ~ including including _l_. ..l. We obtain ~* W' by closing ~W off off under under the the operation operation that that forms forms D D nA EE from from each each formula formula D D in in the the class class closing some and each each formula formula EE that, that, either either is is aa [:]-formula D -formula in in ~, , or or is is of of the the form form [] D ~ FF for and for some W' contains contains only only aa finite finite number number of of equivalence equivalence claim is is that that ~* FF in in the the class. class. The The claim classes. L M - a d e q u a t e set classes. Given Given that that claim claim we we can can construct construct aa finite finite IILM-adequate set by by joining joining to to rv 520 520 G. Japaridze Japaridze and D. de Jongh * , and the set of in w the subformulas subformulas of of a a finite finite set of representatives representatives of of all all equivalence equivalence classes classes in ~*, and finally adding adding all all the the interpretability interpretability formulas formulas combining combining two two members members of of this this finite finite finally set set of of representatives. representatives. It the claim. induction on It remains remains to to prove prove the claim. This This will will be be done done by by induction on the the cardinality cardinality ), the So, we that cardinality of of W ~. . If If that cardinality is is 11 (i.e., (i.e., W9 = -- {..L} {2_}), the result result is is obvious. obvious. So, we can can assume assume the form that 1. We that the the cardinality cardinality is is larger larger than than 1. We note note that that each each element element of of w ~** is is of of the form B W. That B 1\ ^ 0 rn ,,, B B 1\ ^ OCI []C1 1\ ^ ..... . 1\ ^ oCk []Ck,, with with B B 1\ ^ 0 [] ,,, B B from from ~. That 0 [] ,-, B B is is a a member member of of this this conjunction conjunction means means that that in in the the Ci Ci's's all all occurrences occurrences of of B B 1\ ^ 0 [] ~ B B can can be be replaced replaced by by ..L. _l_. Also Also one one will will recognize recognize that that B B 1\ ^ 0 El ~,, B B will will only only be be thrown thrown in in by by the the operation ( B 1\ ... . Replacing operation into into the the Ci Ci in in conjuncts conjuncts of of the the form form ...., -~(B ^ 0 [] ~,, B B 1\ ^ ...). Replacing those those occurrences occurrences of of B B 1\ ^ 0 [] ,-, B B by by ..L _L means means that that one one can can drop drop the the whole whole conjunct conjunct and and keep keep an equivalent equivalent formula. formula. If one drops drops all all those those conjuncts conjuncts containing containing B B/~1\ 0 [] ~, B, then an If one B, then the the resulting resulting formula formula is is of of the the form form B B ^1\ 0 [] ,-, B B 1\ ^ ODI C:ID~ 1\ ^ ..... . 1\ ^ ODm E]Dm with with B B 1\ ^ 0 [] ,-, B B not in the the Di This means the Di been constructed not (relevantly) (relevantly) occurring occurring in Di.. This means that that the Di have have been constructed from -formulas in from the the O []-formulas in 1> (I) and and the the other other elements elements of of W ~. . Thus, Thus, by by the the induction induction hypothesis, hypothesis, there there are are only only a a finite finite number number of of such such Di Di (up (up to to equivalence) equivalence) and and that start start with hence only finite number of equivalence equivalence classes classes of of elements elements of of w ~** that with hence only aa finite number of B B. The the other W, so B 1\ A0 [] ~,, B. The same same holds holds for for each each of of the other elements elements of of ~, so that that the the resulting resulting -l set set is is finite. finite. -~ rv rv rv rv rv ) rv rv rv rv rv rv Proof finite ILM-adequate P r o o f of of theorem t h e o r e m 13.15. 13.15. Take Take some some finite ILM-adequate set set 1> (I) containing containing A A and and some maximal maximal consistent define both some consistent subset subset r F of of 1> (I) containing containing rvA. ~,A. We We define both WI' Wr and and R as the previous This time, holds as as in in the previous proof. proof. This time, however, however, we we let let u S~v apply apply if if (I) holds as well well as as (II') and and (III), u Swv (I) R (II') (III), = (wh * (C) ) * (C) ) and (II') (II') (uh (U)l � c_ ((V)l, and if if (uh (u)t--(W)l (C) H 9T and and (vh ( v ) l -= ((w)l (C) H 9T'' for for some some C, -critical successor C, T, r, T T',' , and and (u)o (u)0 is is a a C C-critical successor of of (w)o, (w)0, then then so so is is (v)o. (v)0. (III) member of (III) each each oA []A Ee (u)o (u)0 is is also also a a member of (v)o, (v)0, That under this definition the Sw will have the properties That under this definition the Sw will have the properties (i)-(iii) (i)-(iii) is is shown shown in in almost the same manner as that the the property required by almost the same manner as before; before; that the Sw Sw has has the property (iv) (iv) required by definition definition 13.5 13.5 is is shown shown as as follows: follows: Suppose that must T')Sw(i�.", T")R (f', a ) . We Suppose that (b,.', (A',T'ISw(A",T")R (F',a). Wem u s t show show (b,.', ( A ' , ~T')R ' ) R (f', ( r ' , aa) ). . That That ' T a, is T' � C a, is immediate. immediate. That That b,.' A' < < F', follows from from b,. A"" < < f' F' combined combined with with the the fact fact that, that, f', follows -formulas are by by (III), (III), O D-formulas are preserved preserved from from A' to to A ' . Naturally, define w I- p iffiff pp Ee (w)o, Naturally, we we again again define w IIFp (w)0, and and it it will will be be sufficient sufficient to to prove prove that, that, - D iff for for each each D D Ee (I), w w IIIFD iff D D Ee (w)o. (w)0. The The only only interesting interesting case case is is the the one one that that D D is is C, i.e., B B r> D C, i.e., we we have have to to show show that that B (w)o Vu(w (u)o 3v(uSwv 1\^ C B >r> C C Ee (w)0 W(w Ru R 1\ ^B B E (u)o v(u C Ee (v)o). (v)0). V l' * W l b,.' * b,." . 1>, {= :: r � --+ Basically as proof for Basically as in in the the proof for IL. IL. ==} ~ : : Assume Assume that that B B r> DC C ~E (w)0, and and that that u is is such such that that w R u and and B B ~E (u)0. Let Let {{ [ODI ] D 1 ,, .. ... ., , ODn} DOn} be be the the set set of of O-formulas []-formulas in in (u)0. By By axiom axiom M M (see (see proposi proposition 1>, (w)0 will with B' tion 2.1 2.1 (d)) (d)) and and the the adequacy adequacy of of (I), will contain contain aa formula formula B B'I> C' with B' and and C' C' ' r> C' respectively DI 1\ Dn and respectively ILM-equivalent ILM-equivalent to to B B 1\ ^ O []D1 ^ ..... . 1\ ^ o []Dn and C C 1\ ^ ODI []D~ 1\ ^ ..... , 1\ ^ OD o D nn. - (w)o, (w)o u (u)o. wRu (u)o. The Logic Logic of of Provability Provability The 521 521 Let the case and (u)0 Let us us just just treat treat the case that that (u)l (uh == (w)l (wh ** (E> (u)o is is an an E-critical E-critical (E) .*TT and successor successor of of (w)0. (w)o . (The (The other other cases cases are are easy, easy, given given our our experience experience with with IL.) IL.) We We 13.13, with with (w)0, (w)o , (u)0 (u)o and and B'C> B' r;, C' C' as as input, input, an an E-critical E-critical can find, find, by by lemma lemma 13.13, can /:l' of of (w)0 (w)o with with both both CC and and [:]D DD E9A' /:l' for for each each [:]D D D E9 (u)0. (u)o. ItIt suffices suffices to to successor A' successor take vv = Given that also in take = . ( uh ) . Given that each each [:l-formula D-formula in in (u)0 (u)o appears appears also in A', /:l' , the the depth depth of A /:l'~ cannot cannot be be larger larger than than the the depth depth of of (u)0. (u)o. Therefore, Therefore, vv E9Wr Wr and and vv fulfills fulfills all all of requirements. -1 requirements. -t Visser from the Visser (see (see Berarducci Berarducci [1990])showed [1990]) showed that, that, from the models models constructed constructed in in the the relation that that is is independent independent of of the the proof, one one can can construct construct models models with with an an SS relation above proof, above world w may have have to world w (see (see also also definition definition 15.4). 15.4) . These These models models may to be be infinite infinite however. however. The The first first arithmetic arithmetic completeness completeness proofs proofs used used these these models models instead instead of of the the finite finite models models constructed in in the the above proof, but but we will not not introduce introduce them them in in this this section, section, since since constructed above proof, we will our 14) uses uses the the finite finite models models directly. directly. our arithmetic arithmetic completeness completeness proof proof (section (section 14) The fixed can be extended to to IL IL and hence to I L M and ILP The fixed point point theorem theorem of of LL can be extended and hence to ILM and ILP (de and Visser [1991]). (de Jongh Jongh and Visser [1991]). Arithmetic completeness of 114. 4. A rithmetic completeness o f ILM ILM We fix aa theory we assume that T the language We fix theory TT containing containing IE1. 1�1 . For For safety safety we assume that T is is in in the language of arithmetic and T is sound, true (in arithmetic and T is sound, i.e., i.e., all all its its axioms axioms are are true (in the the standard standard model model of of arithmetic) , although in fact fact itit is is easy easy to our proof proof of of the of arithmetic), although in to adjust adjust our the completeness completeness 2: 1 -soundness of theorem to the the weaker condition of of El-soundness of T. T. theorem to weaker condition 14.1 . Definition. Definition. The definition of given in section 11 is 14.1. The definition of aa realization realization given in section is extended extended to the the language language of of IILM L M by by stipulating that (A (AC>B)* = Conserv Conserv((rA*7, to stipulating that r;, B)* = A* "', rr BB*. 7"')), , where Conserv Conserv((rA* is an an intensional intensional formalization formalization ((see Chapter IIII of of this this where A' '''7,, rr BB.* 7) "' ) is see Chapter Handbook) of Handbook) of "T "T + + B* B* is is III-conservative IIl-conservative over over T T+ + A*" A*".. If If T T= = PPA, A , then, then, iinn view view of of theorem theorem 12.7, 12.7, the the interpretability interpretability and and II H1lconservativity relations over finite extensions conservativity relations over its its finite extensions are are the the "same" "same" in in all all reasonable reasonable senses, so so we we can can take take Conserv Conserv((rA*7, . 7 ) to to be be a a formalization formalization of of "T "T + B* B* is is A*"', rr BB*"') senses, interpretable in interpretable in T T + + A*" A*".. Below Below we we prove prove the the completeness completeness of of ILM I L M as as the the logic logic of of IIl-conservativity I]l-conservativity over over T T and and thus thus at at the the same same time time the the completeness completeness of of IILM LM as as the the logic logic of of interpretability interpretability over over T T= = PA. PA. The The fact fact that that ILM I L M is is the the logic logic of of interpretability interpretability over over PA P A was was proven proven more more or or less less simultaneously simultaneously and and independently independently by by Berarducci Berarducci [1990] [1990] and and Shavrukov Shavrukov [1988] [1988].. Later, Later, Hajek H~jek and and Montagna Montagna [1990,1992] [1990,1992] proved proved that that ILM I L M is is the the logic logic of of II IIl-conservativity over T T= = 1�1 IE1 and and stronger stronger theories. theories. l-conservativity over 14.2. f-F- ILM A iff 14.2. Theorem. Theorem. ILMA iff for for every every realization realization **,, T T fF- A* A*.. Proof. The ( ====} ) part can be verified by induction on ILM Proof. The (=:~) part can be verified by induction on I L M proofs. proofs. Since Since the the soundness soundness of of L L is is already already known, known, we we only only need need to to verify verify that that if if D D is is an an instance instance of of *. one one of of the the additional additional 66 axiom axiom schemata schemata of of ILM, ILM, then, then, for for any any realization realization **,, T T f~- D D*. All All the the arguments arguments below below are are easily easily formalizable formalizable in in T: T: G. Japaridze Japaridze and and D. D. de de Jongh Jongh G. 522 522 I> B) Axiom (1): (1): D [::](A -+ B) B) -+ -~ (A (At> B).. If If T T f-F A A -+ --+B, B, then then clearly clearly T T+ + B* B* is is Axiom (A -+ conservative over over T T+ + A* A*.. conservative Axiom I> B) I> C) -+ I> C) . Evidently, Axiom (2) (2):: (A (At> B) 1\A (B (Bt>C) ~ (A (At>C). Evidently, the the relation relation of of conservativ conservativity is is transitive. transitive. ity Axiom I> C) 1\^ (B I> C) -+ I> C. It Axiom (3) (3):: (A (At>C) (Bt>C) -+ A A vv B Bt>C. It is is easy easy to to see see that that if if T T+ + C* C* is is (H1-) conservative conservative over over T T+ + A* A* and and T T+ + B* B*,, then then so so is is it it over over T T+ + A* A* Vy B* B*.. (ITI-) A -+ B) . Clearly, 1 -conservative over Axiom I> B) Axiom (4) (4):: (A (At> B) -+ -+ ((0 ~A -~ 0 (}B). Clearly, if if T T+ + B* B* is is IT Ill-conservative over T+ + A* A* and and T T+ + A* A* is is consistent, consistent, then then so so is is T T+ + B* B*.. T Axiom I> A. Suppose Axiom (5): (5): OA ~At>A. Suppose AA is is aa IT Hl!-sentence provable in in T T+ + A* A*.. We We need need l !-sentence provable to * , that to show, show, arguing arguing in in T T+ + (OA) ((}A)*, that then then A• is is true. true. Indeed, Indeed, suppose suppose T T+ + A* A* is is consistent. Then Then it it cannot cannot prove prove aa false false IT IIl!-sentence (by � Ell !-completeness) !-completeness) ,, and and consistent. l !-sentence (by hence/kA must must be be true. true. hence Axiom I> B) Axiom (M): (M): (A (At> B) -+ -~ (A (A 1\ ^ DCI> OCt> B B 1\ ^ DC) DC).. Suppose Suppose T T+ + B* B* is is IT II~-conservative l -conservative over T T+ + A* A* and and A)~ is is aa IT IIl!-sentence provable in in T T+ + B* B* 1\ ^ ((E]C)*. Then T T+ + B* B* over DC)' . Then l !-sentence provable proves DC)* -+ proves ((EIC)* -+ A. )~. But But the the latter latter is is aa IT YIl-sentence and therefore therefore it it is is also also proved proved by by l -sentence and T+ + A* A*.. Hence, Hence, T T+ + A* A* 1\ A (DC)* (DC)* fFA A.. T The following following proof proof of of the the ((r{:= ) part part of of the the theorem theorem is is taken taken from from Japaridze Japaridze The [1994b] and and has has considerable considerable similarity similarity to to proofs proofs given given in in Japaridze Japaridze [1992,1993] [1992,1993] and and [1994b] Zambella Zambella [1992]. [1992]. Just Just as as in in Japaridze Japaridze [1992,1993], [1992,1993], the the Solovay Solovay function function is is defined defined in than provability in terms terms of of regular regular witnesses witnesses rather rather than provability in in finite finite subtheories subtheories (as (as in in Berarducci Berarducci [1990], [1990], Shavrukov Shavrukov [1988], [1988], Zambella Zambella [1992]). [1992]). Disregarding Disregarding this this difference, difference, the function function is is almost the same same as as the the one one given [1992] ' for for both both proofs, proofs, the almost the given in in Zambella Zambella [1992], unlike the ones ones in in Berarducci Berarducci [1990] [1990] and and Shavrukov Shavrukov [1988] finite ILM-models unlike the [1988],, employ employ finite ILM-models rather than rather than infinite infinite Visser-models. Visser-models. Then, by by theorem theorem 13.15, 13.15, there there is finite ILM-model ILM-model J.L1LM A. A. Then, Suppose ~ILM Suppose is aa finite w , IF) If-) in in which which A A is is not not valid. valid. We We may assume that that W W == {{I, (W, R, R, {S~}~ {Sw}w eE w, (W, may assume 1 , .. ...., , l}, l}, W eE W, W, and is the the root root of of the the model model in the sense sense that that 1R for all all 11 ~: and 11 ~WAA.. We We 11 is in the lR w w for i= w define ~ w , ) ): : define aa new new frame frame (W (W',~, RR',~, {S~ {S� }}WEWI w'=wu{o}, W' = W U {O} , R' == RR uU {(0, R' }. . { (O, w) II W E W W} Sb == $1 Sl UU ((1, {(I, w) w ) lw I W ~E W W}} and and for for each each w w eE W, W, S~ S� == S~. Sw ' S~ Observe }wew,) is Observe that that (W', (W', R', R', (S~ {S�}WEWI) is aa finite finite ILM-frame. ILM-frame. Just as as in in section section 3,3, we we are are going going to to embed embed this this frame frame into into TT by by means means of of aa Just Solovay style style function function gg :w : w~ W'~ and and sentences sentences Lim~ Limw for for ww eE W W'~ which which assert assert that that Solovay -+ W w isis the the limit limit of of g.g. This This function function will will be be defined defined in in such such aa way way that that the the following following w basic basic lemma lemma holds: holds: 14.3. emma. 14.3. LLemma. V (a) TT proves proves that that gg has has aa limit limit in in W', W', i.e., i.e., TT Ff- V {Limr { Limr II rr ~E W'}, W'} , (b) (b) IfIfww ~i= u, u , then then TT Ff- -~.., (Limw (Limw n1\ Limu), Limu), (c) (c) IfIfww RR'~u, u , then then TT ++ Limw Limw proves proves that that TT]zJ.L -~ Lim~,, Limu , .., The Provability The Logic Logic of o] Provability 523 523 (d) If If w w '" ~ 0 0 and and not not w w R' R' u, u, then then T T + + Limw Limw proves proves that that T T IF- --, --1Lim~, (d) Lim", v, then that T -conservative over (e) (e) If If U u S� S'~v, then T T + + Limw Lim~ proves proves that T + + Limv Limv is is III H~-conservative over T T + + Lim" Lim~,, (f) (f) Suppose Suppose w w R' R' u u and and V V is is a a subset subset of of W' W ' such such that that for for no no vv Ee V, V, u u Swv S~v ;; V -conservative over then then T T + + Limw Lim~ proves proves that that T T + + V {Limv {Lim, IIvv Ee V} V } is is not not III II~-conservative over T T+ + Lim" Lim~,, (g) (g) Limo Lim0 is is true, true, (h) For each each ii Ee W', W ~, Limi is consistent with T. T. (h) For Limi is consistent with To deduce deduce the the main main thesis thesis from from this this lemma, lemma, we we define define aa realization realization '* by by setting setting To for for each each propositional propositional letter letter p, p, {Limr lI rr EE W pP** = = V V{Limr W,, rr lf-p}. I~-p}. 14.4. 14.4. Lemma. Lemma. For -formula B, For any any w w Ee W W and and any any ILM ILM-formula B, f- B, then (a) imw I-F- B* (a) if if w w lIF-B, then T T + +L Lim~ B*;; (b) (b) if if w w W-B IF B , , then then T T + + Limw Lim~ IF- --, ~B B*. *. Proof. By induction induction on on the the complexity complexity of B.. The The cases cases when when B B is is atomic atomic or or has has Proof. By of B the DC are handled just the form form [:]C are handled just as as in in the the proof proof of of lemma lemma 3.3, 3.3, so so we we consider consider only only the the case case when when B B = = Cl C1 l> D C2 C2.. Assume instead of Assume w w Ee W. W. Then Then we we can can always always write write w wR Rx x and and x x Sw S~ yy instead of w w R' R~xx and (i = establish that both and x x S� S~ yy.. Let Let ai ai = = {{rr II w wR R r, r, rr If-Ci} I~-Ci} (i = 11,, 2). 2). First First we we establish that for for both ii = = 1, 1, 2, 2, ((,) *) V T T+ + Limw Lim~ proves proves that that T T I~q C~ ++ ~ V {Limr {Limr II rr Ee ai} c~i}.. Indeed, Indeed, argue argue in in T T + + Limw Limw.. Since each hypothesis for Since each rr Ee a aii forces forces Ci Ci,, we we have have by by the the induction induction hypothesis for clause clause (a) (a) that that for for each each such such rr,, T T IF- Limr Limr -t --+ Ct C~,, whence whence T T IF- V V {{Lim L i m rr II rr eE a;} ai} -t -+ Ct C~.. Next, Next, V 9 J W'} and, according according to to lemma lemma 14.3(a) 14.3(a),, TI TF-- V {Lim {Lim~r II rr Ee W'} and, according according to to lemma lemma 14.3(d), 14.3(d), T {Limr I] wR r}; at T disproves disproves every every Limr Lim~ with with not not wR w R r; r; consequently, consequently, T T I~- V V{Lim~ w R r}; at the the same same time, time, by by the the induction induction hypothesis hypothesis for for clause clause (b) (b),, Ct C~ implies implies in in T T the the negation negation of of each each Limr Lim~ with with rr W-Ci P~Ci.. We We conclude conclude that that T T IF- q C~ -t -+ V V {Limr {Limr II wRr, w R r, rr If-C;} I~-Ci},, i.e., i.e., {Limr II rr eE a i } . Thus, T T I~q C~ -t ~ V V{Lim~ ai}. Thus, ((,) is proved. proved. Now Now continue: continue" * ) is l w f C l> C2 . Argue in T + Limw . By (a) Suppose (a) Suppose w IF-C1l E> C2. Argue in T + Limw. By ((.), to prove prove that that T T+ + c; C~ * ), to {Limr II rr Ee a2 is is Ill-conservative Hi-conservative over over T T+ + C; Ct,, it it is is enough enough to to show show that that T T+ + V V{Lim~ a2}} is {Limr II rr Ee ad Consider an is III-conservative Hi-conservative over over T T+ + V V{Lim~ al}.. Consider an arbitrary arbitrary U u Ee a all (the (the case case with with empty empty a all is is trivial, trivial, for for any any theory theory is is conservative conservative over over T T + + .i). _[_). Since Since w I I> w lf-C I~-C1 E> C2 C2,, there there is is Vv Ee a2 a2 such such that that u u Sw Sw vv.. Then, Then, by by lemma lemma 14.3(e) 14.3(e),, T T+ + Limv Limv is is {Limr II rr Ee a2} III-conservative HI-conservative over over T T+ + Lim" Lim~.. Then Then so so is is T T+ + V V{Lim~ a2} (which (which is is weaker weaker 524 524 G. Japaridze and D. de Jongh than ). Thus, Thus, for than T T+ + Limv Limv). for each each u u Ee aI C~l,, T T + + V V {{Lim L i m ,r II rr Ee a ~2} is ill-conservative lIi-conservative 2 } is V this implies that T over over T T + + Lim" Lim~.. Clearly Clearly this implies that T+ + V {Limr {Limr II rr Ee a c~2} is il IIl-conservative l -conservative 2 } is V l· over over T T + + V {Limr {Limr II rr eE al C~l}. (b) w WCI ~C1 [> c> C2 C2.. Let Let us us then then fix fix an an element element u u of of al al such such that that u u Sw S~ vv for for (b) Suppose Suppose w no . Argue in T + Limw . no vv Ee a as. Argue in T + Lim~. 2 By lemma By lemma 14.3(f), 14.3(f), T T+ + V {Lim {Lim~r II rr Ee a a2} is not not il 1-Ii-conservative over T T + + Limu Limu.. l -conservative over 2 } is V V Then, neither neither is Then, is it it il Hi-conservative over T T + + V {Limr {Lim~ II rr Ee ad c~1} (which (which is is weaker weaker than than l -conservative over over T + Ci . T ). This T + + Lim,, Lim~). This means means by by (*) (.) that that T T + + c:; C~ is is not not ill-conservative IIl-Conservative over T + C~. -l -~ Now WA, lemma Now we we can can pass pass to to the the desired desired conclusion: conclusion: since since 1l~ZA, lemma 14.4 14.4 gives gives But we we do do have have T T~Z _7Liml Liml But � .., -l according to lemma 14.3(h).. This This ends ends the the proof proof of of theorem theorem 14.2. 14.2. -t according to lemma 14.3(h) T Ik- Lim Limll -+ --+ .., -~ A* A*,, whence whence T T� Y .., -~ Lim Limll :::} =~ T TJz A*.. T � A* Our remaining remaining duty Our duty is is to to define define the the function function 9 g and and to to prove prove lemma lemma 14.3. 14.3. The The recursion ne this recursion theorem theorem enables enables us us to to defi define this function function simultaneously simultaneously with with the the sentences sentences Limw (for (for each each W w Ee W'), W'), which, which, as as we we have have mentioned mentioned already, already, assert assert that that w w is is the the Limw (w, u) ne limit limit of of gg,, and and the the formulas formulas �w,,(y) A ~ ( y ) (for (for each each pair pair (w, u) with with wR'u) wR'u), , which which we we defi define by by �w,, (y) == - 3t > Y y (g(t) = = UA ^ Vz V z ((y y � < z z < t --+ g(z) g(z) = = :1 t > (g(t) < t -+ w)) . 14.5. 14.5. Definition. D e f i n i t i o n . (function (function g) g) We We define define g(O) g(0) = = O. 0. Assume ned for Assume that that g(y) g(y) has has already already been been defi defined for every every yy � ~ x, x, and and let let gg(x) = w. w. (x ) = Then ned as Then g(x g(x + + 1) 1) is is defi defined as follows: follows: (1) wR'u, n w. Then, (1) Suppose Suppose wR'u, n� <~x x and and for for all all z with with n n� ~ z� ~<x x we we have have g(z) g(z) = = w. Then, if if I-xLim" .., �wu(fi) define gg(x (x + F=Limu -+ --+-~ Awu(~),, we we define + 1) 1 ) == u. u. (2) (2) Else, Else, suppose suppose m m� ~ x, x, A A is is a a il H1l !-sentence [-sentence and and the the following following holds: holds: (a) (a) A A has has aa regular regular counterwitness counterwitness which which is is � ~<x, x, (b) (b) I-m l-m Lim" Limu -+ ~ A A,, (c) (c) w Sg(m) uu,, (d) (d) m rn is is the the least least number number for for which which such such A A and and u u exist, exist, i.e., i.e., there there are are no no m m 'I < < m, m, ' world the conditions world u' u' and and il II1l !-sentence [-sentence X ,V satisfying satisfying the conditions (a)-(c) (a)-(c) with with m m',, u' u' and and X ,V substituted substituted for for m, u u and and A ,~.. Then ne g(x 1) = Then we we defi define g(x + + 1) = u. u. (x) . (3) (3) In In all all remaining remaining cases cases g(x g(x + + 1) 1 ) == gg(x). It It is is not not hard hard to to see see that that 9 g is is primitive primitive recursive. recursive. Before Before we we start start proving proving lemma lemma 14.3, 14.3, let let us us agree agree on on some some jargon jargon and and prove prove two two auxiliary auxiliary lemmas. lemmas. When When the the transfer transfer from from w (x ) to to uu == g(x nition 14.5(1), w == gg(x) g(x + + 1) 1) is is determined determined by by defi definition 14.5(1), we we say say that that at at the the moment -move from moment x x+ + 11 the the function function 9 g makes makes (or (or we we make) make) an an R' R'-move from the the world world w w to to the u. If transfer is determined by definition 14.5(2) the world world u. If this this transfer is determined by definition 14.5(2),, then then we we say say that that The Logic Logic of ofProvability Provability The 525 525 takes place place and and call call the the number number m m from from definition definition 14.5(2) 1 4.5 ( 2) the the rank rank of of an S'-transfer S' -transfer takes an this S~-transfer. S'-transfer. Sometimes Sometimes the the S~-transfer S'-transfer leads leads to to aa new new world, world, but but 'mostly' 'mostly ' itit does does this ( u ==))g(x w) , and and then then itit isis not not aa move move in in the the proper proper sense. sense. not, i.e., i.e., (u not, g ( x ++ 11)) == gg(x) ( x ) ((== w), Those S'-transfers S'-transfers which which lead lead to to aa new new world world we we call call S~-moves. S' -moves. As As for for R'-transfers, R' -transfers, Those always lead lead to to aa new new world, world, so so we we always always say say "R'-move" "R'-move" they (by (by irreflexivity irrefiexivity of of RR') they ~) always instead of of "R~-transfer "R'-transfer"'' .. instead In these these terms, terms, the the formula formula A~u(n) tlwu (n) asserts asserts that that starting starting at at or or before before the the moment moment In and until until some some moment moment t,t , we we stay stay at at the the world world w w without without moving moving and and then, then, at at the the nn and we move move directly directly to to u. u. moment t,t, we moment Intuitively, we we make make an an R'-move R'-move from from w w to some uu with with wR'u wR'u in in the the following following Intuitively, to some and up up to to the the present been staying staying at at world world situation: since since some some moment moment nn and situation: present we we have have been w, w , and and just just now now we we have have reached reached evidence evidence that that TT ++ Lim~ Limu thinks thinks that that the the first first (proper) (proper) move which which happens happens after after passing passing moment moment nn (and (and thus thus our our next cannot lead lead move next move) move) cannot directly to to the the world world u; u ; then, then, to to spite spite this this belief belief of of TT ++ Limu, Limu , we we immediately immediately move move directly to uu.. to And the the conditions for an an S~-transfer S'-transfer from from w w to to uu can be described as follows: follows: we And conditions for can be described as we are staying staying at the world are at the world w w and and by by the the present present moment moment we we have have reached reached evidence evidence that that T ++ Lim~ Limu proves proves aa false false IIl[-sentence III !-sentence A. A. This This evidence of two components: T evidence consists consists of two components: (1) which indicates that AA is false, and (2) the rank m m of (1) a a regular regular counterwitness, counterwitness, which indicates that is false, and (2) the rank of the m u , the next the transfer, transfer, which which indicates indicates that that T T + + Limu Lim~ fF- A A.. Then, Then, as as soon soon as as w wS Sg(m) u, the next g( ) remain at there moment moment we we must must be be at at u u (move (move to to u, if if u u f= r w, w, and and remain at w, w, if if u u= = w); w); if if there are choose the the one with the the least are several several possibilities possibilities for for such such a a transfer, transfer, we we choose one with least rank. rank. An An additional additional necessary necessary condition condition for for an an S'-transfer S~-transfer is is that that in in the the given given situation situation an an R'-move R~-move is is impossible; impossible; R'-moves R~-moves have have priority priority over over S'-moves. S~-moves. Note Note that that the the condition condition for for an an R'-move R~-move here here is is weaker weaker than than for for the the function function h h defined -,tlwu defined in in section section 3: 3: T T only only needs needs to to prove prove Limu Lim~ -+ ~ A ~ ( ~(ii) ) . . This This feature feature will will play play aa crucial crucial role role in in the the verification verification of of 14.3(f) 14.3(f).. 14.6. (T 14.6. Lemma. Lemma. (T ft- :) :) For For each each natural natural number number m m and and each each W w Ee W', W', T T + + Limw Limw proves proves that that no no S'-transfer S~-transfer to to w w can can have have rank rank less less than than m m.. Proof. -transfer is P r o o f . Indeed, Indeed, "the "the rank rank of of an an S' S~-transfer is < <m m"" means means that that T T+ + Limw Limw proves proves aa false false (i.e., (i.e., one one with with aa regular regular counterwitness) counterwitness) IIl II1 !-sentence [-sentence AA and and the the code code of of this this proof proof (i.e., (i.e., of of the the T-proof T-proof of of Limw Limw -+ --+ A) A) is is smaller smaller than than m. But But the the number number of of all all III I]1 !-sentences !-sentences with with such such short short proofs proofs is is finite, finite, and and as as T T+ + Limw Lim~ proves proves each each of of them, them, itit also also proves proves that that none none of of these these sentences sentences has has aa regular regular counterwitness counterwitness (recall (recall our our assumptions -j assumptions about about the the formula formula Regwit(x, Regwit(x, y) y) from from section section 12). 12). -~ 14.7. (T 14.7. Lemma. Lemma. (T f~ ::)) If If g(x)R'w, g(x)R'w, then then for for all all yy � <.x, x, g(y)R' g(y)R' w w.. Proof. P r o o f . Suppose Suppose g(x)R' g(x)R' w w and and yy � ~<x. x. We We proceed proceed by by induction induction on on nn = = xx - yy.. If If yy -= x, x, we we are are done. done. Suppose Suppose now now g(y g(y + + l)R' 1)R' w. w. If If g(y) g ( y )= - g(y g(y + + 1) 1),, we we are are done. done. If If not, not, then then at at the the moment moment yy + + 11 the the function function makes makes either either an an R'-move R~-move or or an an S'-move. S~-move. In In the the first first case case we we have have g(y)R' g(y)R' g(y g(y + + 1) 1) and, and, by by transitivity transitivity of of R' R',, g(y)R' g(y)R' w w;; in in the the second second case case we we have have g(y)S� g(y)S~vg(y g(y + + 1) 1) for for some some vv,, and and the the desired desired thesis thesis then then follows follows 526 526 G. Japaridze and D. de Jongh from of lLM-frames nition 13.4) from property property (iv) (iv)of ILM-frames (defi (definition 13.4).. P r o o f of o f lemma l e m m a 14.3. 14.3. In In each each case case below, below, except except in in (g) (g) and and (h), (h), we we reason reason in in T. T. Proof (a):: First First observe observe that that there exists some some zz such such that that for for all all z' z ~;::, > zz, , not not there exists (a) R ' gg(z' ( z ' ++ 1). 1). Indeed, Indeed, suppose suppose this this is is not not the the case. case. Then, Then, by by lemma lemma 14.7, 14.7, gg(z') (z') R' for z, there (z)R' gg(z'). (z') . This nite (or for all all z, there is is z' z' with with g g(z)R' This means means that that there there is is an an infi infinite (or "sufficiently "sufficiently long" long")) chain chain wIR' WlR' W w 22R' R ' .· .·. ·, , which which is is impossible impossible because because W' W' is is finite finite and and R' R' is is transitive transitive and and irreflexive. irreflexive. So, let z. Then So, let us us fix fix this this number number z. Then we we never never make make an an R'-move R'-move after after the the moment moment z. z. We nite number We claim claim that that Sf-moves S'-moves can can also also take take place place at at most most a a fi finite number of of times times (whence (whence it follows follows that that 9 g has has a a limit and this limit is, is, of of course, course, one one of of the the elements elements of of WI) W'). . limit and this limit it Indeed, Indeed, let let x x+ + 11 be be an an arbitrary arbitrary moment moment after after zz at at which which we we make make an an Sf-move, S~-move, and !-sentence >.A with and let let m m be be the the rank rank of of this this move. move. That That is, is, for for some some III IIl[-sentence with a a ::::;~<xx regular regular counterwitness, we have have bmLim~--+ and wS wSg(m)u, where w w= = g(x) g(x) counterwitness, we I-mLimu --+ >.A and g(m)U, where and the next and u u ==gg(x ( x + + 1) 1).. Suppose Suppose we we make make the next Sf-move, S'-move, with with rank rank m', m', at at some some moment moment x' x ' ++ 11,, x' x ' >>xx, , from from the the world world u u to to a a world world vv,, vv -oj ~ uu.. Since Since S Sg(m) is g(m) is reflexive, definition 14.5(2) x', u, u, u, u, A, A, m reflexive, conditions conditions (a)-(c) (a)-(c) of of definition 14.5(2) hold hold for for x', m in in the the roles roles of of x, u, >., A, m m,, respectively, respectively, and and then, then, according according to to condition condition (d) of defi definition 14.5(2),, x, w, w, u, (d) of nition 14.5(2) the only only reason reason for for moving moving to to vv instead instead of of u u - - instead instead of of remaining remaining at at u, that that is is the could could be be that that m m > > m' m' (the (the case case m m = = m' m I is is ruled ruled out out because because Limu Lim~ oj :/: Limv). Limv). Similarly, Similarly, the following will be etc. Thus, Thus, consecutive the rank m the rank m"il of of the following Sf-move S~-move will be less less than than m' m ~,, etc. consecutive Sf-moves S'-moves without without an an R'-move RI-move between between them them have have decreasing decreasing ranks. ranks. Therefore, Therefore, Sf-moves S'-moves can can take take place place at at most most m m times times after after passing passing x. x. (b):: Clearly Clearly 9 g cannot cannot have have two two different different limits limits w w and and u. u. (b) (c): ;::, n, (c): Assume Assume w w is is the the limit limit of of 9 g and and w w R' R' u. u. Let Let n n be be such such that that for for all all x x/> n, (x) = Limu . Deny �wu (fi) - w. w. We We need need to to show show that that T T J.L Y -, -1Limu. Deny this. this. Then Then T T bI- Limu Lim~ --+ ~ -, -1Awu(fi) 9g(x) and, ;::, nn such and, since since every every provable provable formula formula has has arbitrary arbitrary long long proofs, proofs, there there is is x x/> such that --+ -' �wu(fi) nition 14.5(1), that I-x F-x Limu Lim~--+-~ Awe(g);; but but then, then, according according to to defi definition 14.5(1), we we must must have have g(x + + 1) 1) = - u, u, which, which, as as u u oj =fi w w (by (by irreflexivity irreflexivity of of R'), R'), is is a a contradiction. contradiction. g(x 0, w the limit u. If w, then (d): (d): Assume Assume w w oj -~ 0, w is is the limit of of 9 g and and not not wR' wR' u. If u u= = w, then (since (since w 1) = u. This w oj -~ 0) 0) there there is is x x such such that that g(x) g(x) = = vv oj =fiu u and and g(x g(x + + 1) = u. This means means that that at at the the rst case moment moment x x+ + 11 we we make make either either an an R'-move R'-move or or an an Sf-move. S'-move. In In the the fi first case we we have have T Limu --+ easy to the � T IF-Lim~ ~ - ~-, �vu(fi) Ave(g) for for some some n n for for which, which, as as it it is is easy to see, see, the E1l !-sentence !-sentence � !-completeness, T -, Limu . And AT g ) is is true, true, whence, whence, by by �I El!-completeness, T 1b-~Lim~. And if if an an Sf-move S'-move is is vu((fi) taken, taken, then then again again T T IF- -, -~ Limu Lim~ because because T T + + Limu Lim~ proves proves a a false false (with (with a a ::::; ~<x x regular regular counterwitness) III counterwitness) II1 !-sentence. !-sentence. Next, w. Let w. Since Next, suppose suppose u u oj ~: w. Let us us fix fix a a number number zz with with g(z) g(z) = = w. Since 9 g is is primitive primitive recursive, limit recursive, T T proves proves that that g(z) g ( z ) -= w w.. Now Now argue argue in in T T + + Limu Lim~:: since since u u is is the the limit oj uu and of of 9 g and and g(z) g ( z ) == w w oj :/: u u,, there there is is a a number number x x with with x x ;::, >1 zz such such that that g(x) g(x)::/= and gg(x (x + 1) = + 1) = u u.. Since Since not not (w (w = = )g(z)R' )g(z)R' u u,, we we have have by by lemma lemma 14.7 14.7 that that (,) (*) for for each each y with with zz ::::; ~ IInn particular, particular, not not g(x g ( x ))R' R ' uu and and the the transfer transfer from from g(x) g(x) to to g(x g(x + + 1)( 1 ) ( == u u)) can can have have the property been been determined determined only only by by definition definition 14.5(2) 14.5(2).. Then Then (*) (,) together together with with the property (i) (i) The Logic Logic of of Provability Provability The 527 527 of of IL-frames IL-frames and and definition14.5(2c), definition14.5(2c), implies implies that that the the rank rank of of this this S'-move S'-move isis less less than than z, which, which, by by lemma lemma 14.6, 14.6, isis aa contradiction. contradiction. Thus, Thus, TT ++ Lim~ Limu isis inconsistent, inconsistent, i.e., i.e., z, -, Lim~. T ~f- -~ Limu . T (e): S� vv rI- uU (the (the case case vv-= uU isis trivial). trivial) . Suppose Suppose w w is is the the limit limit of of g, g, (e): Assume Assume uU S~ We may may suppose suppose that that A Ae E II1 II I !! and and that that zz isis A . We A is is aa II1-sentence III-sentence and and TT F-zLimv f- zLimv ~-+ A. sufficiently large, large, namely, namely, g(z) g(z) -= w. w . Fix Fix this this z. z . We need to to show show that that TT ++ Lim~ Limu Ff- A. A. sufficiently We need Argue in T is aa regular Argue in T ++ Lim~. Limu . Suppose Suppose not not A. A . Then Then there there is regular counterwitness counterwitness cc for A . Let Let us us fix fix aa number number xx >> z, z, cc such such that that g(x) g(x) -= g(x for A. g(x + + 1) 1) -= uU (as ( as uU is is the the limit limit of g, such such aa number number exists). exists) . Then, Then, according according to to definition definition 14.5, 14.5, the the only only reason reason for for of g, g(x from uu to g(x + + 1) 1) == uU =/ I-=vv can can be be that that we we make make an an S~-transfer S' -transfer from to uu and and the the rank rank of of this this transfer is is less less than than z,z, which, which, by by lemma lemma 14.6, 14.6, is is not not the the case. case. Conclusion: Conclusion: A A (is (is true). true). transfer (f): for each (f) : Assume Assume w w is is the the limit limit of of g, g, wR' WRI uU, , VV c_ <; W W'' and and for each vv eE V, V, not not uu S~ S� v. v. Let Byy primitive Let nn bbee such such that that for for all all zz >1 � n, n, g(z) g(z) = = w. w. B primitive recursiveness recursiveness of of g, g, TT proves proves that n ) == w. A ~ ( ~ ) . . So, ~ ( ~(n) ) i s is aa w . By By definition definition 14.5(1), 14.5(1), TT ++ Lim~ Limu lz J.L --,~6.wu(n) So, as as ~-,A6.wu that gg(( n) V {Limv IIvv Ee V} III -sentence, in in order order to to prove that TT ++ V{Limv not Ill-conservative III-conservative over V} isis not Hi-sentence, prove that over T + Limu, for each T + ~ ( ~(n) ) . . Let Limu , itit is is enough enough to to show show that that for each vv Ee V, V, T + Lim, Limv f-~ -~ -, A 6.wu Let us us fix fix T+ any V. According According to our assumption, not uu S~ S� vv and, by reflexivity reflexivity of S� , uu Iv. any vv eE V. to our assumption, not and, by of S~, :/: v. Argue in in T that 6.wu(n) A ~ ( n ) holds, holds, i.e., i.e., there Argue T ++ Lim,. Limv . Suppose, Suppose, for for aa contradiction, contradiction, that there is such that n <. is the that g(t) g(t) = = uu and and for for all all zz with with n � zz < < t, t, g(z) g(z) = = w. w . As As vv is the limit limit is tt >> nn such there is is t't' >> tt such such that that g(t' g(t' -- 1) 1) Iand at at the the moment of and vv rI- u, ~ v v and moment t't' we we arrive arrive of g9 and u, there at to stay there for for ever. ever. Let Let then then x0 Xo << ... ... < < xk Xk be be all moments in in the the interval interval at v v to stay there all the the moments [t, t'] t'] at at which which R'R' - or or S'-moves S' -moves take take place, place, and and let let u0 Uo == g(xo) Uk == g(xk). g(Xk) . Thus Thus It, g ( x 0 ),, .. .. .., , uk = xk, Xk , uU == u0, Uo , vv == Uk Uk and and tuo, Uk is is the the route route of after departing departing from from tt = = Xo x0,, tt'~= o , ... .. ,. , Uk of g9 after w (at the moment moment t). t). w (at the Now Now let let jj b bee the the least least number number among among 11,, .. ... . ,,kk such such that that for for all all jj � <~ii � ~ k k,, not not Uo uo R' R' Ui ui.. Note Note that that such such a a jj does does exist exist because because at at least least jj = = kk satisfies satisfies the the condition condition u= property (iv) (otherwise, (otherwise, if if ((u = )) Uo u0 R' R' Uk uk (( = = v) v),, property (iv) ofILM-frames of I L M - f r a m e s would would imply imply U u S� S~ v). v). Note Note also also that, that, for for each each ii with with jj � ~/j. j. S'-move. Moreover, Thus, from from the the moment moment Xj xj onwards, onwards, each each move move is is an an S~-move. Moreover, for for each each Thus, the S'-move with jj � ~1Xo x0.. We We have g ( m ) == ue for and, by definition 14.5(2c) some some ee with with 0 0� ~<ee � ~ j. On On the the other other hand, hand, since since consecutive consecutive S' S~-moves decrease which � j. the the rank rank (as (as we we noted noted in in the the proof proof of of (a) (a) above), above), and and since since the the rank rank of of the the SI-move S'-move to to Uk uk cannot cannot be be less less than than n n (lemma (lemma 14.6), 14.6), we we conclude: conclude: for for each each ii with with jj � ~ ii � ~ 528 528 G. Japaridze Japaridze and D. de Jongh contradiction. Limv f-F -~ b.wu (n) . contradiction. Conclusion: Conclusion: T T+ + Limv Aw~(~). (g):: By By (a), (a), as as T T is is sound, one of of the Limw for for w w Ee W' W' is is true. true. Since for no no w w (g) sound, one the Limw Since for do do we we have have wR'w, wR'w, (d) (d) means means that that each each Limw Limw,, except except Limo Lim0,, implies implies in in T T its its own own T -disprovability and false. Consequently, T-disprovability and therefore therefore is is false. Consequently, Limo Lim0 is is true. true. -l (h):: As As 3.2(f) 3.2(f).. --t (h) -, The The proof proof of of theorem theorem 14.2 14.2 is is complete. complete. In In de de Jongh Jongh and and Pianigiani Pianigiani [1998) [1998] this this theorem theorem and and its its extension extension to to an an interpretability interpretability logic logic with with witness witness comparison comparison formulas (Hajek (H~jek and and Montagna [1992]) was was applied applied to to solve solve aa conjecture of Guaspari Guaspari formulas Montagna [1992]) conjecture of [1983].. This This conjecture conjecture stated stated that that those those formulas formulas of of modal modal logic logic that that under under each each [1983) arithmetic realization realization are arithmetic are interpreted interpreted as as E1-sentences El-sentences are are L-equivalent L-equivalent to to disjunction disjunction of O-sentences [:]-sentences (already (already proved proved in in Visser Visser [1995]) [1995]),, and and those those of of modal modal logic logic extended extended of with witness comparison comparison formulas formulas are are R-equivalent R-equivalent to to disjunctions disjunctions which which contain contain with witness as their members as their members conjunctions conjunctions of of witness witness comparison comparison formulas formulas and and O-formulas. D-formulas. A A companion paper is Beklemishev [1993a), companion paper is Beklemishev [1993a], in in which which it it is is shown shown that that the the realization realization of of other other formulas, formulas, i.e., i.e., the the ones ones that that are are not not always always realized realized as as E El-sentences, 1 -sentences, cannot cannot be be restricted restricted to to any any particular particular class class in in the the arithmetic arithmetic hierarchy, hierarchy, thereby thereby improving Guaspari well. improving Guaspari [1983)'s [1983]'s results results as as well. Visser [1990) [1990] showed showed that that IILP L P is is the the interpretability interpretability logic logic for for all all reasonable reasonable Visser SUPEXP. An finitely axiomatizable Ib.o + finitely axiomatizable theories theories that that contain contain IA0 + SUPEXP. An open open problem problem is is the the axiomatization axiomatization of of the the logic logic of of the the principles principles valid valid for for interpretability interpretability in in all all reasonable reasonable r.e. theories. Visser just the intersection of r.e. theories. Visser [1991) [1991] showed showed that that this this logic logic is is not not just the intersection of IILM LM and and ILP. ILP. 1 5 . Tolerance logics 15. T o l e r a n c e llogic o g i c and a n d oother t h e r interpretability interpretability logics 15.1. 15.1. The T h e logics logics of of cointerpretability c o i n t e r p r e t a b i l i t y and and faithful faithful interpretability interpretability Unlike Unlike interpretability, interpretability, no no modal modal axiomatization axiomatization for for the the logic logic of of cointerpretability cointerpretability or or faithful faithful interpretability interpretability (over (over PA P A or or any any other other reasonable reasonable theory) theory) has has been been found found so so far. far. Even Even the the question question of of decidability decidability of of these these logics logics remains remains open. open. However, However, the the logics logics of of weak weak interpretability interpretability and and the the more more general general relations relations of of tolerance 1) have tolerance and and cotolerance cotolerance (see (see section section 111) have been been studied studied thoroughly. thoroughly. Here Here is is aa brief field, which brief history history of of research research in in this this field, which starts starts from from some some digression digression from from the the subject. subject. 15.2. 15.2. The T h e logic logic of of the the arithmetic a r i t h m e t i c hierarchy hierarchy Japaridze decidable propositional logic H HGL Japaridze [1990b,1994aj [1990b,1994a] introduced introduced aa decidable propositional logic G L with with E22 ,, ZEi, infinitely , E1, Ei, infinitely many many unary unary modal modal operators: operators: O D,EI, Z+,E + , ... .. . and and proved proved its its soundness soundness and and completeness completeness with with respect respect of of the the arithmetic arithmetic interpretation interpretation where where oA is understood as formalization of A " is EnA as "A' is [::]A is understood as aa formalization of ""A* is provable provable (in (in PA)", PA)", E~A as "A* is (PA-equivalent "A" is (PA-equivalent to) to) aa En-sentence" E~-sentence" and and E�A E+A as as "A* is (PA-equivalent (PA-equivalent to) to) aa Boolean Boolean The Logic Logic of Provability Provability 529 529 combination of of �n En-sentences". The logic logic has has aa reasonable reasonable axiomatization axiomatization and and Kripke Kripke combination -sentences" . The semantics. semantics. 15.3. 15.3. The T h e logic logic of of ttolerance o l e r a n c e and and its its fragments fragments Ignatiev strengthened the 0 , �l)-fragment of Ignatiev [1990] [1990] (see (see Ignatiev Ignatiev [1993b]) [1993b])strengthened the (([:],E1)-fragment of the of the the arithmetic arithmetic hierarchy by switching switching from from the modal opera operathe logic logic of hierarchy by the unary unary modal » B is tor � E~l to to the the more more general general binary binary operator operator » >>,, where where A A>>B is interpreted interpreted as as tor A* -+ A f-k ((A* "there "there is is aa � E~-sentence ~o such such that that P PA -~ ijJ ~o)) 1\ A ((~o -~ B*)" B*)" (for (for comparison: comparison: ijJ -+ l -sentence ijJ the the interpretation interpretation of of � E~A is nothing nothing but but "there "there is is aa � El-sentence ~o such such that that l A is l -sentence ijJ ijJ -+ A * -+ * ) " ) . He P AA fF- ((A* -~ ~o) ~ A A*)"). He constructed constructed aa logic logic ELH E L H in in this this language, language, called called P ijJ) 1\A ((~o "the logic logic of of � E~-interpolability", and proved proved its its arithmetic arithmetic completeness. completeness. Although Although "the l -interpolability" , and the this, he the author author of of the the logic logic of of �l-interpolability El-interpolability did did not not suspect suspect this, he actually actually had had found the the logic logic of of weak weak interpretability interpretability over over PA, PA, because, because, as as it it is is now now easy easy to to see see found in --, ( A » that PA in view view of of corollary corollary 12 12.8, the formula formula -~(A >> --,B) -~B) expresses expresses that PA + + B* B* is is weakly weakly . 8, the interpretable in in P PA + A A*. interpretable A+ *. We interpretability is (binary) case linear tolerance, We know know that that weak weak interpretability is aa special special (binary) case of of linear tolerance, and the the latter latter is is aa special special (linear) (linear) case case of of tolerance tolerance of of aa tree tree of of theories. theories. JJaparidze and aparidze [1992] gave gave an an axiomatization axiomatization of of the the logic logic T O L of of linear linear tolerance tolerance over over PA, PA, and and [1992] TOL Japaridze [1993] [1993] did did the the same same for for the the logic logic TLR T L R of of the the most most general general relation relation of of Japaridze tolerance for for trees. trees. tolerance All ELH, T TOL TOL All three three logics logics ELH, O L and and TLR T L R are are decidable. decidable. Among Among them them T O L has has the the most TOL most elegant elegant language, language, axiomatization axiomatization and and Kripke Kripke semantics, semantics, and and although although T O L is is TLR, going to just aa fragment just fragment of of T L R , here here we we are are going to have have aa look look only only at at this this intermediate intermediate logic. logic. TOL The The language language of of T O L contains contains the the single single variable-arity variable-arity modal modal operator operator 0 ~:: for for formulas, then This logic logic is any any n, if if A A1,..., Ann are are formulas, then so so is is 0 O(A1,..., A~). is defined defined as as l, . . . , A n ) . This (Al, . . . , A classical logic plus the rule rule --,A/--' the following classical logic plus the ~A/--,O(A) plus the following axiom axiom schemata: schemata: O (A) plus 1. ~ ( 0 , A, 6 ) - ~ (}(C, A A--B,/9) v {}(C, B,/9), 22.. O (}(A)--+ 0 ((A A 1\ A-~0(A)), (A) -+ O --'O ( A)) , 3. 0 ~(C, :D) -+ ~ 0 {}((J, D),, ( 0, 15) ( 0, A, 15) 4. 0 4. ~(C,A,D)-~ (}(C,A,A,.D), ( 0, A, A, 15) , ( 0, A , 15) -+ 0 55.. O 0(A, (}(C)) (}(A/~ -+ O ( A 1\ (}(C)), ( A, 0 0 ( 0)) , ( 0)) --+ 6. 6. 0 ( 0, 0 ( 15)) -+ 0 ( 0, 15) . � 0, 0 (Here ( 0 ) isis identified identified with (Here A A stands stands for for A A1,...,An for an an arbitrary arbitrary n n~>0, (}(()) with T.) -1-.) l , . . . , An for 530 530 G. G. Japaridze and D. de Jongh 15.4. A see Berarducci 15.4. Definition. Definition. A Visser-frame Visser-frame ((see Berarducci [1990]) [1990]) is is aa triple triple (W, {W, R, R, S) S},, where where (W, {W, R) R / is is aa Kripke-frame Kripke-frame for for L L and and S S is is aa transitive, transitive, reflexive reflexive relation relation on on W W such that that R R� c_ S S and, and, for for all all w, W, we we have have w S u R v �~. wRv wry.. such w, u, u, vv EE W, wSuRv A S, If-) A TOL-model TOL-model is is aa quadruple quadruple (W, (W, R, R,S, IF} with with (W, {W, R, R , SS) ) aa Visser-frame Visser-frame com combined with relation Ifbined with aa forcing forcing relation I~- with with the the clause clause w IfIF-O (~(A1 iff there there are are Ul u l ,, .. ... ., , Un un with with ulS u l S . .. .. . SUn Sun such such that, that, w (Al .. .. .., , An) An) iff - Ai . for all all i, wRuii and and U uii IfIF-Ai. for i, WRU Such model is nite. Such aa model is said said to to be be finite, finite, if if W W is is fi finite. 15.5. ((Japaridze Japaridze [1992]) -formula A, A, f-t- TOL 15.5. Theorem. Theorem. [19921) For For any any TOL TOL-formula ToLAA iff iff A A is is valid same is valid in in every every TOL-model; TOL-model; the the same is true true if if we we consider consider only only finite finite TOL-models. TOL-models. ((Japaridze Japaridze [1992]) sound superarithmetic [1992]) Let Let T T be be aa sound superarithmetic theory, theory, arithmetic realization, and let, let, for for ** an an arithmetic realization, ((0 ~ ((AI A 1 ,, .. ... ., , An))* An))* be be interpreted interpreted as as aa natural natural and formalization . . .,T formalization of of "the "the sequence sequence T T + + Ai, A~,..., T + + A� A~ is is tolerant". tolerant". Then, Then, for for any any TOL TOLA iff for every realization *, T fA* . formula formula A, A, fF- TO ToLA iff for every realization *, T FA*. L 15.6. Theorem. 15.6. T heorem. With the the arithmetic arithmetic interpretation in mind, note that that L L is is the the fragment fragment of of TOL TOL With interpretation in mind, note in 1. This in which which the the arity arity of of 0 ~ is is restricted restricted to to 1. This is is because because consistency consistency of of A* A* with with T, expressed T, expressed in in L L by by 0 ~ AA,, means means nothing nothing but but tolerance tolerance of of the the one-element one-element sequence sequence {T + + A*} of theories, expressed in in T O L by by O ~(A). (T A*) of theories, expressed TOL (A) . As for for cotolerance, one can can easily easily show, show, using using theorems theorems 12.7 12.7 and and 12.13 12.13 As cotolerance, one (( i ) {::: iii)) , that ((i) ~ :::} ((iii)), that a a sequence sequence of of superarithmetic superarithmetic theories theories is is cotolerant cotolerant iff iff the the sequence sequence where tolerant. Moreover, where the the order order of of these these theories theories is is reversed reversed is is tolerant. Moreover, it it was was shown shown in in Japaridze [1993] cotolerance - though tolerance - for Japaridze [1993] that that cotolerance though not not tolerance for trees trees can can also also be linear tolerance. tolerance. In particular, aa tree be expressed expressed in in terms terms of of linear In particular, tree of of superarithmetic superarithmetic theories olerant iff theories is is cot cotolerant iff one one of of its its topological topological sortings sortings is. is. Hence, Hence, given given aa tree tree Tr Tr of of modal modal formulas, formulas, cotolerance cotolerance of of the the corresponding corresponding tree tree of of theories theories can can be be expressed expressed in in -:'), where TOL T O L by by O ( }((A s l) V v . .. .. . V vo (~ (A (A~n), where A s I' . . . ' A An:' are are all all the the reverse-order reverse-order topological topological sortings of linear tolerance, can, at sortings of Tr. Tr. Thus Thus TOL, T O L , being being the the logic logic of of linear tolerance, can, at the the same same time, time, be unrestricted) cotolerance be viewed viewed as as the the logic logic of of ((unrestricted) cotolerance over over PA. PA. Just -consistency ((see see defi nition 12.4) Just like like tolerance, tolerance, the the notion notion of of r F-consistency definition 12.4) can can be be generalized generalized to to finite finite trees, trees, including including sequences sequences as as special special cases cases of of trees: trees: aa tree tree Tr -consistent iff there are Tr of of theories theories is is r F-consistent iff there are consistent consistent extensions extensions of of these these theories, theories, of of -conservative over its predecessors the tree. which each one is which each one is r F-conservative over its predecessors in in the tree. Then Then the the corollaries corollaries of of theorems theorems 12.7 12.7 and and 12.13 12.13 generalize generalize to to the the following: following: ((Japaridze Japaridze [1993], 15.7. 15.7. Theorem. Theorem. [1993], PA P A f~- )) For For any any finite finite tree tree Tr Tr of of superarith superarithmetic metic theories, theories, ((a) a) Tr Tr is is tolerant tolerant iff iff Tr Tr is is il rIl-consistent; l -consistent; ((b) b) Tr 1:1 -consistent. Tr is is cotolerant cotolerant iff iff Tr Tr is is El-consistent. Just ILM, in Just as as in in the the case case of of ILM, in the the arithmetic arithmetic completeness completeness theorems theorems for for TOL TOL and essential refl exivity) of and TLR, T L R , the the requirement requirement of of superarithmeticity superarithmeticity ((essential reflexivity) of T T can can be be The Logic of of Provability 531 531 weakened weakened to to nJ1 IE1 � c_ T T if if we we view view these these logics logics as as logics logics of of IT Hi-consistency rather than than l -consistency rather tolerance. tolerance. 15.8. 15.8. Truth T r u t h interpretability i n t e r p r e t a b i l i t y logics logics We want want to to fi finish our discussion discussion of of propositional propositional interpretability interpretability logics logics by by noting noting We nish our that the the closure closure under under modus modus ponens ponens of of the the set set of of theorems theorems of of ILM, or any any other other that ILM, or oA -+ of of the the logics logics mentioned mentioned in in this this section, section, supplemented supplemented with with the the axiom axiom []A --+ A A or or its its equivalent, equivalent, yields yields the the logic logic (in (in case case of of ILM I L M called called ILMW I L M ~)) that that describes describes all all true true principles expressible in in the just as this was principles expressible the corresponding corresponding modal modal language, language, just as this was shown shown to be be the the case case for for L L in section 3. The original sources usually usually contain contain proofs proofs of of both both to in section 3. The original sources versions of of the the arithmetic arithmetic completeness completeness theorems theorems for for these these logics. logics. versions Strannegs [1997] [1997] considers considers infinite infinite r.e. sets of of modal modal formulas formulas of of interpretability interpretability Strannegard Le. sets logic. theorem 5.3 for the logic. He He generalizes generalizes his his theorem 5.3 for the specific specific case case of of interpretability interpretability over over PA PA to the following following theorem. theorem. to the Let e. set Let T T be be aa well-specified well-specified r. r.e. set of of formulas formulas of of interpretability interpretability logic. logic. Then Then T T is is realistic realistic iff iff it it is is consistent consistent with with ILMw I L M ~ .. 15.9. 15.9. Theorem. Theorem. As iinn the the case case of of L (corollary 5.2), 5.2), a a stronger stronger version version of of this this theorem theorem implies implies as as As L (corollary corollary a a uniform uniform version version of of the arithmetic completeness completeness of of ILM I L M with with regard regard to to aa corollary the arithmetic P A. For let us first note the existence PA. For aa further further consequence, consequence, let us first note that that the existence of of Grey-sentences Orey-sentences in .A are in PA, PA, i.e., i.e., arithmetic arithmetic sentences sentences A A such such that that both both PA PA + + A A and and PA PA + +-~A are interpretable interpretable in in PA P A (first (first obtained obtained by by Orey Orey [1961]), [1961]), follows follows immediately immediately from from the the arithmetic arithmetic completeness completeness of of ILM I L M with with regard regard to to PA. PA. In In Strannegard Strannegs ' s terminology terminology c> .p} is this this can can be be phrased phrased as: as: Orey Orey [1961] [1961] showed showed that that the the set set {{T T bc>pp,, T Tb-~p} is realistic. realistic. Orey Orey continued continued by by asking asking what what similar similar sets sets (such (such as as {{T T bc>pp,, T T bC>qq, , T T bC>- ~.(p ( p A1\qq)), , ) , .(T q) } ) are realistic. Let --(T C> b .p) -~p),, .(T -~(T C> b .q -~q), -~(T C> bP p 1\ A q)}) are realistic. Let an an Grey Orey set set be be a a set set of of modal modal .(T C> C) Boolean formulas. formulas formulas of of the the form form (.)(B (-~)(Bb C),, where where B B and and C C are are Boolean formulas. Strannegard Strannegs 's question. can then give can then give the the following following answer answer to to Orey Orey's question. 15.10. 15.10. Theorem. Theorem. with ILM I L M w~ .. with Let e. Grey Let T T be be an an r. r.e. Orey set. set. Then Then T T is is realistic realistic iff iff it it is is consistent consistent 1 6 . Predicate provability logics 16. Predicate provability logics 16.1. 16.1. The T h e predicate p r e d i c a t e modal m o d a l language l a n g u a g e and a n d its its arithmetic a r i t h m e t i c interpretation interpretation The rst order The language language of of predicate predicate provability provability logic logic is is that that of of fi first order logic logic (without (without identity identity or or function function symbols) symbols) together together with with the the operator operator o O.. We We assume assume that that this this language language uses uses the the same same individual individual variables variables as as the the arithmetic arithmetic language. language. Throughout Throughout this this section section T T denotes denotes a a sound sound theory theory in in the the language language of of arithmetic arithmetic containing containing PA. PA. We We also also assume assume that that T T satisfies satisfies the the Lob Lhb derivability derivability conditions. conditions. 532 532 G. Japaridze and D. de Jongh As the previous want to regard each .., P As in in the previous sections, sections, we we want to regard each modal modal formula formula A(PI A(P1,, ...., Pn) n) as , Pn ) by substitution of as aa schema schema of of arithmetic arithmetic formulas formulas arising arising from from A(PI A(P1,, ... ...,Pn) by substitution of arithmetic the predicate predicate letters letters PI> arithmetic predicates predicates P{, P{, ... ...,, P; P,~ for for the P1, ... ...,, P P,n and and replacing replacing D [::l by by Pr PrT(). However, some some caution caution is is necessary necessary when when we we try try to to make make this this approach approach T O . However, ers that precise. In precise. In particular, particular, we we need need to to forbid forbid for for Pt P* to to contain contain quantifi quantifiers that bind bind variables variables occurring occurring in in A. A. 16.2. Definition. Definition. A realization realization for for aa predicate predicate modal modal formula formula A A is is aa function function ** 16.2. A which arithmetic formula (VI , . . . , vv,), which assigns assigns to to each each predicate predicate symbol symbol P P of of A A an an arithmetic formula P* P*(Vl,..., n) , whose whose bound bound variables variables do do not not occur occur in in A A and and whose whose free free variables variables are are just just the the first first variables of of the the alphabetical alphabetical list list of of the the variables variables of of the the arithmetic arithmetic language language if if n n is is nn variables the P. For A, we ne A* the arity arity of of P. For any any realization realization ** for for A, we defi define A* by by the the following following induction induction on on the the complexity complexity of of A: A" •9 in " " , xx,n)))* (XI ' . . . ,,xn), xn ) , in the the atomic atomic cases, cases, (P(XI, (P(Xl,... ) * == P* P*(Xl,... • ers and 9 ** commutes commutes with with quantifi quantifiers and Boolean Boolean connectives: connectives: (VxB)* C)* = (VxB)* = = Vx(B*) Vx(B*),, (B (B -+ -+ C)* = B* ~ C* C*,, etc., etc., B* -+ • DB)* = 9 ((KIB)* = Pr PrT[B*]. T [B* ] . For notation "[]" For an an explanation explanation of of the the notation "[]" see see notation notation 12.2. 12.2. Observe Observe from from this this that that A* A* always always contains contains the the same same free free variables variables as as A. A. We We say say that that an an arithmetic arithmetic formula formula 'P A, if ~o is is aa realizational realizational instance instance of of aa predicate predicate modal modal formula formula A, if 'P ~o= = A* A* for for some some realization realization ** for for A. A. The The main main task task is is to to investigate investigate the the set set of of predicate predicate modal modal formulas formulas which which express express valid valid principles principles of of provability, provability, i.e., i.e., all all of of whose whose realizational realizational instances instances are are provable, provable, or or true true in in the the standard standard model. model. 16.3. 16.3. The T h e situation s i t u a t i o n here here is is not not as as smooth s m o o t h as as in in the t h e propositional p r o p o s i t i o n a l case, case, .. ... . Having Having been been encouraged encouraged by by the the impressive impressive theorems theorems of of Solovay Solovay on on the the decidability decidability of logic, one of propositional propositional provability provability logic, one might might expect expect that that the the valid valid principles principles captured captured by decidability is by the the predicate predicate modal modal language language are are also also axiomatizable axiomatizable ((decidability is ruled ruled out out of of course). course). However, However, the the situation situation here here is is not not as as smooth smooth as as in in the the propositional propositional case. case. The The first first doubts doubts about about this this were were raised raised by by Montagna Montagna [1984J. [1984]. In In fact, fact, it it turned turned out out afterwards afterwards that that we we have have very very strong strong negative negative results, results, one one of of which which is is the the following following theorem theorem on on nonarithmeticity nonarithmeticity of of truth truth predicate predicate logics logics of of provability. provability. 16.4. ((Artiimov Artemov [1985a Suppose T 16.4. Theorem. Theorem. [1985a]) T is is recursively recursively enumerable. enumerable. Then Then J) Suppose (Jor the provability predicate PrT) PrT ) the set Tr predicate modal (/or any any choice choice of o/the provability predicate the set Tr of o/predicate modal formulas formulas all of whose realizational instances are not arithmetic. all o/whose realizational instances are true, true, is is not arithmetic. It It was was later later shown shown by by Vardanyan Vardanyan [1986], [1986], and and also also by by Boolos Boolos and and McGee McGee [1987] [1987] that that Tr Tr is is in in fact fact Il Hi-complete in the the truth truth set set of of arithmetic. arithmetic. l -complete in The Logic of o.f Provability 533 533 Proof 16.4. We P r o o f of of theorem t h e o r e m 16.4. We assume assume here here that that the the arithmetic arithmetic language language contains contains one two-place two-place predicate predicate letter letter E E and and two three-place predicate predicate letters letters A A and and M M,, one two three-place and and does does not not contain contain any any other other predicate, predicate, functional functional or or individual individual letters. letters. Thus, Thus, this language language is is a a fragment fragment of of our our predicate predicate modal modal language. language. In In the the standard standard model model this E(x, y) y),, A(x, A(x, y, z) and and M(x, M(x, y, z) are are interpreted interpreted as as the the predicates predicates x x= : y y,, x x+ + y y= : zz E(x, y, z) y, z) and z, respectively. and x x x x y y= : z, respectively. One One variant variant of of a a well-known well-known theorem theorem of of Tennenbaum Tennenbaum (see (see e.g., e.g., Chapter Chapter 29 29 of of Boolos and Jeffrey Jeffrey [1989]) [1989]) asserts asserts the the existence existence of of an an arithmetic sentence f3 /3 such such Boolos and arithmetic sentence that: that: (1) f3 19 is is true true (("true" here always always means means "true "true in in the the standard standard model" model"),), "true" here (1) (2) any any model model of of/3, with domain domain w co,, E E interpreted interpreted as as the the identity identity relation, relation, f3, with (2) and and A A and and M M as as recursive recursive predicates, predicates, is is isomorphic isomorphic to to the the standard standard model. model. We We assume assume that that f3 ~9 conjunctively conjunctively contains contains the the axioms axioms of of Robinson's Robinson's arithmetic arithmetic Q, Q, including the the identity identity axioms. axioms. Therefore, Therefore, using using standard standard factorization, factorization, we we can can pass pass including from any any model model D D of of/9f3 with with domain co and and such such that E, A A and and M M are are interpreted interpreted from domain w that E, as recursive recursive predicates, predicates, to to a a model model D' D' which which satisfies satisfies the the conditions conditions of of (2) and which which as (2) and (2) can is is elementarily elementarily equivalent equivalent to to D D.. Thus, Thus, (2) can be be changed changed to to the the following: following: f3, with E, A (2') (2') any any model model D D of of ~, with domain domain w co and and E, A and and M M interpreted interpreted as as recursive recursive predicates, predicates, is is elementarily elementarily equivalent equivalent to to the the standard standard model model (i.e., ). (i.e., D D 1= b , -), iff iff , -y is is true, true, for for all all sentences sentences , "),). Let Let C C be be the the formula formula y) Vv D..., E(x, y)) y)) 1\ ^ \:Iwx,, yy ( DE(x, y) y, z)) z)) 1\A (oA(x, y, z) V v D.. [3-,., A(x, A(x, y, x, y, y, zz (DA(x, y, z) \:IVx, (OM(x, y, z) V v D a ~..., M(x, M(x, y, y, z)) z)).. x, y, y, zz (DM(x, y, z) \:IVx, The The following following lemma lemma yields yields the the algorithmic algorithmic reducibility reducibility of of the the set set of of all all true true arithmetic arithmetic formulas theorem, is arithmetic) to formulas (which, (which, by by Tarski's Tarski's theorem, is non nonarithmetic) to the the set set Tr, Tr, and and this this proves proves the theorem. the theorem. 16.5. For 16.5. Lemma. Lemma. For any any arithmetic arithmetic formula formula cp ~o,, cp ~ is is true true if if and and only only if if every every realizational realizational instance instance of of f3 j9 1\ AC C -+ --+ cp ~o is is true. true. Proof. Suppose cp realization for 1\ C* Proof. � ----~ :: Suppose ~ is is true, true, ** is is a a realization for/9f3 1\ AC C --+ + cp ~ and and f3* jg*A C* is is true. We that, since true. We want want to to show show that that cp* ~o* is is also also true. true. It It is is not not hard hard to to see see that, since T T is is consistent consistent and and recursively recursively enumerable enumerable (this (this condition condition is is essential!), essential!), the the truth truth of of C* C* means means that that the the relations relations defined defined on on w co in in the the standard standard model model by by the the formulas formulas E* E*,, domain w that, for A* A* and and M* M* are are recursive. recursive. Let Let us us define define a a model model D D with with domain co such such that, for all all k, m, m, nn E~ w, co, k, D true, D 1= b E(k, E(k, m) m) iff iff E*(k, E*(k, m) m) is is true, D m, n) n) iff n) is true, D 1= b A(k, A(k, m, iff A*(k, A*(k, m, m, n) is true, D m,, nn)) is D 1= b M(k, M ( k , mm,, nn) ) iff iff M* M * ((k, k,m is true. true. 534 534 G. G. Japaridze Japaridze and D. de Jongh Observe that that for which the the realization Observe for every every arithmetic arithmetic formula formula 'Y 7 (for (for which realization '* is is legal) legal),, we we have particular D f3, and es the have D D F= ~ 'Y 7 iff iff 'Y 7"' is is true. true. In In particular D F= ~ #, and thus thus D D satisfi satisfies the conditions conditions of of (2'), i.e., i.e., D D is is elementarily elementarily equivalent equivalent to to the the standard standard model, model, whence (as tp ~ is is true) true) (2'), whence (as D D F= ~ tp ~,, whence whence tp ~*' is is true. true. � : Suppose tp is Let r Suppose ~ is false. false. Let '* be be the the trivial trivial realization, realization, i.e., i.e., such such that that E*(x, E(x, V) y),, A' A*(x, z),, M' M*(x, = M(x, M(x, y, z) z).. Then Then f3' #* = = f3, #, E' (x, y) y) == E(x, (x, y, z) z) == A(x, A(x, y, z) (x, y, z) = ' = tp ~a* = tp ~a and and therefore therefore it it suffices suffices to to show show that that f3 #A A C' C* -+ --+ tp ~ is is false, false, i.e., i.e., that that f3 #A A C' C* is true by (1), and y, is true. true. But But f3 # is is true by (1), and from from the the decidability decidability in in T T of of the the relations relations x x -= y, -1 xx + + y= = z and and x x xy= = zz,, it it follows follows that that C' C* is is also also true. true. Formalizing Formalizing in in arithmetic arithmetic the the idea idea employed employed in in the the above above proof, proof, Vardanyan Vardanyan [1986] [1986] also proved proved that that if if T T is is recursively recursively enumerable, enumerable, then then the the set set of of predicate predicate modal modal also formulas formulas whose whose realizational realizational instances instances are are provable provable in in T T (or (or in in PA) PA) is is not not recursively recursively enumerable and -complete. enumerable and is is in in fact fact II II2-complete. 2 There There is is one one perhaps perhaps even even more more unpleasant unpleasant result result which which should should also also be be mentioned mentioned here. T, the here. For For recursively recursively enumerable enumerable T, the answer answer to to the the question question whether whether aa predicate predicate modal formula expresses expresses aa valid valid provability provability principle, principle, turns turns out out to to be be dependent dependent modal formula on is, on on the the choice choice of of the the formula formula Pr PrT, that is, on the the concrete concrete way way of of formalization formalization of of T , that code of xed the predicate the predicate "x "x is is the the code of an an axiom axiom of of T" T",, even even if if aa set set of of axioms axioms is is fi fixed 's theorems (Artemov Note that (Art~mov [1986]). [1986]). Note that the the proofs proofs of of Solovay Solovay's theorems for for propositional propositional provability logic are provability logic are insensitive insensitive in in this this respect respect and and actually actually the the only only requirement requirement is is that the the three three L6b-conditions Lob-conditions must that must be be satisfied. satisfied. 16.6. still not 1 6 . 6 . . ... . . but b u t still not completely c o m p l e t e l y desperate desperate Against Against this this gloomy gloomy background background one one still still can can succeed succeed in in obtaining obtaining positive positive results results in in two two directions. directions. Firstly, Firstly, although although the the predicate predicate logic logic of of provability provability in in full full generality generality is axiomatizable, some is not not (recursively) (recursively) axiomatizable, some natural natural fragments fragments of of it it can can be be so so and and may may be the choice the formula be stable stable with with respect respect to to the choice of of the formula Pr PrT. T. And And secondly, secondly, all all the the above-mentioned above-mentioned negative negative facts facts exclusively exclusively concern concern recur recursively theories, and sively enumerable enumerable theories, and the the proofs proofs hopelessly hopelessly fail fail as as soon soon as as this this condition condition is is removed. removed. There There are are however however many many examples examples of of interesting interesting and and natural natural theories theories which which are are not not recursively recursively enumerable enumerable (e.g. (e.g.,, the the theories theories induced induced by by w-provability w-provability or or the other strong strong concepts concepts of provability mentioned the other of provability mentioned in in section section 8) 8),, and and it it well well might might be be that that the the situation situation with with their their predicate predicate provability provability logics logics is is as as nice nice as as in in the the propositional propositional case. case. The The main main positive positive result result we we are are going going to to consider consider is is the the following: following: the the "arith "arith's theorems, metic metic part" part" of of Solovay Solovay's theorems, according according to to which which the the existence existence of of aa Kripke Kripke countermodel (with countermodel (with a a transitive transitive and and converse converse well-founded well-founded accessibility accessibility relation) relation) implies arithmetic nonvalidity formula, can implies arithmetic nonvalidity of of the the formula, can be be extended extended to to the the predicate predicate level. level. This This gives gives us us aa method method of of establishing establishing nonvalidity nonvalidity for for a a quite quite considerable considerable class class of of predicate modal formulas. predicate modal formulas. The Logic Logic of o/Provability The Provability 535 535 16.7. Kripke-models K r i p k e - m o d e l s for for the t h e predicate p r e d i c a t e modal m o d a l language language 16.7. A A Kripke-frame Kripke-ffame for for the the predicate predicate modal modal language language is is aa system system M=(W,R,{Dw}~ew), where (W, (W, R) R) is is aa Kripke-frame Kripke-frame in in the the sense sense of of section section 2, 2, {{D~}wew are nonempty nonempty where DW } WEW are sets (("domains of individuals" individuals")) indexed indexed by by elements elements of of W W such such that that if if w wR R uu,, then then sets "domains of Dw Du,, and and aa Kripke-model Kripke-model is is aa Kripke-frame Kripke-frame together together with with aa forcing forcing relation relation IIIF,, D w �g Du which is is now now aa relation relation between between worlds worlds w w E9W W and and closed closed formulas formulas with with parameters parameters which in Dw D~;; for for the the Boolean Boolean connectives connectives and and D El,, IIIF behaves behaves as as described described in in section section 2, 2, and and in we have have only only the the following following additional additional condition condition for for the the universal universal quantifi quantifier: we er: • 9w w II-VxA(x) IhVxA(x) iff iff w w IIhA(a) for all all aa E9Dw D~,, I- A(a) for and a a similar one for for the the existential existential quantifier. quantifier. A A formula formula is is said said to to be be valid valid in in aa and similar one Kripke-model (W, (W, R, R, {D~}w~w, IF}, A is is forced forced at at every every world world W w E9W W.. Such Such a a if A Kripke-model {Dw } WE W , 11) , if model is said said to to be be finite finite if if W W as as well well as as all all Dw D~ are are finite finite.. . model is 16.8. The T h e predicate p r e d i c a t e version v e r s i o n of of Solovay's Solovay's theorems theorems 16.8. For every every predicate predicate modal modal formula formula A, let REFL( REFL(A) denote the the universal universal closure closure For A, let A ) denote of 1\ A {{DB ElB --+ --+ B B[ I [:]B SD},, where where Sb Sb is is the the set set of of the the subformulas subformulas of of A A.. of DB E9Sb} (Artemov 1990]) . For (Art~mov and and Japaridze Japaridze [1987, [1987,1990]). For any any closed closed predicate predicate modal formula A, A, modal formula (a) if A is not with aa transitive converse (a) if A is not valid valid in in some some finite finite Kripke-model Kripke-model with transitive and and converse well-founded accessibility relation, relation, then exists aa realization realization *' for A such that well-founded accessibility then there there exists for A such that T Jz T V A*, A' , (b) if REFL A is is not not valid valid in in such such aa model, model, then there exists realization *' (b) if R E F L ((A) A ) ---+ + A then there exists aa realization for for AA such such that that A* A' is is false. false. 16.9. Theorem. Theorem. 16.9. Proof. We We prove only clause (b) as as an exercise for for the the reader. Proof. prove here here only clause (a), (a), leaving leaving (b) an exercise reader. Some Some details in are in redundant if if we we want want to to prove prove only (a),, but are details in this this proof proof are in fact fact redundant only (a) but they they are of assistance in in passing of assistance passing to to aa proof proof of of (b). (b). w, IF) model with with the above-mentioned prop Assume that (W, R, R, {Dw}w Assume that (W, {Dw}w E9 w, is aa model the above-mentioned prop1 1- ) is erties A isis not not valid. valid. As As before, before, without without loss loss of of generality generality we we may may suppose suppose erties in in which which A that 1 , .. .. .., ,l }I}, , 11 is We suppose is the the root root and and 1JFA. 1 � A. We suppose also also that that DwC_w Dw � w and and that W W== {{I, 0o E9 Dw Dw for for each each w w E9 W. W. Let Let us us define define aa model model (W', (W', R', R', {D~}~ew,, {D�}wE WI , IF'} II- ' ) by by setting setting • W' {O} , 9 w ' : w=u W U {0}, • 9 • R R'' ==R R uU {(O, w) l w E W} , 9 D~ D� -= D1 D l and, Dw , and, for for all all w w E9W, W, D~ D� -= D~, any atomic atomic formula formula P, P, 00 IF'P II- ' P iff iff 11 IFP 1 1- P and, and, ifif w w E9 W, W, w w IF'P il- ' P iff 9 for for any iff w w IFP. II- P. • G. G. Japaridze and D. de Jongh 536 536 We accept the the definitions the Solovay the sentences Limw from We accept definitions of of the Solovay function function hh and and the sentences Lim~ from section the only only additional step is the following: section 33 without without any any changes; changes; the additional step is the following: For {Dw IIxx E9W For each each aa from from D D= - U{Dw W}} we we define define an an arithmetic arithmetic formula formula 'Y %(x) with a (x) with only only x x free free by by setting setting U 'Y%(x) = V V {f3t 3t � ~<xx(( hh(t) ( t ) ==hh(x) (x) = = wWAA- -, ~ 3::Iz z < < tt(h(z) ( h ( z ) ==ww) ) AAx X = t= t + + aa)) l Iaa 9E Dw}. a (x) = Thus, Thus, using using the the jargon jargon from from section section 14, 14, 'Y %(x) says that that we we have have reached reached some some world world a (x) says such such that that aa E9Dw D~,, at at the the moment moment x x we we are are still still at at w w,, and and exactly exactly aa moments moments have have passed since we we moved moved to to this (we assume the first first "move" "move",, to to the the world passed since this world world (we assume that that the world 0, happened happened at at the the initial initial moment 0). We We define define the predicates 'Y ~,� by by 0, moment 0). the predicates W w and for each each 00 # # aa E9D, D, 'Y�(x) 7~(x) = = 'Y %(x), •9 for a (x) , and •9 'Yb(x) D\ {r(ax(x) ~,~)(x) = = '%(x) V{% ) I] a a E9 D \ { {O}}. 0}}. YO (x) Vv - V -, (It is is easy easy to to check check that that the left disjunct disjunct of of 'Yb 7~(x) is redundant; redundant; it it implies implies the the right right (x) is (It the left disjunct.) Since we employ employ the the same same Solovay Solovay function function hh as as in in section 3.2 disjunct.) Since we section 3, 3, lemma lemma 3.2 continues continues to to hold. hold. In In addition, addition, we we need need the the following following lemma: lemma: 16.10. 16.10. Lemma. Lemma. T (i) (i) T fF -, --1(r�(x) (')'~(x) A A 'Y�(x)) "y~(x)) for for all all aa # # b, (ii) T (ii) T fF Limw Limw � --+ 1\ {::Ix 'Y�(x) I a Ee Dw} Dw} for for all all W w E9W', W', (iii) (iii) V T T fF Limw Lim~ � --+ 'v'x( v x ( V {'Y�(x) {7,~(x) I] a a E9Dw}) D~}) for for all all W w E9 W' W'. . Proof. (i) Proof. (i):: The The formulas formulas 'Y %(x) and 'Yb(X) %(x) for for aa # =/-b are are defined defined so so that that each each disjunct disjunct a (x) and of (x) , of 'Y %(x) is inconsistent inconsistent with with each each disjunct disjunct of of 'Yb(X) %(x).. And And the the right right disjunct disjunct of of 'Yb ")'~(x), a (x) is by by definition, definition, is is inconsistent inconsistent with with each each 'Y %(x), ~ O. 0. a (x) , aa # (ii) T+ (ii):: Suppose Suppose aa E9Dw D~ and and argue argue in in T + Limw Lim~.' Since Since W w is is the the limit limit of of h h,, there there is in w is aa moment moment tt at at which which we we arrive arrive in w,, and and stay stay there there for for ever ever (more (more formally: formally: ::Iy < we a) we have have -, --13y < tt (h(y) (h(y) = = w) w) and and 'v'y Vy � >1tt (h(y) (h(y) = - w)). w)). Then, Then, by by definition, definition, �/ %(t + a) a (t + holds, holds, whence holds, whence whence 'Y�(t 7'~(t + + a) a) holds, whence ::Ix 3x 'Y�(x) -),~(x).. And And so so for for each each aa E9Dw Dw.. (iii): x. We (iii): Argue Argue in in T T+ + Limw Lim~.. Consider Consider an an arbitrary arbitrary number number x. We must must show show that that 'Y�(x) Dw ' The definition of that, either 7~(x) holds holds for for some some a a E9Dw. The definition of h h implies implies that, either h(x)R' h(x)R' w w,, or or h(x) w; in both cases h(x) = = w; in both cases we we then then have have Dh Dh(,) c_ Dw D~.' Let Let tt be be the the least least number number such such ( x) � that h(x) , and t. By (and thus that h(t) h(t) = = h(x), and let let aa = = x x- t. By definition, definition, if if aa E9Dh Dh(,) thus a a E9Dw D~),) , (x) (and then then 'Y %(x) holds, whence whence 'Y�(x) 7~(x) holds holds and and we we are are done; done; and and if if aarrt Dh Dh(,), then (the (the (x) , then a (x) holds, .,-t right right disjunct disjunct of) of) 'Yb(x) "y~)(x) holds holds and and we we are are also also done, done, because because 00 E9Dw Dw.' We P, let We now now define define aa realization realization '*.. For For each each n-place n-place predicate predicate letter letter P, let P' P* be be V{Lim~,A'y'~,(vl) A ... A'y'~.(v,)I~,, 9 ,~eD~, wll-'P(ax,...,an)}. 537 537 The The Logic Logic of of Provability Provability Let Let B B be be aa predicate predicate modal modal formula formula with with precisely precisely Xl x t ,, .. ... . ,,xn free. Xn free. Then, Then, for for each each w w E9lV W and and for for all all aI, al, .. .. .. ,, an an Ee Dw Dw,, . . /\A'Y�J xn) --+ (a) (a) if if w w lfI F' (B(al, ' ( B ( a l ,·. .. .·, , an) an),, then then T T f-~- Lim Lim~ (Xl) /\A .... "y~ (xn) --+ B B*; w /\A'Y~/~1 �l (xd *; --+ -,-- B f- Limw (xI) /\ ..... . /\A%Jxn) (b) if i f ww~W'(B(al, ' ( B ( a l , . . .., .a.n,)an) , , then then T TFLim~ /\A%l 3,~1(Xl)A 3'~(xn)--+ B*. (b) *. Proof. Proof. We We proceed proceed by by induction induction on on the the complexity complexity of of B. B. Suppose Suppose B(XI, B ( X l , .. .. .., ,xxn) n) is 'Y�1 (xd is one of the (xn) is atomic. atomic. If If w w If-' IF' B(al B ( a t ,'. .. .. ,. , an) an),, then then Lim Lim~ (xl) /\A .. ... . /\A 'Y 3'~ (xn) is one of the w /\A')"~I �n disjuncts an), then disjuncts of of B B** and and the the desired desired result result is is obvious. obvious. If If w w W' ~ ' B(al B ( a l ,, .. ... . ,,an), then that that formula formula is is not not aa dis disjunct of B B** and, and, according according to to lemma lemma 3.2(b) 3.2(5) and and 16.10(i), 16.10(i), it it implies implies j unct of in T T the the negations negations of of all all the the disjuncts disjuncts of of B B*. in *. Next Next suppose suppose that that B(XI B ( x t , ,. .. .., . , xn) Xn) is is Vy Vy C(y, C(y, Xl X l ,, .. ... ., , Xn) Xn).. If I f- Vy C(y, f- C(b, If W wIFVy C ( y , aa~l ,, ... .. ,. a, nan), ) , then then w w IIF C(b, aa ll ,, .. ... ,. a, an) n ) for for all all bb E9Dw D~.. Then, Then, by by the the induction hypothesis, hypothesis, for for all all bb Ee D D~, induction w, T T fF Limw LimwA'TIb(Y) (V(y, Xl, z l , . ·. .· ,·x,nxn))*. ) ) *. 'Y�n (xn) --+ (C(y, /\ 'Y�(Y) /\Aq/al(Xl) 'Y�1 (xd /\A ..... . /\A")'la,,,(xn)-+ 16.11. 16.11. Lemma. Lemma. Therefore, Therefore, I- Limw T fLim,,, /\ A (V b� (y) I bt, E Dw } ) /\ . . . , xn))*. T ^ 'Y�1 4, (Xl) (x,) /\^.... . . /\ 'Y�JXn) --+ (C(y, (c(,,, Xl, x,,..., Note that that there there is is no no free free occurrence occurrence of of yy in in either either Limw Lim~ or or 'Y�1 711(xd (Xl) /\ A ... .. . /\ A 'Y�JXn 7~, (xn). Note ). Universal quantification quantification over over yy gives gives Universal T 'Y�JXn) --+ 'Y�1 (Xl) /\ ..... . /\h 7'~(xn) T i -f-LLimw i m ~ n V/\YVy [ V (V { 7 ' bb� ( y )(y) Il bb eEDDw ~ } } )) /\ AT'~i(Xl)h --~ Vy(C(y, Xn))*. Vy(C(y, x i , , .. .. .. ,,Xn))*. Xl / H \ (V b� (y) IJbeD,,,} b E Dw} )I VY(V{',/~(y) the conjunct By By lemma lemma 16.10(iii) 16.10(iii),, we we can can eliminate eliminate the conjunct Vy antecedent of and conclude that antecedent of the the above above formula formula and conclude that % - - / iinn the the T fF-Lim~ A'y'a,(xl)A l , . .. .., .X, nXn))*. ))*. T Limw /\ /\ 'Y�JXn) --+ Vy (C(y, xXl, 'Y�1 (Xl) /\ ..... . A~/'~(Xn)--+Vy(C(y, then there the other hand wwW If If on on the other hand ~ Vy V y CC(y, ( y , aal, l , ... .. ,. a, nan) ) , , then there is is bb eEDDw ~ such such that that w WC(b, an). By By the the induction induction hypothesis, hypothesis, w ~C(b, a!, a l , ... .. ,. , an). (C(y, xl, , xn))*. Limw A/\ g/~(y) . . . A/\ 'Y�JXn) (Xl) A/\ ... TT i-f- Lim~ g/~ (xn) "-+-" Xl , .. .. .. ,Xn))*. --+ -, (C(y, 'Y�1 (Xl) 'Y�(Y ) A/\ T'~I contains yy free, /\ 'Y�n (xn) contains Again, neither neither Lim~ Limw nor nor 3'Y�1 free, and and existential existential Again, ' ~ (Xl) ( X l ) /\ A ... .. . A3'~(xn) quantification over yy gives gives quantification over , xn))*. . . . A'/a~ --+ 3:3yy --,' ( (CC(y, f- Lim~ Limw A/\ gY'Y~(Y) :3y 'Y� ( Y) A/\ T'a, Xl, .. .. .. ,xn))*. TT F(xn) --+ ( y , xl, 'Y�1 (Xl) (Xl) A/\ ... /\ 'Y�JXn) According to to lemma lemma 16.10(ii), 16.1O(ii), TT t-f- Lim~ Limw --+ Therefore, According :3y 7~(y). --+ 9y 'Y� ( Y ) . Therefore, . . . A/\T'Y�n T ' ~ ((Xn) X n ) ---+ ~ - -, ' ( V(Vy y CC(y, ( y , xXll ,,. . . , x,nxn))*. ))*. T F-Lim~ f- Limw A3"~l(Xl)A /\ 'Y�1 (Xl) /\ ... . . • Finally, suppose suppose that that BB is is rqC. DC. IfIf ww llf-F DC(a then for for each each uu such such that that Finally, D C ( a ll,,. .. .. ,.a,nan), ) , then and, by by the the induction induction hypothesis, hypothesis, we have have uu IIf-FC(al an ) and, wR'u, we wR'u, C ( a l ,, .. ... ,. , an) T F-Limu AT~,I(X,)A ... Ag/a,,(x,)-+ (C(Xl,...,Xn)) *. G. G. Japaridze Japaridze and and D. D. de de Jongh Jongh 538 538 Therefore, Therefore, T F ( V{Lim~ I wR'u}) A 3,' (Xl)A...A 3/~. (x,)--~ (C(xl,..., xn))*, and, by by the the first first two two Lob Lhb conditions, conditions, and, T F PrT[( V{Lim~, IwR'u}) A ")lal(Xl)A ... A g/'a,(Xn)]-+ (E]C(Xl, . . .,Xn))*. Observe that that the the formulas formulas l%(x) are primitive primitive recursive recursive and and we we have have that that Observe a (X) are T lF, %(x) --+ PrTb PrT[%(x)]; together with with lemma lemma 3.2(d) 3.2(d) this this means means that that T a (X) -+ a (x)] ; together T I-F Limw Lim~ /\A I�' "Y~I(Xl (Xl)) /\A ... .. . /\A I� ")'~. (x.) -+ --~ T n (Xn) im PrT[(( V V{Lim~ wR'u})A "y'a,(Xl) (Xl) /\A . . 9. /\A9 I�JXn)] 3''~.(x,)].. {L u II wR'u} PrT[ ) /\ I�, , . . . , Xn))* . Thus, we we get get T T FI- Limw Lim~A/\ I� '~lal(Xl '~la.(Xn)~ ([-]C(Xl,...,Xn))*. Thus, n (Xn ) -+ (oC(XI ' (Xl)) /\A .. ... . /\A I� If W: OC(aI , . . . , an), then If w wP~[:lC(al,...,a,), then there there is is u u such such that that wR' w R ' uu and and uu ~W: C(al C ( a l ,, .. ... ., a, an). , ) . By By the induction induction hypothesis, hypothesis, the T F Lim, A')/~i(Xl)A ... A ' ) / ~ . ( x , ) ~ - , ( C ( x l , . . . , x , ) ) * . Therefore, Therefore, T F ( C ( X l , . . . , x n ) ) * -~-, (Limu A'y'a,(Xl)A ... AqI~, (x,)), 'Y�JXn))] , · · . , xn)) T FI- PrT[(C(xl PrT[(C(xl,..., x,))']* ] -+ ~ PrT[--(Lim, (Xl)A... (x,))],, PrT["" (Limu /\A-Y',l /�, (Xl) /\ . . . /\A 7'a. T . . . A/\ T'.. 'Y�JXn))] (OC(XI , .. .. .. ,,x.))*. Xn)) * . TT F1- -..., PrT[(x.))I -+ ~ - "" (E]C(xl, 'Y�, (Xl) (Xl) A/\ ... PrT["" (Lim, (Limu A/\ g/,, On the other On the other hand, hand, we we have have (Xn)] --+ -+ 'Y�n (Xn)] (Xl) A/\ . .. .. . A/\ ")/a. T PrTb�, (Xl) T FI- -..., PrT[PrT["" Lim~] Limu] A/\ PrT[')'lal (Xl) A1\ . . . A~la.(Xn))] -,PrT[-(Lim~, /\ 'Y�JXn ))] ..., PrT["" (Limu A/\ T~x 'Y�, (Xl) . . . Oq --+ -+ (}(p (this is realizational instance instance of the principle principle ()p () (p A which is provable (this is aa realizational of the (}p A q) which is provable /\ E]q /\ q) in ) . According we have According to to lemma lemma 3.2(c), 3.2(c) , and and since since TT FI- ")'~(x) 'Y�(x) --~ -+ PrT[7~(x)], PrTb�(x)] , we have in gK). (Xn) --+ -+ TT FI- Limw 'Y�n (x.) (Xl) A/\ . .. .. . A/\ "y~. Limw A/\ "7~, I�, (x,) (Xn)] . . . . A/\ ~'~. ) I�n (x,)]. ] -, PrT[-, Lim~] A PrT[~/al (x,) A . . . /\ PrTb�, (Xl ..., PrT["" Limu /\ Therefore, T F Limw h 7'hi (Xl) A . . . A 'Tlar,(Xn)--~ ~ ( [ - ] C ( X l , . . . ,Xn))*. To finish finish the the proof proof of of theorem theorem 16.9: 16.9: since since A A isis closed closed and and l~ZA, 1 w: A, we we have have by by To lemma 16.11, 16.11, TT FI- Liml Liml ~-+ ~..., A*. A* . By By lemma lemma 3.2(f), 3.2(f) , Liml Liml isis consistent consistent with with T, T, and and lemma consequently -~ -I consequently TT ~J.l A*. A' . The Logic Logic of ofProvability Provability The 539 539 16.12. FFurther 16.12. u r t h e r ppositive o s i t i v e rresults esults One of of the the applications applications of of theorem theorem 16.9 16.9 isis the the following. following. Consider Consider the the fragment fragment One of our predicate predicate modal modal language language which which arises arises by by restricting restricting the the set set of of variables variables to to of our x. In In this this case, case, without without loss loss of of generality, generality, we we may may assume assume that that one single single variable variable x. one every predicate predicate letter letter isis one-place. one-place. Since Since the the variable variable xx isis fixed, fixed, itit isis convenient convenient to to every omit itit in in the the expressions expressions Vx, Vx, P(x), P(x) , Q Q(x) and simply simply write write V, V, p, p, q,q, . . ... . In In fact, fact, omit ( x ) ,, . .. .. . and and V. V. we then then have have aa bimodal bimodal propositional propositional language language with with the the modal modal operators operators []D and we The The modal modal logic logic Lq, Lq, introduced introduced by by Esakia Esakia [1988], [1988], isis axiomatized axiomatized by by the the following following schemata: schemata: . . 1 . all all propositional propositional tautologies tautologies in in the the bimodal bimodal language, language, 1. 2 . the the axioms axioms of of LL for for [3, D, 2. 33.. the the axioms axioms of of $5 S5 for for V, V, i.e., i.e., • � B) B) --+ � (VA � VB), 9V V (A (A --+ (VA --+ VB), • VA � A, 9 VA--+A, • A (3 (3 abbreviates abbreviates -~, V-, ), 9 3A 3A � ~ V V 33 A V-~ ), 44.. D VA []V A -� + VV[ DA, ]A, together ponens, A/ together with with the the rules rules modus modus ponens, A~ DA [3A and and A/VA. A/VA. For For this this logic logic (the (the language language of which is predicate modal modal language) the of which is understood understood as as aa fragment fragment of of the the predicate language) we we have have the following modal completeness completeness theorem: following modal theorem: - Lq A f~-Lq A iff iff A A is is valid valid in in all all finite finite predicate predicate Kripke-models Kripke-models with with aa transitive transitive and and converse converse well-founded well-founded accessibility accessibility relation relation.. 16.13. 16.13. Theorem. Theorem. (Japaridze (Japaridze [1988a,1990a]) [1988a,1990a]) For For any any Lq-formula Lq-formula A, A, In In view view of of the the evident evident arithmetic arithmetic soundness soundness of of Lq, Lq, this this modal modal completeness completeness theorem theorem together together with with the the above above predicate predicate version version of of Solovay's Solovay's first first theorem theorem implies implies the the arithmetic arithmetic completeness completeness of of Lq: Lq: 16.14. For any any Lq-formula Lq-formula A, A, f-Lq F-LqA A iff iff every every realizational realizational instance instance 16.14. Corollary. Corollary. For of of A A is is provable provable in in T. T. Japaridze Japaridze [1988a,1990a] [1988a,1990a] also also introduced introduced the the bimodal bimodal version version Sq Sq of of SS and and proved proved that that f-Sq ~Sq A A iff iff every every realizational realizational instance instance of of A A is is true. true. The The axioms axioms of of Sq Sq are are all all A � A, theorems theorems of of Lq Lq plus plus D []A--+ A, and and the the rules rules of of inference inference are are Modus Modus Ponens Ponens and and A/VA. A/VA. Taking Taking into into account account that that we we deal deal with with aa predicate predicate language, language, the the requirement requirement of of finiteness finiteness of of the the models models in in theorem theorem 16.9 16.9 is is aa very very undesirable undesirable restriction restriction however. however. In In Japaridze Japaridze [1990a] [1990a] aa stronger stronger variant variant of of this this theorem theorem was was given given with with the the condition condition of of finiteness finiteness replaced replaced by by aa weaker weaker one. one. What What we we need need instead instead of of finiteness, finiteness, is is roughly roughly the the following: following: G. Japaridze and D. de Jongh G. 540 540 (1) The The relations relations W w eE W, W, w w RRuu, , aa eEDDw ~ must must be be binumerable binumerable in in T T (see (see defini defini(1) tion tion 12.1), 12.1), and and this this fact fact must must be be provable provable in in T. T. (2) (2) The The relation relation IfIF must must be be numerable numerable in in T T and and T T must must prove prove that that fact. fact. To To defined for be more be more precise, precise, IIIF need need not not be be defined for all all worlds worlds and and all all formulas, formulas, but but only only for which are falsify the for those those which are needed needed to to falsify the formula formula A A in in the the root root of of the the model model (i.e., (i.e., - ..., B - B nor in in some some cases cases we we may may have have neither neither w w IIlFB nor w w 11IF-~ B);) ; T T should should just just prove prove that that w II- B or or w II-C) , IF behaves behaves "properly" "properly",, e.g., e.g., w w lII-B b B ===} ==# w w ~].jL..., - ~ BB, , w w lII-B F B vVC C ===} ==~ ((wlFB wlFC), IIw lllF-..., - ~(B ( B vvCC) ) ===} ==~ (w (wlF-~B and w wlF-~C), w l h B and l l-..., C) , .. .. .. . . (3) T also also must must "prove" "prove" that that the the relation relation R R is is transitive transitive and and converse converse well well(3) T founded. course, well-foundedness founded. Of Of course, well-foundedness is is not not expressible expressible in in the the first first order order language, language, and T T should should somehow somehow simulate simulate aa proof proof of of this this property property of of R. This is is the the case case if, if, R. This and e.g., e.g., T T proves proves the the scheme scheme of of R-induction R-induction for for the the elements elements of of W, W, i.e., i.e., T T f- Vw E W w (Vu ( wR u -Hp(U)) ---+ cp (w) ) ---+ -+ Vw v,, E W w cp(w). We want want to to end end this this section section by by mentioning mentioning one one last last positive positive result. result. Let Let QL QL be be the the We logic logic which which arises arises by by adding adding to to L L (written (written in in the the predicate predicate modal modal language) language) the the axioms axioms and and rules rules of of the the classical classical predicate predicate calculus. calculus. Similarly, Similarly, let let QS QS be be the the closure closure of S S with with respect respect to to classical classical predicate predicate logic. logic. of (Japaridze 1991j). Suppose (Japaridze [1990a, [1990a,1991]). Suppose T T is is strong strong enough enough to to prove prove all true -sentences, and all true ITI Hi-sentences, and A A is is aa closed closed predicate predicate modal modal formula formula which which satisfies satisfies one one of of the the following following conditions: conditions: 16.15. 16.15. Theorem. Theorem. (i) in the some occurrence A, or or (i) no no occurrence occurrence of of aa quantifier quantifier is is in the scope scope of of some occurrence of of 0 [] in in A, (ii) no some other (ii) no occurrence occurrence of of 0 • is is in in the the scope scope of of some other occurrence occurrence of of 0 [] in in A, A, or or (iii) A (iii) A has has the the form form O9n ..L ---+ ~ B. B. Then Then we we have: have: (a) (a) fF QL qt. A A iff iff all all realizational realizational instances instances of of A A are are provable provable in in T, T, (b) (b) f-QS F qs A A iff iff all all realizational realizational instances instances of of A A are are true. true. (Of (b) is (ii) and (Of course, course, clause clause (b) is trivial trivial in in case case (iii).) (iii).) The The proof proof for for the the (ii) and (iii) (iii)fragments fragments in in Japaridze Japaridze [1990a] [1990a] is is based based on on the the above-mentioned above-mentioned strong strong variant variant of of the the 's theorems. 's theorems predicate predicate version version of of Solovay Solovay's theorems. Both Both Vardanyan's Vardanyan's and and Artemov Artiimov's theorems on (i) and on nonenumerability nonenumerability and and nonarithmeticity nonarithmeticity hold hold for for the the (i) and (ii)-fragments (ii)-fragments as as well, well, but 16.15. The but this this is is not not in in contradiction contradiction with with theorem theorem 16.15. The point point is is that that the the use use of of Tennenbaum in the only on Tennenbaum's' s theorem theorem in the proofs proofs of of these these negative negative results results is is possible possible only on assumption recursive enumerability T, whereas theory which assumption of of the the recursive enumerability of of T, whereas no no consistent consistent theory which proves recursively enumerable. there are proves all all the the true true IT IIl-sentences can be be recursively enumerable. Thus Thus there are no no I -sentences can immediate against the that QL immediate objections objections against the optimistic optimistic conjecture conjecture that QL and and QS QS are are complete complete for strong theories for such such strong theories without without any any restriction restriction on on the the language. language. The Logic Logic of of Provability Provability 541 541 117. 7 . Acknowledgements Acknowledgements In the the first first place place we we are are very very grateful grateful to to Lev Lev Beklemishev Beklemishev for for providing providing us us with with In draft for the sections sections 6, and 8 8 in in a a near near perfect perfect state. state. He He also also gave gave extensive extensive aa draft for the 6, 77 and comments on other sections. Sergei Sergei Artemov Art~imov supported supported us us with with section section 10, 10, and and in in comments on other sections. answering some some questions questions for for us. us. Albert Albert Visser Visser was was very very helpful helpful with with comments comments answering and discussions, discussions, answering answering questions, and pointing out mistakes. Giovanni Sambin Sambin and questions, and pointing out mistakes. Giovanni provided valuable valuable comments. Claes Strannegard Strannegs shared his his expertise expertise with with us. us. Joost Joost provided comments. Claes shared Joosten, Rosalie Rosalie Iemhoff Iemhoff and and Eva Eva Hoogland Hoogland found found quite quite a a number number of of inaccuracies inaccuracies Joosten, in stood by in the the manuscript. manuscript. Anne Anne Troelstra Troelstra stood by us us with with advice. advice. Sam Sam Buss Buss was was a a very very helpful helpful editor editor and and careful careful reader. reader. The first first author author was was supported supported by by N.W.O., N.W.O., the the Dutch Dutch Foundation Foundation for for Scientifi Scientificc The Research, working on Research, while while working on this this chapter chapter in in 1992-1993, 1992-1993, and and by by the the National National Science Science Foundation ((grant CCR-9403447)) while while working working on on its its final final version version in in 1997. 1997. grant CCR-9403447 Foundation References References S. S. N. N. ARTEMOV ARTi~MOV [1980] Arithmetically Arithmetically complete complete modal modal theories, theories, Semiotika i Informatika, VINITI, Moscow, Moscow, 14, 14, [1980] pp. 115-133. 115-133. In In Russian, Russian, English English translation translation in: in: Amer. Math. Soc. Soc. Transl. Transl. (2), (2), 135: 135: pp. 39-54, 39-54, 1987. 1987. 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CHAPTER CHAPTER VIII VIII The The Lengths Lengths of of Proofs Proofs Pavel Pavel Pudlak Pudls Mathematical Institute, Academy of of Sciences of o.f the Czech Republic 115 115 67 67 Prague 1, 1, The Czech Republic Contents Contents 1. Introduction 1. I n t r o d u c t i o n .. .. .. .. .. .. . . . . . . . . . . . .. .. .. .. .. .. . . . . . . . . . . . . . . . . . . . 2. Types T y p e s of of proofs proofs and and measures measures of of complexity complexity . . . . . . . . . . . . . . . . . . . . 3. Some 3. Some short short formulas formulas and and short short proofs proofs .. .. . .. . . . . . . . . . . . . . . . . . . . . 4. 4. More More on on the the structure s t r u c t u r e of of proofs proofs .. .. .. .. .. . . . . . . . . . . . . . . . . . . . . . . . . Bounds on on cut-elimination cut-elimination and and Herbrand's H e r b r a n d ' s theorem theorem ................ 5. Bounds 6. Finite Finite consistency consistency statements statements - concrete concrete bounds b o u n d s .. . . . . . . . . . . . . . . . . . 7. 7. Speed-up Speed-up theorems theorems in in first first order order logic logic . . . . . . . . . . . . . . . . . . . . . . . P r o p o s i t i o n a l proof p r o o f systems systems .. .. .. .. .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8. Propositional 9. Lower Lower bounds b o u n d s on on propositional propositional proofs proofs .. .. .. . . . . . . . . . . . . . . . . . . . . 10. 10. Bounded B o u n d e d arithmetic a r i t h m e t i c and and propositional propositional logic logic . . . . . . . . . . . . . . . . . . . . 11. Bibliographical Bibliographical remarks remarks for for further further reading reading .. . . . . . . . . . . . . . . . . . . . . References References .. .. .. .. .. .. . . . . . . . . . . . . .. .. .. .. .. .. .. .. . . . . . . . . . . . . . . . . . . . . . HANDBOOK PROOF H A N D B O O K OF OF P R O O F THEORY THEORY Edited E d i t e d by by S. S. R. R. Buss Buss © 1998 Elsevier (~) 1998 Elsevier Science Science B.V. B.V. All All rights rights reserved reserved 548 549 555 564 573 577 585 590 605 619 627 629 548 548 P. Pudldk 1 1.. Introduction Introduction In In this this chapter chapter we we shall shall consider consider the the problem problem of of determining determining the the minimal minimal complexity complexity of of aa proof proof of of aa theorem theorem in in aa given given proof proof system. system. We We shall shall deal deal with with propositional logic. There propositional logic logic and and first first order order logic. There are are several several measures measures of of complexity complexity of of proof and and there are many many different different proof proof systems. systems. Let Let us us give give some some reasons reasons for for this this aa proof there are research, research, before before we we discuss discuss particular particular instances instances of of the the problem. problem. 11.1. . 1 . Our Our subject subject could could be be called called the the quantitative quantitative study study of of the the proofs. proofs. In In contrast contrast with the the classical proof theory theory we we want want to to know know not not only only whether whether aa theorem theorem has has with classical proof proof but but also also whether whether the the proof proof is is feasible, feasible, i.e., i.e., can can be be actually actually written written down down aa proof or or checked checked by by aa computer. computer. An An ideal ideal justification justification for for such such research research would would be be aa proof that particular theorem proof that aa particular theorem for for which which we we have have only only long long proofs proofs (such (such as as the the four color theorem), four color theorem), or or aa conjecture conjecture for for which which we we do do not not have have any any proof proof (such (such as as P in some reasonable theory P i= ~ NP), AfT~), does does not not have have aa short short proof proof in some reasonable theory (such (such as as ZF). ZF). Presently Presently this this seems seems to to be be aa very very distant distant goal; goal; we we are are only only able able to to prove prove lower lower bounds on on the the lengths lengths of of proofs proofs for for artifi artificial statements, or or for for natural statements, bounds cial statements, natural statements, but similar to but in in very very weak weak proof proof systems. systems. The The situation situation here here is is similar to the the situation situation in in the the study study of of (weak) (weak) fragments fragments of of arithmetic arithmetic and and complexity complexity theory. theory. In In fragments fragments of of only for sentences obtained arithmetic arithmetic we we can can prove prove unprovability unprovability of of rry H ~ sentences sentences only for sentences obtained by diagonalization, and by diagonalization, and in in complexity complexity theory theory we we can can separate separate complexity complexity classes classes also also only when connected only when diagonalization diagonalization is is possible. possible. These These three three areas areas are are very very much much connected and possible to and it it is is not not possible to advance advance very very much much in in one one of of them them without without making making progress progress in in the the others. others. Nevertheless Nevertheless there there are are already already now now some some practical practical consequences consequences of of this this research. research. For we know For instance instance in in first first order order logic logic we know quite quite precisely precisely how how much much cut-elimination cut-elimination increases increases the the size size of of proofs. proofs. In In propositional propositional logic logic we we have have simple simple tautologies tautologies which which have have only only exponentially exponentially long long resolution resolution proofs. proofs. This This is is very very important important information information for for designers designers of of automated automated theorem theorem provers. provers. Another Another reason reason for for studying studying the the lengths lengths of of proofs proofs is is that that information information about about the the size size of of proofs proofs is is very very important important in in the the study study of of weak weak fragments fragments of of arithmetic, arithmetic, namely fragments is namely when when metamathematics metamathematics of of fragments is considered. considered. For For instance, instance, in in bounded bounded arithmetic arithmetic the the exponentiation exponentiation function function is is not not provably provably total. total. Therefore Therefore the cut-elimination theorem is (in fact the cut-elimination theorem is not not provable provable in in bounded bounded arithmetic arithmetic (in fact first first order cut-elimination than elementary elementary increase increase in size of order cut-elimination requires requires more more than in the the size of proofs). proofs). Consequently Consequently we we have have (at (at least) least) two two different different concepts concepts of of consistency consistency in in bounded bounded arithmetic: the usual arithmetic: the usual one one and and cut-free cut-free consistency. consistency. Furthermore relation between bounded arithmetic arithmetic and Furthermore there there is is aa relation between provability provability in in bounded and the certain proof proof systems the lengths lengths of of proofs proofs in in certain systems for for propositional propositional logic. logic. This This seems seems to be the most promising promising way proving concrete to be the most way of of proving concrete independence independence results results for for bounded bounded arithmetic. arithmetic. Finally this Finally this area area is is important important because because of of its its tight tight relation relation to to complexity complexity theory. theory. Actually, lengths of considered as Actually, research research into into the the lengths of proofs proofs should should be be considered as aa part part of of com complexity connections with computational complexity. plexity theory. theory. There There are are two two kinds kinds of of connections with computational complexity. The Lengths of Proofs 549 549 On the the one one hand hand there there are are explicit explicit connections connections such such as as the the fact fact that that a a proof proof system system On for algorithm for for propositional propositional logic logic is is a a nondeterministic nondeterministic algorithm for the the (coNP (coAl7~ complete) complete) set set of of tautologies. tautologies. On On the the other other hand hand there there are are intuitive intuitive connections connections which which are are not not supported supported by by theorems. theorems. For For example example the the relation relation between between Frege Frege systems systems and and extension extension Frege Frege systems systems (see (see below below for for definitions) definitions) for for propositional propositional logic logic is is very very much like the the relation between the measures of based much like relation between the complexity complexity measures of boolean boolean functions functions based on formula size and and circuit on formula size circuit size, size, respectively. respectively. It It is is an an open open problem problem whether whether Frege Frege systems are systems and systems are as as powerful powerful as as extension extension Frege Frege systems and also also it it is is an an open open problem problem whether formulas formulas are are as as powerful powerful as as circuits; circuits; but but we we are are not not able able to to prove prove any any of of two two whether implications implications between between these these apparently apparently related related problems. problems. Some Some people people think think that that the the difficult difficult problems problems in in complexity complexity theory theory such such as as P 7~ = - NP? AfT~? are are essentially essentially logical logical (not (not combinatorial) combinatorial) problems. problems. If If it it is is so, so, then then proof proof theory, theory, and and in in particular particular the the lengths lengths of of proofs, proofs, should should play play an an important important role role in in their their solution. solution. 1.2. the contents 1.2. Now Now we we shall shall briefly briefly outline outline the contents of of this this chapter. chapter. Section Section 22 introduces introduces some some basic basic concepts. concepts. In In section section 33 we we describe describe aa technique technique of of constructing constructing short short formulas for for inductively inductively defi defined concepts. This This technique technique has has various various applications. applications. formulas ned concepts. Section Section 44 contains contains results results about about dependence dependence of of different different measures measures of of complexity complexity of proofs proofs and and a a remark remark on on the the popular popular Kreisel Kreisel Conjecture. Conjecture. In In section section 55 we we shall shall of consider consider the the cut-elimination cut-elimination theorem theorem from from the the point point of of view view of of the the lengths lengths of of proofs; proofs; namely, shall show bound on section 66 we namely, we we shall show a a lower lower bound on the the increase increase of of the the length. length. In In section we shall prove incompleteness theorem finite consistencies. shall prove a a version version of of the the second second incompleteness theorem for for finite consistencies. This This enables enables us us to to prove prove some some concrete concrete lower lower bounds bounds and and speed-up. speed-up. In In section section 77 we we survey survey speed-up speed-up theorems, theorems, namely namely results results about about shortening shortening of of proofs proofs when when a a stronger stronger theory theory is is used used instead instead of of a a weaker weaker one one and and related related results. results. Section Section 88 is is aa survey survey of of the the most most important important propositional propositional proof proof systems. systems. In In section section 99 we we give give aa nontrivial nontrivial example of of a a lower lower bound bound on on the the lengths lengths of of propositional propositional proofs proofs in in the the resolution resolution example system. 10 we system. In In section section 10 we present present important important relations relations between between the the lengths lengths of of proofs proofs in propositional in fragments nal section in propositional logic logic and and provability provability in fragments of of arithmetic. arithmetic. The The fi final section 1111 surveys results which which have surveys especially especially those those results have not not been been treated treated in in the the main main text. text. 2. Types 2. T y p e s of o f proofs p r o o f s and a n d measures m e a s u r e s of o f complexity complexity In In this this section section we we introduce introduce notation notation and and some some basic basic concepts concepts used used in in both both propositional propositional logic logic and and first first order order logic. logic. 2.1. One 2.1. One can can consider consider many many different different formalizations formalizations and and it it is is difficult difficult to to find find aa classification classification schema schema which which would would cover cover all. all. There There is is however however one one basic basic property property which which all all formalizations formalizations of of the the concept concept of of a a proof proof must must satisfy: satisfy: it it must must be be computable computable in polynomial time in polynomial time whether whether aa given given sequence sequence is is a a proof proof of of aa given given formula. formula. Here Here we nite we assume, assume, as as usual, usual, that that proofs proofs and and formulas formulas are are encoded encoded as as strings strings in in aa fi finite alphabet alphabet and and we we identify identify feasible feasible computations computations with with polynomial polynomial time time computations. computations. This This trivial trivial observation observation gives gives us us important important link link to to computational computational complexity. complexity. The The proof general setting setting are just nondeterministic decision procedures proof systems systems in in such such aa general are just nondeterministic decision procedures 550 550 P. Pudldk for the the set set of of tautologies tautologies or or the the set set of of theorems theorems of of aa theory theory in in question. question. More More for specifically, an an upper upper bound bound on on the the size size of of proofs proofs for for aa particular particular proof proof system system gives gives specifically, nondeterministic decision decision procedure procedure with with the the bound bound on on the the running running time time and, and, aa nondeterministic conversely, aa lower lower bound bound on on the the nondeterministic nondeterministic time time complexity complexity is is aa lower lower bound bound conversely, for any any proof proof system. system. for In particular, let TAUT T A U T be be the the set set of of propositional propositional tautologies tautologies in in some some fixed fixed In particular, let complete basis basis of of connectives. connectives. A A propositional propositional proof proof system system is is aa binary binary relation relation complete P(x, y) y) which which is is computable computable in in polynomial polynomial time time and and P(x, T A U T == - 3y 3y P( Since Since the the set set of of propositional propositional tautologies tautologies is is NP-complete, iV'P-complete, we we get get the the following following immediate corollary. corollary. immediate ((Cook Cook and and Reckhow Reckhow [1979]) [1979]) There There exists exists aa proof proof system system for for propositional logic logic in in which tautologies have proofs of of polynomial polynomial length if and and only only propositional which all all tautologies have proofs length if 0o if NP N'P = coNP coN'P,. if 2.1.1. 2.1.1. Theorem. Theorem. This general general concept concept of of a a proof proof system system can can be be generalized even further. further. Firstly, Firstly, This generalized even we can can allow allow randomized secondly, we we can can assume assume that that the the proof proof is is we randomized computations; computations; secondly, not given given to to us, us, but but we we can can access access parts parts of of the the proof proof via via an an oracle. oracle. Usually Usually such such an an not interactive interactive proof proof system system is is presented presented as as a a two two player player game, game, where where we we are are the the Verifier Verifier and the the oracle oracle is is the the Prover. Prover. It It turns turns out out that that the the Verifier Verifier can can check check with with high high and probability that that a a proof proof for for a a given given formula formula exists exists without without learning learning almost almost anything anything probability about the the proof. proof. The The most most striking example is is the the so-called so-called PCP PCP theorem theorem by by Arora Arora about striking example et al. [1992] speaking, they they showed, showed, that proof et al. [1992].. Roughly Roughly speaking, that there there there there are are interactive interactive proof systems, where the the Verifier Verifier needs needs to to check check only only aa constant constant number number of randomly systems, where of randomly selected bits with high high probability the proof proof is is selected bits of of the the sequence sequence in in order order to to check check with probability that that the correct. correct. be proofs Note, however, that these these results concern only Note, however, that results concern only the the question question how how can can be proofs but do do not give new new information information about about the of proofs. proofs. checked but checked not give the lengths lengths of 2.2. turn now to more structured proofs, proofs, which which are are typical typical for logic, while while the the 2.2. We We turn now to more structured for logic, above concepts rather rather belong belong to to complexity complexity theory. proof systems are usually above concepts theory. Such Such proof systems are usually rules. The basic element called aa defined using using aa finite of deduction defined finite list list of deduction rules. The basic element of of aa proof, proof, called proof step, step, or proof line, line, is is aa formula, formula, aa set set of formulas, aa sequence of formulas formulas or proof or aa proof of formulas, sequence of or aa sequent (pair (pair of of sequences sequences of of formulas). formulas) . A A proof is either either aa sequence sequence or or aa tree tree of of proof proof sequent proof is steps such such that that each each step step is is an an axiom axiom or or follows follows from previous ones ones by by aa deduction deduction steps from previous rule. The The complete complete information information about about the the intended intended way way of of proving proving aa given theorem rule. given theorem should each step should also also contain contain the the information information for for each step of of which which rule rule is is applied applied and and to to which previous previous steps steps itit is is applied. applied. However However in in most most cases this does does not not influence influence the the which cases this complexity proofs essentially. complexity of of the the proofs essentially. It is is important important to to realize realize that that when when proof prooflines and deduction deduction rules rules are are determined, determined, It lines and there are are two possible forms forms of of proofs: proofs: the the tree and the the sequence form. In In the the there two possible tree form form and sequence form. tree form, form, aa proof proof line line may be aa premise of an an application application of of aa rule rule only only once, once, while while tree may be premise of The Lengths Lengths of Proofs 551 551 in the the sequence sequence form form it it can can be be used used again again and and again. again. The The trivial trivial transformation transformation from from in the sequence tree form in exponential the sequence form form to to tree form results results in exponential increase increase of of size. size. The The most most important important measure measure of of complexity complexity of of proofs proofs is is the the size size of of a a proof. proof. We We take a a finite finite alphabet alphabet and and a a natural natural encoding encoding of of proofs proofs as as sequences sequences (words) (words) in take in aa finite alphabet. length of finite alphabet. Then Then the the size size of of a a proof proof is is the the length of its its code. code. The The next next one one is is the the number number of of proof proof lines. lines. Trivially, Trivially, the the number number of of proof proof lines lines is is at at most most the the size, size, however, however, a a proof proof may may contain contain very very large large formulas, formulas, thus thus there there is is an an essential essential difference difference between between the the two two measures. measures. Quite often important to to bound bound the Quite often it it is is important the maximal maximal complexity complexity of of formulas formulas in in the the proof. proof. Usually Usually we we consider consider the the quantifier quantifier complexity complexity or or the the number number of of logical logical symbols. symbols. Thus Thus we we get get other other measures. measures. Comparing Comparing the the above above measures measures with with the the complexity complexity measures measures in in computational computational complexity complexity we we see see that that the the size size corresponds corresponds clearly clearly to to time. time. At At first first glance glance it it may may seem that the maximal seem that the maximal size size of of a a formula formula (or (or proof proof line) line) should should correspond correspond to to space, space, but but this this is is not not correct. correct. In In order order to to present present aa proof proof in in a a lecture, lecture, or or to to check check it it on on a a computer we single formula line) at time, we computer we cannot cannot show show a a single formula (proof (proof line) at a a time, we have have to to keep keep the blackboard until until they last time time as the formulas formulas (lemmas) (lemmas) on on the the blackboard they are are used used for for the the last as premises. premises. The The minimal minimal size size of of a a blackboard blackboard on on which which the the proof proof can can be be presented presented is the the right right concept corresponding to to space. space. Note Note that that a a suitable choice of of the the is concept corresponding suitable choice concept of a a proof proof line and rules leads to to linear linear proofs, proofs, where where each each rule rule has has at at most most concept of line and rules leads one (Craig [1957a]). one premise premise (Craig [1957a]). In In such such proofs proofs the the maximal maximal size size of of aa proof proof line line is is the the measure measure corresponding corresponding to to space. space. T. This In the proofs In first first order order logic logic we we consider consider also also the proofs in in aa theory theory T. This means means that that we we can can use use axioms axioms of of T T in in proofs. proofs. Talking Talking about about theories theories is is not not quite quite precise precise here; different here; different axiomatizations axiomatizations give give clearly clearly different different concepts concepts of of proofs proofs and and hence hence the the smallest smallest size size proofs proofs of of a a given given formula formula may may be be different. different. Therefore Therefore we we shall shall use use preferably preferably the the term term axiomatization. axiomatization. 2.3. resp. aa proof 2.3. We We shall shall use use the the following following notation. notation. The The size size of of aa formula formula
{ [[~[[A = / minimal minimal n such such that that A A Sf-nn
otherwise. c~ otherwise. t 00 This enables us This enables us to to write write inequalities inequalities such such as as IIr < II llA + ella + Ir + o(1), which holds in ponens in which holds in the the presence presence of of modus modus ponens in A. A. 552 552 P. PudlO.k Pudldk 2.4. In propositional 2.4. Suppose Suppose that that we we consider consider aa particular particular logical logical calculus. calculus. In propositional logic, logic, this this simply simply means means that that we we fix fix aa set set of of connectives; connectives; in in first first order order logic, logic, this this means means that that we we fix fix aa language language and, and, possibly possibly consider consider some some theory. theory. Then Then we we can can compare compare the the power power of of different different proof proof systems systems with with respect respect to to the the complexity complexity of of proofs. proofs. If If we we consider consider the the size size of of proofs, proofs, then then it it is is quite quite natural natural to to disregard disregard polynomial polynomial differences differences in in proofs. proofs. In In particular particular we we define define PI P1 j "< P P2, if there there exists exists aa polynomial polynomial 2 , if cp, if cp, then p(x) resp. theorem p(x) such such that that for for each each tautology tautology ((resp. theorem)) ~, if ddll :: PI Pl f~ ~P, then for for some some using the [d21 _< p(ldI p(Idll), the norm norm notation: notation: IIcpll I]~llpl < p(lIcpll P(IIr pZ » · 9 cp, ((using dd2, I), dd2:P2 pl :::; 2 : P2 f-~ ~, 2 1 :::; 2 , Id Usually, there exists Usually, if if PI P1 j -<_ P P2, then there exists aa polynomial polynomial time time algorithm algorithm to to construct construct 2 , then dd22 from see Cook from dI dl;; in in such such aa situation situation we we say say that that P P22 polynomially polynomially simulates simulates PI P1,, ((see Cook and , if PI and P and Reckhow Reckhow [1979]). [1979]). We We say say that that PI P1 is is polynomially polynomially equivalent equivalent to to P P2, if P1 and P22 2 polynomially other. polynomially simulate simulate each each other. A well-known theorem A well-known theorem of of Craig Craig states states that that aa theory theory has has aa recursive recursive axiomati axiomatization, zation, if if it it is is recursively recursively enumerable. enumerable. It It is is an an easy easy exercise exercise to to prove prove the the following following modification modification of of the the theorem. theorem. Let Let PI P1 be be an an arbitrary arbitrary proof proof system system for for aa calculus calculus with with the the connective connective of of implication. implication. Then Then there there exists exists aa polynomially polynomially equivalent equivalent calculus calculus P P22 based based on on aa polynomial polynomial time time decidable decidable set set of of axioms axioms and and the the single single rule rule of of modus modus 2.4.1. 2.4.1. Proposition. Proposition. ponens. ponens. 0 [] Consequently consider stronger Consequently one one has has to to consider stronger assumptions assumptions in in order order to to restrict restrict the the class class of of proof proof systems. systems. The The usual usual approach approach is is to to work work with with the the schematic schematic theories theories of of rules and Parikh [1973] Parikh [1973],, where where we we have have aa finite finite set set of of rules and axiom axiom schemas. schemas. 2.5. 2.5. We We shall shall conclude conclude this this section section by by presenting presenting the the most most often often used used proof proof systems systems for consider those mathematical logic, logic, there for first first order order logic; logic; we we consider those used used in in mathematical there are are several several others used others used in in artificial artificial intelligence, intelligence, see see Chang Chang and and Lee Lee [1973] [1973] and and Eder Eder [1992]. [1992]. 2.5.1. 2.5.1. Gentzen Gentzen [1935] [1935] attributes attributes the the following following system system to to Hilbert Hilbert and and Glivenko: Glivenko: Rules Rules (6.1) A, A , AA -+ -+ B B B B (6.2) A -+ (x) . A, A ~ (I)(x) ' where where x x does does not not occur occur m in A, A A -+ -+ 't/y(y) Vy(I)(y) ' -+ (x) r ~ A A . A. (6.3) x does not occur m (6.3) 3xr -+ A'' where 3x(x) -+ A where x does not occur in A. The Lengths of Proofs 55 5533 Axiom Schemas Axiom S chemas (1.1) -+ A (1.1) A A-+ A (1.2) A -+ --+ (B (S -+ --+ A) A) (1 .2) A (1.3) (A -+ -+ (A (A -+ --+ B)) S)) -+ --+ (A (A -+ --+ B) B) (1. 3) (A (1.4) (A (A -+ --+ (B (B -+ --+ e)) C)) -+ ~ (B (S -+ --+ (A (A -+ --+ e)) C)) (1.4) (1.5) (1.5) (A (A -+ ~ B) B) -+ --+ ((B ((B -+ --+ e) C) -+ --+ (A (A -+ --+ e)) C)) (2.1) (A (A A B) B) -+ -+ A A (2.1) (2.2) (A (A A B) B) -+ --+ B B (2.2) -+ (A (2.3) (A -+ --+ B) B ) --+ + ((A ((A -+ --+ e) C)--+ (A -+ -+ (B (B A e) C) (2. 3) (A A -+ --+ (A (A v V B) B) ((3.1) 3 .1) A B -+ --+ (A (A v Y B) B) ((3.2) 3 .2) B ((3.3) (A -+ --+ e) C) -+ ~ ((B ((S -+ --+ e) C) -+ --+ ((A ((A v V B) B) -+ --+ e)) C)) 3 . 3) (A (4.1) (4.1) (A (A -+ --+ B) B)--+ ((A -+ --+-~S)--+-~A) -+ ((A --,B) -+ --,A) (4.2) --,A ~A -+ --+ (A (A -+ --+ B) B) (4.2) (5.1) 'v'x VxO(x) -+ ~(t) (5.1) ( x -+ il> ) il> (t ) (5.1) il> O(t)--+ 3xO(x) (5.1) (t ) -+ 3x il> (x ) term, x, x, yy are stands for for aa term, are variables variables).) . ((tt stands Note that shall refer calculus as We shall refer to to this this calculus as the the Hilbert Hilbert style style calculus. calculus.Note that in in aa system system We schemas or such as as above above we we can can either either say say that that we we have have axiom axiom schemas or that that we we have have axioms axioms such and and allow allow the the substitution substitution rule rule to to be be applied applied only only to to axioms. axioms. We We shall shall consider consider the the power power of of various various proof proof systems systems for for propositional propositional logic logic in in section section 8. 8. The The propositional propositional part part of of the the Hilbert Hilbert style style system system is is aa special special case case of of calculi calculi called called Frege Frege systems. systems. Contrary Contrary to to the the history, history, the the general general substitution substitution rule rule is is not not permitted permitted in in Frege Frege systems. systems. There There are are more more compact compact Hilbert Hilbert style style systems, systems, e.g. e.g. the the one one considered considered by by Hilbert Hilbert --'. As and and Ackermann Ackermann [1928] [1928],' use use only only the the connectives connectives V V and and -~. As we we shall shall see, see, the the unless we propositional propositional parts parts simulate simulate each each other other and and ((unless we use use some some strange strange quantifier quantifier be extended whole systems. rules rules)) this this can can be extended to to the the whole systems. Let Let us us note note that that there there are are natural natural proof proof systems systems for for first first order order logic logic which which have have only and the the quantifi er rules only modus modus ponens ponens as as aa rule rule and quantifier rules are are replaced replaced by by aa finite finite number number of of simple simple axiom axiom schemas, schemas, see see e.g. e.g. Grzegorczyk Grzegorczyk [1974]. [1974]. 2.5.2. 2.5.2. Another Another important important system system has has been been introduced introduced by by Gentzen Gentzen [19 [1935]. The ba ba35]. The sic proof are sequences 'PI sic elements elements of of the the proof are sequents sequents which which are are sequences ~1,, . 9. 9. , 'Pn ~n --t~ 'l/JI r , . . 9. , 'l/Jm era.. Here Here --t~ is is aa syntactical syntactical symbol, symbol, aa different different symbol symbol -+ -~ is is used used for for implication. implication. The The interpretation of with -+ interpretation of such such aa sequent sequent is is 'P ~1I A .. ... . A 'Pn ~n -+ ~ 'l/J rIV V .. ... . V V 'l/Jm Cm ((with --+ standing standing now now for for implication implication).) . The The system system has has aa single single axiom axiom scheme scheme A A --t> A, A, where where A A is is aa formula, assumptions and formula, and and several several rules rules which which have have one one or or two two sequents sequents as as assumptions and one one sequent instance of sequent as as a a conclusion. conclusion. A A proof proof is is aa tree tree of of sequents sequents where where leaves leaves are are instance of the the axiom axiom and and every every other other sequent sequent follows follows from from its its predecessors predecessors by by aa rule. rule. The The tree tree structure structure is is very very convenient convenient for for analyzing analyzing proofs, proofs, but but one one can can also also consider consider sequences sequences 554 554 P. P. Pudlak Pudldk of of sequents sequents as as aa proof. proof. The The most most important important rule rule as as the the cut cut rule rule C~l,...,~k ) ~l,...,~t,~ ~l,...,~k,"Yl,...,"Ym ~,"YI,...,"Ym ) 61,...,(~n ~1,''',~,61,''',6n, )' Observe whole Observe that that for for k k= - ll = - m m = : 0, 0, n n = - 11 we we get get essentially essentially modus modus ponens. ponens. The The whole system is is described in Chapter Chapter I. Gentzen presented presented transformations transformations of of proofs proofs from from system described in I. Gentzen the Hilbert style calculus his sequent sequent calculus vice versa. Eder [[1992] 1992] it the Hilbert style calculus to to his calculus and and vice versa. In In Eder it is is shown that this gives polynomial the systems shown that this in in fact fact gives polynomial simulations simulations of of the systems if if 1. in both 1. in both we we take take tree-proofs tree-proofs or or 2. in in both 2. both we we take take sequence-proofs. sequence-proofs. In shall show simulation of In section section 44 we we shall show that that there there is is also also polynomial polynomial simulation of sequence-proofs sequence-proofs by tree-proofs tree-proofs in the Hilbert Hilbert style style calculus. calculus. Thus Thus the the most most commonly commonly used used systems systems by in the are are polynomially polynomially equivalent. equivalent. The schematic theory. The systems systems above above are are prototypes prototypes of of what what is is called called aa schematic theory. This This concept concept is is aa natural natural extension extension of of the the concept concept of of the the Frege Frege system system used used in in propositional propositional logic. In first first order order logic, logic, however, however, it it is is not not easy easy to to define define precisely precisely such such aa concept concept logic. In especially especially because because restrictions restrictions on on occurrences occurrences of of variables variables in in quantifier quantifier rules rules ((or or axioms 1967] , axioms)) are are needed. needed. For For possible possible definitions definitions of of schematic schematic theories theories see see Vaught Vaught [[1967], 1973] , Krajicek 1989a] , Farmer 1984,1988] and 1994] . Parikh Parikh [[1973], Kraji~ek [[1989a], Farmer [[1984,1988] and Buss Buss [[1994]. Hilbert Hilbert's's e-calculus c-calculus is is based based on on a a different different language. language. Instead Instead of of quantifiers quantifiers it it uses whose meaning is an element which satisfies the formula cp(x) if uses e-terms c-terms tev(x) whose meaning is an element which satisfies the formula ~(x) if (x)
2.6. 2.6. is is the the only only rule rule where where some some structure structure present present in in the the assumptions assumptions is is missing missing in in the the if we conclusion ((if conclusion we disregard disregard the the terms terms).) . Some Some of of the the numerous numerous application application of of the the cut-elimination cut-elimination theorem theorem and and its its proof proof can can be be found found in in Chapters Chapters II and and II II of of this this handbook. handbook. Of Of course course we we have have to to pay pay something something for for it it and and the the price price is is high: high: the the increase increase of of the the size size cannot cannot be be bounded bounded by by an an elementary elementary recursive recursive function, function, Le., i.e., cannot cannot be be bounded bounded by by a a constant constant number number of of iterations iterations of of the the exponential exponential function. function. We We shall shall prove prove such such aa lower lower bound bound in in section section 5. 5. 555 555 The Lengths of Proofs 2.6.1. 2.6.1. There There are are several several theorems theorems which which are are in in aa sense sense equivalent equivalent to to the the cut cutsee Hilbert Hilbert and elimination theorem: theorem: Herbrand's Herbrand's theorem, elimination theorem, Hilbert's Hilbert's E-theorem c-theorem ((see and Bernays Bernays [1934,1939]) [1934,1939]),, semantic semantic tableaux. tableaux. Each Each of of them them can can be be used used to to define define aa concept of of aa proof proof and and the the resulting measures are are closely closely related. Namely, the the known known concept resulting measures related. Namely, transformations transformations give give mutual mutual simulations simulations in in time time bounded bounded by by iterated iterated exponential exponential functions, functions, see see 5.1. 5.1. Thus Thus we we have have two two main main classes classes of of proof proof systems systems for for first first order order ones, and (2) cut-free and the ). logic: (1) logic: (1) the the unrestricted unrestricted ones, and (2) cut-free ((and the equivalent equivalent ones ones). Considering the logic, which Considering the undecidability undecidability of of first first order order logic, which means means that that we we cannot cannot bound the bound the size size of of aa proof proof of of aa formula formula by by any any computable computable junction, function, it it is is quite quite surprising that surprising that the the spectrum spectrum of of natural natural complexity complexity measures measures consists consists essentially essentially of of two this empirical evidence be be supported two elements. elements. Can Can this empirical evidence supported by by aa mathematical mathematical theorem? theorem? 3. S Some 3. o m e short s h o r t formulas f o r m u l a s and a n d short s h o r t proofs proofs In length of In this this section section we we discuss discuss basic basic concepts concepts used used in in the the study study of of the the length of proofs proofs in bounds on in first first order order logic logic and and prove prove some some bounds on the the length length of of proofs. proofs. The The upper upper bounds bounds have have two two applications: applications: firstly firstly they they enable enable us us to to show show big big differences differences in in lengths lengths between different different types types of of proofs, proofs, the the so-called so-called speed-up; speed-up; secondly, secondly, they they are are needed needed between for reductions reductions of of the lower bounds on the the length length of of proofs proofs of of one one set set of of formulas formulas to to for the lower bounds on another one. one. another 3.1. 3.1. We We shall shall use use the the Hilbert Hilbert style style proof proof system system described described in in the the previous previous section section with with the the following following axioms axioms of of equality: equality: x X - -= -X x x x ==y -y- -+ + y y= x=, x, x x ==y A ~ x x= = Z, z, Y y1\=Y = zz --+ 9 = = y YI , 1\ . . . 1\ Xn Xl = Yn -+ (R(xI , . . . , xn) for for each each predicate predicate symbol symbol R, R, and and -+ R (Yb ' . . , Yn )) for for each each function function symbol symbol F. F. 3.2. length of defined by 3.2. The The first first question question that that we we consider consider is is the the length of formulas formulas defined by some examples be considered ) . Let iterating iterating aa certain certain construction construction ((some examples will will be considered below below). Let iP bl) be be aa formula (~(R, a l ,, .. ... ., , ak, ak, bb b l , '. .. .., , bl) formula with with aa specified specified k-ary k-cry predicate predicate symbol symbol and and ( R, aI where ak , bI where aa b l , .. ... ., , ak, b l,, .. .. .., , bl bl are are all all free free variables variables of of iP (I).. Let Let us us abbreviate abbreviate the the strings strings of of need aa sequence variables by variables by a fi and and bb.. Now Now suppose suppose a a formula formula IP ~0(a, is given given and and we we need sequence o (a, bb)) is of formulas formulas IPI, ~1, IP2, ~2,... such that that of . . . such IPn+1 (a, b) == iP ( lPn, a, b) is is provable provable in in first first order order logic. logic. Here Here == ~ denotes denotes the the biconditional. biconditional. (1) P. Pudlak Pudldk 556 In order to understand understand better what is is going going on, let us us write write (I) as as In order to better what on, let (2) (2) (I)(R(Xl),..., R(:~t), a, b), where (Xi ) denote denote particular where R R(~i) particular occurrences occurrences of of R R in in (I) and and Xi xi is is aa string string of of kk bound, bound, . Thus not of ~. not necessarily necessarily distinct, distinct, variables variables of Thus (1) (1) is is better better represented represented by: by: (a, ------ a, (3) The The variables variables bb do do not not change, change, they they are are "parameters" "parameters",, thus thus we we shall shall omit omit them them from from now now on. on. A A trivial trivial solution solution is is to to take take 'Pn ~pn+l(a) to be be equal equal to to ('Pn, (I)(~pn,a) ~).. However However often often we we +l (a) to to be be of of polynomial in fact, in nn)) size need need 'Pn ~ , to polynomial size size and, and, in fact, we we need need aa polynomial polynomial ((in size proof proof of of (3). (3). If If tt > 11,, which which is is usually usually the the case, case, then then mere mere substitutions substitutions lead lead to to exponentially (R, a) exponentially large large formulas. formulas. The The solution solution is is to to replace replace (I)(R, ~) by by an an equivalent equivalent connective is formula, which R only once. once. This formula, in in which R occurs occurs only This is is always always possible possible if if == - as as aa connective is present present in in our our language. language. ((Ferrante Ferrante and and Rackoff Rackoff [1979)) [1979]) Suppose Suppose == - is is present present in in the the language. language. Then, Then, for for every every formula .formula (R, (I)(R, a), 5), there there exists exists an an equivalent equivalent formula formula 0 w9 ((R, R, a) ~),, in in which which R R occurs occurs only only once. once. [~ 3.2.1. 3.2.1. Theorem. Theorem. We shall not here, because We shall not prove prove this this theorem theorem here, because we we want want to to construct construct polynomial polynomial size size formulas formulas not not using using biconditional. biconditional. Several Several people people observed observed that that the the assumption assumption of course about about biconditional biconditional is is essential essential for for Theorem Theorem 3.2.1, 3.2.1, ((of course the the negation negation of of == - is is sufficient If we sufficient too too).) . If we consider, consider, say, say, all all binary binary connectives connectives without without biconditional biconditional and and its ne positive its negation, negation, then then one one can can defi define positive and and negative negative occurrences occurrences of of R R and and it it is is not not possible possible to to replace replace one one by by the the other. other. Therefore Therefore the the theorem theorem fails fails to to hold hold in in this this case. case. Let Let us us consider consider the the construction construction of of formulas formulas satisfying satisfying the the inductive inductive condition condition (1) (1) using using Theorem Theorem 3.2.1. 3.2.1. If If we we disregard disregard the the size size of of variables, variables, i.e., i.e., we we assign assign aa unit clearly get unit cost cost to to each each variable, variable, we we clearly get formulas formulas 'Pn(a) ~n(a) of of linear linear size size by by iterating iterating 'Pn+ l (a) = de/ ('Pn(a) , a) =de: a).. In In order order to to obtain obtain aa polynomial polynomial size size proof proof of of (1) (1) we we prove, prove, for for every every n, 9 (~,, a) -- (I)(~,, a). (4) (4) The (R, a) == a) by The proof proof of of this this formula formula is is obtained obtained from from the the proof proof d d of of w ~(R,a) _= (R, O(R,a) by substituting the proof also of substituting 'Pn ~n for for each each occurrence occurrence of of R R in in d. d. Hence Hence the proof is is also of linear linear size size in in n n.. IInn aa more more precise precise computation computation of of proof proof size, size, we we have have to to take take into into account account the the size size of of variables. variables. After After the the reduction reduction to to one one occurrence occurrence the the inductive inductive condition condition is is = a), (5) where where x ~ is is a a string string of of variables variables bound bound in in ffJ. Clearly, we we cannot cannot use use the the same same string string W. Clearly, x for except in for all all n n ((except in trivial trivial cases cases)) because because of of the the possible possible clashes. clashes. If If we we use use different different The Lengths of of Proofs 557 557 strings n, we n, since strings x9 for for each each n, we get get formulas formulas of of size size of of the the order order n n . . log log n, since the the n-th n-th variable variable can can be be coded coded by by a a word word of of length length O(log O(log n) n).. Alternatively Alternatively we we can can use use just just two strings: strings: one one for for odd and one one for for even even n's. n's. The The resulting resulting formulas formulas are are of of linear linear two odd nn's' s and size but but a a little little unnatural, unnatural, since since one one variable variable occurs occurs in in the the scope scope of of several several quantifiers quantifiers size bounding it. it. Though bounding Though unnatural unnatural it it is is usually usually permitted. permitted. 3.2.2. Now 3.2.2. Now we we prove prove the the existence existence of of polynomial polynomial size size formulas formulas defined defined by by iteration iteration not present in the in in the the case case when when == -_- is is not present in the language. language. 3.2.3. (Solovay [unpublished]) SSuppose uppose ...,~ and 3.2.3. Theorem. Theorem. (Solovay [unpublished]) and at at least least one one of the the -+, V the language. connectives connectives --+, V,, /\ A are are present present in in the language. Let Let ipo(a) qOo(~) and and if.>(R, (~(R, a) ?t) be be given. given. Then possible to , . .. .. such Then it it is is possible to construct construct formulas formulas ipl qol(a) (~),, ip2(a) qo2(?z),. such that that v,,+, (a) - r (6) (6) a) have have polynomial polynomial size size proofs. proofs. Proof. shall use P r o o f . We We shall use only only the the fact fact that that p p -+ qq has has an an equivalent equivalent formula formula in in the the occurs once. once. We (6) and abbreviation of language language where where qq occurs We use use p p == -- qq in in (6) and below below as as an an abbreviation of ) , if both -+ an an equivalent equivalent formula formula in in the the language, language, e.g. e.g. (p (p -+ qq)) /\ A (q (q -+ -4 p p), if both -4 and and /\ A are are present. present. additional trick. The The idea idea of of the the proof proof is is the the same same as as for for the the case case with with == - plus plus an an additional trick. s. If The rst define The trick trick is is to to fi first define the the graph graph of of the the truth truth value value function function for for ipn ~pn's. If fn(a) fn(5) is is the both ipn(a) the truth truth value value function, function, then then both ~ ( 5 ) and and ""ipn(a) -~p~(a) can can be be expressed expressed as as positive positive = 1 statements statements fn(a) A(a)= 1 and and fn(a) A(a)== 00 respectively. respectively. In simpler formulas inessential assumptions assumptions that In order order to to get get simpler formulas we we shall shall use use inessential that aa constant y) is logical axiom. Consider the constant 00 is is in in the the language language and and 3x3y(x 3x3y(x =1= ~ y) is aa logical axiom. Consider the formula formula ' Take Take a a prenex prenex normal normal form form of of it it if.>(R, a) r a) == = yy = = O. 0. Q O ( R ( ~ , ) , . . . , R(~,), a, y), where er prefi x which where Q Q denotes denotes the the quantifi quantifier prefix which bounds, bounds, among among others, others, the the variables variables XI > and Xl , . . . ,,xt, all occurrences occurrences of ~1,... and all of R R in in 8 (9 are are displayed. displayed. Now Now we we define define formulas formulas for for 's. In In order the formulas, be represented the graphs the graphs of of fn(a) f~ (5)'s. order to to simplify simplify the formulas, truth truth will will be represented by by and falsehood different from be the the formula o0 and falsehood by by anything anything different from O. 0. Define Define Wo(a, ~0(a, y) y) to to be formula ipo(a) == ~0(a) - yy = = 00 (7) and and define define Wn+l ~n+l (a, (a, y) y) to to be be the the formula formula QVYI QVy, '. .". vVYz(VzVu(((z yz(WVu(((~, = = XI :~, /\ ^ u ,, = = YI) y,) V v . .·. · · V v (z (~, = = X ~,t /\ ^ u= = Yt) y,))) -+ -~ Wn(z, ~,,(~, u)) u)) -+ 8 e ( (YI y, = = O, o , .·. ·. ,· , Yt y, = = O, o, a, y)), y)), (8) (8) where where Q Q is is as as above, above, Z 2 = - Xi xi is is an an abbreviation abbreviation for for Zl zl = = XiI x/1 /\ A . . . /\ A Zk Zk = = Xik Xik and and are substituted substituted for for R(Xi) R(~/) in in 8 (9.. Note Note that that the the meaning meaning of of the the antecedent antecedent in in Yyi/ -=- 00 are • • • P. Pudlak Pudldk 558 the proof the definition definition is is that that Yi Yi codes codes the the truth truth value value of of CPn( ~n(xi); below we we give give aa formal formal proof Xi ) ; below of this. this. of Since Wn ~n occurs occurs only only once once in in the the recurrence recurrence relation, relation, we we get get wn ~n of of polynomial polynomial Since (0 (n log size size (O(n log n) n) if if we we use use different different variables variables and and linear linear if if we we "recycle" "recycle" variables variables).) . Define Define formulas formulas cpn (o,) = ~.(a) =~: ~.(a,0), 0), df Wn(o" (o" y) Wn ~.+:(a, y) = ~ ~+: (a, y) == - (( <1> ~ ((CPn ~ . , , a) a) == - yy = = 0) 0),, an 1 (o" y) = +l + d/ f3~ n(o" ( a , y) y) = =~f ~ . ( a(o", y) y) == -- (cp ( ~ (na()o,) == - Yy = = 0). 0). df wn Let 3.2.4. 3.2.4. Lemma. Lemma. Let x ~ be be the the string string of of all all free free variables variables of of formulas formulas J.L # and and v; ~,; let let r(J.L) R. Then F(#) and and r(v) F(u) be be obtained obtained by by substituting substituting J.L # and and v u in in r(R) F(R) for for R. Then v~(~(~)x) == = v( ~(~)) -~ r(J.L) r(~) == = r(v) r(~) VX(J.L( x» -+ has has aa polynomial polynomial size size proof proof in in the the size size of of J.L, #, v u and and r F.. The The idea idea of of the the proof proof iiss to to use use induction induction on on the the depth depth of of r F.. o 3.2.5. 3.2.5. Lemma. Lemma. ((i) i ) 3y o(o" 3yqto(5, y ) i is s provable. provable. w y) The The following following formulas formulas have have polynomial polynomial size size proofs. proofs. (o" y); ((ii) ii) Vyan VyoLn+I (a, y) y) -+ -+ 3y 3y~n+l y) ; wn+ 1 (a, + l (o" (CPn, a) == <1> ) 1 ( Vyan+i (5, y) --+ ~n+1(5) 4)(~n, 5);; ((iii) iii) Vyan 1 (o" y) -+ CPn + o, -+ ((iv) iv) Vyan 1 (o" y); 1 (o" y) -+ Vyf3n VyO/n+ 1 (a, y) ~ Vy~n+l (a, y)~" + + ((v) v) V . . . an (a, y) . . . an V... c~n(5, y) -+ -+ V V... c~n+i(fi, y);; where where V V.. .. .. denotes denotes the the universal universal closure. closure. + 1 (a, y) Proof. i ) Use (a) , then 0, if Proof. ((i) Use (7): (7)" if if CPo ~po(fi), then take take yy = = 0, if -,cp( ~ ( 5 o, ) ,) , take take an an arbitrary arbitrary yy =I~ O. 0. ((ii) ii) Similar i ) : to distinguish the Similar as as in in ((i)" to find find yy such such that that w ~n+l (a, y) y) holds holds distinguish the cases cases n+ 1 (o" the first 0, in second any (I)(~pn, and -, -~(I)(~n, fi).. In In the first case case take take yy = = 0, in the the second any yy =I~ O. 0. The The ( CPn, afi)) and <1> <1> ( CPn, a) formulas polynomial size, size, the formulas involved involved are are of of polynomial the number number of of steps steps is is constant, constant, thus thus the the whole proof proof is is polynomial. polynomial. whole ((iii) iii) Assume y) ; in Assume Vya Vy~n+l(a, in particular particular we we have have a c~n+i(fi, 0) which which is is n+ l (o" y); n+ 1 (o" 0) 9 .+l(a, 0 ) - (~(~., a)- 0 = 0). Using Using the the definition definition of of CPn ~n+l this reduces reduces to to the the statement statement +l ((5) o, ) this CPn ~ + +i (1a(a) ) == - <1> ~ ((~CPn, . , aa).) . ((iv) iv) Assume iii) we (CPn, a5)) in Assume Vyan Vyan+l(fi, y).. By By ((iii) we can can substitute substitute CPn ~n+l(5) for <1> (I)(~n, in +1 (o" y) + 1 (o,) for which we assume. Thus we get Vy f3n(o" y) . As we do not have the (o" y) , Vy Vy an an+:(5, y), which we assume. Thus we get Vy fin(a, y). As we do not have the +1 substitution rule, we substitution rule, we must must use use Lemma Lemma 3.2.4 3.2.4 to to estimate estimate the the length length of of the the proof. proof. ((v) v) Assume By the iv) we Assume V V .. .. ..aan n ((o" f , yy) ) . . By the definition definition of of Wo ~0 and and ((iv) we have have also also VV ..... . f3n (o" y) definition (8) (8) we immediately get fin(a, y).. From From definition we immediately get n (Xt , fit) Wn Yt ( wn(xt , Yl) Yl .. .. .. VVyt(~n(Zl, ~n+I(fi, y) == - (Jv Q--Vyl yi) /\ A .. .. .. /\ Aw ~n(xt, flt) -+ -+ + 1 (o" y) . . . , YYtt = --+ 8(Yl O(yi = 0, 0,..., = O, O, o"a, y» y)),, -+ = 559 559 The Lengths Lengths o:f ofProofs Proofs The we can can substitute substitute ~o~ using aa polynomial polynomial size size proof. proof. By By k/.../3n V . . . !3nCa, for y~ ipn(Xi) y) we using (a, y) (2~) for Yi -= 0.O. Thus we we get get Thus -~ o ( ~ ( ~ ) , . . . , ~ ( ~ ) , a, y)). Pushing the the universal universal quantifies quantifies inside, inside, we we get get Pushing -~ o ( v ~ ( ~ ) , . . . , v~(~)a, y)). have polynomial polynomial size size proofs, proofs, thus thus we we get get (ii ) , 3yi~n(2i, 3 YiWn (Xi , yi) Now, by by (ii), Now, Yi ) have ~§ y) - ~ o ( ~ ( ~ ) , . . . , ~ ( ~ ) , a, y). this is is equivalent equivalent to to By definition definition of of 8, By O, this 9 ~§ y ) - ( ~ ( ~ , a) - y = o). and the The that the size use Lemma 3.2.4 The calculation calculation that the proofs proofs are are of of polynomial polynomial size use Lemma 3.2.4 and the same same ideas ideas as as we we have have already already used used before. before. For For instance, instance, the the last last equivalence equivalence is is obtained obtained by by taking taking the the constant constant size size proof proof of of ((~(R, , . . . , R(X a) == _-- YU = = 0) 0) == =- Q ~ 8(R(Xl) O(R(e,),..., R(e,)a, y) t )a, y) iP (R, a) and and substituting substituting ipn ~ for for each each occurrence occurrence of of R R in in the the proof. proof. o To . . al (a, y) To finish finish the the proof proof of of the the theorem theorem we we first first prove prove V k/...al(5, y).. The The proof proof isis identical v) above, !3o(a, identical with with ((v) above, except except that that we we get get V V . . ..& ( ~ , yy) ) directly directly from from Now the the defining defining equation equation (7) (7).. Now we we combine combine the the polynomial polynomial size size proofs proofs of of . . . a2 (a, y) . . . an --~ V k/...a2(5, y ),, .. .. .. ,,VV... ., an(a, an(5, y) y) -+ ~ V k/... c~n+l(a, y) to to obtain obtain aa poly polyVV.... aOZl(a,y) l (a, y) -+ +l (a, y) nomial . . an iii) , we nomial size size proof proof of of V k/... ~ ++1l (a, (a, y) Y).. Then, Then, by by ((iii), we get get aa polynomial polynomial size size proof proof of of . . . . . . . ~ + ~ ( a ) --- ~ ( ~ , a). o 3.2.6. 3.2.6. Suppose Suppose we we allow allow repeated repeated use use of of the the same same variables, variables, hence hence ip~on's are of of n 'S are linear linear size. size. Then Then one one can can easily easily check check that that the the sentences sentences in in Lemma Lemma 3.2.5 3.2.5 have have linear linear size size proofs proofs hence hence (6) (6) has has quadratic quadratic size size proofs. proofs. P. Pudl6.k Pudldk 560 560 consider two applications of application is 3.3. 3.3. We We consider two applications of Theorem Theorem 3.2.3. 3.2.3. The The first first application is to to construct construct a a partial partial truth truth definition. definition. T, aa sufficiently We We shall shall consider consider T, sufficiently strong strong fragment fragment of of arithmetic arithmetic or or set set theory; theory; T. A namely, namely, we we need need to to be be able able to to formalize formalize syntax syntax in in T. A natural natural assumption assumption is is that that the the theory theory T T is is sequential sequential which which means means that that T T contains contains Robinson Robinson arithmetic arithmetic Q Q (see (see II) and -th element Chapter II) and aa formula formula formalizing formalizing the the relation relation "x "x is is the the ii-th element of of y "; we we Chapter only only require require that that there there exists exists an an empty empty sequence sequence and and each each sequence sequence can can be be prolonged prolonged by adding adding an an arbitrary arbitrary element. element. E.g. E.g. in in the the Godel-Bernays Ghdel-Bernays set set theory theory GB G B we we can can by define define the the i-th i-th element element of of the the sequence sequence coded coded by by a a class class X X by by x, {) i) Ee x}. X}. (X)i (x), =d =~se/ {X {~ ;; ((~, Let Let us us stress stress that that it it is is important important to to code code all all elements, elements, it it would would not not suffice suffice to to code, code, say, B. say, only only sets sets in in G GB. Godel number code) of a. By Let r[a] denote the the Ghdel number (the (the code) of a a formula formula a. By a a well-known well-known Let a1 denote (x ) such theorem theorem of of Tarski Tarski [1936], [1936], there there is is no no formula formula r,o ~p(x) such that that T m fk- r,oUal f ( r : l ) ) :::: a <,, - for all all sentences sentences a c~ (it (it is is a a simple simple application application of of the the diagonalization diagonalization lemma). lemma). However However for a, in it construct such classes of sentences c~, it is is possible possible to to construct such aa formula formula for for some some classes of sentences in particular particular for quantifier complexity. for a c~ with with bounded bounded quantifier complexity. We We shall shall need need the the following following particular particular case. ( x) of case. We We would would like like to to define define satisfaction satisfaction for for formulas formulas a a(2) of bounded bounded size size and and a a string x2 of elements. Let (X)i denote coding function T, i.e., string of elements. Let (x)i denote some some coding function in in T, i.e., (X)i (x)i is is the the i-th i-th element element of of the the sequence sequence x x (we (we may may assume assume that that every every element element is is aa code code of of some some sequence) sequence).. We We want want to to construct construct formulas formulas r,o ~n(x, y), n n= - 1, 1, 2, 2 , .. .. .., , such such that that for for every every n (x , y), a(yl, a ( y l , .. .. .., , Y Yn) of depth depth ::; < n, n, n ) of h T ft- r,on Ua1 , x ) :::: a ((x , · · . , (x h) , T - (9) using using a a polynomial polynomial size size proof, proof, (depth (depth 00 are are atomic atomic formulas formulas etc.). etc.). In In fact fact we we need need more: more: we we want want to to have have polynomial polynomial size size proofs proofs of of Tarski Tarski's's conditions conditions for for r,o qon. Tarski's n . Tarski's conditions conditions are are conditions conditions which which define define satisfaction satisfaction by by induction induction on on the the depth depth of of formulas. formulas. For For each each connective connective and and each each quantifier quantifier there there is is one one condition. condition. E.g., E.g., for for 's condition implication condition is implication Tarski Tarski's is --+ .-tl, z) - (<#,.,(rnl, ---> <,:,,(r,.>,l, It It is is assumed assumed that that satisfaction satisfaction for for open open formulas formulas is is easily easily definable definable.. This This is is true true in in our our case, case, since since we we assume assume that that T T is is sufficiently sufficiently strong. strong. Let Let R(x, y) be be a a new new binary binary predicate. predicate. Let Let r,oo ~P0 be be a a formula formula defining defining satisfaction satisfaction for for open open formulas formulas and and let let if:J(R, x, y) y) be r x, be a a formula formula expressing expressing the the following: following: r,oo(x, y) 1. if 1. if x x is is atomic atomic then then ~0(x, y),, 2. if u then R (u, y) 2. if x x is is ..., -~u then ..., -~R(u, 3. if 3. if x x is is u --+ --+ vv then then R(u, R(u, y) --+ ~ R(v, R(v, y) y),, 561 561 The Lengths Lengths of Proofs 4. if if x x is is VZ Vzii u u and and R(u, R(u, y') y~) for for every every sequence sequence y' y~ identical identical with with yy on on all all coordinates coordinates 4. ~= i, i, then then R(x, R(x, y) y),, jj =J etc. etc. for for the the other other connectives connectives and and for for quantifiers. quantifiers. By Theorem Theorem 3.2.3 3.2.3 we we have have polynomial polynomial size size formulas formulas 'Pn(x, ~pn(x, y) y) and and polynomial polynomial size size By proofs proofs of of (10) ~n+l(x,y) - r (10) What little different; What we we need need is is a a little different; namely, namely, we we need need polynomial polynomial size size proofs proofs in in T T of of (11) (11) where where dptn(x) dptn (x) is is a a formula formula saying saying that that x x is is aa formula formula of of depth depth � < n. n. To To prove prove this, this, it it suffices suffices to to prove, prove, using using polynomial polynomial size size proofs proofs in in T, T, (12) (12) To To prove prove (12) (12) we we observe observe that that for for n n = - 00 it it follows follows from from the the definition definition of of �9 and and for for nn >>0 O -+ Vx, y(dptn+ l (x) -+ 'Pn+2 (X, y) == 'Pn+l (x, y)) (13) (13) have polynomial size have polynomial size proofs. proofs. Let Let us us prove prove the the implication. implication. Assume Assume the the antecedent antecedent and We distinguish the cases: x is atomic, x is a negation, xx is and dptn dpt~+l(x). We distinguish the cases: x is atomic, x is a negation, is an an l (X) . + implication implication etc. etc. If If x x is is atomic, atomic, then, then, by by the the definition definition of of �9 and and (10), (10), both both 'P ~+ 2 ((xX,, y) y) n+2 and (x, y) by and 'Pn ~ ++~l (x, (x, y) y) are are equivalent equivalent to to 'Po ~0(x, y).. If If x x is is the the negation negation of of x' x' then then we we have have ((by the definition the definition of of �) r y) - y) and and y)- By By our our assumption assumption we we have have = 'Pn(x' , y), y), since since we we have have also also dptn(x') dptn(x').. Thus Thus
562 562 P. Pudllik Pudldk ((Pudlgk Pudlak [1986]) sequential theory. [1986]) Let Let T T be be aa sequential theory. There There exists exists aa sequence polynomial sequence of of formulas formulas CPn(x, go,,(x, y) y) (of (of polynomial polynomial size) size) and and such such that that there there are are polynomial size size proofs proofs in in T T of of Tarski Tarski's's condition condition for for CPn(x, gon(x, y) y) where where x x is is of of depth depth ::; < n n and and polynomial polynomial size size proofs proofs of of 3.3.1. Theorem. 3.3.1. T heorem. - (x)o) for Qc~ of for of depth depth ::; o [3 3.4. consider another application of 3.4. Now Now we we consider another application of Theorem Theorem 3.2.3. 3.2.3. Let Let T T be be aa fragment fragment of be much of arithmetic, arithmetic, let let S S be be aa function function symbol symbol for for the the successor. successor. Now Now T T can can be much weaker; we we shall shall specify specify the the condition condition that that we we need later. Let Let cp(x) go(x) be formula. We We be aa formula. weaker; need later. say of T say that that cp(x) go(x) is is aa cut cut in in T T of T proves proves cp(O) go(O),, 'ix(cp(x) Vx(go(x) -+ ---+cp(S(x))), go(S(x))), 'ix, Vx, y(x y(~ ::; < Yy /\ A cp(y) ,#(y) -+ --+ cp(x)). ,#(~)). (14) (14) (15) (15) (16) (16) If (15), then If cp(x) go(x) satisfies satisfies only only (14) (14) and and (15), then cp(x) go(x) is is called called inductive. inductive. Let Let cp(x) go(x) be be inductive proves xx + inductive and and assume assume that that T T proves + 00 = - 0, O, x x + + S(y) S(y) = = S(x S(x + + y) y) and and the the associative associative law law for for +. +. Define by Define 'Ij;(x) r by 'Ij;(x) = (17) r --all Vz(go(z) -+ --+ cp(z go(z + + x)) x)).. (17) df 'iz(cp(z) Then one can easily show is Then one can easily show that that 'Ij;(x) r is also also inductive inductive in in T T and and T /\ -+ T r P 'Ij;(x) r A 'Ij;(y) r --+ r'Ij;(x + + y); y); (18) (18) ((this this construction satisfied, we construction is is due due to to Solovay, Solovay, unpublished unpublished).) . If If (18) (18) is is satisfied, we say say that that 'Ij; addition. Assuming little bit r is is closed closed under under addition. Assuming aa little bit more more about about T T and and that that cp(x) go(x) is is aa cut, we is cut. Suppose along cut, we get get that that 'Ij;(x) r is also also aa cut. Suppose that that T T contains contains exponentiation exponentiation 22xx along with with axioms axioms 2200 = = S(O), S(O), 22Ss(~) = 22x~ + + 22x~.. (x) = (19) (19) Then first taking Then we we can can continue continue by by first taking ~(~) =~z r (20) We resp., is We get get that that CPl gol(X) is inductive inductive ((resp., is a a cut cut)) and and (X) is T F- Vx go1(x) --+ go(2x), (21) (21) since -+ cp(x) . Then since T T r k- 'Ij;(x) r --+ go(x). Then we we can can repeat repeat the the construction, construction, since since CP gol(x) is l (X) is inductive, obtain some with inductive, and and we we obtain some CP go2(x) 2 (X) with T F- W ~(~) -+ ~(2 ~~ 563 563 The The Lengths of Proofs etc. etc. Observe Observe that that the the above above construction construction is is schematic, schematic, we we could could have have assumed assumed that that is just just a a second second order order variables variables and and derive derive (21) (21) from from (14) (14)-- (16). (16). More More precisely precisely it it rp~o is R(x) be means means the the following: following: let let R(x) be aa unary unary predicate, predicate, let let Ind(R(x)) Ind(R(x)) (resp. (resp. Cut(R(x)) Cut(R(x)) denote (14) and (15) (and denote the the conjunction conjunction of of (14) and (15) (and (16) (16) resp.). resp.). Then Then there there exists exists aa formula formula (R, x) x) and and aa finite finite fragment fragment of of arithmetic arithmetic To To such such that that W9 (R, Too fF- Ind(R(x)) Ind(R(x)) -+ ~ Ind( Ind(qt(R,x)) V y (w ~ ((R, R , yy) ) -+ ~ R(2 R(2u)). T Y )). w (R, x)) 1\A Vy( Applying Applying Theorem Theorem 3.2.3 3.2.3 to to 4!(R, ~(R, x) x) defined defined by by 4!(R, 9 (R, x) = df we we obtain obtain the the following following theorem. theorem. 1\ To -+ w (R, x) Let Let T T be be aa sufficiently sufficiently strong strong fragment fragment of of arithmetic; arithmetic; suppose suppose inductive (resp. ~Oo(x) is is inductive (resp. is is aa cut) cut) in in T T.. Then Then there there exists exists aa sequence sequence of of formulas formulas rpo(x) such that (x), rp ~o2(x), that for for each each n n rp~oll (x), 2 (X) , .. .. .. ,, such (x) 1\A rpn(2"'))) rpn+ 1 (x) -+ Vx(~On+l(X) "+ (rpn (~On(X) ~On(2X))) IInd(~On+l(X)) nd( rpn+ 1 (x)) 1\A Vx( 3.4.1. 3.4.1. Theorem. Theorem. (resp. (resp. the the formula formula with with Cut Cut instead instead of of Ind) Ind) has has aa proof proof in in T T of of size size polynomial polynomial in in n. n. [] o 3.5. Since cuts containing some 3.5. Since cuts are are quite quite important important in in the the study study of of theories theories containing some part part of shall mention mention aa few of arithmetic, arithmetic, we we shall few basic basic facts facts about about them, them, though though they they are are not not needed and needed in in this this chapter. chapter. More More can can be be found found e.g. e.g. in in Hajek Hs and Pudlak Pudls [1993]. [1993]. In In order order to to obtain obtain aa cut cut closed closed under under multiplication multiplication and and contained contained in in rp(x) ~o(x) one one can . It possible to can apply apply the the trick trick of of (17) (17) to to 'lj;(x) r It is is possible to go go on on and and get get cuts cuts closed closed special under under more more rapidly rapidly growing growing functions, functions, but but not not for for 2'" 2x (unless (unless rp(x) ~o(x) has has some some special properties). properties). There There is is another another way way to to get get such such cuts, cuts, using using which which we we can can better better see see denote n-times n-times iterated what these functions are. Let what these functions are. Let 2� 2~ denote iterated exponential exponential function. function. Let Let w~ (x) be be nondecreasing functions such such that that wn(x) nondecreasing functions 22.nS(",) = Wn (2"') n '' we properties are we assume assume that that these these properties are provable provable in in T. T. Let Let 'lj;n r (x) be be defined defined by by (22) (22) = 'lj;n(x) = df 3y(rpn(Y) 1\ x :::; _< 2� ) . (23) (23) By By construction construction T (24) T f~- rpn(Y) ~On(y) -+ --+ rp(2 ~o(2u~), (24) � ), (x) is hence x) . Also T. hence 'lj;n r (x) is is contained contained in in rp( ~o(x). Also it it is is easy easy to to check cheek that that rpn ~o.(x) is aa cut cut in in T. To To see see that that 'lj;n(x) r (x) is is closed closed under under wn(x) w.(x) just just observe observe that that (Y) . T fi-- x x :::; <_ 22~� -+ --+ w.(x) < wn(2 w.(2u~) 2s(y). wn(x) :::; T � ) = 2� = For instance take For instance take W w2(x) = x22,, then then we we obtain obtain 'lj; r 2 (X) closed closed under under x x 22,, hence hence closed closed 2 (X) = under multiplication. multiplication. under 564 564 P. Pud16k Pudldk P. 4. 4. More M o r e oon n the t h e structure s t r u c t u r e oof f proofs proofs In shall prove In this this section section we we shall prove two two basic basic results. results. First First we we prove prove that that for for the the usual usual calculi for predicate predicate logic logic proofs proofs as as sequences sequences can can be be replaced replaced by by tree-proofs tree-proofs with with calculi for only aa polynomial increase. Thus Thus the the size size measures measures based based on on proofs proofs as as sequence sequence and and only polynomial increase. proofs We shall proofs as as trees trees are are polynomially polynomially related. related. ((We shall sketch sketch a a different different proof proof of of the the same same result result for for some some propositional propositional proof proof systems systems in in section section 8.) 8.) The The second second result result says says that the the depth depth of be bounded root of that of aa proof proof can can be bounded by by aa square square root of its its size, size, provided provided the the proved proof is theory of cation proved sentence sentence has has negligible negligible size size.. The The proof is based based on on the the theory of unifi unification of few other of terms. terms. We We shall shall survey survey aa few other results results which which use use unification, unification, in in particular particular ' s Conjecture Kreisel Kreisel's Conjecture on on generalizations generalizations of of proofs proofs in in arithmetic. arithmetic. ee be equence resp. I rp lissequence Let IIlqo[[ resp. IIII~pl[ tree be the the size size of of the the Krajicek [[1994a]) 1994a] ) Let rp l l tr ((Krajf~ek smallest sequence-proof resp. tree-proof of a provable sentence in the Hilbert style smallest sequence-proof resp. tree-proof of a provable sentence rp qa in the Hilbert style calculus. polynomial p(x) calculus. Then Then there there exists exists aa polynomial p(x) such such that that 4.1. Theorem. Theorem. 4.1. for every provable for every provable sentence sentence rp. ~. I lisequence III1~11 < p p(llcpll ree ::; rp wt~eo ( l rp ~eque~~ ) We consider here only the calculus, but We consider here only the Hilbert Hilbert style style calculus, but the the result result can can be be extended extended to to the the Gentzen Gentzen sequent sequent calculus, calculus, as as there there are are polynomial polynomial simulations simulations for for both both versions versions - tree 1992] . tree and and sequence, sequence, cf. cf. Eder Eder [[1992]. This surprising, since This result result is is quite quite surprising, since there there is is an an obvious obvious similarity similarity between between sequence-proofs circuits on hand, and sequence-proofs and and circuits on the the one one hand, and tree-proofs tree-proofs and and formulas formulas on on the the other hand. In other hand. In circuits circuits the the output output of of a a gate gate can can be be connected connected with with several several other other gates, gates, thus boolean function thus we we can can use use the the boolean function computed computed at at this this gate gate several several times. times. While While aa formula tree, thus node has successor. Similarly formula is is represented represented by by a a tree, thus each each node has at at most most one one successor. Similarly in in a a sequence-proof sequence-proof we we can can use use a a formula formula several several times times as as aa premise premise of of aa rule, rule, while while in in aa tree tree proof proof it it is is allowed allowed only only once. once. It It is is generally generally accepted, accepted, through through still still a a difficult difficult open open problem, problem, that that circuits circuits are are exponentially exponentially more more powerful powerful for for computations computations of of boolean formulas Still the boolean formulas than than formulas. formulas. Still the corresponding corresponding statement statement for for proofs proofs is is false false as see below. as we we shall shall see below. - Proof. P r o o f . We We shall shall first first prove prove the the theorem theorem for for the the propositional propositional calculus. calculus. The The idea idea is is quite quite simple. simple. Let Let (rp ( ~ I1,, .. .. ..,, rp ~n) be a a proof. proof. We We shall shall replace replace this this sequence sequence by by n ) be rpI . . . , ((... . . . (rpi ~Pl,, rpi ~1 1\ A rp2, ~P2,..., ( ~ 1\ A rp2) ~2) 1\ A .. ... ). ) 1\ A CPn. ~Pn. In In this this sequence sequence each each formula formula follows follows from from the previous This sequence not aa proof, thus we the previous one. one. This sequence is, is, however, however, not proof, thus we have have to to insert insert some some proof it such proof trees trees in in it such that that a a leaf leaf of of a a tree tree is is ((... ~ 1\ A rp2) ~2) 1\ A ..... .)) 1\ A rpi ~i and and the the root root is is . . . ((rpi (... ^ ^ . . . ) ^ . . . ) 1\ 1\ 1\ . . ( . (rpi rp2) rpi+! · In order to notation we In order to simplify simplify notation we agree agree to to omit omit parenthesis parenthesis in in expressions expressions like like n . Furthermore a) ")) 1\ ((.".". ((~Pl (( rpi 1\ A rp2) ~o2) 1\ A CP ~3)'" A rp ~n. Furthermore let let us us say say that that aa class class of of sentences sentences has has polynomial polynomial upper polynomial size size tree tree proofs, proofs, abbreviated abbreviated by by pst-proofs, pst-proofs, if if there there is is a a polynomial upper bound on bound on the the size size of of tree tree proofs proofs in in terms terms of of the the size size of of a a formula. formula. We We shall shall use use this this also also for for proofs proofs from from assumptions. assumptions. The The proof proof now now reduces reduces to to the the two two statements statements in in the the following following lemma. lemma. • . . 565 565 The Lengths Lengths of ofProofs Proofs The 4.1.1. LLemma. 4.1.1. emma. an instance instance ofof an an axiom; axiom; (a) aa --~ -+ aa A/3 /\ 13 has has aa pst-proof pst-proofprovided/3 provided 13 isis an (a) . . . A/\ an provided a~ ai --4 -+ 7, is is aj aj .for for an provided an A/\ "~, has has aa pst-proof pst-prooffrom from al a1 A/\ ... (b) 0/,1 a1 A/\ "." . . A/\ o~n (b) some 11 <:::; ii,, jj <:::; n.n. some ( a) The The proof proof of of (a) ( a) isis trivial, trivial, but but since since we we shall shall use use the the same same argument argument PProof. r o o f . (a) an instance instance several times times below below we we shall shall spell spell itit out out at at least least once. once. Suppose Suppose ~13 isis an several thus /3 13 isis r'ljJ(13b . . . ,~k) , 13k) for Consider of an an axiom axiom r'ljJ(P1, . . . ,Pk), , Pk) , thus for some some ~1,...,/3k. 131 , . . . , 13k ' Consider of , Pk) . ItIt isis aa tautology, PH1 -+ -+ Pk+l Pk+ 1 A/\ r'ljJ(Pb . . . ,Pk). tautology, thus thus itit has has aa the following following formula formula Pk+l the tree-proof do. do . Let Let arbitrary arbitrary aa and/31,..., and 13b ' . . , ~/n Then we we obtain obtain aa tree-proof tree-proof 13n be be given. given. Then tree-proof in do. do . simply by by substituting substituting ~131, for PPb PH1 in 13k, aa for of l , ..... ., , ~/k, l , ... .. ,. , Pk+l -+ aa A /\ r'ljJ(13b . .9. , ]3n) 13n) simply of aa -+ . . . Thus the size of this tree-proof will be bounded by c · (Ial + :::; c· + l13kl) 1131 1 ++' " + ]~k]) _< c. Thus the size of this tree-proof will be bounded by c. (]orI + ]]31] 131 131) (]al = O(]c~ -+ c~ A ~/]), where the constant c is determined by do. ) , where the constant is determined by do . = O(la -+ a /\ ( I al ++ I~1) 1 To prove prove the the second second statement statement we we derive derive another another lemma. lemma. (We (We shall shall not not need need (b) ( b) To it in its full strength. ) it in its full strength.) 4.1.2. LLemma. 4.1.2. emma. Let r7r be be aa permutation permutation on on {{I, n}, aab an formulas. formulas. Then Then Let 1 , .. .. .., , n}, l , . .. .. ,. , an OZ1 A OZ2 A . . . A ocn -+ oL~-(1) A 06r(2) A . " A oL~-Cn) (25) (25) has pst-proof, has a a pst-proof. Proof. First we P r o o f . First we prove prove that that (~A]3Aal A a 2 . . . A a n A " f - - ~ ( ~ A f f A a l A a 2 . . . A a n A / 3 (26) has proof. Clearly has aa pstpst-proof. Clearly (� ( ~ /\ ^ 7, -+ -~ (~: /\ ^ 13) Z) -+ ~ (� (~ /\ ^ 'TJn /\ ^ , 7 --+ -~ (~ /\ ^ 'TJ~ /\ ^ 13) Z) (27) (27) have t-proofs for 13, "7, �, have aa ps pst-proofs for any any/3, ~, (~,, 'TJ. r/. Thus Thus start start with with aa pst-proof pst-proof of of ~ A 7 -~ 7A ~, take proofs of take the the pst pst-proofs of - (~ ^ 7 -~ r ^ Z) ~ (~+1 ^ Z -~ ~+1 ^ 7) j equal given given by by (27) (27) for for �j ~j equal equal to to {}5 /\ A 13 ~ /\ A a1 al /\ A .. .. .. /\ Aaj aj and and (~j equal to to {}5 /\ A,7 /\ Aa al1 /\ A .. .. .. /\ Aa ai, i, and and then then apply apply modus modus ponens ponens inferences inferences to to get get (26) (26).. Notice Notice that that (26) (26) shows shows that that also also al A.. "Aai-1A~Aai+I A.. "Aan-1A'7 -~ al A.. "Aai-1A"fAai+l A.. "Aan-1A~ (28) has has pst-proofs. pst-proofs. Since Since (� (~ -+ -* (r ) -+ -* (� (~ /\^ 'TJn -+ -+ (~ /\^ 'TJ) n) 566 566 P. Pudtak Pudldk has proof, we st-proof of has a a pst pst-proof, we get get from from (28) (28) aa p pst-proof of (25) (25) for for any any transposition transposition 7f. 7r. In In order order to to get get it it for for a a general general 7r r just just recall recall the the well-known well-known fact fact that that each each 7r 7r can can be be 0 decomposed decomposed into into a a polynomial polynomial number number of of transpositions. transpositions. [] - Using the Using the same same argument argument as as in in (a) (a) one one can can show show that that � /\ ^ ((3 (/~ -+ ~ ,) v) /\ ^/~(3 -+ -~ �~ /\ ^ ((3 (/~ -+ -~ ,) v) /\ ^/~(3 /\ ^, (29) (29) has pro of for (3 7.. Now has a a pst pst-proof for any any � ~,/3, Now we we can can finish finish the the proof proof of of (b). (b). For For aa given given al A ..... . /\ Aa a~, first move move ai ai -+ --+ , 7 and and aj aj to to the the end end of of the the conjunction conjunction using using Lemma Lemma al /\ n , first 4. 1.2, then (29) to end, and .2 move 4.1.2, then apply apply (29) to add add , 7 at at the the end, and finally, finally, again again using using Lemma Lemma 4.1 4.1.2 move ai and aj ai -+ -~ , 7 and a~ back. back. [] o - , " This finishes the This finishes the proof proof of of the the theorem theorem for for the the case case of of propositional propositional logic. logic. 4.1.3. Now 4.1.3. Now we we sketch sketch how how the the above above argument argument should should be be modified modified in in order order to to get get the the result result for for the the predicate predicate calculus. calculus. We We cannot cannot simply simply take take the the conjunction conjunction of of formulas proof, since since clashes clashes of variables may formulas in in the the proof, of variables may occur occur and and it it is is no no longer longer true true that that i . Therefore with universal universal closures ~ol A .. ... . /\ A CPH ~oi+1 follows from from CPl ~Ol /\ A . . . /\ A CP ~oi. Therefore we we work work with closures CPl /\ I follows of , . . . , CPn . . . cP of the the formulas formulas CPl ~Ol,..., ~on.. Let Let V V... ~o denote denote a a universal universal closure closure of of aa formula formula cpo ~o. Instead Instead of of modus modus ponens ponens and and the the two two quantifier quantifier rules rules we we need need now, now, for for some some formulas formulas 'Ij;, cp, a, (3, . . . 'Ij; V . . . (cp . . . cP; (1) (1) to to derive derive V V... r from from V... (~o -+ --+ 'Irj;) and and V V... ~o; . . . (a . . . (a (3(x)) , where (2) (2) to to derive derive V V... (a -+ --+ Vy(3(y) Vy~(y))) from from V V... (a -+ --+/~(x)), where x x does does not not occur occur in in a; a; . . . (3ya(y) . . . (a (x) -+ (3) , where (3) to to derive derive V V... (3ya(y) -+ --+ (3) ~) from from V V... (a(x) --+/~), where x x does does not not occur occur (3) in (3. in ft. This This can can be be done done as as follows. follows. For For each each particular particular case case of of (1)-(3) (1)-(3) prove prove the the corresponding implication, i.e., corresponding implication, i.e., . . • Vv ... .. . cp . . . (cp v -+ -~ (V (v... (v -+ -+ 'Ij;) r -+ -~ V v ... .. . 'Ij;) r ; Vv ... .. . (a (3 (x) ) -+ ( . -+ -+/~(~)) -~ V v ... .. . (a (~ -+ -+ Vy(3(y)) vy/~(y));; Vv... . . . (a (x) -+ (3) -+ . . . (3ya(y) (3) . (~(~) ~/~) -~ V v... (3y~(y) -+ -+/~). (30) (30) Insert . . . CPt, . . . CP obtained from Insert these these subproofs subproofs in in the the sequence sequence V V... qOl,V V... ~o2,... V.... . CPn ~on obtained from the the 2, . . V original calculus. Thus original proof proof and and then then use use the the proof proof for for the the propositional propositional calculus. Thus the the proof proof reduces reduces now now to to following following lemma lemma whose whose proof proof we we omit. omit. . 4.1.4. 4.1.4. Lemma. Lemma. The The sentences sentences (30) have have pst-proofs. pst-proofs. . o Let Let us us note note that that the the above above formulas formulas (30) (30) are are essentially essentially the the schemas schemas used used to to formalize formalize first first order order logic logic by by a a finite finite number number of of schemas schemas and and the the single single rule rule modus modus ponens. ponens. Thus Thus what what we we actually actually did did above above was was replacing replacing the the quantifier quantifier rules rules by by quantifier logic. quantifier axiom axiom schemas schemas and and applying applying the the result result for for the the propositional propositional logic. The Lengths of of Proofs 567 567 4.2. We shall shall prove prove the the next next result, result, Theorem Theorem 4.2.5, 4.2.5, using using an an estimate estimate on on unifica unifica4.2. We tion. It it directly, tool and tion. It is is also also possible possible to to prove prove it directly, but but unification unification is is aa very very useful useful tool and it it is is natural natural to to express express the the combinatorial combinatorial statement statement that that we we need need using using it. it. Consider terms terms in in aa language language consisting consisting of of variables variables constants constants and and function function Consider symbols. symbols. A A substitution substitution a a is is a a mapping mapping from from the the set set of of variables variables into into the the set set of of terms. We We shall shall write write ta ta for for the the result result of of substitution substitution a a applied applied to to tt which which means means terms. that (x) . A that we we replace replace each each variable variable x x in in tt by by a a(x). A unification unification problem problem is is aa set set of of pairs pairs (t l , tt2), of of terms terms {{(tl, (t3, (t2k-lt2k)}. A substitution substitution a a is is a a unifier unifierifif 2 ) , (t 2k - It2k)}. A 3 , tt 44))', .. ...., ' (t t l O- --- t 2 0 " , . . . ' t 2 k _ l 0" - - t2kO-" We We think think of of a a unification unification problem problem as as a a system system of of equations equations with with variables variables being being unknown terms; terms; however however variables variables may may occur occur in in a a solution solution (=unifier) (=unifier) too. too. A A unifier unifier � unknown is is a a most most general general unifier unifier if if for for every every unifier unifier a a there there exists exists aa substitution substitution 05 such such that that 2(f = = a. a. The The following following result result is is easy easy but but has has very very important important applications, applications, see see �o Chapter Chapter II and and Chang Chang and and Lee Lee [1973]. [1973]. If If there there exists exists aa unifier unifier then then there there exists exists aa most most general general [] D 4.2.1. 4.2.1. Proposition. Proposition. unifier. unifier. We shall use this proposition later. Now we we only only need need to to observe observe that that the the most most We shall use this proposition later. Now general general unifier unifier gives gives the the smallest smallest possible possible solution. solution. terms as As As usual, usual, we we shall shall think think of of terms as rooted rooted trees. trees. We We say say that that the the root root is is in in depth depth t, denoted d(t) is 0, its 0, its sons sons in in depth depth 11 etc. etc. The The depth depth of of aa term term t, denoted by by d(t) is the the maximal maximal depth it; the t, denoted It I , is depth that that occurs occurs in in it; the size size of of t, denoted by by Itl, is the the number number of of subterms subterms (i.e., (i.e., the the nodes nodes in in the the tree). tree). We We say say that that a a subterm subterm ss of of aa term term tt is is in in depth depth d d if if the the root root of ss in in depth depth d d in in tt.. of The The following following lemma lemma iiss the the combinatorial combinatorial substance substance of of the the bound bound on on the the depth depth of of formulas formulas which which we we are are going going to to prove. prove. 4.2.2. 4.2.2. Lemma. Lemma. Let Let � ~ be be aa most most general general unifier unifier of of aa unification unification problem problem {(tl, {(t,, t2 ), (t3 , tt,),..., 4 ) ' . . . , (t2k - b t2k)}. Let = L: Let d d= = m!\X max d(t;), d(ti), 8 S = E It ItiEI and D D = = m!\Xd(t max d(tiE). Then i �1 and i �). Then , t i ii i V D (2 + (d + 8. D � _< ~/(2 + o(I)) o(1))(d + 1) 1)S. Proof. Consider aa P r o o f . Let Let w w be be a a term term ttio~ with the the maximal maximal depth; depth; thus thus d(w) d(w) = = D D.. Consider io � with D branch Let B be the end segments of Bb branch B B in in w w of of length length D D.. Let 1 , .. .. .., , B Br, = l[d-~J, be the end segments of B B of of r , rr = d�I J , 1, 2(d r(d ++ 1) 1) respectively. lengths lengths d d+ + 1, 2(d + + 1), 1 ) , .. .. .., , r(d respectively. 0, there For For each each subterm subterm u u of of any any tti~ with d(u) d(u) > O, there exists exists jj such such i � with that u u occurs occurs in in ttj� i e in in depth depth < d(tj) d(tj).. that 4.2.3. Claim. 4.2.3. C laim. P. Pudltf.k Pudldk 568 To suppose it only in To prove prove the the Claim, Claim, suppose it is is false. false. Then Then u u can can occur occur only in the the part part of of the the (J. Thus terms t e r m s tie which which belongs belongs to to a. Thus we we can can obtain obtain a a smaller smaller unifier unifier by by replacing replacing all all 0 occurrences occurrences of of u u by by aa variable. variable. [] t/E We shall show We shall show that that B1, B1,...,, B Brr have have disjoint disjoint occurrences. occurrences. For For each each Bi take take the the term corresponding to term Wi wi corresponding to the the first first vertex vertex of of Bi Bi and and take take an an occurrence occurrence of of Wi wi in in the the depth depth :::; _< d(tj) in in some some tiE. Then Then the the occurrences occurrences of of D;'s Di's in in these these occurrences occurrences of of w; wi's's must must be be disjoint. disjoint. Thus Thus we we have have . . • d(tj) tjE. +...+ Dl 1 JJ (d(d§+ 1) dd ++l +12+( d2(d + l )+ + . 1) . . ++ . . . + lLd d +� (d+ ( d + l )1). ~· . � L ( [ dd + 21 d· +l�J d D+l1J . · (l�J + 11] ++ 11)) DJ.D · �~2I [l� d+ d + l1 J D D 22 D D 22 D D D . 9(1(1 -- 0(1)). = ~ o(1)). 22 2(d 2(d 2(d++ 1) 1) 2(d++ 1) 1) s Bd S > > IIBII-+-...§ I Br l > > = = > > > > - 0 4.2.4. Suppose that 4.2.4. Suppose that Do A = = (rp (qol,..., qon) is is aa proof proof of of rp qo,, i.e., i.e., rpn ~n = = rp ~.. The The skeleton skeleton I . . . . , rpn) of length where of Do A is is a a sequence sequence of of the the same same length where each each rpi ~i is is replaced replaced by by an an axiom axiom schemas schemas or in Do this step; step; moreover, then there or aa rule rule used used in A at at this moreover, if if a a rule rule was was used, used, then there is is also also information lines to applied. E.g. information about about the the proof proof lines to which which the the rule rule was was applied. E.g. aa formula formula obtained P, j, k) obtained by by modus modus ponens ponens from from formulas formulas rpj qoj and and rpk q0k will will be be replaced replaced by by (M (MP, k).. We shall show We shall show that that for for a a given given formula formula rp ~p and and aa skeleton skeleton E there there exists exists in in aa sense sense aa most unifier most general general proof. proof. This This proof proof will will be be constructed constructed from from a a most most general general unifier for unification problem defining the cation problem for a a unification problem obtained obtained from from L E. In In defining the unifi unification problem assigned assigned to to the the proof proof Do A we we shall shall follow follow Baaz Baaz and and Pudbik Pudl~k [1993], the the idea idea goes goes back back to to Parikh Parikh [1973]. Replace . , rp�) Replace all all atomic atomic formulas formulas in in Do A by by a a single single constant constant c; let let Do' A' = = (rp�, (qo~,..., qo~) be be the the resulting resulting sequence. sequence. The The language language for for the the terms terms in in the the unification unification problem problem will will consists consists of of the the constant constant cc,, distinct distinct variables variables va for for every every subformula subformula f3 ~ of of Do' A ~ and and aa function function symbol symbol for for each each connective connective and and quantifier, quantifier, i.e., i.e., f--; f_~,, f�, f~, h f3 etc. etc. We We shall shall write write the the pairs pairs of of the the unification unification problem problem as as equations: equations: (1) For For each each propositional propositional axiom axiom schema schema used used in in the the proof proof we we add add an an equation equation : is ) , then which it; e.g., which represents represents it; e.g., if if rp qo~ is a c~ --r --+ (f3 (~ --r ~ a c~), then we we add add equation equation E [1993], [1973]. . . v/3 (1) v~: = f_,(~,, f_,(v~,,,)); (2) j ponens, where (2) if if rpi qoi is is derived derived from from rpj qoj and and rpk qok via via modus modus ponens, where rpk qok is is rp qoj --r --+ rpi qoi,, we we add add v~, = y_,(v~,~, v~); 56 9 569 The Lengths of of Proofs (t) (3) if if 'Pi ~i is is an an instance instance of of aa quantifier quantifier axiom, axiom, say say 'Pi ~pi is is (I)(t) --+ -+ 3x(x) 3x~(x),, then then we we (3) add the the equation equation add v~ - f-~(va, f3(v~)); (t) here 0: c~ is is the the formula formula obtained obtained from from (I)(t) by by substituting substituting c for for atomic atomic formulas, formulas, this this here is the the same same formula formula which which we we thus thus obtain obtain from from (I)(x); is (4) in in the the same same way way we we add add equations equations for for quantifier quantifier rules: rules: e.g., e.g., suppose suppose 'P ~jj is is (4) c~ --+ -~/~, ~ : is is 3xo: 3xc~ --+ -~/~ and 'Pi ~i is is derived derived from from ~j by by the the quantifier quantifier rule rule (6.3) (6.3),, then then we we 0: /3, 'P /3 and add equations equations add V",' (va, vp) v~;1 = = f--7 f~(v~, vB),, (5) fi finally we add add nally we (5) (x); 'Pj V", (13(va), vp) v~: = = f--7 f~(fa(v~), v~);; V~[ = T, where T is is obtained obtained from from 'P� ~'~ by by replacing replacing connectives connectives and and quantifi quantifiers by the the where ers by corresponding corresponding function function symbols symbols f_,, . .. -4 ' f~, � , f3, .... Now we we are are ready ready to to prove prove the the result. denote the the depth of of 'P ~ aa formula, formula, Now result. Let Let dp(~) denote where we we consider consider 'P ~ as as aa term term but but we we treat treat atomic atomic formulas formulas as as atoms. atoms. Let Let dp(A), where for aa proof the maximal maximal depth depth of of aa formula formula in in A. for proof A, denote denote the f f 13 , . dp('P) depth dp(fl), fl. 4.2.5. Theorem. Theorem. (Krajf~ek [1989a] [1989a],, Pudhik PudlAk [1987]) [1987]) Let Let fl A be be aa proof proof of of 'P ~ and and 4.2.5. (Krajicek suppose fl A has has smallest smallest possible possible size. size. Then Then suppose )) . dp(fl) 0o ( Vlfll ' (dp('P) + 11/) Proof. Consider aa proof proof A fl ofof ~'P ofof minimum minimum size. size. Let U be be the problem P r o o f . Consider Let H the unification unification problem assigned to A. Clearly Clearly A fl determines determines aa unifier unifier au for for U H in in the the natural natural way. way. Let Let 1: assigned to fl. fl, = be aa most most general unifier of of H. U . We construct aa proof proof rF = 'l/Jn) from from be general unifier We shall shall construct = (Vh ( r , . . . , Cn) ~.. Let . Choose Choose a a small small formula 1: Let ~8 be be the the substitution substitution such such that that au -= ~1:8. formula �~ which which does not not contain any variable variable which occurs in in A, e.g., e.g., 00 = 00 if is in in the does contain any which occurs if itit is the language. language. Consider the the proof i.e., 'Pi ~i,, and and terms terms vv "'~:au and and Vv"'~: .1: . We We Consider the i-th i-th formula formula in in the proof A, i.e., with some some subformulas subformulas replaced v",: u ; this this means means that that v",:~ 1: is have ~ 61:8 = - v~a; is v"': v ~ au with replaced have v ",: by variables. variables. Thus to be be 'Pi subformulas corresponding corresponding to by Thus we we define define 'l/Ji r to ~i with with the the subformulas to variables in in v~ replaced by variables v"': ~1: replaced by ~. �. fl, = fl, 'Pj 4.2.6. Let us consider an example. Suppose Suppose 'Pi been obtained obtained from 4.2.6. Let us consider an example. ~i has has been from ~j by by the the quantifier (6.3) . Suppose Suppose ~Pi 'Pi is is quantifier rule rule (6.3). 3x(P(x) ~--+ (Q(x) (Q(x) --+ 3x(P(x) R(y))) ~--+ R(y), R(y), --+ R(y))) Then vvla is f-4 (13 f-4 ( f ( c)),)) ) c, -4 c, c By By case case (4) (4) of of the the definition definition of of H, U, vv~E v"'i1: has has form form s), ,c . P. P. Pudlak Pudldk 570 570 t s. f-+ (c, s c t for and s. Because most general, and t for some some terms terms t and Because E E is is most general, s is is either either c or or a a variable variable and is either as is either as in in vvv~a or f_.(c, vv~), or a a variable. variable. Let Let us us suppose suppose that that V f-+ ( hf-+ (c, va) , c). Then Then 'l/Ji r is is 3x(P(x) -+ 3x(P(x) --+ �) ~) -+ --->R(y) R(y).. Furthermore Furthermore 'l/Jj Cj must must be be (P(x) (P(x) -+ --> �) ~) -+ --+ R(y). R(y). We needed in We see see that that the the structure structure of of formulas formulas needed in axiom axiom schemas schemas and and rules rules is is preserved. preserved. Note satisfied, since Note that that also also the the restrictions restrictions on on variables variables in in quantifier quantifier rules rules are are satisfied, since � does not should be does not contain contain any any variable variable which which should be bounded. bounded. Finally Finally we we have have also also r = = ~CPn , =~. 'l/Jn = cp o 4.2.7. Now Lemma 4.2.2. 4.2.7. Now we we can can apply apply Lemma 4.2.2. The The terms terms in in U b/ have have constant constant depth depth (where determined by (where the the constant constant is is determined by our our choice choice of of the the proof proof system) system) except except for for the the last equation equation where where we we have have aa term term whose whose depth depth is is equal equal to to dp(~o);j thus thus the the maximal maximal last depth depth is is O(dp(~)). Hence Hence the the maximal maximal depth depth of of aa term term vv S Ei O(dp(cp)). S O(Ei � dp(cp) O(Jdp(cp)S), O(l�I), 4.2.8. 4.2.8. Remarks. R e m a r k s . (1) (1) Clearly Clearly the the theorem theorem holds holds for for aa variety variety of of other other systems. systems. In In particular it particular it holds holds for for every every Frege Frege system, system, (see (see section section 88 for for the the definition) definition).. (2 (2)) In In the the proof proof we we have have actually actually constructed constructed "a "a most most general" general" proof proof r F with with the the same same skeleton skeleton A. To To make make it it more more precise, precise, we we should should allow allow propositional propositional variables variables in order formulas then keep in our our first first order formulas and and then keep the the variables variables Vv~ in r F and and treat treat them them as as a in propositional propositional variables. variables. �. 4.3. 4.3. Now Now we we consider consider the the relation relation of of the the number number of of steps steps to to the the size size and and depth depth of of aa proof. depth is since the the depth depth does does not proof. A A relation relation to to the the depth is easy easy to to obtain, obtain, since not include include information information about about terms. terms. For For instance instance we we can can also also bound bound the the depth depth of of aa most most general general unifier unifier as as follows follows (see (see Krajicek Kraji~ek and and Pudlak Pudl~k [1988]). [1988]). Let fication problem Let E E be be aa most most general general unifier unifier of of aa uni unification problem {(tt, (t2n- t, tt2n)}. )}. Then {(tl, tt 22),) , .. .. ..,, (t2n-1, Then 2n 4.3.1. 4.3.1. Lemma. Lemma. maxd(t,2) <_ E, lt,[. i o Then using aa similar similar proof Then using proof as as above above derive: derive: 571 571 The Lengths Lengths of ofProofs Proofs The If cp has a proof with n steps, then cp has a proof with n steps and depth bounded above by O(n ++ I,pl). Icpl). o(n 4.3.2. TTheorem. (Parikh [1973], [1973], Farmer Farmer [1984], [1984] ' Kraji~.ek Krajicek [1989a]) [1989a]) If qo has a proof 4.3.2. heorem. (Parikh with n steps, then qo has a proof with n steps and depth bounded above by o This result result gives gives aa bound bound on on the the size size of of aa proof proof in in terms terms of of the the number number of of steps, steps, This if we we disregard disregard terms terms or or use use aa language language without without function function symbols. symbols. if It isis more more difficult difficult to to bound bound the the size size of of aa proof proof using using the the number number of of steps steps and and the the It size of of the the formula, formula, ifif we we use use the the usual usual definition definition of of the the size size which which includes includes terms. terms. size The technique technique based based on on unification unification works works only only in in cut-free cut-free Gentzen Gentzen sequent calculi. The sequent calculi. An ordinary ordinary proof proof must be first first replaced replaced by by aa cut-free cut-free proof, proof, which which results results in in aa big big An must be increase. Again Again we the result result without see Krajicek and Pudl~k Pudhik [1988] [1988] increase. we state state the without aa proof; proof; see Kraji~.ek and for for aa more more precise precise bound bound and and aa proof proof (the (the idea idea will will be be also also sketched sketched in in the the proof proof of of Theorem 4.4.1). 4.4.1). Theorem There exists a primitive recursive function F such that for every sentence cp and number n, if cp has a proof with n steps, then it has a proof with size bounded by F(cp, n). 4.3.3. Theorem. 4.3.3. T heorem. There exists a primitive recursive function F such that for every sentence qo and number n, if qo has a proof with n steps, then it has a proof with size 0 bounded by F(qo, n). 0 4.3.4. (Krajicek 4.3.4. PProblem. roblem. (Kraji~ek and and Pudhik Pudlhk [1988], [1988], Clote Clote and and Krajicek Kraji~ek [1993]) [1993]) Can Can bounded by F be be elementary, elementary, i.e., i.e., bounded by aa constant constant time time iterated iterated exponential exponential function function (in (in F I cpl ++ n)? The following following interesting interesting result result of of S. S. Buss Buss shows shows very very nicely nicely that that it it is is hard hard to to The determine rst order determine the the structure structure of of terms terms in in fi first order proofs. proofs. He He proved proved this this theorem theorem for for aa particular particular version version of of aa sequent sequent calculus. calculus. (Buss (Buss [1991b]) [1991b]) Given Given aa number number n n and and aa sequent sequent r F -+ --+ � A,, it it is is 0 not not decidable decidable whether whether r F -+ --~ � A has has aa proof proof with with :::; < nn steps. steps. [] 4.3.5. 4.3.5. Theorem. Theorem. At At first first it it may may seem seem that that this this contradicts contradicts Theorem Theorem 4.3.3, 4.3.3, however however notice, notice, that that Theorem Theorem 4.3.3 4.3.3 does does not not claim claim that that given given aa proof proof of of qo with with n steps, steps, there there must must exist exist aa proof proof of of qo with with size size :::; < F(Iqol, n) and n steps. Consequently Consequently it it is is not not possible possible to to minimize minimize the the size size and and the the number number of of steps steps at at the the same same time. time. For For some some solvable solvable cases cases see see Farmer Farmer [1988]. [1988]. cp F(lcpl, n) and n steps. cp n 4.4. 4.4. Finally Finally we we mention mention aa related related topic topic which which is is very very popular popular in in this this field field and and also also demonstrates demonstrates that that the the structure structure of of terms terms in in first first order order proofs proofs is is rather rather complex. complex. Kreisel Kreisel stated stated the the following following conjecture, conjecture, see see Friedman Friedman [1975] [1975] and and Takeuti Takeuti [1987]: [1987]: Kreisel's K r e i s e l ' s Conjecture C o n j e c t u r e Suppose for a formula qo(x) and a number k, k, one Suppose for a formula cp(x) and a number one n (o)) inin Peano can can prove prove cp(s ~(S"(0)) Peano Arithmetic Arithmetic using using :::; <_ kk steps steps for for every every n. n. Then Then 'v'xcp(x) Vxqo(x) is is provable provable in in Peano Peano arithmetic. arithmetic. P. Pudltik Pudldk 572 sn (o) S Here Sn(0) stands for the term obtained by applying applying the the successor successor function function S n-times n-times Here stands for the term obtained by to to 0. The The statement statement seems seems to to be be also also quite quite sensitive sensitive on on particular particular formalization formalization of of Peano Arithmetic. idea of proof for Peano Arithmetic. We We shall shall sketch sketch the the idea of aa proof for the the case case where where Peano Peano 's Arithmetic arithmetic. The Arithmetic is is replaced replaced by by a a finite finite fragment fragment of of arithmetic. The validity validity of of Kreisel Kreisel's Conjecture for using aa different nite fragments Conjecture for fi finite fragments was was first first proved proved by by Miyatake Miyatake [1980] [1980] using different proof. proof. O. 4.4.1 There primitive recursive 4.4.1.. Theorem. Theorem. There exists exists aa primitive recursive function function G G such such that that for for every every n (o)) has k, n, formula and numbers and n formula cp(x) ~(x) and numbers k, n, if if cp(s ~(Sn(O)) has aa proof proof with with kk steps steps and n > G(cp, G(~, k) k) n (x)) is provable. then then Vxcp(s Vx~(Sn(x)) provable. In logic; note In the the theorem theorem we we use use the the provability provability in in pure pure logic; note that that this this implies implies that that the the theorem theorem is is true true also also for for any any finitely finitely axiomatized axiomatized theory theory T as as we we can can incorporate incorporate finitely many finitely many axioms axioms in in ~. We We need need to to add add only only aa very very weak weak assumption assumption about about T in in 's conjecture. order order to to deduce deduce Kreisel Kreisel's conjecture. T cp. T Let Let T T be be a a finite finite fragment fragment of of arithmetic arithmetic such such that that n (y))) n- l (o) VV 3y(x sSn(y))) Vx(x = 00 V Vx x = S(O) S(O) V V ... .. . V Vx x s sn-l(0) TT I-b Vx(x for n. Then for every every n. Then Kreisel' Kreisel'ss Conjecture Conjecture holds holds for for T T.. Proof-hint. P r o o f - h i n t . By By the the assumption assumption on on T T we we have have n-l (o)) /\A Vxcp(S TT I-~- cp(O) ~(0) /\ A cp(S(O)) ~(S(0)) /\ A .. ... . /\ A cp(s ~(Sn-l(0)) Vx~(Sn(x)) Vx~p(x), n (x)) --+ Vxcp(x), o for for every every formula formula cp(x). ~(x). [3 n (o)) has Proof-idea cp(x) and k, nn be P r o o f - i d e a of of Theorem T h e o r e m 4.4.1. 4.4.1. Let Let ~(x) and k, be given given such such that that cp(s ~(Sn(O)) has be. aa proof proof with with k k steps. steps. We We shall shall see see how how large large n n must must be. 4.4.2. 4.4.2. Corollary. Corollary. :::]y(x = = = = = = -+ First First we we transform transform the the proof proof into into aa cut-free cut-free proof proof in in the the Gentzen Gentzen system. system. By By Corollary Corollary 5.2.2 below, below, the the number number of of steps steps in in aa cut-free cut-free proof proof can can be be bounded bounded by by aa constant which depends only only kk and constant which depends and [~p(x)l. Then Then we we apply apply the the technique technique of of unification. unification. This This time, time, however, however, we we consider consider also also the the terms terms in in the the proof. proof. This This is is done done in in two two stages. stages. First First we we consider consider all all proof-skeletons there are ) . For proof-skeletons of of length length K, ((there are finitely finitely many many). For each each of of them them we we find find aa with respect most general proof most general proof ((with respect to to the the propositional propositional and and quantifier quantifier structure structure)) as as in the the proof proof of Theorem 4.2.5. these proofs find most most general in of Theorem 4.2.5. Then Then for for each each of of these proofs we we find general terms them. This terms which which can can be be used used in in them. This can can also also be be done done using using the the theorem theorem about about aa most general unifier. most general unifier. However, However, now now we we treat treat terms terms Sn(O) in in the the sentence sentence ~(Sn(O)) as as unknown, it is is represented in the unknown, which which means means that that it represented by by aa variable variable in the unification unification problem. problem. If If in in terms terms in in the the most most general general solution solution remain remain variables variables for for terms, terms, we we replace replace them them by by first first order order variables. variables. Thus Thus we we obtain obtain a a proof proof whose whose size size is is bounded bounded by by aa primitive primitive and ~(x), thus thus also also in in kk and and ~(x). Let Let us the bound recursive in K and recursive function function in us denote denote the bound by L. ((This This was by L. was essentially essentially the the idea idea of of the the proof proof of of Theorem Theorem 4.3.3, 4.3.3, except except for for the the treatment of of the the term term Sn(O).) treatment (0) .) 1 K, sn (o) K cp(x), sn cp(x). cp(sn (o)) of Proofs The Lengths of 573 573 Let us us have have aa look look on on what what happens happens with with ~(Sn(0)) cp(S"(O)) in in the the most most general general proof. proof. Let This cp(t) for for some some term term tt which which has has two two properties properties This formula formula isis replaced replaced by by ~(t) L; (1) Itl_ It I � L; (1) (2) ta ta = = S"(O), sn (o) , for for some some substitution substitution a. a. (2) Thus is either or S"(0) Thus tt is either Sin(y) sm (y) for for m m <_ and some some variable variable y, y , or sn (o) and and nn <_ � L. L. � LL and m Hence, ifif we we choose choose nn >> L, L, we we get get aa proof proof of of ~(Sm(y)), cp(s (y)) , with with m m << n. n. Then, Then, applying applying Hence, generalization, we we get get aa proof proof of of rye(Sin(y)) Vycp(sm (y)) with with m m << n, n, which which in in turn turn implies implies generalization, n [] 0 Vxcp(s (x) ) . 4.4.3. 4.4.3. If If we we now now consider consider full full Peano Peano Arithmetic, Arithmetic, we we can can also also perform perform the the first first part part of the the proof. proof. But But in in the the second second part, part, where where we we want to bound bound the the size size of of terms, terms, of want to the proof proof fails. fails. It It is is not not possible possible to to write write the the conditions conditions on on terms terms in in the the form form of of aa the unification problem. problem. Some Some time time ago ago Baaz Baaz proposed proposed aa program program for for proving proving Kreisel's Kreisel 's unification Conjecture. Among Among the the most most important important ideas ideas of of his his are are the the use use of of Hilbert's Hilbert ' s c-calculus c-calculus Conjecture. and semiunification (a (a generalization generalization of of unification) unification).. This far and semiunification This program program has has been been so so far realized only for existential induction induction Baaz Pudl~k [1993]; realized only for aa subtheory subtheory of of existential Baaz and and Pudhik [1993J; the the proof uses uses Herbrand instead of of the the c-calculus. proof Herbrand's' s theorem theorem instead c-calculus. 5. o u n d s on o n cut-elimination cut-elimination aand n d Herbrand's Herbrand's ttheorem heorem 5. B Bounds The undecidability of first logic is fact that cannot The undecidability of first order order logic is caused caused by by the the fact that we we cannot bound the the size size of of aa proof proof in in terms terms of of the the size the proved sentence. Nevertheless bound size of of the proved sentence. Nevertheless it is is still still possible possible to to deduce deduce something something about about the the proof the structure structure of the it proof from from the of the (Fortunately proof proof theoretical theoretical studies studies in direction started started before the formula. (Fortunately formula. in this this direction before the undecidability undecidability was was discovered discovered and and therefore therefore they they were were not not hindered hindered by by this this negative negative ' s c-theorem fact.) The this type theorem, Hilbert fact.) The theorems theorems of of this type are are Herbrand's Herbrand's theorem, Hilbert's c-theorem ' s cut-elimination and and Gentzen Gentzen's cut-elimination theorem. theorem. The The important important consequence consequence for for all all natural natural systems systems is is that that one one can can bound bound the the quantifier quantifier complexity complexity of of the the proof proof in in terms terms of of the the quantifier quantifier complexity complexity of of the the formula. formula. This This is is achieved achieved on on the the expense expense of of lengthening lengthening the the proof, proof, however however the the lengthening lengthening can can be be bounded bounded by by aa primitive primitive recursive recursive function. function. This This raises raises an an interesting interesting question question which which we we are are going going to to deal deal with with in in this this section: section: determine determine the the growth growth rate rate of of this this function. function. These These theorems theorems give give more more information information about about the the structure structure of of proofs. proofs. The The most most important important is is the the cut-elimination cut-elimination theorem, theorem, which which states states that that a a general general proof proof can can always be be replaced replaced by by aa cut-free cut-free proof. proof. Cut-free Cut-free proofs proofs have have the the so-called so-called subformula sub:formula always property, property, which which means means that that all all formulas formulas in in the the proof proof are are subformulas subformulas of of the the proved proved formula cpo Here formula ~p. Here the the concept concept of of being being aa subformula subformula is is slightly slightly weaker: weaker: the the terms terms in in the the subformula subformula may may be be different different from from those those in in cpo ~p. Hence Hence there there are are infinitely infinitely many many nitely subformulas subformulas of of cp, ~, (even (even if if we we do do not not use use function function symbols, symbols, since since there there are are infi infinitely many many variables) variables).. The The three three theorems theorems are are equivalent equivalent in in the the sense sense that that there there are are easy easy proofs proofs of of one another one. More important one from from another one. More important the the simulations simulations are are polynomial, polynomial, or or at at most most exponential ((depending depending on particular proof exponential on particular proof systems) systems).. Hence, Hence, if if we we are are satisfied satisfied with with 574 574 P. Pudlak Pudldk precision up up to to an an exponential function, it it is is sufficient sufficient to to give give bounds to one aa precision exponential function, bounds only only to one of of them. them. 5.1. 5.1. Let Let us us consider consider the the important important specific specific case case of of the the relation relation of of the the Herbrand Herbrand theorem theorem and and the the cut-elimination cut-elimination theorem. theorem. An An easy easy extension extension of of the the cut-elimination cut-elimination theorem theorem is is the the Midsequent Theorem. It It states states that that each each proof proof of of aa formula formula in in the the prenex prenex form form can can be be transformed transformed into into aa proof proof where where there there is is aa sequent sequent above above which no quantifier is used used and and below which only quantifier rules are used. used. which no quantifier rule rule is below which only quantifier rules are This This can, can, in in fact, fact, be be easily easily constructed constructed from from aa cut-free cut-free proof. proof. An An easy easy analysis analysis of of the see Hajek the midsequent midsequent shows shows that that it it is is essentially essentially aa Herbrand Herbrand disjunction disjunction ((see H~jek and and Pudlak 1993,Chapter V]) Recall that Pudl~k [[1993,Chapter V]).. Recall that aa Herbrand disjunction is is aa disjunction disjunction of of term term instances instances of of a a Herbrand variant of of a a formula, formula, where where the the Herbrand Herbrand variant variant is is obtained by omitting the ers, starting obtained by systematically systematically omitting the quantifi quantifiers, starting from from the the outermost, outermost, and bounded variable and replacing replacing each each universally universally bounded variable x x by by F(Yl, F ( y l , .. .. .., , Yk) Yk),, where where F F is is aa new . . . , Yk new function function symbol symbol and and Yl, Yl,..., Yk are are the the free free variables variables of of the the current current formula. formula. A A midsequent midsequent does does not not contain contain these these new new function function symbols, symbols, but but the the dependencies dependencies among among the the occurrences occurrences of of variables variables allow allow us us to to replace replace variables variables by by such such terms terms while while preserving preserving the the propositional propositional validity validity of of the the disjunction. disjunction. Now Now suppose suppose we we are are given given a a Herbrand Herbrand disjunction. disjunction. First First replace replace the the maximal maximal terms Herbrand function terms whose whose outermost outermost function function symbol symbol is is aa Herbrand function symbol symbol by by distinct distinct variables. variables. Then Then omit omit disjunctions disjunctions and and interpret interpret it it as as aa sequent. sequent. It It has has aa propositional propositional proof proof in in the the sequent sequent calculus. calculus. Now Now each each sequent sequent provable provable in in the the propositional propositional sequent sequent calculus exponential size. size. Thus calculus has has a a proof proof of of at at most most exponential Thus we we get get the the upper upper part part of of the the sequential sequential proof. proof. The The lower lower part part is is obtained obtained by by applying applying quantifier quantifier rules rules in in aa suitable suitable order. This possible due order. This is is possible due to to the the structure structure of of the the Herbrand Herbrand disjunction. disjunction. The The number number of bounded by of the the proof proof lines lines with with quantifier quantifier rules rules is, is, of of course, course, bounded by the the number number of of variables. 1987] and 1993,Chapter variables. For For more more details details see see Takeuti Takeuti [[1987] and Hajek H~jek and and Pudlak Pudl~k [[1993,Chapter V, section section 5]. 5]. V, Midsequent Theorem. Herbrand variant Herbrand disjunction 5.2. shall use 1. Note 5.2. We We shall use the the Hilbert Hilbert style style system system of of Chapter Chapter 1. Note however however that that when when no no restrictions restrictions are are posed posed on on the the complexity complexity of of formulas formulas in in the the proof proof the the Hilbert Hilbert style style and calculi are increase of size. By and Gentzen's Gentzen's sequent sequent calculi are equivalent equivalent up up to to aa polynomial polynomial increase of size. By Theorem Theorem 4.1 4.1 it it is is true true even even if if we we take take proofs proofs in in aa tree tree form form in in one one of of them them and and in in a a sequence sequence form form in in the the other other one. one. We We shall shall start start with with an an upper upper bound bound to to cut-elimination. cut-elimination. Suppose Suppose aa sentence sentence qa has has aa proof proof of of size size n n and and depth depth dd (i.e., (i.e., each the proof each formula .formula in in the proof has has logical logical depth depth at at most most d). d). Then Then qa has has aa cut-free cut-free proof proof of o/ 0 size 22~)(d 1"-1 size 0 (d)) .. cp 5.2.1. 5.2.1. Theorem. Theorem. cp The The proof proof can can be be found found in in Chapter Chapter I. I. 5.2.2. 5.2.2. Corollary. Corollary. 22 onO(l~l.v~)" ( l IfI / ~ has has aa proof proof of o/ size size n, n, then then ~ has has aa cut-free cut-free proof proof of o/ size size cp cp 575 575 The Lengths Lengths of ofProofs Proofs The D [] This follows follows from from Theorems Theorems 5.2.1 5.2.1 and and 4.2.5. 4.2.5. PProof. r o o f . This Now we we consider consider aa lower lower bound. bound. The The proof proof will will be be easy easy since since we we have have already already Now developed the the theory theory of of definable definable cuts. cuts. developed There exists a sequence of sentences such that 'l/;n has a proof of size p(n), n = . . . , where p is a fixed polynomial, and there is no cut-free proof of'l/;n with less than proof-lines for n = 5.2.3. TTheorem. '1/;2, . such that r 5.2.3. heorem. There exists a sequence of sentences r'1/;1 , ~/)2,... 1 , 22,, . . . , where p is a fixed polynomial, and there is no has a proof of size p(n), n = 1, 1, 22 ..... . cut-free proof of r with less than 22� o proof-lines for n = 1, . . . . Proof. Consider the the following following very very weak weak fragment fragment of of arithmetic. arithmetic. It It has has the the constant constant P r o o f . Consider 0, the the successor successor function function S(x), addition addition + and and exponentiation exponentiation 22x=.. It It has has axioms axioms of of 0, 3, and and the the following following mathematical mathematical axioms: axioms: equality, say say those those considered considered in in section section 3, equality, S(x), + O+x =x x ++ (y(y ++ z)z) == (=(x ++ y)y) ++ z,z, x + s(~) S(x) = s(~ S(x + y), s(o), 2° = S(O), 0+x=x + = + y), 20 = 2s(x) = 2= + 2=. l(x) Furthermore Furthermore the the theory theory contains contains a a unary unary predicate predicate symbol symbol I(x) with with interpretation interpretation "an initial segment include the "an initial segment of of integers integers without without the the last last element" element".. Thus Thus we we also also include the I is inductive: axioms saying that axioms saying that I is inductive: 1(0) I(0) l(x) I(x) --+ I(S(x)). I(S(x)). (31) (31) --+ n t Let Let us us call call this this theory theory A. A. For For a a natural natural number number n and and aa term term t we we denote denote by by n (t) the EE~(t) the term term defined defined inductively inductively by by E~ = t, (En(t)) . n+ l (t) =- 22(E"(t)). EEn+l(t) En (o) 'l/;n n (o))), (A A A -+ I(E I(En(0))), :33 . .... (/\ In In particular particular the the value value of of En(0) is is 2� 2~. Now Now we we define define r by by . --+ where where /\ AA A denotes denotes the the conjunction conjunction of of the the axioms axioms of of A A and and :33 .. .. .. denotes denotes the the existential existential closure. closure. 5.2.4. 5.2.4. Claim. Claim. 'l/;Cnn 's's have have polynomial polynomial size size proofs. proofs. 576 576 P. Pudldk shall use Proof. P r o o f . We We shall use Theorem Theorem 3.4.1 3.4.1.. By By this this theorem theorem there there exists exists aa sequence sequence of of o(x) equal with IP o (x) , IPl (X) , . . . with formulas formulas IP ~0(x), qal(X), (x), IP2 qo2(x),.., qo0(x) equal to to I(x) and and I(x) (32) (S(X))), (O) /\A 'v'X( ~i+1(0) VX(~i+l(X) -'ff IPi+l (/9i+l(~(X))), IPi+l (X) -+ IPi+l (33) 'v'X( (2"')), Vx(~Pi+l (x) -+ -+ IPi qoi(2")), (33) IPi+l (X) having 1, ..... . . . Combining having polynomial polynomial size size proofs proofs in in A A for for ii = - 0, 0,1, Combining (33) (33) for for ii = = 0, 0 , . .. .., , nn-l -1 we we get get aa polynomial polynomial size size proof proof of of Vx(qo~(x)--+ I(E~(x))) i I(En (o)) in A. half of with the in A. The The first first half of (32) (32) for for i + + 11 = -- n together together with the last last sentence sentence give give aa polynomial A, hence polynomial size size proof proof of of I(E~(O)) in in A, hence a a polynomial polynomial size size proof proof of of r in in first first 0 order order logic. logic. [] 'l/Jn Let tt bebe aa closed closed term Let [3)3 bebe aa Let term of of A A with with value value in in IN IN equal equal to to m m.. Let conjunction conjunction of of term term instances instances of of axioms axioms of of A A such such that that I(t) [3 I(t) [3 contains isis provable provable in in first first order order logic. logic. Then Then 13 contains at at least least m m term term instances instances of of the the axiom axiom I(x) -+-+ I(S(x)) . 5.2.5. Claim. 5.2.5. C laim. -+ -+ Proof. P r o o f . W.l.o.g. W.l.o.g. we we may may assume assume that that all all the the terms terms in in [3 /3 are are closed closed (otherwise (otherwise [3 contains substitute substitute 0) 0).. Suppose Suppose/3 contains fewer fewer m m occurrences occurrences of of the the axiom. axiom. Consider Consider the the [3. By values values of of terms terms t such such that that I(t) -+ -+ I(S(t)) occurs occurs in in/3. By the the pigeonhole pigeonhole principle principle there there is is an an i0 < m which which is is not not the the value value of of any any such such a a term. term. Assign Assign truth truth values values to the atomic atomic subformulas [3 -+ truth value to the subformulas of of/3 -+ I(t) as as follows: follows: assign assign an an identity identity a a truth value according in natural natural numbers, numbers, and according to to its its interpretation interpretation in and assign assign I(t) the the value value TRUE TRUE bigger. Thus if if the the value value of of tt is is less less then then or or equal equal to to i0 and and FALSE FALSE if if it it is is bigger. Thus all all the the instances giving this [3, while while I(t) instances of of axioms axioms of of A A get get the the value value TRUE TRUE giving this value value also also to to/3, 0 gets gets FALSE. FALSE. Thus Thus/3[3 -+ ~ I(t) cannot cannot be be provable. provable. D I(t) I(S(t)) I(t) I(t) io I(t) I(t) Now Now we we derive derive the the lower lower bound. bound. Let Let aa cut-free cut-free proof proof d d of of l/J r n be be given given.. Let Let All the quantifier rules rules of rules of denote 33 ..... . (AA (/\ A --+ II(E~(0))). (En (0))) . All 'Y-y denote the quantifier of d d are are the the rules of io t -+ 3-introduction 3-introduction applied applied to to a a term term instance instance of of 'Y ")' or or aa term term instance instance of of a a formula formula obtained obtained from from 'Y T in in this this way. way. Let Let d' be be the the proof proof obtained obtained by by applying applying the the same same rules rules to initial segments to initial segments of of d but but omitting omitting the the quantifier quantifier rules. rules. We We have have to to omit omit also also the the contractions 3, since will not contractions applied applied to to formulas formulas with with 3, since such such formulas formulas will not appear appear in in the the new new proof proof d'. Thus Thus the the end end sequent sequent of of d' d' is is aa sequent sequent --+ ~ 1 7 11,,.. .. ..,, 'Yk 7k where where 'Y 9'i's are i 'S are term instances be a term instance /\ A). term instances of of 'Y. 7. Let Let 'Y 7ii be aii -+ -+ I(En(O)), (where (where a cqi is is aa term instance of of/~ A). Then Then d d'. d' I(En (o)), n 0)) a-1l /\^ . .. .. . /\ ak I(E I ( E " ( (o)) -+ d is Since in original proof is aa tautology. tautology. By By Claim Claim 5.2.5, 5.2.5, k k � _> 2� 2~ . Since in the the original proof d all all 'Y, 7 , .. ...., , 'Yk 7k must must eventually eventually merge merge into into one one formula, formula, d d must must contain contain at at least least kk � _> 2� 2o proof-lines. proof-lines. o D The Lengths of of Proofs 577 577 Let us us note note that that the the above above proof proof can can be be applied applied directly directly to to Herbrand Herbrand theorem theorem Let too. Namely, Namely, the the above above argument argument also also shows shows that that any any Herbrand Herbrand disjunction disjunction for for 1/Jn Cn too. must have have at at least least 220 disjuncts. must � disjuncts. 5.3. The question question whether whether mathematical mathematical reasoning reasoning as as represented represented by by Zermelo Zermelo5.3. The Fraenkel set set theory theory is is consistent consistent has has intrigued intrigued aa lot lot of of mathematicians mathematicians and and philoso philosoFraenkel phers. The The approach approach of of finitists finitists is is to to discard discard it it as as meaningless meaningless and and ask ask instead instead phers. whether there there is is aa feasible feasible proof proof of of contradiction contradiction from from our our axioms axioms of of set set theory. theory. We We whether shall say say more more about about this this modified modified question question in in the the next next section. section. Now Now we we only only want want shall to show show that that there there are are theories, theories, not not quite quite unnatural, unnatural, which which are are inconsistent inconsistent but but in in to which no no feasible feasible proof proof of of contradiction contradiction exists. exists. Such Such theories theories have have been been considered considered which by several several researchers researchers including including Parikh Parikh [1971], [1971], Dragalin Dragalin [1985], [1985], Gavrilenko Gavrilenko [[1984] and by 1 984] and Orevkov [1990]; [1990]; the the first first and and the the most most infl influential was the the paper paper of of Parikh. Parikh. Orevkov uential was Let Let T T be be any any fragment fragment of of arithmetic arithmetic (it (it can can be be even even the the set set of of all all true true sentences sentences in the the standard standard model) model).. Let Let tt be be aa closed closed term term whose whose value value m m is is so so large large that that no no in 1 00 . proof of of size size m m can can be be ever ever constructed. constructed. Note Note that that tt can can be be quite quite simple, simple, say say 221~176 proof Extend T T to to T' T ~ by by adding adding axioms axioms Extend I(0),, 1(0) l(x) I(S(x)) , ~I(2~ . ""1(2�) -+ Clearly T' Clearly T ~ is is not not consistent. consistent. We We shall shall show, show, however, however, that that there there is is no no feasible feasible contradiction in in T' T ~.. contradiction Suppose we can can derive derive a a contradiction in T' T ~ of of size size less less than than n. Then, by the the Suppose we contradiction in Then, by bound on cut-elimination, cut-elimination, there there is is a a cut-free cut-free proof proof of of contradiction contradiction of of size size less less than than bound on A To fragment To To we have have such of --+) ..., 22�~ . This This means means that such aa proof proof of -~/~ To,, for for aa finite finite fragment that we be aa Skolemization Skolemization of of To. To . Then the proof of ---+ ..., A A T1 Tl is of T' T'.. Let T1l be Then the proof of -~ ~ at most of Let T is at most polynomially larger than each sentence sentence has has a a polynomial proof from from its its polynomially larger than 22�o (since (since each polynomial size size proof Skolemization). Thus by by taking also t, only slightly slightly larger larger than we get get Skolemization) . Thus taking m, hence hence also t , only than n, n, we to the the open open theory TI . Then Then we use the "interpretation" an upper ~ to theory T1. we use the same same "interpretation" an upper bound bound 22� argument as the lower lower bound to show show that that such such a a proof cannot exist. argument as in in the bound proof proof above above to proof cannot exist. Let us us note closure properties properties such as I(x) A I(y) l(x) /\ l(y) -+ Let note that that we we can can add add also also other other closure such as l(x + y) y) and and the the same same for take tt aa little little larger, larger, since since we we can can I(x for multiplication, multiplication, if if we we take using small small formulas formulas and (see 3.5). interpret such such aa theory theory in in TT'~ using interpret and short short proofs proofs (see 3.5). 6. i n i t e cconsistency onsistency sstatements t a t e m e n t s - - cconcrete o n c r e t e bbounds ounds 6. F Finite We have have already already remarked remarked that there are almost no no concrete concrete examples examples of of sentences sentences We that there are almost for which which one one can can prove nontrivial bounds bounds on on the the length length of of proofs. proofs. There There is, is, however, however, for prove nontrivial T does does not not prove prove one exception; exception; namely, namely, the the sentences sentences expressing expressing that that aa theory one theory T contradiction using using aa proof proof of of length length _< ::; n; n ; (we (we shall shall say say that that the the theory theory TT is is aa contradiction consistent n) . consistent up up to to n). These are are not not real real mathematical mathematical theorems, theorems, which which would would be be interesting interesting for for These an ordinary ordinary mathematician, mathematician, but but they they are are very very interesting interesting for people who who study study an for people 578 578 P. Pudlak Pudldk foundations of of mathematics. We shall prove bounds bounds on on the the length of such such aa statement statement foundations mathematics. We shall prove length of could be called a finite (or, (or, if in in the the theory theory T T itself. itself. This This could be called if you you prefer prefer the the word, word, feasible) version of of the the second second Ghdel Theorem. Furthermore, Furthermore, these these bounds bounds (especially the version Godel Theorem. (especially the lower applications. lower bounds) bounds) have have interesting interesting applications. ajinite feasible) 6.1. We 6.1. Formalization F o r m a l i z a t i o n of of syntax. syntax. We shall shall derive derive aa strengthening strengthening of of the the second second Godel Incompleteness Ghdel Incompleteness Theorem Theorem and and some some speed-up speed-up results. results. We We shall shall try try to to avoid avoid the the boring subject possible. However boring subject of of the the formalization formalization of of syntax syntax as as much much as as possible. However we we have have to the classical to say say something something about about it, it, since since the classical way way of of formalizing formalizing syntax syntax cannot cannot be be used used here. here. 6.1.1. First 6.1.1. First we we need need a a more more efficient efficient way way of of representing representing numbers numbers by by terms. terms. The The n (o) cannot classical numerals s Sn(0) cannot be since their their length is already already greater greater than than n, be used, used, since length is classical numerals while while we we want want to to bound bound the the lengths lengths of of proofs proofs by by aa polynomial polynomial iinn In] - the the length length of of the binary representation representation of of n. Thus Thus we we define the n-th numeral 11 n_n_as as follows. follows. If If the binary define the n-th numeral In l n. kk . n, nn == ;=E~ 0 2'a;, I}, then 2iai, a; ai E {O, {0,1}, then n is is the the closed closed term term . . .) )) , !!o _a_0+ + 2 2.· (!h (_al + + 22.· (!h (a2 + + ' ". . . (!! ( ~k--1l + + 22.' !!k) a_k)...))), 11 i=0 where 1 1 -= S(O) S(0),, 2 2= = 1 1+ + 1. 1. where We numbers. A suitable one-to-one We need need also also to to represent represent sequences sequences by by numbers. A suitable one-to-one mapping mapping from {O, I1}* } * onto from {0, onto IN IN is is given given by by ( a 0 , . . . , an)~-+ E 2i( a' + 1). a, A A formula formula cp ~ is is first first represented represented as as a a 00 -- 11 sequence sequence a, then then we we take take the the number number m m which codes a as Godel number number of cpo We cp1 for which codes as the the Ghdel of ~. We shall shall use use the the symbol symbol r[~] for such such aa Godel number of Ghdel number of cp ~.o a 6.1.2. Suppose 6.1.2. Suppose that that we we want want to to formalize formalize aa concept concept which which can can be be represented represented as as aa k If R subset T, then subset R R � C_ lN INk.• If R is is formalized formalized by by aa formula formula p(xl,..., Xk) in aa theory theory of of T, then k) in we clearly need we clearly need that that p(Xl, . . . , X ( n l , . . . , n k ) E R r T ~- p ( n l , . . . , n_n_k). This alone is usually not sufficient. The This alone is usually not sufficient. The key key property property for for our our proof proof is is that that the the above above formula length. As important concept, formula has has a a proof proof of of polynomial polynomial length. As it it is is an an important concept, we we shall shall define define it it precisely. precisely. k and 6.1.3. Definition. Let 6.1.3. Definition. Let an an axiomatization axiomatization of of aa theory theory T T be be fixed, fixed, let let R R � c_ lN INk and let be aa formula. T, if let p(xl,..., , xk) formula. We We say say that that p polynomiaUy numerates R R in in T, if for for k ) be holds: R(nl,..., R( some some polynomial polynomial p and and every every n l , .. ... ., , nk E IN, IN, the the following following holds: nk) iff iff k! . T f~ p ( n , .. ...., , Ilk) n_k) by by a a proof proof of of length length :::; _< p(lnll,..., , ]nkl). T p(Xl, . . . X p p(l1, nl , p polynomially numerates n l , . . . , nk) p(lnll, . . . In ) It turns out that, for sufficiently strong T, the numerable It turns out that, for aa sufficiently strong theory theory T, the polynomially polynomially numerable relations are just the relations are just the NP AlP relations. relations. The The Lengths Lengthsof of Proofs Proofs 579 579 6.1.4. Theorem. T h e o r e m . The The following following are are equivalent equivalent 6.1.4. (1) R isi NP; J 7,, (1) (2) R R is is polynomially polynomially numerable numerable in in Robinson Robinson arithmetic arithmetic Q Q.. (2) Since (2) (2) => ~ (1) (1) is is trivial trivial for for any any finitely finitely axiomatized axiomatized theory theory T, the the same same theorem theorem Since holds for for any any finite finite consistent consistent extension extension of of Q. Q. holds Before Before we we sketch sketch the the proof proof of of the the converse converse implication, implication, we we state state aa lemma lemma whose whose proof we we defer defer to to section section 6.3.4. 6.3.4. proof 6.1.5. For 6.1.5. Lemma. Lemma. For every every bounded bounded formula formula cp(x) ~(x),, with with xx the the only only free free variable, variable, there exists exists aa polynomial polynomial pp such such that that there If1 IA00 + + Exp Exp f~ Vxcp(x) Vx~(x) implies that that for for every every nn EE IN, implies Q f- cp(n,) Q by aa proof proof of of length length :::; < p(log p(log n) n).. by This f10 + This lemma lemma allows allows us us to to replace replace Q Q by by IIA0 + Exp Exp in in the the proof proof of of the the implication implication (1) => ~ (2) (2).. If If we we are are proving proving some some property property of of a a concept concept formalized formalized by by aa f1 A00 formula formula (1) Q, but in f10 + in IIA0 + Exp, Exp, then then this this statement statement may may not not be be provable provable in in Q, but each each numeric numeric instance instance has has a a polynomial polynomial proof. proof. Thus Thus for for instance instance we we are are free free to to use use commutative commutative and associative associative laws. laws. and Proof-sketch R EE Np be given. given. We formalize P r o o f - s k e t c h of of Theorem T h e o r e m 6.1.4. 6.1.4. Let Let an an R JV'T~ be We formalize computations machine defining R. Thus R(( nn ll ,, .. ... ., , n~) nk) is is equivalent equivalent to to the computations of of aa Turing Turing machine defining R. Thus R the existence of string ss whose length is is bounded by aa polynomial in IInd, existence of aa 00 - 11 string whose length bounded by polynomial in n l l , ·. .. .., , Ink Inkll accepting computation). computation). satisfies aa certain property (namely, (namely, ss codes and which which satisfies certain property codes an an accepting and particular bits states that that each each c particular This This property property states bits of of ss have have one one of of some some particular particular forms, forms, where cc is some constant. For aa given there are are polynomially polynomially many many such such conditions. is some constant. For given s, s , there conditions. where Xk, y) y) such formula, where stand for nk and and Denote by aa(xI Denote by ( x l ,, ... .. ,. , xk, such aa formula, where X for nnI, l , ... .. ,. , nk Xn stand X Il,, .. ...., , xn holds for for some some true, then then a(nl,...,n_n_k,m) a(ZlJ , . . . , Ilk , m) holds nk) isis true, the string string s.s. If If RR(n yy for for the ( n ll,, ..... ,. , nk) m, whose whose length length is is bounded bounded by by aa polynomial polynomial in in IInnll ll,, .. .. .., , Inkl. number m, To prove prove Ink l . To number Ilk, m) by aa polynomial polynomial proof proof in in Q, Q, transform transform itit into into statements statements about about single single m) by aa(n.I, ( n l , .'. , . n_~, these bits of of the the string string encoded encoded by by m. m. Since Since the the string string really really witnesses R(( nnl, nk ) , these bits witnesses R l , ..... ., , nk), 3ya(n.I' · · · ,n_k, elementary statements statements are are true, true, hence hence provable. provable. Finally Finally derive derive 3ya(nl,... elementary y) ' Ilk, y) 0 from aa(n, Ilk, m). m) . Thus Thus 3ya(xl,..., 3ya(xI , . . . , xn) xn) polynomially polynomially numerates numerates R. R. from ( n ll,>... .. ,. , n_k, [] . Now we we apply apply Theorem Theorem 6.1.4 6.1.4 to to the the provability provability predicate. predicate. Suppose Suppose aa theory theory TT isis Now NP, resp. resp. 7P~,, set set of of axioms. axioms. Let Let R(x, R( x, y) y) denote denote that that xx is is aa proof proof of of yy given by by an an AlP, given also. By By Theorem Theorem 6.1.4 6.1 .4 there there isis aa formalization formalization in T. T . Then Then RR isis in in A/P, NP , resp. resp. PP, , also. in of this this relation, relation, such such that that every every true true numeric numeric instance instance has has aa polynomial polynomial proof proof ProofT of PrOofT " Iml < n" can also be polynomially numerated, we get the in Q. Since the relation in Q. Since the relation "lml < n" can also be polynomially numerated, we get the following following corollary: corollary: 580 580 P. Pudl6.k Pudldk
6.1.6. 6.1.6. Corollary. Corollary. Recall Recall that that 1II~ll _< nn is is a a convenient convenient notation notation for for the the statement statement that that there there exists exists I
ConT(x) --dr -~ PrT(x, [_0 = !I), the consistency the length the consistency of of T T up up to to the length x. x. 6.2. 6.2. Now Now we we are are ready ready to to prove prove the the main main lemma lemma of of the the lower lower bound. bound. Let Let T T be be aa sufficiently sufficiently strong strong fragment fragment of of arithmetic. arithmetic. Let Let f f : 9IN ]N -+ --~ be polynomial time be aa polynomial time computable, computable, increasing increasing function. function. Suppose Suppose that that
6.2.1. 6.2.1. Lemma. Lemma. IN ]N nn = l iT)) = 0 O ((yJ (II (ll ConT(11.) COnT(n)lIT)) Hence Hence if if f f can can be be extended extended to to an an increasing increasing function function defined defined on on positive positive real real numbers, numbers, then then we we can can write write the the conclusion conclusion as as = l l iT -- n(J [] CoaT(n)lIT a ( f --1 (n)). (n)). II ConT(11.) Proof.
6.2.2. 6.2.2. Claim. Claim. < n -+ ~ Iio 111= � < g(n) g(n),, where ~ h ~ g(n) g(~) = = O(J(n) O(/(n)) ) . 11.0. = lilT 1115(_~)11= 1 8(11.) II T � To To prove prove the the claim claim assume assume (34) (34) = _< n. n. 1I]5(n)llT 1 8(11.) IIT � (35) (35) Substituting Substituting 11. n in in (34) (34) we we get get ( Pr (11., r8(11.)1) log n ) o~ 1) . l iT = lira(n) == = .., ~Pr(n, [~(n)])ll~ = ((log~) 118(11.) (36) (n) . IIII Pr(~, Pr(11., r[5(n)])ll~ < f1(~). § (11.) l ) IIT � (37) (37)
The Lengths of of Proofs 581 581 Thus we we get get from from (35) (35),, (36) (36) and and (37) (37) Thus il T = IIIQI 0=- llilT -- O(n O(n + + (logn) (logn) O(l ~ )+ + f(n)) f(n)) = -- O(f(n)), O(f(n)), 0 which proves the claim. [] which proves the claim. Since the the assumption assumption of of the the lemma lemma is is formalized formalized in in T, T, the the claim claim can can also also be be Since proved in in T, T, thus thus we we get get proved T ConT ( fg (x)l ). (38) T f~ Pr PrT(x, --4 -, ~COnT(rg(x)]). (38) 6 (x) 1 ) � T (x, fr~(x)]) Since T T is is consistent, consistent, the the claim claim implies implies in in particular particular that that 116( II~(n)llT n. Thu Thuss it it Since n.) li T > n. (Observe that the proof follows using Con suffices ) suffices to to upper-bound upper-bound 116( II~(n)l[T using II COnT(n)[IT" (Observe that the proof follows ( ) II li II T n. T' n. T very much much the the structure structure of of the the proof proof of of the the second second Gi:idel Gbdel Incompleteness Incompleteness Theorem.) Theorem.) very First substitute substitute n. n in in (38), (38), thus thus we we obtain obtain First l ( fg (n._)l) )l ) � -, ) 1 ) li T = IIII Con 6 (n.)1)11 ConT(rg( ~ Pr( Pr(_n, - (log 0og n)) O( ) . n., fr6( Combining it it with with (36) (36) we we get get Combining l n) O( IIII Con COnT(rg( = (log (logn) ~ ), T ( fg (n.)l)l)) � 6 (n.) li T = hence hence n) O(l n) O~ (l) :::: li T - (log )l ) li T :::: IIII Con ConT(rg( > 116( 116(n.) )IIT(logn) >- n n - - (log (logn) ~ ). T ( fg (n._)l)llT 1 o( ) n) ~ , the Since g(n) Since 9(n) = = O(f(n)) O(f(n)) and and IlIIF9(n_)l = g(n)IIT g(n)llT = = (log (logn) the conclusion of the the fg (n.)l = conclusion of 0 lemma follows. follows. [:3 lemma -- (Friedman [1979], 6.2.3. 6.2.3. Theorem. Theorem. (Friedman [1979], Pudllik Pudl~k [1986]) [1986]) Let Let T T be be aa sufficiently sufficiently strong strong fragment of of arithmetic arithmetic axiomatized axiomatized by by an an NP AfT9 set of axioms. Then there there exists exists €e > 00 fragment set of axioms. Then such that that for for all all n, n, such II Con ( _)ll > o P r o o f . by by Corollary Corollary 6.1.6 6.1.6 and [] Proof. and Lemma Lemma 6.2.1. 6.2.1. With more additional can reduce about the With aa little little more additional work work one one can reduce the the assumption assumption about the strength the condition is possible precise lower lower 2 Q. Q. Also Also itit is possible to to give give aa more more precise strength to to the condition TT _D bound by by improving improving the the bound bound in in Corollary Corollary 6.1.6. bound 6.1.6. The The best best lower lower bound bound has has been been proved Pudls [1987]. In that we considered first order order logic proved in in Pudllik [1987] . In that paper paper we considered first logic augmented augmented ' s C-rule, with Rosser's to introduce names for with Rosser C-rule, which which allows allows to introduce names for objects objects whose whose existence existence has proved. Formally Formally it it means from 3xcp(x) 3xgg(x) for for a a new has been been proved. means that that we we can can derive derive ~(c) cp(c) from new c. (This (This apparently to shorten shorten some some proofs, proofs, but but we we are not able able to to constant constant c. apparently enables enables to are not prove aa speed-up speed-up of of this this calculus calculus versus versus the the ordinary one.) For For such calculus we prove ordinary one.) such aa calculus we obtained obtained aa lower lower bound bound f~(n/(log D(n/(log n)2). nf) . 6.3. Now Now we we turn turn to to the the upper upper bound. bound. Recall Recall that that in in section section 33 we we proved that for 6.3. proved that for aa sequential theory theory T, T, there there exists exists aa sequence sequence of of formulas formulas ~n CPn which which define define satisfaction satisfaction sequential for for formulas formulas of of depth depth nn -= 1, 1 , 2, 2, .... . . . . Moreover Moreover ~r,(ro,], x) = a ( ( x ) , , . . . , (X)r,) (39) (39) and and Tarski's Tarski ' s conditions conditions have have polynomial polynomial size size proofs. proofs. The The following following is is an an immediate immediate consequence. consequence. P. Pudldk Pudl6k P. 582 582 6.3.1. LLemma. (1) For For every every axiom axiom a0 of of T, T, dp(a) dp(o) ~:::; n, n, TT proves proves w<,:>.(r<:,<-i. 'v'xCPn ( fol , ~) x) 6.3.1. emma. (1) using aa polynomiaUy polynomially long long proof. proof. using (2) (2) For For every every n, n, TT proves proves that that any any axiom axiom of of depth depth ~:::; nn isis true true and and the the truth truth of of formulas of of depth depth ~:::; nn isis preserved preserved by by every every rule, rule, furthermore furthermore these these proof proof are are bounded bounded formulas by aa polynomial polynomial in in n. n. by Proof. The The first first part part follows follows directly directly from from (39). (39). For For part part (2), (2) , let let us us consider consider only only Proof. modus ponens. ponens. Thus Thus we we need need aa proof proof of of modus 'v'X, y ( f dp(x --+ y) :::; 111 /\ 'v'zcpn (x, z) /\ 'v'zCPn (x --+ y, z) --+ 'v'zCPn (Y, z)) . (40) We know Tarski's condition condition We know that that Tarski's [@(x .->. v) <_ al --+ -> v,z) - z) -..+ <,<::,,.,,(v,z))) has proof, thus has aa polynomial polynomial proof, thus also also (40) (40) has has aa polynomial polynomial proof. proof. 6.3.2. T Theorem. 6.3.2. heorem. o ((Pudls Pudhik [1986]) [1986]) Let Let T axiomatized by T be be aa sequential sequential theory theory axiomatized by aa finite set of finite set of axioms. axioms. Then Then ConT(n) = n ~ Proof. be given. Let o(x) be the the following following formula Proof. Let Let nn be given. Let a(x) be formula 'v'y, z("y z( "y is depth :::; size :::; /\ Vy, is a a proof proof of of depth ~ 11 n and and size < x" x"A A"z y" --+ 'v'vcpn Vv~n (y, v)). "z is is aa formula formula of of y" /\ (Y , v)). Lemma Lemma 6.3.1 6.3.1 implies implies that that o(Q) a(0) and and 'v'x(o(x) Vx(a(x) --+ --+ o(S(x)) a(S(x)) have have polynomial polynomial size size proofs. proofs. Thus proof of Thus by by proving proving o(Q) a(0),, 0(1), a ( 1 ) , .. ...., , 0(11) a(n) one one by by one one we we get get aa polynomial polynomial proof of 0(11) a(n).. hand, by fQ = On On the the other other hand, by (39) (39),, we we have have 'v'v"'CPn Vv-~n (([0 - 11 11,, v), v), also also by by aa polynomial polynomial proof. proof. Thus long proof Thus we we have have aa polynomially polynomially long proof that that aa proof proof of of length length :::; _< n n does does not not contain contain 0 the the formula formula Q 0= = 1. 1. [:] This This theorem theorem has has been been proved proved also also for for some some theories theories which which are are not not finitely finitely axiomatized, axiomatized, namely namely for for theories theories axiomatized axiomatized by by aa certain certain kind kind of of axiom axiom schemas. schemas. These These results results include include the the theories theories Peano Peano Arithmetic Arithmetic and and Zermelo-Fraenkel Zermelo-Fraenkel set set theory. theory. Furthermore for finitely axiomatized sequential theories it is possible to improve Furthermore for finitely axiomatized sequential theories it is possible to improve the bound to the bound to O(n) O(n).. This This improvement improvement is is based based on on the the following following ideas. ideas. 's Firstly, Firstly, by by counting counting more more precisely, precisely, it it is is possible possible to to prove prove that that (39) (39) and and Tarski Tarski's cP) . conditions definition for conditions for for the the truth truth definition for formulas formulas of of depth depth dd have have proofs proofs of of size size O( O(d2). Secondly, Secondly, by by Theorem Theorem 4.2.5, 4.2.5, aa proof proof of of contradiction contradiction of of length length n n can can be be transformed transformed into into aa proof proof of of depth depth O(,.fii) O ( x / ~ . . Thus Thus we we need need the the truth truth definition definition only only for for such such aa depth depth and and hence hence the the auxiliary auxiliary formulas formulas have have linear linear size size proofs. proofs. Finally, Finally, one one can can use use aa shorter shorter way way to to prove prove 0(11) a(n).. This This is is because because of of the the following following lemma, lemma, which which gives gives us us aa proof proof even even much much shorter shorter than than 0 O(n). (n) . 583 583 The Lengths of ofProofs The 6.3.3. LLemma. 6.3.3. emma. a(x) a(O) ^1\ wVx (~(~) (a(x) ~ .(s(~))). a(S(x))) . .(0) Suppose TT D_ 2Q Q and and a(x) isis aa formula formula such such that that TT proves proves Suppose -+ Then Then ) ( lIa(n) IIT (log n?2) I1~(~-)11~ == o0 ((logn) (thus have proofs proofs polynomial polynomial in in log log n). n). a(n) have (thus a(n_) al a Proof. First define subcut a ~ of of a by by P r o o f . First define aa subcut ~'(~) =~ vy < ~ ( y ) . all a Then take subcut a" of of a which is closed closed under addition and and multiplication. multiplication. This This Then take aa subcut which is under addition is easy, if the integers in T satisfy the laws laws of ring; in 3.5 we have sketched sketched aa is easy, if the integers in T satisfy the of aa ring; in 3.5 we have possible definition definition of such an an a". This This is is more more technical technical for alone, so so we the possible of such for Q Q alone, we refer refer the Then, in in order order to to prove prove c~"(n), prove reader to to Nelson Nelson [1986]. reader [1986]. Then, prove c~"(t) inductively inductively for for n) such such subterms subterms and and they they have have length length O(log the subterms subterms of the of n. There There are are O(log O(log n) O(log n) n).. 0 Finally use the fact that TT b a"(x) ~-+ a(x). Finally use the fact that D all. n. all(n), all(t) f- all(x) a(x). 6.3.4. Lemma 6.3.4. Proof-sketch P r o o f - s k e t c h of of L e m m a 6.1.5. 6.1.5. We We shall shall use use aa similar similar idea idea in in the the proof proof that we still owe to the reader. that we still owe to the reader. First length of �o . An First we we consider consider the the length of proof proof of of ~(n) in in IIA0. An easy easy model-theoretical model-theoretical argument argument shows shows that that the the assumption assumption of of the the lemma lemma implies implies f- Vx(3y I�o zzx0 ~ w ( 3 y = = 2% 2~ -+ -~ ~(~)) 1jJ(x) inin I�o IA0 such such that that I�o Vx(1jJ(x) -~ 3syy = Ia0 f-e w(r for for some some kk E E IN. IN. Take Take aa cut cut r -+ Then Then we we have have = 2%). 2~). -+ f- Vx(1jJ(x) ~ lemma lemma above. above. �o has To To get get the the theorem theorem for for Q Q,, use use the the well-known well-known fact fact that that IIA0 has an an interpretation interpretation 0n in in Q Q (this (this is is an an unpublished unpublished result result of of Wilkie, Wilkie, for for aa proof proof see see Nelson Nelson [1986]). [1986]). 6.4. The 6.4. Some S o m e applications a p p l i c a t i o n s of of the t h e bounds. bounds. The lower lower bound bound can can be be used used to to show show some some strengthenings strengthenings of of the the second second G6del Ghdel Incompleteness Incompleteness Theorem. Theorem. While While the the original original theorem theorem says says only only that that T T is is consistent consistent with with its its formal formal inconsistency inconsistency (cf. (cf. Chapter Chapter II) II),, we we shall shall show show that that T T is is consistent consistent with with aa statement statement saying saying that that there there is is aa short short proof proof of of contradiction. contradiction. We We have have two two such such results. results. 584 584 P. P. Pudltik Pudldk 6.4.1. and axiomatized 6.4.1. Corollary. Corollary. (Pudlak (Pudls [1985]) [1985]) Let Let T T ;2 D Q q be be consistent consistent and axiomatized by by an an NP set of hold: A l P set of axioms. axioms. Then Then the the following following hold: (1) if if I I is is aa cut cut in in T, T , then then (1) T T + + 3x(I(x)&..., 3 x ( I ( x ) & - ~ CConT(x)) onT(x)) is is consistent, consistent, (2) (2) if if ~(3(x) ( x ) is is aa bounded bounded arithmetical arithmetical formula fo r m u la such such that that (3(n.) ~ ( n ) is is true true for f o r every every nn EE IN, IN, then that then there there exists exists k k E E IN IN such such that k T T + + 3x((3(x)&..., 3 x ( ~ ( x ) & ~ CCon o n ( x(x ~ ) ))). . k is xx. . xx . . .. ... . .. x). is consistent, (x k is x). is consistent, (x -------- kk --tti im m eess Proof. P r o o f . (1) (1) Suppose Suppose that that the the statement statement is is false. false. Then Then -~r T T f~- I(x) I(x) -t --+ ConT(x). ConT(x). But, IIT = log n) 2 ) , whence But, by by Lemma Lemma 6.3.3, 6.3.3, we we have have that that III(n.) IlI(n)llT = O(( O((logn)2), whence also also 2 contradiction with II ConT(n.) liT = n) 2)) which n°, cc > II ConT(n)lIT = O((log O((log n) which is is aa contradiction with [I ConT(n)IIII � --> n~, > O. 0. II ConT(n.) (2) We We need need the the following following lemma. lemma. (2) 6.4.2. 6.4.2. Lemma. Lemma. ~(~) = = ~ (n), ~ + n_ m + m ·. n. ~ = = m m ·. ~n s(n.) s(n), m+ n. == m + ~n,, m have have proofs proofs of of size size polynomial polynomial in in log log n n and and log log m m.. The The proof proof is is easy, easy, if if we we assume assume ring ring operations, operations, otherwise otherwise we we have have to to work work iinn aa suitable cut. suitable cut. [] 0 One arithmetical terms. One consequence consequence is is that that the the same same holds holds for for arbitrary arbitrary arithmetical terms. This This can be used that, for bounded formula (3(x) in Q, there can be used to to show show that, for a a bounded formula/?(x) in the the language language of of Q, there exists exists aa polynomial polynomial PI Pl such such that that II (3 (n.) l i T �~ PI IIZ~(~)IIT Pl (n), (n), (41) (41) whenever is true. whenever (3(n.) ~(_n)is true. The The second second consequence consequence is is that that there there is is aa polynomial polynomial P p22 such such that that k = n~lr~ n k l i T ::; (42) lIn. tin~ = < Pp~(k, log n). ~). (42) 2 (k, log (Hint: -I = (Hint: prove prove n. n .. n. n = = n n 22,, n. n .. n n 22 = = n3, n3, . . . ,, n. n .. n n kk-1 = n n kk.) .) We (3(n.) is We continue continue with with the the proof proof of of (2). (2). Assume Assume that that/?(n) is true true for for every every n n E E IN. IN. Then Then we we have have (41) (41) for for all all n n.. Clearly Clearly • • • II C ~ I1~ _< O(llZ(~)I1~ + IIW(Z(~) ~ C o n T ( x k) lit + I1~k = nkllr), The Lengths Lengths of Proofs 585 585 ]1Con ( ')ll 00 be be such such that that for for every every rr Jl C~ (44) (44) >-- r~. Let Let dd be be the the degree degree of of n n in in P3(n, p3(n, k, k, m m).) . Take Take kk so so that that kc ke > d. d. Now Now suppose suppose (2) (2) fails fails for k, thus for k, thus k Vx(~(x) -~ Con COnT(Xk)). TT f-~ Vx T (x )) . ( {3 (x) -+ Let Let m be be the the length length of of this this proof. proof. Take Take n n so so large large that that p3(n, k, m) < nkke.. P3(n, << Then, by Then, by (43) (43),, k Con(nk)lJT <--P3(n, p3(n, k, m)) < nn kk6,, k, m IIJl Con (n ) liT � which is is aa contradiction with (44) (44).. which contradiction with << D 6.5. Let bound on II Con 6.5. Let us us observe observe that that also also the the upper upper bound on II COnT(n)lJT can be be used used to to T (nJ I l T can obtain obtain interesting interesting corollaries. corollaries. This This is is because because the the upper upper and and the the lower lower bounds bounds are are quite the calculus C-rule, hence quite close, close, especially especially in in the the case case of of the calculus with with the the C-rule, hence the the results results that upper bound that we we used used in in the the proof proof of of the the upper bound cannot cannot be be substantially substantially improved. improved. There There are are two two such such results. results. One One is is Theorem Theorem 3.2.3, 3.2.3, where where more more precise precise calculations calculations O( y'n) on give bound O(n2). give aa bound O(n2). The The second second one one is is the the bound bound O(v/-~) on the the depth depth of of the the shortest shortest proof to our proof of of aa fixed fixed size size formula. formula. Due Due to our bounds, bounds, the the first first result result cannot cannot be be proved proved The first second for log n) for which is for aa function function which is oo(n2/(log n)2), while while the the second for oo(v/-~ n).. ((The first ( y'n// log ( n2 / ( log n)2), statement is order logic logic without without the statement is true true also also for for first first order the C-rule, C-rule, see see Pudllik Pudls [1987].) [1987].) However However we we feel feel that that it it should should be be possible possible to to find find direct direct arguments arguments showing showing even even the the sharp bounds ~(n 2) and and r2(y'n) ~ ( v ~ ) .. sharp bounds r2(n2) Further applications of bounds will Further applications of the the lower lower bounds will be be shown shown in in the the next next section. section. 7. Speed-up 7. S p e e d - u p ttheorems h e o r e m s in i n first f i r s t order o r d e r logic logic proof The The speed-up speed-up phenomenon phenomenon is is aa situation, situation, where where we we have have two two systems systems ((proof systems, systems, theories theories)) such such that that some some theorems theorems have have much much shorter shorter proofs proofs in in one one of of them. them. After After the the problem problem of of proving proving lower lower bounds bounds on on proofs proofs of of concrete concrete statements, statements, this this is is the the second second most most interesting interesting problem. problem. Note Note that that in in the the intuitive intuitive relation relation between between complexity complexity of of computations computations and and complexity complexity of of proofs, proofs, speed-up speed-up theorems theorems should should correspond correspond to to separations separations of of complexity complexity classes. classes. We 5, where We have have already already encountered encountered aa speed-up speed-up theorem theorem in in section section 5, where we we showed showed that can be be much with cuts that cut-free cut-free proofs proofs can much longer longer than than the the proofs proofs with cuts or or proofs proofs in in aa Hilbert shall consider consider such Hilbert style style calculus. calculus. We We shall such questions questions about about propositional propositional logic logic in in sections sections 88 and and 9. 9. In In this this section section we we shall shall talk talk about about two two most most important important speed-up speed-up P. Pud16k Pudldk 586 phenomena speed-up caused phenomena in in first first order order logic. logic. The The one one is is the the speed-up caused by by having having a a stronger stronger theory. theory. The The second second one one appears appears when when we we prove prove Pr PrT(~) in T T instead instead of of
IIOIIT f(llllT+), for .for every every sentence sentence
Proof. a f-~- derive a, To prove axioms T ) . Thus prove a c~ V V
Let Let T T be be aa recursively recursively axiomatized axiomatized theory theory containing containing Robinson Robinson ArithmeticQ ArithmeticQ.. Then Then any any proper proper extension extension of of T T has has arbitrary arbitrary recursive recursive speed-up speed-up over over T T.. 7.1.2. 7.1.2. Corollary. Corollary. Proof. that Q P r o o f . This This follows follows from from the the fact fact that Q is is essentially essentially undecidable undecidable.. o E3 The concerns the steps. The following following result result of of Statman, Statman, improved improved by by Buss, Buss, concerns the number number of of steps. We without aa proof. We state state it it without proof. The Lengths Lengths of Proofs The of Proofs 587 587 7.1.3. heorem. (Statman 7.1.3. TTheorem. (Statman [1981], [1981], Buss Buss [1994]) [1994]) Let Let TT be be aa theory theory axiomatized axiomatized by by sentence undecided undecided by by TT and and such such finite number number of of axiom axiom schemas. schemas. Suppose Suppose ~a isis aa sentence aa finite is consistent. consistent. Then Then TT ++ ~a has has an an infinite infinite speed-up speed-up with with respect respect to to the the that TT ++ --,a that ~ is number of of steps steps over over T, T, i.e., i.e., there there exists exists an an infinite infinite set set 9 of of sentences sentences and and aa kk such such number that that k V~ E (~ T J- Ol ~'steps (fl, but there there isis no no mm such such that that but TT ~'f-s�eps Vcp EE (I) cp, V~ tmeps~, o We shall We shall give give some some intuition intuition about about this this theorem theorem by by an an example example due due to to Baaz. Baaz. Suppose Kreisel's Kreisel ' s Conjecture Conjecture holds holds for for T. T. Let Let ~(x) cp(x) be be aa formula formula such that Suppose such that 1 . TVVxcp(x); T If Vxcp(x) ; 1. 2. Vn T 2. Vn T f-F ~(n). CP (ll } A typical typical example example of of such such aa formula for some A formula is is ConT(x). ConT(x) . Then, Then, clearly, clearly, for some constant constant cc T T + + Vxcp(x) Vx~(x) fFs~teps ~(n), �teps cp(n), for every every n, n, since the term term n n into ~(x).. If, for since we we only only need need to to substitute substitute the into cp(x) If, however, however, for for some m some m T f-�eps cp(n) T ~-~t~ep~ ~(n),, Conjecture. Thus for every we would get TT Ff- Vx~(x) Vxcp(x) by by Kreisel's Kreisel ' s Conjecture. Vxcp(x) has for every n n,, we would get Thus T T + + Vx~(x) has T. infinite speed-up over infinite speed-up over T. 7.2. 7.2. Next Next we we shall shall show show that that the the lower lower bound bound on on the the length length of of proof proof of of Con COnT(n) in T (n) in T (Theorem (Theorem 6.2.3) 6.2.3) can can be be used used to to obtain obtain speed-up speed-up when when T T is is extended extended to to T T+ + Con ConT T T or or when when Pr(cp) Pr((p) is is used used instead instead of of cpo ~. Let Let T T be be a a sufficiently sufficiently strong strong fragment fragment of of arithmetic. arithmetic. We We want want to to use use sentences sentences Con ConT(f(n)) for fast-growing fast-growing functions functions f f. . Such Such functions functions needn needn't' t be be representable representable T U (n)) for by T. So by terms terms in in T. So we we take take the the sentence sentence 3y 3y (cp(n, (~(n, y) y) V V ConT(cp)) ConT(~)) (45) (45) instead, f. Then instead, where where cp ~ defines defines the the graph graph of of f. Then we we need need two two conditions conditions to to be be satisfied satisfied T, i.e., ); 1. ff isis aa provably 1. provably total total function function in in T, i.e., T T fF Vx3!ycp(x, Vx3!y~(x, yy); polynomially numerates 2.2. cp~ polynomially numerates the the graph graph of of f f.. Let Let us us make make aa simple simple observation. observation. For For every every recursive recursive function function f f ,, there there exists exists aa recursive recursive function function 9g such such that that 1. Vn E IN f(n) :5 g(n) , 2. 2. the the graph graph of of gg is is aa polynomial polynomial time time computable computable relation. relation. 7.2.1. 7.2.1. Lemma. Lemma. 588 588 P. Pudldk Pudldk P. Proof. Proof. Let Let M M be be aa Turing Turing machine machine for for ff.. Then Then one one can can construct construct aa Turing Turing machine machine M' M ~which which on on input input nn prints prints 22mm in in binary, binary, where where m m is is the the number number of of steps steps of of M M on on We take take gg to to be be the the function function computed computed by by M' M ~.. 0[:1 nn.. We If If T T is is sufficiently sufficiently strong, strong, Lemma Lemma 7.2.1 7.2.1 can can be be formalized formalized in in T. T. Thus Thus polynomial polynomial numerability is is not not an an essential essential restriction. restriction. We We shall shall abbreviate abbreviate (45) (45) by by Con ConT(f(n)). numerability T (J (:a) ) . Let Let T T be be aa sufficiently su]flciently strong strong theory. theory. Let Let ff be be aa provably provably total total increasing recursive recursive function function in in T T whose whose graph graph has has aa polynomial polynomial numeration numeration in in T. T. increasing Then there there exists exists aa 6r > > 00 such such that that Then 7.2.2. 7.2.2. Theorem. Theorem. II COnT(f(~))llT >_ f(~)~, while 1.1. IIIConT(J( COnT(Y(n))llr+con~ = O(logn) O(logn);; :a)) IIT+conT = 2. fConT(J(:a))l ) li T = 2. IIIIPrT( PrT([ConT(Y(n__))q)llT = O(log O(logn). n) . Thus Thus in in both both cases cases we we get get aa speed-up speed-up by by any any provably provably total total recursive recursive function function of of T. T. Proof. (x, y) Proof. Lower Lower bound. bound. Let Let rp qo(x, y) be be the the formula formula which which polynomially polynomially numerates numerates f (x) = = y. Thus we we want want to to bound bound f(x) y. Thus 3y( rp (:a, Yy)/x ) 1\ ConT(y)) liT. 11113y(~(~, COnT(y))[IT. Let m -= f(n). f(n) . Clearly Clearly Let m 3! yrp (:a, y) m) --+ 3lye(n, y)A1\ qo(n, --+ ConT(m). ConT(m). rp (:a, m) Thus Thus (46) (46) ConT (m) lIT liT :::; []II ConT(m) ConT ( Y)) liT + y) 1\ ConT(Y))llT m) liT ++ K, 113y(rp (:a, y)A + 113!Y~(n, y) liT ++ Ilk(n, K, Ilrp(:a, m)/IT 11 3 ! yrp(:a, Y)IIT 113y(~(n, where K K isis the the length length of of the the proof proof of of (46). (46) . The The proof proof of of (46) (46) depends depends only only linearly linearly where on on the the lengths lengths of of nn and and m, m, thus thus K K == O(log O(log m). m) . Similarly Similarly 11 3 ! yrp(:a, y)lIT O(log n), n), Y) IIT == O(log 113!y~(n, on since f- Vx3!yqo(x,y). 'v'x3! yrp (x, V) . Finally Finally we we have have aa bound bound (log (log m)~ m) O( l) on since we we assume assume TT Fby polynomial polynomial numerability. numerability. Thus, Thus, using using Theorem Theorem 6.2.3 6.2.3 we we have have m) IIT by II~(n, IIrp (:a, m)liT (log m)~O( l ) , y) 1\ COnT(y))IIT m" -< ConT (m) liT _< 113y(rp (:a, Y)A me ConT(y)) liT ++ (logm) :::; 113Y(qo(n, :::; IIII COnT(m)llT which which gives gives the the lower lower bound. bound. Upper bound bound (1). (1). Recall Recall that that ConT ConT denotes denotes Vx ConT (x) and and that that we we assume assume TT ~f Upper 'v'x ConT(x) Vx3!yqo(x, 'v'x3! yrp (x, y). y) . Again, Again, the the proof proof of of y) A1\ ConT(y)) ConT(y)) --+ 3y(~(n, 3y(rp(:a, y) y) --+ Vx 'v'x3!yrp(x, y) ConT (x) A1\ VxS!y~(x, 'v'xConT(x) The Lengths of Proofs 589 589 depends of n, depends only only linearly linearly on on the the length length of n, thus thus 113 ( , ( ) Con ( y
(47) ( f3y ( l 7.2.4. 7.2.4. Theorem. Theorem. we have have we (1) (1) (2) (2) (Hajek, �o + (H~jek, Montagna Montagna and and Pudlak Pudl~k [1993]) [1993]) Let Let T T = - IIAo +n ft.. Then Then nO(l) ; II PrT 2!!.)l ) li T = p r ~ ((rf3 3 yy( (yy = = 22~)1)11~ = ~o(~); 113 ( = 2 2!!. ) liT = y y = 2~o)11~ n) . 113y(y = n(2 rt(2~). o [] 7.3. It 7.3. Speed-up Speed-up of of GB GB over over ZF ZF.. It is is well-known well-known that that GB GB proves proves the the same same set set of of formulas formulas as as ZF ZF.. (We (We consider consider ZF ZF with with the the axiom axiom of of choice.) choice.) Therefore Therefore it it is is very very interesting interesting to to find find out out if if the the set set formulas formulas have have proofs proofs of of approximately approximately the the same both theories. no; in same length length in in both theories. The The answer answer is is no; in fact fact there there is is aa nonelementary nonelementary speed-up speed-up as as for for cut-elimination. cut-elimination. This This seems seems to to be be typical typical for for results results obtained obtained from from cut-elimination where aa direct not known) cut-elimination or or Herbrand Herbrand theorem theorem (and (and where direct proof proof is is not known).. This This result result is is based based on on the the following following lemma lemma due due to to Solovay Solovay (unpublished); (unpublished); aa similar construction construction was considered by similar was considered by Vopi'mka Vop6nka (unpublished). (unpublished). 590 590 7.3.1. Lemma. Lemma. 7.3.1. P. P. Pudl6.k Pudldk There I(x) in There is is aa cut cut I(x) in GB G B such such that that GB 'v'x(I(x) -+ G B f~ Vx(I(x) --+ ConZF(x)) COnzF(x)).. It is is outside outside of of the the scope scope of of this this chapter chapter to to give give aa proof proof of of this this lemma. lemma. Let Let us us only only It very idea. One very briefly briefly describe describe the the main main idea. One can can construct construct aa sort sort of of inner inner model model of of ZF ZF in in G B where where the the universe universe of of sets sets is is some some cut. cut. This This model model is is constructed constructed along along with with aa GB satisfaction relation for relation is satisfaction relation for it. it. Since Since the the satisfaction satisfaction relation is defined defined by by aa formula formula with with class quantifiers, quantifiers, we we cannot cannot use use induction induction to to show show the the consistency consistency of of ZF ZF.. Instead Instead class we we only only show show that that the the segment segment of of numbers numbers x x such such that that there there is is no no contradiction contradiction of of 0 length � _< x x is is closed closed under under successor. successor. But But this this is is exactly exactly what what we we need. need. [] length 7.3.2. Pudhik [1986]) 7.3.2. Theorem. T h e o r e m . ((Pudl~k [1986]) O( l) ; (2n.) 11 GB = (1) (1) IIII ConzF COnZF(2n_)IiGB = n nO0); constant ce > (2) (2n.) ll zF = (2) ]lII ConzF COnZF(2~)IIZF = (2 (2,,) ~, for for sorne some constant > o. O. n ) <, P r o o f . To To prove prove (1) (1) we we need, need, by by Lemma Lemma 7.3.1, 7.3.1, only only ttoo have have aa short short proof proof of of I(2 I(2a) Proof. n.) in in G G BB.. The The bound bound I(2n ) IIGB = = n nO ~ (l) II]]I(2.)]]GB 0 follows 3.4.1. The second part follows from from Theorem Theorem 3.4.1. The second part is is contained contained in in Theorem Theorem 7.2.2. 7.2.2. [] 2 A bound IIII ConzF(2J II GB = A more more precise precise computation computation gives gives a a bound COnZF(2a)IIGB = O(n O(n2),) , which which vn . An upper implies implies a a lower lower bound bound on on the the speed-up speed-up of of GB G B over over ZF ZF of of the the form form 2n 2n(~). An upper ( ) bound Vn) complementing the above result was proved by Solovay [1990]. Let bound 2o 2o(v~) complementing the above result was proved by Solovay [1990]. Let ( us bounds can us observe observe that that such such bounds can be be used used to to show show that that the the estimate estimate on on the the depth depth of of formulas proof, Theorem Theorem 4.2.5, formulas in in a a proof, 4.2.5, is is asymptotically asymptotically optimal. optimal. 8. Propositional 8. Propositional pproof r o o f systems systems In In this this section section we we consider consider some some concrete concrete propositional propositional proof proof systems. systems. There There are are several reasons these systems. they are systems which several reasons for for studying studying these systems. Firstly Firstly they are natural natural systems which are are used good approximations used for for formalization formalization of of the the concept concept of of aa proof, proof, in in fact, fact, they they are are good approximations of especially resolution, of human human reasoning. reasoning. Some Some systems, systems, especially resolution, are are also also used used in in automated automated theorem proving. proving. Therefore theorem Therefore it it is is important important to to know know how how efficient efficient they they are. are. Secondly Secondly they they are are suitable suitable benchmarks benchmarks for for testing testing our our lower lower bound bound techniques. techniques. Presently Presently we we are are able able to to prove prove superpolynomial superpolynomial lower lower bounds bounds only only for for the the weakest weakest systems; systems; we we shall shall give give an an example example of of a a lower lower bound bound in in section section 9. 9. Thirdly Thirdly there there are are important important connections connections between between provability provability in in some some important important theories theories of of bounded bounded arithmetic arithmetic and and the the lengths lengths of of proofs proofs in in these these propositional propositional proof proof systems. systems. This This will will be be the the topic topic of of section section 10. 10. 8.1. Frege 8.1. Frege Frege systems systems and and its its extensions. extensions. Frege systems systems are are the the most most natural natural calculi calculi for for propositional propositional logic logic and and they they are are also also used used to to axiomatize axiomatize the the propositional propositional part Hilbert style part of of the the first first order order logic logic in in the the Hilbert style formalizations. formalizations. We We have have used used aa particular special case particular special case of of a a Frege Frege system system for for presenting presenting results results on on the the lengths lengths of of proofs order logic. proofs in in first first order logic. of Proofs The Lengths of 8.1.1. 8.1.1. 591 591 To define define aa general general Frege Frege system, system, we we need need the the concept concept of of aa Frege Frege rule rule.9 A A To such that that the the , Pn)}, ~CP(Pb Pn)) , such Frege rule rule isis aa pair pair ({~I(Pl,... ({ CPl (Pb ' " ,, PPn), CPk(Pl, . . . ,Pn)}, Frege ~ ) , ." . . ", ~k(Pl,... ( P l , .. ... ., , Pn)), implication qOlA...A~Pk -+ ~cP is is aa tautology. tautology. We We use use PPb Pn to to denote denote propositional propositional implication l , .. ... ., , Pn CPl l\ . . . I\CPk --+ variables. Usually Usually we we write write the the rule rule as as variables. CPb ' . . , ~k CPk (~I, '' 9 , cP When using using the the rule, rule, we we use use actually actually its its instances instances which which are are obtained obtained by by substituting substituting When Pbl , .' .". , , P~. Pn' A A Frege Frege rule rule can can have zero assumpassump arbitrary formulas formulas for for the the variables arbitrary variables P have zero axiom schema. schema. AA Frege Frege proof proof isis aa sequence sequence of of formulas formulas tions, in in which which case case itit is is an an axiom tions, such that that each each formula formula follows follows from previous ones ones by by an an application application of of aa Frege Frege rule rule such from previous from aa given set. from given set. 8.1.2. 8.1.2. Definition. Definition. A A Frege Frege system system FF isis determined determined by by aa finite finite complete complete set set of of connectives B connectives B and and aa finite finite set set of of Frege Frege rules. rules. We We require require that that FF be be implicationally implicationally complete for the of formulas formulas in in the the basis complete for the set set of basis B. B. Recall Recall that implicationally complete complete means means that that whenever whenever an an implication implication 'l/Jl r 1\ A that implicationally -+ r'l/J is is aa tautology, from 'l/Jb r 1 6 .2. . , 'l/Jk . Rules Rules such . . 1\ r'l/Jk -+ tautology, then then r'l/J is is derivable derivable from such as as ..../~ modus ponens ponens and and cut cut ensure ensure that that the the system system is is implicationally implicationally complete complete whenever modus whenever it it is is complete. complete. An example example of of aa Frege is the the propositional the proof proof system system An Frege system system is propositional part part of of the considered in in section section 2; 2; itit has has 14 14 axiom and one one rule rule with with two two assumptions assumptions considered axiom schemas schemas and (modus ponens) (modus ponens).. Note 8.1.3. 8.1.3. Note that that in in an an application application of of a a Frege Frege rule rule (in (in particular particular also also in in axioms) axioms) we rule, however we substitute substitute arbitrary arbitrary formulas formulas for for the the variables variables in in the the rule, however we we are are not not allowed allowed to to substitute substitute in in an an arbitrary arbitrary derived derived formula. formula. It It is is natural natural to to add add such such a a rule. rule. The The rule rule is is called called the the substitution substitution rule rule and and allows allows to to derive derive from from CP(Pb T ( P l , .. ... ., P, Pk k )), , with with 1 propositional propositional variables variables PI p l ,, .. ... ., , Pk pk,, any any formula formula of of the the form form cp ~ (('l/Jb r . . . , 'l/Jk) Ck).. 1 8.1.4. A 8.1.4. Definition. Definition. A substitution substitution Frege Frege system system SF SF is is a a Frege Frege system system augmented augmented with substitution rule. rule. with the the substitution 8.1.5. 8.1.5. The The extension extension rule rule is is the the rule rule which which allows allows to to introduce introduce the the formula formula Pp -==~ pcP, , where where p p is is aa propositional propositional variable variable and and cP ~ is is any any formula formula and and the the following following conditions conditions hold: hold: 1. when introducing introducing P p == - cP, ~, P p must must not not occur occur in in the the preceding preceding part part of of the the proof proof 1. when or in cP orin ~;; 2. 2. such such aa P p must must not not be be present present in in the the proved proved formula. formula. 11in In fact fact Frege Frege used used this this rule rule originally originally and and the the idea idea of of axiom axiom schemas schemas was was introduced introduced by by von von Neumann Neumann later. later. 592 592 P. Pudl6.k Pudldk If If == - is is not not in in the the basis, basis, we we can can use use an an equivalent equivalent formula formula instead, instead, e.g. e.g. (p (iv -7 --+ cp) ~p) 1\ A (~p -7 --+ p) p).. This This rule rule is is not not used used to to derive derive aa new new tautology tautology quickly, quickly, as as it it is is the the case case of of (cp Frege purpose is long formulas. Frege and and substitution substitution rules, rules, but but its its purpose is to to abbreviate abbreviate long formulas. 8.1.6. An 8.1.6. Definition. Definition. An extension extension Frege Frege system system EF E F is is aa Frege Frege system system augmented augmented with with the the extension extension rule. rule. 22 8.1.7. shall address 8.1.7. The The first first question question that that we we shall address is: is: how how does does the the lengths lengths of of proofs proofs depend on aa particular of the the basis and the depend on particular choice choice of basis of of connectives connectives and the Frege Frege rules. rules. If If the the , it is fairly easy to prove that they basis basis is is the the same same for for two two Frege Frege systems systems Fl F1 and and F F2, it is fairly easy to prove that they 2 are are polynomially polynomially equivalent. equivalent. We We have have used used this this argument argument already already in in the the previous previous sections. Let sections. Let for for instance instance qOl,..., ~Pk cP F2 is implication ally complete, complete, there be be aa rule rule in in Fl F1.. Since Since F2 is implicationally there exists exists a a proof proof 7r r of of cp q0 , . . . , CPk from CPI ~Pl,..., ~k in in F F2. To simulate simulate an an instance instance of of this this rule rule obtained obtained by by substituting substituting from 2 • To some it, we some formulas formulas into into it, we simply simply substitute substitute the the same same formulas formulas in in r7r.. Thus Thus we we get get only only linear linear increase increase of of the the size. size. If the the two two bases bases are are different, different, the the proof proof is is not not so so easy, easy, but but the the basic basic idea idea is is simple. simple. If One One uses uses aa well-known well-known fact fact from from boolean boolean complexity complexity theory theory that that aa formula formula in in one one complete complete basis basis can can be be transformed transformed into into an an equivalent equivalent formula formula in in another another complete complete basis basis with with at at most most polynomial polynomial increase increase in in size, size, in in fact, fact, using using aa polynomial polynomial algorithm. algorithm. This, This, of of course, course, does does not not produce produce a a proof proof from from aa proof, proof, but but one one can can show show that that it it suffices suffices to to add add pieces pieces of of proofs proofs of of at at most most polynomial polynomial size size between between the the formulas formulas to to get get one. one. Details Details are are tedious, tedious, so so we we leave leave them them out. out. The The same same holds holds for for substitution substitution Frege Frege and and extension extension Frege Frege systems. systems. Thus Thus we we have: have: (Cook (Cook and and Reckhow Reckhow [1979), [1979], Reckhow Reckhow [1976)) [1976]) Every Every two two Frege Frege systems systems are are polynomially polynomially equivalent, equivalent, every every two two substitution substitution Frege Frege systems systems are are polynomially polynomially equivalent, equivalent, and and every every two two extension extension Frege Frege systems systems are are polynomially polynomially 0 equivalent. equivalent. D 8.1.8. Theorem. 8.1.8. T heorem. 8.1.9. still leaves classes, moreover 8.1.9. This This still leaves three three classes, moreover each each can can be be considered considered also also in in the the tree size, which tree form form and and we we can can count count the the number number of of steps steps instead instead of of the the size, which gives gives altogether these cases reduce to only three, altogether twelve twelve possibilities. possibilities. We We shall shall show show that that these cases reduce to only three, if if we we identify identify polynomially polynomially equivalent equivalent ones ones (namely, (namely, Frege, Frege, extension extension Frege Frege and and the the number number of of steps steps in in substitution substitution Frege). Frege). The The question question about about aa speed speed up up of of sequence sequence versus versus tree tree proofs proofs has has been been solved solved modus ponens. in Theorem 4.1 in Theorem 4.1 for for Frege Frege systems systems which which contain contain modus ponens. The The same same holds holds for for extensions extensions of of such such Frege Frege systems systems by by extension extension and and substitution substitution rules. rules. We We shall shall return return to to this this question question below. below. Now Now we we shall shall consider consider the the remaining remaining ones.' ones.' Let Let us us first first consider consider the the relation relation of of substitution substitution Frege Frege systems systems and and extension extension Frege Frege systems. systems. 2Here 2Here II deviate deviate slightly slightly from from the the literature literature where where the the name name extended extended Frege Frege system system is is used, used, which, I think, is rather ambiguous. ambiguous. 593 593 The Lengths Lengths of of Proofs Proofs The (Dowd [1985], [1985], Krajicek Kraji~ek and and Pudlcik Pudls [1989]) [1989]) Every Every substitution substitution (Dowd Frege system system is is polynomially polynomially equivalent equivalent to to every every extension extension Frege Frege system. system. Frege 8.1.10. Theorem. Theorem. 8.1.10. P r o o f . By By the the theorem theorem above, above, we we can can assume assume that that both both systems systems have have the the same same Proof. language. language. First we we show show aa polynomial simulation of of an an extension extension Frege Frege system system by by aa 11.. First polynomial simulation substitution Frege Frege system. system. Let Let an an extension extension Frege Frege proof proof of of aa tautology tautology 'l/J r be be given, given, substitution let let PI Pl == = 'P qol,...,pm - 'P ~om be all all formulas formulas introduced introduced by by the the extension extension rule rule listed listed in in m be I , . . . , Pm == the the order order in in which which they they were were introduced. introduced. By By Theorem Theorem 8.1.8 8.1.8 we we can can assume assume w.l.o.g. w.l.o.g. that our our systems systems contain contain suitable suitable connectives connectives and and suitable suitable Frege Frege rules. rules. Using Using an an that effective version version of of the the deduction deduction theorem theorem (whose (whose easy easy proof proof we we leave leave to to the the reader) reader) effective we get, get, by by aa polynomial transformation, aa proof proof of of we polynomial transformation, pl == - 'PI ~ol 1\ A .. ... . 1\ Apm = 'P ~om ~ 'l/J r m -+ PI Pm == (48) (48) which does does not not use use the the extension extension rule rule (i.e., (i.e., a a Frege Frege proof) proof).. Now Now apply apply the the substitution substitution which rule to to (48) (48) with with the the substitution substitution Pm r-+ ~ 'P ~m. Thus we we get get rule m . Thus Pl PI == : 'P ~ 1I 1\ A .. . .. . 1\ A Pm P r o -1l == -- 'P 1\ ~ mm-l -1 A 'Pm ~ == :-- 'Pm qOm -+ --~ 'l/J. ~D. (49) (49) From (49) (49) we we get get by by a a polynomial polynomial size size Frege Frege proof proof From PI Pl == -- 'PI qO1 1\ A .. .. .. 1\ A PmPro-1l == -- 'Pm-l qPm - 1 -+ --~ 'l/J. ~) . We repeat repeat the the same same until until we we get get a a proof proof of of 'l/J. r We 2. The The polynomial polynomial simulation simulation of of substitution substitution Frege Frege systems systems by by extension extension Frege Prege 2. systems not so so simple. simple. The proof is is the have systems is is not The idea idea of of the the proof the following. following. Suppose Suppose we we have simulated substitution Frege ~oj which which is is derived derived by by a a substitution substitution from from simulated aa substitution Frege proof proof until until 'Pi Then we we can can derive derive qoj 'Pi by repeating the the previous previous part part of of the qoi,, say qoj = = qoi(P/(~). by repeating the say 'Pi 'Pi (p/a) . Then 'Pi replaced by by 5. a. However, However, repeating the new Frege proof proof with variables variables p t5 replaced repeating this, this, the new Frege proof proof with would grow grow exponentially. The trick to prove the formulas formulas of substitution would exponentially. The trick is is to prove the of the the substitution proof not not for for aa particular substitution, but for the the substitution, substitution, where the Frege particular substitution, but for where the Frege proof proof fails fails for for the first time. In reality the proof proof does not fail fail (we (we start with a a real proof the first time. In reality the does not start with real proof), but it then we we could we can proof), but it enables enables us us to to argue: argue: "if "if itit failed, failed, then could go go on, on, hence hence we can go on in in any which propositional go on any case". case" . Now Now the the problem problem is is for for which propositional variables variables should should the the proof fail. Therefore Therefore we an extra extra set each formula formula of the proof fail. we introduce introduce an set of of variables variables for for each of the substitution proof. The at some step will will be defined using using variables variables substitution Frege Frege proof. The variables variables at some step be defined at at the the following following steps steps of of the the proof. proof. As As this this nesting nesting may may result result in in an an exponential exponential growth, growth, we we introduce introduce them them using using the the extension extension rule. rule. Now we we shall shall argue argue formally. formally. W.l.o.g. W.l.o.g. we can assume assume that that the the substitution substitution Frege Frege Now we can 'Pm ) system has has only only modus modus ponens ponens and and axiom axiom schemas schemas as as Frege Frege rules. rules. Let Let (qol,..., ('PI , . . . , qPm) system be aa substitution substitution Frege proof. Let Let i5 p == ((PI be all all propositional propositional variables variables of of be Frege proof. P l,, .. .. .., , Pn) Pn) be the proof. Take sequences iii of of length length nn consisting consisting of of new new distinct distinct variables, for the proof. Take sequences qi variables, for be (Pi(P/qi), 'Pi (P!iJi ) , for mj thus thus ii -= 11,, . ... ., , m for ii -= 11,, . ... ., , m; m-- 1, 1 , and and denote denote by by q-m q-;" -= P. p. Let Let r'l/Ji be · Let 13i be a sequence of n formulas defined as follows: Cm ~m. Let/?j be a sequence of n formulas defined as follows: = 'P m 'l/Jm - 594 594 P. PudlrJ.k Pudldk P. qoj isis an an axiom axiom or or isis derived derived by by modus modus ponens ponens then then fJj ~j = = tli ~;; 1.1. ifif CPj q0j isis derived derived by by substitution substitution from from CPi ~i,, namely namely CPj qoj = = CP~i(/~/5), then fJj ~j 22.. ifif CPj i(p/a) , then a(p!tli). = The extension extension Frege Frege proof proof will will start start by by introducing introducing The q == ,t- (Wi /\ A -,'l/Ji 1 /\ qi,1 + ^ f3i+ l ,I) Vv ... .. . Vv (Wm-I /\A -''l/Jm /\A f3m,I) , where W ~jj is is 'l/J r I /\A .. .... /\A 'l/J Cj. We have have to to introduce introduce these these formulas formulas in in the the order order where j . We ii -=m -m - 1,1 ,.. .. .. ,,11.. Then we we add add polynomial polynomial size size proofs proofs of of Then ~j-1 A ~r ~ r ~ el(all'j), (50) (50) for ii < < jj.. To To prove prove (50) (50) we we first first derive derive for - 1 /\ A -, ~Dj --+ qi,1 qi,t == - f3 ~j,t j ,l 'l/Jj -+ WI'T~jj -I from the the axioms axioms introducing introducing qi,1 qi,t and and then then successively successively construct construct proofs proofs of of the the from corresponding statements statements for for subformulas subformulas of of 'l/J r i. corresponding Now Now we we derive derive 'l/J r l , . . . , 'l/Jm era.. Suppose Suppose we we have have proved proved 'l/Jl, ~31, .. .. .. ,, 'l/J C jj- l-" I . Consider Consider three three cases. cases. qoj is is an an axiom. axiom. Then Then 'l/J Cjj is is also also an an axiom. axiom. 11.. CPj 2. CPj modus ponens. 2. qoj was was derived derived from from CPu, ~ , CPv ~,,, u, v < < jj,, CPu ~p~ = - CPv ~, -+ --+ CPj ~j by by modus ponens. First First derive W ~j-1. Then, using using (50) (50) with with ii = = u, v, we we get get derive j - I . Then, u, v -,'l/Jj -+ u, v, 'l/Ju (/Jj ) /\ 'l/Jv (/Jj ) . Since Since = r we get we get Cj(Zj), -''l/Jj -+ 'l/Jj (/Jj ) . As we we are are considering considering the the case case of of modus modus ponens, ponens, /~j /Jj == ~ili, , hence hence we we have have derived derived As -~r -,'l/Jj --+ -+ Cj, 'l/Jj , whence whence we we get get Cj 'l/Jj immediately. immediately. was derived derived from from q0i, 3. qoj CPi , ii << jj by by substitution. substitution. Then Then Cj 'l/Jj isis just just r'l/Ji (ij;j/Jj ) , thus 3. thus CPj was (50) (50) gives gives Wj -I /\ -,'l/Jj -+ 'l/Ji == 'l/Jj and 'l/Jj-I . and we we get get Cj 'l/Jj easily easily from from r'l/JI , . . . , Cj-1. Finally recall recall that that Cm 'l/Jm isis the the conclusion conclusion of of the the proof proof ~m, CPm , thus thus we we have have the the Finally simulation. simulation. D 0 The simulation simulation of of extension extension Frege Frege systems systems by by substitution substitution Frege Frege systems systems was was The [1979]. The The other other simulation simulation has has aa simple simple shown already already inin Cook Cook and and Reckhow Reckhow [1979]. shown Namely by by "higher order" order" proof proof based based on on aa relation relation to to bounded bounded arithmetic. arithmetic. Namely "higher Theorem 10.3.6 10.3.6 below, below, itit suffices suffices to to prove prove the the reflection reflection principle principle for for substitution substitution Theorem which isis easy. easy. This This was was observed observed independently independently by by Dowd Dowd [1985] [1985] Frege system system inin SSi Frege 1, , which and and Kraji~ek Krajicek and and Pudl~k Pudlak [1988]. [1988]. The Lengths of Proofs 595 595 8.1.11. It 8.1.11. It is is an an open open problem problem whether whether Frege Frege systems systems can can simulate simulate extension extension and and substitution substitution Frege Frege systems. systems. We We conjecture conjecture that that the the answer answer is is no. no. It It seems, seems, though though it is is not not supported supported by by any any mathematical mathematical result, result, that that the the relation relation of of Frege Frege systems systems to to it extension Frege Frege systems systems is is the the same same as as the the relation relation of of boolean formulas to to boolean boolean extension boolean formulas circuits circuits in in complexity complexity theory. theory. It It is is generally generally accepted accepted that that it it is is unlikely unlikely that that formulas formulas can polynomial increase can simulate simulate circuits circuits with with only only polynomial increase in in sizej size; the the uniform uniform version version of of Pj both this this conjecture conjecture is is NC Arc 11 i= =/=P; both conjectures conjectures are are also also open. open. 8.1.12. 8.1.12. We We shall shall now now consider consider the the number number of of steps steps in in these these systems. systems. It It turns turns out out that that the the number number of of steps steps in in Frege Frege and and extension extension Frege Frege systems systems is, is, up up to to aa polynomial, polynomial, the the size size of of extension extension Frege Frege proofs. proofs. If If cp ~ can can be be proved proved by by aa proof proof with with n n steps steps in in an an extension extension Frege Frege system, with nn steps system, then then it it can can be be proved proved by by aa proof proof with steps in in aa Frege Frege system; system; namely, namely, we we can can omit omit the the extension extension rule rule from from the the given given system. system. 8.1.13. Lemma. 8.1.13. L emma. Proof-sketch. the extension P r o o f - s k e t c h . Omit Omit every every instance instance of of the extension rule rule p p == - cp ~ and and at at the the same same 0 time time replace replace all all occurrences occurrences of of the the variable variable p p by by cpo ~. [] There There exists exists aa polynomial polynomial f(x) f (x) such such that that for for every every tautology tautology cp and and every every extension extension Frege Frege proof proof of of cp ~ with with n n steps, steps, there there exists exists an an extension extension Frege Frege size is ::; f(lcpl + pproof oo/ of o/ cp whose size The introduce propositional The idea idea of of the the proof proof is is to to introduce propositional variables variables for for each each relevant relevant subformula proof and with these sub formula of of the the proof and work work with these variables variables instead instead of of formulas. formulas. A A subformula subformula is is relevant, relevant, if if it it is is in in constant constant depth depth from from the the root root of of some some formula formula in in the the proof, determined by proof, where where the the constant constant is is determined by the the Frege Frege system. system. We We leave leave out out further further 0 details. details. [] From From these these two two lemmas lemmas we we get get immediately: immediately: (Statman (Statman [1977] [1977],, Cook Cook and and Reckhow Reckhow [1979]) [1979]) For For every every tautology cp, the tautology ~, the minimal minimal number number of of steps steps of of aa proof proof of of cp ~ in in aa Frege Frege system, system, the the minimal minimal number number of of steps steps of of aa proof proof of of cp ~ in in an an extension extension Frege Frege system system and and the the minimal minimal 0 size size of of aa proof proof of of cp ~ in in an an extension extension Frege Frege system system are are polynomially polynomially related. related. [:] 8.1.15. 8.1.15. Theorem. Theorem. minimal number 8.1.16. 8.1.16. It It can can be be shown shown that that the the minimal number of of steps steps in in aa proof proof of of a a tautology tautology in in an an extension extension Frege Frege system system can can be be exponentially exponentially larger larger than than in in aa substitution substitution ubarjan [1975], Frege Frege system system Tsejtin Tsejtin and and C Cubarjan [1975], Krajicek Kraji~ek [1989b]. [1989b]. Consider Consider the the tautology tautology 2n 2n n times -~. -, ) (p V -,p -'. It ~2-(p -~p),, where where -, _~2- denotes denotes 22n-times It is is not not very very hard hard to to show show that that the the number number of of steps steps needed needed to to prove prove it it in in aa Frege Frege system system is is n(2 ~(2 nn)) (it (it is is also also possible possible to to prove prove it it by by defining defining aa winning winning strategy strategy for for Adversary Adversary in in the the game game below). below). Thus Thus the the 2n (n) .. On minimal size size of of an an extension extension Frege Frege proof proof for for -, _12-(p (p V -,p) -~p) must must be be 22!l n(n) On the the other other minimal hand hand it it can can be be proved proved using using only only O(n) O(n) steps steps in in aa substitution substitution Frege Frege system. system. This This is is 2k +l 2k q from based possible to q in based on on the the fact fact that that it it is is possible to derive derive qq -+ ~ -, -~2k+lq from qq -+ --+ -, -~2kq in constant constant - 596 596 P. Pudltik Pudldk 2k number number of of steps: steps: first first derive derive ..., ~2kqq -+ ~ the the transitivity transitivity of of implication. implication. ...,2 k+l 2k ~2k+lqq by by substitution substitution q f-t ~-~ -~2kq, and then then use use q, and ..., 8.2. The 8.2. A A game. game. The following following game game was was devised devised as as an an approach approach to to proving proving lower lower bounds on bounds on the the lengths lengths of of propositional propositional proofs. proofs. So So far far we we were were not not able able to to get get new new lower bounds result it; in lower bounds result using using it; in fact fact it it is is even even not not so so easy easy to to interpret interpret the the known known lower lower bounds using bounds using this this game. game. However However the the game game can can be be used used at at least least to to prove prove something something about about the the structure structure of of propositional propositional proofs. proofs. 8.2.1. shall call call the game. The 8.2.1. We We shall the game game Prover-Adversary Prover-Adversary game. The game game is is determined determined by a a complete complete set set of of propositional propositional connectives connectives B. There are are two two players players Prover Prover and and B. There by Adversary. to prove proposition cP Adversary. The The aim aim of of Prover Prover is is to prove a a proposition ~o and and the the aim aim of of Adversary Adversary is pretend that, that, for is to to pretend for some some assignment, assignment, the the formula formula cP ~o can can have have value value 0 (=false). (=false). The The game game starts starts with with Prover's Prover's asking asking cp ~o and and Adversary Adversary answering answering 00,, and and then then Prover Prover asks other propositions them. The asks other propositions and and Adversary Adversary assigns assigns values values to to them. The game game ends ends when when there simple contradiction there is is aa simple contradiction in in the the statements statements of of the the Adversary Adversary which which means means the the following. Suppose we consider propositions connectives B. following. Suppose we consider propositions in in aa basis basis of of connectives B. Then Then simple contradiction contradiction means means that that for for some connective 0o E B, B, and and propositions propositions some connective aa simple CPI, . . . , CPk . . · , ~o~, CPk, o(CPb . . . , CPk) ~ol,..., ~ok,, Adversary Adversary has has assigned assigned values values to to CPb ~Ol,..., o(~ol,..., ~o~) and and they they do do 0; e.g., cP, 11 to not not satisfy satisfy the the truth truth table table of of o; e.g., he he assigned assigned 0 to to ~o, to 'ljJ r and and 11 to to cP ~o1\ A 'ljJ. r We We define define that that aa proposition proposition cP ~o is is provable provable in in this this game, game, if if Prover Prover has has a a winning winning strategy. strategy. A A natural natural measure measure of of complexity complexity of of such such proofs proofs is is the the minimal minimal number number of of rounds to convict rounds needed needed to convict any any Adversary. Adversary. It the Prover-Adversary sound and It is is easy easy to to prove prove that that the Prover-Adversary game game as as aa proof proof system system is is sound and complete, proof system complete, (however (however it it does does not not satisfy satisfy the the definition definition of of aa propositional propositional proof system 2.5). 2.5). To To prove prove the the soundness, soundness, suppose suppose cP ~o is is not not aa tautology. tautology. Then Then Adversary Adversary can can simply evaluate cp[a] = o. To simply evaluate the the propositions propositions on on an an input input aa for for which which ~o[a] = 0. To prove prove the the cP, including completeness, all subformulas completeness, let let Prover Prover ask ask all subformulas of of ~o, including the the variables. variables. The The most most interesting interesting fact fact about about the the Prover-Adversary Prover-Adversary game game is is the the relation relation of of the the number number of of rounds rounds in in the the game game to to the the number number of of steps steps in in aa Frege Frege proof. proof. The The minimal minimal number number of of rounds rounds in in the the Prover-Adversary Prover-Adversary game needed minimal number number of game needed to to prove prove cP ~o is is proportional proportional to to the the logarithm logarithm of of the the minimal of steps in aa Frege steps in Frege proof proof of of cpo ~o. More More precisely, precisely, for for every every basis basis B B and and every every Frege Frege system system F, F, there there are are constants constants Cb Cl, C2 c2 such such that that for for every every tautology tautology cP ~o,, log kk rounds (i) F, then proved in (i) if if it it has has aa proof proof with with kk steps steps in in F, then it it can can be be proved in � g CI cl log rounds and and (ii) proved in (ii) if i / iit t can can be be proved proved in in rr rounds, rounds, then then it it can can be be proved in F F in in kk steps steps with with log log k k� _< Cc2r. 2r . 8.2.2. 8.2.2. Proposition. Proposition. Proof. · . . , CPk P r o o f . 1. 1. Let Let a a Frege Frege proof proof of of cP ~o bbee given, given, say say CPb ~ol,..., ~ok,, with with CPk ~Ok= ----cpo ~o. Consider Consider conjunctions conjunctions r = (... ^ ^...) ^ 597 597 The Lengths of Proofs If If Adversary Adversary tries tries to to be be consistent consistent as as long long as as possible, possible, Prover Prover needs needs only only a a constant constant numbers numbers of of questions questions to to force force him him to to assign assign 11 to to an an axiom. axiom. Thus Thus he he can can force force value value 11 for for 'tfJ r I ' Also Also he he needs needs only only a a constant constant number number of of questions questions to to get get 00 for for 'tfJ O. Then Ck, since 'Pk ~k H ~ 0. Then he he uses uses binary binary search search to to find find an an ii such such that that 'tfJ ri H ~-~ 11 and and k , since 'tfJ O. This ( log k) rounds. A r HI H ~ 0. This takes takes O O(log k) rounds. A constant constant number number of of rounds rounds is is needed needed to to O. Suppose i. For get get 'PHI ~i+l H ~-~ 0. Suppose 'PHI ~i+l was was derived derived from from 'Pi" ~ i l , .. .. .., , 'Pi/ ~iz,, ill i l , ." . . ., , il it :::; _ i. For each each or to of of these these premises premises it it takes takes only only :::; < log log ii rounds rounds to to force force 11 ((or to get get an an elementary elementary binary search contradiction contradiction),) , since since 'tfJ ri H ~ 11 ,, -- use use binary search again. again. Once Once the the premises premises got got ll's' s and the the conclusion Prover needs needs only only a a constant constant number number of of questions questions to to force force an an 0, Prover and conclusion 0, elementary elementary contradiction. contradiction. 2. Let 2. Let a a winning winning strategy strategy for for Prover Prover be be given, given, suppose suppose it it has has rr rounds rounds in in the the r which, as worst worst case. case. We We construct construct a a sequent sequent calculus calculus proof proof of of 'P ~ of of size size 220( ~ ) , which, as we we know, know, can can be be transformed transformed into into a a Frege Frege proof proof with with at at most most polynomial polynomial increase; increase; we we shall in more shall consider consider this this transformation transformation in more details details below. below. P, let , " " at , tt :::; Consider Consider a a particular particular play play P, let al C~l,...,at, <_ rr be be the the questions questions asked asked by by Prover, where where we we have have added added ((or removed)) negations, negations, if if Adversary Adversary answered answered 00 ((in Prover, or removed in particular particular al C~l is is ""'P) ~ ) . . Thus Thus al ~1 1\ A ..... . 1\ A at c~t is is false, false, hence hence --+ --+ ..., - ~aall,, .. .. .., , ...,at ~c~t is is a a true true sequent. Moreover, Moreover, as as easily easily seen, seen, it it has has aa proof proof with with constant constant number number of of lines, lines, sequent. since since there there is is a a simple simple contradiction contradiction in in the the statements statements al a l ,, .'. .. ,. , at at.. The The proof proof of of 'P is taking proofs such sequents is constructed constructed by by taking proofs of of all all such sequents and and then then using using cuts cuts eliminating eliminating 'P. This successively successively all all formulas formulas except except of of ~. This is is possible possible due due to to the the structure structure of of the the possible possible plays. plays. Namely, Namely, 1. 1. for for each each play play P P there there is is another another play play pI P~ in in which which all all the the questions questions and and answers answers are the last last question question of are the the same same except except for for the the answer answer which which corresponds corresponds to to the of P; pI P; P~ may may be be longer longer than than P; P; 2. 2. for for every every two two plays plays P, P, pI, P', if if they they have have the the same same questions questions up up to to the the i-th i-th one, one, say say al a l ,l .". . ," ai c~i,, then then they they have have the the same same answers answers up up to to the the ii - 1-st 1-st one one and and different different i-th i-th answer. answer. Finally observe Finally observe that that the the number number of of such such sequents sequents is is at at most most 22rr,, which which gives gives the the 0 bound. bound. [] Let Let us us note note that that the the proof proof constructed constructed from from the the game game has has aa very very special special structure. structure. Firstly Firstly it it is is in in a a tree tree form; form; secondly, secondly, it it is is like like a a dual dual to to cut-free cut-free proofs, proofs, since since it it uses uses everywhere everywhere only only the the cut cut rule, rule, except except for for the the leaves leaves of of the the proof proof tree. tree. Let Let us us note note also also that that we we can can characterize characterize the the size size of of proofs proofs in in Frege Frege systems systems in in a a similar way, logarithm of similar way, we we have have only only to to add add the the logarithm of the the maximal maximal size size of of a a query query to to the the cost cost of of the the play. play. - 8.2.3. We to the the problem about the the relation 8.2.3. We return return to problem about relation of of the the lengths lengths of of proofs proofs as as sequences and lengths use the the proof sequences and lengths of of proofs proofs as as trees. trees. We We would would like like to to use proof obtained obtained by by transforming transforming a a general general proof proof into into the the Prover-Adversary Prover-Adversary game game and and then then back back to to aa proof. proof. The The resulting resulting proof proof has has the the number number of of steps steps polynomial polynomial in in the the original original number number of of steps steps k k and and it it is is in in a a tree tree form, form, but but it it is is a a sequent sequent proof. proof. We We shall shall analyze analyze its in order its transformation transformation into into a a Frege Frege proof proof in order to to see see that that the the tree tree structure structure can can be be preserved preserved also also in in the the Frege Frege form. form. 598 598 P. Pud/6.k Pudldk First First we we shall shall assume assume that that we we have have aa Frege Frege system system Fo F0 with with suitable suitable rules. rules. Let Let -+ ..., a l , . . . , ~c~t ..., at be -~c~1,..., be a a sequent sequent on on the the leaf leaf of of the the sequent sequent proof. proof. We We shall shall replace replace it it by by (... v v --,o,t. (51) (51) . . . ((~c~il ..., ai l V We We shall shall start start with with proofs proofs of of such such sequents. sequents. We We know know that that some some ((... V )V aic ' where is aa constant -~c~i2) V ..., ~c~io, where c is constant determined determined by by the the basis basis that that we we use, use, is is aa ..., ai2 ) ."". .") tautology. tautology. Moreover Moreover this this tautology tautology has has aa constant constant size size proof, proof, as as it it comes comes from from aa simple simple contradiction. contradiction. Hence Hence it it also also has has a a constant constant size size tree tree proof. proof. To To get get (51) (51) we we have using aa tree have to to add add the the remaining remaining disjuncts disjuncts using tree proof. proof. It It is is quite quite easy, easy, if if we we use use the the ideas ideas shown shown in in section section 4. 4. Let k) , thus k) . Let us us note note that that tt = = O(log O(logk), thus also also the the proof proof of of (51) (51) is is O(log O(logk). The The rest rest of of the the proof proof is is the the same same as as in in the the sequent sequent case, case, provided provided that that we we have have a a cut cut rule rule in in the the form form A vVB B, , C C vV- -...,, BB A A A vvCC " If If we we have have aa general general Frege Frege system system F, F, we we have have to to simulate simulate each each application application of of a a rule rule of of Fo F0 by by several, several, however however a a constant constant number, number, of of rules rules of of F F.. Structurally Structurally it it means means that that we we replace replace each each node node with with its its in-going in-going edges edges of of the the original original tree tree by by a a constant constant size tree. Thus tree; but size tree. Thus in in general general we we get get aa larger larger tree; but the the point point is is that that the the new new tree tree has has depth k. Hence depth also also O(log O(log k) k),, thus thus it it has has size size also also polynomial polynomial in in k. Hence have have proved: proved: 8.2.4. (Krajicek 8.2.4. Theorem. Theorem. (Kraji~ek [1994a]) [1994a]) For For every every Frege Frege system system there there exists exists aa polynomial p(x) p( x) such polynomial such that that for for every every tautology tautology cp qo ttree ee ssequence equence ) . (I I cP i steps < r II cp I i steps II~oll,tep,, -< - P p(ll ' f " III steps ,,., II I" o 8.3. Resolution. The 8.3. R esolution. The most most important important propositional propositional calculus calculus for for automated automated theorem theorem proving proving is is the the resolution resolution system. system. It It is is fairly fairly easy easy to to implement implement and and there there is is aa variety heuristics there variety of of heuristics there that that one one can can try try in in the the proof proof search. search. The follows. Suppose The idea idea can can be be simply simply explained explained as as follows. Suppose that that we we want want to to prove prove aa tautology DNF. Thus tautology which which is is aa DNF. Thus it it suffices suffices to to derive derive aa contradiction contradiction from from its its negation, negation, which which is is a a CNF, CNF, say say AiE A/e/~i. This is is the the same same as as to to derive derive aa contradiction contradiction from from the the l Oi . This set set {oihEI {hi}iet.. If If we we think think of of disjunctions disjunctions as as obtained obtained by by applying applying the the set set operator operator of of disjunction disjunction to to a a set set of of variables variables and and its its negations, negations, then then we we need need only only a a single single rule rule the set. the cut. cut. The The contradiction contradiction then then would would be be the the disjunction disjunction of of an an empty empty set. In In the the usual usual terminology terminology we we call call variables variables and and negated negated variables variables literals; literals; the the disjunctions disjunctions are are represented represented simply simply as as sets sets of of literals literals and and they they are are called called clauses, clauses, the the cut cut rule rule is is called called resolution. resolution. As As we we are are proving proving a a contradiction contradiction from from assumptions, assumptions, we we rather rather talk talk about about a a refutation refutation than than a a proof. proof. Thus Thus a a resolution resolution refutation refutation of of aa set set of C, the of clauses clauses C C is is a a sequence sequence starting starting with with the the clauses clauses of of C, the following following clauses clauses are are derived derived by by resolution resolution and and the the last last clause clause should should be be 0. 0. The The Lengths Lengths of of Proofs Proofs 599 599 ' s, 8.4. Extended Though 8.4. E x t e n d e d resolution. resolution. Though aa lot lot of of interesting interesting tautologies tautologies are are DNF DNF's, we prove also also others. others. There we would would like like to to be be able able to to prove There is is aa natural natural way, way, in in which which we we can can extend extend the the resolution resolution system system to to be be able able to to talk talk about about arbitrary arbitrary formulas; formulas; namely, namely, we we introduce introduce variables variables for for formulas formulas and and add add the the defining defining clauses. clauses. Formally -,} and Formally extended extended resolution resolution for for the the basis basis {t\, {A, V, -~) and variables variables Pb p t , .' .". , , Pn Pn is is resolution resolution augmented augmented with with the the clauses clauses obtained obtained from from the the CNF CNF's's of of %~ -- Pi, q-~, -- ~q~,, q~,~^~,~ -- q~,x A q~,2, q~,xv~,2 -- q~x V q~,2, ' s are for for all all formulas formulas in in the the language language {Pb { p t , .' .". , , Pn, pn, t\, A, V, -'} 9},, where where qq's are some some new new distinct distinct variables. == -'q", is -,q",}. variables. E.g., E.g., q�", q~v--~qv is replaced replaced by by the the two two clauses clauses {q�"" {q~v, q",} qv} and and {-,q�"" {~q~v,-~qv}. We We define define it it for for other other bases bases similarly. similarly. While resolution resolution is systems, the extended resolution While is much much weaker weaker than than Frege Frege systems, the extended resolution system system is is polynomially polynomially equivalent equivalent to to extension extension Frege Frege systems. systems. The The simulation simulation of of extension the same extension Frege Frege system system by by extended extended resolution resolution is is based based on on essentially essentially the same idea idea as as Lemma Lemma 8.1.14. 8.1.14. 8.5. Intermediate 8.5. Bounded Bounded depth depth Frege Frege systems. systems. Intermediate between between the the resolution resolution system system and and the the Frege Frege systems systems are are bounded bounded depth depth Frege Frege systems. systems. They They are are very very important see section they are the strongest important for for bounded bounded arithmetic, arithmetic, see section 10. 10. Also Also they are the strongest systems systems for for which which we we are are able able to to prove prove exponential exponential lower lower bounds. bounds. Consider {t\, V, -,}. ne inductively Consider formulas formulas in in basis basis {A, ~}. We We defi define inductively classes classes � ~ii and and II Hii of of such such formulas. formulas. �o ~0 and and IIo II0 are are just just literals. literals. A A formula formula
do do,, depth depth dd and + 11 systems separated in in such such aa way way using using tautologies tautologies of of depth depth � _< do do.. On On the the other other hand hand there there is is aa sequence sequence of of tautologies tautologies of of depth depth 33 which which have polynomial size (unbounded depth) have polynomial size (unbounded depth) Frege Frege proofs, proofs, but but only only exponentially exponentially large large depth depth dd Frege Frege proofs proofs for for every every constant constant dd (see (see Buss Buss [1987], [1987], Krajicek, Krajffzek, Pudhik Pudl~k and and Pitassi, Beame Woods Woods [1995] [1995],, Pitassi, Beame and and Impagliazzo Impagliazzo [1993] [1993] and and Beame Beame et et al. al. [1992]). [1992]). The The tautologies tautologies express express a a very very simple simple theorem theorem-- the the pigeonhole pigeonhole principle. principle. We We shall shall prove prove aa lower clauses expressing the pigeonhole lower bound bound for for resolution resolution refutations refutations of of sets sets of of clauses expressing the pigeonhole principle principle in in the the next next section. section. 600 600 P. Pudldk mentioned that 8.6. Propositional P r o p o s i t i o n a l sequent sequent calculus. calculus. We We have have already already mentioned that sequent sequent proof systems proof systems are are polynomially polynomially equivalent equivalent to to the the Frege Frege system system that that we we considered considered in in the fi first part of of the the chapter, chapter, hence hence to to all all Frege Frege systems. systems. Thus Thus it it remains remains to to mention mention the rst part the the cut-free cut-free propositional propositional sequent sequent calculus. calculus. Since Since we we know know about about nonelementary nonelementary speed-up order logic, speed-up in in the the case case of of first first order logic, it it is is not not surprising surprising that that there there is is aa speed-up speed-up also and, trivially, also for for propositional propositional logic. logic. The The speed-up speed-up is is exponential exponential Takeuti Takeuti [1990], [1990], and, trivially, cannot be larger. (slightly worse) cannot be larger. An An exponential exponential speed-up speed-up (slightly worse) follows follows also also from from the the speed-up of bounded depth speed-up of unbounded unbounded depth depth Frege Frege system system versus versus bounded depth Frege Frege system system (using fact that that aa cut-free cut-free proof proof of of aa bounded depth tautology tautology is is also also bounded bounded (using the the fact bounded depth depth).. depth) 8.7. The deduction system 8.7. Propositional P r o p o s i t i o n a l natural n a t u r a l deduction. deduction. The natural natural deduction system is is essentially additional rule essentially aa Frege Frege system system with with an an additional rule which which allows allows to to prove prove an an implication implication cp -+ --+ 't/J r by by taking taking cp ~ as as an an assumption assumption and and deriving deriving r't/J. The The fact fact that that this this rule rule can can be be simulated simulated in in aa Frege Frege system system is is called called the the deduction deduction theorem theorem and and the the rule rule is is called called the the deduction Mutual simulations natural deduction deduction rule. rule. Mutual simulations of of the the sequent sequent calculus calculus and and natural deduction were were shown shown by by Gentzen Gentzen [1935] [1935] and and they they are are actually actually polynomial polynomial simulations, simulations, see see Eder Eder [1992]. [1992]. The The power power of of the the deduction deduction rule rule has has been been investigated investigated in in more more detail detail by by Bonet Bonet and and Buss Buss [1993]. [1993]. 8.8. Quantified It Quantified propositional proof systems. systems. It seems seems unlikely unlikely that that there there is simulate all is aa proof proof system system for for propositional propositional logic logic which which can can polynomially polynomially simulate all other other proof Pudhik [1989] proof systems systems (see (see Krajicek Kraji~ek and and Pudl~k [1989] for for the the relation relation of of this this question question to to problems problems in in computational computational complexity) complexity).. Thus Thus it it is is interesting interesting to to look look for for stronger stronger and How can and stronger stronger proof proof systems. systems. How can one one construct construct aa system system stronger stronger than than extension extension Frege possible way Frege systems? systems? One One possible way is is to to extend extend the the expressive expressive power power of of the the language language used used in in the the proofs proofs and and the the most most natural natural extension extension is is to to take take quantified quantified propositional propositional formulas. 3 formulas.3 The The language language of of quantified quantified propositional propositional logic logic consists consists of of quantified quantified propositional propositional formulas binding some formulas which which are are usual usual propositional propositional formulas formulas with with quantifiers quantifiers binding some propositional propositional variables. variables. The The semantics semantics of of such such formulas formulas is is clear. clear. E.g., E.g., the the following following is is aa quantified quantified propositional propositional tautology tautology Vp, Vp, q:Jr((p q3r((p -+ --+ q) q) -+ --+ (p (p -+ --+ r) r) A (r (r -+ -+ q)). q)). As As aa logical logical calculus calculus we we simply simply modify modify either either aa Hilbert Hilbert style style or or Gentzen Gentzen sequent sequent first first order calculus. Again, Again, we 1) of order calculus. we give give only only an an example. example. Consider Consider the the axiom axiom schema schema (5. (5.1) of section section 22 (t) 9 (t)-~ 3x~(x).. -+ :Jx(x) We propositional logic stands, the We use use this this schema schema in in quantified quantified propositional logic as as it it stands, the only only point point is is that that now now there there is is no no distinction distinction between between terms terms and and subformulas. subformulas. So So precisely precisely stated stated cp(p) be propositional formula it it is is as as follows. follows. Let Let ~(p) be aa quantified quantified propositional formula with with aa free free variable variable p p 3It is interesting interesting that a quantified quantified propositional propositionalcalculus calculus was was introduced introduced by Russell Russell [1906J [1906] as as a "theory of implication". "theory Lengths of of Proofs The Lengths 601 601 and let let r'If; be be any any quantified quantified propositional propositional formula, formula, then then the the following following formula formula is is an an and axiom axiom cp(p/'lf;) -+ 3xcp(x). It is all of It is interesting interesting to to investigate investigate the the proof proof systems systems for for all of quantified quantified propositional propositional logic, but but we we would would also also like like to to know, know, if if such such systems enable us us to to prove ordinary logic, systems enable prove ordinary propositions faster. faster. This This seems as quantified quantified propositional propositional formulas formulas propositions seems plausible, plausible, as can have stronger can define define functions functions in in 7~S'P.ACE, PSPAC£ , thus thus very very likely likely they they have stronger expressive expressive power power than than ordinary ordinary propositional propositional formulas. formulas. But But even even ifif this this were were true, true, itit would would not necessarily imply that, that, say, the quantified quantified propositional propositional sequent calculus has has not necessarily imply say, the sequent calculus shorter for some we cannot shorter proofs proofs for some propositional propositional tautologies. tautologies. Also Also we cannot exclude exclude that that the the quantified propositional propositional sequent calculus is is stronger, stronger, but but at at the the same same time time quantified quantified sequent calculus quantified propositional propositional formulas formulas of of polynomial polynomial size size define define the the same same functions functions as as ordinary ordinary propositional formulas of polynomial size. The only relation that we know for sure propositional formulas of polynomial size. The only relation that we know for sure is it polynomially simulates substitution is that that it polynomially simulates substitution (hence (hence also also extension) extension) Frege Frege systems, systems, [1990] . see Kraji~ek Krajicek and and Pudl~k Pudlak [1990]. see Important applications in bounded arithmetic arithmetic were by Dowd [1979] and Important applications in bounded were found found by Dowd [1979] and KrajiSek further applications applications in in bounded bounded arithmetic, arithmetic, see see [1990] ; for for further Krajicek and and Takeuti Takeuti [1990]; section 10 and and Krajicek Pudlak [1990]. section KrajiSek and and Pudl~k A related related question is, how is the the propositional part of of the the first first order A question is, how strong strong is propositional part order calculus. us consider consider the the Hilbert Hilbert style of section 2. If If we had calculus. Let Let us style calculus calculus of section 2. we had only then itit is is just Frege system. if we only propositional propositional variables, variables, then just a a Frege system. However, However, if we have some predicate, predicate, say P(x) , then can code propositional variable have some say P(x), then we we can code the the propositional variable Pi using the first first order order variable variable xi Xi as P(Xi). This This enables enables us us to to code code all all quantified quantified using the as P(xi). formulas, thus thus we at least the power of the the quantifi ed propositional propositional propositional formulas, we get get at least the power of quantified propositional calculus. calculus. However, However, using using aa suitable suitable representation representation we we can can simulate simulate arbitrarily arbitrarily strong strong propositional propositional proof proof system, system, see see 10.4.1 below. below. 8.9. 8.9. "Mathematical" " M a t h e m a t i c a l " proof p r o o f systems. systems. Let Let us us have have look look at at the the problem problem about about the the length length of of proofs proofs in in propositional propositional logic logic from from the the point point of of view view of of complexity complexity theory. theory. The The set set of of propositional propositional tautologies tautologies is is aa coNP-complete coN:P-complete sets, sets, say say L L.. A A proof proof system system is aa relation relation R(x, R(x, y) computable in in polynomial polynomial time time such such that that is y) computable Xx EE LL == y). - 3yR(x, 3yR(x,y). A x, y) -complete A proof proof of of X x is is aa yy such such that that R( R(x, y).. Thus Thus we we can can take take an an arbitrary arbitrary coNP coN:P-complete set set and and an an arbitrary arbitrary R R for for it it and and ask ask what what are are the the lengths lengths of of such such proofs. proofs. We We shall shall consider consider three three examples examples of of such such calculi. calculi. 8.9.1. In rst example 8.9.1. The T h e Haj6s Haj6s calculus. calculus. In the the fi first example the the set set L L consists consists of of graphs graphs which cannot be colored by three colors. Haj6s [1961] has proved that every which cannot be colored by three colors. Haj6s [1961] has proved that every such such graph can can be be obtained obtained as as follows. follows. graph 1. Start 1. Start with with K K4, the complete complete graph graph on on four four vertices, vertices, and and apply apply the the following following 4 , the operations: operations: P. P. Pud16k Pudldk 602 T Quantified propositional Quantified propositional calculus calculus Extension Extension Frege; Frege; Substitution Substitution Frege; Frege; steps steps in in Frege Frege l Frege Prege system; system; Sequent Sequent calculus; calculus; Natural Natural Deduction Deduction / Cutting Cutting planes planes T Bounded depth Frege Resolution Cut-free sequent calculus Figure 1: The Figure 1: The hierarchy hierarchy of of propositional propositional proof proof systems systems 603 603 The The Lengths Lengths o] ofProofs Proofs 2. edge/vertex edge/vertex introduction: introduction: add add any any new new vertices vertices and and any any new new edges edges to to aa 2. constructed graph; constructed graph; 3. join: join: ifif G1 with disjoint disjoint sets sets and G2 G2 has has already already been been constructed, constructed, G1 G1 and and G2 G2 with G 1 and 3. of vertices, vertices, (al, (al , bl) bd an an edge edge in in G1, G1 , (a2, (a2' b2) �) an an edge edge in in G2, G2 , then then construct construct aa new new of deleting the the edges edges (al, (al , b~), graph by by contracting contracting al a1 with with a2, a2 , deleting and adding adding (a2, b2) b2) and b1) , (a2, graph the edge (bl , b2) ; the edge (bl, b2); 4. ccontraction: contract any any two two non-adjacent non-adjacent vertices vertices in in aa constructed constructed graph. graph. 4. o n t r a c t i o n : contract On the the other other hand hand itit isis quite quite easy easy to to prove prove that that no no graph graph obtained obtained in in this this way way isis On 3-colorable. A A pproof of the the fact fact that that G G isis not not 3-colorable 3-colorable in in the the Hajds Hajas calculus calculus isis aa 3-colorable. r o o f of sequence where where K4 K4 isis used used as as an an axiom, axiom, the the three three rules rules above above are are used used to to construct construct sequence theorem asserts asserts that that this this calculus calculus G. Hajds' Hajas ' theorem new graphs graphs and and where where the the last last graph graph is is G. new is complete complete for for graphs graphs which which are are not not 3-colorable. 3-colorable. is Surprisingly Pitassi Pitassi and and Urquhart Urquhart [1992] [1992] have have shown shown that that the the Hajds Hajas calculus calculus is is Surprisingly polynomially equivalent equivalent to to extension extension Frege Frege systems. systems. This This means means that that polynomially 1. 1 . there there is is aa polynomial polynomial time time computable computable function function which which to to each each tautology tautology cp and its its extension extension Frege Frege proof proof dd assigns assigns aa graph graph G G and and aa proof proof hh in in the the Hajhs Hajas and calculus that that G G isis not not 3-colorable; 3-colorable; calculus 2. and there is is aa polynomial polynomial time computable function function which 2. and vice vice versa, v e r s a , there time computable which to to each each graph G G and proof hh in calculus that G is 3-colorable assigns assigns and aa proof in the the Hajhs Hajas calculus that G is not not 3-colorable graph tautology cp and its its extension extension Frege Frege proof proof dd . . aa tautology ~ and This shows that the concept of extension Frege is quite quite robust robust and that it This shows that the concept of extension Frege systems systems is and that it will be very hard to prove that there is no polynomial bound on shortest proofs will be very hard to prove that there is no polynomial bound on shortest proofs in in the the Hajas Hajhs calculus. calculus. 8.9.2. 8.9.2. Nullstellensatz. Nullstellensatz. over over finite finite fields. fields. Let Let The The second second example example are are systems systems of of algebraic algebraic equations equations fl(Xl,''',Xn) : 0 ((52) 52) f~(xl,...,x,) = 0 's Nullstellensatz be be aa system system of of algebraic algebraic equations equations over over aa field field F. F. The The famous famous Hilbert Hilbert's Nullstellensatz says that (52 ) does not have a solution in F (the algebraic closure says that (52) does n o t have a solution in T (the algebraic closure of of F) F) iff iff there there exist exist polynomials polynomials 91 gl (Xl ( x l ,, .. .. .. ,, Xn) x n ) ,, .. .. .. ,, 9 gm i n(Xl ( x 1 ,, ... .. ,. , Xn) x n ) such such that that mm L E g ,9 ( xi l(Xl , . . .,, x. ,. ).f ,i (xn) x l , .fi . . ,(x1 x , ~ ,) " " Xn) i= i=1l = = 11 (53 (53)) in in the the ring ring of of polynomials polynomials over over F. F. We We shall shall make make some some additional additional assumptions. assumptions. We We shall shall assume assume that that F F is is finite finite and 52 ) can and that that equations equations ((52) can have have solutions solutions only only in in F. F. The The last last condition condition can can be be ensured ensured by by adding adding equations equations TIa [ I a eEFF(X (Xi i -- aa )) = - - 00 for for ii = = 11, ,.. .. .. ,, nn.. Then Then such such sets sets of of unsolvable unsolvable systems systems of of equations equations are are coNP-complete. coAfP-complete. Furthermore Furthermore we we shall shall assume assume that polynomials polynomials are are given given as as sums sums of of monomials. monomials. Then Then (53 (53)) can can be be decided decided in in that 604 604 P. Pudltik Pudldk polynomial time, we can can expand expand the the sum sum of of products into aa sum sum of of monomials monomials polynomial time, since since we products into where where the the number number of of monomials monomials is is polynomial polynomial in in the the number number of of monomials monomials of of polynomials polynomials ggii and and j; fi,, ii = - 1, 1 , .. .. .., , n. n. Thus Thus we we can can think think of of the the system system of of polynomials polynomials gg 1,l , .. .. .., , ggm as aa proof proof that that (52) (52) is is unsolvable. unsolvable. (Let (Let us us remark remark that that in in this this special special case case m as the proof proof of of the the Nullstellensatz Nullstellensatz is is easy, easy, so so this this proof proof system system is is not not based based on on aa deep deep the result.) result.) This This system system is is not not known known to to be be equivalent equivalent to to another another proof proof system, system, it it is is weaker weaker than Frege systems and sequence of than Frege systems and there there are are superpolynomial superpolynomial lower lower bounds bounds for for aa sequence of unsolvable main application this approach unsolvable systems. systems. The The main application of of this approach is is in in the the works works of of Beame Beame et al. [1996] al. [1996/1997] et al. [1996] and and Buss Buss et et al. [1996/1997], proving independence of of counting counting principles principles ' proving independence in rst proved in bounded bounded depth depth Frege Frege systems systems and and in in bounded bounded arithmetic arithmetic (this (this was was fi first proved by by Ajtai Ajtai [1994b) [1994b] using using a a different, different, and and very very deep deep proof) proof).. A introduced in A related related system, system, called called the the polynomial polynomial calculus, calculus, 44 was was introduced in Clegg, Clegg, Edmonds and Impagliazzo [1996). Edmonds and Impagliazzo [1996]. In In this this system system we we derive derive equations equations sequentially sequentially using additions additions and using and multiplications multiplications by by arbitrary arbitrary polynomials. polynomials. Alternatively, Alternatively, it it is is just just equational equational calculus calculus with with no no variables variables allowed. allowed. For For aa given given bound bound d d on on degree degree of of polynomials polynomials occurring occurring in in the the proof, proof, the the system system is is stronger stronger than than the the Nullstellensatz Nullstellensatz system. still decidable polynomial time, time, if system. If If d d is is a a constant, constant, it it is is still decidable in in polynomial if there there is is aa proof proof of polynomial from of aa given given polynomial from a a given given set set of of polynomials. polynomials. Finally Finally we we consider consider aa proof proof system system which which uses uses ideas ideas of of linear linear programming programming which which was introduced in W. Cook, was introduced in W. Cook, Coullard Coullard and and Tunin Turgn [1987]. [1987]. 8.9.3. Cutting 8.9.3. C u t t i n g plane proof ssystem. y s t e m . This This system system is, is, in in aa sense, sense, an an extension extension of of resolution; resolution; in in particular particular it it is is also also a a refutation refutation system system for for aa set set of of clauses. clauses. However, However, instead instead of of clauses clauses we we use use linear linear inequalities inequalities which which adds adds power power to to the the system. system. A A proof proof line line is is an an expression expression (54) (54) alpl + . . . + anpn >_ B, B. For where allow also where a1 a l ,, .. ... ., , an, a,, B B are are integers. integers. We We allow also expressions expressions of of the the form form 00 2: _> B. For aa given identically and given clause clause C C we we represent represent literals literals Pi pi identically and ""Pi ~Pi by by 11 - PPi i . . Let Let II f l ,, ... .. , , fk fk be be the linear terms the linear terms expressing expressing the the literals literals of of C. C. Then Then we we represent represent C C by by the the expression expression . k + + . .. .. +. + j ' kfk 2: > 1. 1. h (Of (Of course, course, to to get get an an expression expression of of the the form form (54), (54), we we have have to to collect collect the the constant constant terms side; also terms on on the the right right hand hand side; also we we collect collect constant constant and and other other terms terms after after each each application of application of aa rule.) rule.) The The axioms axioms and and derivation derivation rules rules are are 11.. axioms all translations the clauses question and axioms are are all translations of of the clauses in in question and the the expressions expressions Pi pi 2: _> 0, 0, Pi 2: -Pi _> - 11; ; 22.. addition: a d d i t i o n : add add two two lines; lines; 3. multiplication: 3. m u l t i p l i c a t i o n : multiply multiply aa line line by by a a positive positive integer; integer; 44Another Another name proposed for this calculus is the Groebner Groebnerproof system. of Proofs The Lengths of 605 605 4. division: division- divide divide aa line line (54) (54) by by aa positive positive integer integer cc which which divides divides evenly evenly al a l l, . .. .., , ak 4. and round-up round-up the the constant constant term term on on the the right right hand hand side, side, i.e., i.e., we we get get and al an I + . . . + -Pn -P c c +...+ -c . � rBl VB1. (Note (Note that that on on the the left left hand hand side side we we have have integers, integers, thus thus rounding rounding up up is is sound.) sound.) A contradiction contradiction is is obtained, obtained, when when we we prove prove 00 > 11.. A We suggest suggest to to the the reader, reader, as as an an easy easy exercise, exercise, to to check check that that this this system system simulates simulates We resolution. Goerdt Goerdt [1991] [1991] proved proved that that Frege Frege systems systems polynomially polynomially simulate simulate the the resolution. cutting plane plane proof proof system. system. Furthermore, Furthermore, Buss Buss and and Clote Clote [1996] [1996] proved proved that that the the cutting cutting plane plane system system with with the the division division rule rule restricted restricted to to the the division division by by 22 (or (or any any cutting other constant constant > 11)) polynomially polynomially simulates simulates the the general general system. system. Recent Recent success success in in other proving exponential exponential lower lower bounds bounds on on the lengths of of cutting cutting plane plane proofs proofs (see section proving the lengths (see section 9.3) gives gives us us also also interesting interesting separations. separations. The The cutting cutting plane plane proof proof system system cannot cannot be be 9.3) simulated by by bounded depth Frege Frege systems systems as as it it proves proves the the pigeonhole pigeonhole principle principle (see (see simulated bounded depth Cook, Coullard and Tunin Turin [1987]) [1987]) using using polynomial polynomial size size proofs. proofs. The The cutting plane Cook, Coullard and cutting plane proof system system does does not not polynomially simulate bounded bounded depth depth Frege Frege systems systems Bonet, Bonet, proof polynomially simulate Pitassi and and Raz Raz [1997a] [1997a],, Krajicek Kraji~ek [1997a], [1997a], Pudhik Pudl~k [1997]. [1997]. Pitassi � 9. L o w e r bounds b o u n d s on o n propositional p r o p o s i t i o n a l proofs proofs 9. Lower In this this section section we we give give an an example example of of aa lower lower bound proof in in propositional propositional logic. logic. In bound proof Our lower lower bound bound will will be be an an exponential exponential lower lower bound bound on on the the size size of of resolution resolution proofs proofs Our of the the pigeonhole pigeonhole principle. principle. The The first first such such bound bound for for unrestricted unrestricted resolution resolution was was of proved by Haken [1985].. Unfortunately Unfortunately his his proof proof cannot proved by Haken [1985J cannot be be generalized generalized to to stronger stronger systems, (at nobody has doing it). it) . Therefore Therefore we shall apply systems, (at least least nobody has succeeded succeeded in in doing we shall apply a a technique of Ajtai Ajtai [1994a], [1994a], which which he bounded depth Frege systems. case technique of he used used for for bounded depth Frege systems. The The case which can can be considered as as aa depth depth one system, is simpler than than of resolution, resolution, which be considered one Frege Frege system, is simpler of for depths and serve as more advanced advanced results. for larger larger depths and thus thus can can serve as aa good good introduction introduction to to more results. 9.1. A general general m ethod. Before we consider consider the we shall 9.1. A method. Before we the concrete concrete example, example, we shall present a a general lower bound proofs, which can be be applied applied to to some some present general framework framework for for lower bound proofs, which can existing proofs proofs and, and, maybe, for some proofs. A general existing maybe, can can be be also also used used for some new new proofs. A general description of is going going on on in in lower bound proofs proofs is is always always useful, useful, since, since, when when description of what what is lower bound proving aa lower lower bound, we are things (the (the short proving bound, we are working working with with nonexisting nonexisting things short proofs proofs whose whose existence existence we we are are disproving) disproving) and and therefore therefore itit is is difficult difficult to to give give any any intuition intuition about about them. them. The basic basic idea idea of of our our approach approach is is as as follows. follows. Suppose Suppose that that we we want to show show that that The want to a2, . . . ,am) , am ) is is not not aa proof proof of of c~. a. Let Let LL be be the the set set of of subformulas subformulas of of c~1, aI, a2,... a2, . . . ,am , am (c~1, (aI , c~2,... and and c~. a. L L isis aa partial partial algebra algebra with with operations operations given given by by the the connectives. connectives. Suppose Suppose that that and aa homomorphism homomorphism A A :: L L --+ -+ B B such such that that A(c~) A(a) ~=I- lB. lB. we have have aa boolean boolean algebra algebra B and we a cannot cannot be among c~1,... al l . . . ,c~m, , am , since since A(qo) A(ip) == 1B 1B for for every every axiom axiom and and this this is is Then c~ Then be among preserved Frege rules. a is is aa tautology tautology preserved by by Frege rules. In In this this form form the the method method cannot cannot work: work: ifif c~ (and Therefore we have to (and we we are are interested interested only only in in tautologies), tautologies) , then then A(c~) A(a) -= lB. lB. Therefore we have to B Pudldk P. Pud16k 606 modify it. it. We We take take only only some some subsets subsets L Lii � C_L L and and AAii :: L Lii ---+ --+ B Bii for for different different boolean boolean modify algebras algebras B Bi. .i Now we we shall shall describe describe this this method method in in details. details. Let Let Now (p~,..., v~(p~,..., , ' " , Ppt),..., k(Pl, ' " , Pp~) CPv~l (Pl i) , . . . , CP i) (P! , · · . ,,Pl) Pi) cp(/9(Pl,''' be be aa Frege Frege rule rule R. R. We We shall shall associate associate with with it it the the set set LR LR of of all all subformulas subformulas of of ~Pl,..., ~kk and and cp ~.o If If l, ' . . , CP CP l (¢! , . . . , ¢l k (¢! , . . . , ¢l ~1(r r ) , . . . , CP ~(r r ) CP (¢! , . . . , ¢l ) cp~(r162 is an an instance instance of of R, R, we we associate associate with with it it the the set set is LLR(g ) ; a(pl R( ,j)) = (tPl ....... = LR LR(r C t )= - {a( ( a ( r¢l, . . . , ¢i Ct); a ( p t ,, .. ... ., ,pPtl)) E LR}. LR). ...tPtl Let B B be be aa boolean algebra. A A homomorphism homomorphism A A : 9L LR(5) --+ B B is is a a mapping which Let boolean algebra. mapping which R(,j) ---+ maps connectives connectives onto onto corresponding corresponding operations operations in in B B,, i.e., i.e., maps z ( .cp) ~) = = 'BA( ~.~(~) A( cp) z(vcp V v¢ r)= = A( ~(v) v , A( ~(r¢ ) A( cp) VB etc. etc. The following following lemma lemma formalizes formalizes our our method. method. The Let Let (a (at,! , a2, a 2 , .' .. .., , am am)) be be aa Frege Frege proof proof using using aa set set of of assumptions assumptions SS.. Suppose Suppose the the following following conditions conditions are are satisfied: satisfied: the proof we have 1. every .formula of the proof we have aa boolean boolean algebra algebra B Bii and and an an element element 1. For For every formula ai ai of bbii E Bi. Bi . Furthermore, Furthermore, ifif ai ai EE S, S, then bi == 11B~. Bi . then bi 2. For For every every instance instance of proof we have aa boolean boolean algebra algebra BR(g BR(,j)) 2. of aa rule rule R( R(r"f) of of the the proof we have AR(,j) : 9LR(g) ---+ BR(g). BR(,j) . LR(,j) --~ and homomorphism AR(g) and aa homomorphism For every formula ai ai of the proof proof and and and and every every instance instance of rule R( 3. For every formula of the of aa rule R(r"f) where where we have have an /\'i.R(,j) :" B LR(,j) ' we ai EE LR(r BR(,j)) so ai an embedding embedding ai,R(g) Bii ---+ --+ BR(g so that that /\' ai,R(5)(bi) i.R(,j) (bi ) == AR(,j) (ai ) Then Then 9.1.1. 9.1.1. Lemma. Lemma. ' bt = 1 s l , . . . , b m = lB,,. The proof proof of of this this lemma lemma is is based based on on the the following following observation: observation: The 9.1.2. emma. 9.1.2. LLemma. A rule is any boolean boolean algebra. A Frege Frege rule is sound sound in in any algebra. Proof. Suppose for for some some assignment assignment of of values values from B we we get get the the value value 1B lB for for P r o o f . Suppose from B the assumptions assumptions but but aa value value bb << 1B lB for for the the conclusion. conclusion. Take Take aa homomorphism homomorphism the l } such /\, :: B B --+ such that that a(b) /\'(b) == 0. Then we we get get aa contradiction contradiction with with the the soundness soundness O. Then ---+ {0, {O, 1} 0 of for the [:] of the the rule rule for the algebra algebra {0, {O, 1I }. }. The Lengths Lengths of of Proofs Proofs 607 607 P r o o f oof f LLemma e m m a 9.1.1. 9.1.1. We We shall shall use use induction. induction. If If al cq E ES S,, then then bbtl = = IB 1B1, otherwise Proof l ' 0therwise c~t is is an an instance instance of of aa logical logical axiom, axiom, say, say, R(-¢) R(r (a (a rule rule without without assumptions). assumptions). Thus, Thus, al by 1.2, AAR(~7)(c~t ( (al) = IB Hence by Lemma Lemma 9. 9.1.2, ) 1BR(r ). Hence R .,&) R(,f) ' = = ~l,R(g)(bt) BR(.,&))." "'l, R(.,&) (bd == AAR(~7)(~t R("&) (ad) = I1BR(~7 Since Since "'l, ~t,R(~) an embedding, embedding, bbll = IB 1Bt. The induction induction step step is is similar. similar. , . The R("&) isis an = 0 [:] It may may seem seem at at the the first first glance glance that that it it does does not not make make sense sense to to talk talk about about boolean boolean It algebras B Bi,i , since since we we can can simply simply take take all all of of them them to to be be 4-element 4-element algebras, algebras, and and thus thus algebras isomorphic. It It turns turns out, out, however, however, that that in in applications applications nontrivial nontrivial boolean boolean algebras algebras isomorphic. Bii appear appear quite quite naturally. naturally. They They have have to to reflect reflect the the properties properties of of the the tautologies tautologies that that B we consider. consider. we Let us us remark remark that that this this is is only only one one possible possible interpretation interpretation of of some some lower lower bound bound Let proofs and and there there are are other other interpretations. interpretations. In In particular particular other other interpretations interpretations are are proofs based on on the the idea idea of of forcing, forcing, due due to to Ajtai Ajtai [1994a] [1994a],, and and partial partial boolean boolean algebras, algebras, due due to to based Krajf~ek [1994bJ; [1994b]; let let us us note note that that using using partial partial boolean boolean algebras algebras one one can can characterize characterize Krajicek (up to to aa polynomial) polynomial) the the length length of of proofs proofs in in Frege Frege systems. systems. (up Another tool which which we we shall shall use use are are random random restrictions. restrictions. They They have have been been Another tool successfully applied applied for for proving proving lower lower bounds on bounded depth boolean boolean circuits circuits successfully bounds on bounded depth and later later also also for for lower lower bounds on proofs proofs in in bounded depth Frege Frege systems systems by by and bounds on bounded depth Ajtai [1994a,1990,1994bJ, [1994a,1990,1994b], Beame et al. al. [1992] [1992], Beame and Pitassi Pitassi [1996] [1996],, Bellantoni, Bellantoni, Ajtai Beame et Beame and ' Pitassi and and Urquhart Urquhart [1992] [1992], Krajicek [1994a], Krajicek, Krajf~ek, Pudlak Pudl~k and and Woods Woods [1995J [1995] Pitassi ' Kraji~ek [1994aJ, and and Pitassi, Pitassi, Beame Beame and and Impagliazzo Impagliazzo [1993J. [1993]. The The idea idea is is to to assign assign more more or or less less 's and randomly randomly OO's and 1l's's to to some some variables. variables. Then Then many many conjunctions conjunctions and and disjunctions disjunctions became constant constant and and thus thus the the circuits, circuits, or or the the formulas, formulas, can can be be simplified. simplified. For For became the to construct boolean algebras algebras with the the reduced reduced formulas formulas itit is is then then much much easier easier to construct boolean with the required required properties. properties. 9.2. An exponential lower lower bound bound on pigeonhole principle principle in in resolution. resolution. 9.2. An exponential on the the pigeonhole Let D D and be disjoint with cardinalities cardinalities IDI = nn ++ 11 and We shall shall Let and R R be disjoint sets sets with IDI-and IRI I R I= - n. We consider the proposition PHPn stating that there is no 11 - 11 mapping D onto onto PHPn stating that there is no mapping from from D consider the proposition R. weaker proposition proposition than "there is mapping of of D D into into R. (This (This is is aa weaker than just" just: "there is no no 11 -- 11 mapping thus the lower bound denote by R" , thus D, ii E R R", the lower bound is is a a stronger stronger statement.) statement.) We We denote by Pij, R Pij , ii EE D, propositional variables that ii maps the alleged propositional variables (meaning (meaning that maps onto onto jj in in the alleged mapping). mapping) . We We and "" we shall shall use use the the true shall consider consider clauses clauses made made of Pijj and ~Pij, furthermore we true clause clause shall of Pi Pij , furthermore T and the the false, false, or or empty, clause _l_. .l . PHPn PHPn is is the the negation negation of of the the following following set set of of T and empty, clause clauses clauses VjERPij , for E D; D; VjERPij, for ii E ViER Pij , for jj EE R; R; VieRPij, for for jjEEDD j, kk EERR, , jj~=I-k ;k ; -~pijV-~pik, , , j, V ""Pik, for ""Pij for -~pjiV-~p~i, , jj~=Ik , k , ii EERR. . ""Pji V ""Pki , for j,j, kk EEDD, Let M M be be the the set of partial partial one-to-one one-to-one mappings mappings D D ~-+ R. R. We We shall consider Let set of shall consider boolean algebras determined by by subsets subsets TT C_ �D DU as follows. follows. Let Let VT Vr be be aa boolean algebras determined T] < U R, R, IITI < n, n, as subset such that that subset of of partial partial matchings matchings gg such 608 608 P. Ptldltf.k Pudldk 1. T � c_ dom( dom(g)U rng(g); 1. T (g) j g ) u rng v 2. \f(i,j) 2. V(i,j) E e gg(i ( i eE T vTj e TJ) .E T) . The The boolean boolean algebra algebra associated associated with with T T is is P(VT) P(VT),, the the boolean boolean algebra algebra of of subsets subsets of of VT Yr.. In relation of In order order to to be be able able to to assign assign a a value value to to aa clause clause � A in in P(VT) P(VT),, aa certain certain relation of T to to � A must must be be satisfi satisfied. We define A is is covered covered by by T T if if T ed. We define that that � 11.. P ij EE � Pij /k =} ::~ ii E E T T vv jj E E Tj T; 2. �==} Tj/\E TJ .E T. 2 . -'P - T ij~ jEE A ~ iiEET A The T. Suppose Suppose � T, then The clauses clauses T T and and ..1 _1_are are covered covered by by any any set set T. A is is covered covered by by T, then the the value T) is value of of � A in in P(V P(VT) is the the set set bT =={ g{ge VETVTj ; g ( i ) = jjf ofor r s osome m e pPij ijeE A ,� , or or g(i) g(i) =I7~ jj f for o r ssome o m e -'Pij ~ p i jEe A� b� g(i) l j ) =Ior or g g -- l ((j) 7L ii for for some some ' -'Pij A}. Pij Ee �}. The The following following can can be be easily easily checked: checked: 9.2.1. 9.2.1. Lemma. Lemma. then b� bT = 1p(v 1P(VTT )).. then If Pn and If � A is is one one of of the the clauses clauses of of PH PHPn and T T covers covers � A,, ITI ITI < n, n, [] o Suppose T is aa natural mapping AT Suppose T � C_ TI, T', ITII IT'I < < n, n, then then there there is natural mapping )~T,T," P(VT) -+ --+ P(VT,) P(VT,) ,T' : P(VT) defined defined by by ~,~,(b) = {g' y~,, :.3g b(g � c_ gdin )}. . AT {gl Ee VT 3g Ee b(g ,T,(b) = Let Let T T � C_ TI, T', ITII IT'I < n n.. Then Then AT )~T,T' i8 an an embedding embedding of of the the boolean boolean ,T' is algebra algebra P(VT) P(VT) into into P(VT') P(VT,).. 9.2.2. Lemma. 9.2.2. L emma. Proof. All properties injective. This P r o o f . All properties are are trivial trivial except except for for the the following following oone: n e : AT ,~T,T' is injective. This ,T' is property property follows follows from from the the fact fact that that each each 9 gE E VT VT can can be be extended extended to to aa 9 g'' E E VT, VT, which which 0 holds n. holds due due to to the the fact fact that that ITII IT'I < n. [] Consider an instance of Consider an instance of the the cut cut rule rule rP V v Pij pij � A V V 'Pij --'Pij rF VVA� Suppose Suppose we we have have chosen chosen P(VTJ P(VT~),, ii = = 11,, 2, 2, 33 as as the the boolean boolean algebra algebra for for r rvp~j, zXV v-,p~j VPij , � 'Pij and r and r V V� A respectively. respectively. Then Then we we choose choose P(VT) P(VT) with with T T = = T T11 U UT T22 U U Ta Ta for for this this rule. rule. We We only only have have to to ensure ensure that that ITI ITI < < n n.. Since Since T T covers covers all all subformulas subformulas involved, involved, we we T) . The can can define define their their values values in in P(V P(VT). The condition condition that that this this is is a a homomorphism homomorphism of of ' and and V V is is easy easy to to check. check. By By Lemma Lemma 9.2.2 9.2.2 we we have have also also the the necessary necessary embeddings. embeddings. The The simplest simplest way way to to ensure ensure ITI ITI < < nn for for the the rules rules is is to to choose choose the the covering covering sets sets of of the e.g. Pp~H V . . . Pnn ) , the formulas formulas of of size size < < n/3. This This is is not not always always possible possible (take (take e.g. VP p22 22 VV...Pnn), therefore we we apply apply random random restrictions. restrictions. therefore ' s and Suppose Suppose we we assign assign OO's and 1l's' s to to some some variables variables Pi; pq and and leave leave the the other other as as they they are. are. If If we we do do it it for for all all formulas formulas in in a a proof, proof, the the resulting resulting sequence sequence will will be be a a proof proof again. ..1, so again. However However some some initial initial clauses clauses may may reduce reduce to to _1_, so we we cannot cannot argue argue that that ..1 _k 609 609 The The Lengths Lengths of of Proofs Proofs cannot cannot be be derived derived from from them them by by a a short short proof. proof. Therefore Therefore the the restrictions restrictions must must reflect reflect the the nature nature of of the the tautology tautology in in question. question. Let Let g gE EM M be be a a partial partial one-to-one one-to-one mapping. mapping. We We shall shall associate associate with with 9 g the the partial partial assignment assignment defined defined by by PO; -+ --+ 11 Pi pij; -+ --+ 00 Pi P i j; -+ - + Pi PO; Pi if if (i (i,, jj)) E E gg;j if if ii E e dom( dom(g) or jj E E rng( rng(g), g ) or g) , otherwise. otherwise. but but (i, (i, j) j) ¢. r gg;j Given a a clause clause � A,, we we define define �g Ag to to be be Given 1. 1. T T if if some s o m e Pi Pij; E E � A is is mapped mapped to to 11 or or some some Pi P0; such such that that ""Pi ~Pij; E E � A is is mapped mapped to to 0, 0, 2. otherwise otherwise it it is is the the clause clause consisting consisting of of all all literals literals which which are are not not O. 0. 2. Let R'' == R Let us us denote denote by by D' D ' == D D - - dom( dom(g), R -- rng rng(g), n ' == IR' IR'II .. Clearly Clearly �g Ag is is a a (g ) , nl g) , R D', Jj EE R'. RI. If clause clause with with variables variables Pi P0, E D', If � A is is a a clause clause of of PH P H PPn ~ then then �g Ag is is either either ; , ii E T T or or becomes becomes aa clause clause of of PHPn, PHPn, (on (on D' D ~ and and R'). R~). Denote Denote by by Mn,n" M,,,,, the the set set of of all all partial one-to-one one-to-one mappings mappings of of size size n n - n' n ~.. The The following following is is the the key key combinatorial combinatorial partial lemma lemma for for the the proof proof of of the the lower lower bound. bound. 1 /3J , let 9.2.3. Let 9.2.3. Lemma. Lemma. Let nl n' = = ln [nl/3J, let � A be be an an arbitrary arbitrary clause. clause. Then Then for for aa 9g E chosen with E Mn,n" Mn,n', chosen with uniform uniform probability, probability, the the probability probability that that �g Ag can can be be covered covered by by 1 i aa set set of of size size < < i�nl n is is at at least least 1 - 2enl/3~ where where c~ > 00 is is aa constant. constant. We shall use We shall use the the following following simple simple estimate. estimate. 9.2.4. Let I AI I = A� 9.2.4. Lemma. Lemma. Let a, a, b, b, l1 � < n, n,A C_ { l1,, . . .., ,nn} } , l, A = aa.. Take Take aa random random E B � C_ { 1 , .. .. . , , n}, n}, lEI [B[ = = b, with uniform uniform probability. probability. Then Then {I, b, with ( ) . eab Prob(IA n EII � l) � Prob(IAMB >_l) < \ --:;;z nl ] l Proof. Proof. Prob(IA Prob(IA n ME BII � _>l) < _ = ~ �AA {al, ... ,ad { a l ..... at}C () Prob(a E,, .. ... ., , aatt EE E) Prob(all E EB B) a . !!.. . b - 11 . . b - l + 1 .. . l nn nn --1 1 nn --l +l1+ 1 D 610 610 P. Pudlak Pudldk l 1 Proof n'J . Let P r o o f of of Lemma L e m m a 9.2.3. 9.2.3. Let Let us us denote denote by by l1 = = l[~n'J. Let � A be be given. given. We We shall shall simplify simplify the the situation situation by by replacing replacing each each -'Pij "~Pij E E � A by by 'j V V Pi pi,J v V V Pij pij,.' · V i'i'r#i ij'r'#j This operation commutes with the restriction and This operation commutes with the restriction and the the new new clause clause is is covered covered by by assume that that � the old old one T, ITI ::; < ll,, iff iff the one is, is, since since £e < < n' n' - 2. 2. Thus Thus we we can can assume A contains contains only is determined only positive positive literals. literals. Such Such aa � A is determined by by the the graph graph E E = {{(i,j);pij A}.. (i, j); Pij Ee �} Let Let 2/ 3 nn2/3 a = 40 · 40 From shall omit From now now on on we we shall omit the the integer integer part part function function and and assume assume that that all all numbers numbers are only inessential are integers. integers. This This introduces introduces only inessential errors. errors. Furthermore Furthermore denote denote by by a ~" ~ . A }. A= = {j {j E E R; R; degEU) degE(j) � _ 2a 2a}. We We shall shall consider consider two two cases. cases. Case Case 1: 1" IAI IAI � _> 2a. 2a. We We shall shall show show that that in in this this case case � Agg = = T 7- with with high high probability. probability. First estimate IA IA n First we we estimate M rng( rng(g)I. Note that that rng( rng(g) is aa random random subset subset of of R R of of size size g ) is g ) I . Note Hence we apply n- n' n',, thus thus also also R' R' = - R\ R \ rng( rng(g) is a a random random subset subset of of size size n'. n'. Hence we can can apply g) is Lemma 9.2.4. Lemma 9.2.4. Prob(IA Prob([A n M rng( rng(g)[ < a) a) = Prob(IA Prob([A n MR R'[' I � > [IAI A [- a) a) g) I < I AI -a ( 2 2e < (elZlnX/3 )lal-a e)a eIA l n l/3 < < n(IAI - a) n(IAI-a) -< nn-~ 2 /3 The bounded by The probability probability that that � Agg is is not not T 7- is is bounded by ) ( (' ( ) a •" ) Prob v'j EE A Prob (Vj mn M rng( rng(g)((g-l(j),j) E)) ::; < g ) ((g -I U ) , j) �~ E) Prob(IA Prob(IA n n rng( rng(g)l a) + + g ) I < a) Prob Vj EE A n rng(g ) ((g - I U) , j) �r E) IA n rng(g ) I � Prob(Vj ANrng(g)((g-~(j),j) E) II IAnrng(g)l > aa). . ( ) The estimated by The second second term term can can be be estimated by ( (55) (55) (56) ) max Prob (Vj Vj EE A n rng(g ) ((g - I U), j) �r E) Il A n rng(g ) = C max Prob Anrng(g)((g-~(j),j) Anrng(g)= C),, C�A. CC_A, ICI2:a ICl>a thus thus it it suffices suffices to to consider consider a a fixed fixed such such C C and and bound bound the the probability. probability. Let Let C C = = l I I the of , , ) j . , , vertices . . U as . , . think one ) ; , U } {jl {jl, h j 2 , .· . . , Jlcl}; j2),... ' g g-l(Jlcl) chosen one by by I gg - t ((h) lcl as chosen I Cl think of the vertices gg-X(jl), one independently, except that that they they must be different. one independently, except must be different. Prob (id = . . . , g (it ) = ( g - I Ut+I ) , jt+ l) �r E Prob ((g-~(jt+,),jt+~) E II g g(il) = jb jl,...,g(it) = jjr)t = = I it } 1 < 1 -_ degEU ) = 11 - IE ] E -- l (UJ t+ t +ll)) - {il { i l ,, .. ... . ,,it}l < 1 -_ 22aa - tt. < 1 d e g E ( Jt+t +l l )-- tt = < 1 . nn ++l -1--t t nn ++l 1 nn ++l 1 - ( ) - _ - 611 611 The The Lengths Lengths of of Proofs Proofs = l (jt), jt) � Thus ei is Thus the the probability probability that that (g(g-l(jt),jt) r E E for for all all tt = 11,, .. .. .., , lICI is a )a (1 - �) (1 - 2an +- 11 ) . . . (1 - 2a -n I+C1I + 1 ) - (1 - _ n+1 n+1 /3 , this expression is e -n(nl/3 ) . The first term of (56) is Since n�l rv � and aa ,.., nn 22/a, estimated in (55) and is even smaller. Thus in Case 1 the probability is 1 _ e-n(nl/3 ) as required. <(1 < - 2a ) ( 1 n+l Since W4-i 1 - 7~~ a 1 and 2a-1) n+l < <(1 ( l _ 2 a - ]C] + l ) n+l "'" - a n + l )" " this expression is e -n(~a/a). The first term of (56) is rv estimated in (55) and is even smaller. Thus in Case 1 the probability is 1 - e -n(~l/3) as required. 2a. In we cover Case Case 2: 2: IAI ]A] < < 2a. In this this case case we cover f}.g A g by by the the set set (A (R'\A) n '). (A n NR R')') U U (E-1 (E-I(R'\A) ND D'). We We need need only only to to estimate estimate the the probability probability that that the the size size of of the the two two sets sets in in the the union union is is small. small. We We shall shall use use Lemma Lemma 9.2.4 9.2.4 again. again. nl/3 /6 ( 12e ) nl/3 ( IA~n R)' I > � < (e2an~/a '''/3/6 /6 e2an 1 /3 ~"l/Z/6 n(nl/3 ) (57) = rob P Prob IA a R'I > < n:n-~-]/6] = 40nl/3 = ee--n("l,3).. (57) 1 1 40n /3 2 n . n /3/6 ( ) ( ) = ( �) To To estimate estimate the the second second set, set, first first observe observe that that n 22/3 /3 1 /3 .. 22 -� 1R'I ·" 2a -~) <_ P = e . fa- . (n1 /3 + 1) nl/3 /6 1/3/6 ) + 1)n in--(11nl/3/6 (n /6 3e n(n n(n 11/3 + 1) 1)) nl/3 nl/3/6 /3 + 3e . = e- n(nl/3 ) 1 ' 10 (n + 1)n /3 (58) (58) since since the the term term in in the the parentheses parentheses converges converges to to � ~ < < 11.. By By (57) (57) and and (58) (58) we we get get the the D required 2. required bound bound in in Case Case 2. [:3 Now the lower Now we we are are ready ready to to prove prove the lower bound bound which which was was originally originally proved proved by by Haken Haken [1985] [1985] with with aa better better exponent exponent than than we we give give here. here. 3e 9.2.5. Theorem. Pn has 9.2.5. T h e o r e m . (Haken (Haken [1985]) [1985]) Every Every resolution resolution proof proof of of PH PHPn has size size at at least least l 3 / 26n~/3, where where c~ > 00 is is aa constant. 2w constant. = l / 3 is Proof. Proof. Suppose Suppose aa proof proof of of size size < < 2w 2~nl/3 is given. given. Take Take aa random random gg E E Mn" Mn,, n' n' 1 /3 J . Then, ln Lemma 9.2.2, Lnl/3J. Then, by by Lemma 9.2.2, for for every every formula formula f}. A of of the the proof proof the the probability probability that that l/3 .. Thus n' is 2enl/z f}.g Ag is is not not covered covered by by aa set set of of size size < < � -~ is at at most most 2w Thus we we have have positive positive n t probability probability that, that, for for some some g9 E E Mn Mn,n,, all formulas formulas are are covered covered by by sets sets < < � ~ . . Hence Hence ,n" all there there is is at at least least one one such such aa g9-. Consider Consider the the proof proof restricted restricted using using such such aa gj 9; it it is is aa derivation derivation of of ..1 _L from from clauses clauses n t of Pn, . Choose of PH PHPn,. Choose aa covering covering set set of of size size < < � ~ for for each each clause clause in in this this proof. proof. Then Then take take 612 612 Pudldk P. Pudlak boolean algebras algebras P(V P(VT) for clauses clauses and and for for each each application application of of the the rule rule as as described described boolean T) for above. As As we we have have observed, observed, the the clauses clauses of of PH PHPn, get value value 11 in in their their boolean boolean Pn' get above. algebras. Now Now we we can can apply apply Lemma Lemma 9.1.1. 9.1.1. The The conclusion conclusion should should be be that that ..1 _1_gets gets algebras. also 11.. But But ..1 _1_gets gets the the value value 00 by by the the definition definition of of the the boolean boolean algebras. algebras. also l/3 .. Hence the the proof proof must must have have size size ;::: _> 2W 2~nl/3 [:] Hence 0 9.3. Lower Lower bounds b o u n d s based b a s e d on on effective effective interpolation interpolation theorems. We We are are going going 9.3. to discuss discuss an an approach approach which which is is not not based based on on such such ad ad hoc hoc proofs, proofs, but but instead instead it it uses uses to some general general theorems theorems interesting interesting in in their their own own right. These theorems theorems are are versions versions some right. These of the the interpolation interpolation theorem, theorem, aa classical classical result result of of Craig Craig [1957a,1957b] [1957a,1957b],, see see Chapter Chapter II.. of The interpolation interpolation theorem theorem has has aa first first order order logic logic version version and and aa propositional propositional version. version. The Recently Recently some some strengthenings strengthenings of of the the propositional propositional interpolation interpolation theorem theorem have have been been successfully applied applied to to prove prove lower lower bounds bounds on on the the length length of of propositional propositional proofs. proofs. successfully The propositional propositional interpolation interpolation theorem theorem states states that that for for aa given given propositional propositional The tautology iP(p, tautology ~(p, ij) ~) -+ --+ w(p, ~(p, r) ~),, where where p, p, ij, q, rr are are disjoint disjoint strings strings of of propositional propositional variables, variables, there there exists exists aa formula formula I(p) I(p),, which which contains contains only only the the common common variables variables p, such that ~(p, ij) ~) -+ -+ I(p) I(p) and and I(p) I(p) -+ ~ W(p, ~(p, r) ~) are are also also tautologies. tautologies. Such Such aa that both both iP(p, p, such formula I I(p) is called an interpolantof ~(p, (t) -+ ~ W ~(P, ~).. The The proof proof of of this statement formula (p) is called an interpolant of iP (p, ij) (p, r) this statement is trivial: Take the the quantified quantified boolean formula 3x 32 iP(p, r (or Vx V2 w(p, ~(p, x) 2)); clearly, is trivial: Take boolean formula x) (or ) ; clearly, it interpolates interpolates iP(p, O(p,~) --+ w(p, ~(i~,f). As any any boolean boolean function function can can be be defined defined by by an an it ij) -+ r) . As ordinary propositional formula, ordinary propositional formula, there there is is aa propositional propositional formula formula I(p) I(i~) equivalent equivalent to to 3x iP(p, x) . Craig gave gave constructive constructive proofs proofs of of his his theorems, i.e., he he showed showed how how to to construct Craig theorems, i.e., construct an interpolant I(p) an interpolant I(p) from from aa proof proof dd of of iP(p, O(p, ij) q) -+ --+ w(p, ~(i~, r) ~).. Thus Thus the the complexity complexity of of I(P) I(p) depends of the the proof This led led Krajicek Kraji~ek [1994a] [1994a] to propose aa depends on on the the complexity complexity of proof d. d. This to propose method can be stated as as follows: follows: suppose we can can method of of lower lower bounds bounds proofs proofs whose whose idea idea can be stated suppose we -+ ~(i~, W(p, ~) f) does does not not have have aa simple simple interpolant, interpolant, then then itit cannot cannot have show that that iP(p, show ~(i~, ij) ~) ~ have simple proof. aa simple proof. Another relationship relationship of of interpolation complexity theory theory Another interpolation theorems theorems to to questions questions in in complexity had earlier been considered considered by by Mundici Mundici [1984], [1984], but did not consider the the lengths lengths of of had earlier been but he he did not consider proofs. proofs. 9.3.1. original proof proof of of Craig based on on cut-elimination, so the constructed 9.3.1. The The original Craig was was based cut-elimination, so the constructed interpolant can large. His proof can be used used to to get get aa good good bound bound interpolant can be be exponentially exponentially large. His proof can be on we have consider aa on interpolants interpolants for for cut-free cut-free sequent sequent propositional propositional proofs, proofs, but but we have to to consider different measure of of the the complexity complexity of of interpolants. interpolants. The The new new idea idea is is that that we we can can look look different measure at an an interpolant interpolant as as aa boolean boolean function function and and then then we we can can apply apply any any of of the the measures measures at of complexity complexity of of boolean boolean functions. functions. Here Here the the right right measure measure is is the the size size of of the the smallest smallest of circuit computing circuit computing the the boolean boolean function. function. 9.3.2. heorem. 9.3.2. TTheorem. sequent sequent (Krajicek [1997a]) [1997a]) Let Let dd be be aa cut-free cut-free proof proof with lines of (Kraji~ek with kk lines of aa 613 613 The Lengths of of Proofs where e. no where p, p, ij, q, fr are are disjoint disjoint sets sets of of propositional propositional variables variables (i. (i.e. no ij(t occurs occurs in in the the consequent consequent and and no no f~ occurs occurs in in the the antecedent). antecedent). Then Then it it is is possible possible to to construct construct aa an an j (p, 1') interpolant I(p) interpolant I(p) of of A A~i � (~i(P, (1) -+ -4 V Vjj \II 9j(p, ~) which which is is aa boolean boolean circuit circuit of of size size kk O ~ ( l ) .. i (p, ij) The original one The proof proof is is essentially essentially the the original one of of Craig Craig [1957a,1957b] [1957a,1957b].. The The idea idea is is to to construct construct interpolants interpolants for for each each sequent sequent in in the the proof proof successively successively starting starting with with the the initial initial sequents sequents and and going going down down to to the the end end sequent. sequent. As As the the proof proof is is cut-free, cut-free, each each sequent only formulas only variables ij, or only variables sequent contains contains only formulas containing containing either either only variables p, p, q, or only variables 0 p, f, so p, r, so it it makes makes sense sense to to talk talk about about an an interpolant interpolant for for it. it. [] The The reason reason for for using using circuit circuit size size is is because because we we consider consider proofs proofs in in the the sequence sequence form. polynomial size form. For For tree-like tree-like proofs proofs we we actually actually get get a a polynomial size formula formula as as an an interpolant. interpolant. 9.3.3. Suppose (p, ij) -+ \II (p, f) i.e., 9.3.3. Suppose we we have have an an interpolant interpolant I(p) I(i~) for for � (I)(i~,~) 9(p,~) i.e., the the (p, 1') implications implications � r (p, ij) (1) -+ -4 I(P) I(p) and and I(P) I(p) -+ -4 \II 9(p, f) are are true. true. Let Let aa truth truth assignment assignment a to the the variables variables p p be be given. given. Then Then either either -,�(p, -~(I)(/~,a) 5) or or \II 9(5, f) is is true. The interpolant interpolant to ( a, 1') true. The I(a) is true, can holds, namely, can be be used used to to decide decide which which of of the the two two possibilities possibilities holds, namely, if if I(~) is true, (a, f) true, otherwise both -~(I)(/~, �(p, a) then then \II 9(5, ~) is is true, otherwise -,�(a, -~(I)(5,f) ~) is is true. true. (It (It is is possible possible that that both 5) and (a, 1') true, in and \II 9(5, ~) are are true, in which which case case I(a) I(5) could could be be true true or or false.) false.) Thus Thus there there is is an an f3(p, 1') be aa valid alternative alternative way way of of looking looking at at interpolant: interpolant: Let Let o:(p, a(p, ij) ~) V V fl(p, ~) be valid disjunction; disjunction; an interpolant interpolant is is aa procedure procedure which which produces produces one one of of the the two two disjuncts disjuncts which which becomes becomes an p. aa tautology tautology after after assigning assigning given given truth truth values values to to p. We We can can look look at at the the interpolation interpolation theorem theorem even even more more abstractly abstractly (see (see Razborov Razborov [1994]) [1994]).. Let Let A A and and B B be be disjoint disjoint NP AfT~ sets. sets. Then Then we we can can define define the the set set of of input input strings A, resp. B, by strings a~ of of length length n n which which are are not not in in A, resp. not not in in B, by aa polynomial polynomial size size formula formula n (P, ij) B, see an(P, q),, resp. resp. f3n(P, fl,~(p, f) ~) (a (5 is is not not in in A A if if Qn(a, an(a, ij) ~) is is aa tautology, tautology, similarly similarly for for B, see O: next next section) section).. Since Since A A and and B B are are disjoint disjoint i.e., i.e., the the complements complements cover cover all all inputs, inputs, the the f3n (ft, tautology. If disjunction disjunction Qn an (p, (P, ij) ~) V V~n (fi, 1') ~) is is aa tautology. If we we have have aa polynomial polynomial time time computable computable set 0, then set C C which which separates separates A A from from B B i.e., i.e., A A� C_ C, C, C Cn MB B = - O, then we we have have aa polynomial polynomial 's (a, 1') time time decision decision algorithm algorithm for for finding finding a a true true disjunct disjunct from from Qn(a, an(f, ij) ~) V V f3n ~n(fi, ~).. Cook Cook's theorem theorem implies implies that that then then there there exists exists also also aa polynomial polynomial size size circuit circuit Cn(p) Cn(P) for for this this problem. Clearly, interpolant for problem. Clearly, Cn Cn (p) (P) is is an an interpolant for Qn an (p, (p, ij) ~) V V f3n ~n (p, (P, 1') ~).. -, 9.3.4. interesting application application of 9.3.4. The The most most interesting of the the effective effective interpolation interpolation is is in in the the case case of of resolution. resolution. (Krajicek (Kraji~ek [1997a]) [1997a]) Let Let dd be be aa resolution resolution proof proof of of the the empty empty clause from clauses A I, Bj (p, f), sets of clause from clauses A~(p, (t),, ii E E I, Bj(p, f), jj E E J g where where p, p, ij, q, fr are are disjoint disjoint sets of i (P, ij) propositional to construct propositional variables. variables. Then Then itit is is possible possible to construct aa circuit circuit C(p C(p)) such such that that for for 0-1 assignment every 0-1 every assignment a~t for for p p 9.3.5. 9.3.5. Theorem. Theorem. C(a C(a)) = = 00 C(a C(5)) = = 11 * ~ * ::~ A A~(a, (1),, ii E E II are are unsatisfiable, unsatisfiable, and and i (a, ij) Bj (a , 1'), Bj(~t, ~),jj E EJ g are are unsatisfiable; unsatisfiable; ). the the size size of of the the circuit circuit C C is is bounded bounded by by O(ldl O(]d[). P. P. Pudlak Pudldk 614 614 Moreover, one can Moreover, one can construct construct aa resolution resolution proof proof of of the the empty empty clause clause from from clauses clauses Ai(a, 0, respectively (a, 1'~), ) , jj EE JJ if Ai(a, q), ~t), ii E E II if if C(a) C(a) = = O, respectively from from Bj Bj(5, if C(a) C(a) = = 11,, whose whose size size is the size is at at most most the size of of d. We shall shall sketch sketch two two proofs proofs of of this this theorem. theorem. The The idea idea of of the the first first one, one, due due to to We Krajicek Kraji~ek [1997a], [1997a], is is to to reduce reduce it it to to Theorem Theorem 9.3.2. 9.3.2. This This looks looks strange, strange, as as resolution resolution proofs proofs consist consist only only of of cuts cuts and and we we know know that that cut-elimination cut-elimination does does not not work. work. The The trick eliminate cuts replacing them trick is is to to eliminate cuts by by replacing them by by conjunctions. conjunctions. For For each each initial initial clause clause Sl Sl V .. ... . V Sk sk,, where where Si si are are literals, literals, first first prove prove the the sequent sequent k Our goal --+ 1\�= denote by Ai=l si, S1, s l , . . . , , Sk s~;i we we denote by 8; ~7 the the literal literal complementary complementary to to si si.' Our goal is is 1 8;, to derive derive aa sequent only of to sequent consisting consisting only of such such conjunctions conjunctions obtained obtained from from initial initial clauses clauses -Thus we we want want to to replace the refutation refutation proof by a a proof of the --+ .9 .9 .9 ,, 1\�= replace the proof by proof of the A i =k I 1 8;, 8 i , .. . . . . . Thus corresponding tautology (DNF) shall not succeed, we corresponding tautology (DNF).. We We shall not quite quite succeed, we have have to to add add also also conjunctions conjunctions of of the the form form Ssii 1\ A 8;, ~ , which which are, are, however, however, false, false, hence hence do do not not influence influence interpolants interpolants at at all. all. In In transforming transforming the the resolution resolution proof proof into into aa cut-free cut-free sequent sequent proof proof we we follow follow the the given given resolution resolution proof, proof, but but instead instead of of applying applying cut cut with with some some cut cut literal literal Si si,, we we introduce introduce Si si 1\ A 8;. ~7. Thus Thus aa general general sequent sequent in in the the proof proof will will consist consist of of conjunctions conjunctions of of 's and 's, conjunctions negated negated literals literals of of Ai(p, Ai(~, q) ~)'s and Bj(p, Bj(i~, r) ~)'s, conjunctions of of complementary complementary literals literals Si si 1\ A 8; ~ and and single single literals. literals. The The single single literals literals of of the the sequent sequent are are just just the the literals literals of of the the corresponding corresponding clause clause in in the the resolution resolution proof. proof. In In the the last last sequent, sequent, as as in in the the resolution resolution proof, proof, the the single single literals literals will will be be eliminated eliminated and and we we are are left left only only with with conjunctions conjunctions ' s and (p, r) 's and of of negated negated literals literals of of the the initial initial clauses clauses Ai(p, Ai(i~, q) q)'s and Bj Bj(p, ~)'s and conjunctions conjunctions of literals. Since of complementary complementary literals. Since conjunctions conjunctions of of complementary complementary literals literals are are false, false, an an interpolant interpolant for for this this sequent sequent is is also also an an interpolant interpolant for for the the sequent sequent without without them. them. So So we we can can apply apply Theorem Theorem 9.3.2 9.3.2 to to get get an an interpolant interpolant for for this this sequent sequent which which is is an an 0 I, Bj interpolant (p, q) interpolant for for Ai A~(p, ~),, ii E E I, Bj (p, (i~,r), ~), jj E EJ J in in the the sense sense of of the the theorem. theorem. [] • . . . The The idea idea of of the the second second proof proof (Pudlak (Pudl~k [1997]) [1997]) is is to to construct construct aa refutation refutation proof proof I, or (a, r), truth assignment. assignment. either from q), ii EE I, either from Ai(a, Ai(~, q), or from from Bj Bj(~, ~), jj E EJ J for for every every given given truth If If there there is is a a polynomial polynomial time time algorithm algorithm for for constructing constructing such such aa proof, proof, then then there there is is deciding which the two unsatisfiable, hence one one also also for for deciding which of of the two sets sets is is unsatisfiable, hence also also aa polynomial polynomial size size circuit. circuit. Substitute Substitute the the truth truth assignment assignment a fi into into the the initial initial clauses clauses and and discard discard those those which which contain contain a a literal literal which which is is true true under under the the truth truth assignment assignment a ~ and and delete delete the the substitution the .1 / produced produced by by the substitution from from the the others. others. Then Then we we follow follow the the proof. proof. What What we we want want is is to to never never mix mix variables variables q ~ with with variables variables r. 4. So So when when we we should should resolve resolve along along aa variable ri we it, since variable qi qi or or ri we do do it, since this this will will not not produce produce aa mixed mixed clause. clause. However, However, if if we we should should resolve resolve along along some some Pi Pi,, we we must must do do something something else. else. Now Now we we simply simply take take the the clause clause which which corresponds corresponds to to an an original original clause clause where where the the literal literal Pi pi,, resp. resp. Pi ~ , , is is false false under a. This under the the truth truth assignment assignment ~. This clause clause will will be be aa subclause subclause of of the the next next original original clause, clause, hence hence we we can can continue continue and and eventually eventually obtain obtain an an empty empty clause clause.. Since Since variables variables q and mixed, the the new will split split into and rf are are never never mixed, new proof proof will into at at least least two two disconnected disconnected parts. parts. We We can can backtrack backtrack which which initial initial clauses clauses are are actually actually needed needed to to get get the the empty empty clause clause The Lengths of of Proofs 615 615 (they must must be be of of the the same same kind) kind) and and take take only only that that component component as as the the new new proof. proof. 0 [] (they A closer closer analysis analysis of of this this proof proof shows shows that that we we can can use use the the directed directed graph graph of of the the A proof as as the the graph graph for for the the circuit, circuit, provided provided we we take take suitable suitable connectives. connectives. So So the the proof relation between between the the proof proof and and the the circuit circuit is is very very close. close. relation One can can also also easily easily show show that that we we have have to to use use circuits circuits instead instead of of formulas, formulas, unless unless One formulas are are as as powerful powerful as as circuits circuits (which (which most most researcher researcher doubt) doubt) Krajicek Kraji~ek [1994a]. [1994a]. formulas Namely, for for every every circuit circuit we we can can write write aa tautology tautology stating stating that that the the computation computation is is Namely, unique. We We use use variables variables P p for for the the input input values values of of the the circuit, variables ij~ for for the the circuit, variables unique. values at at the the gates gates in in the the first first computation computation and and variables variables f~ for for the the values values of of at at the the values gates in in the the second second computation. computation. The The tautology tautology asserts asserts that that if if the the output output value, value, say say gates qk,, in in the computation is is 11,, then then the output value value rk rk in in the the second second computation computation the first first computation the output qk is also also 11.. Clearly, Clearly, any any interpolant interpolant of of this this tautology tautology computes computes the the same same function function as as is the circuit. circuit. On On the the other other hand, hand, the the tautology tautology has has aa resolution resolution proof proof of of linear linear size. size. the 9.3.6. In In order order to to apply apply Theorem Theorem 9.3.5 9.3.5 we we need need to to have have good good lower lower bounds bounds on on the the 9.3.6. size of of circuits circuits computing computing some some explicitly explicitly defined defined boolean boolean functions. functions. Presently Presently all all size the known known lower lower bounds bounds for for explicitly explicitly defined defined functions functions are are only only linear. linear. Fortunately Fortunately the there is version of of the the theorem which can can be be combined with currently currently known known lower lower there is aa version theorem which combined with bounds. Quite Quite surprisingly surprisingly a a very very mild condition on on the the clauses clauses implies that the the bounds. mild condition implies that interpolating circuits can be be constructed monotone. A A monotone monotone boolean boolean circuit circuit is is interpolating circuits can constructed monotone. circuit in in the the basis basis {A, {A, V, V, O, 0, I1}, i.e., a a circuit circuit whose whose gates gates are are monotone boolean } , i.e., monotone boolean aa circuit functions. functions. 9. 3. 5 9.3.7. Theorem. (Krajicek [1997a]) [1997a]) Assume Assume that in Theorem Theorem 9.3.5 9.3.7. Theorem. (Krajff:ek that clauses clauses as as in are given. Suppose all variables occur in ij), ii EE II only only are given. Suppose moreover moreover that that either either all variables pp occur in Ai(P, Ai(P, (1), positively all variables variables pp occur E J J only only negatively, exists occur in in Bj(p, Bj(p, ~), 1') , jj E negatively, then then there there exists positively or or all 9.3. 5 which the conclusion conclusion of Theorem 9.3.5 which is C satisfying of Theorem is moreover moreover monotone. monotone. aa circuit circuit C satisfying the The proof proof of of this this theorem theorem is is obtained obtained by by inspection inspection of of either either of of the of The the proofs proofs of 0 Theorem [::1 Theorem 9.3.5. 9.3.5. There are well-known exponential lower bounds for for the monotone circuit circuit complex There are well-known exponential lower bounds the monotone complexity of of explicit explicit boolean boolean functions. alone would would not not suffice suffice to to get get an an exponential exponential ity functions. This This alone lower bound bound on on resolution resolution proofs. By another another lucky lucky coincidence coincidence the the lower lower bounds bounds on on lower proofs. By monotone more: they actually show of disjoint disjoint NP sets monotone circuits circuits give give more" they actually show that that some some pairs pairs of AfT~ sets cannot cannot be be separated separated by by monotone monotone circuits. circuits. In particular particular such such aa lower lower bound bound can can be be derived for tautologies tautologies related related to to the the In derived for (p, (t) ij) denote denote aa set of clauses expressing that that the the graph graph clique problem. Let Let Cliquen, Cliquen,~k (p, clique problem. set of clauses expressing with vertices coded coded by by pp has has aa clique clique of of size size at at least least kk coded coded by by ~. ij. The The variables variables pp with nn vertices represent represent edges edges of of the the graph graph and and the the variables variables ~ij represent represent the the graph graph of of aa one-to-one one-to-one function from aa k-element vertices of take function from k-element set set into into the the set set of of vertices of the the graph. graph. Formally, Formally, we we take 616 P. Pudtak Pudldk variables variables Pi Pi,j, < ii < jj :::; < n, n, qqi,r, < ii :::; <_ n, n, 1 :::; <_ rr :::; <_ kk and and clauses clauses i,r , 11 :::; ,j , 11 :::; V i ,r V ii q qi,r qi ,r' V V -' ~qi,r' qi' ,r'~ VV Pi,i' i,r V -'q ~qi,r V -' ~qe,r Pi,e qi ,r -~qi,r -' for for alI all I1 :::; < rr :::; _ k; k; for for alI a l l Il _:::;i Let Let Color Colorn,t(p, ~) denote denote aa set set of of clauses clauses expressing expressing that that the the graph graph with with n n vertices vertices n ,l (p, f) coded by mapping from coded by P p is is l-colorable. l-colorable. The The variables variables r~ code code aa mapping from the the set set of of vertices vertices of of that no the graph into size ll such the graph into aa set set of of size such that no edge edge is is mapped mapped on on aa single single point. point. This This can can be expressed by clauses as be expressed by aa similar similar set set of of clauses as above. above. containing aa k-clique If If kk > ll,, the the set set of of graphs graphs containing k-clique is is disjoint disjoint with with the the set set of of l-colorable graphs (a clique needs needs at hence the l-colorable graphs (a clique at least least kk colors) colors),, hence the two two sets sets of of clauses clauses Clique Clique~,k(p, q) and and Colorn Color~,~(p, ~) cannot cannot be be satisfied satisfied simultaneously. simultaneously. For For suitable suitable ,I(P, f) n,k (p, ij) parameters it it has has been been shown shown that that these these sets sets of of graphs graphs cannot cannot be be separated separated by by small small parameters monotone monotone circuits. circuits. 9.3.8. (Razborov 9.3.8. Theorem. Theorem. (Razborov [1985], [1985], Alon Alon and and Boppana Boppana [1987]) [1987]) Let Let ll < kk and and Vfi :::; 1:g n . Then -clique < 8sl~gn" Then every every monotone monotone circuit circuit which which outputs outputs 11 on on graphs graphs with with aa kk-clique -colorable graphs and and 00 on on l1-colorable graphs has has size size 2n(vq) . D• 2f!(vi) . 9.3.9. set of 9.3.9. Corollary. Corollary. (Krajicek (KrajfSek [1997a]) [1997a]) Any Any resolution resolution refutation refutation of of the the set of clauses clauses Clique D (p, ij) U Colorn l(p, r) has size Cliquen,k(p, q) O Colorn,t(p, f) has size 2 n(vq) . [] n ,k , 2f!(0) . Using this combinatorial technicalities, Using this approach approach we we do do not not avoid avoid combinatorial technicalities, since since the the proof proof of of Theorem Theorem 9.3.8 9.3.8 is is nontrivial. nontrivial. Its Its advantage advantage is is that that an an exponential exponential lower lower bound bound on on the the length length of of resolution resolution proofs proofs is is easily easily accessible accessible to to those those who who already already know know lower lower bounds on on the size of monotone circuits. bounds the size of monotone circuits. 9.3.10. 9.3.10. Another Another advantage advantage of of this this approach approach is is that that it it can can be be applied applied to to cutting cutting plane the random plane proofs, proofs, where where the random restriction restriction method method does does not not seem seem to to work. work. The The version version of of Theorem Theorem 9.3.5 9.3.5 for for cutting cutting plane plane proofs proofs is is almost almost identical. identical. There There are are two two versions versions of of the the monotone monotone case, case, Theorem Theorem 9.3.7, 9.3.7, for for cutting cutting plane plane proofs. proofs. The The first first one one (Bonet, (Bonet, Pitassi Pitassi and and Raz Raz [1997a], [1997a], Krajicek Krajf~ek [1997a]) [1997a]) gives gives monotone monotone boolean circuits, boolean circuits, but but requires requires that that the the coefficients coefficients in in the the proof proof are are polynomially polynomially bounded bounded by by its its size size (put (put otherwise, otherwise, the the size size of of the the monotone monotone circuit circuit is is bounded bounded not not only only by by the the number number of of lines lines but but also also by by the the size size of of the the coefficients). coefficients). The The second (Pudhik [1997]) second version version (Pudl~k [1997]) works works without without any any restriction restriction on on the the coefficients, coefficients, but boolean circuit. but the the interpolating interpolating circuit circuit is is not not an an ordinary ordinary monotone monotone boolean circuit. We We have have arbitrary real to which are and compute with arbitrary to consider consider circuits circuits which are monotone monotone and compute with real numbers. numbers. Again fortunately, Again fortunately, the the known known proofs proofs of of the the lower lower bounds bounds for for monotone monotone boolean boolean circuits circuits can can be be easily easily extended extended to to the the more more general general model model (Pudlak (Pudls [1997] [1997],, Haken Haken and Cook [n.d.]). bound can and Cook [n.d.]). In In particular, particular, an an exponential exponential lower lower bound can be be proved proved for for the the clauses clauses Clique Cliquen,k(p, q) U (J Colorn Colorn,t(p, ~) presented presented as as inequalities inequalities in in the the cutting cutting plane plane n ,k (p, ij) ,l (p, r) proof proof system. system. The Lengths Lengths of Proofs 617 617 9.3.11. At 9.3.11. At first first this this approach approach to to lower lower bounds bounds looked looked very very promising. promising. Unfortu Unfortunately, became clear clear very soon that cannot be extended much nately, it it became very soon that it it cannot be extended much further further beyond beyond resolution. resolution. We We do do not not know, know, if if an an effective effective interpolation interpolation theorem theorem in in the the style style of of Theorem Theorem 9.3.5 9.3.5 holds holds for for bounded bounded depth depth Frege Frege systems systems and and we we rather rather think think it it does does not not hold hold even even for for such such weak weak proof proof systems systems (cf. (cf. Krajicek Krajfhek [1997a] [1997a] for for some some arguments). arguments). For have strong For Frege Frege systems systems we we have strong evidence evidence that that it it does does not not hold. hold. Namely Namely one one can can prove that that such such aa theorem theorem does does not not hold hold for for Frege Frege systems systems using using the the widely widely accepted accepted prove conjecture that that factoring factoring of of integers integers is is not not in in polynomial polynomial time. time. conjecture (Bonet, (Bonet, Pitassi Pitassi and and Raz Raz [1997b]) [1997b]) There There exists exists aa sequence sequence of of r) which tautologies tautologies of of the the form form an (p, (P, if) ~) V V i3 ~n (p, (p, ~) which have have polynomial polynomial size size Frege Frege proofs, proofs, but but for which which there there is is no no sequence sequence of of polynomial polynomial size size interpolation interpolation circuits, circuits, provided provided that that for 0 factoring not in in polynomial factoring of of integers integers is is not polynomial time. time. [] 9.3.12. 9.3.12. Theorem. Theorem. Instead Instead of of proving proving this this theorem theorem we we shall shall explain explain in in general general terms terms the the rather rather surprising connection surprising connection between between propositional propositional calculus calculus and and cryptography. cryptography. The The basic basic one-way function, concept concept of of cryptography cryptography is is the the one-way function, which which is, is, roughly roughly speaking, speaking, aa function function which which can can be be easily easily computed computed (in (in polynomial polynomial time) time) but but whose whose inverse inverse function is is hard.5 hard. 5 It It is is not not known known if if such such functions functions exist; exist; in in fact, fact, we we even even do do not not function know know how how to to prove prove their their existence existence assuming assuming P 7~ =I ~: NP. AfT~. We We do do know, know, however, however, that that one-way function function exists exists iff iff there there exist exist disjoint disjoint NP AfT) sets sets which which cannot cannot be be separated separated aa one-way in the arithmetical theorems by by aa set set in in 7P. ~. We We shall shall see see in the next next section section (see (see 10.3) 10.3) that that arithmetical theorems of of certain certain logical logical complexity complexity can can be be translated translated into into aa sequence sequence of of propositional propositional tautologies. tautologies. Furthermore Furthermore for for each each first first order order theory theory we we can can construct construct aa propositional propositional proof proof system system where where the the translations translations of of such such theorems theorems have have polynomial polynomial size size proofs. proofs. Now, if if we we have have aa pair pair of of disjoint disjoint NP AfT~ sets sets A, A, B B which which cannot be separated separated by by aa cannot be Now, include this this set set in in 7P, ), we we can can take take aa theory theory T T in in which which this this fact fact is is provable provable (just (just include statement Hence in the propositional propositional proof statement as as an an axiom). axiom). Hence in the proof system system P P derived derived from from T can prove the tautologies A, B. the other T we we can prove the tautologies derived derived from from A, B. On On the other hand, hand, polynomial polynomial time time interpolation interpolation for for P P would would give give us us a a separating separating set set for for A, A, B B as as noted noted above. above. (For talking about polynomial time (For sake sake of of simplicity simplicity we we are are talking about polynomial time algorithms algorithms instead instead of of polynomial polynomial size size circuits; circuits; the the distinction distinction between between the the two two concepts concepts is is not not essential essential for for our argument.) argument.) our A A weaker weaker version version of of Theorem Theorem 9.3.12, 9.3.12, which which gave gave the the result result only only for for extension extension Frege Frege systems, systems, was was originally originally proved proved by by taking taking the the conjectured conjectured one-way one-way function function X mod xx r-+ ~ ggX mod n n and and proving proving in in Si S 1,, which which is is aa theory theory associated associated with with extension extension Frege Frege proof systems, proof systems, that that the the corresponding corresponding pair pair of of NP A/'7~ sets sets is is disjoint disjoint (see (see Krajicek Krajf~ek and and Pudhik Pudls [1998] [1998] for for aa full full proof) proof).. 9.4. Other The restrictions (exemplified (exemplified in 9.4. O t h e r lower lower bounds. bounds. The method method of of random random restrictions in section xed depth section 9.2) 9.2) has has been been extended extended by by Ajtai Ajtai [1994a] [1994a] to to any any fi fixed depth Frege Frege system. system. It bounds. In It gives, gives, however, however, only only slightly slightly superpolynomial superpolynomial lower lower bounds. In order order to to get get 5For practical practical cryptography one needs avemge; here consider only needs hard in the average; here we consider only the worst complexity. case complexity. 618 618 Pudldk P. Pudlak exponential lower lower bounds bounds one one needs needs aa more more substantial substantial change change in in which which the the concept concept exponential of covering covering sets sets is is replaced replaced by by certain certain decision decision trees trees and and aa Switching Switching Lemma, Lemma, of of the the of type [1986], type used used by by Yao Yao [1985] [1985] and and Hastad Hs [1986], is is applied applied to to reduce reduce the the depth depth offormulas; of formulas; see Pudlak and Woods [1995] and Pitassi, Beame and see Beame Beame et et al. al. [1992] [1992], Krajicek, ' Krajf~ek, Pudl~k and Woods [1995] and Pitassi, Beame and Impagliazzo [1993]. [1993]. Impagliazzo Let Let us us define define at at least least the the concept concept of of the the decision decision tree tree which which is is used used in in these these bounds bounds for PH PHPn. We use use the the same same notation notation as as above. above. Such Such aa tree tree is is aa labelled labelled rooted rooted tree, tree, Pn. We for where the the vertices vertices are are labelled labelled by by elements elements of of D DU U R, R, except except for for the the leaves, leaves, which which are are where labelled by by 00 -- reject, reject, and and 11 - accept; accept; the the edges edges are are labelled labelled by by pairs (i,j), D, j ) , ii EE D, labelled pairs (i, jj EE R. D, resp. R. We We require require that that for for aa nonleaf nonleaf vertex vertex v with with aa label label ii E E D, resp. ii E E R, R, the the outgoing (i, j), outgoing edges edges are are labelled labelled by by (i, j), resp resp (j, (j, i) i),, one one edge edge for for every every jj which which does does not not occur on on the the path path leading leading to to v. Consequently, Consequently, the the edge edge labels labels on on every every branch branch are are occur independent, i.e., i.e., they they form form aa partial partial one-to-one one-to-one mapping. mapping. independent, v v. In the the lower lower bound bound proof proof we we assign assign to to each each formula formula the the boolean boolean algebra algebra of of all all In subsets of of leaves leaves of of such such aa tree tree and and the the value value of of the the formula formula ).( A(~) is the the subset subset of of subsets cp) is leaves labelled labelled by by 11.. leaves The intuitive intuitive meaning meaning of of this this concept concept is is the the following. following. We We think think of of truth truth values values The of of the the propositional propositional variables variables Pi Pi,j,j as as given given by by some some imaginary imaginary one-to-one one-to-one mappings mappings from D D onto onto R. fact, in in aa nonstandard nonstandard model model with with n n infinite, there are are such such R. In In fact, infinite, there from external mappings. The decision tree enables enables us us to to decide decide in in aa natural natural way way if if such such external mappings. The decision tree mapping is is accepted or not. Then all all the the boolean boolean algebras algebras defined by trees trees can can aa mapping accepted or not. Then defined by be single one one which the boolean algebra of subsets of be embedded embedded into into aa single which is is the boolean algebra of subsets of one-to-one one-to-one R. Put mappings mappings from from D D onto onto R. Put otherwise otherwise our our logic logic is is aa logic logic of of one-to-one one-to-one mappings mappings from D D onto onto R. from R. not the sequence for for which which one lower bounds bounds PHPn is not the only only sequence one can can prove prove exponential exponential lower PHPn is on bounded bounded depth depth Frege proofs. Another Another such ARn - the parity on Frege proofs. such sequence sequence is is P PAP~ the parity principle PARn , n odd, expresses expresses that of cardinality cannot be be principle - - where where PAP~, n odd, that a a set set of cardinality n n cannot partitioned into into pairs. pairs. Similarly, consider the counting principle principle COUNTp,~ partitioned Similarly, one one can can consider the counting COU NTp,n which expresses that aa set set of of size size n, n, nn not not divisible divisible by p, cannot partitioned into into which expresses that by p, cannot be be partitioned blocks [1990] has has shown PAP~ not have polynomial size size p. Ajtai Ajtai [1990] shown that that P ARn does does not have polynomial blocks of of size size p. PHPm as similar bounded depth proofs, even if we use instances bounded depth proofs, even if we use instances of of PHPm as premises, premises, and and similar independence for the the counting counting principles [1994b], independence results results have have been been proved proved for principles by by Ajtai Ajtai [1994b] ' Beame Beame et et al. al. [1996], [1996], Buss Buss et et al. al. [1996/1997]. [1996/1997]. Together with exponential Together with exponential lower lower bounds bounds for for cutting cutting plane plane proof proof systems systems and and degree lower lower bounds bounds for for the the polynomial polynomial calculus, calculus, these these are are the the strongest strongest results results so so degree 2 ) lower far. Frege system for tautologies n(n2) lower bound bound for tautologies far. For For unrestricted unrestricted Frege system we we have have only only an an ~(n 2n (p V such as as _~2n(p .., V ~p). ..,p) . The The proof proof is is based based on on the the claim claim that that all all subformulas subformulas of of this this such tautology tautology must must occur occur essentially essentially (i.e., (i.e., in in aa constant constant depth) depth) in in the the proof. proof. This This is is essentially the the same idea as as in in Claim Claim 4.2.3, 4.2.3, see 8.1.16. Apart from this this rather rather essentially same idea see also also 8.1.16. Apart from simple proof proof we we do do not not have have anything anything for for Frege Frege and and stronger stronger systems. systems. simple The Lengths of Proofs 619 619 1 0 . Bounded 10. B o u n d e d arithmetic a r i t h m e t i c and a n d propositional p r o p o s i t i o n a l logic In In this this section section we we shall shall show show an an important important relation relation between between the the lengths lengths of of proofs proofs of of propositional propositional tautologies tautologies and and provability provability in in fragments fragments of of arithmetic. arithmetic. By By this this connection connection certain certain arithmetical arithmetical formulas formulas can can be be translated translated to to aa sequence sequence of of propositions, propositions, and and if if the the formula formula is is provable provable in in some some theory, theory, then then the the propositions propositions have have small in some some propositional small (e.g., (e.g., polynomial polynomial size) size) proofs proofs in propositional proof proof system system associated associated with with the the theory. theory. Surprisingly, Surprisingly, there there are are pairs pairs of of such such aa theory theory T T and and aa propositional propositional proof system system P P where where both both the the theory theory T T and and the the proof proof system system T T are are quite quite natural. natural. proof In In this this situation situation we we can can think think of of T T and and P P to to be be just just two two facets facets of of aa single single concept, concept, where P. This where T T is is aa uniform uniform version version of of the the nonuniform nonuniform P. This is is just just another another parallel parallel to to boolean uniform model boolean circuit circuit complexity, complexity, where where the the uniform model is is the the Thring Turing machine machine and and the the nonuniform model is boolean circuits. nonuniform model is aa sequence sequence of of boolean circuits. The this relation independence results. The main main application application of of this relation is is in in showing showing independence results. If If we we could on strong could prove prove superpolynomial superpolynomial lower lower bounds bounds on strong propositional propositional proof proof systems, systems, then independence results bounded arithmetic then we we could could show show interesting interesting independence results in in bounded arithmetic such such as as unprovability of unprovability of NP AfT) = coNP. coN'7). There practical use There is is also also practical use of of this this relation relation which which is is necessary necessary to to take take into into account account even be fairly even if if you you are are not not interested interested in in first first order order theories. theories. It It might might be fairly difficult difficult to to find find and and describe describe short short proofs proofs of of some some tautologies tautologies directly, directly, while while in in aa bounded bounded arithmetic arithmetic we the corresponding we can can often often see see easily easily that that the corresponding first first order order formula formula is is provable. provable. This This was was to disprove used, e.g. in used, e.g. in Pudlak Pudls [1991] [1991], to disprove aa conjecture conjecture saying saying that that formulas formulas expressing expressing ' Ramsey's Ramsey's theorem theorem in in propositional propositional logic logic do do not not have have polynomial polynomial size size proofs proofs in in Frege Frege systems. to prove systems. Similarly, Similarly, it it is is possible possible to prove the the existence existence of of aa polynomial polynomial simulation simulation of of a a proof proof system system P P by by aa proof proof system system Q Q by by proving proving the the reflection reflection principle principle (see (see Q. In polynomial simulation below) below) for for P P in in aa theory theory associated associated with with Q. In such such aa way way the the polynomial simulation of of substitution substitution Frege Frege by by extension extension Frege Frege system system was was discovered discovered by by Dowd Dowd [1985] [1985] and and Kraj~ek and and Pudlak Pudl~k [1989]. [1989]. Krajicek This This subject subject requires requires some some familiarity familiarity with with fragments fragments of of arithmetic arithmetic considered considered in in bounded bounded arithmetic. arithmetic. The The reader, reader, who who does does not not know know that that subject subject should should consult consult Chapter and Chapter II II Buss Buss [1986] [1986],, Hajek Hs and Pudlak Pudls [1993] [1993] or or Krajicek Krajf~ek [1995]. [1995]. 10.1. translations of bounded formulas 10.1. There There are are basically basically two two translations of bounded formulas into into propositions. propositions. They They are are determined determined by by the the particular particular way way in in which which we we represent represent truth truth assignments. assignments. A A truth truth assignment assignment is is a a finite finite sequence sequence a ~ of of O's O's and and l's. l's. We We can can code code it it either either by by aa subset finite segment binary representation representation is subset of of aa finite segment of of integers integers or or by by aa number number whose whose binary is la. We lfi. We start start with with the the simpler simpler one. one. 10.2. Lo(a) be 10.2. First F i r s t translation. t r a n s l a t i o n . Let Let L0(~) be the the language language of of arithmetic arithmetic with with nonlogical nonlogical l-ary relations. symbols symbols Q 0,, S, S, +, § ',. , _:::; augmented augmented with with a a second second order order variable variable a c~ for for/-ary relations. We We consider consider the the class class �o(a) A0(~) of of bounded bounded formulas formulas in in the the language language Lo(a) L0(~).. Assume Assume that that we we use use the the same same connectives connectives in in the the first first order order language language and and the the propositional propositional 1\, V calculus calculus and and they they include include A, V;j moreover moreover we we shall shall assume assume that that we we have have propositional propositional constants .1, T constants _L, T in in propositional propositional logic logic and and that that the the propositional propositional variables variables are are indexed indexed P. Pud/ak Pudldk 620 by/-tuples of nonnegative nonnegative integers. integers. by l-tuples of Let Let ()0 E E .6.o(a) A0(c~) be be aa formula formula with with k k free free variables. variables. Then Then for for each each sequence sequence nI, nl, .. .. .. ,, nk nk of of nonnegative nonnegative integers integers we we define define aa propositional propositional formula formula (0)~1 ..... ~, inductively inductively as as follows. follows. 1. for 1. for terms terms s(nl s ( n l ,, ." . . ", n nk), ( n l ,, .". . ", nk), nk), we we define define k ), tt(nI = (s(nI ( s ( n , , , .. .. .. ,, nnk) k) = = t(nI t ( n l , , ". . . , n n kk)) ) ) nl, m ..... n, = = ddf.l I -L (( resp. = T T),) , ... ,nk if (resp. true) if s(nI, s ( n , , .. ... ., , nk) nk) = = t(nI, t ( n i , .. ... ,. , nk) nk) is is false, false, (resp. true);; we we use use the the same same definition definition for =; for ::; _< in in place place of of--; 2. for Xk), . . . ,,tt(xl,... t,(XI , . . . ,,Xk), Xk) , we for terms terms tl t l ((XI x l ,, ... .. . ,,Xk),... we define define ... ,ii,," (XI , . . . , Xk), . . . , t,(XI , . . . , Xk))) nt ,..... (a(tl n, nk = = dd/ y Pi P i ,t ,..... ... ,nk where , . . . , t,t , (nI nk) ;; where il i , ,, .. ... ., , i, it are are the the values values of of tt,l (nI, ( n , , .. ... ., , nk) nk),..., ( n l ,, .' .. .., , nk) 3. propositional connectives connectives are are translated translated identically, identically, e.g., e.g., 3. propositional ( 0 , A O 2 ) n , ..... nk = d r (O1)n, ..... nk A ( 0 2 ) h i ..... nk; 4. ers are 4. bounded bounded quantifi quantifiers are translated translated to to long long disjunctions disjunctions and and conjunctions, conjunctions, thus thus (~Y ~ 8 ( X l ' ' ' ' ' X k ) O(Xl'''''Xk'Y))nl .....nk =dr () (XI , . . . , Xk, Y)) nl,..... ~ , ,OoVV .' ". v. V ((o(~,,...,~,y))~, ~,,~ ((o(~,,...,~,y))~, ,m, ' ...,nk () (XI , . . . , Xk , Y ))nl,..... ...,nk, where m is the value of s( nI, . . . , nk) ; in the case of bounded universal where m is the value of s ( n l , . . . , nk); in the case of bounded universal quantifier quantifier the the propositional propositional formula formula is is defined defined dually. dually. 10.2.1. Let (x) be pigeonhole principle 10.2.1. Example. Example. Let ()0(x) be the the formula formula expressing expressing the the pigeonhole principle for for the the binary binary relation relation a c~ (for (for sake sake of of simplicity simplicity we we use use aa little little stronger stronger form form than than in in section section 9) 9)": 3u < s ( ~ ) w < ~ ( ~ ( u , v)) v 3u,, u~ < s(~)3~ < ~(u, # u~ ^ ~(u,, ~) ^ ~(u~, ~)). translation (()(x)) For For aa given given n, n, the the translation (0(x)} nn has has form: form: j V V V /\ A 'Pi ~p,~ v ii V V i1 ,i2 /\ Pil ,j /\ Pi2,j , V V 8~,1,,~ ^p,i,J ^p,2J, il,i 2$n+1 j$ il,i2 where where 85il,iz denotes .1 _L if if il it = = ii22 and and denotes denotes T T otherwise. otherwise. The The constants constants T T and and .1 _1_ it,i2 denotes can be easily can be easily eliminated; eliminated; namely, namely, the the formula formula is is equivalent, equivalent, using using a a polynomial polynomial size size bounded bounded depth depth Frege Frege proof, proof, to to V V /\ A 'Pi ~Pijj V V ii V V Pil,j Api2,j. il,i2 621 621 The Lengths of o:fProofs We the propositional We have have obtained obtained the the usual usual form form of of the propositional formula formula expressing expressing the the pigeonhole pigeonhole principle. principle. Let Let us us observe, observe, which which is is quite quite clear clear from from the the example, example, that that the the translation translation is is formula of of polynomial polynomial size size in in the the indices indices n n l ,, ... .. ,. , nk nk and, and, moreover, moreover, the the depth depth is is aa formula bounded bounded by by a a constant, constant, namely namely by by the the depth depth of of the the first first order order formula. formula. �o(a) denote �o with with the (a) Let Let I IAo(a) denote I IA0 the induction induction schema schema extended extended to to all all �o A0(a) formulas. formulas. 10.2.2. ((implicit implicit in �o (a) proves proves 10.2.2. Theorem. Theorem. in Paris Paris and and Wilkie Wilkie [1985] [1985])) If If IIA0(a) 'VVXl , " " , Xk) VXkO(Xl,..., Xk) ,, where where 8(XI O(Xl,... Ao(OL),, then then there there exists exists aa polynomial polynomial Xk) Ee �o(a) " " Xk) XI ..... . 'VXk8(Xb pp and and aa constant constant dd such such that that the the propositions propositions (8(XI (O(x~,... ~k have have Frege Frege proofs proofs Xk)) nl ,.....,nk ' . . . ,,xk))~ of of size size ::; <_p(nl p ( n l ,, .". . ," nk) nk) and and depth depth ::; <_ d. d. .•. Proof-sketch. P r o o f - s k e t c h . Suppose Suppose 'V Vx~...VxkO(x~,...,xk) is provable provable in in I�o(a) IA0(a), , let let XI . . . 'VXk8(Xb . . . , Xk) is n l ,, ..... ., n, nk k be be given. given. By By cut cut elimination elimination in in the the sequent sequent calculus calculus formalization formalization of of nl we have have a a free-cut-free free-cut-free proof proof of of this this sentence. sentence. From From this this proof proof we we get get a a proof proof IIA0(a), �o (a) , we of , ' . . , ak) (a) formulas. of the the sequent sequent --t -+ 8(al O(al,..., ak) which which contains contains only only �o A0(a) formulas. Starting Starting at at the the ' . . , ak), ak) , we rst order bottom, i.e., bottom, i.e., with with --t -+ 8(ab O(al,..., we shall shall gradually gradually translate translate the the fi first order proof proof into into a a propositional propositional proof. proof. The The structural structural and and propositional propositional rules rules are, are, of of course, course, translated translated identically. identically. Consider induction rule Consider an an instance instance of of the the induction rule A(b) A(b),, r F --t --+ � A,, A(S(b)) A(S(b)) , r A( A(O), --+ �, A, A(t) A(t) ' Q) , F --t where translated the where we we have have already already translated the part part of of the the proof proof from from the the lower lower sequent sequent on. on. Suppose Suppose that that in in the the course course of of translation translation we we have have assigned assigned numbers numbers mb m l , ... .. ., , mr mr to to the the free , . . . , bbr, (bl , . . . , bbr)))m, free variables variables of of the the lower lower sequent. sequent. Observe Observe that that (A(bl (A(bl,..., mrr Tl tt(bl,..., r ))) ml ,..... ... ,m is ' . . , br, bTl bbr+l)}m, mr) ' is equal equal to to (A(bb (A(bl,..., mr,m~+,, where mr mr+l is the the value value of of t(mb t ( m l , .'. .. ,. , mr). r+l))ml ,..... ... ,m + l is r,mr + l ' where We indices mb mr, m, m, m We take take mr mr translations translations of of the the upper upper sequent sequent with with indices m l , .". . " ,mr, m = (m stands free variable 00,, .. .. .., , mr mr - 11 (m stands for for the the free variable b). b). The The translation translation of of the the lower lower sequent sequent follows applying rr -- 11 cuts. follows from from them them by by applying cuts. The The quantifier quantifier rules rules for for bounded bounded quantifiers quantifiers are are treated treated similarly. similarly. Eventually Eventually we we reach reach initial initial sequents sequents which which are are translated translated to to initial initial sequents sequents in in propositional propositional logic. logic. o O This This theorem theorem can can be be used, used, as as mentioned mentioned above, above, to to construct construct short short bounded bounded depth depth Frege Frege proofs, proofs, but, but, what what is is more more interesting, interesting, also also to to prove, prove, for for instance instance that that the the �o (a) . pigeonhole pigeonhole principle principle for for a a free free second second order order variable variable a a is is not not provable provable in in IIA0(a). The model theory The first first proof proof of of this this independence, independence, by by Ajtai Ajtai [1994a] [1994a],, was was based based on on model theory and bound on propositional pigeonhole and a a lower lower bound on the the length length of of proofs proofs of of the the propositional pigeonhole principle principle was was derived derived as as aa corollary. corollary. Nowadays Nowadays it it is is clear clear that that the the right right and and simpler simpler way way is is to to prove prove the the lower lower bound bound for for propositional propositional logic logic first, first, see see Beame Beame et et al. al. [1992], [1992], Pitassi, Pitassi, Beame Beame and and Impagliazzo Impagliazzo [1993] [1993] and and Krajicek, Kraji6ek, Pudlak Pudls and and Woods Woods [1995] [1995].. Let Let us us mention mention by by passing passing another another parallel parallel with with computational computational complexity. complexity. The The results theories augmented results for for theories augmented with with an an extra extra free free second second order order variable variable are are alike alike to to 622 622 P. Pudlcik Pudldk the oracle oracle results results in in complexity complexity theory. theory. The The "absolute" "absolute" results, results, e.g. e.g. unprovability unprovability of of the the the pigeonhole pigeonhole principle principle for for �o-formulas A0-formulas in in I�o IA0,, are are beyond beyond present present means, means, as as well well as unrelativized unrelativized separation separation results results in in computational computational complexity complexity theory. theory. as 10.2.3. theories where 10.2.3. The The same same translation translation can can be be applied applied to to second second order order theories where we we log n) O(I) on have also second order order axioms. axioms. For For Uf Utt we we get get aa bound bound 22((l~176 on the the size size of of have also true true second I bound on Frege Frege proofs proofs (with (with aa (log (log n n)} O ~ ( ) bound on the the depth) depth);; for for Vl V11,, Krajicek Kraji~ek [1994b] [1994b] gives gives aa polynomial on the size of extension Frege polynomial bound bound on the size of extension Frege proofs. proofs. 10.3. translation we 10.3. Second Second translation. t r a n s l a t i o n . For For the the second second translation we consider consider the the language language L2 introduced by L2 of of the the theories theories 32 $2 and and T2 T2 introduced by Buss Buss [1986], [1986], see see Chapter Chapter II. II. This This language language extends extends Lo L0 by by Lx L x// 22JJ ,, x x ##yy, , Ixl. The interpretation interpretation of of these these function function symbols symbols is is Ix! - The obtain faster Ix[ = = f!Og [log2(x x # yy = = 2Ixl·lvl 21xl'lul.. The The # # function function is is used used to to obtain faster growth growth Ixl ) 1 , x# 2 (X ++ 11)], (log x) , Pp aa polynomial. rate terms, namely rate of of terms, namely 2P 2p(l~ polynomial. This This means means that that the the lengths lengths of of the the numbers numbers increase increase polynomially, polynomially, which which renders renders formalization formalization of of polynomial polynomial time time computations possible. The computations possible. The Ixl Ix I function function is is used used to to define define sharply sharply bounded bounded quantifiers quantifiers w:x � < Itl, \/ 3x � < IItl1,, where only polynomially where tt is is aa term. term. The The basic basic property property is is that that there there are are only polynomially many many t I , since elements elements x x less less than than or or equal equal than than I[tl, since the the outermost outermost function function in in this this term term is, is, essentially, essentially, the the logarithm. logarithm. The The class class II� II~ consists consists of of formulas formulas of of L L22 which which contain contain only only sharply sharply bounded bounded quantifiers quantifiers and and strong strong bounded bounded quantifiers quantifiers (positive (positive occurrences occurrences of of universal universal bounded bounded and and negative negative occurrences occurrences of of existential existential bounded); bounded); the the other other classes classes II�, I1~, E� ~ are are defined defined similarly. similarly. ( X l , . . . , Xk} We propositional translations We want want to to define define propositional translations of of aa m II~ formula formula rp ~O(Xl,..., Xk).. The The ... ,n translation will be translation will be denoted denoted by by [[~O(Xl,...,Xk)]n, n~" Now we we index index the the translation translation rp (X l , . , xk}] n l ,..... k ' Now with ally with strings strings of of integers integers again, again, but but the the meaning meaning is is that that we we express express proposition propositionally XI I � . . . ,,Xk) Xk} holds l , . . . ,,xk Xk with that that the the sentence sentence rp(Xl, ~O(Xl,... holds for for all all x Xl,... with IIXll < nl, n ~ , .. ... ., , IXkl [Xkl � <_ nk nk.' The intuition behind The intuition behind the the translation translation is is the the following. following. We We identify identify truth truth assignments assignments with (binary representations Since the with (binary representations of) of) numbers. numbers. Since the terms terms are are polynomial polynomial time time computable functions, computable functions, we we can can express express atomic atomic first first order order formulas formulas by by polynomial polynomial size size propositions. propositions. Sharply Sharply bounded bounded quantifiers quantifiers are are translated translated to to polynomial polynomial size size disjunctions disjunctions and and conjunctions. conjunctions. The The strong strong bounded bounded quantifiers quantifiers are are represented represented by by sequences sequences of of propositional propositional variables; variables; this this is is aa correct correct interpretation, interpretation, since, since, by by for all definition, definition, aa propositional propositional tautology tautology must must be be satisfied satisfied/or all truth truth assignments. assignments. A A formal formal definition definition is is fairly fairly involved, involved, thus thus most most authors authors do do not not give give aa full full definition, definition, and and we we shall shall also also only only sketch sketch how how to to resolve resolve some some technical technical problems problems of of the the definition. definition. First First consider consider an an atomic atomic formula, formula, say, say, with with only only one one free free variable, variable, s(x) - tt(x). s(x} = (x} . Let be given ally that Let n n be given for for which which we we want want to to express express proposition propositionally that the the sentence sentence holds for propositional variables s(x) = = t(x} t(x) holds for all all x x with with Ixl Ixl � _ n n.. Ideally Ideally we we would would take take propositional variables s(x} PP = (p), T; (p} , ii = - ((Pl p l ,, .· .·. ,· p, Pn) n ) representing representing such such numbers numbers and and formulas formulas 0'; a,(p), Ti(P), -- 1, 1 , .. .. .., , m m,, " The Lengths of of Proofs 623 623 = nn~O ( l ) ) , representing representing the the bits bits of of s(x) s(x) resp. resp. t(x) t(x),, and and defi define ne ((m m= [s(x ) = = t(x)] n =dI 1\ A ao,(P) - TT,(p). j (p). j (p) = ii=l,...,rn = l ,... ,m There are are such such formulas formulas of of polynomial polynomial size size for for each each of of the the basic basic functions, functions, hence hence There by composing composing them them we we get get polynomial polynomial size size formulas formulas for for all all terms. terms. Probably Probably one one by can use use these these formulas, formulas, but but it it would would require require to to find find short short extension extension Frege Frege proofs proofs can of basic basic properties properties of of these these functions, functions, which which is is by by no no means means obvious obvious for for such such aa of formalization. Therefore, Therefore, instead instead of of it, it, we we take take the the natural natural circuits circuits for for the the functions functions formalization. and introduce introduce propositional propositional variables variables for for the the functions functions computed computed at at the the vertices vertices of of and the circuits. circuits. Then Then the the translation translation will will be be an an implication implication with with the the antecedent antecedent being being the the conjunction conjunction of of simple simple clauses clauses relating relating the the values values of of the the vertices vertices of of the the circuits circuits and and the consequent being being consequent 1\ A qqij = ~ rri,j , ii=l,...,n = l ,... ,n where qi and and rrii are are the the propositional propositional variables variables for for the the outputs outputs of of the the circuits for s(x) s(x) where circuits for and t(x) t(x) respectively. respectively. Thus Thus the the translation translation will will have have aa polynomial polynomial number number of of extra extra and variables which which do do not not code code bits bits of of the the numbers numbers representing the free free variables variables of of variables representing the the first first order order formula. formula. For For such such a a formalization formalization it it is is much much easier easier to to prove prove the the basic basic the properties of of the the translation translation properties As explained explained above, above, the the strong strong bounded bounded quantifiers quantifiers are are simply simply omitted, omitted, (except (except As that the the bounds bounds on on the the variables variables are are left left as as aa part part of of the the formula formula)) and and the the sharply sharply that bounded quantifi quantifiers are translated translated using using disjunctions disjunctions and and conjunctions. conjunctions. Consider Consider bounded ers are for instance instance a a formula formula O starting with aa sharply sharply bounded bounded quantifier quantifier followed followed by by aa starting with for universal bounded quantifier, say universal bounded quantifi er, say y, z), It(x)IVz s(x , y) y) ~(x, z) , I Vz ~:S s(x, cp (x , y, 33yy <:S It(x) where cp is an an open open formula. formula. We want to to define define the the translation translation [ replace where qo is We want [O]n" We first first replace ] n ' We the quantified variable by the numerical instances, then translate translate the numerical instances, and and then the quantified variable yy by (i :S __ It(x) It(z)[ l -+ cp (x, i, z,)), Zi )) , V V i=O ..... n) 1 ,···,lt( i--o It(n)l where Zl t(n) 1 are are new new distinct distinct variables. variables. where z0,..., Zo , . . . , Zlt(n)l After it should should not difficult for for the the reader to go go on and handle handle After this this example example it not be be difficult reader to on and more complex complex cases. cases. more 10.3.1. 82 is aa theory theory based on aa finite finite number number of of basic basic open open axioms with induction induction 10.3.1. S~ is based on axioms with for bounded for bounded formulas formulas of of the the form form cp(Q) ^1\ Vx( cp ( Lx/2J ) -+ cp(x)) --+, Vxcp (x) . The most most important important fragment fragment of of bounded bounded arithmetic arithmetic 82 is the the theory theory Sz 8i1 where where The $2 is the the induction induction schema schema is is restricted restricted to to E~ E� formulas. formulas. This This theory theory is is adequate adequate for for formalization formalization of of polynomial polynomial time time computations, computations, see see Buss Buss [1986]. [1986] . Furthermore Furthermore itit is is related to Frege proof related to extension extension Frege proof systems: systems: P. P. Pudl6k Pudldk 624 624 . . . ,,Xk), Xk), Buss [1986]) Si proves 10.3.2. Theorem. T h e o r e m . (Cook (Cook [1975], [1975], Buss [1986]) If IfS~ proves VXl VXl ..... . VXkCP(Xb VXk~(Xl,... 10.3.2. where . . . , Xk) where CP(Xb ~p(Xl,..., Xk) E E ITt lib,, then then there there exists exists aa polynomial polynomial pp such such that that the the propositions propositions 0 [CP(Xb ... ,nk [(/9(X1,..., Xk)]nln l ,..... nk have have extension extension Frege Frege proofs proofs of of size size ::; < p(nb p ( n l , .. ... ., , nk) nk).. [] ' . . , xk)] The proof of similar to The proof of this this theorem theorem is is similar to the the proof proof of of Theorem Theorem 10.2.2, 10.2.2, but but much much more involved more involved due due to to the the difficulties difficulties with with the the basic basic axioms. axioms. 10.3.3. 10.3.3. This This theorem theorem naturally naturally rises rises the the question: question: is is extension extension Frege Frege proof proof system system the know; it the weakest weakest system system for for which which we we can can prove prove this this theorem? theorem? We We do do not not know; it is is possible possible that that one one can can construct construct some some pathological pathological counterexample, counterexample, but but there there is is another another reason reason for for associating associating extension extension Frege Frege systems systems with with Si S~,, which which we we shall shall consider consider next. next. Following Following Krajicek Krajf6ek and and Pudllik Pudls [1990] [1990],, we we shall shall define define aa natural natural relation relation between between theories theories and and propositional propositional proof proof systems. systems. 10.3.4. 10.3.4. Definition. Definition. (1) (1) For For aa propositional propositional proof proof system system P P we we denote denote by by RF R F NN(P) (P) (the (the reflection reflection principle principle for for P) P) the the VITt VII~ sentence sentence Vd, u((d u((d:: P P fF- u) u) -+ -~ Taut(u)), Taut(u)), Vd, where Taut(x) Taut(x) is is aa ITt lib formula formula defi defining the set set of of propositional propositional tautologies. tautologies. Note Note where ning the that that dd : 9P P f~ u u (d is aa P P proof proof of of aa proposition can be be written written as as aa �t 2b1 formula, formula, (d is proposition u) u) can since since it it is is aa polynomial polynomial time time computable computable predicate. predicate. T, if cp(x) E ITt (2) A (2) A propositional propositional proof proof system system P P simulates simulates aa theory theory T, if for for every every ~p(x) II~ T fF- Vxcp(x) Vx~(x) T => =~ S� [CP(x)] lyl ) ' S~ fF- Vy3d Vy3d (d (d": P P fF-[~(X)llyl). T, if (3) A (3) A propositional propositional proof proof system system P P is is associated associated to to aa theory theory T, if P P simulates simulates T T and and T T fF- RFN(P) RFN(P).. Probably in (2) you Probably in (2) you expected expected rather rather aa statement statement like like in in Theorems Theorems 10.2.2 10.2.2 and and 10.3.2. condition (2) (2) is 10.3.2. In In fact fact the the condition is stronger: stronger: by by Buss's Buss's Theorem Theorem 11.3.2, II.3.2, the the provability provability of of such such aa IT II22 statement statement in in Si S~ implies implies that that it it can can be be witnessed witnessed by by aa polynomial polynomial time time computable computable function. function. Thus, Thus, in in particular, particular, the the P P proofs proofs of of [cp(x)] [~(x)]n's must be be of of n 's must polynomial (2) means there is polynomial size. size. So So (2) means that that there is aa polynomial polynomial bound bound on on the the lengths lengths of of P P proofs proofs of of [cp(x)] [~(x)]n's provably in in aa weak weak theory. theory. n 's provably Let us is equivalent the consistency Let us also also note note that that RF R F NN(P) ( P ) is equivalent to to the consistency of of P P assuming assuming some "mild conditions" conditions" on some "mild on P. P. We We shall shall denote denote by by G G the the quantified quantified propositional propositional proof proof system system based based on on the the sequent calculus, see sequent calculus, see 8.8. 8.8. Let Let G; Gi denote denote the the subsystem subsystem of of G G obtained obtained by by imposing imposing the the restriction restriction of of at at most most ii alternations alternations of of quantifiers quantifiers in in each each formula formula of of aa proof. proof. Let Let G; G* denote denote G; Gi where where we we allow allow only only tree-like tree-like proofs. proofs. The The following following theorem theorem gives gives some some known known pairs pairs of of aa proof proof system system associated associated to to aa theory theory (for (for definitions definitions of of the the theories theories see see Chapter Chapter II) II).. 625 625 The The Lengths Lengths of of Proofs Proofs 10.3.5. ((Cook Cook [1975] 10.3.5. Theorem. Theorem. [1975],, Krajicek Kraji~ek and and Takeuti Takeuti [1990] [1990],, Krajicek KrajiSek and and Pudlak Pudls [1990]) [1990]) The The following following are are pairs pairs of of aa theory theory and and aa proof proof system system associated associated to to it: it: 0[] T� , Gi) (S~,, extension extension Frege) Frege),, (S�, (S~, Gn G*) for .for ii � >_11,, ((T~, G,) for for ii � >_11,, (Ui (U~,, G) G).. (Si Note Note that that for for Ui U~ we we have have two two related related systems, systems, depending depending on on which which translation translation we take. take. Further Further results results of of this this type type were were proved proved in in Clote Clote [1992] [1992].. we Next Next theorem theorem shows shows that that under under reasonable reasonable conditions conditions the the associated associated propositional propositional proof system system is is determined determined up up to to polynomial polynomial simulation. simulation. proof 10.3.6. ((Krajf~ek Krajicek and 10.3.6. Theorem. Theorem. and Pudlak Pudl~k [1990]) [1990]) Let Let P P be be aa propositional propositional proof proof system associated associated to to aa theory theory T. T. Suppose Suppose T T contains contains Si S 1 and and the the following following is is provable provable system in in Si S~ :: P P simulates simulates extension extension Frege Frege systems systems and and it it is is closed closed under under modus modus ponens. ponens. Then P P polynomially polynomially simulates simulates any any propositional propositional proof proof system system for for which which T T proves proves the the Then reflection principle. principle. reflection Thus, e.g. by Thus, e.g. by Theorem Theorem 10.3.5, 10.3.5, extension extension Frege Frege systems systems and and Gi G~ are are polynomially polynomially equivalent. equivalent. Proof. Proof. Suppose Suppose T T fF RFN(Q) R F N ( Q ) . . Let Let PQ(x, pQ(x, y) y) be be the the ITt II~ formula formula which which defines defines the the reflection principle, principle, i.e., i.e., reflection pq(d, pQ(d, u) u) == -- dd:: Q Q fFu u -t -+ Taut(u). Taut(u). By By the the assumptions assumptions Si S~ fF Vz(P Vz(P fF [PQ(x, [pQ(x, y)l Y)]z)" We now now we we argue argue in in the the theory theory Si S 1.. z ) . We Thus we we have have Thus P P fF [x [x:: Q Q fF yy -t --+ Taut(y)] Taut(y)]~.z . is [x: [x : Q Q Ff- y]z [Taut(y)] z and and PP is closed under under Since Ix: [x : Q f- y Taut(y)]zz is yl z -t Since Q F y -t --+ Taut(y)] ~ [Taut(y)]~ is closed modus ponens, we we get get modus ponens, PFf-[ x[x: Q: FQyf-] ~yl z P We have also also We have -t ~ P f- [Taut(y)l z . PF[Taut(y)]~. P f-F [Taut(y)]z [Taut(y)l z P -t --~ P P fF y, y, P Ff- [x P [x :: Q Q Ff- y]z yL -+ -t P P Ff- y. y. since itit is true already already for for extension extension Frege Frege systems systems (we (we leave leave this this claim claim without without aa since is true proof). have obtained Thus we we have obtained in in S~ Si proof) . Thus witnessing theorem theorem itit means means that that one one can can construct construct Back in in the the real real world, world, by Buss 's witnessing Back by Buss's in in polynomial polynomial time time aa proof proof of of ~cp in in PP from from aa proof proof of of [d: [d : Q Q Ff- ~P]n in P. P. CP] n in Now suppose suppose that that we we are are given given aa proof proof dd of of ~cp in in Q. Q. Substituting Substituting the the numbers numbers Now which encode encode dd and and ~cp we we get get aa true true variable-free variable-free propositional.formula propositional formula [d [d ": QQ Ff- :t:l which ~]n. n. formulas always always have have polynomial polynomial size size proofs proofs even even in in aa Frege Frege system. system. Thuswe Thus we Such formulas Such get [] 0 proof of of ~p cp in in polynomial polynomial time. time. get aa PP proof The meaning meaning of of this this theorem theorem isis that that the the proof proof system system associated associated to to aa theory theory TT is, is, The from from the the point point of of view view of of T, T, the the strongest strongest proof proof system, system, i.e., i.e., stronger stronger systems systems may may be be inconsistent. inconsistent. Let Let us us state state itit formally: formally: 626 626 P. PudlO.k Pudldk 10.3.7. 10.3.7. Corollary. Corollary. Under the the same assumptions as as in in Theorem Theorem 10. 10.3.6, if T T fiUnder same assumptions 3. 6, if NP AlP = coNP, coAf P, then then P P is is polynomially polynomially bounded. bounded. Proof. Since the complete, the Proof. Since the set set of of propositional propositional tautologies tautologies is is coNP coAfT)-complete, the assumption assumption T T f~ NP AlP = = coNP coAf7~ means means that that - T T fF- 'v'x(a(x) Vx(a(x) == - Taut(x)), Taut(x)), (59) (59) for nes aa polynomially for some some a(x) a(x) E E I;� E b.. So So aa defi defines polynomially bounded bounded propositional propositional proof proof system system bounded quantifi ers) . The Q (proofs (proofs are are the the witnesses witnesses for for the the existential existential bounded quantifiers). The sentence sentence Q (59) implies implies T Hence, by (59) T fF- RF R F NN(Q) ( Q ) . . Hence, by Theorem Theorem 10.3.6, 10.3.6, P P polynomially polynomially simulates simulates Q. Q. 0 But But if if Q Q is is polynomially polynomially bounded, bounded, then then also also P P must must be. be. El As corollary will will be As we we believe believe that that NP AfP =I 5r coNP, coA/'7), we we expect expect that that the the corollary be used used in in the the contrapositive contrapositive form. form. Let Let us us state state the the nicest nicest special special case case of of it it (proved (proved directly directly by Wilkie Wilkie in in 1987, 1987, unpublished; unpublished; as as observed observed in in Krajicek Krajfhek and and Pudllik Pudl~k [1989] [1989] it it also also by follows follows from from results results of of Cook Cook [1975] [1975] and and Buss Buss [1986]). [1986]). 10.3.8. If 10.3.8. Corollary. Corollary. If extension extension Frege Frege proofs proofs are are not not polynomially polynomially bounded, bounded, then then 0 S~ does does not 3/'7~ = : coNP coAf ~ . . [] not prove prove NP Si 10.4. Optimal The 10.4. O p t i m a l proof p r o o f systems s y s t e m s and and consistency consistency statements. statements. The second second translation can link between translation can be be used used to to show show aa link between aa fundamental fundamental problem problem about about the the lengths of of proofs proofs of of fi finite consistency statements statements and and the the existence existence of of an an optimal optimal lengths nite consistency propositional proof propositional proof system. system. Furthermore Furthermore there there is is aa statement statement from from structural structural complexity problems. A I} * is complexity theory theory which which is is equivalent equivalent to to these these problems. A set set Y Y � C_ {O, {0, 1}* is n is bounded bounded by called sparse, called sparse, if if for for every every n n,, the the size size of of Y Yn gl {O, {0, l} 1}n is by aa polynomial. polynomial. 10.4.1. 10.4.1. Theorem. T h e o r e m . (Krajicek (grajfhek and and Pudllik Pudls [1989]) [1989]) The The following following are are equivalent: equivalent: 1. There exists exists aa consistent consistent finitely finitely axiomatized axiomatized theory theory T T 2 D_Si S 1 such such that that for for every every 1. There consistent finitely consistent finitely axiomatized axiomatized theory theory S S IICons( )IIT = n ~ 2. 2. There There exists exists an an optimal optimal propositional propositional proof proof system, system, i.e., i.e., aa propositional propositional proof proof propositional proof system system P P such such that that for for every every propositional proof system system Q Q II lIQ- li il for for every every tautology tautology cp. ~o. 3. -set X 3. For For every every coNP co.hf P-set X there there exists exists aa nondeterministic nondeterministic Turing Turing machine machine which which accepts and uses polynomial time subset Y accepts X X and uses only only polynomial time on on every every sparse sparse subset Y � CX X ,, Y Y E EP 7).. The 1. and 2. is The proof proof of of the the equivalence equivalence of of 1. and 2. is based based on on the the following following two two construc constructions. If tions. If T T is is an an optimal optimal theory theory in in the the sense sense of of 1., 1., we we take take aa propositional propositional proof proof system defined by: system P P defined by: dd": P df dd": T P f~ cp ~o == --dl T fF- Taut( Taut(~_). if)' Lengths of of Proofs The Lengths 627 627 If, on on the the other other hand, hand, P P is is an an optimal optimal propositional propositional proof proof system, system, we we take take the the theory theory If, T defined defined by: by: T T =df =~f S� S1 + + RFN(P). RFN(P). T D[Z] We omit omit the the rest rest of of the the proof. proof. We Given Given aa propositional propositional proof proof system system P P which which is is not not polynomially polynomially bounded, bounded, we we can can produce, produce, using using this this theorem, theorem, aa sequence sequence of of tautologies tautologies which which surely surely do do not not have have polynomial size size proofs proofs in in P. P. Unfortunately, Unfortunately, the the tautologies tautologies will will be be rather rather complex complex polynomial artificial statements, statements, thus thus not not amenable amenable to to aa combinatorial combinatorial analysis. analysis. However, However, as as artificial noted by by Krajicek Krajf~ek [1995], [1995], one one can can use use the the polynomial reductions, by by which which NP Af:P noted polynomial reductions, completeness results results are are proved, proved, to to turn turn these these tautologies tautologies into into simple simple combinatorial combinatorial completeness statements. For For instance instance one one can can construct construct aa sequence sequence of of nonhamiltonian nonhamiltonian graphs, graphs, statements. such that that there there are are no no polynomial polynomial size size proofs proofs in in P P of of the the tautologies tautologies expressing expressing such that the the graphs graphs are are nonhamiltonian. nonhamiltonian. Thus Thus the the problem problem reduces reduces to to finding finding aa class class that of nonhamiltonian nonhamiltonian graphs graphs for for which which it it is is difficult difficult to to prove prove in in P P that that they they are are of nonhamiltonian. nonhamiltonian. 11. Bibliographical r e m a r k s for f o r further f u r t h e r reading reading 1 1 . Bibliographical remarks In this this section section we we shall shall give give aa few few more more references references which which have have not not been been mentioned mentioned In in text. This in the the main main text. This should should serve serve to to the the reader reader who who is is interested interested in in the the history history of of the subject subject or or who who wants wants to to learn learn more more about about it. it. Our Our aim aim is is not not to to complete complete the the the list of of references references about about results results on on the the lengths lengths of of proofs, proofs, rather rather we we want want to to partially partially list complement the the above above presentation presentation which which concentrated concentrated on on methods methods used used in in this this complement research research area. area. Thus, Thus, in in particular, particular, we we shall shall not not repeat repeat results results described described above. above. Probably oldest recorded recorded paper paper on the subject subject is is GSdel Probably the the oldest on the Godel [1936]. [1936]. In In this this two-page abstract stated the the result that there the lengths lengths two-page abstract he he stated result that there is is aa speed-up speed-up between between the of proofs of formulas order and § 1-st order arithmetics. To quote quote 1-st order arithmetics. To of proofs of formulas provable provable in in i-th i-th order and ii + him: The transition to the logic of of the results in transition to the logic the next next higher higher type type not not only only results in certain certain him: The previously unprovable unprovable propositions provable, but it becoming becoming possible possible previously propositions becoming becoming provable, but also also in in it to the proofs proofs already The length to shorten shorten extraordinarily extraordinarily infinitely infinitely many many of of the already available. available. The length of proofs is considered considered to steps and speed-up is r for any of proofs is to be be the the number number of of steps and the the speed-up is ¢>(n) for any function ¢>r "computable" "computable" in in the lower system. system. There no proof proof given paper. function the lower There was was no given in in the the paper. For aa full statement see see Buss [1994]. For full proof proof of of this this statement Buss [1994]. Another important important writing Godel which which was was discovered only aa few Another writing of of GSdel discovered only few years years ago, ago, is is the the letter letter by by GSdel Godel [1993]. [1993]. In In that that letter letter he he posed posed the the question question whether whether one one can can decide in in linear, linear, quadratic, quadratic, etc. etc. time time in in nn whether whether aa given given formula formula has has aa proof proof of of decide length (= (= number number of of symbols) symbols) n. n. Now Now we we know know that that this this problem problem is is Af:P-complete. NP-complete. length See Buss Buss [1995a] [1995a] for for aa discussion discussion and and aa proof proof of of an an unproven claim of of GSdel. Godel. See unproven claim Looking at at the the literature literature itit seems seems that that the the subject subject lay dormant for for several several decades. decades. Looking lay dormant think that that many many people people thought thought about about problems problems on on the the lengths lengths of of proofs, but the the II think proofs, but things that that they they actually could prove prove did did not not look look interesting interesting enough, enough, especially especially when when things actually could compared with with other other fancy fancy topics topics like like set set theory. theory. Furthermore some basic basic concepts concepts compared Furthermore some were missing (one (one of of such such crucial crucial things things was the distinction distinction between polynomial size size were missing was the between polynomial 628 628 P. Pudldk Pudlak P. and exponential exponential size). size) . This This can can be be documented documented by by aa remark remark of of Kreisel Kreisel [1967,page [1967,page and 241] , who who mentions mentions aa conversation conversation with with GSdel G6del where where GSdel G6del asked asked the the question question of of 241], what are are the the lengths lengths of of proofs proofs of of finite finite consistency consistency statements. statements. No No paper paper had had been been what written about about itit until until Friedman Friedman [1979], [1979], but but he he did did not not consider consider itit to to be be worth worth written publishing. publishing. At the the early early stages, stages, Georg Georg Kreisel Kreisel was was one one of of the the main main proponents proponents of of this this field. field. At His student student Statman Statman [1978] [1978] determined determined the the increase increase of of the the lengths lengths of of proofs proofs in in His cut-elimination and and Herbrand's Herbrand ' s Theorem. Theorem. Another Another of of his his students, students, Baaz Baaz (see (see Baaz Baaz cut-elimination and Pudl~k Pudhik [1993], [1993], Baaz Baaz and and Zach Zach [1995]), [1995]), made made significant significant progress progress in in Kreisel's Kreisel's and Conjecture. As seen seen on on Kreisel's Kreisel's Conjecture, Conjecture, Kreisel Kreisel was more interested interested in in positive positive Conjecture. As was more results in in the the sense sense of of deriving deriving more more information information from the proofs proofs than than just just the the mere mere results from the fact that the the statement statement is is true. true. Logic Logic should should help help mathematicians mathematicians to to get get more more or or fact that better results, results, rather rather than than only only to to show show impossibilities impossibilities of of certain certain proofs, see e.g. e.g. better proofs, see Kreisel [1990]. [1990]. From From this of view, of the the greatest greatest successes successes in in proof proof theory theory Kreisel this point point of view, one one of was the Luckhardt [1989], [1989], deriving deriving explicit explicit bounds bounds on was the result result of of Luckhardt on approximation approximation of of algebraic numbers by by rational rational numbers numbers (Roth's (Roth's theorem), using Herbrand's Herbrand ' s theorem. algebraic numbers theorem), using theorem. Originally in the lengths of of proofs mainly on Originally the the interest interest in the lengths proofs was was based based mainly on philosophical philosophical and With the new practical and methodological methodological considerations. considerations. With the advent advent of of computers computers aa new practical proving. The The main main tool tool in in automated reason reason appeared: appeared: automated automated theorem theorem proving. automated theorem theorem proving for first first order see e.g. e.g. Chang Lee [1973]. proving is is the the resolution resolution system system for order logic, logic, see Chang and and Lee [1973]. For For us, us, theoreticians, theoreticians, most most of of the the papers papers are are too too much much applied, applied, however however there there are are several several results results which which are are important important also also for for theory. theory. Such Such aa notable notable result result is is the exponential lower bound for propositional regular resolution of the exponential lower bound for propositional regular resolution of Tsejtin Tsejtin [1968]. [1968]. The The question question about about the the efficiency efficiency of of proof-search proof-search strategies strategies are are often often nontrivial nontrivial mathematical mathematical problems, problems, let let us us mention mention at at least least some some results results of of this this type type Baaz Baaz and and Leitsch Leitsch [1992,1994]. [1992,1994]. There There are are several several books books about about the the complexity complexity of of logical logical calculi, calculi, e.g. e.g. Eder Eder [1992]; [1992]; they they deal deal mainly mainly with with the the first first order order logic. logic. The The next next important important stimulus stimulus was was the the rise rise of of complexity complexity theory. theory. The The lengths lengths of of proofs proofs is is just just one one of of several several research research areas areas which which combine combine logic logic and and complexity complexity theory. theory. Another one, which Another one, which is is closely closely related related to to it, it, is is the the complexity complexity of of logical logical theories. theories. The The problem problem is is how how efficiently efficiently can can we we decide decide if if aa sentence sentence is is provable provable in in aa given given decidable decidable theory theory T T (e.g., (e.g., Presburger Presburger arithmetic). arithmetic). Note Note that that an an upper upper bound bound on on the the lengths lengths of of proofs proofs in in T T gives gives an an upper upper bound bound on on aa nondeterministic nondeterministic procedure procedure for for decidability. decidability. Often Often this this bound bound is is not not very very far far from from the the best. best. We We refer refer the the reader reader to to the the surveys surveys Rabin Rabin [1977] [1977] and and Compton Compton and and Henson nenson [1990]. [1990]. We We can can say say that that the the research research into into complexity complexity of of proofs proofs really really started started with with the the seminal seminal paper paper of of Parikh Parikh [1971] [1971] which which introduced introduced several several important important concepts concepts and and proved proved basic basic results results about about them: them: speed-up speed-up for ]or first first order order theories, theories, theories theories which which are are inconsistent inconsistent but but are are consistent consistent for .for practical practical purposes, purposes, and and bounded bounded arithmetic. arithmetic. Soon Soon after after it, it, he he published published aa basic basic result result on on Kreisel's Kreisel's Conjecture Conjecture in in Parikh Parikh [1973]. [1973]. He He proved proved that that the the conjecture conjecture is is true, true, if if we we take take Peano Peano arithmetic arithmetic with with + + and and xx as as ternary ternary relations relations instead instead of of function function symbols. symbols. That That proof proof has has been been aa paradigm paradigm for for ' s Conjecture. all all subsequent subsequent proofs proofs of of instances instances of of Kreisel Kreisel's Conjecture. After After that that several several people people started started to to work work on on these these subjects. subjects. One One of of the the most most The The Lengths Lengths of of Proofs Proofs 629 629 influential researchers researchers in in this this fi field has been been Orevkov. Orevkov. We We shall shall mention mention only only the the influential eld has most important important papers papers of of the the many many that that he he published. published. Orevkov Orevkov [1982] [1982] gave gave aa most different proof proof of of the the lower lower bounds bounds on on the the lengthening lengthening of of proofs proofs in in cut-elimination cut-elimination different and Orevkov Orevkov [1986] [1986] gave gave more more precise precise upper upper bounds. bounds. Orevkov Orevkov [1987b] [1987b] introduced introduced and explicitly the the concept concept of of the the skeleton skeleton and and Orevkov Orevkov [1987a] [1987a] proved proved several several results results explicitly ' s Conjecture. related to to Kreisel Kreisel's Conjecture. All All these these results, results, and and many many more, more, are are covered covered in in related Orevkov [1993] [1993].. Orevkov There are are more more results results on on the the complexity complexity of of first first order order proofs. proofs. Of Of those those that that There we have have not not presented presented yet, yet, let let us us mention mention the the dissertation dissertation of of Ignjatovic Ignjatovid [1990]. [1990]. He He we proved aa nonelementary nonelementary speed speed up up between between Primitive Primitive Recursive Recursive Arithmetic Arithmetic and and /I:.o I~0.. proved Currently the the most most active active area area is is propositional logic and and bounded bounded arithmetic. arithmetic. Currently propositional logic The fundamental fundamental paper paper is is Cook Cook [1975] [1975],, where where aa relation relation of of the the lengths lengths of of proofs proofs in in The propositional logic logic and and provability provability in in arithmetic arithmetic was was considered considered for for the the first first time. time. propositional The most most influential influential papers papers in in bounded bounded arithmetic arithmetic after after Parikh Parikh [1971] [1971] were were written written The by Paris Paris and and Wilkie; Wilkie; let let us us mentioned mentioned at at least least the the Paris Paris and and Wilkie Wilkie [1985] [1985] paper paper by on counting counting problems problems which which influenced influenced very very much much research research on on the the complexity complexity of of on propositional logic. logic. The The basic book on on bounded bounded arithmetic arithmetic is is due due to to Buss Buss [1986] [1986].. propositional basic book Another fundamental paper is by by Ajtai Ajtai [1994a], [1994a], where where he he introduced introduced the the method method Another fundamental paper is of random random restrictions restrictions into into propositional propositional logic, logic, which which had had already already been been used used in in of complexity theory. theory. This This development development has has been been partially partially described described in in this this chapter chapter complexity and also also in in Chapter Chapter II; much more more can can be be found found in in the the monograph by Krajfbek and II; much monograph by Krajicek [1995],, which which covers covers the the whole whole area area in in detail detail except except for for the the most most recent recent results. As [1995] results. As this being finalized, obtained on this manuscript manuscript is is being finalized, new new exciting exciting results results are are being being obtained on the the polynomial calculus calculus by by Razborov Razborov [n.d.] [n.d.], Krajicek [1997b] and and Riis Riis and and Sitharam Sitharam polynomial ' Kraji~ek [1997b] [1997].. [1997] A cknowledgments Acknowledgments II would would like Sam Buss helping me with the the preparation preparation of the like to to thank thank Sam Buss for for helping me with of the manuscript and suggesting and Jan Jan Kraji~ek for checking checking the the manuscript and suggesting several several improvements improvements and Krajicek for manuscript. article was was supported supported by grant #A1019602 manuscript. The The preparation preparation of of the the article by grant #A10l9602 of Academy of Sciences of of the Czech Republic Republic and the cooperative cooperative research of the the Academy of Sciences the Czech and the research grant INT-9600919/ME-103 of the the U.S. U.S. National National Science Science Foundation Foundation and and the the Czech Czech grant INT-9600919/ME-103 of Republic Ministry Ministry of Republic of Education. Education. R eferences References M. AJTAI AJTAI [1990] pigeonhole principle, in: Feasible Feasible Mathematics: Mathematics: A A Mathematical Mathematical Sciences [1990] Parity and the pigeonhole Institute Workshop Workshop held in Ithaca, New New York, June June 1989, 1989, S. R. Buss Buss and P. J. Scott, Institute eds., Birkh~iuser, Birkhauser, Boston, Boston, pp. 1-24. Combinatorica, 14, pp. 417-433. 417-433. 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The Lengths Lengths of of Proofs Proofs The 637 637 R. L. L . VAUGHT VAUGHT R. On axiomatizability axiomatizability by by aa schema, schema, Journal of Symbolic Logic, Logic, 32, 32, pp. pp. 473-479. 473-479. [1967] [1967] On A. C.-C. C.-C. YAO YAO A. Separating the the polynomial polynomial time time hierarchy hierarchy by by oracles, oracles, in: in: Proceedings Proceedings of £6th Annual Annual [1985] of the 26th [1985] Separating IEEE Computer Computer Society, Society, Piscataway, Piscataway, Symposium on Foundations of Computer Science, IEEE New Jersey, Jersey, pp. pp. 1-10. 1-10. New This Page Intentionally Left Blank CHAPTER CHAPTER IX IX A Proof-Theoretic Proof-Theoretic Fr Framework A amework for Logic Progr Programming Logic amming Gerhard J~iger Ger hard Jager fiir Informatik 'lind und angewandte Mathematik, Universitiit UniversitSt Bern Institut fUr Neubriickstrasse 110, Neubruckstrasse 0, CH-3012 Bern, Switzerland jaeger@iam, unibe, ch [email protected] Robert F. Sts Robert F. Stark ]iir Informatik, Universitiit Universit~t Freiburg Institut fUr Rue Faucigny 2, CH-1 CH-1700 700 Fribourg, Switzerland robert. [email protected] robert, staerk @unifr. ch Contents Contents 1. IIntroduction 1. ntroduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2. Basic Basic notions notions .. .. .. .. .. .. . . . . . . . . . . . . .. .. .. .. .. .. .. .. .. . . . . . . . . . . . . . . . 3. Some Some model-theoretic model-theoretic properties properties of of logic logic programs p r o g r a m s .. . . . . . . . . . . . . . . . . 4. Deductive Deductive systems systems for for logic logic programs p r o g r a m s .. . . . . . . . . . . . . . . . . . . . . . . . 5. SLDNF-resolution S L D N F - r e s o l u t i o n .. .. .. .. .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . P a r t i a l i t y in in logic logic programming p r o g r a m m i n g .. . . . . . . . . . . . . . . . . . . . . . . . . . . 6. Partiality 7. 7. Concluding Concluding remark remark . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References References .. .. .. .. .. .. . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . H A N D B O O K OF O F PPROOF R O O F THEORY THEORY HANDBOOK E d i t e d by by S. S. R. R. Buss Buss Edited © 1998 Elsevier 9 1998 Elsevier Science Science B.V. B.V. All All rights rights reserved reserved 640 641 650 655 661 672 678 679 640 640 G. J~iger Jager and R. St~irk Stark G. 1 . IIntroduction 1. ntroduction The purpose purpose of of this this article article isis to to present present one one specific specific proof-theoretic proof-theoretic framework framework The for for first first order order logic logic programming, programming, but but of of course course itit is is not not claimed claimed that that our our approach approach is the the only only possible possible one. one. However, However, we we hope hope to to succeed succeed in in providing providing aa perspicuous perspicuous is and and satisfactory satisfactory explanation explanation of of the the most most central central concepts concepts in in this this area, area, where where our our emphasis is is put put on on aa deductive deductive and and procedural procedural point point of of view. view. emphasis The basic basic principles principles of of logic logic programming, programming, its its history, history, and and its its relationship relationship to to The the programming programming language Prolog are are well well presented presented in in many other publications publications the language Prolog many other (cf. e.g. e.g. Apt Apt [1990], [1990] , Doets Doets [1994] [1994] and and Lloyd Lloyd [1987]) [1987]) so so that that we we can can omit omit details. details. A A (cf. first important important distinction distinction is is between between definite definite logic logic programs programs which which are are based based on on so so first called definite definite Horn Horn clauses, clauses, and and extensions extensions thereof thereof which which provide provide means means for for treating treating called negative information. negative information. For definite definite logic quite simple. For logic programs programs the the situation situation is is quite simple. We We have have the the straightstraight forward observation observation that that aa closed closed atomic atomic formula A is is valid in the Herbrand forward formula A valid in the least least Herbrand of aa definite P if if and and only if A A is is aa logical logical consequence of P. model MR definite logic logic program program P only if consequence of model Mp of P. Moreover, the least least fixed fixed point point of the immediate consequence operator operator T Moreover, M MR is the of the immediate consequence Tpp p is introduced in in van van Emden and Kowalski Kowalski [1976]. [1976]. T also provides provides the link to to the the introduced Emden and Tpp also the link so SLD-resolution, which the standard proof procedure procedure for for definite definite logic so called called SLD-resolution, which is is the standard proof logic programs in suitable calculi. programs and and equivalent equivalent to to direct direct proofs proofs in suitable sequent sequent calculi. Although formulated formulated in in aa very definite logic are Although very restricted restricted language, language, definite logic programs programs are computationally complete in in the sense that that they all recursively computationally complete the sense they can can represent represent all recursively enumer enumerable relations. Nevertheless definite programs programs do able relations. Nevertheless definite do not not adequately adequately reflect reflect the the paradigm paradigm of programming programming in in logic there is is no of logic since since there no way way to to express express negative negative information. information. The standard to introduce introduce negative negative information programming The standard method method to information into into logic logic programming environments Clark's famous environments is is by by Clark's famous negation negation as as failure failure rule. rule. The The result result of of adding adding this will be this rule rule to to SLD-resolution SLD-resolution is is called called SLDNF-resolution SLDNF-resolution and and will be described described below below in in detail. detail. Negation Negation as as failure failure has has aa strong strong procedural procedural character character and and is is easy easy to to ed with, implement. implement. On On the the other other hand, hand, negation negation as as failure failure must must not not be be identifi identified with, for for example, example, classical classical or or intuitionistic intuitionistic negation, negation, and and its its exact exact logical logical meaning meaning is is quite intricate. The Shepherdson [1992] quite intricate. The survey survey articles articles Apt Apt and and Bol Bol [1994] [1994] and and Shepherdson [1992] are are dedicated logical environment environment of negation in in logic logic programming dedicated to to the the logical of negation programming and and are are aa good guide field. Jager concerned with good guide to to the the relevant relevant literature literature in in this this field. J~ger [1989] [1989] is is concerned with the the treatment treatment of of negative negative information information by by means means of of so so called called default default operators operators and and axiomatic axiomatic extensions. extensions. Our Our article article focuses focuses on on the the interpretation interpretation of of logic logic programs programs (with (with negation) negation) as as deductive deductive systems systems and and provides provides aa natural natural reconstruction reconstruction of of logic logic programming programming in in terms terms of of traditional traditional proof proof theory. theory. By By following following this this line line we we can can exploit exploit the the close close interplay interplay between between proof proof search search and and computation computation and and can can profit profit from from the the fact fact that that proof proof theory theory gives gives more more insight insight into into the the procedural procedural behavior behavior of of logic logic programs programs than than most most model-theoretic model-theoretic approaches. approaches. The The paper paper consists consists of of five five major major parts. parts. We We begin begin in in Section Section 22 with with introducing introducing the the basic basic syntactic syntactic and and semantic semantic notions. notions. This This is is essentially essentially aa repetition repetition of of standard standard terminology terminology including including threethree- and and four-valued four-valued structures. structures. A Proof-Theoretic Proof-Theoretic Framework ~rameworkfor Logic Logic Programming 641 641 The The backbone backbone of of Section Section 33 is is formed formed by by the the general general theory theory of of inductive inductive definability definability for for threethree- and and four-valued four-valued structures structures plus plus the the notions notions of of adequate adequate structure structure and and envelope generated generated by by aa logic logic program. program. This This machinery machinery is is used used to to introduce introduce in in the the envelope fastest fastest possible possible way way that that part part of of model model theory theory which which will will be be needed needed later. later. The The aim aim of of Section Section 44 is is to to set set up up deductive deductive systems systems for for logic logic programs. programs. We We introduce introduce sequent sequent calculi calculi with with additional additional program program rules, rules, consider consider their their identity-free identity-free subsystems and prove shown to subsystems and prove cut-elimination cut-elimination for for them. them. In In addition addition they they are are shown to be be sound and and complete complete with with respect respect to to the the semantics semantics introduced introduced before. before. sound In Section 55 we we study Starting point In Section study SLDNF-resolution. SLDNF-resolution. Starting point is is the the negation negation as as failure rule which is is carefully carefully integrated integrated into into the the resolution process. Modes Modes and and failure rule which resolution process. mode then introduced input/output behavior mode assignments assignments are are then introduced in in order order to to specify specify the the input/output behavior of logic logic programs. programs. They They provide provide aa powerful powerful tool tool for for setting setting up up large large and and natural natural of syntactically nable classes syntactically defi definable classes oflogic of logic programs programs for for which which SLDNF-resolution SLDNF-resolution is is shown shown to be procedure. to be a a sound sound and and complete complete proof proof procedure. Partiality is is considered considered in in the the last last section. section. We We show show how how aa simple simple syntactic syntactic Partiality transformation to regard transformation makes makes it it possible possible to regard logic logic programs programs (with (with negation) negation) as as aa system of of closure closure conditions conditions of of simultaneous simultaneous positive positive inductive inductive definitions definitions so so that that system the proof theory theory of of inductive becomes immediately to logic logic the proof inductive definitions definitions becomes immediately applicable applicable to programming. with aa brief indication of programming. This This section section concludes concludes with brief indication of the the importance importance of of induction principles for proving properties about logic logic programs with presenting induction principles for proving properties about programs and and with presenting an adequate adequate formal formal basis for such such activities. activities. an basis for 2 2.. Basic B a s i c notions notions 2.1. 2.1. Syntactic Syntactic framework framework In will deal with countable countable first order languages languages C with equality In the the following following we we will deal with first order s with equality which consist of which consist of the the following following basic basic symbols: symbols: . . . ) and 11.. Count ably many (u, v, v, w, W, U1 Countably many free free variables variables (u, ul,, VI Vl,, WI, Wl,...) and countably countably many many bound variables , . . .) ; (x, y, Z, bound variables (x, z, Xl, xl, YYl, Zl,...); 1 , Zl 2. countable 2. one one or or more more O-ary 0-ary function function symbols symbols ((= - constants) constants) and and an an arbitrary arbitrary countable number number of of function function symbols symbols of of finite finite arities arities greater greater than than 0; 0; 3. 3. the the symbols symbols = - for for equality equality and and ir for for inequality; inequality; (R, S, , . . . ) of 4. ably many 4. count countably many relation relation symbols symbols (R, S, T, T, Rl, R1, Sl S1,, Tl T1,...) of every every finite finite arity arity greater than 0; greater than 0; 5. the the symbol symbol - for for the the formation formation of of complementary complementary relations; relations; 5. 6. 6. the the propositional propositional constants constants T T and and .1.. l , , the the propositional propositional connectives connectives V V and and 1\ A and V. and the the quantifiers quantifiers :3 3 and and V. As notation we As auxiliary auxiliary symbols symbols we we have have parentheses parentheses and and commas. commas. To To simplify simplify the the notation we do relation symbols. symbols. Apart do not not denote denote the the equality equality and and inequality inequality symbols symbols as as relation Apart from from the function and relation the basic basic vocabulary the function and relation symbols, symbols, the vocabulary of of all all languages languages which which we we will consider first order will consider is is the the same. same. Each Each of of our our first order languages languages is is thus thus determined determined by by its its function and and relation relation symbols. symbols. function 642 642 C. G. Jager J~ger and R. Stark The terms terms (a, b, c, c, d, d, aI, the language The The (a, b, al, b1, bl, Cl Cl,, d1, d l , . . . )) of of the language C L: are are defined defined as as usual. usual. The literals (L, . . . ) of all expressions , . . . , an) and R(a1 , . . . , an) literals (L, M, M, L1, L1, MI M,...) of C s are are all expressions R(a1 R(a,,..., an) and R(al,..., an) so so , . . . , an) an) are that that R R is is an an n-ary n-ary relation relation symbol symbol of of sC; the the literals literals R(a1 R(al,..., are called called positive, positive, and literals R(al , . . . , an) called negative; and the the literals R(al,..., aN) are are called negative; the the positive positive literals literals are are sometimes sometimes atomic formulas ..1, (a also denoted as also denoted as atoms. atoms. The The atomic formulas of of C s are are the the literals literals plus plus T T,, A_, (a = = b) b) and (a #(A, B, C1, . . . ) of and (a r b) b).. The The formulas formulas (A, B, C, C, A!, A1, B1, B1, C1,...) of C s are are generated generated as as follows: follows: 11.. If If A A is is an an atomic atomic formula formula of of sC, then then A A is is an an C s formula. formula. 2. If 2. If A A and and B B are are C s formulas, formulas, then then (A (A V V B) B) and and (A (A 1\ A B) B) are are C L: formulas. formulas. 3. If A(u) A(u) is is an an C s formula, formula, then then 3xA(x) 3xA(x) and and 'v'xA(x) VxA(x) are are C s formulas. formulas. 3. If The The vector vector notation notation if V is is used used as as shorthand shorthand for for aa finite finite string string V V1,..., V~ whose whose length length ! , . . . , Vn will be specified will be specified by by the the context. context. We We write write A[it] A[g] to to indicate indicate that that all all free free variables variables of of A A come come from from the the list list a; g; analogously, analogously, a[a] a[g] stands stands for for aa term term with with no no variables variables different different a) and a( a) may other free from from a. g. The The formulas formulas A( A(g) and the the terms terms a(~) may contain contain other free variables variables besides a. besides g. We We denote denote the the set set of of all all free free variables variables of of the the formula formula A A by by var(A) vat(A).. The The universal universal closure closure of of a a formula formula A A is is denoted denoted by by 'v'(A) V(A) and and its its existential existential closure closure by by . • • 3(A).. 3(A) So for Z: C formulas. So far far we we have have no no negation negation for formulas. However, However, it it can can be be easily easily introduced introduced by by means the complementary the law and de means of of the complementary relations, relations, the law of of double double negation negation and de Morgan's Morgan's ,A of laws. laws. The The negation negation ---A of an an C Z: formula formula A A is is inductively inductively defined defined as as follows: follows: 11.. If C, then If R R is is an an n-ary n-ary relation relation symbol symbol of of E, then we we set set -,R(ii) := := R(ii) ~R(d) R(d) and and -,R(ii) --R(d) := R(ii). := R(d). 2. 2. For For the the other other formulas formulas we we have have -,T ~ T := ..1, _L, -,(a := (a #~(a = = b) b):= r b), ,(A VV B) (-,A 1\A -,B), B ) : := = (~A ~B), ---(A -,3xA(x) := 'v'x-,A(x), -,BxA(x) "= Vx-,A(x), -,..1 = T, --1 :"= T, := (a (a = -,(a -~(a #r b) b)"= = b), b), -,(A - ( A 1\ A B) B ) ":= = ((-,A -A V V -,B), ~B), -NxA(x) := 3x-,A(x) -~VxA(x) "= Bx-,A(x).. Logical implication Logical implication (A (A -+ --+ B) B) and and logical logical equivalence equivalence (A (A t+ ~ B) B) are are defined defined as as usual. usual. In In the the following following we we shall shall omit omit parentheses parentheses whenever whenever the the meaning meaning is is evident evident from from the the context. will often rank. context. The The complexity complexity of of formulas formulas will often be be measured measured in in terms terms of of their their rank. The m(A) of 2.1.1. 2.1.1. Definition. Definition. The rank rank rn(A) of an an C s formula formula A A is is inductively inductively defined defined as as follows: follows: := O. 11.. If If A A is is an an atomic atomic formula, formula, then then m(A) rn(A) := 0. 2. V C) 2. If If A A is is aa formula formula (B (B Y C) or or (B (B 1\ A C) C) so so that that m(B) rn(S) = = m m and and m(C) rn(C) = = n, n, then then m(A) := m ax (m , n rn(A) max(m, n )) ++ 11.. 33.. If =n If A A is is a a formula formula 3xB(x) 3xB(x) or or 'v'xB(x) VxB(x) so so that that m(B(u)) rn(B(u)) = = nn,, then then m(A) rn(A) ::= n ++l l. . Terms Terms and and formulas formulas without without free free variables variables are are called called closed. closed. The The equality equality formulas formulas the C which do (E, . . . ) of (E, E1, E1,...) of C s are are the L: formulas formulas which do not not contain contain relation relation symbols; symbols; the the A ProofTheoretic Framework Proof-Theoretic Framework for Logic Logic Programming 643 643 <1,1> (1, 1) <1,o> , 0) (1 (0, 1) (0, 0) Figure Figure 11": The The information information ordering ordering on on FOUR. FouR. positive positive formulas formulas of of .c s are are the the .c L formulas formulas which which do do not not contain contain negative negative literals. literals. Observe, however, however, that that equations equations ((aa = - b) b) and and inequations inequations (a (a =1= =/=b) b) are are not not considered considered Observe, as as literals literals in in our our terminology terminology so so that that positive positive formulas formulas may may contain contain equations equations and and inequations. inequations. Following on [1988], Following Shepherds Shepherdson [1988], aa language language .c s is is called called finite finite if if its its set set of of function function nite languages symbols symbols is is finite, finite, otherwise otherwise it it is is called called infinite. infinite. Thus Thus fi finite languages with with at at least least one function function symbol symbol of of positive arity have have an an infinite infinite number of closed closed terms. terms. l1 one positive arity number of The The Herbrand Herbrand universe universe U.c UL of of .c L is is the the collection collection of of all all closed closed terms terms of of s.c. By By our our assumptions U.c contains assumptions on on .c L we we know know that that UL contains at at least least one one element. element. 2.2. 2.2. Two-valued, T w o - v a l u e d , three-valued t h r e e - v a l u e d and a n d four-valued four-valued structures structures Classical true) and false) . On Classical logic logic just just employs employs two two truth truth values values tt ((true) and ff ((false). On the the other hand, recent other hand, recent research research in in logic logic programming programming indicates indicates that that aa third third truth truth value value u undefined) and contradictory) have u ((undefined) and aa fourth fourth truth truth value value cc ((contradictory) have their their natural natural place place ((cf. cf. e.g. Mycroft [1984] Fitting [1985] Kunen [1987, e.g. Lassez Lassez and and Maher Maher [1985] [1985],, Mycroft [1984], Fitting [1985] and and Kunen [1987, ' 1989]). 1989]). We We follow follow the the presentation presentation of of Fitting Fitting [1991]; [1991]; similar similar approaches approaches are are due due to to Belnap Belnap [1977] FOUR :-:= {0, {O, 1I}} x {0, {O, I1}. }. [19771 and and Ginsberg Ginsberg [1987]. [1987]. The The set set of of truth truth values values is is the the set set FOUR If FOUR is If (x (x,, yy)) E E FOUR is assigned assigned to to some some statement statement A, A, then then xx represents represents the the degree degree of of evidence evidence against usual truth evidence for for A A and and yy the the degree degree of of evidence against A. A. The The usual truth values values can can be be embedded into this this framework = (0, embedded into framework by by setting setting tt := (1, (1, 0) 0> and and ff ::= (0, 1) 1>;; the the third third truth truth FOUR aa binary value value u u is is represented represented by by (0, (0, 0) 0> and and cc is is (1, (1, 1) 1>.. On On FOUR binary relation relation is is defined defined by by <Xl, y~) [:: (X2, Y2> :r Xl _< X2 and Yl -- Y2 where on the right hand where the the relation relation :S _< on the right hand side side is is the the usual usual ordering ordering relation relation of of the the natural natural numbers see Fig. Fig. 1). numbers ((see 1). This This partial partial ordering ordering is is sometimes sometimes denoted denoted as as information information- ordering. ordering. On FOUR one defines which will be used On FOUR one defines the the following following operations operations which will be used below below to to 11Hence Hence "finite" "finite" refers refers to the number number of function symbols symbols and not to the number number of closed closed terms. G. G. Jager Jiiger and and R. R. Stark Stiirk 644 644 interpret the logical interpret the logical connectives: connectives: -(x, = (x, y) ::= , Y2 ) : = , YI) + (xl,yl) + (X2 (x2,y2):= (XI . = (Xl,, YYI)" (x2,' YY2) (Xl I ) (X2 2 ) ::= := Eiei(x~, Yi/:= iEI(Xi, Yi) L rliei(xi, y i / :: = = iEI (Xi, Yi) Il , x) , (y,x), (y max(Xl , x2 ) , min( min(yl,Y2)/, ((max(xl,x2), YI , Y2 ) , min(xl , X2 ) , max( ((min(xl, x2), max(yYI1,, YY2)), 2) , ax{xi : ii EE I}, (max{x/: I}, min{ min{yi I}/,) , (m Yi :: ii EE I} in{xi :: ii EE I}, max{ Yi : ii EE I} (min{xi I},max{y/: I}/.) . (m Observe {t, f} Observe that that the the sets sets {t, f},, {t, {t, f, f, c} c} and and {t, (t, f, f, u} u} are are closed closed under under these these operations. operations. This FOUR This is is not not the the case case for for the the limit limit of of elements elements of of F o u r which which is is defined defined by by taking taking the the pointwise pointwise maxima: maxima: lirn~eI(xi, Yi):= (max{x/: i E I},max{y/: i E I}). It It is is clear, clear, however, however, that that all all these these operations operations are are monotone monotone on on FOUR FOUR with with respect respect to to the the relation relation � E.. 2.2.1. 2.2.1. Definition. Definition. 1. A 1. A lour-valued f o u r - v a l u e d structure structure oot 9Y~ for for C s consists consists of of aa non-empty non-empty domain domain l!)Y~I together lootl together oot(J) and with with assignments assignments ~)Y~(f) and oot(R) ffJ~(R) to to all all function function symbols symbols I f and and relation relation symbols symbols R R of of C s so so that that (a) lootlI to (a) oot(J) ffJl:(f) is is an an n-ary n-cry function function from from 103~ to lootl In[ if if I f is is n-ary, n-cry, (b) (b) ffJl(R) oot(R) is is an an n-ary n-cry function function from from lootl [ffJl:[ to to FOUR F o u r if if R R is is n-ary. n-cry. 2. An 2. An upper upper three-valued three-valued structure s t r u c t u r e for for C s is is aa four-valued four-valued structure structure for for C s so so that that oot(R) do relation symbol the the functions functions ~)91:(R) do not not take take the the value value u u for for any any relation symbol R R of of C. s 3. 3. A A lower lower three-valued three-valued structure s t r u c t u r e for for C s is is aa four-valued four-valued structure structure for for C s so so that that the the functions functions oot(R) ~}Y~(R)do do not not take take the the value value cc for for any any relation relation symbol symbol R R of of C. s 4. 4. A A two-valued two-valued structure structure for for C s is is aa four-valued four-valued structure structure for for C s so so that that the the functions functions oot(R) ff2(R) do do not not take take the the values values ce and and u u for for any any relation relation symbol symbol R R of of C. s For by For aa four-valued four-valued C s structure structure oot if2 one one introduces introduces the the language language C[oot] s by adding adding to to C s ootl . Yet new new constants constants ~m for for all all m m E l]ff2]. Yet in in order order to to simplify simplify notation notation we we often often write write mn] . The A [[ml m l , '. .. . ., . , m mn] instead of of A[ml A [ ~ I ,' .. ... ., , mn]. The value value of of each each closed closed expression expression of of C[oot] s A n] instead is now inductively is now inductively defined defined as as follows: follows: 2.2.2. Let 2.2.2. Definition. Definition. Let oot ff)t be be aa four-valued four-valued C s structure. structure. We We assign assign to to each each closed term aa and closed formula aa value closed term and closed formula A A of of C[oot] s value oot(a) !)Y~(a) E lootl 1if2] and and aa value value oot(A) ff2(A) E E FOUR. FouR. 1. If 1. If aa is is the the term term m ~ for for some some element element m m of of lootl Iff)~],, then then oot(a) ff2(a) ::= m. = m. 2. for 2. If If aa is is the the term term l f ( (aI, a l , . ... ., a. n, an) ) for some some n-ary n-cry function function symbol symbol I f and and terms terms : = oot(J) al al,, .. .. .. ,, aan n ,, then then oot(a) !)Y~(a):= ~ ( f ) ( f f(oot(al) J ~ ( a l ) , , .. .. .. oot(an)). ffJ~(an)). A Proof-Theoretic Proof-Theoretic Framework Frameworkfor Logic Logic Programming Programming 645 645 3. If A A is is the the formula formula T T,, then then 9Yt(A) :-- tt;; if if A A is is the the formula formula (a (a = - b) b),, then then 3. If OO1(A) := { 9= [ t, t, 001 (A) := ff2(A) [ f, if if OO1(a) ~:(a) = = OO1(b), ~Yl:(b), if OO1(a) i' OO1(b). 4. 4. If If A A is is the the formula formula R(aI R ( a l ,, .. ... ,. , a an) for some some n-ary n-cry relation relation symbol symbol R R and and terms terms n ) for := OO1(R) a l ,, ... .. ,. a, na,n , then then OO1(A) ffJ~(A):= ~ ( R ) ((OO1(al) ~ ( a l ) ,". . .' ," ~OO1(a ( a n )n))). . aI 5. B and 5. If If A A is is the the formula formula ..., ~B and B B the the formula formula T T,, an an equation equation or or a a positive positive literal, literal, then then OO1(A) ~ ( A ) " =:= -OO1(B). -~(B). C) , then C) . 6. If If A A is is aa formula formula (B (B V V C), then OO1(A) ff2(A) := := OO1(B) ~:(B) + + OO1( ~:(C). 6. 7. OO1(A) := 9Jr(B). OO1(B) · OO1(C). 7. If If A A is is a a formula formula (B (B 1\ A C) C),, then then ffJt(A)'= ffJt(C). 8. 8. If If A A is is a a formula formula 3xB(x) 3xB(x),, then then OO1(A) ffJt(A) := "= EmEI!)]! ~mel~l1 OO1(B(m)) 9Yt(B(m)).. 9. : = TImEI!)]!1 9. If If A A is is a a formula formula VxB(x) YxB(x),, then then OO1(A) ff2(A)'= YImel~l OO1(B(m)) 9'A(B(m)).. Obviously t , ff,, cc}} for t , ff,, u} Obviously one one has has OO1(A) 9Jr(A) E e {{t, for all all upper upper three-valued three-valued 001, ffJt, OO1(A) ffJt(A) E e {{t, u} for {t, f} for all all lower lower three-valued three-valued 001 ffJt and and OO1(A) ff2(A) E {t, f} for for all all two-valued two-valued 001. 9Yr. Hence Hence these these three-valued three-valued and and four-valued four-valued structures structures are are natural natural generalizations generalizations of of the the two-valued two-valued case. Observe Observe that that equality equality is is always always handled handled as as the the usual usual two-valued two-valued identity. identity. If If A A case. is an an equality equality formula formula of of s£[001] , then then 9Jr(A) - tt or or 9Yt(A) = f, also for for three-valued three-valued is OO1(A) = OO1(A) = f, also 001. and and four-valued four-valued structures structures 9Yr. If £ and formula If 001 ffJt is is aa four-valued four-valued structure structure for for/:: and A A is is aa closed closed £[001] s formula so so that that x, y) then we (A) for y; hence OO1(A) = 9Jr(A) = ((x, y),, then we often often write write 0011st ~Jtlst(A) for xx and and 0012 ~Jt2nd(A) for y; hence we we have have nd (A) for (A) ) . st (A), 0012nd OO1(A) 9~t(A) = = (0011 (~Jtlst(A), ~Jt2nd(A)}. The The Herbrand Herbrand structures structures for for £ s are are the the £ s structures structures so so that that the the domain domain of of these these structures structures is is the the set set U Uez and and the the function function symbols symbols have have their their obvious obvious interpretations interpretations Herbrand structure over over Ue. UL. Hence Hence every every Herbrand structure 001 ffJt is is characterized characterized by by the the interpretation interpretation of symbols. In of its its relation relation symbols. In the the following following we we write write ile ~/z for for the the two-valued two-valued Herbrand Herbrand f, and structure structure for for £ s which which interprets interprets each each relation relation symbol symbol as as identically identically f, and 3ile 3ill to to denote denote the the lower lower three-valued three-valued Herbrand Herbrand structure structure for for £ s which which interprets interprets each each relation relation symbol symbol as as identically identically u. u. There There is is a a natural natural notion notion of of extension extension on on the the four-valued four-valued £ s structures structures which which is is obtained obtained by by lifting lifting the the above above defined defined relation relation � E on on FOUR F o u r pointwise pointwise to to the the four-valued four-valued £ /2 structures: structures" 2.2.3. Definition. be four-valued 2.2.3. D e f i n i t i o n . Let Let 001 ~ and and 1)1 9~ be four-valued structures structures for for £ Z: which which have have the the same same universe universe and and the the same same interpretations interpretations of of the the function function symbols. symbols. 1)1 9~ is is called called an an extension ( R( m)) � ( R( m)) for extension of of 001 ~Y~if if we we have have 001 9~:(R(~)) E 1)1 9~(R(rh)) for all all relation relation symbols symbols R R of of £ s and and m r5 E 10011 I~1.. In In this this case case we we write write 001 if2 � E 1)1. 9~. The relation � ordering on The relation E is is a a partial partial ordering on the the four-valued four-valued structures structures for for s£, and and 3ile 3ill is is the the least least Herbrand Herbrand structure structure with with respect respect to to this this ordering. ordering. In In addition addition it it is is easy easy to to see see that that the the £ s formulas formulas are are monotone monotone with with respect respect to to � E in in the the sense sense of of the the following following remark. remark. Jager and and R. R. Stdrk Stark G. Jiiger G. 646 646 2.2.4. RRemark. Let 92 !lJt and and 91 91 be be four-valued four-valued s£ structures. structures. Then Then we we have have for for all all 2.2.4. e m a r k . Let closed s£[!lJt] formulas formulas A" closed A: ~ ===} !lJt E[; ~91 9Jr 91 (A) . ffJt(A) !lJt(A) E [; 91(A). As aa special special case case this this means means that that as as soon soon as as an an E[92] £[!lJt] formula formula has has obtained obtained aa value value As or ff in in aa lower lower three-valued three-valued structure structure 92, it will will keep keep this this value value in in all all lower lower tt or !lJt , it three-valued extensions extensions of of 92. !lJt. three-valued 2.3. FFour-valued versus ttwo-valued 2.3. o u r - v a l u e d versus w o - v a l u e d sstructures tructures In this this section section we we introduce introduce the the extension extension s£" of of aa first first order order language language E£ and and show show In that the the four-valued four-valued structures structures for for s£ can can be be identified identified with with two-valued structures that two-valued structures for £" . for/:~. 2.3.1. Definition. Let Let £" be the which results results from 2.3.1. Definition. s be the first first order order language language which from s£ by by replacing each relation relation symbol symbol R replacing each R of two new of s£ by by two new independent independent relation relation symbols symbols R+ and R Rwhich are are of of the the same same arity arity as as R. R. R + and - , , which Hence each relation symbol R R of corresponds to pair (R+ R-) relation symbols Hence each relation symbol of s£ corresponds to a a pair (R +,, R - ) of of relation symbols of £" . Four-valued for R R can be associated two-valued of/:~. Four-valued interpretations interpretations for can therefore therefore be associated to to two-valued interpretations for R R++ and Rinterpretations for and R - .. 2.3.2. Let !lJt be be aa four-valued four-valued structure 91 aa two-valued 2.3.2. Definition. Definition. Let 92 structure for for s£ and and 91 two-valued structure structure for for £" s . 1. !lJt" and agrees 1. 92~ is is the the two-valued two-valued s£" structure structure which which has has the the same same universe universe as as !lJt 92 and agrees with !lJt relation symbols with 92 on on the the interpretation interpretation of of the the function function symbols; symbols; for for relation symbols R± R+ of and m set of £" s and rh E 6 1!lJt1 1921 we we set : = (!lJt !lJt" (R+ ) (m) := 1st (R(m)) , 1 !lJt1st (R(m)) ) , !lJt" (R- ) (m) : = (!lJt2nd (R(m)) , 1 !lJt2nd (R(m )) ) . := - - 2. 91° 2. 91<>is is the the four-valued four-valued £ s structure structure which which has has the the same same universe universe as as 91 91 and and agrees agrees symbols; for relation symbols with with 91 91 on on the the interpretation interpretation of of the the function function symbols; for relation symbols R R of o f /£: and and m n5 E 6 1911 1911we we set set 9I~ := (gllst(R+(rh)),9Ii,t(R-(r5))). The The previous previous two two constructions constructions are are inverse inverse to to each each other other in in the the strongest strongest possible possible sense. sense. We We have have for for all all four-valued four-valued £ s structures structures !lJt 9~ and and all all two-valued two-valued £" s structures structures 91 91 that 91° ) " = that (!lJt"t (92~)o = = !lJt 92 and and ((91o)~ _ 91. 91. Hence Hence it it is is perfectly perfectly legitimate legitimate to to identify identify the the four-valued four-valued structures structures for for the the language language £ s with with the the two-valued two-valued structures structures for for the the extension extension £" s of o f /£. : . In In view view of of the the following following remark remark it it is is possible possible to to identify identify the the lower £" structures lower three-valued three-valued £ s structures structures with with the the two-valued two-valued/J structures which which satisfy satisfy the the uniqueness uniqueness condition condition that that all all R+ R + and and RR - are are interpreted interpreted as as disjoint disjoint relations. relations. A A Proof-Theoretic Proof-TheoreticFramework Frameworkfor Logic Logic Programming Programming 647 647 2.3.3. Remark. Remark. Let 9Jt 93t be be aa four-valued four-valued .c s structure. structure. Then Then 9Jt 9JI is is upper upper three threeLet 2.3.3. )) = valued (R+ (rii) V valued if if and and only only if if 9Jtu 93t~(R+(r5) Y RR - (rii (r5)) - tt for for all all relation relation symbols symbols R R of of .c s and and all all iii ) t\R- (iii r5 E e 19Jt 199~1; analogously, 9Jt 93t is is lower lower three-valued three-valued if if and and only only if if 9JtU(R ffJt~(R + + (rii (rh)AR(rh)) -- ff )) = 1 ; analogously, for relation symbols 9Jt 1 . for all all relation symbols R R of of .c s and and all all iii r5 E 119I[I. Based on .c to Based on the the extension extension of of the the language language/: to the the language l a n g u a g e /.cU J we we now now translate translate every every .c into .cU and A Z: formula formula A A into ZJ formulas formulas A A+ + and A-- as as follows: follows" := A 1. If 1. If A A is is an an atomic atomic equality equality formula, formula, then then A A+ + := A and and AA- := : - --.A. -~A. 2. .c, then a) 2. If If A A is is of of the the form form R(a) R(g) for for some some n-ary n-cry relation relation symbol symbol of of/:, then A A+ + := := R R+ + ((~) - ( a) . and := R and AA- := R-(~). relation symbol 3. If 3. If A A is is of of the the form form R(a) R(d) for for some some n-ary n-cry relation symbol of of .c, s then then A+ A + := := RR-(g) (a) and and AA - : =:= R+(a) R + (~).. 4. (B V A++ := := (S (B++ V C+) := ((B4. If If A A is is of of the the form form (B Y C) C),, then then A y C +) and and AA- := B - t\ A C-) C - ) .. If A A is is of of the the form form ((B A C) C),, then A+ + ::= (B + t\ A C+) C +) and and AA- ::= (B- V V C-) C - ) .. 55.. If B t\ then A = (B+ = (B6. = 3xB+ (x) and 6. If If A A is is of of the the form form 3xB(x) 3xS(x),, then then A+ A + :"= 3xB+(x) and AA- := := VxBVxB-(x). (x) . 7. (x) , then then A+ (x) and (x) . 7. If If A A is is of of the the form form VxB YxB(x), A + := "= VxB+ VxB+(x) and AA- := := 3xB3xB-(x). This This means means that that the the .cU s formula formula A A+ + is is obtained obtained from from the the .c s formula formula A A by by changing changing + (a) and all positive positive literals literals R(d) A into into R R+(6) and all all negative negative literals R(g) in in A A into into literals R(a) all R(a) in in A + (a) R(a) ; Ain --.A by R-(d); A- is is obtained obtained from from A A by by replacing replacing all all positive positive literals literals R(a) R(~)in-~A by R R+(g) and all negative literals literals R(a) R(d) in ~A by by RR-(~). If there there are are axioms axioms available available which which in --.A (a) . If and all negative - (a) are express (it) , then express that that the the formulas formulas R R-(g) are the the negations negations of of the the formulas formulas R+ R+(g), then one one may may identify identify A+ A + with with A A and and AA- with with --.A. --A. .cU formulas. 2.3.4. 2.3.4. Remark. R e m a r k . If If A A is is an an .c Z: formula, formula, then then A A+ + and and AA- are are positive positive/::~ formulas. The the close The following following remark remark shows shows how how the close connection connection between between four-valued four-valued/:.c struc structures .cU structures tures and and two-valued two-valued/:~ structures extends extends to to arbitrary arbitrary .c s formulas. formulas. 2.3.5. .c[I)tO] 2.3.5. Remark. R e m a r k . We We have have for for all all two-valued two-valued .cU ZJ structures structures I)t 92 and and all all closed closed/:[fit ~ formulas (A+), I)t = (l)tlst formulas A A that that I)tO(A) 92~ (921,t(A+), 921st(A-)/. lst (A- » . 2.4. 2.4. Logical Logical consequences consequences An An .c s theory theory is is aa (possibly (possibly infinite) infinite) set set of of .c s formulas. formulas. By By Th "l'h It- A A we we express express that that the the formula formula A A can can be be deduced deduced from from the the theory theory Th Th by by the the usual usual axioms axioms and and rules rules of of first first order order predicate predicate logic. logic. If If 9Jt 93t is is a a two-valued two-valued structure, structure, A[it] A[~7] is is an an .c s formula formula and and Th Th an an .c s theory, theory, then then J) = . Then we we define define as as usual: usual: A[it] A[~7] is is valid valid in in 9J 93q: if 9Jt(A[iii ffJt(A[r5]) = tt for for all all iii r5 E e 19Jt1 193q:1. Then we we t if and write Th is is valid in 9J all elements call call 9Jt ff.rt aa model model of of A[a] A[~7] and write 9Jt 93t F ~ A[a] A[~7].. Th valid in 93tt if if all elements of of Th Th are are valid valid in in 9Jt. 93t. Then Then we we call call 9Jt ffYta a model model of of Th Th and and write write 9Jt ff)I F ~ Th. Th. A[it] A[~7] is is aa logical logical consequence of Th if models of Then we Th F consequence of Th if A[a] A[~7] is is valid valid in in all all models of Th. Th. Then we write write Th ~ A[a] A[~7].. The The usual usual completeness completeness result result for for first first order order logic logic states states that that derivability derivability is is equivalent equivalent to to logical logical consequence, consequence, Le., i.e., that that Th Th I~- A A if if and and only only if if Th Th F ~ A A for for all all s theories theories Th Th and and/:: formulas A. A. .c .c formulas 648 648 G. G. Jager J~ger and R. Stark This form This form of of logical logical consequence consequence is is based based on on two-valued two-valued structures. structures. Special Special forms forms of of consequences consequences of of logic logic programs programs with with respect respect to to certain certain three-valued three-valued and and four-valued four-valued structures structures will will be be introduced introduced in in Section Section 3.1. 3.1. 2.5. 2.5. Clark's C l a r k ' s equational e q u a t i o n a l theory theory Unification plays Unification plays a a major major role role in in practically practically all all implementations implementations of of logic logic program programming ming environments. environments. In In general general the the most most simple simple form form of of unification unification is is employed employed which which treats treats two two closed closed expressions expressions as as equal equal if if and and only only if if they they are are syntactically syntactically identical. identical. The The corresponding corresponding unification unification theorem theorem goes goes back back to to Robinson Robinson [1965J [1965] and and states states the the existence existence of of an an algorithm algorithm which which for for any any two two expressions expressions produces produces an an idempotent idempotent most general general unifi unifier if they they are are unifiable unifiable and and otherwise otherwise reports reports the the nonexistence nonexistence of of aa most er if unifier. unifier. Space Space does does not not permit permit to to go go into into details, details, and and only only the the basic basic terminology terminology can can be be repeated. repeated. An An £ s substitution substitution B0 is is aa finite finite set set {ut/ { u l / aaIl ,, .. ... ., , un/an} u~/a~} of of bindings bindings so so that that the £ s terms terms aaii are are different different from from the the variables variables Ui ui for for 11 :::; _< ii :::; _< n n and and Ui ui is is different different the :::; ii < < jj :::; n. We B, a, T, B01,al, , . . . for the from from Uj uj for for 11 _< _< n. We shall shall use use O,a, T1,... for substitutions; substitutions; the I , aI , TI empty denoted by empty substitution substitution is is denoted by c. The The instance instance ZB ZO of of an an expression expression Z Z and and aa substitution substitution B0 = = {Ut/a { u l / a l, l , .. ... ., , un/an} u~/a~} is is the replacing each occurrence of the expression expression obtained obtained from from Z Z by by simultaneously simultaneously replacing each occurrence of the the variable addition, an variable Ui ui in in Z Z by by the the term term aaii (i (i = = 11,, .. .. .., , n) n).. In In addition, an expression expression Zl Z1 is is called called aa variant variant of of the the expression expression Z2 if if there there exist exist substitutions substitutions a a and and T so SO that that Zw 2:1a = -- Z2 and l. and Z2T = =Z Z1. Let {al = Let S $ be be the the set set of of equations equations {al = bl b l,, .. .. ..,,aann = = bn} bn}.. A A unifier unifier of of S $ is is an an £ substitution B0 with identical for s substitution with the the property property that that aiB aiO and and biB biO are are identical for 11 :::; < ii :::; < n n.. This This unifier most general general if substitution T unifier is is most if for for any any other other unifier unifier a a of of S $ there there exists exists aa substitution so composition of general) unifier SO that that a a is is the the composition of B0 and and T T,, i.e., i.e., a a = -- B1". OT. A A (most (most general) unifier of of (most general) the the two two atoms atoms R R (( aal, l , .. ...., , an an)) and and R(bI R ( b , ,, .. ...., , bn) bn) is is aa (most general) unifier unifier of of the the set set of = bl of equations equations {a {all = b l,, .. .. .., , a ann = = bn} bn}.. For For further further unexplained unexplained notions notions we we refer refer to to Apt Apt [1990], [1990], Doets Doets [1994J [1994] and and Lloyd Lloyd [1987J. [1987]. 's equational Clark Clark's equational theory theory GETc CETL (cf. (cf. Clark Clark [1978]) [1978]) may may be be understood understood as as the the axiomatic axiomatic counterpart counterpart of of this this form form of of unification. unification. The The theory theory GETc CET~ depends depends on on the the language language £ s and and comprises comprises the the following following equality equality axioms axioms (E1) (El) and and (E2) (E2).. First First we we have have (E1) /\ .. ... . /\ = bn , (al = (El) ...~(al = bl bl A A an an = bn)) for bn} is able. an, bbIl,, .. .. .., , bbnn so for all all £ s terms terms aab l , .. ...., , an, so that that {a {all = = bl, b l , .. .. .., , an an = = bn} is not not unifi unifiable. The The second second group group of of axioms axioms states states (E2) . . a/\nan (E2) (al ( a l ==b bl l A/\. . .. A = b= n ) bn) - - + -+ c = dc = d provided bn} is able with er B0 and provided that that {al {al = = bl b l,, .. .. .., , aann = = bn} is unifi unifiable with a a most most general general unifi unifier and cB cO and and dB dO are are syntactically syntactically identical. identical. A A four-valued four-valued structure structure rot if2 is is called called an an equational equational structure structure if if the the universal universal closures closures of (E2) are of the the equality equality axioms axioms (E1) (El) and and (E2) are true true in in rot. ~ . ilc 12~ is is an an equational equational structure structure and ed by and is is sometimes sometimes called called the the standard standard model model of of GETc CETL.. This This is is justifi justified by the the obvious obvious A Proof-Theoretic Framework for Logic Programming 649 649 fact fact that that every every model model of of CETe CETL contains contains an an isomorphic isomorphic copy copy of of the the standard standard model. model. CET~ does does not not contain contain equality equality assertions assertions for for relation relation symbols. symbols. Hence, Hence, for for example, example, CETe formula of of the the form form aa - bb /\ A R(a) R(a) -+ --+ R(b) for some some unary unary relation relation symbol symbol R R is is not not aa formula R(b) for derivable derivable from from CETe CETL.. ' cev [1971] A result result of of Mal Mal'cev [1971] states states that that CETe CETL is is complete complete if if C s is is an an infinite language. A infinite language. Observe, however, however, that that CETe CET~ is is in in general general not not complete complete for for finite finite languages. languages. Let Let C, s Observe, for example, example, be be a a language language with with a a constant constant aa and and no no other other function function symbols. symbols. Then Then for LtL is is a a model model of of 'v'x( Vx(xx - a) a) but but CETe CETL does does not not prove prove this this equality equality formula. formula. In In order order Ue to to obtain obtain completeness completeness also also for for finite finite languages languages C s one one has has to to strengthen strengthen CETe CETL by by the the so so called called domain domain closure closure axiom axiom DCAe DCAL,, = = := = f(fi)), DCAe W 3y(x =/(Y-0), DCAc := 'v'x Vx~/3g(x f Ee which belongs to which says says that that every every element element of of the the universe universe belongs to the the range range of of some some function function 'cev [1971] symbol symbol of of sC. Then Then the the following following theorem theorem follows follows for for example example from from Mal Mal'cev [1971] or or Shepherdson [1988] Shepherdson [1988].. 2.5.1. Theorem. 2.5.1. Theorem. have have the the equivalence equivalence Let Let E E be be aa closed closed equality equality formula formula of of the the language language C s . Then Then we we CETe E C E T L Ff-- E {:=> -: ;- Ue L i LF l E E, , provided provided that that C s is is infinite. infinite. On On the the other other hand, hand, if if C s is is finite, finite, then then one one can can only only show show that that Ue CETL + + DCAe DCA~ fF- E E r t2L F ~ E E.. CETe {:=> 2.6. 2.6. Logic Logic programs p r o g r a m s and and their their completions completions Finally the article: logic Finally the stage stage is is set set for for introducing introducing the the central central object object of of this this article: logic programs. programs. What What we we simply simply call call a a logic logic program program here here is is sometimes sometimes denoted denoted as as aa cf. e.g. normal or normal or general general logic logic program, program, in in contrast contrast to to definite definite logic logic programs programs ((cf. e.g. Apt Apt [1990] [1990],, Doets Doets [1994] [1994] and and Lloyd Lloyd [1987]). [1987]). Goals possibly empty Goals (G, (G, H, H, Gi, G1, Hi, H1,...) in the the language language C Z: are are finite finite ((possibly empty)) sequences sequences . . . ) in of of C Z: literals. literals. The The empty empty goal goal is is denoted denoted by by 0 O.. A A program program clause clause in i n /C: is is an an expression expression of the the form form of A A ::-GG so so that that A A is is a a positive positive literal literal of of C. s The The atom atom A A is is the the head head and and the the sequence sequence of of the body clause is literals literals G G the body of of the the clause. clause. If If the the body body of of a a program program clause is empty, empty, we we simply simply C is (C, PC) PC) which write write A A instead instead of of A A : -- 0 O.. A A logic logic program program in in/:: is aa pair pair (s which consists consists of of aa first C and first order order language language/:: and aa finite finite set set PC PC of of program program clauses clauses in in C. s Suppose Suppose that that P P is is a a logic logic program, program, R R is is an an n-ary n-cry relation relation symbol symbol and and that that there there th clause are R( . . . ) so are m m clauses clauses in in P P whose whose heads heads are are of of the the form form R(...) so that that the the iith clause is is of of the the form form : R( R(ai,l[~, ai,nM) - Li Li,IM, Li,~(i)[~ ,dv] , ·. .. .. ,, Li ,k(i) [V] ai,l [V] , .. .. .. ,, ai ,n [V] ) :: - G. Jager Jagerand andR. R. Stark Stark G. 650 650 and has has k( k(i) literals in in its its body. body. Then Then the the definition definition form form of of R R with with respect respect to to P P isis and i) literals defined to be be the the formula formula defi ned to DR[uI, . . . , un] := k ( i» n !/Jx ({& (Uj = ai,j [X] ) 1\ (& Li,j [x]) . rn m n k(i) i=l - j=l The i) == 00 are The special special cases cases m m= = 00 and and k( k(i) are included included by by interpreting interpreting empty empty disjunctions disjunctions as ..1 3_ and and empty empty conjunctions conjunctions as as T. T. as In Section Section 44 we we will will introduce introduce deductive deductive systems systems for for logic logic programs, programs, and and in in these these In systems so so called called program program rules rules are are associated associated to to the the program program clauses. clauses. However, However, systems from aa declarative declarative point point of of view view aa logic logic program program P P is is often often identified identified with with the the theory theory from consisting consisting of of all all formulas formulas V~(DR[:~'] -+ -+ R R(~)) so that that each each DRIP7] is is the the definition definition ( )) so form of of R R with with respect respect to to P. P. Other Other schools schools in in the the model-theoretic model-theoretic approach approach to to form logic programming programming argue argue that that the the intended intended meaning meaning of of aa logic logic program program P P is is better better logic reflected by the the so so called called Clark Clark completion completion of of P P,, in in which which the the implications implications of of the the refl ected by previous formulas formulas are are replaced replaced by by equivalences equivalences (cf. (cf. Clark Clark [1978]). previous More formally, formally, let let P P be be aa logic logic program program in in .c s and and assume assume that that the the definition definition form form More of each each relation relation symbol symbol R R of of .c s is is the the formula formula DRIP7]. Then Then we we call call of 'v'X(DR[X] x DR[i1] [1978]). DR[i1] . V~(DR[~] ++ n(~)) the P. The the completed completed definition definition of of R R with with respect respect to to P. The completion completion of of P P is is the the .c Z: theory theory comp(P) definitions of comp(P) which which consists consists of of CETc CETc plus plus the the completed completed definitions of all all relation relation symbols symbols of/:. of .c. 3.. Some Some m odel-theoretic pproperties r o p e r t i e s of o f logic l o g i c pprograms rograms 3 model-theoretic There are are some some central central model-theoretic model-theoretic properties properties of of logic logic programs which are are There programs which crucial for our crucial for our proof-theoretic proof-theoretic approach, approach, in in particular particular from from the the point point of of view view of of providing aa semantic semantic platform platform and and motivation motivation of of the the following following steps. steps. We We will will now now providing recall recall these these results results and and present present them them in in aa form form tailored tailored for for our our later later applications. applications. 3.1. d e q u a t e sstructures tructures 3.1. A Adequate We start start with with structures structures which which are are adequate adequate to to logic logic programs. programs. Informally, Informally, We adequate structures structures are are structures structures which which reflect reflect the the meaning meaning of of aa logic logic program program in in adequate the sense sense that that the the information information content content of of the the definition definition form form isis inherited inherited to to the the the corresponding corresponding relation. relation. 3.1.1. Definition. Definition. Let Let PP be be aa logic logic program program in in s.c. AA four-valued four-valued equational equational 3.1.1. structure adequate toto PP ifif structure 93t VR for for s.c isis called called adequate ff2(DR[rh]) E ffJt(R(r5)) DR[U] for all all rh m EE 192tl IVRI and and all all relation relationsymbols symbols RR of ofs.c plus plus their their definition definition form form DR[g] with with for respect respect toto P. P. A ProofTheoretic Framework Proof-Theoretic Frameworkfor .for Logic Logic Programming 651 651 This definition definition implies implies that that a a two-valued two-valued equational equational structure structure oot 9Yt for for C E is is adequate adequate to to This logic program program P P if if and and only only if if oot ffJt is is a a model model of of comp(P) comp(P).. Moreover, Moreover, Remark Remark 3.2.6 3.2.6 aa logic below below describes describes the the relationship relationship between between three-valued three-valued models models of of comp(P) comp(P) in in the the sense sense of of Fitting Fitting [1985] [1985] and and Kunen Kunen [1987] [1987] and and three-valued three-valued structures structures which which are are adequate adequate to to P. P. 3.1.2. 3.1.2. Definition. D e f i n i t i o n . Let Let P P be be a a logic logic program program in in C s and and A[iI] A[g] an an C s formula. formula. 1. A[iI] -adequate consequence 1. A[g] is is called called a a 44-adequate consequence of of P P if if oot1st(A[m]) ~Utlst(A[rn]) = = 11 for for all all four fouroot l . valued equational C structures oot which are valued equational s structures ffJt which are adequate adequate to to P P and and all all m rh E E lIffJtl. Then Then we we write write P P F ~44 A[iI] A[g].. 2. 2. A[iI] A[~7] is is called called an an upper upper consequence consequence of of P P if if oot1st(A[m]) ffJhst(A[nh]) = = 11 for for all all upper upper three-valued three-valued equational equational C s structures structures oot ffJt which which are are adequate adequate to to P P and and all all m oot l . Then r5 E e lIffJtl. Then we we write write P P FL> ~z~ A[iI] A[~7].. lower consequence 3. A[~7] is is called called a a lower consequence of of P P if if oot1st(A[m]) ffJtlst(A[rh]) = = 11 for for all all lower lower three three3. A[iI] valued oot l . valued equational equational C s structures structures oot ffJt which which are are adequate adequate to to P P and and all all m n5 E E lIffJtl. Then Then we we write write P P F'il ~ v A[iI] A[~7].. 4. A[iI] -adequate consequence 4. A[g] is is called called a a 22-adequate consequence of of P P if if oot1st(A[m]) ffJtl,t(A[nh]) = = 11 for for all all two twovalued equational which are adequate to all m oot l . valued equational C s structures structures oot ffJt which are adequate to P P and and all n5 E E lIff)tl. Then Then we we write write P P F2 ~2 A[iI] A[~7].. Since Since the the two-valued two-valued structures structures which which are are adequate adequate to to a a logic logic program program P P agree agree with with the the models models of of comp(P) comp(P),, it it is is obvious obvious that that a a formula formula A A is is a a 2-adequate 2-adequate consequence consequence of completion of of P P if if and and only only if if it it is is a a logical logical consequence consequence of of the the completion of P. P. 3.2. 3.2. Envelopes E n v e l o p e s generated g e n e r a t e d by b y logic logic programs programs A structure oot providing some partial information A four-valued four-valued structure ffJt can can be be viewed viewed as as providing some partial information about intended scope interest, and about the the intended scope of of interest, and a a logic logic program program P P as as aa means means of of modifying modifying this structure oot [P] , which call the P-envelope of this information information oot 9~t to to a a new new structure 9Yt[P], which we we call the P-envelope of oot. ffJt. 3.2.1. Let 3.2.1. Definition. Definition. Let oot ffJt be be a a four-valued four-valued C s structure structure and and P P aa logic logic program program in the P-envelope which has in sC. Then Then the P-envelope oot[P] ffJt[P] of of oot ff)t is is the the C s structure structure which has the the same same universe with oot symbols; if universe as as oot ffJt and and agrees agrees with ffJt on on the the interpretation interpretation of of the the function function symbols; if R P, then R is is a a relation relation symbol symbol of of C s and and DR[iI] DRIp7] its its definition definition form form with with respect respect to to P, then we we set (R) (m) := oot(DR[m]) oot l . set oot[P] ffJt[P](R)(rh):= ffJt(Dn[nh]) for for all all m n5 E e l19Jtl. It definition that It follows follows from from this this definition that the the P-envelope P-envelope of of aa two-valued two-valued structure structure is is two-valued two-valued and and that that of of an an upper upper or or lower lower three-valued three-valued structure structure is is upper upper or or lower lower three-valued, respectively. In general it extension of three-valued, respectively. In general it is is not not the the case case that that oot[P ffJt[P]] is is an an extension of oot, but property is ffJt, but at at least least the the following following property is given. given. 3.2.2. Let 3.2.2. Remark. Remark. Let oot ~ and and !.Jl fit be be four-valued four-valued/:C structures structures and and assume assume that that P P is is aa logic logic program program in in sC. Then Then we we have: have: oot 9 / t _[; !.Jl 9t ==> ~ oot[P] 9 / t [ P ] _[; !.Jl[P] 9tIP].. 5 G. G. Jager J@er and R. Stark StSrk 66522 Hence the formation Hence the formation of of envelopes envelopes is is monotone. monotone. Making Making use use of of envelopes, envelopes, it it is is now now an an easy task task to easy to characterize characterize those those structures structures which which are are adequate adequate to to aa logic logic program program P. P. VJt 3.2.3. Let 3.2.3. Remark. Remark. Let ff~ be be a a four-valued four-valued equational equational structure structure for for C s and and assume assume that logic program only if that P P is is a a logic program in in C. s Then Then ffY~is is adequate adequate to to P P if if and and only if ffY~[P] � _E 9)t. VJt VJt[P] VJt. There close relationship There is is a a close relationship between between four-valued four-valued and and lower lower three-valued three-valued adequate adequate structures. structure which structures. Every Every four-valued four-valued structure which is is adequate adequate to to a a logic logic program program P P extends extends a a lower lower three-valued three-valued structure structure which which is is invariant invariant under under the the formation formation of of its its P-envelope P-envelope and and thus thus adequate adequate to to P P by by the the previous previous remark: remark: 3.2.4. 3.2.4. Proposition. P r o p o s i t i o n . Let Let P P be be aa logic logic program program in in C s and and VJt ~ aa four-valued four-valued C L struc strucP. Then ture ture which which is is adequate adequate to to P. Then there there exists exists aa lower lower three-valued three-valued structure structure IJ1 9l for for C L soso that that IJ1 9I � E VJt ffJ~ and and IJ1[P] 9~[P] = - 1J1. 9t. Proof. P r o o f . Let Let K K be be the the nonempty nonempty set set of of all all lower lower three-valued three-valued structures structures for for C s which which are K, � are extended extended by by VJt. 9)t. Then Then ((K, K) is is a a complete complete partial partial ordering. ordering. In In addition, addition, the the operation operation which which maps maps an an element element of of K K to to its its P-envelope, P-envelope, which which belongs belongs to to K K as as well, is well, is monotone monotone according according to to Remark Remark 3.2.2. 3.2.2. Therefore Therefore there there exists exists a a structure structure IJ1 9t 0 as claimed in as claimed in the the assertion. assertion. [] ) We have for all logic programs in and for all formulas that is a 4-adequate consequence of if and only if is a lower consequence of i.e., i.e., 3.2.5. 3.2.5. Corollary. Corollary. We have for all logic programs P P in C s and for all C s formulas that A A is a 4-adequate consequence of P P if and only if A A is a lower consequence of P, P, P~4A [1985] [1987] ~, ,~ P~v A. Fitting Fitting [1985] and and Kunen Kunen [1987] use use slightly slightly different different definitions definitions and and introduce introduce the the notion model of notion of of aa three-valued three-valued model of the the completion completion comp(P) of of aa logic logic program program P. P. Then Then it it is is obvious obvious that that one one has has the the following following correspondence: correspondence: comp(P) VJt 3.2.6. Let 3.2.6. Remark. Remark. Let P P be be a a logic logic program program in in C s and and if2 aa lower lower three-valued three-valued model of completion comp(P) if structure for structure for sC. Then Then 9)t is is aa three-valued three-valued model of the the completion if and and only only if if ff2[P] = = 9/1:. VJt[P] VJt. VJt comp(P) 3.3. 3.3. Least Least adequate a d e q u a t e sstructures tructures Standard techniques Standard techniques of of the the theory theory of of inductive inductive definitions, definitions, as as presented, presented, for for the means show that that all all logic example, in Moschovakis example, in Moschovakis [1974], provide provide the means to to show logic programs programs P P lower three-valued) three-valued) structures P. These have have least least ((lower structures which which are are adequate adequate to to P. These structures structures are are generated generated by by iterating iterating the the formation formation of of P-envelopes P-envelopes through through aa sufficiently sufficiently large large initial initial segment segment of of the the ordinals. ordinals. If If 9Yt is is a a four-valued four-valued structure structure for for C L: then then 3VJt 39/1:is is the the lower lower three-valued three-valued structure structure for for C s which which has has the the same same universe universe and and the the same same interpretation interpretation of of all all function function symbols symbols as as 9)t and and interprets interprets each each relation relation symbol symbol as as identically identically u. u. A A family family (ffJ~i : i E E I) I) of of [1974], VJt VJt (VJti : i 653 653 A Proof-Theoretic Framework for Logic Programming based on !.m 3!.mi = four-valued four-valued structures structures for f o r /.c: is is based on a a four-valued four-valued structure structure ~ if if 3ffJ~i = 3!.m 3931:for for all all ii E E /. I. Now be aa non-empty family of Now let let (ff)l:i :: i E E /) I) be non-empty family of four-valued four-valued structures structures for for .c s which which are based based on on aa four-valued four-valued .c s structure structure 931:. Then Then the the limit Iliei ffJ~i of of this this family family is is are the the four-valued four-valued structure structure SJt fit for for .c s so so that that 33fit - 3!.m 3ff2 and and (!.mi i limit UiEI !.mi SJt = fft(R)(r5) SJt(R)( m) :=:= llimff2i(R)(rh) i� !.mi(R)(m) !.m lI.. This for relation symbols for all all relation symbols R R of of .c s and and m r5 E E l]gJl: This implies implies that that the the degree degree of of evidence evidence for for (against) (against) a a positive positive literal literal R(~) ff2~, if if it it is is 11 in in some some !.mi, ff)l:~, R(a) isis 11 inin Ui]l~eiE/ !.mi, and 0 otherwise. !.m. and 0 otherwise. P !.m 3.3.1. Let 3.3.1. Definition. Definition. Let P be be aa logic logic program program in in .c s and and if2 a a four-valued four-valued .c s struc structure. Then we we define by recursion on the the ordinals the following following four-valued four-valued structures structures ture. Then define by recursion on ordinals the for for .c: s +l := J9Jt], J!.m� ~[~p := "--" 3!.m, 3~)~, ~ff~p+l .= J�[P], 3ff~p[P], 3 ~ P := U U J!.m}, 39~p J� ~<~ for A. Then is the for all all limit limit ordinals ordinals A. Then J!.mp ~ [ ~ p is the limit limit of of all all J!.m}, 39~p for for �~ E E On; On; i.e. i.e.,, J!.mp :=:= UU J!.m},. �EOn ~On := � J'f, J� J!.mp, It I;;; that there there exists exists an that It follows follows that that J� 3~ E ; J!.m� 3ff2~p for for c~ ::; < / 3f3 and and that an ordinal ordinal -y so so that J!.mj, least such called the closure ordinal ordinal ofof PP 3ff)~ - J!.mp. 3ffJl:p. The The least such ordinal ordinal is is called the three-valued three-valued closure with !.m. with respect respect to to 9Y~. Further, Blair [1982] Further, standard standard results results about about inductive inductive definability definability (cf. (cf. e.g. e.g. Blair [1982] and and Moschovakis three-valued closure Moschovakis [1974]) [1974]) show: show: (i) (i) The The three-valued closure ordinal ordinal of of P P with with respect respect to to f g for 3U.c 312L is is less less than than or or equal equal to to the the first first non-recursive non-recursive ordinal ordinal we1 for all all logic logic programs programs (definite) logic P; for every every ordinal _ wf cg there there exists exists a a (definite) logic program program P P whose whose P; (ii) (ii) for ordinal a ::; If the lower structure 3U.c write 3~ and If !.m ~ is is the lower three-valued three-valued Herbrand Herbrand structure 311s, then then we we simply simply write and ~p instead instead of of 3ff~p and and 3ffJ~p, respectively. respectively. Jp I Ct = Ct W W K K three-valued ordinal with three-valued closure closure ordinal with respect respect to to 3U.c 311L is is Ct a.. 3.3.2. This 3.3.2. Example. Example. This example example shows shows that that the the three-valued three-valued closure closure ordinal ordinal of of a a logic to 3U.c logic program program with with respect respect to 311c can can be be greater greater than than w. Let Let .c s be be a a language language with with aa constant 0, aa unary constant 0, unary function function symbol symbol f and and two two relation relation symbols symbols R R and and S. Let Let P be be the the logic logic program program in in .c s which which consists consists of of the the following following clauses: clauses: f S. P R(J(u)) n(f(u)) : - R(u) n(u) S(J(u)) S(f(u)) ::-- S(u) S(u) S(u) S(u) :: - R(v) It It is is easy easy to to see see that that the the three-valued three-valued closure closure ordinal ordinal of of P P with with respect respect to to 3U.c 3IlL is is w+ i.e., J't> + w, i.e., 3~+ +~w = = Jp. 3p. One One simply simply has has to to consider consider the the definition definition forms forms DR[U] DR[u] and and Ds Ds[u] and S S with with respect respect to to P: P: [u] ofof RR and f (x) /\ A R(x)) R(x)) and and ::I 3xx (u (u = = f(x) f (x) /\ A S(x)) S(x)) v V ::I3xx R(x). R(x). ::I3xx (u(u =-- f(x) :- 654 654 G. G. Jager JSger and R. Stark StSrk J� Trivial induction induction on on the the ordinals ordinals shows the structures structures :iffJ~p and and the the structure structure Trivial shows that that the :iMp are are lower lower three-valued. three-valued. Furthermore, Furthermore, if if aa four-valued four-valued s structure structure 9~ is is adequate adequate then :iff~p [;:; to to P P and and 3ffYt - 39~, then _ 9~ for for all all a a E E On. In In view view of of Remark Remark 3.2.3 we we therefore obtain the therefore obtain the following following theorem. theorem. Jv.np 3v.n = 3!J1, J� !J1 On. .c !J1 3.2.3 3.3.3. Let 3.3.3. Theorem. Theorem. Let P P be be aa logic logic program program in in .c f~ and and v.n ffJ~ aa four-valued four-valued structure structure for lower three-valued for .c. L. Then Then Jv.np :iffJ~p is is aa lower three-valued structure structure which which is is adequate adequate to to P P.. In In addition, to PP and addition, iiff !J1 9~ is is aa four-valued four-valued structure structure for for .c L which which is is adequate adequate to and satisfies satisfies 3ffJ~ 39~, then then we we have have Jv.np :iMp [;:; ~ !J1. ~[~. 3v.n == 3!J1, This This theorem theorem implies, implies, in in particular, particular, that that the the structures structures Jp :JR are are the the least least four-valued four-valued P. Some Some simple Herbrand Herbrand structures structures which which are are adequate adequate to to the the logic logic programs programs P. simple reflections reflections on on the the high high logical logical complexity complexity of of the the structures structures Jp, :IF, which which is is reflected reflected K by the the fact fact that that the the three-valued three-valued closure closure ordinals ordinals may may be be as as large large as as w wf oK,, make make it it by clear that the least least adequate only be clear that the adequate structures structures can can only be of of limited limited use use for for a a procedural procedural approach to to logic logic programming. programming. It It follows, follows, for for example, example, that that in in general general the the collection collection approach of of all all closed closed formulas formulas true true in in :ip is is not not even even first first order order definable. definable. We We agree agree with with Kunen Kunen [1987] that that the the "procedural "procedural content" content" of of aa logic logic program program P P is is better better approached approached by finite stages by the the finite stages (:i~ 9n n < < w w)) of of :IF- Jp [1987] (Jp : Jp. 3.4. 3.4. The T h e finite finite stages stages of of least least adequate a d e q u a t e structures structures This This thesis thesis is is also also supported supported by by the the following following observations. observations. Taking Taking up up an an idea idea of of assigns to Kunen [1987] and Kunen and Shepherdson Shepherdson [1988] one one assigns to each each .c s formula formula A and and natural natural number number n n an an equality equality formula formula E~(A), which which depends depends on on the the given given logic logic program program P. P. [1987] [1988] Ep (A), 3.4.1. 3.4.1. Definition. Definition. A Let Let P P be be a a logic logic program program in in .c s . For For every every n n < < w w and and every every formula A A we the equality formula E E~(A) by induction induction on on n. .cs formula we define define the equality formula p (A) by 1.1. IfIf AA isis an (A) := A. an atomic atomic equality equality formula, formula, then then E E~(A) := A. p 2. 2. If If R R is is a a relation relation symbol symbol of of .c s and and D DR[g] the definition definition form form of of R R with with respect respect R [it] the n. to P, then to P, then we we set set E�(R( E~ _1_, E~ R (a)) := := 1-, _1_, E�(..., a)) :=:= 1-, l 1 :- E E~(DR[~]), EE~+I(R(~)) p (DR[a]), EE~+I(-~R(~)):p (...,DR [a]). �+ (R(a)) := �+ (""R(a)) := EE~(~DR[~]). 3.3. The The propositional propositional connectives connectives and and quantifiers quantifiers are are dealt dealt with with in in the the obvious obvious way: way: B) ::= - E E~(A) B) ::=- E E~(A) EE~(A p (B), p (A) /\A EE~(B), p (A /\A B) p (A) VV EE~(B), p (B), EE~,(A p (A VV B) := 3xE 3xE~,(A)(x), xEp (A)(x). EE~(3xA(x)) p (V'xA(x)) :=:= V'VxE~(A)(x). p (3xA(x)) := p (A)(x), EE~(VxA(x)) Now Now the the lemma lemma below below reduces reduces truth truth of of A A in in the the finite finite stages stages Jv.n� :iffJt~ of of the the lower lower three-valued structures structures Jv.np three-valued :iMp to to two-valued two-valued validity validity of of the the formulas formulas E E~(A). p (A). ItIt isis essentially proved essentially proved by by main main induction induction on on n n and and side side induction induction on on the the rank rank of of the the formulas formulas involved. involved. 655 655 A Proof-Theoretic Pro@ Theoretic Framework for for Logic Programming Programming Let Let P P be be a a logic logic program program in in £, s and and !JJt ffJ~ a a four-valued four-valued £, s structure. structure. Then have for closed £,s formulas A and Then we we have for all all closed formulas A and all all n n <w w": 3.4.2. 3 . 4 . 2 . Lemma. Lemma. J!JJt � (A) = 3ffJC~p(A) = tt � -: - ~ E�(A) E~(A). . J3ff~p 9Jtj, F This lemma may be be combined combined with with Theorem Theorem 2.5.1. 2.5.1. As As aa result result we we have have aa reduction reduction This lemma may of of truth truth in in the the finite finite stages stages J� 3~ to to purely purely equational equational reasoning. reasoning. Let Let P P be be aa logic logic program program in in s£', A A an an £, s formula formula and and n n < w w.. Then Then we we have have the the equivalence equivalence 3.4.3. 3.4.3. Theorem. Theorem. J�(A) ?Pp(A) = = tt ~, ,~ � CETs IF- E�(A) E ~ ( A ) ,, CETc. provided provided that that £, s is is infinite. infinite. On On the the other other hand, hand, if if £, s is is finite, finite, then then one one only only obtains obtains that that J�(A) :I~p(A) = = tt -: "� CETc. CETc + + DCAc. DCAL It- E�(A). E~(A). 4. Deductive 4. D e d u c t i v e systems s y s t e m s for f o r logic l o g i c programs programs After the the preceding preceding semantic semantic considerations considerations we we will now approach approach logic logic program programAfter will now ming procedural way. ming in in aa more more deductive deductive and and procedural way. Traditionally, Traditionally, aa logic logic program program is is often often regarded possible to regarded as as aa set set of of axioms. axioms. Alternatively, Alternatively, however, however, it it is is also also possible to replace replace this this programs-as-theories interpretation interpretation of of logic logic programs programs by by aa programs-as-deductivesystems paradigm paradigm (cf. (cf. e.g. e.g. Hallniis Halln~is and and Schroeder-Heister Schroeder-Heister [1990] [1990],, Jager J~iger [1994] [1994], ' Schroeder-Heister [1991] conceptually closer Schroeder-Heister [1991], Stark St~irk [1991,1994a]) [1991,1994a]) so so that that one one is is conceptually closer to to aa ' procedural understanding understanding of procedural of logic logic programming. programming. We begin with introducing aa calculus R(P) for P, which We begin with introducing calculus TO(P) for each each logic logic program program P, which will will provide are provide the the widest widest framework framework for for the the following following considerations. considerations. The The systems systems R(P) 7r are designed the model designed for for aa proof-theoretic proof-theoretic treatment treatment and and form form aa link link between between the model theory theory of -resolution) of logic logic programming programming and and very very specific specific proof proof procedures procedures (like (like SLDNF SLDNF-resolution) suited will also R(P) which suited for for implementations. implementations. Later Later we we will also study study subsystems subsystems of of the the 7r which arise arise naturally naturally in in the the context context of of logic logic programming. programming. programs-as-theories systems programs-as-deductive 4.1. 4.1. The The calculi calculi R(P) TO(P) The The following following deduction deduction systems systems are are presented presented in in aa Tait-style Tait-style manner. manner. Accord Accordingly, ingly, the the axioms axioms and and derivation derivation rules rules are are formulated formulated for for finite finite sets sets of of £, Z: formulas formulas which which are are interpreted interpreted disjunctively. disjunctively. The The capital capital Greek Greek letters letters r, F, A A,, II, H, �, E , .. ... (pos (possibly sibly with with subscripts) subscripts) denote denote finite finite sets sets of of £, s formulas, formulas, and and we we write write (for (for example) example) r, A {A, B}. r, F, A A,, A, A, B B for for the the union union of of F, A and and {A, B}. Given Given aa four-valued four-valued structure structure !JJt ffJr for for £' s, we ( r) for we sometimes sometimes simply simply write write !JJt 9Jt(F) for the the truth truth value value of of the the universal universal closure closure of of the the disjunction t . We disjunction of of the the formulas formulas in in r F according according to to !JJ YJ~. We say say that that r F is is valid in in !JJt ffJr if if ~ ( r( r) ) = : tt.. !JJt The for The systems systems R(P) 7r for logic logic programs programs P P are are extensions extensions of of the the usual usual Tait Tait calculus calculus for for predicate predicate logic logic (cf. (cf. e.g. e.g. Tait Tait [1968]) [1968]) by by adding adding equality equality axioms axioms and and so so called called . valid G. Jager and R. Stark 656 program which take care of program clauses clauses in the program rules rules which take care of the the program in P. P. Altogether Altogether we we have have the following axioms and and rules. following five five classes classes of of axioms rules. I. Logical I:- formulas Logical axioms. a x i o m s . For For all all atomic atomic/:: formulas A" (LI) (L1) F,-~A, A, (L2) (L2) F, T. T. The The axioms axioms (LI) (L1) are are often often called called identity axioms. They They will will play play an an important important role role later. later. II. Equality II. E q u a l i t y axioms. a x i o m s . For For all all I:Z: terms terms al a l ,, .. ... ., a, an, n , bI b l,, .. ...., b, bn n so so that that the the set set of of equations {a bn} is equations {all = = bI b l,, .. .. .., , an an = = bn} is not not unifiable: unifiable: A: r, -,A, A, r, identity axioms . (EI) (El) r, # # bn. c, d For For all all I:s terms terms aI a l ,, .. ... ., , an, an, bI b l,, .. .. .., , bn, bn, c, d so so that that {al {al = - bI b l,, .. .. .., , an an = = bn} bn} is is unifiable unifiable with with a a most most general general unifier unifier ()0 and and cO is is syntactically syntactically identical identical to to dO: c() (E2) (E2) d(): F, al ~ b l , . . . , an ~ bn, c = d. A B, u III. III. Logical Logical rules. rules. For For all all I:s formulas formulas A and and B, all all I:s terms terms a a and and all all free free variables variables u which do occur in which do not not occur in F, VxA(x): r, V'xA(x): r,F, AA r,F, BB (1\), r, BB V2) , F, r,F, AA A1\BB (A), r,F, AA VvBB ((V2), r,F, A(a) r,F, A(u) A(a) A(u) , (3) V'). 3xA(x) (3), r,F, V'VxA(x) xA(x) ((V). r,F, 3xA(x) IV. rules. For IV. Cut C u t rules. For all all I:Z: formulas formulas A: A: r,F, AA r,r, -,A ~A (cut) . rF The The formulas formulas A A and and -,A ~A are are called called the the cut cut formulas formulas of of this this cut; cut; the the rank rank of of aa cut cut is is r,F, AA VI ) , r,F, AA VVBB ((Vl), the the rank rank of of its its cut cut formulas. formulas. R V. P. For definition form V. Program P r o g r a m rules rules for for P. For every every relation relation symbol symbol R of of I:Z: and and its its definition form Dn[~7] with with respect respect to to the the logic logic program program P P and and all all I:s terms terms ~ we we have have the the following following positive positive and and negative negative program program rules: rules: DR[U] r,F, DR[ii] DR[ ] (+R), r,r, R(ii) ( +R) , r, ii (-R). One should should emphasize emphasize that One that the the program program rules rules are are impredicative impredicative in in the the sense sense that that the the rank rank of of the the main main formula formula of of the the premise premise of of such such aa rule rule is is in in general general greater greater than than the the rank corresponding conclusion. rank of of the the main main formula formula of of the the corresponding conclusion. R(P) is Based Based on on these these axioms axioms and and rules rules of of inference inference derivability derivability in in TO(P) is introduced introduced R(P) ~_n F expresses in in the the standard standard way. way. The The notation notation TO(P) expresses that that F is is provable provable in in R(P) Ti(P) by by a a proof proof whose whose length length and and cut cut complexity complexity are are bounded bounded by by n n and and r, r, respectively. respectively. f-� r r 657 657 Proof-Theoretic Framework Framework .for for Logic Programming Programming AA Proof-Theoretic .c .c 4.1.1. Definition. Definition. Let Let PP be be aa logic logic program program in in s and and Fr aa finite finite set set of of s formulas. formulas. 4.1.1. Then we we define define 7~(P) R(P) F-~ I-� Fr for for all all n, r << w by by induction induction on on n. Then 1. of TO(P), 1. If If rr isis an an axiom axiom of R(P) , then then we we have have 7r R(P) ~n I-� Fr for for all all n, r << w. 2. If If TO(P) R(P) F-~' I-�i Fi ri and and ni < n for for every every premise premise Fi ri of of aa logical logical rule, rule, aa program program 2. rule have 7~(P) then we we have R(P) f-~ I-� Fr for for rule or or aa cut cut of of 7~(P) R(P) whose whose rank rank is is less less than than r, then the conclusion conclusion Fr of of that that rule. rule. the n, r ni n n. n, r r, n We write write RR(P) I-r Fr ifif there there exists exists an an n << w so so that that RR(P) I-� F. r. Similarly, Similarly, nR(P) I- Fr We ( P ) t-r ( P ) t-~ ( P ) ~so that that 7~(P) R(P) t-r I-r F. r. is written written ifif there there exists exists an an rr << w so is 4.1.2. emark. Each to the 4.1.2. R Remark. Each TO(P) R(P) is is aa deductive deductive system system which which corresponds corresponds to the of the the logic logic program P; i.e., i.e., we we have all logic logic programs programs PP completion comp(P) of completion program P; have for for all in s and and all all L formulas formulas A" A: in .c comp(P) .c R ( P ) ~I-AA R(P) ~, '; {::=} comp(P) F c o m p ( P ) ~ AA. . The notation notation R(P) R ( P ) ~-0 means that cut-free proof of F (P).. The 1-0 Fr means that there there exists exists aa cut-free proof of r in in R R(P) Later we will say more about cut elimination elimination in in TO(P) R(P) and Later we will say more about cut and related related systems. systems. For For the we only TO(P).. Since the moment moment we only mention mention aa partial cut elimination theorem for for R(P) Since the the ranks of the main main formulas formulas of of the the logical equality axioms axioms and and the the program program rules rules ranks of the logical and and equality of R(P) 7~(P) are proved by standard techniques, techniques, as as presented presented for for example example in of are 0, 0, itit is is proved by standard in Girard or Takeuti Takeuti [1987]. [1987] . Girard [1987b], [1987b], Schutte Schiitte [1977] [1977] or partial cut elimination theorem 4.1.3. 4.1.3. Theorem. Theorem. formulas: formulas: We for all all logic all finite finite sets r of .c We have have for logic programs programs P P in in .c s and and all sets F of f_, R(P) r R ( P ) IF-F ~ ==} R(P) R ( P ) ~h I F r. . 4.2. 4.2. Identity-free I d e n t i t y - f r e e derivations d e r i v a t i o n s in in R(P) TO(P) Now Now we we turn turn to to the the subsystems subsystems of of the the calculi calculi R(P) TO(P) which which deal deal with with identity-free identity-free with lower derivations derivations only. only. They They are are of of great great importance importance in in connection connection with lower three threevalued valued structures structures which which are are adequate adequate to to logic logic programs programs P P and and are are also also related related to to SLDNF -resolution. SLDNF-resolution. If R(P) under If we we consider consider the the system system T~(P) under the the aspect aspect of of lower lower three-valued three-valued logic, logic, we we make the the following following observation: observation: (i) (i) The The identity axioms of of R(P) 7~(P) are are not not valid valid in in the make identity axioms the sense equality axioms valid in sense of of lower lower three-valued three-valued structures; structures; (ii) (ii) the the equality axioms of of R(P) R ( P ) are are valid in all all lower lower three-valued three-valued equational equational structures; structures; (iii) (iii) if if every every premise premise of of a a propositional propositional rule, R{P) rule, quantifier quantifier rule rule or or cut cut rule rule of of R ( P ) is is valid valid in in aa lower lower three-valued three-valued structure structure rot, 9Yt, then then the the conclusion conclusion of of that that rule rule is is also also valid valid in in rot. ff/t. This This means means that that m besides besides the the identity identity axioms axioms and and the the program program rules r u l e s- all all axioms axioms and and rules rules of of R{P) T~(P) are are correct correct in in the the lower lower three-valued three-valued sense. sense. Moreover, Moreover, if if rot ffJt is is aa lower lower three-valued three-valued structure structure which which is is adequate adequate to to aa logic logic program program P, P, then then also also the the program program rules rules are are valid valid in in rot. ffJt. Hence Hence from from the the point point of of view view of of lower lower three-valued three-valued logic logic only only the the identity identity axioms axioms create create some some problems. problems. For For this this reason reason it it is is very very pleasing pleasing that that just just deleting deleting the the identity identity axioms axioms yields yields an an interesting interesting subsystem subsystem of of R{P) 7~(P).. 658 658 G. Jiiger Jager and and R. R. St6rk Stark G. 4.2.1. Definition. Definition. Let Let PP be be aa logic logic program program in in sC and and Fr aa finite finite set set of of sC formulas. formulas. 4.2.1. Then we we define define 7r R{P) H-~ II-� Fr for for all all n, r << w by by induction induction on on n. Then 1. If If Fr is is an an axiom axiom of of Tr R{P) but but not not an an identity identity axiom, axiom, then then we we have have Tr R{P) ~-~ II-� rr 1. for all all n, r << w. w. for 2. If If 7~(P) R{P) ~II-�i Fi r and 2. and ni << n for for every every premise premise Fi ri of of aa logical logical rule, rule, aa program program rule ( P ) whose rule or or aa cut cut of of nR{P) whose rank rank isis less less than than r, then then we we have have TO(P) R{P) H-~ II-� Fr for for the conclusion conclusion Fr of of that that rule. rule. the n, r n, r i n. ni n r, The shorthand shorthand notations notations Ti(P) R{P) H-r II-r Fr and and "R(P) R{P) HII- Fr will will be be used used in in the the obvious obvious The sense. Hence Hence 7~(P) 'R.{P) HII- Ff means means that that Ff is is derivable derivable in in 7r 'R.{P) by by aa proof proof which which does does sense. not make make use use of of the the identity-axioms. identity-axioms. not 4.2.2. R Remark. The omission omission of of the the identity identity axioms axioms does does not not affect affect the the equality equality 4.2.2. e m a r k . The A: formulas. We for all all logic logic programs programs in in sC and and all all equality equality formulas formulas A: formulas. We have have for CETt:. R{P) A. T~(P) IItt-A. One conclude that that the the identity-free identity-free derivations in 'R.{P) One can can immediately immediately conclude derivations in 7~(P) are are correct correct with respect to the structures which are adequate P. In In with respect to the lower lower three-valued three-valued structures which are adequate to to P. Stark [1991] [1991] also the corresponding completeness result is proved. St~irk also the corresponding completeness result is proved. The The approach approach presented there makes of techniques techniques similar similar to to Schiitte's Schiitte ' s deductive chains (see presented there makes use use of deductive chains (see Schiitte [1977]) to Girard's Schiitte [1977]) and and is is dual dual to Girard's proof proof of of the the completeness completeness of of the the cut-free cut-free rules rules of [1987b] . Semantic elimination can of the the sequent sequent calculus calculus in in Girard Girard [1987b]. Semantic cut cut elimination can be be derived derived from the proof the following from the proof of of the following completeness completeness result. result. C E T L ~f-AA 4.2.3. Theorem. 4.2.3. T heorem. las A A:: las ~� Let C. Then Let P P be be aa logic logic program program in in f_,. Then we we have have for for all all C f~ formu formuR{P) A Ti(P) IIH-A {::: z ,.:::} P P ~FV' v A . A. It It is is possible possible tt,)o tie tie identity-free identity-free derivations derivations in in 'R.{P) T~(P) very very closely closely to to finitely finitely iterated iterated program program envelopes envelopes of of three-valued three-valued structures. structures. Given Given aa lower lower three-valued three-valued C s struc structure ture VJt ffJt and and aa logic logic program program P P in in sC, we we define define for for all all n < w w:: ffYt~ ::= 9Jt and 9Jt~,+1 ::= ffJt~[P].. VJtj, = VJt and VJtpH = VJtp[P] This This (iterated) (iterated) formation formation of of envelopes envelopes is is the the semantic semantic counterpart counterpart of of the the program program rules. rules. Therefore Therefore aa formula formula proved proved in in n steps steps is is true true in in the the n-times n-times iterated iterated envelope. envelope. n n 4.2.4. Let 4.2.4. Theorem. Theorem. Let P P be be aa logic logic program program in in C L and and VJt ff~ aa lower lower three-valued three-valued equational equational structure structure for for C ~, so so that that VJt 91t � U_VJt[P] 9~[P].. Then Then we we have have for for all all finite finite sets sets of of C f~ formulas formulas and and all all n, n, rr < w w": R{P) � VJtp{f) n(p) II-� H-7 r r ~ ~,(r) = = tt.. This This assertion assertion iiss verified verified by by simple simple induction induction on on n. Together Together with with Theorem Theorem 4.2.3 4.2.3 it it implies implies the the following following corollary, corollary, which which was was first first obtained obtained by by Kunen Kunen [1987] [1987] by by aa purely purely model-theoretic model-theoretic proof proof based based on on ultrapower ultrapower constructions. constructions. The The above above theorem theorem and and its its corollary corollary are are quite quite remarkable remarkable since since the the structures structures 9.JtJ, 9~p are are in in general general not not adequate adequate to to P P (cf. (cf. Example Example 3.3.2) 3.3.2).. n. 659 659 Proof-Theoretic .for Logic Programming A ProofTheoretic Framework for Let P P be be aa logic logic program program in in C, L, A A aa closed closed C L formula formula and and 9Jt Let lower three-valued three-valued equational equational structure structure for .for C L so so that that 9Jt 9Y~ � E_ 9Jt[P] 9~[P].. Then Then P P t=v ~7 A A aa lower implies that that there there exists exists an an n n < w so so that that !JJ1 9~p(A) implies 1, (A) = tt.. 4.2.5. Corollary. Corollary. 4.2.5. = In aa next next step step we we compare compare identity-free identity-free derivability derivability iinn R(P) 7~(P) with with truth truth iinn the the finite finite In stages of of Jp :JR.. For For this this purpose purpose we we make make use use of of some some notions notions and and results results which which have have stages been presented presented at at the the end end of of Section Section 3.4. 3.4. been 4.2.6. Lemma. Lemma. 4.2.6. and all n < w w": and all n Let Let P P be be aa logic logic program program in in C. ~,. Then Then we we have have for for all all C L formulas formulas A A 7~(P) If--H--~E~(A), A. R(P) --,E� (A) , A. n The proof proof of of this this lemma lemma by by main main induction induction on on n and and side side induction induction on on the the rank rank of of A A The is aa matter matter of of routine. This relationship relationship between between the the formulas formulas A A and and Ep(A) E~,(A) permits permits is routine. This the integration integration of of Theorem Theorem 3.4.3 3.4.3 into into the the context context of of R(P) 7~(P) and and provides provides soundness soundness the and completeness completeness with with respect respect to to the the family family (Jp (:J~ :"n < < w w). and ). n 4.2.7. Let PP bebe aa logic 4.2.7. Theorem. Theorem. Let logic program program in in C L and and A A aa closed closed C L formula. formula. Then Then we have have the the equivalence we equivalence :J~p(A) = = tt L Jp(A) n(:w n<w -: :-::::} {::: T~(P) If--H- A, A, R(P) provided provided that that C ~, is is infinite. infinite. On On the the other other hand, hand, if if C ~, is is finite, finite, then then one one only only obtains obtains that that 3~(A) = = tt L Jp(A) n<w -,' .'-::::} {::: n ( P ) If--H- (DeAL: (DCAL -+ ~ A). A). R(P) n P r o o f . We assume first first that that sC is If Jp(A) :J~(A) -= tt for some n < Proof. We assume is infinite. infinite. If for some < w, then then we can Theorem 3.4.3 Remark 4.2.2 4.2.2 and and conclude conclude that that nR(P) ( P ) ~~(A).. we can apply apply Theorem 3.4.3 and and Remark If--- E Ep(A) The lemma and cut yield yield the the assertion. assertion. On On the the other other hand, if we we have The previous previous lemma and aa cut hand, if have R(P) HIf--- A, then Theorem 4.2.4 implies implies 2[~,(A) Jp(A) == tt for some n << w. w . In In the the second second T~(P) A, then Theorem 4.2.4 for some case C L: is is finite. we have have only observe that that the closure axiom axiom DCAL case finite. Then Then we only to to observe the domain domain closure DeAL: D is valid valid in in all all Herbrand structures and and can [:] is Herbrand structures can proceed proceed as as before. before. Theorem and Theorem approach to an interesting Theorem 4.2.3 4.2.3 and Theorem 4.2.7 4.2.7 provide provide aa proof-theoretic proof-theoretic approach to an interesting result which exhibits exhibits the connections between between statements statements with with respect the result which the close close connections respect to to the fifinite nite stages stages of of ~p and lower lower three-valued three-valued consequences Its first first part part is is originally originally Jp and consequences of of PP.. Its due Kunen and due to to Kunen and proved proved in in Kunen Kunen [1987]. [1987]. n 4.2.8. C Corollary. 4.2.8. orollary. have the equivalence have the equivalence Let be a logic program in and an formula. Then we P be a logic program in f~ C and AA an LC formula. Then we Let P L 3~p(A) Jp(A) == tt ~<~ n<w {:::,~::::} ~, A, P ~t=v P v A, provided that is infinite. On the other hand, if is finite, then one only obtains that provided that sC is infinite. On the other hand, i / sC is finite, then one only obtains that L :I~p(A) Jp(A) == tt ~{:::::::} PP ~t=v (DeAL: ~-+ A). A) . v (DCAL nn<w (w G. G. Jager Jiiger and R. Stark StSrk 660 660 4.3. Cut elimination 4.3. Cut elimination for for identity-free identity-free derivations derivations We end end this this section section with with aa theorem theorem which which states states that that all all identity-free identity-free derivations derivations We in in R(P) 7~(P) can can be be transformed transformed in in identity-free identity-free derivations derivations without without cuts. cuts. 4.3.1. 4.3.1. Theorem. Theorem. We We have have for for all all logic logic programs programs P P and and finite finite sets sets r F of of C f_, formulas: formulas: R(P) T~(P) ItH- r F ====> =~ R(P) T~(P) It-o H-o r. F. Proof. P r o o f . This This proof proof is is based based on on the the following following idea: idea: We We interpret interpret R(P) T~(P) with with respect respect to its its identity-free identity-free derivations derivations into into aa ramified ramified system system R*(P) 7~*(P),, eliminate eliminate the the cuts cuts to in in R*(P) T~*(P) and and then then go go back back from from R*(P) 7~*(P) to to R(P) TO(P).. (P) by infinite family ,... Each relation symbol Each relation symbol R R of of C s is is replaced replaced in in R* 7~*(P) by an an infinite family RO, R ~ Rl R1,... of stratified stratified relation relation symbols. symbols. Given Given a a natural natural number number n, n, we we obtain obtain the the n-translation n-translation of R*(P) by of the the C s formula formula A A into into the the formula formula A A nn of of 7~*(P) by replacing replacing all all relation relation symbols symbols R R of in A A by by R R nn.. Then Then it it is is easy easy to to see see that that the the definition definition of of rank rank can can be be modified modified so so that that in rn(D~(g)) < rn(Rn+l(~)) (1) (1) for for all all relation relation symbols symbols R, R, all all terms terms a ~ and and all all natural natural numbers numbers n; n; the the rank rank of of disjunctions, quantified formulas disjunctions, conjunctions conjunctions and and quantified formulas is is defined defined from from its its subformulas subformulas as as before. before. The The ramified ramified system system R*(P) 7~*(P) contains contains the the logical logical axiom axiom (L2), (L2), the the equality equality axioms axioms (E1) the logical logical rules rules and and the (El) and and (E2) (E2),, the the cut cut rules rules of of R(P) 7~(P),, all all formulated formulated for for the the language identity axioms instead of language of of R*(P) 7~*(P).. The The identity axioms of of R(P) 7~(P) are are omitted, omitted, and and instead of the the program rules rules we all natural natural numbers numbers m and program we have have for for all and n n so so that that m < n n the the following following stratified stratified program program rules: rules: r, ""D�[a] F, ~D~[~] r, D�[a] F, D~[~] r, ...,Rn (a) · If finite set then � called n-suited If r F is is a a finite set of of C s formulas, formulas, then A is is called n-suited for for r F if if it it results results from from r F by replacing each occurrence of by replacing each occurrence of a a relation relation symbols symbols R R in in r F by by Rm R m for for some some m m so so that that nn :S _< m m.. Then Then the the following following is is proved proved by by straightforward straightforward induction induction on on n: n" * (P) f-b �. R (2) R(P) T~(P) fbnn r F and and � A is is n-suited n-suited for for r F ====> ~ T~*(P) A. (2) r, r, Rn (a) Because (1) it Because of of (1) it is is further further possible possible by by standard standard techniques techniques to to eliminate eliminate all all cuts, cuts, i.e., i.e., R*(P) 7~*(P) fF- r F ====> } R*(P) f-o r 7~*(P)F-0 F (3) (3) for (E1) and for all all finite finite sets sets r F offormulas offormulas of of R*(P) 7~*(P).. Note, Note, that that the the equality equality axioms axioms (El) and (E2) (E2) are nite set are closed closed under under cuts. cuts. Finally, Finally, if if r F is is a a fi finite set of of formulas formulas of of R*(P) 7~*(P),, then then we we write write ro relation symbols F ~ for for the the set set of of C s formulas formulas which which results results from from r F by by replacing replacing all all relation symbols Rm Rm by R. Then induction on by R. Then an an easy easy induction on the the length length of of the the derivation derivation shows: shows: R* (P) f-o 7~*(P) ~0 r F ====> ==~ R( P) f-o ro. 7~(P)~0 F~ (4) (4) From From (2)-(4) (2)-(4) we we can can conclude conclude that that all all cuts cuts can can be be eliminated eliminated from from the the identity-free identity-free 0 derivations in R(P) derivations in 7~(P).. [] A Proof-Theoretic Proof-Theoretic Framework Frameworkfor Logic Logic Programming Programming 661 661 A corresponding corresponding result result for for derivations derivations which which make make use use of of identity identity axioms axioms is is in in general general A not not possible. possible. However, However, there there are are syntactically syntactically definable definable classes classes of of logic logic programs programs P P example call-consistent call-consistent and and stratified stratified programs) programs) so so that that the the systems systems R{P) T~(P) ((for for example permit the the elimination elimination of of all all cuts. cuts. Unfortunately Unfortunately space space does does not not permit permit to to say say more more permit about this this interesting interesting direction of research. research. about direction of 5 5.. SLDNF-resolution SLDNF-resolution The The well-known well-known SLD-resolution SLD-resolution 22 is is a a sound sound and and complete complete proof proof procedure procedure for for definite logic cf. e.g. Prolog as definite logic programs programs ((cf. e.g. Lloyd Lloyd [1987]). [1987]). Prolog as well well as as many many other other rela relational programming languages languages are are based based on on some some form form of of SLD-resolution. SLD-resolution. However, However, tional programming SLD-resolution SLD-resolution does does not not permit permit to to derive derive negative negative information. information. To To overcome overcome this this restriction, restriction, the the negation negation as as failure failure rule rule NF NF has has been been introduced introduced in in Clark Clark [1978] [1978].. 5.1. 5.1. Negation N e g a t i o n as as failure failure Intuitively, closed) negative Intuitively, one one wants wants aa ((closed) negative literal literal -,A ~A to to be be derivable derivable from from aa logic logic program possible attempts program P P by by means means of of NF NF if if and and only only if if all all possible attempts to to derive derive A A from from P P fail nitely many steps. This fail after after fi finitely many steps. This is is often often reflected reflected as as follows: follows: 1. If 1. If the the atom atom A A finitely finitely fails, fails, then then -,A -~A succeeds. succeeds. 2. 2. If If A A succeeds, succeeds, then then -,A - A fails. fails. There There exist exist various various versions versions of of SLDNF-resolution, SLDNF-resolution, many many of of which which are are discussed discussed in in the the survey survey article article Apt Apt and and Bol Bol [1994] [1994] and and in in Shepherdson Shepherdson [1989] [1989].. For For notational notational simplicity and rather simplicity we we restrict restrict ourselves ourselves in in the the following following to to one one particular particular ((and rather general) general) form form of of SLDNF-resolution. SLDNF-resolution. In In order order to to have have aa compact compact form form of of notation notation let let *9 be be aa new new symbol symbol which which does does not not belong C. Goal belong to to the the language language •. Goalforms forms of of C L are are all all expressions expressions of of the the form form G G1, G22 I , **,, G so so that that G G11 and and G G22 are are arbitrary arbitrary goals goals in in sC. Thus, Thus, goal goal forms forms are are finite finite sequences sequences of of literals symbol *9 in which *9 occurs occurs exactly exactly once. ] is a goal form and literals and and the the symbol in which once. If If Gl G[,] is a goal form and * H then G[H] results from G[] is H aa goal, goal, then G[H] is is the the goal goal which which results from Gl G[,] by replacing replacing *9 by by H; H; G[] is *] by the ] . the goal goal which which we we obtain obtain if if we we delete delete the the symbol symbol *9 from from G[ G[,]. * The The following following definition definition is is based based on on Kunen Kunen [1989] [1989].. For For aa given given logic logic program program P P in C, we R{P) consisting of in L, we define define a a set set R ( P ) consisting of pairs pairs formed formed from from goals goals and and substitutions substitutions R{P) G, ()) in i n /C, : , and and a a set set F{P) F ( P ) consisting consisting of of C s goals. goals. We We write write G GR ( P ) ()0 for for ((G, 0) E e R{P) R ( P ) .. R{P) G G R ( P ) ()0 means means that that the the goal goal G G succeeds succeeds from from the the program program P P using using SLDNF SLDNFresolution resolution and and returns returns answer answer ()0. G G E E F{P) F ( P ) , , on on the the other other hand, hand, means means that that G G finitely finitely fails fails from from P P using using SLDNF-resolution. SLDNF-resolution. . 5.1.1. C. Then R{P) 5.1.1. Definition. Definition. Let Let P P be be a a logic logic program program in in L. Then the the sets sets R ( P ) and and F F ((P) P) are are defined defined to to be be the the least least sets sets which which are are closed closed under under the the following following conditions conditions for for all all positive literals A. positive literals A. (RO) (a0) 0 o R(P) R ( P ) rc . 2SLD 2SLD stands for 'linear resolution for definite clauses clauses with selection selection function' G. J~iger Jager and R. Stiirk Stark G. 662 662 P -H G[A], a A G[H]a R{P) T (var{Aa) var(G[H]a)) var(G[H]aT) G[A] R{P) (J (J aT G[A]. ( R2) If If A A EE FF{P), A isis closed closed and and G[] GO RR{P) (J, then then G[-~A] G[....,A] RR{P) (J. (a2) (P), A ( P ) 0, ( P ) 0. (El) of all so that ( F1 ) Let Let P(A) F{A) bebe the the collection collection of all clauses clauses of of PP which which have have aa variant variant so that its its head each element element of/5(A) exists aa variant A. IfIf for for each of F{A) there there exists variant head is is unifiable unifiable with with A. most B ( P ) for B: :- - gH containing containing no no variables variables from from G[A] G[A] soso that that G[g]a G[H]a EE FF{P) for aa most of A A and and B, I!, then then G[A] G[A] EE F(P). F{P). general unifier unifier aa of general ( F2) If If A A RR{P) and there there exists exists aa substitution substitution aa so so that that AOa A(Ja isis identical identical to to A, (F2) ( P ) 0(J and A, then G[-,A] G[....,A] eE F(P). F{P). then (R1) ( R1 ) Let Let BB: -: H be be aa variant variant of of aa program program clause clause of of P which which contains contains no no varivari and let let a be be aa most most general general unifier unifier of of A and and B. B. ables from from the the goal goal G[A], and ables If If G[H]a R ( P ) T and and (var(Aa) \\ var(G[H]a))Mn var(G[H]aT) = 0, 0, then then G[A] R ( P ) 0 for for the the restriction restriction 0 of of aT to to the the variables variables of of G[A]. Condition (R0) ( RO) says says that that the the empty empty goal goal returns returns the the identity identity substitution. substitution. In In Condition rule ((R1), R1) , an an atom selected from from the the goal goal G and program clause clause of of P is is rule atom A is is selected and aa program chosen. clause is it does does not not contain from G. chosen. The The clause is renamed renamed so so that that it contain any any variables variables from If the head head of the clause new goal derived from If the of the clause is is unifiable unifiable with with A, then then aa new goal is is derived from G by by replacing the the selected atom A with with the the body body of of the the clause clause and and applying applying the the most most replacing selected atom to the the new goal. If returns an substitution T, unifier a to general unifier general new goal. If the the derived derived goal goal returns an answer answer substitution then of a and to the the variables of the original goal goal G, then the the composition composition of and r, restricted restricted to variables of the original is Rule (F1) ( F1 ) is rule (R1). ( R1 ) . An atom A is goal G is returned. returned. Rule is dual dual to to rule An atom is selected selected from from the the goal and goal from fails, then then G fails. fails. The R2 and ( F2 and if if every every derived derived goal from G using using A fails, The rules rules ((R2) and (F2) switch from success to failure and vice versa. versa. Usually, Usually, it required in in rule rule (F2) ( F2 switch from success to failure and vice it is is required that is closed. closed. In In this this article article we work with with an an asymmetric asymmetric version version of of that the the atom atom A is we work SLDNF -resolution. The The reason reason for for choosing choosing this this variant -resolution is is that that SLDNF-resolution. variant of of SLDNF SLDNF-resolution it exactly what Theorem 5.3.5. it is is exactly what is is needed needed for for the the Completeness Completeness Theorem 5.3.5. A a G A, A a T, G P G. G T, G, G ) ) ) A G A A 5.1.2. This 5.1.2. Example. Example. This example example explains explains why why the the condition condition on on variables variables in in rule R1 ) of rule ((R1) of the the previous previous definition definition is is necessary. necessary. Let Let P be be the the following following logic logic program: program: P R( R(y(u)) J {u)) S{u, S(u, v) : - R{u) R(u) Obviously, f{v)}. IfIf we Obviously, we we have have R{u) R(u) R{P) R ( P ) {u/ (u/f(v)}. we omit omit the the variable variable condition, condition, then then we we v) R{P) f{v)} asas well. But the the answer answer {u/ f{v)} isis not not most most general obtain obtain S{u, S(u, v) R ( P ) {u/ (u/f(v)} well. But (u/f(v)} general {u/ f{w)}. for for S{u, S(u, v). v). A A most most general general answer answer is is (u/f(w)}. The The definition definition of of the the sets sets R{P) R ( P ) and and F{P) F(P) is is very very much much directed directed towards towards an an actual actual implementation implementation of of SLDNF-resolution, SLDNF-resolution, and and as as aa consequence consequence we we are are willing willing to to accept accept aa rather perspicuous formulation, rather un unperspicuous formulation, which which is is partly partly caused caused by by the the fact fact that that one one has has to to deal deal with with most most general general unifiers unifiers and and variants variants of of clauses. clauses. Later Later we we will will prove prove soundness soundness and and completeness completeness of of SLDNF-resolution. SLDNF-resolution. In In order order to to simplify simplify the the proofs proofs of of these going to these results, results, we we are are now now going to present present aa theoretically theoretically more more transparent transparent approach approach to to SLDNF-resolution SLDNF-resolution by by means means of of sets sets Y(P) and and N(P). They They correspond correspond to to the the sequent sequent calculus calculus Sp Sp of of Buchholz Buchholz [1992]. [1992]. Y{P) N{P). 663 663 A ProofTheoretic Framework for Proof-Theoretic for Logic Programming P Y(P) 5.1.3. Let 5.1.3. Definition. Definition. Let P be be aa logic logic program program in in sC. Then Then the the sets sets Y(P) and and the least N(P) are are defined defined to to be be the least sets sets of of C s goals goals which which are are closed closed under under the the following following conditions conditions for for all all positive positive literals literals A and and goal goal forms forms G[.]" (YO) 0 (Y0) ~ E e V(P). (Yl) (Y1) Let Let B B ':-- H be be aa program program clause clause from from P and and let let a be be an an C s substitution substitution so so that that then G[A] Ee Y(P). A = = B Ba. If If G[Ha] E e Y(P), then literals only. (Y2) Let G be be the the goal goal ~A1,...,--Ak which which consists consists of of negative negative literals only. If If Y2 Let Ai Ai E E N(P) for for all all 11 ::; < ii ::; < kk,, then then G E e Y(P). (Nl) (N1) If If Ga[HT] E E N(P) for for all all program program clauses clauses B B ':-- H of of P and and all all C s substitutions substitutions a and and T so SO that that Aa = B ST, then then G[A] E e N(R). N2 If (N2) If A E e Y(P), then then G[-,A] E e N(P). N(P) A Gl*]: Y(P). H P a A a. G[Ha] Y(P), G[A] Y(P). ,AI, . . . , ,Ak ( ) G N(P) G Y(P). Ga[HT] N(P) H P a T Aa T, G[A] N(P). ( ) A Y(P), G[,A] N(P). The The sets sets Y(P) Y(P) and and N(P) N(P) have have several several natural natural closure closure properties. properties. remark remark we we list list three three of of those those which which will will be be needed needed later. later. In In the the following following Let Let P P be be aa logic logic program program in in C E and and a a an an C s substitution. substitution. G Y(P), then Y(P). then Ga Ga E e Y(P). G N(P), then then Ga Ga E e N(P). N(P). G[H] Y(P) ifif and and only only if if G[] G[] E e Y(P) Y(P) and and H H E e Y(P). V(P). The relationship between The relationship between the the sets sets Y(P) Y(P) and and N(P) N(P) and and R(P) R(P) and and F(P) F(P) is is rather rather intricate. A intricate. A first first and and easy easy observation observation shows shows that that Y(P) Y(P) and and N(P) N(P) comprise comprise the the previously R(P) and lemma, which previously defined defined R(P) and F(P) F(P) in in the the sense sense of of the the following following lemma, which is is proved proved by by induction induction on on the the definition definition of of R(P) R(P) and and F(P). F(P). 5.1.5. Lemma. 5.1.5. L e m m a . Let P P be be aa logic logic program program in in C s and and G G be be aa goal goal in in C s . 1. R(P) 1. If /]" G GR ( P ) 00, , then then GO GO E e Y(P). Y(P). 2. 2. If /f G G e F(P), F(P), then then G G e N(P) N(P). . The The sets sets Y(P) Y(P) and and N(P), N(P), however, however, are are not not "equivalent" "equivalent" to to the the sets sets R(P) R ( P ) and and F(P). F(P). 5.1.4. 5.1.4. Remark. Remark. 11.. If If G E e V(P), 2. If 2. If G E e N(P), 3. G[H] Ee Y(P) 3. E E The lemma is The following following example example shows shows that that the the converse converse of of the the previous previous lemma is not not true true in in general. general. 5.1.6. 5.1.6. Example. Example. Let Let C s be be aa language language with with two two constants constants a and and bb and and two two relation relation symbols S. Consider the logic program in C which consists of the R and symbols R and S. Consider the logic program P in s which consists of the following following clauses: clauses: a R R(a) S(b) : - ,R (u) Then R(b) Ee N(P), -~R(b) S(b) Ee Y(P). But R b Ee Y(P) and Then we we have have R(b) and S(b) But there there is is no no substitution 00 so substitution so that that S(v) R(P) O. 0. P () N(P), , ( ) Y(P) Y(P). S(v) R(P) The F(P). The sets sets Y(P) Y(P) and and N(P) N(P) are are more more general general than than the the sets sets R(P) R(P) and F(P). Therefore Therefore + (P) and we define for logic program program P and we define for every every logic P two two sets sets of of goals, goals, S S+(P) and S S -- ((P), P ) , so so that that SLDNF-resolution R(P) and SLDNF-resolution in in the the sense sense of of R(P) and F(P F(P)) and and "derivability" "derivability" in in the the sense sense of of + (P) and Y(P) and and N(P) N(P) do do agree agree for for goals from S S+(P) and S S -- ((P). P). Y(P) goals from 664 664 G. Jager and R. Stark 5.1.7. be aa logic logic program program in in C. pair (S ++,, S-) goals 5.1.7. Definition. Definition. Let Let P P be Z:. A A pair S-) of of sets sets of of goals is is called called safe for for P P if if the the following following conditions conditions are are satisfied: satisfied: (A1) If If a a is is an an/:: substitution and and G E S+ S+,, then then Ga E S+ S+.. (AI) C substitution ((A2) Let B B ' -: - H H be be a a program program clause clause of of P P and and a a be be an an C /:: substitution substitution so so that that A2 ) Let A = =B Ba. If G[A] E S+ S +,, then then G[Ha] E S + +.. A a. If (A3) Let , . . . , ..,Ak which literals only. (A3) Let G be be the the goal goal ..,AI -~A1,...,--A~ which consists consists of of negative negative literals only. If If + , then G E SS+, then Ai 5- for all 1 _ i _< k. A/isis closed closed and and Ai E Sfor all k. ((B1) BI ) If is an If a a is an C Z: substitution substitution and and G E SS-,, then then Ga Ga E SS-.. (B2) Let Let B B ' -: - H H be be a a program program clause clause of of P P and and a a be be an an C /:: substitution so that that (B2) substitution so A a. If G[Ha] E SA = =B Ba. If G[A] E SS-,, then then G[Ha] S-.. (B3) G[..,A] E S(83) If If G[-,A] S-,, then then A E S+ S +.. safe 1�i� Obviously, P, and Obviously, (0, (0, 0) 0) is is safe safe for for P, and if if (st, (S +, SS~-)iei is aa family family of of safe safe pairs pairs for for P, P, then then ; )iEI is also for P. P. Therefore also the the union union (U (U/e1 S+, UiEI U/e1 S;) S~-) is is safe safe for Therefore every every program program P P has has aa iEI st, + (P), Slargest pair, called called (S largest safe safe pair, (S+(P), S - ((P) P)).. + (P) and The The complements complements of of the the sets sets S S+(P) and SS - ((P) P ) consist consist of of goals goals which which always always flflounder. ounder. They nition. We the They have have the the following following inductive inductive defi definition. We say say that that aa goal goal of of the the goal with an input clause clause of there exists exists aa form form G[H]B G[H]O is is a a resolvent of of the goal G[A] with an input of P P if if there variant P so er of variant B B :: - H H of of a a clause clause of of P so that that ()0 is is aa most most general general unifi unifier of A and and B. B. resolvent 5.1.8. Let 5.1.8. Definition. Definition. Let P P be be a a logic logic program program in in C. L. Then Then the the sets sets Fl F s + (P) and and ned to to be the least F F sl - (P) are are defi defined be the least sets sets of of sC goals goals which which are are closed closed under under the the following following conditions conditions for for all all positive positive literals literals A: A" . . . , ..,Ak 1. Let Let G be be the the goal goal ..,AI, --A1,..., --A~ which which consists consists of of negative negative literals literals only. only. If If Ai A/ is for at is non-ground non-ground or or Ai E FC F s (P) for at least least one one 1 < i < kk,, then then G E Fl+ Ft~ + (P) (P).. 2. If P, then 2. If H H E Fl+ F~ + (P) (P) and and H H is is a a resolvent resolvent of of G with with an an input input clause clause from from P, then 1. 1�i� G G E E Ft+ F s (P) . 3. (P) , then G[..,A] E F.e 3. If If A E F.e Fs + +(p), then G[~A] F s - (P) . 4. P, then then 4. If If H H E FC F s (P) and and H H is is a a resolvent resolvent of of G with with an an input input clause clause from from P, G a E F.eF t - ( P(P) ). . + (P) and + (P) is It It is is easy easy to to see see that that S S+(P) is the the complement complement of of F.e F/~+(P) and that that SS - ((P) P ) is is the the FC (P) . complement of complement of F•- (P). Let be a logic program in and be an goal. Then we have: 5.1.9. 5.1.9. Lemma. L e m m a . Let P P be a logic program in C s and G be an C f_. goal. Then we have: + 1. G E S+(P) G rt~g F.e 1. S+(P) � ~ F s (P) , 2. G rtr FC 2. G E SS - ((P) P) � ~ F s (P) . The "derivation" of The next next lemma lemma is is aa lifting lifting lemma. lemma. A A "derivation" of aa goal goal G(} GO in in Y(P) Y ( P ) is is lifted lifted down substitution aa down to to an an SLDNF-resolution SLDNF-resolution proof proof of of the the goal goal G which which returns returns aa substitution so induction on so that that Ga is is more more general general than than G(}. GO. The The lemma lemma is is proved proved by by induction on the the definition of definition of Y(P) Y ( P ) and and N(P) N ( P ) ; ; cf. cf. Stark St~irk [I994b]. [1994b]. 665 665 Proof- Theoretic Framework Framework for for Logic Logic Programming Programming AA Proof-Theoretic Let P be a logic program in and be an goal. 1. 1. IfIf GO GO EE YY((P) P ) and and GG EE SS++(P), (P), then then there there exist exist s.c substitutions substitutions aa and and TT SO so that that G RR(P) and GaT GaT = GO. GO . G ( P ) aa and 2. /IfG N(P) and GG eE SS--((P), then GG eE FF(P). 2. f G eE N ( P ) and P ) , then (P). 5.1.ID. LLemma. 5.1.10. emma. .c and GG be an f_. .c goal. Let P be a logic program in f_. = Is isis important important to to observe observe that that this this lemma lemma does does not not remain remain true true ifif we we restrict restrict rule rule (F2) (F2) Is of Definition Definition 5.1.1 5.1.1 to to closed closed atoms atoms A. of A. f 5.1.11. EExample. Let s.c be be the the language language with with one one unary unary function function symbol symbol f 5.1.11. xample. Let and one one unary unary relation relation symbol symbol R, R, and and let let P be be the the logic logic program program which which consists consists and of the the bodyless bodyless clause clause R(f(u)) only. only. Then Then we we have have R(f(u)) eE Y ( P ) and and thus thus of Since the the goal goal -~R(f(u)) belongs belongs to to S - ( P ) , the the goal goal -~R(f(u)) -,R(f(u)) eE N ( P ) . Since must be be in in F ( P ) according according to to the the previous previous lemma. lemma. Indeed Indeed this this is is true, true, because because must If rule rule (F2) ( F2) of of Definition 5.1.1 were restricted to to closed closed atoms, atoms, then R ( f (u)) R ( P ) rc. If Definition 5.1.1 were restricted then we would not have E we would not have -~R(f(u)) E F ( P ) . ...,R (f(u)) N(P). F(P) R(f(u)) R(P) P R(f(u)) ...,R(f(u)) R(f(u)) Y(P) S- (p), ...,R (f(u)) ""R(f(u)) F(P). 5.2. M Mode 5.2. o d e aassignments ssignments Some concerning the Some results results concerning the completeness completeness of of SLDNF-resolution SLDNF-resolution are are formulated formulated and and proved proved with with reference reference to to the the sets sets S+(P) and and S - ( P ) . In In general, general, however, however, it it is is not not decidable decidable whether whether a a goal goal belongs belongs to to one one of of these these sets. sets. In In fact, fact, the the set set G E E S+(P)} is is m II~-complete. In In this this section section we we will will therefore therefore introduce introduce {{(G, (G, P ) " : G something something like like a a syntactically syntactically decidable decidable "approximation" "approximation" of of S + (P) and and S- (P) which which covers covers most most practically practically relevant relevant cases. cases. . . . for Modes ((a, fl, %, 0'.0, co, (30, rio, ,0, 70,...) for n-ary n-cry relation relation symbols symbols R R are are n-tuples n-tuples 0'., (3" (Xl ( x l,, .. .. .., x, Xnn)) so so that that xi E {out, {out, log, log, in}. in}. If If xi = in in,, then then the the ith ith argument argument of of R R is is called called an an input argument. argument. If If X; xi = = out, out, then then the the ith ith argument argument of of R R is is called called an an output argument. argument. Otherwise, Otherwise, if if X; xi = log, log, then then the the ith ith argument argument of of R R is is called called aa logical argument. argument. . . . ,,xn). Xn) . Then Let Let A be be the the atom atom R R ((aI a l , .. ... ., , an) and 0'. a be be the the mode mode (Xl, (xl,... Then the the set set n and of of input variables of of A with with respect respect to to 0'. a is is the the set set of of variables variables which which occur occur in in input input arguments. arguments. The The set set of of output variables is is defined defined analogously: analogously: S+ (P) P) Modes input S- (P). S+ (P)} S+ (P) ) Xi Xi = S- (P) output logical = A input variables A a) output variables n },}, in(A, O'.) a ) ::= = in(..., in(~A, [J{ var(ai) war(hi)": 11 ::; <_ii ::; <_n, n, x~ = = iin in(A, A, O'.)a) ::== U{ out(A, A , O'.)a ) ::== U{ out(A, O'.) a ) ":= - out(..., out(-,A, [J{ var(ai) var(ai)": 11 ::; _ X; = J.L J.L J.L modes modes are are used used in in positive positive calls calls and and negative negative modes modes are are used used in in negative negative calls. calls. G. Jager Jdger and R. Stark Stark 666 666 A the set A mode mode assignment assignment/zp, is is extended extended in in aa natural natural way way from from the set of of relation relation symbols symbols to literals. If R( it) , we to the the set set of of all all literals. If A A is is the the atom atom R(~), we set set + (A) p, := p,+ (R), /z+ (A) :=/z+ (R), p,(A) :=/z-(R), := p,- (R), /z-(A) + (.A) p, := p,- (R), /z+ (-~A) :=/z-(R), p,-(.A) := p,+ (R) /z-(~A) :=/z+ (R).. A A program program must must be be written written in in such such aa way way that that in in aa computed computed answer answer the the output output variables variables are are contained contained in in the the input input variables. variables. Thus Thus in in aa certain certain way way aa mode mode assignment data flow assignment reflects reflects the the data flow of of aa logic logic program. program. Consider Consider aa clause clause A" - B 1 , . . . , Bin, ~C1, 999 -~Cn and suppose suppose that that A A is is called called with with some some input input values. values. Then Then these these values values are are passed passed and to computes some output values which are passed to B . This value , and B to B B1, and B1 computes some output values which are passed to B2. This value 2 1 1 the passing is continued until until one passing is continued one reaches reaches .C1 -~C1.• Finally, Finally, if if the the tests tests C1, C1,..., , Cn Cn fail, fail, the output values to A output values output values of of Bm Bm are are returned returned to A and and are are the the output values of of A. A. Formally, Formally, this this can can be be expressed expressed by by the the following following three three definitions. definitions. • • • 5.2.1. C. We 5.2.1. Definition. Definition. Let Let/zp, be be aa mode mode assignment assignment for for/:. We call call aa program program clause clause A A ' :-- L1, L1,..., Lnn p,-correct ~z-correct if if the the following following conditions conditions are are satisfied: satisfied: • . .,L (Cl) p,+ (A) there p,+ ( L 1 ) , , (3n Ee/z+(Ln) P,+ ( Ln) so (C1) For For all all modes modes a c~ E e/z+(A) there exist exist (31 fll E e/z+(L1),...,/~n so that that for for all 11 :'S all < ii :'S < n n:: (a) (a) out(A, out(A, ac~)) � C in(A, in(A, a) ~) U U U{ U{ out(Lj out(Lj,, (3j) ~j) :" 1 :::; < jj :::; <_n, n, Lj Lj positive positive }} (b) in( Li (3i ) � in(A, a) U{ out(Lj, (b) in(L,,/?,) C_ in(A, c~) U U U{ out(Lj, (3j) ~j) : 91 :::; < jj < i,i, Lj Lj positive positive }} if if L Lii is is positive, positive, (c) a) U U{ U{ out(Lj , (3j) :" 11 :::; (c) var(L var(Li)i ) � C_ in(A, in(A,c~)U out(Lj,~j) < jj :::; < n, n, Lj nj positive positive}} if if L nii is is negative, negative, - (L1), . . . , (3n Ee P, - (Ln) so (C2) (A) there (C2) For For all all modes modes a ce E e p,/z-(A) there exist exist (31 f l l eE p, /z-(L1),...,fin /z-(Ln) so that that in(L in(A, a) in(L,,i , (3 ~,) C_ in(A, c~) for for alI all I1 :::; _< ii _< n. i) � :::; n. • . • m ' In In this this definition definition condition condition (CI) (C1) means means that that for for all all positive positive modes modes a ce of of the the head head A A there literals L; body so there exist exist positive positive modes modes (3 ~ii of of the the literals Li in in the the body so that that we we have: have: (a) (a) Every Every output variable (with respect respect to a) is input variable output variable of of A A (with to c~) is an an input variable of of A A (with (with respect respect (3;); (b) to a) or to c~) or an an output output variable variable of of some some positive positive L; Li (with (with respect respect to to/~i); (b) every every input input (with respect to (3 (with respect variable variable of of aa positive positive L L,i (with respect to ~,) is an an input input variable variable of of A A (with respect i ) is to a) or (3j ) which to ce) or an an output output variable variable of of some some positive positive Lj Lj (with (with respect respect to to/~j) which is is left left every variable of is an of L Li; (c) every variable of of a a negative negative L Lii is an input input variable variable of of A A (with (with respect respect to to a ce)) i ; (c) or ) . In particular, if or an an output output variable variable of of some some positive positive Lj Lj (with (with respect respect to to (3j flj). In particular, if the the body of p,-correct if body of the the clause clause is is empty, empty, then then the the clause clause is is/z-correct if and and only only if if out(A, out(A, a) c~) is is aa subset p,+ (A) . in(A, a subset of of in(A, c~)) for for all all modes modes a c~ E E/z+(A). 5.2.2. Let C. Then 5.2.2. Definition. Definition. Let p, # be be aa mode mode assignment assignment for for/::. Then aa logic logic program program P P in C is called p,-correct in/2 is called #-correct if if all all clauses clauses of of P P are are p,-correct. #-correct. 667 667 A Proof-Theoretic Proof-Theoretic Framework for .for Logic Programming For clauses clauses and and programs programs the the notion notion of of correctness correctness will will be be sufficient sufficient for for our our purposes. purposes. For In In the the case case of of goals goals we we need need more more and and introduce introduce the the sets sets of of Jl-correct p-correct and and Jl-closed p-closed goals, thus thus obtaining obtaining an an analogue analogue to to the sets S S+ + (P) (P) and and SS - (P) (P);; see see Proposition Proposition 5.2.5 5.2.5 goals, the sets below. below. Ll , Jl-correct (31 i n: , (3n Jl L in(Li, (3i) out(Lj, (3j) i, Lj Li (G2) (G2) var(Li) var(Li) � c_ U{ U { out(Lj, out(Lj, (3j) flj) . I1 � < jj � < n, n, Lj Lj positive positive }} if if Li Li is is negative. negative. 5.2.4. Definition. Let 5.2.4. Definition. Let Jl # be be a a mode mode assignment assignment for for s£. A A goal goal L1, L 1 , . . . , , Ln Ln is is called called p-closed if if there there exist exist modes modes (31 ~t E9 Jl# - ((L L 11 )) ,, .. ... . ,,/~ #-(Ln) that in(Li, in(Li,~i) Jl-closed {3n E9 Jl-(L {3i) == 00 n ) soso that for all all I1 � < ii � < n. n. for 5.2.3. 5.2.3. Definition. Definition. Let Let Jl # be be a a mode mode assignment assignment for for sC. A A goal goal L 1 , .. .. .., , Ln Ln is is called called + ( ) so that + (L 1 ), E9 #+(L~)so p-correct if if there there exist exist modes modes /~1 E9 Jl #+(L1),...,/3n that for for all all I1 � _ i � _ n: n (GI) (G1) in(Li, ~i) � C_ U{ U { out(Lj, flj) :. 1I � <_jj < i, Lj positive positive }} if if Li is is positive, positive, . • . : • • • A goal goal which which consists consists of of aa single single atom atom A A is is Jl-correct #-correct if if and and only only if if there there exists exists a a mode mode A + ( A ) so (3 0. IfIf aa goal /? E9 Jl p+(A) so that that in(A, fl) = = 0. goal G G is is Jl-closed p-closed and and contains contains the the negative negative literal -~A, then the goal goal A A is is Jl-correct. p-correct. literal ,A, then the in(A, (3) Let Jl be a mode assignment and be a Jl-correct program in Then we have: the goal is Jl-correct, then the goal is Jl-closed, then 5.2.5. 5.2.5. Proposition. Proposition. Let # be a mode assignment and P P be a p-correct program in C s . Then we have: 1. If 1. If the goal G G is p-correct, then G G E ES S+ + (P) (P).. 2. 2. If If the goal G G is p-closed, then G G E ES S --((P) P).. - be Proof. P r o o f . Let Let D D+ + be be the the set set of of all all Jl-correct p-correct goals goals and and D 13be the the set set of of all all Jl-closed p-closed goals. goals. - ) is Then P. Therefore Then it it is is easy easy to to see see that that (D (D + +,, D D-) is safe safe for for P. Therefore we we have have D D+ + � C_ S S+ + (P) (P) 0 nition of and and D D-- � C_ SS - ((P) P ) by by the the defi definition of S S+ + (P) (P) and and S S -- ((P) P ) .. [] decidable whether whether logic It is is decidable logic programs programs are are correct correct and and whether whether goals goals are are correct correct or or It closed closed with with respect respect to to aa given given mode mode assignment assignment Jl. #. Hence Hence we we have have gained gained aa lot lot since since the to make the previous previous proposition proposition enables enables us us to make use use of of results results based based on on the the sets sets S S+ + (P) (P) and and S S -- ((P) P ) provided provided that that the the syntactic syntactic criteria criteria of of correctness correctness and and closedness closedness with with respect ed. Moreover, respect to to suitable suitable mode mode assignments assignments are are satisfi satisfied. Moreover, there there are are well-known well-known classes special cases classes of of logic logic programs programs which which are are special cases of of programs programs that that are are correct correct with with respect canonical mode assignment. respect to to some some canonical mode assignment. 5.2.6. Example E x a m p l e (Definite (Definite logic logic programs) p r o g r a m s ) .. Definite 5.2.6. Definite logic logic programs programs are are pro programs literals. If grams which which contain contain no no negative negative literals. If we we set set + (R) := Jl#+(R) og) } and and Jl# --((R) := {{} R ) ::= ( l og, . . . , llog>} og)} = {{ 668 668 G. Jager J~iger and R. Stark 5.2.7. Example E x a m p l e (Allowed (Allowed logic logic programs) p r o g r a m s ) . . Often Often aa program program clause clause is is called called 5.2.7. allowed allowed if if every every variable variable of of the the clause clause occurs occurs also also in in aa positive positive literal literal of of its its body; body; aa program program is is called called allowed allowed if if it it consists consists of of allowed allowed clauses clauses only. only. This This class class of of programs programs is of of interest interest since since Cavedon Cavedon and and Lloyd Lloyd [1989] [1989] and and Kunen Kunen [1989] [1989] could could show show that that is SLDNF-resolution is complete complete for for allowed allowed programs. programs. If If we we set set SLDNF -resolution is R )) : := = { <{ o(out, u t , . .. .., .o,uout) t > } } and and J1 # --((RR)): =: = { <{l(log, o g , . ... ., I. o, log)} g>} J1# ++ (( R for every every relation relation symbol symbol R R of of .c, s then then it it is is easy easy to to see see that that allowed allowed programs programs and and for allowed clauses clauses are are J1-correct. #-correct. Hence Hence the the completeness completeness results results due due to to Cavedon, Cavedon, Lloyd Lloyd allowed and Kunen Kunen also also follow follow from from our our more more general general completeness completeness results results below. below. and In the the following following we we present present several several examples examples which which show show the the usefulness usefulness of of mode mode In assignments and and indicate indicate that that the the programs programs used used in in practice practice are are always always correct correct with with assignments respect respect to to some some mode mode assignment. assignment. We We use use the the Prolog Prolog notation notation for for lists. lists. The The constant constant n i l is is denoted by [][] and and the the term term cons(u, cons(u, v) v) is is written written as as [ulv] [ulv ].. A A term term of of the the form form nil denoted by cons (ub cons(u2, cons(ul, cons(u2,.., cons(un, nil) n i l ) . .. .. ). )» is is written written as as rUb [Ul, U2, U 2 , . ·. .· · ,, un] Un].. . . . cons(un, 5.2.8. 5.2.8. Example. E x a m p l e . The The concatenation concatenation of of two two lists lists U ull and and U2 u2 to to aa list list Ua u3 is is described described by the the following following definite definite program: program: by append([], u, u, u) u) append([], append([vlu~], u2, [vlua]) [~lu~]) :: - append(ub append(u1, U2, u2, ua) u3) append([v lul], U2, According to to Example Example 5.2.6 5.2.6 these these program program clauses clauses are are correct correct with with respect respect to to the the According following trivial trivial mode mode assignment assignment J1: #: following - (append) = (append) = = {{ (log, (log, log, log, log) log>}},, J1 #-(append) = {{ (log, (log, log, log, log)} log>}.. J1#++ (append) For this trivial mode-assignment is For applications, applications, however, however, this trivial mode-assignment is not not sufficient. sufficient. One One is is interested with as output arguments interested in in having having modes modes with as many many output arguments as as possible. possible. The The clauses clauses of of the append relation relation are with respect respect to assignment #: W the append are also also correct correct with to the the following following mode mode assignment #+(append) = {(in, in, out>, { (in, in, (out, out, out, in)}, in) } , out) , #-(append) Jog, log) (log, log, log) }. }. J1- (append) == {{ Examples of #-correct are (a, are constants constants):) : Examples of J1-correct goals goals are (a , b, b , c, c , dd are and append(ul' U2, [a, c, d]). d]). append([a, b], append(u1, u2, [a, b, b, c, append([a, b], [c, [c, d], d], u) u) and The first first goal goal is is used used to concatenate two two lists, lists, the the second second one one to decompose aa list. list. The to concatenate to decompose 5.2.9. EExample. The actual actual reason reason to to introduce introduce mode mode assignments assignments are programs 5.2.9. x a m p l e . The are programs with nested nested negation. negation. The The subset subset relation relation between between lists lists Ul Ul and and u2 U2 isis described described by by with the the following following program: program: not subset (U l ' u2) subset (Ul' uU2) subset(ul, 2 ) :: -- ~--, notsubset(ul, U2) member(v, Ul), --, member(v, member(v, u2) U2) nnotsubset o t s u b s e t ( u(ul ,l , uU2) 2 ) :: -- member(v, Ul) , -~ member(u, member(u, [u]v]) [ulv]) member( - member(u, ember( , w) member(u,, [vlw]) :: - 669 669 .for Logic Programming A Proof-Theoretic Framework for These clauses clauses are are correct correct with with respect respect to to the the following following mode mode assignment assignment W ft" These + (subset) - (subset) (in, in) }, -- {{}, /L ft-(subset) -- {{}, (in, in)}, /L#+(subset) + (notsubset ) ---- {{}, /L#-(notsubset) {}, (in, in)}, /L#+(notsubset) (notsubset) ---- {(log, }, + (member) - (member) -- {{}, = {(log {}. (out, in) } , /L#-(member) /L#+(member) }. Although the the two two clauses clauses for for the the member member relation relation are are definite, definite, one one cannot cannot take take the the Although canonical mode mode assignment assignment of of Example Example 5.2.6 5.2.6 for for member member.. canonical 5.3. Soundness S o u n d n e s s and a n d completeness c o m p l e t e n e s s of of SLDNF-resolution SLDNF-resolution 5.3. L 1 , Ln L /\ . . . /\ LnB L1 , , Ln ...,:3 (Ll /\ . . . /\ Ln) SLDNF-resolution SLDNF-resolution can can be be justified justified in in the the following following sense: sense: If If aa goal goal L 1 , . . . ,, Ln returns some some answer answer B0 using using SLDNF-resolution, SLDNF-resolution, then then the the formula formula L10 returns I B A . . . A L,,O is aa lower lower three-valued three-valued consequence consequence of of the the logic logic program; program; if if aa goal goal L 1 , . . . , Ln fails fails is finitely using using SLDNF-resolution, SLDNF-resolution, then then the the formula formula -~3(L1 A . . . A L~) is is aa lower lower finitely three-valued consequence consequence of of the the logic logic program. The proof proof of of these these two two facts facts follows follows three-valued program. The from Clark Clark [1978]. [1978]. from . • • . • • 5.3.1. Theorem T h e o r e m (( Soundness S o u n d n e s s of of SLDNF-resolution S L D N F - r e s o l u t i o n )). . 5.3.1. and G be the I:~C goal L 1 , . . . , , Ln. Then we have: 1. ( P ) 0, P F'l ~ 7 V(L10 A . . . A L,,O). 1. If G R R(P) B, then P F ((P) P ) , , then P P F'l ~ v -~3(L1 A . .. A L,,). 22.. /f G Ee F and G be the goal L1 , Ln. Then we have: If G then V(LIB /\ . . . /\ LnB). . ., :3(L1 /\ . . . /\ Ln). then If G Let P P be be aa logic logic program program Let . • . The converse converse of of this this theorem, theorem, the the completeness completeness of of SLDNF-resolution, SLDNF-resolution, does does not not hold hold The for arbitrary arbitrary programs programs and and goals. goals. In In the the next next example example we we present present aa logic logic program program for and aa goal goal for for which which the the converse of the the first first assertion of the the previous previous theorem is not and converse of assertion of theorem is not true. true. c 5.3.2. E x a m p l e . Let Let I:Z: be language with with a a constant constant c and and two two unary relation 5.3.2. Example. be the the language unary relation symbols R R and and S. P be program in which consists following S. Let Let P be the the logic logic program in Z: I:- which consists of of the the following symbols clauses: clauses: R(c) R(c) - R(u) S(u) :: R(u) S(u) : - ...,R (u) since PP ~F'l = cVx cVx =IS(u) R R(P) Then PP F'l Then ~ v Vx v Vx(x Vx(x = 7~ c) c).. But But we we do do not not have have S(u) ( P ) C~;; Vx S(x) S(x) since {u/c} . Observe we only only have have S(u) S(u) RR(P) Observe that that the the goal goal S(u) S(u) does does not not belong belong to to we ( P ) {u/c}. sS§+ (P) . can prove completeness of of SLDNF-resolution, SLDNF-resolution, however, however, for arbitrary logic logic propro One prove completeness for arbitrary One can + (P) or grams P ) . . Below and all all goals goals which which belong belong to to SS+(P) or SS-- ((P) Below we we will will generalize generalize grams PP and which he he proved proved for for allowed allowed programs programs and and allowed goals aa theorem allowed goals theorem of of Kunen Kunen [1989] [ 1989] which only. Instead of of working working in in aa language language which which contains contains infinitely infinitely many many constants constants only. Instead (as Kunen Kunen does), does) , we we consider consider structures structures which which are are generated generated from terms of of the the (as from all terms language. language. all G. G. Jager Jiiger and and R. R. Stark Stark 670 670 9Jl .c free term structure for .c 9Jl .c .c 9Jl(f)(al " ' " an ) = f(al, . . . , an ). at, . . . , an .c J9Jl� 5.3.3. 5.3.3. Lemma. L e m r n a . Let Let P P be be aa logic logic program program in in .c f_. and and 9Jl 9Jt aa free free term term structure structure for for .c. f_.. IfIf GG isis the the s.c goal goal L Lx, Lk, then then we we have have for'all for'all natural natural numbers numbers nl, nl, .. .. .. ,, nk: nk " 1 , . . . ,, Lk, 1. IfG k, then 1. /f G E e S+(P) S + (P) and and J9Jl'}1(Li) 29Jt~p~(Li) = = tt for for all all ii so so that that 11 $ < ii $ < k, then G G E e Y(P) Y(P).. 2.2. IfIf GG EE SS --( (P) P ) and and iifffor for each each .c f_, substitution substitution () 0 there there exists exists an an ii so so that that 11 $ < ii $ < k k and and J9Jl'}1(Li()) 39Jt~p~(LiO) = = f, f, then then G G E E N(P) N(P). . has the the A four-valued four-valued structure structure 9Jr for for/2 is is called called aa free term structure for f_. if if it it has A following (i) The all terms, following properties: properties: (i) The domain domain 19J11 19Ytl of of 9Jr is is the the set set of of all/2 terms, including including those those which which contain contain variables; variables; (ii) (ii) for for all all n-ary n-cry function function symbols symbols f f of of/2 and and all all /2 terms terms a l , . . . , an we we have have 9 ~ ( f ) ( a l , . . . , an) = f ( a l , . . . , an). The proof proof of of the the completeness completeness of of SLDNF-resolution SLDNF-resolution is is based based on on the the following following The lemma lemma which which relates relates the the finite finite stages stages 39Jt~ to to SLDNF-resolution. SLDNF-resolution. . • • Proof. ne the P r o o f . For For the the sake sake of of this this proof proof we we defi define the body-length body-length of of aa program program clause clause to to be be the the number number of of literals literals in in its its body. body. Now Now let let r be be a a natural natural number number which which is is larger P. Such bound always larger than than the the body-length body-length of of every every clause clause from from P. Such an an upper upper bound always exists nite. Assertions exists since since logic logic programs programs are are fi finite. Assertions 1I and and 2 are are proved proved by by simultaneous simultaneous induction induction on on the the natural natural number number r n' + + .-... . + + r n~ . There There are are four four cases. cases. + (P) ; 39Jt~p~(Li) = tt for 1. G EE SS+(P); k; Lj is Case 1. for all all 11 $ _< ii $ _< k; is positive positive for for some some 11 $ k: We positive. Since _ jj $ <_ k: We can can assume assume that that L1 is is positive. Since 3ffJt~1(L1) = tt,, nl is is greater greater definition of than than 00 and and by by the the definition of 3ffJt~x there there exists exists aa clause clause B ' - M 1 , . . . , Me from from P P and substitution a so and 39Yt~p~-X(M1a /\ and aa substitution so that that Lx = B a and A . .. .. . /\ A Mta) = = tt.. By By (A2) (A2) (P) . Since of of Definition Definition 5.1.7 5.1.7 the the goal goal M l a , . . . , Mta, L 2 , . . . , Lk belongs belongs to to S+ S+(P). Since r 2 rn 1 rn k . J9Jl'}1(Li) = Lj Ll J9Jl�1 (L1) = n l J9Jl�1 B : - Mt, . . . , Mi Ll = Ba J9Jl�I- I (Mw Mia) MIa, . . . , Mia, L2 " ' " Lk Case G a g " r nl-l h - r n2 h - " " " W r nk < r n l -l- r n2 -k- " " " W r n k , MIa, . . . , Mia, L2 , . . . , Lk J9Jl'}1 (Li()) = Ll a Ml T, . . . , MiT, L2 a, . . . , Lka . . . , MiT, L2a, . . . , Lka i J9Jl�1 (Ll a()) = 1 J9Jl�1 - (MjT()) we we obtain obtain by by the the induction induction hypothesis hypothesis that that the the goal goal M l a , . . . , Mta, L 2 , . . . , Lk is is in in Y(P) Y ( P ) . . By By (YI) (Y1) we we obtain obtain that that G E e Y(P) Y(P).. substitution ()0 it Case 2. G E E SS - ((P) P ) ; ; for for every every .c s substitution it is is 39Jt~p~(LiO) - ff for for some some 11 $ k; Lj is _ i $ _ k; is positive positive and and nj > > 00 for for some some 11 $ _< jj $ _ k: k" We We can can assume assume that that L1 is is positive positive and and 00 < < nl. Let Let B ' - M 1 , . . . , Mt be be aa clause clause from from P P and and a and and T T be be substi substitutions tutions so so that that L l a = - BT. We We have have to to show show that that the the goal goal M I T , . . . , MtT, L 2 a , . . . , Lka is BI) and is in in N(P) N ( P ) . . Note Note that that by by ((B1) and (B2) (B2) the the goal goal M M 1I T, T , . . . , MIT, L 2 a , . . . , L~a belongs belongs By assumption, to to SS - ((P) P ) . . Let Let ()0 be be a a substitution. substitution. By assumption, there there exists exists an an 11 $ _< i $ < kk so so that that 3ffYt~p~(LiaO) = = ff.. If If i = 11,, then then YffJt~pl(LlaO) = ff and, and, by by the the definition definition of of the the finite finite $ jj $ f. Since stages stages 39Jt~, there there exists exists aa 11 _< < Cg so so that that 39Jt~1-1(MjT0) = - f. Since G Case 2. G i Lj nj nl. B : - M1, . . . , Mi L1 a BT. i= J9Jl'}1 (Lia()) J9Jl�, g. r nl-l W r n2 W . . . -t- r nk < r nl + r n2 W . . . W r n k , MIT, . . . , MiT, L2a, . . . , Lka a i Li J9Jl'}1(Ai) = we we can can apply apply the the induction induction hypothesis hypothesis and and obtain obtain that that M I T , . . . , MtT, L 2 a , . . . , L k a is is in in N N ((P) P ) . . Since Since the the clause clause B " - M 1 , . . . , Mt and and the the substitutions substitutions a and and T T are are chosen that G EE N(P) chosen arbitrarily, arbitrarily, we we obtain obtain that N ( P ) by by (NI). (N1). (P) ; YgJt~(Li) = tt for Case 3. 3. G E e S+ S+(P); for all all 11 $ _< i $ <__ k; k; Li is is negative negative for for all all 11 $ _< i $ _< kk": Then Then Li is is of of the the form form --Ai for for ali all I1 $ <_ i $ < k k and and therefore therefore 3ffJt~,~(Ai) = f. f. B : - M1 , , Mi G J9Jl'}1(Li) = i -,Ai • • • Case i G Li 671 671 A Pro@ Proof-Theoretic Framework .for for Logic Logic Programming Programming A Theoretic Framework Ai Ai (P). Case G Y(P). B JWl']1(LiB) Li i JWl']1(Li) -,A. A S+ (P) A Y(P). G N(P) ( A3 ) we we know know that that Ai isis closed closed and and Ai EE SS-- ( P ) . Hence Hence Case 22 Because of of condition condition (A3) Because implies Ai EE N ( P ) , and and by by (Y2) (Y2 ) we we obtain obtain that that G eE Y ( P ) . implies 4. G EE S - ( P ) ; for for some some for every every sC substitution substitution 0 itit isis :IffJ~fi(L~O) == ff for Case 4. 11 <_ kj ni == 00 for for all alI 1I <� i _� kk so so that that Li isis positive: positive: We We apply apply the the assumption assumption � i _� k; to the the identity identity substitution substitution ~c and and obtain obtain an an 11 _� i _� kk so so that that :JffJ~p~(Li) -= f.f. The The to literal have A E literal L~ must must be be negative, negative, i.e., i.e., of of the the form form -~A. By By (B3) ( B3) we we have E S+(P) and and 3ff2~(A) = have G E = t. t . By By Case I1 we we obtain obtain that that A EE Y ( P ) . Therefore Therefore we we have E N(P) 0 because of of (N2). ( N2) . because [-1 Ai N(P), Case G S-(P); i nj Lj JWl']1(A) Case i Theorem 4.2.3, 4.2.3, Theorem Theorem 4.2.4, 4.2.4, Lemma Lemma 5.1.10 5.1.10 and and the the previous previous lemma lemma yield yield the the Theorem completeness of SLDNF-resolution SLDNF-resolution for for goals goals from from SS++ (P) and completeness of and S - ( P ) . (P) S-(P). P be logic program in and let G be the goal L1, . . . , Lk . Then we have: Lk B), then there exist substitutions G S+ (P) and P FV' V(L1 B (J and T so that G R(P) (J and G(JT GB. Lk ), then 2. and PP ~FV'v -~3(L1 -,3(L1 A/\ ..... . A/\ Lk), then GG EE F(P). 2. If If G G E E SS-- ((P) P ) and F(P). + 3. If 3x A(x), A ( x ) , then there exists that If the the atom atom AA isis inin SS+ (P) (P) and and PP ~FV'v 3x then there exists aa term term tt soso that A(u) RR(P) A(u) ( P ) {u/t}. {u/t}. -,A), then FV'v 3(A) ~. the atom atom A P ) and and P P ~ 3(A) V y V( V(-,A), then either either 4. IfIf the A isis inin SS++ (P) (P) and and inin SS-- ((P) there exists anan Cs substitution R(P) there exists substitution (J a so so that that A A R ( P ) (J a or or A A E E F(P). F(P). Proof. Wl bebe aa free free term term structure structure for the first first assertion P r o o f . Let Let if2 for sC. To To prove prove the assertion we we assume assume + (P) and and PP ~FV' V(L1 B A/\ ..... . /\A LkO). LkB). By Theorem 4.2.3 4.2.3 there there exists exists aa that G that G E E S S+(P) 7 V(L10 By Theorem Since J� natural number number n n so so that that n'R(P) (P) H 39~p I;;; E ; JWl� ~lffJ~lp and and natural H--'~n V(L~O V(L1B /\^ . .. . . /\A i~O). Lk B). Since JWl� B1 /\A .. ... . /\A LkB)) :Jff)~]~ = = J�[P], :Jff~p[P], we we obtain obtain JWl'],(V(L 3ffJ~(V(L10 LkO)) = tt by by Theorem Theorem 4.2.4. 4.2.4. From From Lemma 5.1.10 there exist the the previous previous lemma lemma we we conclude conclude that that GB GO E E Y(P). Y ( P ) . By By Lemma 5.1.10 there exist substitutions substitutions (J a and and T T so SO that that G G R(P) R ( P ) (J a and and G(JT Ga'r = GB. GO. Now Now we we turn turn to to the the second second assertion assertion and and assume assume that that G G is is an an element element of of S S -- ((P) P) and first case and P P FV' ~ v -,3(L1 -~3(L1 /\ A .. ... . /\ AL Lk). As in in the the first case there there exists exists a a natural natural number number n n so so k ). As that that JWl'],(-,3(Ll 3ffJt~(-~3(il /\ A .. ... . /\ AL Lk)) This means means that that JWl'],(V(-,L 3ffJt~,(V(-,il1 V Y .. ... . V V -,L -~Lk)) k )) f.f. k)) t.t. This Hence, Hence, for for every every substitution substitution B0 there there exists exists an an 11 � _< ii � _< k k so so that that JWl'],(LiB) 3ff2fl~p(LiO) = f. f. By By the the previous previous lemma lamina it it follows follows that that G G E E N(P). N ( P ) . By By Lemma Lemma 5.1.10, 5.1.10, we we obtain obtain that that GG EE F(P). F(P). 5.3.4. TTheorem of SSLDNF-resolution) Let P be aa logic 5.3.4. h e o r e m ((Completeness C o m p l e t e n e s s of L D N F - r e s o l u t i o n ) .. Let C, and let G be the f_. C goal L1, . . . , Lk. Then we have: program in E, . . . A/\ LkO), then there exist LC substitutions 1. 1. If If G EE S+(P) and P ~ v V(L~O A/\ ... = GO. a and T SO that G R ( P ) a and CaT = = = = - = - The similar way. The third third and and the the fourth fourth assertion assertion are are proved proved in in aa similar way. = = 0 El Proposition Proposition 5.2.5 5.2.5 yields yields in in addition addition the the completeness completeness of of SLDNF-resolution SLDNF-resolution for for mode mode assignments assignments #. Jl. Let Let P P be be aa logic logic program program in in C s which which is is correct correct with with respect respect to to aa mode mode assignment assignment Jl # and and let let G G be be the the C s goal goal L1, L1, .. .. .. ,, L Lk. Then we we have: have: k . Then 1. If 1. If the the goal goal G G is is Jl-correct #-correct and and P P FV' ~ 7 V(L1B V(L10 /\ A .. ... . /\ AL LkO), then there there exist exist kB), then s substitutions substitutions (J a and and T ~- so so that that G G R(P) R ( P ) (J a and and G(JT G a y = GB. GO. C 2. 2. If If the the goal goal G G is is Jl-closed #-closed and and P P FV' ~ 7 -,3(L1 -~3(L1 /\ A .. .. .. /\ AL L~), then G GE E F(P). F(P). k ), then 5.3.5. 5.3.5. Theorem. Theorem. = Jager and Stark 672 672 G. G. J~iger and R. R. Stiirk p,+(R), then for every closed term a there If P Vx3y R(x, y) and exists aa closed closed term term bb so so that that R( R(a, R(P exists a, v)v) R( P)) {v{v/b). / b} . If the the atom atom A A is is p,-correct #-correct and and p,-closed #-closed and and P P F'V ~ v 3(A) 3(A) VV V( V(-~A), then either either 4~.. If ....,A), then there there exists exists an an C s substitution substitution a so so that that A A R(P) R ( P ) a or or A A E F(P). F(P). t) Ee #* (R), then for every closed term a there 3. 3. If P F'V ~ Vx3y R(x, y) and (in, (in, ou out) (J (J 6.. Partiality P a r t i a l i t y in i n logic l o g i c programming programming 6 The The omission omission of of the the identity identity axioms axioms has has the the effect effect of of disconnecting disconnecting aa relation relation symbol symbol R from from its its complement complement il, R, and and thus thus an an adequate adequate framework framework for for discussing discussing the the procedural procedural aspects aspects of of logic logic programs programs and and for for SLDNF-resolution SLDNF-resolution is is provided. provided. However, the the identity-free identity-free derivation derivation in in the the calculi calculi 7~(P) are are extremely extremely weak weak and and However, sometimes sometimes considered considered unnatural. unnatural. In In addition, addition, on on the the semantical semantical side side we we have have to to deal deal with additional additional truth truth values values in in order order to to obtain obtain aa decent decent model model theory theory for for identity-free identity-free with derivations. derivations. Now Now we we want want to to further further the the conceptual conceptual clarity clarity and and present present an an alternative alternative ap approach to to logic logic programming programming which which is is based based on on two-valued two-valued logic logic only. only. To To this this end end proach we we introduce introduce aa form form of of partiality partiality into into logic logic programming: programming: We We present present the the partial partial completion completion compS(P), the the corresponding corresponding deductive deductive system system O(P) and and the the inductive inductive extension of logic logic programs programs P. extension ind'(P) of These These formalizations formalizations are are also also discussed discussed in in Jager J~iger [1994] and and Stark St~irk [1996]. Further Furthermore, similar concepts are studied in Drabent and more, similar concepts are studied in Drabent and Martelli Martelli [1991] and and Van Van Gelder Gelder and Schlipf Schlipf [1993]. and R R(P) comp�(P), ind�(P) 8(P) P. [1994J [1991J [1993J . [1996J. 6.1. The T h e partial p a r t i a l completion c o m p l e t i o n of of logic logic programs programs 6.1. The syntactic syntactic framework for defining defining the the partial partial completion completion of of logic The framework for logic programs programs is provided by the the extension extension /2~ of/2 of introduced introduced in Section 2.3. Given Given this this in in ~2 is provided by in Section of closure rich language, rich language, the the partial partial completion completion of of aa logic logic program program P just just consists consists of closure and R- plus plus some basic equality and freeness conditions for the relations relations R + and conditions for the some basic equality and freeness axioms. axioms. C C� C R- P 2.3. R+ 6.1.1. Definition. Definition. Let Let PP be be aa logic logic program program in in sC. Then Then the the partial partial completion completion 6.1.1. the sC� theory theory which which consists consists of of the the following following axioms. axioms. comp� (P) ofof PP isis the compS(P) CETe. plus plus equality axioms for all relation relation symbols symbols R R±+ ofof sC�: 1. equality axioms for all 1. CETL (al - bl A . . . A an -- bn A R+(al,...,an)) ~ R+(bl,...,bn). 2. R C For all all relation relation symbols symbols R of of s and and their their definition definition forms forms DR[u] DR[U] with with respect respect 2. For to to P: V~(D+[~] --+ R+(~)) and V~(D~[~] ~ R-(~)). P: R+ RP. 9.n 9.n(R+), and R- are are closed closed with with respect respect to to the the formulas express express that that the the relations relations R + and These These formulas partial definition definition forms forms D Dti and D~ DR of of P. If If ff)t isis aa model model of of the the partial partial completion completion partial + and and ifif we we know, know, in in addition, addition, that that the the relations relations ffJ~(R-) are are the the program P and of aa program of (set-theoretic) complements complements of of the the relations relations ff)l(R+), then then the the ffJ~(R+) describe describe fixed fixed (set-theoretic) P 9.n(R-) 9.n(R+) 673 673 A Proof-Theoretic Framework for .for Logic Programming P. points of the system points of the system of of inductive inductive definitions definitions which which is is associated associated to to P. But But in in general general we do do not not know know whether whether 9Jt(R ffJt(R + +)) and and 9Jt(R-) ffJt(R-) are are complementary complementary so so that that the the prefix prefix we "partial" is is in in place. This is is similar to the the distinction distinction between between truth truth definitions and "partial" place. This similar to definitions and partial truth truth definitions definitions as as for for example example in in Feferman Feferman [1991]. [1991]. partial The relation symbols partial The equality equality axioms axioms for for the the relation symbols of of .c} s have have to to be be added added in in the the partial case possible to case since since it it is is not not possible to derive derive them them from from the the other other axioms. axioms. This This is is different different from the completions comp(P) of of logic and the the calculi calculi ~ ( P ) since there logic programs programs P and since there from the completions the the equality equality axioms axioms for for relations relations can can always always be be proved. proved. Because Because of of the the close close relationship relationship between between the the four-valued four-valued structures structures for for/: and and the the two-valued two-valued structures structures for for s and and Remark Remark 2.3.5 2.3.5 in in particular, particular, the the following following is is obvious. obvious. comp(P) P R(P) £ £� 6.1.2. Let 6.1.2. Remark. Remark. Let P P be be a a logic logic program program in in £ s and and 9Jt 9Yt aa four-valued four-valued equational equational structure for for/:. Then 9Jt 9Jr is is adequate adequate to to P P if if and and only only if if 9Jt� 9J~ is is a a ((two-valued) model £. Then two-valued) model structure of of comp comp �~(P). (P). Hence Hence we we obtain obtain a a theorem theorem which which reduces reduces the the 4-adequate 4-adequate consequences consequences of of aa logic logic program P to to logical consequences of of the the partial partial completion of P; in in view view logical consequences completion compS(P) of program of Corollary Corollary 3.2.5 3.2.5 this this is is also also true true for for lower lower consequences consequences of of P. of comp� (P) P; P. 6.1.3. Let 6.1.3. Theorem. Theorem. Let P P be be aa logic logic program program in in £ s and and A A an an £ s formula. formula. Then Then we we have have the the following following two two equivalences: equivalences: P P ~4 A ~ comp~(P) ~ A + r P ~ v A. The completion reduces completion if The partial partial completion reduces to to the the Clark Clark completion if we we add add the the two two further further (i)) for (i) V (R+ (i) axioms axioms Vi(R+ V~(R+(~) V RR - (i)) (~)) and and Vi..., V~--(R+ (~) /\ h RR-(~)) for all all relation relation symbols symbols R R of of/:. They (i) is is equivalent They express express that that each each RR-(~) equivalent to to the the negation negation of of R+ R + (i) (:~).. The rst of The fi first of these these axioms axioms is is aa kind kind of of totality totality assertion, assertion, stating stating that that at at least least one one (a) or (a) has SLDNF -resolution this of of R+ R+(~) or RR-(~) has to to be be true. true. From From the the point point of of view view of of SLDNF-resolution this means that R(a) means that R(6) succeeds succeeds or or fails. fails. However, However, this this is is generally generally not not the the case, case, so so that that this this totality totality assertion assertion is is rejected. rejected. (a) The The second second axiom axiom is is aa uniqueness uniqueness condition condition which which means means that that R+ R + (a) (g) and and RR-(~) must must not not both both be be true. true. This This corresponds, corresponds, in in the the context context of of SLDNF-resolution, SLDNF-resolution, to to the the statement statement that that it it is is not not possible possible that that R(a) R(g) succeeds succeeds and and fails. fails. This This is is correct, correct, but but the previous theorem theorem makes partial completion the previous makes it it clear clear that that the the logical logical power power of of the the partial completion is is not not increased increased by by adding adding this this axiom. axiom. £. 8(P) 6.2. 6.2. The T h e calculi calculi O(P) It It is is now now easy easy to to set set up up deductive deductive systems systems which which correspond correspond to to the the partial partial completion completion of of programs. programs. As As in in Section Section 44 we we work work in in extensions extensions of of the the Tait Tait calculus calculus 8(P) is for the only for predicate predicate logic; logic; the only significant significant difference difference between between :R(P) and and O(P) is the the way way in which the programs into the the calculi. in which the programs are are incorporated incorporated into calculi. The The systems systems O(P) for for logic logic programs programs P in in £ s comprise comprise the the logical logical axioms, axioms, the the equality equality axioms, axioms, the the logical logical rules rules and and the the cut cut rules rules of of 7~(P), but but all all formulated formulated for for/:~ R(P) 8(P) P R(P), £� G. G. Jager Jbger and R. Stark Stiirk 674 674 £. £� instead the following the relation instead of of s In In addition addition we we have have the following equality equality axioms axioms for for the relation symbols symbols of of L:~ and and the the following following partial partial program program rules. rules. £� . al, . . . , am , b1, £� bm , Equality , E q u a l i t y axioms a x i o m s for for the t h e relation r e l a t i o n symbols s y m b o l s of of/:~. If If a l , . . . , a m , b~,...,bm, . . . , en . . . , dn c~,..., cn and and d~,..., dn are are/:~ terms, terms, R± R + is is aa n-ary n-cry relation relation symbol symbol of of L:~ so so that that er ()0,, the the set set {a {all = - b l , . . . , ,a m = = bin} is is unifiable unifiable and, and, for for the the most most general general unifi unifier } B are ) () and (dl, . . . , dn)O identical, then R± (cl , . . . , en R+(cl,..., Cn)O and R± R+(dl,..., are identical, then we we have: have: Cl, d1, b1, • • • £� am bm } dn • • . F, a~ ~ b~,..., am r b,~, R'-'-~(cl,... ,Cn), R+(d~,... ,dn). P. £ Partial P a r t i a l program p r o g r a m rules rules for for P. We We have have for for every every relation relation symbol symbol R R of of s and and its its definition form definition form DR[Uj DR[g] with with respect respect to to the the logic logic program program P and and for for all all s terms terms a ~:: P £ r, -) . F, Dli[aj D [g] (R (R-). - (a) r, R R-(a) r, D� [iiJ (R+) F, (R+ ) ,' r, R+ (a) R(P) There There is is an an important important difference difference between between the the program program rules rules of of 7~(P) and and the the partial partial program D�[iIJ and program rules rules of of O(P)" The The formulas formulas D+[~7] and Dli[uj D~[~7] in in the the premises premises of of the the partial partial program relation symbols program rules rules of of O(P) are are positive positive in in all all relation symbols whereas whereas the the formulas formulas DR[uj DR[g] and DR[uj in and ..., ~DR[~7] in the the premises premises of of the the program program rule rule of of TO(P) may may contain contain positive positive and and negative occurrences of negative occurrences of R R and and other other relation relation symbols. symbols. Let Let P be be aa logic logic program program in in s and and r F aa finite finite set set of of s formulas. formulas. Then Then O(P) I-� t-') r F and and O(P) II-� ~ r F is is defined defined for for all all n, r < < w in in analogy analogy to to 7~(P) I-� t-~ r F and and TO(P) II-� ~ r F ((cf. cf. Section Section 44).) . We will also r, O(P) I-F- r, We will also make make use use of of the the abbreviated abbreviated forms forms O(P) I-r ~-r F, F, O(P) II-r H--rr F and and O(P) IIH-- r, F, in in the the same same sense sense as as before. before. 8(P): 8(P) P 8(P) 8(P) R(P) £� R(P) £ n, r 8(P) 8(P) 8(P) R(P) 8(P) 8(P) 6.2.1. 6.2.1. Remark. R e m a r k . It It is is obvious obvious that that O(P) is is aa Tait-style Tait-style formalization formalization of of the the partial partial completion completion of of P; i.e., i.e., we we have have for for all all logic logic programs programs P in in s and and all all/:~ formulas formulas A: A" Pi 8(P) O(P) ~- A A I- <==} r P £ � (P) F~ A. compS(P) comp £� Because of possible by Because of the the positivity positivity of of the the partial partial program program rules rules it it is is possible by standard standard proof-theoretic eliminate all Further, the proof-theoretic techniques techniques to to eliminate all cuts. cuts. Further, the identity identity axioms axioms can can be be removed removed as as well well if if we we restrict restrict ourselves ourselves to to the the positive positive fragment fragment of of the the system system O(P). 8(P). 6.2.2. 6.2.2. Theorem. T h e o r e m . Let Let P P be be aa logic logic program program in in £. f~. Then Then w wee have have for for all all finite finite sets sets r F ofof £f_.~� formulas .formulas and and all all finite finite sets sets � A of of positive positive £ ffl� formulas: .formulas: 1. 1. 8(P) O(P) IF- r F ==v 8(P) O(P) 1-0 ~-o r, F, 2. 8(P) 1-0 � 8(P) 11-0 2. O(P) F-o A ==~ O(P) ~-o � A.. => => The The following following shorthand shorthand notation notation will will bbee used used from from now now on: on" If If r F iiss the the set set then rF ++ stands stands for the corresponding {A1,..., of s formulas, formulas, then for the corresponding set set {At { A +,, .. .. .., A +} {Al , . . . An} of of relationship between of L:~ formulas. formulas. Then Then the the relationship between the the identity-free identity-free and and cut-free cut-free deriva derivais obvious: in the of the in 7~(P) and tions in and a(P) is obvious: they they are are identical identical in the sense sense of the following following tions lemma. lemma. £� , An } £ R(P) 8(P) , A�} A ProofTheoretic Framework for Logic Programming Proof-Theoretic 6.2.3. Lemma 6.2.3. L e m m a .. .cs formulas: formulas: 675 675 We We have have for for all all logic logic programs programs P P in in .c s and and finite finite sets sets r F of of R( P) It-o 8(a(P)P) It-o T~(P) H-o r F ~ ~-o rr ++.. {=:} Together with with Theorem Theorem 4.3.1 4.3.1 which which states states cut cut elimination elimination for for identity-free identity-free derivations derivations Together in 7~(P) we we therefore therefore obtain obtain the the following following result result about about the the relationship relationship between between 7~(P) in and and O(P). R(P) 8(P). R(P) Let Let P P be be aa logic logic program program in in .c s and and r F be be aa finite finite set set of.c of s formulas. formulas. Then we we have the following following equivalences: equivalences: Then have the r. R(P) r r 8(P) T~(P) It-o H-oF O ( P ) ~f- Fr++ < :- R(P) 7~(P) Ittt-F. This This means means that that the the identity-free identity-free and and the the identityidentity- and and cut-free cut-free derivations derivations in in R(P) 7~(P) correspond exactly exactly to to the the positive positive fragment fragment of of 8( O(P). The following following side side remark remark refers refers correspond P). The to cut-free cut-free derivations derivations in T~(P) which which permit permit identity-axioms. identity-axioms. to in R(P) 6.2.5. Let TOT) consist 6.2.5. Remark. Remark. Let ((TOT) consist of of the the following following sets sets of of .c� s formulas formulas which which 6.2.4. 6.2.4. Corollary. Corollary. {=:} {=:} express express that that all all pairs pairs (R (R + +,, R-) R-) are are total total in in the the sense sense that that at at least least one one of of the the two two R-- ((it) is true: formulas formulas R ~ ) oor r R+ R + (a) (~)is true: ((TOT) TOT) F, R-(g), R + (~). P .c It-o r+ . r Then Then one one immediately immediately has has for for all all logic logic programs programs P in in s and and all all finite finite sets sets F of of E-formulas" .c-formulas: TOT) H-o F +. n ( P ) bo F {=:} < :- O(P) + + ((TOT) R(P) f-o r 8(P) The calculus is, The general general role role of of cuts cuts and and cut-free cut-free derivations derivations in in the the sequent sequent calculus is, for for example, in Girard and Girard, Lafont and example, analyzed analyzed in Girard [1987b] [19875] and Girard, Lafont and Taylor Taylor [1989] [1989].. Similar Similar results results about about the the identity-free identity-free derivations derivations in in the the sequent sequent calculus calculus are are contained contained in in Hosli [1994]. HSsli and and Jager Js [1994]. This This article article also also studies studies the the close close dualities dualities between between cut-free cut-free and identity-free derivations. and identity-free derivations. 6.3. 6.3. The T h e inductive inductive extension e x t e n s i o n of of logic logic programs programs The partial completions logic programs The partial completions of of logic programs are are comparatively comparatively weak weak theories. theories. They They are are not not powerful powerful enough enough to to prove prove many many interesting interesting properties properties of of logic logic programs programs and, and, for instance, the clearer, consider for instance, the equivalence equivalence of of logic logic programs. programs. To To make make this this point point clearer, consider the the following following two two examples. examples. 6.3.1. 6.3.1. Example E x a m p l e (( Termination T e r m i n a t i o n )). . We We use use the the same same notions notions as as in in Example Example 5.2.8 5.2.8 and and let let P1 be be the the logic logic program program which which consists consists of of the the following following clauses: clauses: list ([]) list([]) list list([ ([ulv]) l ]) :: - list li t(v) member(u, member(u, [ulv]) [ulv]) member(u, [vlw]) [vlw])::member(u, w) w) member(u, - member(u, PI (v) 676 676 G. Jager J6ger and R. Stark St6rk G. Suppose, that that we we want want to to prove prove that that for for every every term term aa and and every every list list bb the the goal goal Suppose, member(a, b) b) either either succeeds succeeds or or fails fails using using SLDNF-resolution. SLDNF-resolution. By By our our previous previous results results member(a, we we know know that that this this is is equivalent equivalent to to the the statement statement that that the the partial partial completion completion of of P1 proves the the formula formula proves Pl + (u, - (u, v)). l i s t ++( (v) v ) -+ ~ (member (member+ (u, v) v) V V member member-(u, v)). list However, it it is is easy easy to to see see that that this this is is not not possible possible without without making making use use of of some some form form However, of induction. induction. of 6.3.2. 6.3.2. Example E x a m p l e (Equivalence). (Equivalence). Now Now we we define define the the addition addition of of natural natural numbers numbers in two two different different ways: ways: by by recursion recursion on on the the first first argument argument and and by by recursion recursion on on the the in second argument. argument. Let Let P2 be be the the following following logic logic program: program: second nat(0) nat (0) ~at(~(u)) :: - nat ~at(u) nat(s(u)) (u) addl (O, (0, u, u, u) u) addl ~ddl(~(u),, ~, ~(~)) :: - addl addl(~, v, w) ~) addl(s(u) v, s(w)) (u, v, add2 (u, 0, 0, u) u) add2(u, ~dd2(u, s(v) ~(~),, s(w)) ~(~)) :: - add2(u, add2(u, ~, ~) add2(u, v, w) It would would be be nice nice if if one one could could show show that that both both definitions definitions have have the the same same input/output input/output It behavior. Unfortunately, Unfortunately, this this is is not not possible possible in in compS(P2); for for example, example, the the following following behavior. formula is is not not provable provable there: there: formula P2 comp�(P2); n~t+(u) ^ .~t+(~) -~ (~ddl+(~, ~, ~) ~ add2+(~, ~, ~)). In ciencies we partial completion suited In order order to to overcome overcome these these defi deficiencies we add add to to the the partial completion suited forms of induction. induction. The following. Suppose we are are given given aa logic forms of The basic basic idea idea is is the the following. Suppose we logic program in .c s which contains the relation symbol symbol R0,..., P~.. Then we collect collect all all program P in which contains the relation Ro, . . . , R" Then we positive formulas formulas Dt positive D ~ [x] [~] and and DR. D ~ [~] and consider them as as the the definition definition clauses clauses of of [x] and consider them aa simultaneous simultaneous inductive inductive definition the relations relations R Rt, ,...,R R;;, R;; in the definition SID SID of of the +, RO R0-,..., +, R~ in the sense of, for sense of, for example, example, Moschovakis Moschovakis [[1974]. 1974] . The partial completion expresses that that these relations are this simul simulThe partial completion expresses these relations are closed closed under under this does not say that the relations relations Rt taneous inductive not say that the R+ taneous inductive definition. definition. However, However, compS(P) does and R; are fixed fixed points, let alone least fixed fixed points of SID. SID. The next step step is is therefore therefore and R~- are points, let alone least points of The next to add further induction induction principles principles which which enforce the relations relations Rt to add further enforce the P~,, RO R o ,, . ... ., , R +, R;; R~ R;;, to to be be least least fixed fixed points. points. For notational convenience convenience we we have to introduce introduce some some shorthand shorthand notations: notations: Let Let For notational have to P be be aa logic logic program program in in s which which contains contains the the relation relation symbols symbols R0,..., Ro, . . . , P~. R" . Then Then we we write write closed(P) for for the the formula formula P comp� (P) . P closed(P) .c n n Rt (x)) /\ VZ /)~(VZ R; (x))). (DR. [x] --+ (Dt [x] --+ V'x (D~[Z] -+ R+(Z))A -+ R~-(Z))). � (V'x (D~[Z] i=O (CLOSURE) (Ct,osuRP,) i--O closed(P) comp�(P). is provable in compS(P). Now Now suppose suppose further further that that we we have for Obviously closed(P) is Obviously provable in have for relation symbol symbol P~ R; two two s formulas formulas Ai(g) Ai(it) and and Bi(~7) Bi(it) with with distinguished distinguished free free each each relation .c� 677 677 A Proof-Theoretic Framework Frameworkfor Logic Logic Programming Programming closed (P, Ji+ Jiclosed(P) Rt(ii) Ai (ii) R;(ii) Bi (ii) n. sub (Ji+, Ji-, (R t + ((si!) --+ Ai( Ai (~)) (R:,; ((~) Bi (5c'))). i!) --+ Bi(X))). i!)) 1\A VV~i! (R i�=O (V(V~i! (R 6.3.3. 6.3.3. Definition. Definition. Let Let P P be be a a logic logic program program in in .c. s Then Then the the inductive inductive extension extension ind'(P) of of P P is is the the/:~ theory which which consists consists of of comp�(P) compS(P) and and comprises comprises the the following following ind�(P) {} theory variables i1 variables g = = UI t t l ,,. .. .. . ,, Urn Urn,, provided provided that that Ii; R / iis s m-ary. m-cry. Then Then closed(P, R+/.4, ) /X, R - //BB) is the the formula formula which from closed(P) by by simultaneously simultaneously replacing replacing each each oc ocis which results results from currence currence of of R+(g) by by Ai(~) and and R~-(~) by by Bi(~) for for 00 < _ i < _ n. As As additional additional abbreviation abbreviation we we write write sub(R + X, .4, .~-, B) B) for for n -+ -+ i=0 additional additional axioms axioms - /B) -+--+ sub + /X, JiR-IB) (Ji++, X, closed(P, closed(P, Ji R+I.4, sub(R .4, Ji R-,-, B) B) (MINIMALITY) (MINIMALITY) .c� for B with for all all s formulas formulas X A and and/9 with a a suitable suitable number number of of distinguished distinguished free free variables. variables. ind�(P) From From the the point point of of view view of of inductive inductive definitions definitions ind'(P) is is an an extremely extremely natural natural theory. theory. We We can can show show in in ind'(P) that that for for each each relation relation symbol symbol R of of P the the relation relation symbols symbols R + and and R - are are least least fixed fixed points points of of the the simultaneous simultaneous inductive inductive definition definition which which corresponds corresponds to to P in in the the sense sense described described above. above. This This means, means, in in particular, particular, that that in ind'(P). Further induction induction on on all all R + and and R - is is available available in Further one one can can prove prove in in the the inductive inductive extension extension of of P that that R + and and R - have have no no elements elements in in common. common. Although Although strong strong induction induction principles principles are are added added to to compS(P), the the theory theory ind'(P) is is a a conservative conservative extension extension of of compS(P) with with respect respect to to positive positive s formulas. formulas. The The proof theorem is proof of of the the following following theorem is obvious obvious from from the the elementary elementary theory theory of of inductive inductive definitions. definitions. R+ R P RP R+ Rind�(P). P R+ Rcomp�(P), comp�(P) .c� 6.3.4. 6.3.4. Theorem. Theorem. .c�ffl formulas formulas A: A" ind�(P) ind�(P) Let Let P P be be aa logic logic program program in in .c. L. Then Then we we have have for for all all positive positive ind'(P) F- A ~ compS(P) F- A. The sound The combination combination of of the the previous previous result, result, Theorem Theorem 6.1.3, 6.1.3, Theorem Theorem 5.3.1 5.3.1 ((soundness completeness of ness of of SLDNF-resolution SLDNF-resolution)) and and Theorem Theorem 5.3.5 5.3.5 ((completeness of SLDNF-resolution SLDNF-resolution)) provides for the provides a a powerful powerful framework framework for the analysis analysis of of logic logic programs. programs. We We show show this this by by continuing the introduced above. continuing the discussion discussion of of the the programs programs P1 and and P2 introduced above. First termination in induction is First we we turn turn to to the the question question of of termination in Example Example 6.3.1 6.3.1.. Since Since induction is available, available, it it is is easy easy to to see see that that we we have have PI P2 + (x, + (y) --+ (member - (x, y))). � (Pd f-F-VxVy(list ind y) V member ind'(P1) VxVy(list+(y) (member+ (x, y)V member-(x, y))). (1) (1) Now Now choose choose arbitrary arbitrary terms terms a and and bb so so that that list l i s t ( b(b) ) succeeds succeeds using using SLDNF SLDNFresolution, we can resolution, i.e., i.e., list(b) l i s t ( b ) R(P) R ( P ) c. Then Then we can conclude conclude that that member(a, member(a, b) b) either either -+ a succeeds succeeds or or fails fails using using SLDNF-resolution SLDNF-resolution by by the the following following argument: argument: In In view view of of Theorem 5.3.1 Theorem 6.1.3 Theorem 5.3.1 and and Theorem 6.1.3 we we have have compS(P1) F- list+(b). (2) 678 678 G. G. Jager Jhger and R. Stark Sthrk ind�(Pl ) ind�(Pl ) Since ind'(P1) is is an an extension extension of of comp�(Pt) compS(P1),, we we obtain obtain from from (1) (1) and (2) that Since and (2) that ind'(P1) + (a, b) - (a, b) proves proves member member+(a, b) V V member member-(a, b).. Hence Hence the the previous previous theorem theorem implies implies compS(P1) F member+(a, b) V member-(a, b). (3) (3) Applying Applying Theorem Theorem 5.3.5 5.3.5 and and Theorem Theorem 6.1.3 6.1.3 with with the the mode mode assignment assignment of of Ex Example that either ample 5.2.6 5.2.6 yields yields that either member(a, member(a,b) for some substitution a a or or b) R ( P ) aa for some substitution member(a, b) b) E E F(P). In In other other words, words, the the goal goal member(a, member(a, b) b) succeeds succeeds or or fails fails using using member(a, SLDNF-resolution. SLDNF -resolution. After After the the treatment treatment of of termination termination of of logic logic programs programs we we come come back back to to the the problem problem of of the the equivalence equivalence of of the the logic logic programs programs in in Example Example 6.3.2. 6.3.2. It It is is easy easy to to verify verify that that we we have have R(P) F(P). i~d~(P~) F wvyvz(~t+(~)A ~t+(y)-+ (~ddl+ (z, y, z) ~ ~dd2+(~,y, z))). (4) Let be terms terms so that the the goals goals nnat(a) at (a) and ( b) succeed Let a, a, bb and and c be so that and nat nat(b) succeed using using SLDNF SLDNFresolution. resolution. Since Since the the program program P2 is is definite, definite, we we can can use use the the trivial trivial mode mode assignment assignment of Example Example 5.2.6, 5.2.6, and and by by the the same same argument argument as as above above we we can can conclude conclude that that the the goal goal of add 1 (a, b, b, cc)) addl(a, b, cc)) succeeds succeeds using using SLDNF-resolution SLDNF-resolution just just in in case case that that the the goal goal add2(a, add2(a, b, succeeds. succeeds. The The two two relations relations add1 addl and and add2 add2 do do not not only only have have the the same same behavior behavior with with respect respect to share the to success success but but also also share the same same behavior behavior with with respect respect to to failure. failure. We We first first observe observe that that (5) (add1 + (x, y, z) V add1 ind~(P2) f-I- 'v'x'v'y'v'z VxVyVz(nat+(x) -+ (addl+(x,y,z) addl-(x,y,z))) (5) (nat + (x) -+ - (x, y, z))) P2 ind � (P2 ) and and + (x, y, z) V add + (y) -~ (add2 � (p2 ) f-F 'v'x'v'y'v'z(nat (6) ind~(P2) VxVyVz(nat+(y) (add2+(x,y,z) add2-(x,y,z))). ind T (x, y, z))). � (P2 ) proves - (x)) for Lines (5), (6) (:l) /\ R Lines (4) (4),, (5), (6) and and the the fact fact that that ind ind'(P2) proves 'v'x-.(R+ V%-~(R+(%)A R-(%)) for any any -+ relation yield that that relation R R of of £, s yield i~d~(P~) F WVyW(~t+(~) ^ ~t+(y) ~ (~ddl-(~, y, z) ++ ~dd2-(~, y, z))). From this we b, cc)) fails using SLDNF-resolution From this we can can conclude conclude that that the the goal goal add1(a, addl(a, b, fails using SLDNF-resolution c) fails if aa,, bb and at(a) and if if and and only only if if the the goal goal add2(a, add2(a, b, b, c) fails if and c are are terms terms so so that that n nat(a) and succeed. Thus, at(b) succeed. nnat(b) Thus, the the relations relations add1 addl and and add2 add2 have have the the same same behavior behavior with with respect respect to to success success and and failure. failure. 7. Concluding Concluding remark remark In In discussing discussing the the foundations foundations of of logic logic programming programming it it is is often often possible possible to to distin distinguish guish between between three three levels levels of of abstraction: abstraction: I.I. Declarative Declarative semantics. semantics. Semantical Semantical considerations considerations about about logic logic programming programming are are of often guided ten guided by by the the attempt attempt of of constructing constructing suitable suitable minimal minimal models models of of logic logic programs. programs. However, minimal models models of However, in in gen�ral general the the logical logical complexity complexity of of minimal of logic logic programs programs is is very very high high and and the the corresponding corresponding semantics semantics is is noneffective. noneffective. Proof-Theoretic Framework Framework for for Logic Logic Programming Programming AA Proof-Theoretic 679 679 II. Proof Proof theory. theory. ItIt deals deals with with the the development development and and analysis analysis of of deductive deductive systems systems for for II. proofs as as computations computations paradigm. paradigm. In In the the logic programs programs and and isis often often directed directed to to the the proofs logic ideal case case there there isis aa close close connection connection between between the the proof proof theory theory and and the the procedural procedural ideal aspects of of logic logic programming programming in in the the sense sense that that query-answering query-answering mechanisms mechanisms can can aspects be interpreted interpreted as as formal formal proofs proofs and and suitable suitable formal formal proofs proofs can can be be transformed transformed into into be successful computations. computations. successful III. Procedural Procedural semantics. semantics. ItIt isis concerned concerned with with the the general general principles principles behind behind the the III. implementations of of logic logic programming. programming. Since Since today today most most procedural procedural approaches approaches implementations to first first order order logic logic programming programming are are based based on on some some form form of of SLDNF-resolution, SLDNF-resolution, the the to distinguished role role of of this this concept concept is is evident. evident. distinguished In our our article article we we followed followed this this general general pattern. pattern. The The results results we we presented presented can can be be In roughly summarized summarized as as follows. follows. If If PP is is aa "decent" "decent" logic logic program program and and A A aa closed closed roughly atom, then then the the following following assertions are equivalent: equivalent: atom, assertions are A is is true least adequate structure ~[p of P. ((i) i) A true in in the the w-segment w-segment of of the the least adequate Herbrand Berbrand structure 'Jp of P. A is true in all structures structures which which are are adequate P. (ii ) A (ii) is true in all adequate to to P. ((iii) iii) A A is identity and and cut-free cut-free provable provable in deductive system R(P) . is identity in the the deductive system 7~(P). is derivable derivable by by SLDNF-resolution. SLDNF-resolution. (iv) A (iv) A is Furthermore, by aa simple simple syntactic transformation it possible to to associate each Furthermore, by syntactic transformation it is is possible associate to to each logic program program P system of inductive definitions �(P) so logic P a a system of positive positive inductive definitions ind ind'(P) so that that for for closed closed (P) is atoms atoms A A derivability derivability from from ind� ind'(P) is equivalent equivalent to to each each of of the the four four assertions assertions above. above. In (P) which In addition addition induction induction principles principles are are available available in in ind� ind'(P) which make make it it possible possible to to prove prove properties properties about about logic logic programs. programs. In In this this sense sense we we hope hope that that we we could could provide provide aa proof-theoretic proof-theoretic framework framework for for logic logic programming. programming. Of Of course course there there exist exist other other proof-theoretic proof-theoretic approaches approaches to to logic logic programming programming which which we we did did not not mention mention at at all, all, and and we we conclude conclude this this article article with with mentioning mentioning two two of of them. them. 's linear cf. e.g. Some Some interesting interesting activities activities in in this this area area start start off off from from Girard Girard's linear logic logic ((cf. e.g. Girard 1987a]) and Girard [[1987a]) and study study the the connections connections between between logic logic programming programming and and linear linear logic. logic. Another Another important important area area in in the the general general field field of of logic logic programming programming deals deals with with 1991] higher higher order order logic logic programming, programming, and and we we refer refer the the reader reader for for example example to to Miller Miller [[1991] and 1992] for and Pfenning Pfenning [[1992] for further further reading. reading. References References K. K. R. R. APT APT [1990] [1990] Logic Logicprogramming, programming, in: in: Handbook Handbookof o] Theoretical Theoretical Computer Computer Science, Science, Volume Volume B, B, J. 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North-Holland, Amsterdam. Amsterdam. [1987] Proof Theory, North-Holland, [1987] A . VAN VAN GELDER GELDER AND AND J. J . S. S. SCHLIPF SCHLIPF A. Commonsense axiomatizations axiomatizations for for logic logic programs, programs, J. of Progmmming, 17, 17, pp. pp. 161161[1993] of Logic Programming, [1993] Commonsense 195. 195. CHAPTER XX CHAPTER Types Types in Logic, Logic, Mathematics M athematics and and Programming Programming Robert L. L. Constable Constable Robert Computer Science Department, Cornell Cornell University Computer Science Department, 14853, USA Ithaca, New York 1~853, Contents Contents 1. Introduction 1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2. Typed Typed logic logic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3. Type 3. Type theory theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4. Typed Typed programming programming languages languages . . . . . . . . . . . . . . . . . . . . . . . . . . . 5. Conclusion 5. Conclusion .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6. Appendix Appendix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References References .. .. .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . HANDBOOK H A N D B O O K OF OF PROOF P R O O F THEORY THEORY Edited Edited by by S. S. R. R. Buss Buss © 1998 1998 Elsevier Elsevier Science Science B.V. B.V. All All rights rights reserved reserved 684 684 692 726 754 766 768 773 R. Constable R. 684 684 1 . IIntroduction 1. ntroduction Proof theory theory and and computer computer science science are are jointly jointly engaged engaged in in aa remarkable remarkable enterenter Proof prise. Together Together they they provide provide the the practical practical means means to to formalize formalize vast vast amounts amounts of of prise. mathematical knowledge. knowledge. They They have have created created the the subject subject of of automated reasoning mathematical and and aa digital digital computer computer based based proof technology; these these enable enable aa diverse diverse community community of mathematicians, mathematicians, computer computer scientists, scientists, and and educators educators to to build build aa new new artifact artifact of aa globally globally distributed distributed digital digital library library of of formalized formalized mathematics. mathematics. II think think that that this this artifact artifact signals signals the the emergence emergence of of aa new new branch branch of of mathematics, mathematics, perhaps perhaps to to be be called called Formal Mathematics. The theorems theorems of of this this mathematics mathematics are are completely completely formal formal and and are are processed The processed digitally. They They can can be be displayed displayed as as beautifully beautifully and and legibly legibly as as journal journal quality quality digitally. mathematical text. text. At At the the heart heart of of this this library library are are completely completely formal formal proofs proofs mathematical created with with computer computer assistance. assistance. Their Their correctness correctness is is based based on on the the axioms axioms and and rules rules created of various of various foundational foundational theories; theories; this this formal formal accounting accounting of of correctness correctness supports supports the the highest known standards to formally formally relate highest known standards of of rigor rigor and and truth. truth. The The need need to relate results results in in different topic in in proof proof theory theory and foundations of of different foundational foundational theories theories opens opens a a new new topic and foundations mathematics. mathematics. Formal theories are Formal proofs proofs of of interesting interesting theorems theorems in in current current foundational foundational theories are very very large objects. Creating Creating them the speed capacities of large rigid rigid objects. them requires requires the speed and and memory memory capacities of modern expressiveness of modern software. Programs modern computer computer hardware hardware and and the the expressiveness of modern software. Programs fill in in tedious tedious detail; detail; they many kinds kinds of called theorem provers fill called they recognize recognize many of "obvious "obvious inference," and and they find long chains of and even inference," they automatically automatically find long chains of inferences inferences and even complete complete subproofs or or proofs. The study of these these theorem theorem provers and the the symbolic symbolic algorithms subproofs proofs. The study of provers and algorithms that make make them them work part of of the the subject reasoning. This that work is is part subject of of automated automated reasoning. This science science and and the the proof proof technology technology built built on on it it are are advancing advancing all all the the time, time, and and the the new new branch branch of mathematics that will have methods, surprises of mathematics that they they enable enable will have its its own own standards, standards, methods, surprises and and triumphs. triumphs. This This article article is is about about the the potent potent mixture mixture of of proof proof theory theory and and computer computer science science behind automated behind automated reasoning reasoning and and proof proof technology. technology. The The emphasis emphasis is is on on proof proof theory theory topics topics while while stressing stressing connections connections to to computer computer science. science. Computer Computer science science is is concerned concerned with with automating automating computation. computation. Doing Doing this this well well has possible to has made made it it possible to formalize formalize real real proofs. proofs. Computing Computing well well requires requires fast fast and and robust robust hardware hardware as as well well as as expressive expressive high high level level programming programming languages. languages. High High level level lan languages guages are are partially partially characterized characterized by by their their type systems; i.e., i.e., the the organization organization of of data types expressible expressible in in the the language. language. The The evolution evolution of of these these languages languages has has led led to to type type systems systems that that resemble resemble mathematical mathematical type type theories theories or or even even computationally computationally effective effective set set theories. theories. (This (This development development underlines underlines the the fact fact that that high high level level programming programming is is an an aspect aspect of of computational computational mathematics.) mathematics.) This article will focus mainly on relating data types and mathematical types. The The connection connection between between data data types types and and mathematical mathematical types types in in the the case case of of formal formal mathematics mathematics and and automated automated reasoning reasoning is is even even tighter tighter than than the the general general connection. connection. Here Here is is why. why. To To preserve preserve the the highest highest standards standards of of rigor rigor in in formalized formalized mathematics mathematics built built with with computer computer assistance assistance (the (the only only way way to to produce produce it) it),, it it is is necessary necessary to to reason reason proof technology; automated reasoning Formal Mathematics. theorem provers types data types and mathematical types. type systems; data This article will focus mainly on relating Types Types 685 685 about about programs programs and and computations. computations. This This is is what what intuitionists intuitionists and and constructivists constructivists do do at aa very very high high level level of of abstraction. abstraction. So So as as the the programming programming languages languages for for automating automating at reasoning become become more more abstract abstract and and expressive, expressive, constructive constructive mathematics mathematics becomes becomes reasoning directly relevant relevant to to Formal Formal Mathematics Mathematics and and to to the the "grand "grand enterprise" enterprise" of of building building it it directly using theorem theorem provers. provers. We We will will see see that that connections connections are are quite quite deep. deep. using It turns turns out out that that proof proof technology technology is is relevant relevant to to other other technologies technologies of of economic economic and and It strategic importance. importance. For For instance, instance, the the type checkers in in commercial commercial programming programming strategic languages like like ML ML are are actually actually small small theorem theorem provers. provers. They They check check that that arguments arguments languages to aa function function match match the the type type of of the the function function (see (see section section 3). 3). Industrial Industrial model to checkers systematically systematically search search for for errors errors in in the the design design of of finite finite state state systems, systems, such such as as hardware circuits circuits or or software software protocols. protocols. More More general general tools tools are are program verification hardware systems. These These combine combine type type checkers, checkers, model model checkers, checkers, decision decision procedures, procedures, and and theorem theorem provers provers that that use use formalized formalized mathematics. mathematics. They They are are employed employed to to prove prove that programs programs have have certain certain formally formally specified specified properties. properties. Such Such proofs proofs provide provide the the that highest levels levels of of assurance assurance that that can can be be given given that that programs programs operate operate according according to to highest specifications. There There are are also also software software systems systems based based on on proof proof technology technology which which specifications. synthesize synthesize correct correct programs programs from from proofs proofs that that specifications specifications are are realizable. realizable. We We will will examine the the proof proof theory theory underlying underlying some some of of these these systems. systems. examine My My approach approach to to the the subject subject comes comes from from the the experience experience of of designing, designing, studying, studying, and and using using some some of of the the earliest earliest and and then then some some of of the the most most modern modern of of these these theorem theorem provers. provers. Currently Currently my my colleagues colleagues and and II at at Cornell Cornell are are working working with with the the system system we we ,, l ) . We in Constable call "new pearl call Nuprl Nuprl (("new pearl").1 We call call it it a a proof development system in Constable et et al. [1986] al. [1986],, but but some some call call it it a a problem solving environment (PSE) (PSE) or or aa logicalframework (LF). From From another another point point of of view view it it is is a a collaborative mathematics environment, c.f., c.f., (LF). Chew Whatever Nuprl Chew et et al. al. [1996]. [1996]. Whatever Nuprl is is called, called, II am am concerned concerned with with systems systems like like it it and and their evolution. evolution. We examine the the logical logical features to aa variety of current current their We will will examine features common common to variety of systems similar kind, kind, such such as as ACL2, ACL2, Alf, Alf, Coq, Coq, HOL, IMPS, Isabelle, Kiv, LA, systems of of aa similar HOL, IMPS, Isabelle, Kiv, LA, Mizar, NqThm NqThm and Otter. So So while while II will refer to to Nuprl from time time to time, Lego, Mizar, Lego, and Otter. will refer Nuprl from to time, most of general and apply to 21st century most of the the ideas ideas are are very very general and will will apply to the the systems systems of of the the 21st century as well. Before saying saying more more about about the the article, article, let let me put the the work work into historical as well. Before me put into historical perspective. Doing Doing this this will will allow allow me me to to state state my my goals goals more more exactly (especially after after perspective. exactly (especially each each topic topic of of Section Section 1.1). 1.1). type checkers checkers systems. model program verification proof development system problem solving environment logicalframework collaborative mathematics environment, Begriff Grundgesetze sschrift and the the ground ground was was cleared cleared to to provide provide aa firm firm foundation foundation for mathematics.22 Frege, and Frege, for mathematics. In Principia Principia Mathematica, Mathematica, Whitehead Whitehead and and Russell Russell [1925-27] [1925-27J revised revised Frege's Frege ' s flawed flawed In From BegriffHistorical 1875-1995. From 1.1. H 1.1. i s t o r i c a l pperspective e r s p e c t i v e on on aa ggrand r a n d eenterprise n t e r p r i s e 1875-1995. [1879J onwards onwards until until Grundgesetze [1903], [1903], logic logic was was re-surveyed re-surveyed by by Gottlob Gottlob sschrift [1879] architectural plans, plans, and and then then using using these these plans, plans, Hilbert Hilbert [1926] [1926J laid laid out out aa formalist formalist architectural We have have released released Version Version 4.2, 4.2, see see http://www.cs.cornell.edu/Info/Projects/NuPrl/nuprl.html. http://www.cs.comell.edu/lnfo/Projects/NuPrl/nuprl.html. 11We Version 55 and and "Nuprl Light" Light" will will be be available available at at this this World World Wide Wide Web Web site site in in 1999. 1999. Version 2 BegrifJsschrijt ("concept ("concept script") script") analyzed analyzed the the notion notion of of aa proposition proposition into into function function and and 2Begri~sschrift argument, introduced introduced the the quantifiers, quantifiers, binding, binding, and and aa theory theory of of identity. identity. This This created created the the entire entire argument, predicate calculus. calculus. Grundgesetze Grundgesetze presented presented aa theory theory of of classes classes based based on on the the comprehension comprehension predicate principle and and defined defined the the natural natural numbers in terms terms of them. 686 686 R. Constable Constable program program to to build build the the completely completely formal formal theories theories which which would would be be used used to to explain explain and justify justify the the results results and mathematics. His His program program would would defend defend and and methods methods of of mathematics. mathematical mathematical practice practice against against critics critics like like Brouwer Brouwer who who saw saw the the need need to to place place the the foundation pilings pilings squarely squarely on on the the natural natural numbers numbers and and build build with with constructive constructive foundation methods. 33 methods. Hilbert called for training some some himself, Hilbert called for workers, workers, training himself, and and began began with with them them the the task task which compelling and which proved proved to to be be so so compelling and attractive attractive to to many many talented talented mathematicians mathematicians like Neumann, Herbrand, Herbrand, Gentzen, like Church, Church, von von Neumann, Gentzen, Skolem, Skolem, Turing, Turing, Tarski, Tarski, Godel, GSdel, and and many more. deep into into the the bedrock to explore explore the the foundation foundation site, site, Kurt Kurt GSdel many more. Boring Boring deep bedrock to Godel [1931] limitations to to the [1931] unexpected unexpected limitations the planned planned activity. activity. It It could could never never be be completed completed as envisioned envisioned by by Hilbert.4 Hilbert. 4 His His surprising surprising discovery discovery changed changed expectations, expectations, but but the the as tools Godel tools GSdel created created transformed transformed the the field field and and stimulated stimulated enormous enormous interest interest in in the the enterprise. enterprise. More More remarkable remarkable discoveries discoveries followed. followed. Within two two decades, decades, computer computer science science was was providing providing new new "power "power tools" tools" to to realize realize Within in software software the the formal formal structures structures needed needed to support mathematics. mathematics. By By 1960 computer in to support 1960 computer hardware hardware could could execute execute programming programming languages languages like like Lisp, Lisp, c.f. c.f. McCarthy McCarthy [1963], [1963], designed designed for for the the symbolic symbolic processing processing needed needed to to build build formal formal structures. structures. Up Up in in the the scaffolding scaffolding computer computer scientists scientists began began to to encounter encounter their their own own problems problems with with "wiring "wiring and and communications," communications," control control of of resource resource expenditure, expenditure, design design of of better better tools, etc. 1970's poised tools, etc. But But already already even even in in the the 1970's poised over over the the ground ground like like aa giant giant drilling rig, the structures supported supported still deeper penetration drilling rig, the formal formal structures still deeper penetration into into the the bedrock bedrock designed designed to to support support mathematics mathematics (and (and with with it it the the mathematical mathematical sciences sciences and and much much of our technical of our technical knowledge). knowledge). The The theory theory of of computational computational complexity, complexity, arising arising from from 's P Hartmanis and Stearns led to like Cook Hartmanis and Stearns [1965], [1965], led to further further beautiful beautiful discoveries discoveries like Cook's P = N N PP problem, problem, and and to to a a theory theory of of algorithms algorithms needed needed for for sophisticated sophisticated constructions, constructions, and and to to a a theory theory of feasible mathematics (see (see Buss Buss [1986], [1986], Leivant Leivant [1994b,1994a,1995]) [1994b,1994a,1995]),, and and to to ideas ideas for for the the foundations of computational mathematics. By 1970 the By 1970 the value value of of the the small small formal formal structure structure already already assembled assembled put put to to rest rest the the nagging nagging questions questions of of earlier earlier times times about about why why mathematics mathematics should should be be formalized. formalized. The The existing economic benefit engineering, just Leibniz dreamed, existing structure structure provided provided economic benefit to to engineering, just as as Leibniz dreamed, Frege Frege foresaw, foresaw, McCarthy McCarthy planned planned [1962], [1962], and and many many are are realizing. realizing. Even Even without without the the accumulating accumulating evidence evidence of of economic economic value, value, and and without without counting counting the the immediate immediate utility utility of of the the software software artifacts, artifacts, scientists scientists in in all all fields fields recognized recognized that that the the discoveries discoveries attendant attendant on on this this "grand "grand enterprise" enterprise" illuminate illuminate the the very very nature nature of of knowledge create and results of knowledge while while providing providing better better means means to to create and manage manage it. it. The The results of this enterprise because all scholars and this enterprise have have profound profound consequences consequences because all scholars and scientists scientists are are in the business in the business of of processing processing information information and and contributing contributing to to the the accumulation accumulation and and dissemination dissemination of of knowledge. knowledge. The construction of on; it kind The construction of the the foundational foundational structure structure goes goes on; it is is forming forming aa new new kind ofJeasible mathematics foundations of computational mathematics. 31 nitistic analysis 3I refer to Hilbert's formalist formalistprogram programfounded on a fifinitistic analysis of formal formal systems systems to prove their consistency reasoning as a (possibly their consistency and to justify justify non-constructive non-constructive reasoning (possibly meaningless) meaningless) detour detour justifi ed by the consistency justified consistency of a formal formal system. system. 4Godel consistency is not sufficient 4GSdel showed showed that consistency sufficient to justify the detour because because there are formulas formulas of number theory theory such such that that both both P p can added (P an of number P and and ..., ~P can be be consistently consistently added an unprovable unprovable formula). formula). 687 687 Types of of place, place, like like a a biosphere biosphere made made out out of of bits. bits. We We might might call call it it aa "cybersphere" "cybersphere" since since it it encloses . . . which encloses the the space space we we call call "cyberspace." "cyberspace." Many Many people people now now live live in in this this space space.., which supports supports commerce commerce and and recreation recreation as as well well as as scholarship scholarship and and science. science. It is is in in the the context context of of this "grand enterprise" enterprise" that that II have have framed framed the the article. article. II see see It this "grand it the major it concerned concerned with with two two of of the major modes modes of of work work in in assembling assembling the the formal formal structures structures logical analysis construction. II will - - logical analysis and and algorithmic algorithmic construction. will briefly briefly mention mention the the aspects aspects of of these these activities activities that that II treat treat here. here. Logical Logical analysis. analysis. When When looking looking back back over over the the period period from from 1879 1879 to to now, now, we we see see that that the the formal formal analysis analysis of of mathematical mathematical practice practice started started with with logical logical language. language. Frege Frege [1879] [1879] said: said: "To "To prevent prevent anything anything intuitive intuitive from from penetrating penetrating [[into an argument argument]] un uninto an noticed, noticed, II had had to to bend bend every every effort effort to to keep keep the the chain chain of of inferences inferences free free of gaps. gaps. In attempting to to comply comply with requirement in strictest of In attempting with this this requirement in the the strictest possible way obstacle. This possible way II found found the the inadequacy inadequacy of of language language to to be be an an obstacle. This defi ciency led ... deficiency led me me to to the the present present ideography ideography... Leibniz, Leibniz, too, too, recognized recognized n and and perhaps perhaps overrated overrated the the advantages advantages of of ad adequate . . . calculus philosophicus.., . . . was equate notation. notation. His His idea idea of of aa... was so so gigantic gigantic that that the the attempt attempt to to realize realize it it could could not not go go beyond beyond the the bare bare preliminaries. preliminaries. The The enthusiasm enthusiasm that that seized seized its its originator originator when when he he contemplated contemplated the the immense mankind that the calculus immense increase increase in in intellectual intellectual prover prover of of mankind that [[the calculus would would bring bring]] caused caused him him to to underestimate underestimate the the difficulties difficulties .. .. ... . But But even even if if this this worthy worthy goal goal cannot cannot be be reached reached in in one one leap, leap, we we need need not not despair despair of of aa slow slow step step by by step step approximation." approximation." So So Frege Frege began began with with very very limited limited goals goals and and took took what what he he characterized characterized as as "small "small steps" (like (like creating creating all all of of predicate predicate 10giC!). logic!). He He did did not not include include aa study study of of computation computation steps" and and its its language; language; he he limited limited his his study study of of notation notation to to logical logical operators, operators, and and he he ruled ruled out out creating creating a a natural natural expression expression of of proofs proofs or or classifying classifying them them based based on on how how "obvious" "obvious" they they are. are. In In addition, addition, Frege Frege focused focused on on understanding understanding the the most most fundamental fundamental types, types, natural natural numbers, numbers, sequences, sequences, functions, functions, and and classes. classes. He He adopted adopted aa very very simple simple approach approach to to in his the the domain domain of of functions, functions, forcing forcing them them all all to to be be total. He He said said ((in his Collected Papers) "the sign always have "the sign a + + bb should should always have aa reference, reference, whatever whatever signs signs for for definite definite objects objects ' may b '." Principia took took aa different may be be inserted inserted in in place place of o f ' 'aa' and and ''b'." different approach approach to to functions, functions, introducing introducing types, types, but but also also it it excluded excluded from from consideration consideration an an analysis analysis of of computation computation or or natural natural proofs proofs or or the the notational notational practices practices of of working working mathematics. mathematics. It It too too developed developed only only basic basic mathematics mathematics with with no no attempt attempt to to treat treat abstract abstract algebra algebra or computational computational parts parts of of analysis. analysis. or Principia Mathematica, the the monumental monumental work work of of Whitehead Whitehead and and Russell Russell [1925[192527] 27],, was was indeed indeed the the first first comprehensive comprehensive rendering rendering of of mathematics mathematics in in symbolic symbolic logic. logic. ' s celebrated 1931 paper Godel GSdel's celebrated 1931 paper "On "On Formally Formally Undecidable Undecidable Propositions Propositions of of Principia Mathematica and and Related Related Systems" Systems" begins: begins: "The "The development development of of mathematics mathematics toward toward greater greater precision precision has has led, led, as as is known, to it, so is well well known, to the the formalization formalization of of large large tracts tracts of of it, so that that one one calculus philosophicus a total. Principia Collected Papers) Principia Mathematica, Mathematica Principia 688 688 R. Constable Constable can nothing but rules. The can prove prove any any theorems theorems using using nothing but aa few few mechanical mechanical rules. The most most comprehensive comprehensive formal formal systems systems that that have have been been set set up up hither hither to to are are hand and the the system system of of Principia Mathematica (PM) (PM) on on the the one one hand and the the Zermelo-Fraenkel . . . on Zermelo-Fraenkel axiom axiom system system of of set set theory theory.., on the the other." other." Principia presents presents a a logic logic based based on on types types and and derives derives in in it it aa theory theory of of classes, classes, while ZF ZF set set theory theory provided provided informal informal axioms and, and, logic logic was was incidentaP incidental. 5 Principia while deals with with the deals the topics topics that that II find find fundamental fundamental in in my my own own work work of of "implementing "implementing mathematics" mathematics" in in the the Nuprl Nuprl system, system, Constable Constable et et al. al. [1986] [1986].. Thus Thus much much of of what what II say say here is here is related related to to PM. P M. Indeed, Indeed, in in a a sense sense Nuprl Nuprl is is aa modern modern style style Principia suitable suitable for for computational computational mathematics. mathematics. Hilbert introduced Hilbert introduced a a greater greater degree degree of of formalization, formalization, essentially essentially banishing banishing seman semantics, and tics, and he he began began as as aa result result to to deal deal with with computation computation in in his his metalanguage. metalanguage. But But he took aa step he took step backwards backwards from from Principia in in terms terms of of analyzing analyzing when when expressions expressions are are meaningful. He meaningful. He reduced reduced this this to to an an issue issue of of parsing parsing and and "decidable "decidable type type checking" checking" of of formulas semantic judgments formulas as as opposed opposed to to the the semantic judgments of of Principia. ' t until It It wasn wasn't until Gentzen Gentzen that that the the notion notion of of proofs proofs as as they they occur occur in in practice practice was was ' t until analyzed, first in in natural deduction and then in analyzed, first and then in sequent calculi. It It wasn wasn't until Her Herbrand, Godel, Church, brand, GSdel, Church, Markov, Markov, and and Turing Turing that that computation computation was was analyzed analyzed and and not not until until de de Bruijn Bruijn that that the the organization organization of of knowledge knowledge (into (into aa "tree "tree of of knowledge" knowledge" with with ' s Automath explicit explicit contexts) contexts) was was considered. considered. De De Bruijn Bruijn's Automath project project also also established established links to computing, like links to computing, like those those simultaneously simultaneously being being forged forged from from computer computer science. science. Recently Recently Martin-Lor Martin-LSf has has widened widened the the logical logical investigation investigation to to reintroduce reintroduce aa se semantic mantic approach approach to to logic, logic, to to include include computation computation as as part part of of the the language, language, and and to to make make manifest manifest the the connections connections to to knowledge. knowledge. Martin-Lor Martin-LSf [1983,p.30] [1983,p.30] says: says: "to "to have have proved proved = -- to to know know = - to to have have understood, understood, comprehended, comprehended, grasped, grasped, or or seen. seen. It It is is now now manifest, manifest, from from these these equations, equations, that that proof proof and and knowledge knowledge are are the the same. same. Thus, Thus, if if proof proof theory theory is is construed, construed, not not in in ' s sense, Hilbert Hilbert's sense, as as metamathematics metamathematics but but simply simply as as the the study study of of proofs proofs in in the the original original sense sense of of the the word, word, then then proof proof theory theory is is the the same same as as theory theory of of knowledge ..." knowledge..." We position where consider all We are are now now in in a a position where mathematical mathematical logic logic can can consider all of of these these elements: truth, and elements: an an analysis analysis of of the the basic basic judgments judgments of of typing, typing, truth, and computational computational equality; equality; an an analysis analysis of of natural natural proofs proofs and and their their semantics; semantics; the the integration integration of of compu computational concepts tational concepts into into the the basic basic language; language; an an analysis analysis of of the the structure structure of of knowledge knowledge and and its its role role in in practical practical inference; inference; and and classification classification of of inference inference according according to to its its computational complexity. computational complexity. We will attempt logic that We will attempt aa consideration consideration of of logic that takes takes all all this this into into account account and and is is linked to computing practice, linked to computing practice, and and yet yet is is accessible. accessible. II begin begin the the article article with with an an account account of logic that naturally to of typed typed logic that relates relates naturally to the the Automath Automath conception. conception. The The connection connection is is discussed explicitly explicitly in Section 2.12. discussed in Section 2.12. Principia Mathematica Principia axioms Principia Principia Principia natural deduction Principia. sequent calculi. 55principia Principia is is not not formal formal in in the the modern modern sense. sense. There There are are semantic semantic elements elements in in the the account account which Wittgenstein [1953,1922] Hilbert made a point of formalizing which Wittgenstein [1953,1922] objected objected to. Hilbert formalizing logic, logic, and we follow that purely purely formal follow in that formal tradition. 689 689 Types The article article stresses stresses the the nature nature of of the the underlying underlying logical logical language language because because that that The is so so basic basic - - everything everything else else is is built built upon upon it. it. The The structures structures built built are are so so high high that that is small change change in in the the foundation foundation can can cause cause aa large large movement movement at at the the top top of of the the aa small structure. So So any any discoveries discoveries that that improve improve the the foundation foundation for for formal formal mathe mathe '1tics atics structure. are among among the the most most profound profound in in their their effect. effect. As As it it is, is, we we are are standing standing on on the the shoulders shoulders are of giants. giants. II take take the the time time in in section section 22 to to review review this this heritage heritage that that is is so so crucial crucial to to of everything else. else. everything A l g o r i t h m i c construction. c o n s t r u c t i o n . Computer Computer science science completely completely transformed transformed the the "grand "grand Algorithmic enterprise." First First it it introduced introduced computational computational procedure procedure and and procedural knowledge, enterprise." and it it gradually gradually widened the scope scope of of its its successes. It could could check check formulas formulas and and and widened the successes. It synthesize them; later later it it could check proofs proofs and and synthesize synthesize them. them. In In all all this, this, the the synthesize them; could check precision that that Godel GSdel referred referred to to reached reached new new levels levels of of mechanical precision. The The vast vast precision change of of scale scale from from processing processing aa few few hundred hundred lines lines by by hand hand to to tens tens of of thousands thousands change by machine machine ((now hundreds of of millions millions)) caused caused aa qualitative qualitative change change and and created created now hundreds by new fi fields like Automated Automated Deduction Deduction and and Formal Formal Mathematics Mathematics in in which which formalisms formalisms new elds like became usable. became usable. The success success of of procedural procedural knowledge knowledge created created the the questions questions of of relating relating it it to to declar declarThe ative knowledge, knowledge, aa question question at at the the heart heart of of computer computer science, science, studied studied extensively ative extensively in in the database database area, area, also also in in "logic "logic programming" and in in AI. It is is a a question question at at the the programming" and AI. It the ' s work heart of of AD, AD, McAllester McAllester [[1989], as is is clearly clearly seen seen in in Bundy Bundy's work on on proof plans heart 1989] , as From the the AI AI perspective, perspective, one one can can see see this this impact impact as as reintroducing reintroducing "mind" "mind" [[1991]. 1991 ] . From and "thought" "thought" into into the the enterprise enterprise McAllester McAllester [[1989]. From the the logical logical perspective perspective 1989] . From and ' s notion one can can see see this as reintroducing reintroducing the the study study of of intension and and Frege Frege's notion of of sense one this as into logic. logic. As As Jean-Yves Jean-Yves Girard Girard put put it it in in Girard, Girard, Taylor Taylor and and Lafont Lafont [[1989,p.4]: into 1989,p.4] : procedural knowledge, mechanical precision. proof plans intension sense "In recent recent years, during which the algebraic algebraic tradition tradition has flourished, "In years, during which the has flourished, the tradition was not of of note the syntactic syntactic tradition was not note and and would would without without a a doubt doubt have disappeared in one or two more for want want of any issue issue have disappeared in one or two more decades, decades, for of any or methodology. methodology. The The disaster disaster was was averted because of of computer computer science science or averted because y n t a x- which which posed some very very important - that that great great manipulator manipulator or or ssyntax posed some important theoretical theoretical problems." problems." Computer produced new new high high level for expressing expressing algorithms. algorithms. Computer science science produced level languages languages for These programming languages languages such such as as ML ML (for Meta LanLan These have have evolved evolved to to modern modern programming ( for Meta designed to to help help automate automate reasoning. reasoning. ML and its its proposed proposed extensions extensions have have guage) designed guage) ML and data types types that that the the type type system system resembles resembles aa constructive constructive theory theory such aa rich rich system system of of data such mathematical types. types. We discuss this this observation observation in in section section 3. 3. Our Our concern concern for for the the of We discuss of mathematical relationship between data relationship between data types types and and mathematical mathematical types types is is aa reason reason that that II will will talk talk so much about 2. so much about typed typed logic logic in in section section 2. Computer science science also also created created aa new new medium for for doing doing m mathematics the digital digital Computer a t h e m a t i c s- the This affects affects electronic medium medium now now most most visible visible through through the the World World Wide Wide Web. Web. This electronic every aspect For example, syntax can every aspect of of the the enterprise. enterprise. For example, the the "surface" "surface" or or concrete concrete syntax can be be disconnected from the abstract syntax, syntax, and we can disconnected from the abstract and we can display the the underlying underlying terms terms in in aa large large variety To take universal quantifier, variety of of forms. forms. To take aa trivial trivial point, point, the the typed typed universal quantifier, "for "for medium display 690 690 R. Constable Constable all A" can : A. or all x x of of type type A" can be be displayed displayed as as ' all, • hypertext Mathematica hyperprooJs, digital mathematics libraries. algorithmic phase 's to 60's was Stage 11.. From From the the late late 50 50's to the the late late 60's was an an algorithmic phase during during which which Stage the the basic basic symbolic symbolic procedures procedures and and low low level level data data structures structures were were discovered discovered and and improved. unification, and algorithms were improved. The The basic basic matching, matching, unification, and rewriting rewriting algorithms were coded coded and method and and tested tested in in the the resolution resolution method and various various decision decision procedures. procedures. We We learned learned the the extent extent and and value value of of these these basic basic symbolic symbolic procedures, procedures, and and they they were were made made very very efficient in systems al. [1984]) there are efficient in systems like like Otter Otter (Wos (Wos et et al. [1984]) and and Prolog. Prolog. Now Now there are links links being being formed formed with with other other non-numerical non-numerical computing computing methods methods from from symbolic symbolic algebra. algebra. A A deep deep branch branch of of computational computational mathematics mathematics has has been been formed, formed, and and communities communities of of scientists using the scientists are are using the tools. tools. Stage 2. The Stage 2. The 70's 70's saw saw the the creation creation of of several several systems systems for for use use in in writing writing more more reliable reliable software. software. These These were were called called program verifiers, e.g. e.g. the the Stanford Stanford Pascal Pascal Verifier Verifier (Igarashi, (Igarashi, London London and and Luckham Luckham [1975]) [1975]) and and Gypsy Gypsy (Good (Good [1985]) [1985]) were were targeted targeted to to Pascal PL/CV (Constable, Pascal and and NqThm NqThm (Boyer (Boyer and and Moore Moore [1979]) [1979]) for for pure pure Lisp, Lisp, PL/CV (Constable, Johnson (Gordon, Milner Milner Johnson and and Eichenlaub Eichenlaub [1982]) [1982]) for for aa subset subset of of PL/I, PL/I, and and LCF LCF (Gordon, and and Wadsworth Wadsworth [1979]) [1979]) for for aa higher higher order order functional functional programming programming language. language. These These systems could also implementing logics of computable functions (hence systems could also be be seen seen as as implementing (hence LCF) or LCF) or programming logics. The The motivation motivation (and (and funding) funding) for for this this work work came came from from computer science. science. The computer The goal goal was was to to "prove "prove that that programs programs met met their their specifications." specifications." This This technology technology drew drew on on the the algorithms algorithms from from the the earlier earlier period, period, but but also also contributed contributed new new techniques techniques such such as as congruence closure (Kozen (Kozen [1977] [1977],, Nelson Nelson and and Oppen Oppen [1979], [1979], Constable, Constable, Johnson Johnson and and Eichenlaub Eichenlaub [1982]) [1982]) and and new new decision decision procedure procedure such such as as Arith Arith (Chan (Chan [1982]) [1982]) and and SupInf SupInf (Bledsoe (Bledsoe [1975], [1975], Shostak Shostak [1979]) [1979]) and and the the notion notion of of theorem theorem proving proving tactics (from (from LCF). LCF). During During this this period period there there also also appeared appeared systems systems for for checking formalizing formalizing mathematics mathematics such such as as Automath Automath (de (de Bruijn Bruijn [1970], [1970], Nederpelt, Nederpelt, Geuvers Geuvers and and Vrijer Vrijer [1994]) [1994]) and and FOL FOL (Weyrauch (Weyrauch [1980]). [1980]). program verifiers, programming logics. logics oj computable Junctions congruence closure checking tactics 6This 6This capability capability can bbee explored explored at www.cs.comell.edu/lnfo/Projects/NuPrl. www.cs.cornell.edu/Info/Projects/NuPrl. the Nuprl Nuprl project project home h o m e page page 691 691 Types ' s the ' s and Stage Stage 3. 3. In In the the 80 80's and 90 90's the so-called so-called program program verification verification idea idea was was refined refined aa great deal. The Larch Larch (Guttag, (Guttag, Horning Horning and and Wing Wing [1985]) [1985]) system system was was multi-lingual, multi-lingual, great deal. The and and it it represents represents the the change change in in emphasis emphasis from from "verification" "verification" to to property checking. The LCLint system result of this evaluation The LCLint system is is aa result of this evaluation (see (see www.larch.lcs.mit.edu). www.larch.lcs.mit.edu). The The prover is is seen seen as as aa bug detector or "falsification system." system." The The LCF LCF system system spawned spawned or "falsification prover HOL (Gordon Nuprl (Constable HOL (Gordon and and Melham Melham [1993]), [1993]), Nuprl (Constable et et al. al. [1986]), [1986]), Isabelle Isabelle (Paulson (Paulson [1994]) and and many many others.7 others. 7 Nuprl, in turn, turn, spawned spawned others others like like Coq Coq (Coquand (Coquand and and [1994]) Nuprl, in Huet Huet [1988]), [1988]), Lego Lego (Pollack (Pollack [1995]) [1995]),, PVS PVS (Owre (Owre et et al. al. [1996]), [1996]), and and Nuprl-Light Nuprl-Light (Hickey (Hickey [1997]). [1997]). This period period saw saw also also the the second second generation generation efforts efforts to to build build computer computer systems systems This to mathematics. The Mizar checking (Trybulec [1983]) to help help create create formalized formalized mathematics. The Mizar checking effort effort (Trybulec [1983]) included included starting starting the the Journal of Formalized Mathematics (see (see www.mcs.anl.govjqed). www.mcs.anl.gov/qed). The The Nuprl Nuprl system system focused focused on on tools tools for for synthesizing synthesizing proofs proofs and and will will be be discussed discussed in in this e.g. Alf, this article, article, but but numerous numerous other other systems systems were were created, created, e.g. Alf, Coq, Coq, HOL, HOL, IMPS, IMPS, Isabelle, Isabelle, PVS, PVS, STeP STeP by by Constable Constable et et al. al. [1986], [1986], Paulin-Mohring Paulin-Mohring and and Werner Werner [1993], [1993], Farmer Owre, Rushby Farmer [1990] [1990],, Farmer, Farmer, Guttman Guttman and and Thayer Thayer [1991] [1991], Owre, aushby and and Shankar Shankar ' [1992], Owre et al. [1996], Bj0rner Bjcrner et et al. [1996], and and others. Some of of these these were were [1992] al. [1996], al. [1996], others.s8 Some ' Owre et integrated systems systems with with special special editors, library of of theories theories and and various various "logic "logic integrated editors, aa library ' s is engines." The toward more engines." The 90 90's is seeing seeing aa move move toward more modular modular and and open open systems, systems, and and in in ' s HOL-Nuprl HOL-Nuprl effort the 21st 21st century the century we we will will see see cooperative cooperative systems systems such such as as Howe Howe's effort [1996b]. [1996b]. This introduce some behind tactic-oriented This article article will will introduce some of of the the theory theory behind tactic-oriented theorem theorem proving and will relate proving and will will show show examples examples of of modern modern synthesized synthesized proofs. proofs. It It will relate this this to to developments developments in in typed typed programming programming languages. languages. property checking. bug detector Journal ofFormalized Mathematics pure 1.2. Outline. Section 1.2. Outline. Section 22 covers covers Typed Typed Logic. Logic. The The development development moves moves from from aa pure logic propositions to numbers, logic of of typed typed propositions to aa calculus calculus with with specific specific types-N types--N the the natural natural numbers, cartesian cartesian products products of of types, types, lists over over any any types, types, functions functions from from one one type type to to another, another, sets ned by sets built built over over aa type, type, and and types types defi defined by imposing imposing aa new new equality equality on on our our existing existing type . . . so type.., so called called quotient quotient or or congruence congruence types. types. The exposition exposition is is very very similar similar to to the the way way logic logic is is presented presented in in Nuprl, Nuprl, but but instead instead The of of deriving deriving logic logic from from type type theory, theory, we we start start with with logic logic in in the the spirit spirit of of Principia. We We also also present present logic logic in in the the manner manner of of Automath, Automath, so so one one might might call call this this logic logic "AutoMathematica." It It contrasts contrasts with with the the polymorphic predicate calculus of of LCF LCF (Gordon, [1979]) and (Gordon, Milner Milner and and Wadsworth Wadsworth [1979]) and HOL HOL (Gordon (Gordon and and Melham Melham [1993]) [1993]).. Section Section 33 covers covers Type Type Theory. Theory. This This is is essentially essentially an an introduction introduction ttoo Martin-Lb'f semantics using using an an axiomatic axiomatic theory theory close close to to his his Intuitionistic Intuitionistic Type Type Theory Theory circa circa 1982-88 and 1982-88 and its its Nuprl Nuprl variants variants (Martin-LM (Martin-Lhf [1982,1984,1983]) [1982,1984,1983]).. My My approach approach is is very very "expository" since there accounts in "expository" since there are are many many accessible accessible accounts in the the books books of of Nordstrom, Nordstrom, Petersson Petersson and and Smith Smith [1990], [1990], Martin-LM Martin-Lhf [1984], [1984], Thompson Thompson [1991] [1991],' the the articles articles of of Backhouse Backhouse [1989], [1989], Martin-LM Martin-Lhf [1982] [1982], Allen Allen [1987a] [1987a],, and and the the theses theses of of Allen Allen [1987b] [1987b], ' ' lists "AutoMathematica." semantics Principia. polymorphic predicate calculus Marlin-Lof 77A A comprehensive comprehensive history history of theorem theorem proving proving up to 1970 1970 can be found found in Siekmann Siekmann and Wrightson [1983]; [1995] for recent developments. [1983]; see MacKenzie MacKenzie [1995] developments. 8The SThe web site www-formal.stanford.edu/cJt/ARS/systems.html www-formal.stanford.edu/clt/ARS/systems.html at Stanford lists these systems. systems. 692 692 R. R. Constable Constable Palmgren Palmgren [1991,1995b], [1991,1995b], Poll Poll [1994] [1994], Setzer [19931,, Rezus Rezus [1985], [1985], Helmink Helmink [1992] [1992]which which ' Setzer [1993] provide provide technical technical detail. detail. II use use aa small small fragment fragment to to illustrate illustrate the the theory theory and and present present 's non-type Stuart Stuart Allen Allen's non-type theoretic theoretic account account [1987a] [1987a] as as well well as as aa semantics of proofs as objects. Many Many of of the the ideas ideas of of type type theory theory are are discussed discussed for for the the impredicative theories based on on Girard Girard's's system system F F (see (see Girard, Girard, Taylor Taylor and and Lafont Lafont [1989]). [1989]). There There is is aa based large large literature literature here here as as well well Coquand Coquand and and Huet Huet [1988], [1988], Reynolds Reynolds [1974], [1974], Luo Luo [1994], [1994], Poll [1994] [1994].. Dependent Dependent types types and and universes universes are are in in this this section. section. These These types types can can then then Poll be added added to to the the Typed Typed Logic. Logic. So So in in aa sense, sense, this this section section extends extends Typed Typed Logic. Logic. be Section Section 44 covers covers Typed Typed Programming. Programming. Here Here II explain explain the the notion notion of of "partial "partial types" types" common in in programming programming and and relate relate them them to to type type theory. theory. This This account account is is expository, expository, common designed designed to to make make connections. connections. But But II discuss discuss the the recursive recursive types types that that Constable Constable and and Mendler [1985] [1985] introduced introduced which which are are closely closely related related to to the the subsequent subsequent accounts accounts for for Mendler Coq Coq and and Alf Alf of of Coquand Coquand and and Paulin-Mohring Paulin-Mohring [1990], [1990], Dybjer Dybjer [1994] [1994].. II then then discuss discuss aa new new type type constructor constructor due due to to Jason Jason Hickey Hickey [1996a] [1996a] - - the the very dependent type. These These recursive and and very very dependent dependent types types can can be be added added to to the the type type theory theory and and hence, to the the recursive hence, to Typed Logic. Logic. So So this this section section too too can can be be viewed viewed as as extending extending the the logic logic of of Section Section 2. 2. Typed This section section provides provides the the theoretical theoretical basis basis for for understanding understanding tactic-oriented tactic-oriented proving, proving, This but there there is is no no space space to to treat treat the the subject subject further. further. but semantics of proofs as impredicative theories objects. very dependent type. 1.3. Highlights. Highlights. Section Section 22 stresses stresses aa typed typed presentation presentation of of the the predicate predicate calculus calculus 1.3. because because we we deal deal with with aa general general mechanism mechanism for for making making the the judgment judgment that that aa formula formula Nuprl, but is is meaningful. This This is is done done first first in in the the style style of of Martin-LOf Martin-Lbf as as expressed expressed in in Nuprl, but 's use II also mention the also mention the essential essential novelty novelty in in Nuprl Nuprl's use of of direct computation rules rules which which go go beyond beyond Martin-Lars Martin-Lbf's inference inference rules rules by by allowing allowing reasoning reasoning about about general general recursive recursive functions the Y functions (say (say defined defined by by the Y combinator). combinator). This This is is aa powerful powerful mechanism mechanism that that is is not widely not widely known. known. are sensible sensible for classical Section 22 also also stresses stresses the notion that that proof expressions are Section the notion for classical logic (see Constable [1989]). This separates an important origins logic (see Constable [1989]). This separates an important technique technique from from its its origins in constructive constructive logic. logic. So So the both constructive constructive as well in the account account of of Typed Typed Logic Logic covers covers both as well as classical classical logic. logic. It It also tactics. as also lays lays the the ground ground work work for for tactics. Section 33 features features aa very simple fragment fragment of of type type theory which illustrates illustrates the the Section very simple theory which major design design issues. issues. The The fragment fragment has has aa simple simple (predicative) (predicative) inductive inductive semantics. semantics. major We keep extending extending itit until until the the semantics semantics requires requires the the insights insights from from Allen Allen [1987a]. [1987a]. We keep The treatment treatment of of recursive recursive types types in in Section Section 44 is is quite quite simple simple because because itit deals The deals only only partial types. types. It It suggests suggests that just as as the the notion of program program "correctness" "correctness" can can with with partial that just notion of be nicely nicely factored factored into into partial partial and and total total correctness, correctness, the the rules rules for for data data types types can can also also be be be factored factored this this way. way. Table 11 shows shows how how similar similar concepts concepts change change their their appearance appearance as as they they are are situated situated Table in in the the three three different different contexts. contexts. meaningful. direct computation proof expressions 2. y p e d llogic ogic 2. T Typed typed language. Ordinary mathematical mathematical statements statements are are usually usually expressed expressed in in aa typed language. Ordinary Consider the the trivially trivially true true proposition: proposition: "if "if there there is is aa rational rational number number q whose whose Consider 693 693 Types Typed Typed Logic Logic terms with with binding binding terms definitional definitional equality equality Type Type Theory Theory terms with with binding binding terms computation rules rules computation Pr°Pi Typei Vi Typed Typed Programming Programming terms with with binding binding terms observational equivalence equivalence & & observational bisimulation bisimulation content of proofs proofs proofs as as objects objects programs proofs content of programs (partial types) types) Propi Typei or Type (partial or Ui judgments equality equality functions functions equality equality judgments equality propositions propositions equality nontriviality representation integrity integrity consistency nontriviality representation consistency (internal & external) external) (internal & inductive definitions definitions termination standard inductive standard models models termination conditions conditions element synthesis synthesis proof element program proof synthesis synthesis program synthesis synthesis proof data tactic-tree proof proof derivation tactic-tree derivation proof data type type & & tactics tactics Type Table 1" 1: Type Type concepts concepts Table n, absolute value is less than the the reciprocal natural number number n, then then for for every absolute value is less than reciprocal of of any any natural every natural number number n, there there is real number number r whose absolute value value is also less less than than natural is aa real whose absolute is also 1/n." Symbolically Symbolically we we express express this this as as follows for N the the natural natural numbers, numbers, Q follows for IQl the the lR the reals: and ]R rationals, and rationals, the reals: n, lin." r N 3q:lQl. 'v'n:N. (Iql (Iql << lin) 3r:lR. (Irl (Irl << lin) :::lq.Q. Vn'N. l/n) => 'v'n:N. vn-N. 3r-R. l/n). . We can abstract relation Ir < l/n, lin, and speak of relation LL on N x It(lR We can abstract on on the the relation Irll < and speak of any any relation on 1N and recognize that IQl Q is is a a subtype subtype of of lR ]R to obtain and recognize that to obtain 3q:lQl. L (n, q) =>=~ 'v'Vn'N. n :N. 3r:lR. (n, r) . 3q'Q. 'v'n:N. Vn'SI. L(n,q) 3r'R. L L(n,r). Abstracting further, A, A' and Abstracting further, we we know know that that for for any any types types A,W and B B where where A' A ~ is is aa subtype true subtype of of A, A, say say A' A~� _ A A the the following following is is true 3a:A'. x :B. x :B. 3a" A'. 'v' Vx" B. L(x, L(x, a) a) =~ 'v' Vx" B. 3a:A. 3a" A. L L (a, (a, x) x) .. This relation This last last statement statement is is an an abstraction abstraction of of *9 with with respect respect to to 1Ql, Q, N, 51, lR ]R and and the the relation 1In. It It is is these these purely purely abstract abstract typed typed propositions propositions that that we we want want to to study study in in the the IIrlrl < lin. * => beginning. beginning. We We want want to to know know what what statements statements are are true true regardless regardless of of the the types types and and the the exact exact propositions. propositions. We We are are looking looking for for those those properties properties of of mathematical mathematical propositions propositions that that are are invariant invariant under under arbitrary arbitrary replacement replacement of of types types and and propositions. propositions. Here Here logic logic is is presented presented in in aa way way that that relates relates it it closely closely to to the the type type theory theory of of section section 33 and and the the programming programming language language of of section section 4. 4. Essentially, Essentially, we we will will be be able able to to lay lay one one presentation presentation on on top top of of the the others others and and see see aa striking striking correspondence correspondence as as Table Table 11 suggests. suggests. This This goal goal leads leads to to aa novel novel presentation presentation of of logic logic because because of of the the role role of of explicit explicit typing typing judgments. judgments. We We begin begin now now to to gradually gradually make make these these ideas ideas more more precise. precise. 694 694 R. Constable Constable R. 2.1. Propositions Propositions 2.1. proposition declarative sentences Relation Use of of the the word word proposition in in logic logic refers refers to to an an idea idea that that isis R e l a t i o n tto o ssentences. e n t e n c e s . Use new in in this this century. century. The The English English usage usage isis from from Russell's Russell's reading reading of of Frege. Frege. To To explain explain new in some some natural natural propositions itit isis customary customary to to talk talk first first about about declarative sentences in propositions language such such as as English. English. AA sentence sentence isis an an aggregation aggregation of of words words which which expresses expresses language 1 , then then the the complete thought, thought, and and when when that that thought thought isis an an assertion, assertion, e.e. g. g. 00 << 1, aa complete of the the sentence. sentence. We are sentence isis declarative. declarative. The The thought thought expressed expressed isis the the sense of sentence We are interested in in the the conditions conditions under under which which we we can can assert a sentence or or judge it to be interested true. Logic isis not not concerned concerned directly directly with with the the nature nature of of natural natural language language and and sentences. sentences. Logic It isis aa more more abstract abstract subject. subject. The The abstract abstract object object corresponding corresponding to to aa sentence sentence isis It As Church Church [1960] [1960] says says "... " . . . aa proposition proposition as as we we use use the the term, term, is is an an aa proposition. As abstract object object of of the the same same general general category category as as class, class, number number or or function." function." He He says says abstract proposition whether not itit is is expressed expressed in in aa that any of truth-value is that any concept concept of is aa proposition whether or or not natural natural language.9 language. 9 This definition from from Frege [1903] as explained by will suffice even This definition Frege [1903] as explained by Church Church [1956] [1956] will suffice even for the varieties varieties of of constructive logic we we will will consider. for the constructive logic consider. We We can can regard regard truth-values truth-values themselves as as abstractions abstractions from more concrete concrete relationship, namely that themselves from a a more relationship, namely that we we know know evidence for for the of aa formula; formula; by by forgetting forgetting the the details details of we come evidence the truth truth of of the the evidence, evidence, we come to to the the notion notion of of a a truth-value. We We say say that that the the asserted asserted sentence sentence is is true. Thus, Thus, when when we we judge judge that that a a sentence sentence or or expression expression is is a a proposition, proposition, we we are are saying saying that that we we know know what truth, that what counts counts as as evidence evidence for for its its truth, that is, is, we we understand understand what what counts counts as as a a proof proof of of it. it. It It is is useful useful to to single single out out two two special special propositions, propositions, say say T 1- for for a a proposition proposition agreed agreed to without further to be be true, true, accepted accepted as as true true without further analysis. analysis. We We can can say say it it is is aa canonical true concrete proposition proposition, aa generalization generalization of of the the concrete proposition 00 = = 00 in in N. N. We We say say that that T T is is atomically atomically true. true. Likewise, Likewise, let let ..1 _l_ be be aa canonically canonically false false proposition, proposition, aa generalization generalization of of the the idea idea 00 = - 11 in in N; N; it it has has no no proof. proof. sense assert a sentence judge it to be true. proposition. truth-value truth-value. true. canonical true proposition, Prop. The T h e category c a t e g o r y of o f propositions, p r o p o s i t i o n s , Prop. In In order order to to relate relate this this account account of of typed typed logic logic to to the the type type theory theory of of section section 33 and and to to programming programming section section 4, 4, II would would like like to to consider consider the the collection collection of of all all propositions propositions and and refer refer to to it it as as Prop. 1~ But But we we know know already already from from Principia that that the the concept concept of of proposition proposition is is indefinitely extensible. Here Here is is how how Whitehead Whitehead and and Russell Russell put put the the matter. matter. ""... . . . vicious . . . [arise] vicious circles circles... [arise] from from supposing supposing that that aa collection collection of of ob objects jects may may contain contain members members which which can can only only be be defined defined by by means means of of the the collection collection as as aa whole. whole. Thus, Thus, for for example, example, the the collection collection of of propositions will will be be supposed supposed to to contain contain aa proposition proposition stating stating that that 'all 'all propositions propositions are are either either true true or or false.' false.' It It would would seem, seem, however, however, that that such such aa statement statement Principia Prop. lO indefinitely extensible. propositions 9There 9There are, are, of of course, course, opposing opposing views; views; Wittgenstein Wittgenstein [1953,1922] [1953,1922]was was concerned concerned with with sentences sentences or or formulas formulas as as isis Quine Quine [1960], [1960],and and we we access access propositions propositions through through specific specific formulas. formulas. lOIn l~ topos topos theory theory Prop Prop and and the the true true propositions propositions form form the the subobject subobject classifier, classifier, n, f~, T T see see Bell Bell [1988], [1988], MacLane MacLane and and Moerdijk Moerdijk [1992], [1992], LaIllbek Lambek and and Scott Scott [1986]. [1986]. Types Types 695 695 could not be be legitimate legitimate unless unless 'all 'all propositions' propositions' referred referred to to some already could not some already definite collection, which definite collection, which it it cannot cannot do do if if new new propositions propositions are are created created by by statements shall, therefore, statements about about 'all 'all propositions.' propositions.' We We shall, therefore, have have to to say say that that statements 'all propositions' statements about about 'all propositions' are are meaningless. meaningless. By By saying saying that that a a set set has 'no total,' mean, primarily, has 'no total,' we we mean, primarily, that that no no significant significant statement statement can can be be made made about about 'all 'all its its members.' members.' In In such such cases, cases, it it is is necessary necessary to to break break up up our smaller sets, total. This our set set into into smaller sets, each each of of which which is is capable capable of of a a total. This is is what what the the theory theory of of types types aims aims at at effecting." effecting." predicative So we we must must be be very very careful careful about about introducing introducing this this notion. notion. There There are are predicative So approaches to to this which lead lead to to level restrictions as as in in Principia and and allow allow writing writing approaches this which Propi as as the the "smaller "smaller collections" collections" into into which which Prop is is broken broken (Martin-Lof (Martin-LSf [1982] [1982],' Constable et al. [1986]). 11 11 There There are are impredicative approaches approaches that that allow allow Prop but but Constable et al. [1986]). Ultimately we restrict restrict the the logic logic in in other other ways ways (Girard, (Girard, Taylor Taylor and and Lafont Lafont [1989]). [1989]). Ultimately we will will impose impose level level restrictions restrictions on on Prop and and relate relate it it to to another another indefinitely indefinitely extensible extensible concept concept which which we we denote denote as as Type Type.. These These restrictions restrictions and and relationships relationships will will occupy occupy us us in in section section 3. 3. From From the the beginning beginning II want want to to recognize recognize that that we we intend intend to to treat treat collections collections of of propositions types, some some will be propositions as as mathematical mathematical objects objects - - some some collections collections will will be be types, will be too "comprehensive" to to be We will collections categories or or too "comprehensive" be types. types. We will call call these these large large collections classes or or kinds or or large types to to use use names names suggestive suggestive of of "large "large collections." collections." The The use use of of Prop as as a a concept concept in in this this account account of of logic, logic, while while not not common common in in logic logic texts, texts, provides provides an an orientation orientation to to the the subject subject which which will will gradually gradually reveal reveal problems problems that that II think think are are both both philosophically philosophically and and practically practically important. important. So So II adopt adopt it it here. here. We We recognize recognize at at the the onset onset that that Prop will will not not denote denote all propositions. propositions. We We will will be be developing developing an an understanding understanding of of how how to to talk talk about about such such open-ended open-ended notions notions in in a a meaningful will confirm meaningful way. way. This This development development will confirm that that at at the the very very least least we we can can name them. The The issue issue is is what what else else can can we we say? say? level restrictions Principia Prop impredicative Prop ProPi Prop categories classes kinds large types Prop Prop all name them. The analysis of T h e ccategory a t e g o r y of o f propositional p r o p o s i t i o n a l functions. f u n c t i o n s . Frege's Frege's analysis of propositions propositions into into functions logic. It functions and and arguments arguments is is central central to to modern modern logic. It requires requires us us to to consider consider the the notion notion of of a a propositional function. Frege Frege starts starts with with concrete concrete propositions propositions such such as as < 11,, then then abstracts abstracts with with respect respect to to a a term term to to obtain obtain a a form form like like 00 < < x x which which o0 < denotes denotes a a function in x. x. In In Principia notation, notation, given given a a proposition proposition ¢Ja Ca,, ¢Jx Cx is is the the ambiguous ambiguous value, value, and and the the propositional propositional function function is is ¢Jx r ; so so 00 < < 11 can can be be factored factored as as ¢J1 Also, in in Principia aa type r and and abstracted abstracted as as 00 < < x 5.. Also, type is is defined defined to to be be the the range range of of significance write the significance of of a a propositional propositional function; function; we we might might write the type type as as type(x.¢Jx) type(x.r . So So the this type might be be abstracted abstracted to to the function function maps maps this type to to Prop. For For example, example, �i1 < < �1 might �!X << �5,1 . ; itit isis not meaningful for x = O . not meaningful for x = 0. In In the the logic logic presented presented here, here, propositional propositional functions functions also also map map types types T T to to Prop. Given of propositional Given a a type type T T we we denote denote the the category category of propositional functions functions over over T T as as T T -+ --+ manner of Prop. Instead Instead of of using using 00 < < x x,, we we denote denote the the abstractions abstractions in in the the manner of Church Church propositional function. function in Principia Principia Prop. Prop. Prop. 11 In topos theory this leads to the Grothendieck tapas 11in topos which is definable in our predicative type theory af 3. of section 3. 696 696 R. Constable ..\(x. x) . with lambda < x). The The details details of of the the function function notation notation will will not not with lambda notation, notation, A(x. 00 < concern concern us us until until section section 2.9. 2.9. It It suffices suffices now now to to say say that that given given a a specific specific proposition proposition such the individuals such as as 00 < < 11,, we we require require that that the individuals such such as as 0, 0,11 be be elements elements of of types, types, here here iN. The function function maps maps N iN to to Prop, thus thus an an element element of of the the category category (N (iN -+ Prop). N. The Given Given a a function function P in in (T -+ --+ Prop) and and t in in the the type type T, then then P(t) denotes denotes the the application application of of P to to t just just as as in in ordinary ordinary mathematics. mathematics. Prop, P (T Prop) P t t T, P( t) Prop) . Types. T y p e s . Types Types are are structured structured collections collections of of objects objects such such as as natural natural numbers, numbers, N, iN, or or pairs pairs of of numbers numbers N N x • N iN or or lists lists of of numbers, numbers, etc. etc. In In the the section section on on type type theory theory we we will will present present specific specific concrete concrete types, types, here here we we treat treat the the notion notion abstractly. abstractly. We We think think of of the the elements elements of of the the types types as as possibly possibly given given first first without without specifying specifying the the type; type; they they might be be built built from from sets sets for for example example or or be be the the raw raw data data in in a a computer computer (the (the bytes) bytes) might or objects or or be be physical physical objects or be be given given by by constructions. constructions. Even Even when when we we are are thinking thinking of of constructing method specified constructing objects objects according according to to a a method specified by by the the type type (as (as for for the the natural natural numbers based based on on zero zero and and successor) successor),, still still we we imagine imagine the the object object as as existing existing without without numbers type information information attached attached to to it, and thus thus objects objects can can be be in in more more than than one one type. type. type it, and The critical point judging that that T is The critical point about about judging is a a type type is is knowing knowing what what it it means means for for an an expression expression to to be be an an element element of of T. This This is is what what we we know know when when we we show show that that T is is a a type. type. Assertions Assertions about about objects objects require require their their classification classification into into types. types. Moreover, Moreover, we we need need occurring in a proposition relative or hypothetical classification because an object relative or hypothetical classification because an object t occurring in a proposition will be may may be be built built contingently contingently from from other other objects objects xi of of type type Ti, and and will be in in some some type type understand these judge T. We We understand these relative relative type type membership membership judgments judgments as as follows. follows. To To judge that that t[x] is is a a member member of of T2 provided provided x is is a a member member of of 7'1 means means that that for for any any object object tl of of type type TI, tits] is is an an object object of of T2. We We write write this this as as x : T1 f~ t[x] E 7'2. We We extend to n assumptions extend the the notation notation to assumptions by by writing writing Xl : T ~ , . . . , xn :Tn fF- t E T for for xi distinct identifiers. identifiers. (We distinct (We write write t[xl,... ,xn] only only when when we we need need to to be be explicit explicit about about the the variables; variables; this this notation notation is is a a second order variable.) We We give give concrete concrete examples examples of of this judgment discuss them them at length in this judgment below below and and discuss at length in section section 3. 3. We We don don't' t treat treat this this judgment judgment as as an an assertion. assertion. It It summarizes summarizes what what must must hold hold for for aa statement statement about about t to to make make sense. sense. It It is is not not the the kind kind of of expression expression that that is is true true or or false. false. For For example, example, when when we we say say 00 E N, iN, we we are are thinking thinking that that 00 is is aa natural natural number. number. This "0 is_a iN," N," and This is is a a fact. We We are are thinking thinking "0 and it it does does not not make make sense sense to to deny deny this, this, thinking thinking that that 00 is is something something else. else. In In order order to to consider consider propositional propositional functions functions of of more more than than one one argument, argument, say say y) and and express y) as P(x, y) y),, we we explicitly explicitly construct construct pairs pairs of of elements, elements, (x, y) express P(x, y) as The pair y) belongs the Cartesian So our P((x, y)) y)).. The pair (x, y) belongs to to the Cartesian product product type. type. So our types types have have at at least least this this structure. structure. T T. T t Ti, T. t[xJ T2 x TI tl TI , t[t l J T2 . x : TI t[xJ E T2 . n Xl : TI, . . . ,xn : Tn t E T Xi t[XI, . . . ,xnJ second order variable.) Xi t E is_a fact. P(x, P( (x, (x, (x, P(x, A, B Cartesian product, A B. a E A bEB ordered pair (a, b) A B. b) E A 10f ((a, b)) a 20f ((a, b)) b. An A A n Definition. D e f i n i t i o n . If If A, B are are types, types, then then so so is is their their Cartesian product, A x x B. If If a E A and belongs to a, b) E A x and b E B then then the the ordered pair (a, b) belongs to A x • B. We We write write ((a, • B B.. We We write write lof ((a, b)) = a and and 2of ((a, b)) = b. Let Let A n denote denote A x . .. .. . x • A taken taken n times. times. 697 697 Types Prop. We use use membermember Propositions are are elements elements of of the the large large type type (or (or category) category) Prop. We Propositions ship judgements judgements ship Xl : TI, ... , Xn : Tn P Prop , FI- PP EE Prop Prop toto define define them. them. T xl :T1,...,xn :Tn FI- P E E Prop Atomic propositions propositions are are or an an abbreviation abbreviation or Atomic constants: constants: TT for for aa canonically canonically true true one one and and 3_ ..L for for aa canonically canonically false false one, one, or or propositional propositional variables variables or or applications applications of of propositional propositional function function variables variables in in the the large large type (T--+ -+ Prop) for for some some type type T. Compound Compound propositions propositions are are built built from from the the type V, =v or the the logical logical operators operators 3:3 and logical connectives connectives &, V, and V. logical => or The membership membership rules rules are are that that if , FI- F E E Prop and and T, ~I- G EE Prop, then then The if T I- (F op G) EE Prop for for op aa connective; connective; and and ifif xl : 7'1, ...,x~ : Tn ~I- F EE Prop, T, ~then TT'~ FI- (Qxi : Ti. F) EE Prop where where Q is is aa quantification quantification operator operator and and where where T"~ then is obtained obtained by by removing removing the the typing typing assumption assumption xi : Ti. is The usual usual names names for the compound compound propositions propositions are: are: The for the (T Prop) T. &, (F op G) Prop op (QXi : Ti . F) Prop proposition proposition G) (F && G) (F vV G) G) G) (F =~ => G) F(t) F(t) Vx:A.F(x) Vx:A.F(x) 3x:A.F(x) :3x:A.F(x) is is is is is is is is is is is is (F (F (F V. F Prop G Prop, Xl : Tl, ... , Xn : Tn F Prop, Q Xi : Ti. English equivalent English equivalent operator operator name name F and G G F and F or G F or G F implies implies G G F F at at t for for all all x of of type type A, F(x) for for some some x of of type type A, F(x) conjunction conjunction disjunction disjunction implication implication instance instance universal universal quantification quantification existential existential quantification quantification F t x x A, F(x) A, F(x) With definitions in With these these definitions in place place we we can can write write examples examples of of general general typed typed proposi propositions. tions. We We use use definiendum definiendum == == definiens definiens to to write write definitions. definitions. F G Definition. Definition. For For any any propositions propositions F and and G define define =>..L) -~ F = - (F (F=~.i.) (F (F v::G) (G=:~ F) => F) G) - = (G (F r G) G) = = ((F ((F r G) G)& (F :=~G)). G)). (F & (F For P P:: ((A ((A x B) B) -+ -~ Prop) Prop) let let P(x, P(x, y) y ) = = P((x, P((x, y)). y)). For Examples. E x a m p l e s . Let Let P P EE (A (A -+ Prop), Prop), Qi Qi EE (B (B -+ -4 Prop), Prop), R R EE ((A ((A xx B) B) -+ Prop) Prop) and and C C EE Prop. Prop. B. Q(y) Vx:A. Vy:B. Yy:B. (P(x) (P(x) & & Q(y)) Q(y)) r Vx:A. Yx:A. P(x) P(x) & & Vy: Vy:B. Q(y) 11.. Vx:A. 2. 2. :3x:A.P(x) 3x :A.P(x) & & :3y: 3y :B. B. Q(y) Q(y) => =~ :3z:A 3z :A xx B. S. R(z) R(z) 3. 3. -Nx:A. ~Vx: A. P(x) P(x) r :3x:A. Bx :A. -,P(x) -~P(x) 4. 4. Vx:A. Vx: A. (C (C =v P(x)) P(x)) r (C (C => =~Vx:A. Vx: A. P(x)) P(x)) y) 3y:S. Vx:A. R(x, R(x,y) Vx:A. :3y:B. 3y:S. R(x, R(x,y) B. Vx:A. y) =>=~Vx:A. 5.5. :3y: ¢:: ¢:} ¢:: => == -+ -+ ¢:} => ¢:} ¢:} 698 698 R. Constable Constable Judgments and proofs 2.2. Judgments P Knowledge arises arises when when we we judge judge aa proposition proposition to to be be true. true. A A proposition proposition P Knowledge becomes an an assertion when when we we judge judge it it to to be be true true or or assert assert it. it. In In Principia this this becomes judgment is is called called an an assertion and and is is written written IF- P . judgment Normally we we assert assert a a proposition proposition when when we we know it to be true, but but people people also also make make Normally so-called "blind "blind assertions" assertions" which which are are made made without without this this knowledge knowledge but but happen happen to to so-called be true true because because someone someone else else knows knows this this or or the the person person "speaking "speaking blindly" blindly" discovers discovers be it later. later. (These (These "blind "blind assertions" assertions" normally make aa person, person, especially especially students students in in it normally make exams, anxious.) anxious.) exams, Assertions are are not not the the only only form form of of knowledge. knowledge. We We follow follow Per Per Martin-Lof Martin-LSf [1982] [1982] Assertions and speak speak also also of of judgments judgments of of the the form form P E Prop that that we we discussed discussed above. above. These These and are typing judgments. They They also also convey convey knowledge knowledge and and need need to to be be made made evident, evident, are but we we consider consider them them as as a a different different category category of of knowledge knowledge from from assertions. assertions. Indeed, Indeed, but we see see that that knowing knowing these these judgments judgments is is part part of of knowing knowing truth truth judgments judgments because because we we must must know know that that the the object we are are asserting is a a proposition proposition before before (or as) we we asserting is (or as) we object P we judge it to to be be true. true. 12 12 judge it In 2.1 we In section section 2.1 we treated treated this this notion notion in in the the background background with with the the notation notation P E Prop, P E A -+ ~ Prop, without without explaining judgments in in general. In most most logic logic explaining judgments general. In textbooks this judgment judgment is is reduced reduced to to a a question question of of correct correct parsing parsing of of formulas, formulas, i.e., i.e., textbooks this is made made syntactic and and is is thus thus prior prior to to truth. truth. But But we we follow follow Russell Russell and and Martin-Lof Martin-LSf is in believing it to to be be a a semantic semantic notion notion that that can can be be treated treated in in other other ways ways when when we we in believing it formalize a a theory. theory. formalize For predicate calculus, For the the predicate calculus, we we leave leave most most of of these these typing typing judgments judgments implicit implicit and and adopt the the usual usual convention convention that that all all the the formulas formulas we we examine examine represent represent well-formed well-formed adopt propositions. In the the full be checked checked syntactically, propositions. In full typed typed logic logic this this condition condition cannot cannot be syntactically, and judgments must and explicit explicit typing typing judgments must be be made. made. In general, to to make judgment is is to to know know it for it to be In general, make aa judgment it or or for it to be evident. It It does does not not make sense for a a judgment to be evident without without aa person However, make sense for judgment to be evident person knowing knowing it. it. However, we recognize that there are propositions which are are true true but but were were not not known known at at a a we recognize that there are propositions which previous time. time. So So at at any any time time there there are are propositions propositions which are not not known, known, but will previous which are but will be asserted in the future. future. be asserted in the assertion assertion P. know it to be true, Principia P E Prop typing judgments. P PE Prop, P E A Prop, syntactic evident. T r u e propositions p r o o f s . One most interesting interesting properties of a a propo propoTrue propositions and and proofs. One of of the the most properties of Here is concrete way way to to say say sition is true. Here sition is is whether whether the the thought thought it it expresses expresses is is aa more more concrete this. To grasp the the sense sense of of aa proposition proposition is is to to understand understand what what counts counts as as evidence evidence this. To grasp for its truth. truth. To know whether whether aa proposition proposition is is true, true, we we must must find find evidence evidence for for it. it. for its To know 13 Trying activity. 13 Trying to to find find evidence evidence is is aa mathematical task, and and itit is is an an abstract abstract activity. The for an The evidence evidence for an assertion assertion is is called called aa proof. Proving Proving aa proposition proposition is is the the way way true. mathematical task, proof 12In can be Martin-Lors theorem these judgments are prior prior to assertion; in Nuprl they can 12In Martin-LSf's simultaneous simultaneous with assertion. 13But the consequences of finding aa proof or aa disproof can be very concrete and very significant. someone to test aa device in the belief that that it is safe For instance, instance, aa purported purported proof proof might cause someone to do so. If IT the proof is flawed, flawed, the device device might destroy something of great value. 699 699 Types 14 When of of knowing knowing it. it.14 When we we judge judge that that an an expression expression isis aa proposition, proposition, we we specify specify what what counts as as aa proof proof of of it. it. For For the the propositions propositions T, T, _1_ --L (P&Q), (P vV Q), 3x: A. P(x) itit counts we could could cite cite an an axiom, axiom, that that is, is, we we say say itit isis is easy easy to to specify specify this. this. To To prove prove TT we is As we we said said before, before, _1_ --L has has no no proofs. proofs. To To prove prove (P&Q) we we prove prove P and and self-evident. As To prove prove (P (P vV Q) we we either either prove prove PP or or prove prove Q. To To prove prove 3x: A. P(x) we we prove Q. To prove exhibit an an element element a of of A and and prove prove P(a). exhibit self-evident. Q. (P&Q), (P Q), 3x : A. P(x) (P&Q) P Q. 3x: A. P(x) Q) a A P(a) . To To say say what what itit means means to to prove prove (P (P =~� Q)Q) we we need need to to understand understand logical logical concon hypothetical proof. proof. We write this this as as sequence or or what what isis the the same same in in this this article, article, hypothetical sequence We write hypothetical hypothetical judgment judgment PP b-f- QQ which which reads, reads, assuming assuming PP isis true, true, we we can can prove prove that that Q Q is. W What this means means is is that that ifif we we have proof pp of of PP, , then then we we can can build build aa proof proof q of is. h a t this have aa proof of Q using using p. Q To discuss discuss nested nested implications implications such such as as PP =~ � (Q � P) we need need to understand To (Q =~ P) we to understand Pn ~-f- QQ which which means means that that assuming assuming we we hypothetical judgments judgments of of the the form form PPI, hypothetical 1 , .. .. .., , Pn have proofs of Pi, P; , we we can can find find aa proof proof q of of Q. Q. proofs Pi of have To prove prove \Ix Vx:A.P(x) which says To :A.P(x) we we need need the the hypothetical hypothetical judgment judgment x: x: AA f-~- P(x) P(x) which says type A, can find find aa proof proof of P(a) . Combining forms that given given any any aa of that of type A, we we can of P(a). Combining these these two two forms Hn k-f- PP where of hypothetical judgments judgments we will need can be be of hypothetical we will need to to consider consider HI, H 1 , .. .. .., , Hn where Hi Hi can A. either proposition or as x: either aa proposition or a a type type declaration declaration such such as x:A. This This account account of of provability provability gives gives what what we we call call a a semantics semantics of of evidence. evidence. Depending Depending on P(x) we on the the interpretation interpretation of of the the functionality, functionality, judgments judgments like like P P fb- Q Q or or x x :: AA f~- P(x) we q p. q can and constructive in this can explain explain both both classical classical and constructive logic logic in this way. way. These These ideas ideas will will become become clearer proceed. clearer as as we we proceed. In In general, general, there there is is no no systematic systematic way way to to search search for for a a proof. proof. Indeed, Indeed, the the notion notion of of proof proof we we have have in in mind mind is is not not for for any any fixed formal formal system system of of mathematics. mathematics. We We are are interested interested mainly mainly in in open-ended open-ended notions. notions. Like Like the the concept concept of of aa proposition, proposition, the the concept concept of of aa proof proof is is inexhaustible inexhaustible or or open or or creative. By By G6del Ghdel [1933] [1933] and and Tarski Tarski [1956] [1956] we we know know that that for for any any consistent consistent closed non-trivial formal system, we we can systematically enlarge enlarge it, it, namely namely by by adding adding aa rule rule asserting asserting its its consistency consistency or or can systematically giving giving aa definition definition of of truth truth for for that that system. system. For For the the collection collection of of pure abstract abstract propositions, propositions, there there is is aa systematic systematic way way to to search search for for aa proof. proof. If If we we want want to to describe describe this this procedure, procedure, then then it it is is necessary necessary to to have have aa representation of of propositions propositions as as data data that that lets lets us us access access their their structure. structure. We We can can do do this this with with the the inductive inductive definition definition of of propositions propositions given given next, next, but but it it is is more more natural natural to to build build aa representation representation that that directly directly expresses expresses the the linguistic structures that that we we use use when when writing writing and and manipulating manipulating formulas. formulas. This This is is the the traditional traditional approach approach to to the the study study of of logic logic and and provability. provability. We We take take it it up up in in the the subsections subsections on on Formulas Formulas and and Formal Formal Proofs. Proofs. fixed open creative. closed non-trivial formal system, pure representation linguistic structures 14When we we need need to to present present evidence evidence for for aa typing typing judgment, judgment, we we will will incorporate incorporate that that into into our our 14When proofs proofs as as well welland and speak speak of of proving proving aa typing. typing. One One might might want want to to give givethis this aa special special name, name, such such as as derivation,but but in in Nuprl Nuprl we we use use the the term term proof. proof. These These typing typing "proofs" "proofs" never never have have interesting interesting aa derivation, computational computational content. content. R. R. Constable Constable 700 700 2.3. 2.3. Pure P u r e propositions propositions The propositional The traditional traditional approach approach to to studying studying logic logic is is to to first first isolate isolate the the propositional calculus then the rst-order predicate calculus and and then the fi first-order predicate calculus, calculus, and and to to study study completeness completeness results these calculi, calculi, since results for for these since completeness completeness results results tie tie together together the the semantic semantic and and are covered proof-theoretic concepts. proof-theoretic concepts. These These topics topics are covered by by defining defining restricted restricted formal formal languages. languages. We We will will take take this this approach approach to to typed typed logic logic starting starting in in section section 2.4 on on formulas. formulas. But But many many of of the the concepts concepts can can also also be be presented presented by by aa mathematical mathematical analysis analysis of propositional functions of propositional functions without without resort resort to to linguistic linguistic mechanisms mechanisms and and formulas. formulas. There There is is one one outstanding outstanding technical technical problem problem about about the the propositional propositional calculus calculus which which benefits benefits from from this this direct direct analysis, analysis, namely namely aa functional functional approach approach to to complete completeness ness of of the the intuitionistic intuitionistic propositional propositional calculus. calculus. The The most most widely widely known known completeness completeness theorem theorem for for this this calculus calculus is is based based on on Kripke Kripke semantics semantics (Kripke (Kripke [1965]); [1965]); this this account account of of propositional propositional calculus calculus semantics, semantics, while while illuminating illuminating and and technically technically elegant, elegant, and and even even constructively constructively meaningful, meaningful, is is not not faithful to to constructive constructive semantics. semantics. In In particular, particular, it it is is not not based based on on the the type-theoretic type-theoretic semantics semantics we we offer offer in in part part 3. 3. Providing Providing a a completeness result completeness result for for a a constructively faithful semantics semantics is is an an open open area, area, and and it it seems seems to to me me that that Martin-Lors Martin-Lhf's inductive inductive semantics semantics has has created created new new opportunities opportunities here here to to produce produce definitive definitive results. results. The The key key to to a a constructive constructive semantics semantics might might be be a a careful careful study study of of a a functional functional approach approach to to propositions propositions that that allows allows us us to to express express the the functional functional uniformity of proofs that is is central central to to completeness. completeness. I15 As a a start, start, we can try try to understand the the basic basic 5 As we can to understand that concept concept of of a a pure pure propositional propositional function function without without resort resort to to formal formal languages. languages. Consider Consider a a propositional propositional function function of of one one argument, argument, P => =v P. This This can can be be understood understood as as a a function function from from Prop to to Prop. is The The function function P => =~ (Q (Q => =v P) in in variables variables P, Q Q is is a a two-argument two-argument function, function, most most naturally naturally from from Prop to to (Prop --+ --+ writing the Prop) as as would would be be clear clear from from writing the function function as as 2.4 faithful constructively faithful uniformity ofproofs Prop) Prop Prop. 16 P, P P. P P) Prop (Prop )"(P. ( Q. P A(P. ).. A(Q. P =v (Q (Q =v P))). P))). We We could could also also think think of of this this as as a a mapping mapping from from Prop Prop x • Prop Prop to to Prop Prop if if we we took took pairs pairs as (writing )..A(z. (z. lof(z) => lof(z))) as arguments arguments (writing lof(z)=~ (2of(z) (2of(z)=v lof(z))) in in Prop Prop 22 --+ Prop). Prop). For For ease will consider propositional functions ease of of analysis, analysis, we we will consider propositional functions from from the the Cartesian Cartesian power, power, Prop Prop n~,, into into Prop. Prop. The The constants constants T T and and ..1 I are are regarded regarded as as zero-ary zero-ary functions, functions, and and for for convenience convenience define define Propo Prop ~ = 11 for for 11 the the unit unit type. type. Then Then f(x) f (x) = T T and and f(x) f (x) = _l_ are are in in Propo Prop ~ --+ Prop. Prop. The propositional functions The idea idea is is to to define define the the pure pure propositional functions inductively inductively as as a a subtype subtype of of Prop Prop nn --+ --+ Prop Prop constructed constructed using using only only constant constant functions, functions, simple simple projections projections like like projI' (PI , . . . , PPn)n ) = PiPi and &, V, proj~(P1,..., and the the operations operations &, =~ lifted lifted up up to to the the level level of of functions. functions. V, => Each &, VV,, =~ can Each connective connective &, can be be lifted lifted to to the the functions functions Prop Prop nn --+ Prop, Prop, namely namely given given f f and and g, g, define define (J ( f oop p g)(P) g)(P) = f(P) f ( P ) oop p g(P) g(P) where where P 15 E E Prop Prop nn.. For For example, example, => => => = --+ = => --+ = = = ..1 --+ 15Lauchli 15L~iuchli [1970] [1970] tries to express express this uniformity uniformity using using permutations. 16We 16We will will deal deal later with with the issue issue of equality equality on Prop, Prop, which which seems seems necessary necessary to talk about functions. 701 701 Types f(P, P h(P, (P Pn g(P, P)). P) f h Q) = if if f ( P , QQ) ) = = P and and g(P,Q) = (Q (Q � =v P) then then f � =~ 9g is is aa function function h such such that that h(P, Q) Q) = = (P � =v (Q (Q � ::~ P)). We We can can now now define define the the general general abstract abstract propositional propositional functions functions of of n variables variables call call the class class P . as as the the inductive inductive subset subset of of Prop" -+ -+ Prop whose whose base base elements elements are are the the the constant and and projection projection functions, functions, constant n Propn Prop (P) = T C.l CT(P) C• (P) CT p r o j ~ ( P )= - Pi Pi where where/5 ( P l1,,.. .. ..,, Pn) P , ) and and 11 :::; _ ii :::; < n. n. P = (P proji(P) Then Then given given f, f, g E E Pn P , and and given given any any lifted lifted connective connective op, op, we we have have (J ( f oop p g) g) EE Pn P,. · Nothing else else belongs belongs to to Pn. P , . When When we we want want to to mention mention the the underlying underlying type, type, we we write write Nothing n Prop).. Let 7), as as P(Prop P(Prop" ~ Prop) Let P 7~ - U U PPn; , ; these these are are the the pure pure propositions. propositions. Note Note Pn n=O n Prop) isis inductively that that P P = U U PP(Prop ( P r o p " --+ Prop) inductively defined. defined. The The valid valid elements elements of of P P are are n=O n , f(P) those functions functions f f E E P 7~ such such that that for for f f E Pn 7~, and and P 75 any any element element of of Prop Prop", f(7 )) is is those true. Call Call these these True(P) True(7)).. true. = T = = 1-J- 9 = oo 00 -+ 117 7 = = oo 00 .--0 -+ n--0 E Using these these concepts concepts we we can can express express the the idea idea of of a a uniform uniform functional functional proof. proof. The The Using simplest approach approach is is probably probably to to use use a a Hilbert style axiomatic axiomatic base. base. If If we we take take simplest Hilbert style Heyting's or or Kleene's Kleene's axioms axioms for for the the intuitionistic intuitionistic propositional propositional calculus, calculus, then then we we Heyting's can define define ProvableH(P) inductively. inductively. The The completeness completeness theorem theorem we we want want is is then then can ProvableH(P) True(P) - ProvableH(P) Provableg(7)). . True(P) We We can can use use the the same same technique technique to to define define the the pure pure typed typed propositional propositional functions. functions. n First we we need need to to define define pure pure type type functions functions T T as as a a subset subset of of Type Type" --+ Type Type for for First We take take n n � _> 11 since since there are as as yet yet no no constant constant types types to to include. include. nn = 1,1, 2,2, .. ...... We there are An example example is is t(A, t(A, B) B) - A A x x B B.. Next Next we we define define the the typed typed propositional propositional functions functions An pp": t(T) t(T) --+ -+ Prop. Prop. = = -+ = In general general we whose inputs inputs are are n-tuples In we need need to to consider consider functions functions whose n-tuples of of the the type type (tl (T) Prop) (tn(T) Prop) and whose We do topic further further here, but when we and whose output output is is aa Prop. Prop. We do not not pursue pursue this this topic here, but when we (tl(T)--~ -+ Prop) •x . . . •x (t,(T) ~ -+ Prop) examine the the proof for typed we will will see see that it offers offers aa simple examine proof system system for typed propositions propositions we that it simple way to provide abstract for pure pure typed typed propositions that use rules for for the way to provide abstract proofs proofs for propositions that use only only rules the connectives and quantifiers - say say a a pure proof. There There are are various various results results suggesting suggesting connectives and quantifiers that if if there any proof These that there is is any proof of of these these pure pure propositions, propositions, then then there there is is aa pure proof. These for this typed version version of of the the predicate predicate calculus. calculus. We not are completeness results for are this typed We will will not prove them prove them here. here. pure proof completeness results pure proof 2.4. ormulas 2.4. FFormulas P r o p o s i t i o n a l ccalculus. a l c u l u s . Consider Propositional Consider first first the the case case of of formulas formulas to to represent represent pure propositions. propositions. The The standard standard way to do do this this is is to to inductively inductively dede the way to the pure fine The base base case case includes includes fine aa class class of of propositional formulas, PropFormula. The propositional formulas, PropFormula. 17Since we do not not study study any mapping of formulas to pure propositions, not worried about propositions, I have not relating elements of Pn Pn and Pro, Pm , n < m, m , by coherence conditions. R. Constable Constable R. 702 702 Constants {T, ..l}, Variables {P, Q, R, PI , QI,R1, } . F, G PropFormulas, (F G) , (F G) , (F G) . PropFormula. F, PI , . . . , Pn n P PropFormulasn , (Propn Prop) IPi] = proj~ projj [Pi] [F] && i[G] G)] = [F] [(F && G)] I(F al G)] = [FI [(F vV G)] [F] vV [G] [G] [(F [G].] . [[(F ( F ==> ~ GG)] ) ] = I[FJ F I ==> ~iG For variable P Pi,i , corresponds the projection projection function proj~(P) = Pi. For each each variable corresponds to to the function projj(P) Pi. Say Say valid iff [F] isis aa valid valid pure pure proposition. proposition. that FF is that is valid iff IF] the propositional propositional constants, constants, Constants --= {T, _L}, and and propositional propositional variables, variables, the These are are propositional propositional formulas. formulas. The The Variables = These = {P, Q, R, P1, Q1, R1, . . .}. inductive case case is is inductive If F, G are are PropFormulas, then then so so are are (F & G), (FVV G), and Nothing If and (F =~ => G). Nothing else is is aa PropFormula. else We can assign assign to to every every formula formula aa mathematical mathematical meaning meaning as as aa pure pure proposition. proposition. We can Given aa formula be the the propositional propositional variables variables occurring occurring in in itit (say (say let P 1 , . . . , Pn be formula F , let Given be the the vector vector of of them. them. Define Define aa map map from from n variable variable ordered from from left left to to right); right); let let t5 be ordered into (Prop ~ -+ Prop) inductively inductively by by propositional formulas, formulas, PropFormulasn, into propositional . • • = Boolean formulas. If consider aa single-valued single-valued relation relation from propo B o o l e a n vvalued a l u e d formulas. If we we consider from propositions to their truth values, taken as Booleans, Booleans, then then we an especially simple sitions to their truth values, taken as we get get an especially simple B (P, tt) . la == {tt, ff} and and let let BB :Prop xx ]B la -+ that P {:} semantics. Let Let ]B semantics. -~ Prop such such that r B In classical classical mathematics mathematics one one usually usually assumes of aa function like like b, In assumes the the existence existence of say b : P r o p --+ ]B where in lB. But since since b is not a a computable say -+ la where P r{:} b(P) -= tt in la. But is not computable function, of describing the situation be used used in in constructive function, this this way way of describing the situation would would not not be constructive mathematics. Instead Instead we we could talk about about "decidable "decidable propositions" propositions" or mathematics. could talk or "boolean "boolean propositions. propositions."" b : Prop {tt, ff} : Prop P b(P) tt Prop b P (P, tt). function b , BoolProp v) :" Prop l P r (v in Jan BoolProp = = {(P, { (P, v) Prop x x la IBIP (v = tt tt in ]B)} Then there there is Then is a a function function b b E E BoolProp BoolProp -+ la ]~ such such that that P P r (b(P) ( b(P) = tt tt in in la) ]IS).. If If we we interpret interpret formulas formulas as as representing representing elements elements of of pure pure boolean boolean propositions, propositions, then la. An then each each variable variable P; Pi denotes denotes an an element element of of B. An assignment assignment a a is is aa mapping mapping of of variables into into la, is, an la. Given variables ]B, that that is, an element element of of Variables Variables -+ ]~. Given an an assignment assignment a a we we can can compute compute aa boolean boolean value value for for any any formula formula F. F . Namely Namely Value(F, Value(F, a) a) = if if F F is is aa variable, variable, then then a(F) a(F) if if F F is is (F (F1l Oop then Value(F Value(F1,1 , a) a) bop bop Value(F Value(F2, 2 ' aa)) P FF2)2 ) then where corresponding to where bop bop is is the the boolean boolean operation operation corresponding to the the propositional propositional operator operator op op == {:} = {:} -+ = in in the the usual usual way, way, e. e. g. g. P P tt tt ff ff tt tt ff ff Q tt tt tt ff ff ff ff Q PP ~=>b bQQ Vb Q PVb P P &b &b Q P tt tt ff ff ff ff ff ff tt tt tt tt tt tt ff ff tt tt tt tt ff ff tt tt = 703 703 Types Typed formulas. To To define define typed typed propositional propositional formulas, formulas, we we need need T y p e d ppropositional r o p o s i t i o n a l formulas. the notion notion of of aa type type expression, expression, aa term, term, and and aa type context because because formulas formulas are are built built the in in aa type type context. context. Then Then we we define define propositional propositional variables variables and and propositional-function propositional-function variables which which are are used used along along with with terms terms to to make make atomic atomic propositions propositions in in aa context. context. variables From these From these we we build build compound compound formulas formulas using using the the binary binary connectives connectives &, &, V, V, =~, => , and and We let let op denote denote any any binary binary connective connective and and the typed quantifiers Vx:A, 3 x ::AA. . We the denote either either of of the the quantifiers. quantifiers. Qx:A denote type context typed quantifiers \Ix : A, ::Ix op Qx: A type variables, variables, then then Ai Ai are are type type expressions. expressions. Type expressions. Let Let A1, AI, A2,... A2 , bebe type T y p e expressions. TI, T2 T2 are are type type expressions, expressions, then then so so is is (TI T2 ). If T1, If (T1 x T2). • . • Nothing else else is is aa type type expression expression for for now. now. Nothing Xl, X2 , terms. Terms. Let Xl, x2,.., . . . be be individual individual variables variables (or (or element element variables); variables); they they are are terms. T e r m s . Let If If s, t are are terms, terms, then then so so is is the the ordered ordered pair pair (s, t). Nothing Nothing else else is is aa term term for for now. now. s, t (s, t). Xi, i type assumption Tl, . . . , Tn Xi typing context. . ,n are type type expressions and xi, i = are Typing If T1,...,T~ are T y p i n g ccontexts. o n t e x t s . If expressions and = 11,, .. . .. , n are individual variables, variables, then then xi :: Ti 1'; is and the the list list distinct individual is aa type assumption and is aa typing context. We let T, T', Tjj 7j denote denote typing typing contexts. contexts. xl :T1,... ,x~ :T~ is We let T, T', distinct Xl : Tl, . . . , Xn : Tn T y p i n g judgments. j u d g m e n t s . Given can assign assign types types to to terms terms built Typing Given aa typing typing context, context, T, T, we we can built the variables the context. context. The The judgment judgment that that term type T in in context from the from variables in in the term t has has type context T writing T is is expressed expressed by by writing T T ~f-t E T . t T t E T. t, second-order we need need to to be explicit about about the the variables variables of of TT and use aa second-order If If we be explicit and t, we we use variable t[xl,..., xn] and and write write variable t[XI, . . . , xnJ Zl : T ~ , . . . , z , :T, ~- t[z~,... ,z,] ~ T t When When using using aa second-order second-order variable variable we we know know that that the the only only variables variables occurring occurring in in t are variables of are xi. We We call call these these variables of t free variables. Later, we Later, we give give rules rules for for knowing knowing these these judgments. judgments. Just Just as as we we said said in in section section 2.2, 2.2, it be noted that t E T is it should should be noted that is not a proposition; it it is is not an an expression expression that that has has truth value. value. We ordered pair We are are saying saying what what an an ordered pair is rather rather than than giving giving a a property property aa truth of it. So is giving telling us of it. So the the judgment judgment t E T is giving the the meaning meaning of of t and and telling us that that the the expression expression t is is well-formed or or meaningful. meaningful. In In other other presentations presentations of of predicate predicate logic logic these these judgments judgments are are incorporated incorporated into into the the syntax syntax of of terms, terms, and and there there is is an an algorithm algorithm to to check check that that terms terms are are meaningful meaningful before before one one considers considers their their truth. truth. We We want want aa more more flexible flexible approach approach so so that that typing typing judgments judgments need need not not be be decidable. decidable. We . . . denote We let let P1, P2,... denote propositional variables, writing writing Pi E Prop, for for proposi propositional tional function function variables, variables, writing writing Pi E (T -+ -4 Prop) for for T aa type type expression. expression. If If T T ~f- t e T and and P e (T -+ --+ Prop), then then P(t) is is an an atomic formula in the Note, we context T T with with the the variables variables occurring occurring in in t free; it it is is an an instance of of P. Note, we abbreviate abbreviate P ( ( t l , . . . , tn)) by by P(tl,... ,tn). If If t is is aa variable, variable, say say x, then then P ( x ) i is s Xi. t free variables. t E T not a proposition; not is tET t t well-formed PI, P2 , propositional variables, Pi E Prop, Pi E (T Prop) T t E T P E (T Prop) , P(t) atomic formula in the context t free; instance P. P ((tl , . . . , tn )) P(tl, . . . , tn ) . t x, P(x) 704 704 R. Constable Constable arbitrary instance arbitrary value P Pi E Prop, G op Qx: A (F G) (F op opG) x. an arbitrary instance or or arbitrary value of of P with with free free variable variable x. A A propositional propositional an variable, variable, Pi E Prop, is is also also an an atomic atomic formula. formula. If T,, and If F F and and G are are formulas formulas with with free free variables variables X, x, fj y respectively respectively in in contexts contexts T and if er, then if op is is a a connective connective and and Qx:A a a quantifi quantifier, then immediate subfor is {x} U {fj} in is a a formula formula with with free free variables variables {~} U (~} in context context T T and and with with immediate subforand G; mulas F F and mulas G; Qv :T.F Qv:T.F is : A removed; is a a formula formula in in context context T' 7" where where T' 7" is is T T with with vv:A removed; this this formula formula has has leading leading binding operator operator Qv Q v :: AA with with binding binding occurrence occurrence of of vv whose whose scope scope is is F F ,, and and its its free free binding variables and all all free in F variables are are {x} {~} with with vv removed, removed, and free occurrences occurrences of of vv in F become become bound bound by is F. by Qv Qv :: A; A; its its immediate immediate subformula subformula is F. A A formula formula is is closed closed iff iff it it has has no no free free variables; variables; such such a a formula formula is is well-formed well-formed in in an an empty empty context, context, but but its its subformulas subformulas might might only only be be well-formed well-formed in in a a context. context. A A subformula subformula G G of of a a formula formula F F is is either either an an immediate immediate subformula subformula or or aa subformula subformula of of aa subformula. subformula. PI : A Prop, P2 : B Prop, P3 : A B 3y:B. P3(x, P3(x,y) ~ (�x:A. (3x:A. PI(x) P~(x) & & �y 3y :: B. B. P P2(x))) x:A. �y:B. y) ::::} 2 (x))) (\l(Vx:A. is A. �y: B. P3(x, is a a closed closed formula. formula. \Ix: Vx:A. 3y:B. P3(x, y) y) is is an an immediate immediate subformula subformula which which is is also also closed, but B. P3(x, y) isis not A; this closed, but �y: 3y:B. P3(x, y) not closed closed since since it it has has the the free free variable variable x: x:A; this latter latter formula A. formula is is well-formed well-formed in in the the context context x: x:A. The The atomic atomic subformulas subformulas are are PI(X), P~(x), P P2(Y), and P3((x P3((x,, y)) y)) which which are are formulas formulas in in 2 (y), and the (x, y) EE AA x BB isis used the context context x:A, x:A, y:B, y:B, and and the the typing typing judgment judgment x:A, x:A, y:B y : B fF- (x,y) used to (x, y))). to understand understand the the formation formation of of P3(x, P3(x, y) y) (which (which is is an an abbreviation abbreviation of of P3( P3((x, y>)). Examples. for P1 : A -+ E x a m p l e s . Here Here are are examples, examples, for --+ Prop, P2 : B -+ --+ Prop, P3 : A x B -+ --+ Prop. Prop. 2.5. 2.5. Formal F o r m a l proofs proofs There are many ways to to organize e. g. g. natural There are many ways organize formal formal proofs proofs of of typed typed formulas, formulas, e. natural deduction, the deduction, the sequent sequent calculus, calculus, or or its its dual, dual, tableaux, tableaux, or or Hilbert Hilbert style style systems systems to to name aa few. choose aa sequent calculus presented presented in name few. We We choose sequent calculus in a a top-down top-down fashion fashion (as (as with with tableaux) tableaux).. We We call call this this a a refinement logic (RL). (RL). The The choice choice is is motivated motivated by by the the advantages advantages sequents sequents provide provide for for automation automation and and display. display, 1is8 Here Here is is what what aa simple simple proof proof looks looks like like for for A E Type, P E A -+ --+ Prop; only only the the relevant relevant hypotheses hypotheses are are mentioned rst time mentioned and and only only the the fi first time they they are are generated. generated. refinement logic A E Type, P E A Prop; HOL; PVS uses 18This lSThis is the mechanism mechanism used used in Nuprl Nuprl and HOL; uses multiple conclusion conclusion sequents. 705 705 Types Types 1. 11.1 .1 11.1.1 .1.1 Vx": A. (Vy": A. P ( y ) => =v 3x 3x :" A. A. P(x)) P(x)) A. P(y) f-F- Vx A. (Vy Vy": A. A. P ( y ) =v 3x 3 x " : A. P(x) xx": AA f-F- Vy P(y) A. P(x) P(y)) ff :" (Vy (Vy :" A. A. P ( y ) ) fF- 3x 3x :" A. A. P(x) P(x) P ( x ) fF- 3x 3x :" A. A. P(x) P(x) ll :" P(x) f-F- P(x) P(x) 11.1.1.1 .1.1.1 1.1.1.2 1.1.1.2 f-~ xx EE AA f-t - xxEEA A => 11.1.2 . 1 .2 The The schematic schematic tree tree structure structure with with path path names names is is by by VR VR by by =>R =~R by VL f iwith byV L oon nfw thx x 3R x by by hyp hyp ll by by hyp hyp x x by by hyp hyp x x by by 3R with with x f-F-GG I 1. GI 1. HI H1 ft-G1 I 1.1 1.1 H H22 fF- G G22 // \\ 11.1.1 . 1 . 1 H3 11.1.2 . 1.2 H //3 fF- G G22 //22 fF- G3 G3 // \\ 11.1.1.2 . 1 . 1 .2 H3 11.1.1.1 . 1 . 1 . 1 H3 /-/3 fF- G4 G4 /-/3 fP G3 G3 Sequents. S e q u e n t s . The The nodes nodes of of a a proof proof tree tree are are called called sequents. sequents. They They are are aa list list of of hypotheses hypotheses separated separated by by the the assertion assertion sign, sign, ft- (called (called turnstile turnstile or or proof proof sign) sign) followed followed by by the the conclusion. conclusion. A A hypothesis hypothesis can can be be a a typing typing assumption assumption such such as as x x ::AA for for A A aa type type or or aa labeled labeled assertion, assertion, such such as as ll:: P(x) P ( x ) . . The The label label l1 is is used used to to refer refer to to the the hypothesis hypothesis in in the the rules. rules. The The occurrence occurrence of of x x in in x x ::AA is is an an individual individual variable, variable, and and we we are are assuming assuming that an object that it it is is an object of of type type A. A. So So it it is is an an assumption assumption that that A A is is inhabited. inhabited. Here Here is is aa sequent, sequent, Xlxl :" HI, G H 1 , .. ... ., x, X ~ n :" H H~n fF-G Hi Xi Xi where type and where H~ is is an an assertion assertion or or a a type and xi is is either either aa label label or or a a variable variable respectively. respectively. The The xi are are all all distinct. distinct. G is is always always an an unlabeled unlabeled formula. formula. We We can can also also refer refer to to the the hypothesis hypothesis by by number, number, 11 .. ... .n , and and we we refer refer to to G as as the the O-th 0-th component component of of the the sequent. sequent. We We abbreviate abbreviate a a sequent sequent by by fI /~ ~ G for for fI /~ = = (xl " H 1 , . . . ,x~" H~); sometimes sometimes we we write write x9 9fI F- G. G : f- G. n, f- G G (Xl : HI, . . . , Xn : Hn ) ; Rules. R u l e s . Proof Proof rules rules are are organized organized in in the the usual usual format format of of the the single-conclusion single-conclusion sequent sequent calculus. appear in table shortly. explain now this table. calculus. They They appear in a a table shortly. We We explain now some some entries entries of of this table. There There are are two two rules rules for for each each logical logical operator operator (connective (connective or or quantifier) quantifier).. The The right right Constable R. Constable R. 706 706 rule for an an operator operator tells tells how how to to decompose decompose aa conclusion conclusion formula formula built built with with that that rule for for an an operator operator tells tells how how to to decompose decompose such such aa formula formula operator, and and the the left rule for operator, when itit isis on on the the left, left, that that isis when when itit isis aa hypothesis. hypothesis. There There are are also also trivial trivial rules rules for for when the constants constants TT and and _1_ .1. and and aa rule rule for for hypotheses. hypotheses. So So the the rules rules fit fit this this pattern pattern and and the are named named as as shown. shown. are left rule Left Left & & V V => V :J 3 T T .1. _L Right Right &R &R vRI vRI VRr vRr =>R =>R VR VR 3R :JR TR TR &LL & vL vL ~=>L L VL VL :JL 3L - - .1.L _kL hyp Xi Hi by HI, by hyp xi Xl :" H Xl 1 , .. ... . ,, xXnn ": /Hn I n IF Hi l l :::; i :::; n The v R rule rule splits two, VR1 VRI and and VRr. vRr. To I- PP V Q we we can can prove prove The VR splits into into two, To prove prove fI /~ F VQ fI I-F P we can I- Q. Q . The The first first rule rule is is V Rl because we try try to to prove prove the the left /~ P or or we can prove p r o v e /fI ~ F V R1 because we disjunct. We pronounce these these somewhat awkwardly as as "or "or right right left" left" and and "or "or right disjunct. We pronounce somewhat awkwardly right right." right." The have obvious The other other rules rules all all have obvious pronunciations, pronunciations, e. e. g. g. "and "and left" left" for for &L, &L, "and "and right" &R, etc. etc. Some names, such right" for for &R, Some of of them them have have ancient ancient names, such as as ex falso libet for for "false left" left" (meaning ). In "false (meaning "anything "anything follows follows from from false" false"). In Nuprl Nuprl we we use use any for for .1.L. _LL. Sometimes Sometimes vL V L is is called called "cases" "cases",, and and instead instead of of "by "by v VL L I" l" we we might might say say "by "by cases cases on on I" for ." VL on l" for I1 a a label. label. For For VL on I1 with with a a term term t we we might might say say "instantiating "instantiating I1 with with tt." The similar to The rule rule =>L =~L is is similar to the the famous famous modus ponens rule rule usually usually written written as as left ex falso libet any t modus ponens A A AA ~ B=> B B B In In top top down down form form it it would would be be A, A~=> by =>L ~L A,A B FBB I- B by A, A, B B FI-BB A A FI-AA Some Some of of the the rules rules such such as as VL VL and and :JR 3 R require require parameters 9 For For example, example, to to decompose decompose Vx :: T.P(x) T.P(x) as as aa hypothesis, hypothesis, we we need need aa term term t E E T T.. So So the the rule rule is is VL VL on on t. For For Vx : T. P(x) 3x:T. P(x) as as aa goal, goal, to to decompose decompose it, it, we we also also need need aa term term tt E E Tj T; the the decomposition decomposition :Jx generates generates the the subgoal subgoal P(t) P(t).. parameters. t t. 707 707 Types Pv -.P -.-.P P proof Magic rule. These These rules rules do do not not allow allow us us to to prove prove the the formula formula PV-,P nor nor ~--P =~ =:} P M a g i c rule. nor nor any any equivalent equivalent formula formula.9 If If we we add add one one of of these these formulas formulas as as an an axiom axiom scheme scheme by then we we can can prove prove the the others. others. We We can can also also prove prove them them by by adopting adopting the the proof by then rule contradiction rule H H t- P by by contradiction 1-..1 H H,, - - P F-_L contradiction I- P contradiction -. P -.P My base arguments these formulas axiom scheme scheme P P VV --P My preference preference is is to to base arguments for for these formulas on on the the axiom called the the law law of excluded middle middle because because these these arguments arguments have have aa special special status status in in called of excluded relating logic logic to to computation computation and and because because this this law law is is so so important important in in philosophical philosophical relating foundational discussions. discussions. In In the the organization organization II adopt, adopt, this this is is the the only only rule rule which which and and foundational does sequent pattern does not not fit fit the the sequent pattern and and itit is is the the only only rule rule not not constructively constructively justifiable justifiable as we we will will see see later. later. II sometimes sometimes call call the the rule rule "magic" "magic" based based on on the the discussion discussion of of as justification to to follow. follow. justification Justifications. rule names names and and parameters parameters to important J u s t i f i c a t i o n s . The The rule to them them make make up up aa very very important part of the the proof called the of the step. We think of the part of proof called the justification of the inference inference step. We can can think of the justifi cation as an operator on sequents which which decomposes decomposes the into aa justification as an the goal goal sequent sequent into subgoal sequents. This format for for the the justification justification reveals reveals that that role graphically. subgoal sequents. This format role graphically. justification operator on sequents x : H I1.Hl I- 2"H~GG G1 1.H1 ~ GI r(x; t) by by r(2; t-) k.H~ F- Gk For example For example HH I~- (P (P V V Q) Q) by by vRl VRl 1.H I- P 1.HF-P _ The labels of The justification justification takes takes the the variables variables and and labels of x 2 plus plus some some parameters parameters tt and and generates generates the the k k subgoals subgoals H Hii IF- G Gi. The hypothesis hypothesis rule rule generates generates no no subgoals subgoals and and so so i . The terminates terminates a a branch branch of of the the proof proof tree. tree. Such Such rules rules are are thus thus found found at at the the leaves. leaves. By By putting putting into into the the justifications justifications still still more more information, information, we we can can reveal reveal all all the the links links between between a a goal goal and and its its subgoals. subgoals. To To illustrate illustrate this this information, information, consider consider the the =:} &LL rule. =v L L rule rule and and the the & rule. H, [I, ff :" (P (P =:} ~ Q), Q), J J I~- G G by by =:} =rL on o n ff 1. I-~ P 1. H, [ t , ff " (P (P =:} ~ Q), Q),J J P 2. 2. H, [ - If, f ': ((P P ~ Q), Q ) , J, J , yy:Q ' Q I~- G G : =:} H,pq - G by H, pq": P P& &Q Q IFby &L &L H,pq:P Q,p:P, q:Q, jJ I-F-G H, pq.P & &.Q,p.P,q.Q, _ 708 708 R. Constable R. y, If If the the =~ * RR justification justification provided provided the the label label y, then then all all the the information information for for genergener ating ating the the subgoal subgoal would would be be present. present. If If the the &L &L rule rule provided provided the the labels labels p, p, q then then the the data isis present present for for generating generating its its subgoals subgoals as as well. well. So So we we will will add add this this information information to to data form aa complete justification. form Notice Notice that that these these labels labels behave behave like like the the variable variable names names xi Xi in in the the sense sense that that we we can can systematically systematically rename them them without without changing changing the the meaning meaning of of aa sequent sequent or or aa justification. They They act act like like bound variables in in the the sequent. sequent. The The phrase phrase new new u, u, v v in in aa justification. justification allows allows us us to to explicitly explicitly name name these these bound bound variables. variables. justification complete justification. rename bound variables rules. Sequents Sequents as as defined on lists of formulas, formulas, so so SStructural t r u c t u r a l rules. defined here here are are based based on lists of the rules rules for for decomposing decomposing on on the the left must refer to the the position position of of the the formula. formula. the left must refer to This is is indicated indicated by by supplying supplying aa context context around the formula, formula, typically typically of of the the form form This around the H, fl, x ' FF,, JJ FI-- G. G . The The cut rule rule specifies specifies the the exact exact location location at at which which the the formula formula is is to to does the the same. same. be introduced introduced into into aa hypothesis hypothesis list, list, and and thin does be By combining applications applications of can be moved (exchanged) By combining of cut and and thin, hypotheses hypotheses can be moved (exchanged) or contracted. contracted. The The so-called so-called structural rules are are included included among these rules. or among these rules. x: cut thin cut thin, structural rules 2.6. PProof 2.6. r o o f eexpressions x p r e s s i o n s and a n d ttactics actics C o m p l e t e justifications. j u s t i f i c a t i o n s . If If there there is enough information information in genComplete is enough in aa justification justification to to gen erate the subgoals, subgoals, then the tree tree of of justifications justifications and and the the top top goal goal can can generate erate the then the the whole proof. Moreover, Moreover, the the tree tree of of justifications be combined combined into into aa single the whole proof. justifications can can be single "algebraic describing the the whole tree stripped "algebraic expression" expression" describing whole proof. proof. Indeed, Indeed, the the proof proof tree stripped of of the sequents sequents is just aa parse parse tree tree for for this this expression. expression. the is just If we we present present the the rules rules annotated annotated If the justifications justifications in in the the right right way way we we can can read read the (c.f. Reps by by them them as as an an attribute grammar (c.f. Reps and and Teitelbaum Teitelbaum [1988] [19SS],, Reps Reps [1982], [1982], Griffin Griffin [1988a]) [1988a]) for for generating generating an an expression expression describing describing the the proof proof called called aa proof the case case of the =~L *L and let p and expression. Consider Consider the of the and &L &L rules rules again. again. Suppose Suppose we we let and 9 subgoals, then g denote denote proof proof expressions expressions for for the the subgoals, then generate attribute grammar proof p expression. Gbby x5c": fl, H, f f :" (P (P * =~ Q), Q), J J I-t-G y *L =~L on on f f I-- P by P by p(x) p(~) -, y : g(x, y) g(x, y) by --, y " Q Q I-~- G G by g(~, y) If If we we think think of of the the proof proof expressions expressions p(x) p(~) and and g(~, y) as as being being synthesized up up from from the subtrees, then the subtrees, then the the complete complete proof proof information information for for the the goal goal sequent sequent is is synthesized *L with new =~L on on f f from from p(x) p(~) and and from from g(x, g(~, y) y)with new yy y Organizing Organizing this this into into aa more more compact compact expression expression and and recognizing recognizing that that y is is aa new new bound bound variable, variable, aa suggestive suggestive expression expression is is *L(Jj =~L(f; p(x) p(~);j yy .g(x, .g(~, y)) y)) 709 709 Types g x, bound label Here . ( y) y) to Here we we use use the the "dot "dot notation" notation" y y.g(2, to indicate indicate that that yy is is aa new new bound label in in the proof proof expression expression g(2, y) y).. The The dot dot notation is used used with with quantifiers quantifiers as as in in \Ix Vx:A. F notation is : A. F the to : A from to separate separate the the binding binding operator operator \Ix Vx:A from the the formula formula F F .. Likewise, Likewise, in in the the lambda lambda notation, A A(x.b), the dot dot is is used used to to indicate indicate the the beginning beginning of of the the scope scope of of the the binding binding notation, x ) , the of of x x.. In In the the case case of of &L &L,, the the rule rule with with proof proof expressions expressions looks looks like like g(x, ( b . H, zz'P&Q G by by &L &L in in zz with with new new u, u, vv : P&Q }- G x5c": ii, 2" H, u:p, u'P, v:Q v'Q f~G G by by g(x, g(2, u, u, v) v) x:ii, f- A A compact compact notation notation is is &L(z; u, v. g(x, u, v)) u, v Here Here u, v are are new new labels labels which which again again behave behave like like bound bound variables variables in in the the proof proof expression. expression. The will be be the The justification justification for for P V V ~ P will the term term magic(P). This This is is the the only only justification justification term term that that requires requires the the formula formula as as aa subterm. subterm. With logic as With this this basic basic typed typed predicate predicate logic as aa basis, basis, we we will will now now proceed proceed to to add add specific types, lists, functions, number of of specific types, namely namely natural natural numbers, numbers, lists, functions, sets sets over over aa aa number type, type, and and so-called so-called quotient quotient types. types. Each Each of of these these shows shows an an aspect aspect of of typed typed logic. logic. Note, Note, in in these these rules rules we we are are presupposing presupposing that that P, Q, and and the the formulas formulas in in ii /~ are are well-formed according to well-formed according to the the definition definition of of a a formula formula and and that that the the type type expressions expressions are are also accordance with also well-formed well-formed in in accordance with the the typing typing rules. rules. As As we we introduce introduce more more types, types, it typing judgments it will will be be necessary necessary to to incorporate incorporate typing judgments as as subgoals. subgoals. The The Nuprl Nuprl logic logic of of Constable al. [1986] relies on subgoals from Constable et et al. [1986] relies on such such subgoals from the the beginning beginning so so that that the the caveat caveat just stated stated for this table just for this table of of rules rules is is unnecessary unnecessary there. there. P op magic(P) . P, Q, Tactics. Tactics. Complete Complete justifications justifications will will generate generate the the entire entire proof proof given given the the goal goal formula formula because because the the rule rule name, name, and and labeling labeling formation formation and and parameters parameters are are enough enough data data to to generate generate subgoals subgoals from from the the goals. goals. So So the the subgoals subgoals are are computable computable from from the the part part of cation that of the the justifi justification that does does not not include include the the proof proof expression expression for for the the subproofs subproofs to automate (the (the synthesized synthesized expressions). expressions). This This fact fact suggests suggests aa way way to automate interactive interactive proof generation. Namely, proof generation. Namely, a a program program called called aa refiner, takes takes aa goal goal and and aa complete complete justification justification and and produces produces the the subgoals. subgoals. Nuprl Nuprl works works this this way. way. Nuprl and Wadsworth Nuprl also also adapts adapts tactics tactics from from LCF LCF (Gordon, (Gordon, Milner Milner and Wadsworth [1979]) [1979]) into notion of into the the proof proof tree tree setting setting to to get get aa notion of tactic-tree proof (Allen (Allen et et al. al. [1990], [1990], Basin Basin and and Constable Constable [1993], [1993], Griffin Griffin [1988b]). [1988b]). In In this this setting setting the the justifications justifications are are called primitive primitive refinement combined using using procedures called refinement tactics. tactics. These These can can be be combined procedures called called tacticals. For For example, example, if if a a refinement refinement ro generates generates subgoals subgoals G1,..., Gn when when applied applied to sequent then the compound refinement tactic written THENL to sequent Go, then the compound refinement tactic written ro T H E N Lh [ r l;; .. ... . ;;rn]] executes executes ro, then then applies applies ri to to subgoal subgoal Gi generated generated by by ro. There are many tacticals (c.f. Jackson [1994a] , Constable et al. [1986]); There are many tacticals (c.f. Jackson [1994a], Constable et al. [1986]); two two basic REPEAT. The ORELSE tactical basic ones ones are are ORELSE ORELSE and and REPEAT. The ORELSE tactical relies relies on on the the idea idea that that refinement might might fail fail to to apply, apply, as as in in trying trying to to use use &R &R on on an an implication. implication. In In aa refinement to decompose ro ORELSE ORELSE r l , if if ro fails fails to decompose the the goal, goal, then then rl is is applied. applied. refiner, tactic-tree proof tacticals. To Go, ro, r1 , ro Ti To Gi ro. T1 G1 , . . . , Gn To rn 710 710 R. R. Constable Constable Table T a b l e of o f justification j u s t i f i c a t i o n operators operators Right(R) Right(R) Left(L) Left(L) H, xx'P&Q, by &L(xj &L(x;u, . g(u , v)) u, vv.g(u,v)) f- G by : P&Q, JJ~-G fl, fI, xx": P&Q, P&Q, uu :"P, P, vv :"Q Q,, JJ f-F-G G by by g(u g(u,, v) v) 1.1. fl, & v i-I, xx ': P , JJ Ff-- G bby y VL(Xj VL(x;u. gl(u);v, gr(v)) (v)) (U)j V . gr U . g, PVVQQ, fl, i-I, xx ': P Q,u'P, JFG by by g, gt(u) (U) f- G u : P, J PVv Q, 1.1. fl, i-I, x ': P Q,v'Q, JFG by by gr gr(v) (v) f- G PVv Q, 2. fl, v : Q, J fl f-b P&Q P&Q by by &R(Pj &R(p; q) q) P fl f-F-P 1.1./~ P by byp 2.2 . fl / L f-b QQ by by qq fl /~ f-F-P P vV Q Q by by VR1(P) Vnl(p) fl f-F-P 1.1./~ P by bypP fl /~ f-~-P P vV Q Q by by VRr(q) VRr(q) f- Q by 1. 1 . fl /~Q by qq :} /L, xx": P P :} =~ Q, J ,7 f-F- G by :}L(xj =~L(x;pj p; yy.. gg(y)) (y)) fl, 1. fl, [I, x ': PP :} =~ Q, J j f-F-P P by P p 2./~, P :} =~ Q, J, J, yy'Q by g(y) : Q f-F- G by fl, xx": P 2. fl /~ f-I- P P :} =~ Q Q by by :}R(x. =~R(x. q(x)) q(x)) P f-b Q fl, H, xx :: P Q by by q(x) q(x) 'V i-I, xx": 'Vz Vz:A.P(z), JF-f- G by 'VL(xj VL(x;a;y.g(y)) aj y. g(y)) : A. P(z), J fl, [-I, x" Vz'A. P(z), J J f-F-a EG A A 1.1. fl, : A. P(z), x: 'Vz 2. fl, H, x x": 'Vz Vz'A. P(z), J, J, yy'P(a) F- G by by g(y) : P(a) f: A. P(z), p(z)) new : A.P(z) by fl [-I f-F-'Vz Vz'A.P(z) by 'VR(z. VR(z.p(z)) new w w fl, i-I, w w": A A f-~- P(w) P(w) by by p(w) p(w) 3 i-I, x x": 3z 3z": A. P(z), 3L(xj u, f- G by 3L(x; P(z), jJ Fg(u , v)) u, v. v . g(u, fl, new new u, u, v by g(u,v) H, G by J f- G g(u , v) 3z :A. P(z), u:A,v'P(u), x : 3z'A.P(z), fl, x" u : A, v : P(u), JF- 3R(ajp) by 3R(a; P(z) by fl f-~- 3z 3z": A. A. P(z) p) /~, f- G any (x) by any(x) .1, JJ ~x: _l_, G by fl, x" f- T true fl Fby true T by /~ .1 T i f- Hi Hi xXll ' :HHl 1 ,,... .. ,. x, xn n ' H: Hn n F- H~ Hi 11.. fl H Ff-- a EEAA by pp P(a) by f- P(a) fl F2. H 2. by hyp Xi ii == 1,..., 1, . . . , nn hYP xi by distinct) are distinct) (recall xi Xi are (recall cut cut c) /~, cut(x.g (x)j c) by cut(x.g(x); fl, jJ bf- G G by i-I, g(x) by g(x) G by x : C, JJ bf- G fl, x'C, [-I, yc c fl, JJFf-' CCb by @ ii C@ Assert C Assert J. in fl, J. locates CC in/~, where ii locates where thin thin [-I, by g9 G by f- G x : P, JJ Ffl, x'P, Thin ~@ ii Thin in fl, J.J. locates xx-:PP in/~, where ii locates where [-I, b yby g 9 fl, JJbf-G G Magic: Magic: H magic(P) by magic(P) ..,p by f- PPVV --,P H ~- 711 711 Types We We will will use use tacticals tacticals to to put put together together compound compound justifications justifications when when the the notation notation seems seems clear clear enough. enough. 2.7. 2.7. Natural N a t u r a l numbers numbers One One of of the the most most basic basic mathematical mathematical types types is is N, IN, the the natural natural numbers. numbers. This This type type is formed formed by by the the rule rule H H fF- N IN E E Type. Type. The The type type is is inductively inductively defined defined by by the the rules rules is which which say say that that 00 E E N, IN, and and if if n n E E N IN then then suc(n) suc(n) EE N. IN. The The typing typing judgments judgments we we need need are are H tIN H f- 0O EE N type_of_zero H fF- suc( suc(n) IN type_of..successor type_of_successor H n) EE N H H Ff-- nnEEI NN To ned we To express express the the fact fact that that N 1N is is inductively inductively defi defined we use use the the rule rule of of mathematical mathematical induction. In induction. In its its unrestricted unrestricted form, form, this this essentially essentially says says that that nothing nothing else else is is aa member member of of N IN except except what what can can be be generated generated from from 00 using using suc. suc. But But the the form form of of the the rule rule given given here does not all propositional here does not quantify quantify over over all propositional functions functions on on N, IN, so so it it is is not not aa full full statement statement of of the the principle. principle. Suppose Suppose P P": (N (IN x • A) A) -t ~ Prop, Prop, then then (u, i, x)) po; u, i, p. P(n,~) by ind(n; ind(n;po;u,i, p,(u,i,~)) : N f-F- P(n, x) by x~'[-I, : H, nn'iN x~'H,n'IN : H, n : N f-F- P(O) P(O) by by Po Po x) by u, x) . (u, i,i, x) i-I, n n": N, IN, u" iN, ii": P( P(u, ~) fF- P(suc(u) P(suc(u), , ~) by p ps(u, ~) x~ :"H, u : N, _ Arithmetic. A r i t h m e t i c . When When we we display display proofs proofs of of arithmetical arithmetical propositions, propositions, we we will will assume assume that automatic proof proof procedure procedure which will prove quantifier free that there there is is an an automatic which will prove any any true true quantifier free conclusion and < So for conclusion in in a a sequent sequent involving involving 0, 0, suc(n) suc(n),, + +,, - , ., , * , = - and <.. So for example, example, here here are are some some arithmetic arithmetic facts facts in in this this category category - oo < < x, y y < < suc(z) f- y y *9 x9 < < suc(z uc(z *9 x). Although Although there there is is no no proof proof procedure procedure with with this this power power (the (the problem problem is is undecid undecidable), able), there there are are good good arithmetical arithmetical proof proof procedures procedures for for restricted restricted arithmetic arithmetic (Arith, (Arith, see see Boyer Boyer and and Moore Moore [1988], [1988], Church Church [1960]) [1960]) and and linear linear arithmetic arithmetic (SupInf, (SupInf, see see Chan [1982] Chan [1982], Shostak Shostak [1979], [1979], Bledsoe Bledsoe [1975]). [1975]). We We refer refer the the interested interested reader reader to to the the ' citations details. The citations for for details. The use use of of Arith Arith allows allows us us to to present present proofs proofs in in aa form form close close to to that that of of Nuprl Nuprl (Constable (Constable et et al. al. [1986], [1986], Jackson Jackson [1994c]). [1994c]). Here Here are are two two proofs proofs to of one not. (11) . . , Iln) of the the same same trivial trivial theorem, theorem, one one inductive, inductive, one not. We We write write Arith Arith(/1,..., n ) to show are used For readability show which which labeled labeled hypotheses hypotheses are used by by the the proof proof procedure. procedure. For readability we we intentionally parts of the justification, intentionally elided elided parts of the justification, using using -. --. f-t- 'v'x y) by Vx": N. IN. 3y 3y": N. IN. (x (x < y) by 'v'R(x, VR(x, ___ ___)) by R(s(x); ___ 3 x" IN k 3y" IN. (x < y) by 3R(s(x); ___)) 3y : N. (x < y) : N fx f-F- xx < suc(x) by suc(x) by Arith Arith . 712 712 R. Constable The complete proof )) . The complete proof expression expression is is VR(x. VR(x. 3R(suc(x); 3R(suc(x); Arith Arith)). : N. 3y : N. (x y) by Vx:N. 3y:N. (x < < y) by VR(x. VR(x. - - - -) -) f-F Vx 3y:N. (x < < y) y) by ind(x; i n d ( z ; -- - -; ; )) by xx:N : N f-F 3y : N. (x by 3y:: N(O N(0 < < y) by 3R(suc(0); 3R(suc(O); Arith Arith)) f-F 3y y) : N. (suc(u) 3y:N. < y) y ) Ff- 3y 3y:N. (suc(u) < < y) y) by by 3L(i; 3L(i; yo, y o ,- - -) xx:N, : N, uu:N, : N, ii:: 3y : N. (u (u < : N(suc(u) < o) f-F 3y , yo yo:N,l:(u < YYo) 3y:N(suc(u) < y) y) by by 3R(suc(yo); 3R(suc(yo);-- - -) xx ::NN,, uu: N: N, : N, l : (u < by (l ) f-F (suc(u) (suc(u) < < suc(Yo)) suc(yo)) by Arith Arith(/) -- The The complete complete proof proof expression expression is is o, 1.l. 3R(suc(yo); ith(l))))). VR(x. u, i.i. 3L(i; VR(x. ind(x; ind(x; 3R(suc(0) 3R(suc(O); ; Arith Arith);) ; u, 3L(i; YYo, 3R(suc(yo); Ar Arith(l))))). The The following following example example will will provide provide another another compact compact proof proof expression. expression. It It shows shows that integer integer square square roots exist without Magic ((O'Leary et al. al. [1995]) [1995]).. First First we we that roots exist without using using Magic O'Leary et 2. specify these these roots. roots. Let Let Root(r, Root(r, n) - = rr 22 ::; < n n< < (r (r + + 1) 1)2. n) == specify Theorem. Theorem. f-F Vn Vn": N. N. 3r 3r": N. N. Root(r, Root(r, n) n).. by R TTHEN ind new new uu by V VR H E N ind base base case case 1. 1. n n ':NN f-F 3r 3r": N. N. Root(r, Root(r, O) O) by by 3R 3R 00 THEN THEN Arith Arith induction induction case case 11.. nn ':NN 22.. uu": 3r 3r :"N. N. Root(r, Root(r, n) n) f- 3r : N. N. Root(r, Root( , suc(n)) by by 3L 3L on on u u new new ro, ro, vv 3. ro : N 3. ro'N 44.. vv": Root(ro, Root(ro, n) n) f-F 3r 3r": N. N. Root(r, Root(r, suc(n)) suc(n)) by label dd THENA by cut (ro + + 1) 1)22 ::; _ suc(n) suc(n) V V suc(n) suc(n) < < (ro (ro + + 1) 1)22 with with label THENA Arith. Arith. cut (ro ((This This rule rule generates generates two two subgoals. subgoals. The The "auxiliary "auxiliary one" one" is is to to prove prove the the cut cut formula. Arith to formula. That That subgoal subgoal can can be be proved proved by by Arith, Arith, so so we we say say THENA T H E N A Arith to indicate indicate this. this.)) ((The The "main" "main" subgoal subgoal is is this this one. one.)) 5. (ro + + 1) 1)22 ::; _ suc(n) suc(n) V V suc(n) suc(n) < < (ro (ro + + 1) 1)22 5. dd": (ro f-F 3r Root(r, n) 3r": N. N. Root (r, n) by by V V LLoon nd d ((This This is is case case analysis analysis on on the the cases cases in in hypothesis hypothesis 5.) 5.) 6. 6. (ro (to + + 1) 1) 2~ ::; < suc(n) suc(n) f-F- 3r 3r :"N. N. Root(r, Root ( r, n) n) by 1) THEN by 3R(ro 3R(ro + + 1) THEN SuplnJ SupInf ((Since Since r� r o2 ::; < n n < < (ro (ro + + 1) 1)22,, from from (ro (ro + + 1) 1)22 ::; _< suc(n) suc(n) we we know know (ro (ro + + 1) 1)22 ::; _ 2 2 suc(n) + 1l < suc(n) < < ((ro ((ro + + l) 1) + + 1) 1)2 since since n n+ < (ro (ro + + l) 1)2 + + 11.. The The SupInf SupInf procedure procedure can can find find this this proof. proof.)) Types 713 713 6. 6. suc(n) suc(n) << (ro (ro ++ 1) 1 )22 3r : N. Root(r, Root(r, n) n) FI- 3r:N. by 3Rro 3Rr0 TTHEN Arith by H E N Arith ( Since ro r;2 _� nn we we know know immediately immediately that that ro r; <_� (Since suc(n) .) The proof proof expression expression corresponding corresponding to this is is The to this VR(n. ind(n; ind(n; 3R(0; 3R(O; Arith); Arith) ; VR(n. cut(d. YL(d; V L(d; 3R(ro 3R(ro ++ 1; SupInf); SuplnJ) ; n, u. u. 3L(u; 3L(u; ro, ro, v.v. cut(d. n, 3R(ro; Arith)); 3R(ro; Arith)); Arith)))). Arith)))). In the appendix, In the appendix, section section 6.2, 6.2, we we consider consider another another simple simple arithmetic arithmetic example example and and show aa complete complete Nuprl Nuprl proof. proof. show 2.8. Lists Lists 2.8. Sequences are are basic basic forms forms of of construction construction in in mathematics, often written written Sequences mathematics, often ( a l ,, .. ... ., a, an) n ) . . With With the the widespread widespread use use of programming languages languages we we have have come come of programming (aI to several data data types associated with with sequences sequences as as distinct to distinguish distinguish several types associated distinct types. types. There There arrays, and and sequences, they have properties. We We are lists, are lists, arrays, sequences, and and they have different different mathematical mathematical properties. first look at lists. lists. first look at If A type, then then so so is is A A list. write an rule like A isis aa type, list. We We could could write an informal informal rule like this this If A E Type Type A E · A list EE Type A list Type" The elements an A A list list are nil and and (a.1) (a.l) where where aa EE A and lI EE A A list. list. If If A The elements of of an are nil A and A is is N, 51, these are are lists: lists: these nil (1. (1. (2. (2. nil)) nil)) (2. (2. nil) nil) All All lists lists are are built built up up from from nil nil and and the the operation operation of of pairing pairing an an element element of of A A with with aa list. list. The The typing typing rules rules are are fI I-~ nil list type_oj _nil nil E EA A list type_of_nil fI (a. I)l) EE A H I~ (a. A list list type type_of_cons -oj _cons fI [ - II-F aa EEAA fI R IF I1 E EA A list list Equality Equality on on lists lists is is given given by by these these rules. rules. fI I-F (h. (h. t) t) = = (h'. (h'. t') t') in in A A list list by by list list - eq eq fI H FI- hh ==h h' ~ i ninAA fI H IF tt = = t' t ~in in A A llist ist fI I-~ nil nil = = nil nil in in A A list. list. by by nil nil - eq eq For every type A , A list is an inductive type with base For every type A, A list is an inductive type with base case case of of nil nil.. The The inductive inductive character character of of the the type type is is given given by by its its induction induction rule rule stated stated below below for for P P E E (A (A list list x T T => =~ Prop) Prop) 714 714 R. Constable x2": fl, h, t,t, i.i. p(h, o; h, H, ll": A A list list IF P(l, P(1, x) 2) by by lisLind list_ind (l; (l; P Po; p(h, t,t, i,i, x)) 2)) x2.: fl I-F P(nil, H P(nil, x) 2) by by Po po x2 ": H fl,, ll' :AAl i list, st, h h ': AA,, t t' A: A l i slist, t , ii ':PP(t, ( t , 2x) ) I~ P((h. P ( ( h . tt) ) , ,2x) ) by b y pp(h, ( h , tt,, i ,i,2x) ) We can define the usual head-and-tail functions by induction. We can define the usual head-and-tail functions by induction. Compound(x) list. xx = (h. t) Compound(x) == = = 3h 3h": A. A. 3t 3t": A A list. = (h. t) in in A A list list.. _ IF Define Define 'v'x Vx:: A A list. list. (x (x = = nil nil in in A A list list V V Compound(x)) Compound(x)) by 'v' VR by R A list list IF (x (x = = nil nil in in A A list list V V Compound(x)) Compound(x)) xx:: A by by lisLind list_ind I-F nil nil = = nil nil in in A A list list V V Compound(x) Compound(x) by by VRl YRl list, hh :: A list IF (h. (h. t) t) = = nil nil in in A A list list V V Compound(h. Compound(h. t)) t)) xx :: AA list, A,, tt::AA list by VRr VRr by I-F 3h' A. 3t' t') in 3h':: A. 3t':: A A list. list. (h. (h. t) t) = = (h'. (h'. t') in A A list. list. by by 3R 3R h h THEN THEN 3R 3R tt We We can can also also prove prove Yx": A A list. list. 3!h 3!h": A. A. 3!t 3!t": A A list. list. (--,(x (~(x = = nil) I-F 'v'x nil) =~ => = (h. (h. t) t) in in A A list) list) xx = where 3! expresses see the where 3! expresses unique unique existence existence ((see the end end of of section section 2.9) 2.9).. 2.9. Functions 2.9. F unctions The The function function type type is is one one of of the the most most important important in in modern modern mathematics. mathematics. As As we we have have noted, noted, Frege Frege patterned patterned his his treatment treatment of of logic logic which which we we are are following following on on the the concept concept of of a a function. function. In In some some ways ways this this type type represents represents the the divide divide between between abstract concrete mathematics. abstract and and concrete mathematics. By By quantifying quantifying over over functions functions we we enter enter the the realm realm of of abstract abstract mathematics. mathematics. Indeed, Indeed, the the very very notion notion of of obtaining obtaining aa function function from from an is called an expression expression is called abstraction. abstraction. The beginning of The day-to-day day-to-day notation notation for for functions functions at at the the beginning of the the century century was was that that 's notation, sin (x) in " . Russell one notation, ¢>i: one wrote wrote phrases phrases like like "the "the function function sin(x) in x x or or eX e x in in x x". Russell's Ck,, 's lambda and lambda notation, notation, A)~x.e x.ex~ ,, brought brought flexibility notation, creating and Church Church's flexibility to to the the notation, creating aa indicate the single name binding operator single name for for the the function function with with a a binding operator (~) to indicate the arguments. arguments. (A) to The modern working books ((used used in The modern working notation notation in in mathematical mathematical articles articles and and books in Bourbaki Bourbaki for for example example)) is is x x t-t ~ bb for for a a function function with with argument argument x x and and value value given given by by the the expression expression bb in in x x;; for for example example x x t-t ~ x x for for the the identity, identity, x x t-t ~ eX e x for for the the exponential. exponential. As As we we did did for for propositional propositional functions, functions, we we will will adopt adopt the the lambda lambda notation notation in in the the form form A(X. A(x. b) b) for for x x t-t ~-+ b. b. In In Nuprl Nuprl one one can can display display this this in in a a variety variety of of ways, ways, including including => b. b. The xx t-t ~-+ bb or or bi: b~ or or fun fun x x =~ The important important points points are: are: •9 There There is is an an operator operator name, name, lambda lambda that that distinguishes distinguishes functions. functions. Their Their canoni canonical cal value value is is A(X A(x.. b) b).. •9 A x. bb is or formula A binding binding phrase, phrase, x. is used used to to identify identify the the name name of of the the argument argument ((or formula parameter body of parameter),) , x x,, and and the the body of the the function. function. Types Types 715 715 • 9 The The usual usual rules rules about about binding binding phrases phrases apply apply concerning concerning bound bound variables, variables, scope, scope, and and a-equality. a-equality. Essentially the the only way to to use use a a function function is is to to apply apply it it to to an an argument. Informal Essentially only way argument. 1199 Informal notation for (a) or for applying applying aa function function J f to to an an argument argument aa is is to to write write J f(a) or J faa or or even even to show show the the substitution substitution of of "actual" argument for for the the "formal" "formal" one one as as in in sin(a) or or to "actual" argument We adopt adopt an an operator operator name name to to remind remind ourselves ourselves that that application application is is a a distinct distinct eeaa.. We operation. So So we we write write ap(J; ap(f; a) a).. But But again, again, Nuprl Nuprl can can display this this anyway anyway the the user user J(a) or fa or even f. a or f@a . pleases, e.g. as pleases, e.g. as f(a) or fa or even f. a or f@a. One One of of the the major major discoveries discoveries from from aa systematic systematic study study of of function function notations, notations, especially the the lambda lambda calculus calculus and and combinatory combinatory calculus calculus and and later later programming programming especially languages, languages, is is that that rules rules for for formally calculating with with functions functions can can be be given given indepen independently meaning, especially dently of of their their meaning, especially independently independently of of types. types. The nitional equality" The rules rules for for calculation calculation or or for for "defi "definitional equality" can can be be expressed expressed nicely nicely as as evaluation rules. rules. Here Here is is the the so so called called "calLby "call_by_name" evaluation rule. rule. evaluation -Ilame" evaluation Jf {. .\(x. b) b) b[z/x] {. c ap(J; a) {.,1,cc ap(/; a) The _value" rule The "calLby "call_by_value" rule is is this this f {. .\(x. b) a {. a' b[a'/x] {. c p(Y; a) {.$ c ap(J; Closed .\(x . .\(y. Closed expression expression functions functions like like 1 I == = = .\(x. A(x. x) x) or or K g == = = A(x. A(y. x)) x)) are are called called combinators; these these two two are are "polymorphic" "polymorphic" in in that that we we can can compute compute their their values values regardless the form regardless of of the form of of the the input. input. Thus Thus ap(.\(x. ap(A(x, x); x); K) K) {. $K g and and ap(.\(x. ap(A(x, x); x); 0) 0) {. $ O0,, and ap(K; 1I)) {. $ .\(x. A(x. 1) I).. and Other functions .\(z.add(loJ(z); 20J(z))) only be Other functions like like .\(z.lof(z)) A(z.lof(z)) or or A(z.add(lof(z); 2of(z))) can can only be reduced reduced to specific form, to values values on on inputs inputs of of aa specific form, and and others others like like .\(x.suc(x)) A(x.suc(x)) or or .\(x. A(x. 44/x) /x) reduce reduce to to meaningful meaningful values values (typed (typed values) values) only only on on specific specific inputs. inputs. For For example, example, ap(.\(z.lof(z) ap(A(z.lof(z);; 0) 0) {. $ 10f(0) lof(0) but but 10f(0) lof(0) is is not not a a canonical canonical value value let let alone alone aa sensible sensible value. pair(O; 0)) value. In In the the case case ap(.\(x. ap(A(x, suc(x)); suc(x));pair(O; 0)) the the result result of of evaluation evaluation is is the the value value suc(pair(O; 0)) this value 0)),, but but this value has has no no type. type. Typing T y p i n g functions. functions. The The space space of of functions functions from from type type A A to to type type B B is is denoted denoted A the range A --+ --+ B B.. The The domain type type is is A A,, the range (or (or co-domain) co-domain) is is B B.. The The typing typing rule rule for for functions functions is is intuitively intuitively simple. simple. We We say say that that .\(x. A(x. b) b) E EA A --+ ~ B B provided provided that that on on each each a) EE B input input aa E EA A,, ap(.\(x. ap(A(x, b) b);; a) B.. This This judgment judgment is is usually usually made made symbolically symbolically by by that bb EE B the form assuming assuming x x E EA A and and judging judging by by typing typing rules rules that B.. This This is is the form of of typing typing judgment judgment we we adopt. adopt. So So the the typing typing rule rule has has the the form form fI /~ fk .\(x. A(x. b) b) E EA A --+ -+ B B by by fun_type fun_type fI, x : A fF-- b E B [-t,x:A B 19 19Although Although if functional functional equality is defined defined intensionally, intensionally, then it is also also possible possible to analyze analyze their structure. Of course, course, function function can also also be passed passed as data. 716 716 R. Constable More More generally, generally, given given an an expression expression f f we we allow allow H /~ fF f f E EA A -+ ~ B B by by fun_type fun_type H x) EE B H ,, xx::AA fF ap(jj ap(f;x) B In judging that In the the course course of of judging that an an expression expression tt has has aa type type T, T, we we allow allow replacing replacing tt by definition ally equal by any any term term t' t ~ that that is is definitionally equal or or by by a a term term t' t ~ that that tt evaluates evaluates to. to. So So if if tt T. In in T T and and tt .j.. $ t' t',, then then tt E E T. In the the logic logic over over (A (A -+ --+ B) B) we we add add the the rule rule for for function function in equality equality H /~ ft- f f = = 9 g in in A A -+ --+ B B by by extensionaLequalityR extensional_equalityR H, A fF ap(Jj ap(f; x) ap(g; x) in B B H, xx:: A x) = ap(gj x) in ap(gj b) /~, - 9 g in in A A -+ --+ B B ft-- ap(Jj ap(f; a) a) = ap(g; b) in in B B by by extensionaLequalityL extensional_equalityL H, ff = f-~ -aa EE AA Here Here is is Cantor's Cantor's interesting interesting argument argument about about functions functions based based on on the the method method of of diagonalization. rules for (See the diagonalization. It It illustrates illustrates the the rules for functions. functions. (See the appendix appendix for for a a Nuprl Nuprl proof.) proof.) Definition. Call ff in : B. f(g(y)) Definition. Call in (A (A -+ --+ B) B) onto onto iff iff 3g 3 g :: ((B B -+ --+ A) A) such such that that Vy Vy:B. f ( g ( y ) ) == yy in in B B.. Cantor Cantor shows shows that that for for inhabited inhabited types types A A with with two two distinct distinct elements elements there there is is no no -essentially because function function from from A A onto onto (A (A -+ --+ A) A)--essentially because (A (A -+ --+ A) A) is is "too "too big" big" to to be be enumerated A. We enumerated by by A. We state state the the condition condition on on A A using using functions. functions. We We require require that that A such there there is is a a function function diff diff E EA A -+ --+ A such that that diff(x) diff (x) =1= ~ x x for for all all x x in in A A.. The The theorem theorem is is Cantor's Cantor's Theorem. Theorem. : A. diff(x) 3 e : A -+ (A is onto) onto) (3 diff (A -+ A) (3 diff ::(A A).. Vx Vx:A. dill(x) =1= r x x in in A) A) :=} =v (..., (~3e:A (A -+ --+ A). A). ee is Proof. THEN :=}R P r o o f . by by :=}R =~R THEN ==~R 11.. 33 diff A. diff(x) dill: : (A (A -+ A). A ). Vx Vx:: A. dill(x) =1= ~ xx in in A A 2. 3e : A -+ 2. 3e:A --+ (A (A -+ --+ A) A).. ee is is onto onto f-l. FA_ Next Next use use 3L 3L on on 22 TTHEN H E N unfold unfold "onto" "onto" TTHEN H E N 3L 3L 2. ee:A : A -+ -+ A) A) 2. ~ (A (A-+ 3. gg:: (A (A -+ --+ A) A) -+ ~ A A A -+ 4. Vh:(A -~ A) A).. e(g(h)) e(g(h))== hh in in ((A --+ A) A) 4. Vh : (A -+ Next A -+ .2 Vx : A. diff(x) Next 3L 3L on on 11 to to replace replace 11 by by 1.1 1.1 diff diff ::A -~ A A,, 11.2 Vx:A. dill(x) =1= ~ xx in in A A Let Let ho ho == == ).(x. A(x. diff(e(x)(x))) diff (e(x)(x))) Now Now VL VL on on 44 with with ho h0 5. = ho 5. e(g(ho)) e(g(ho))= ho in in A A -+ --+ A A Let by extensionaLequalityL Let d d == == g(ho) g(h0),, by extensional_equalityL ho(d) in 6. (d) = ho(d) 6. e(d) e(d)(d) in A A Types 717 717 Now evaluate evaluate ho(d) to to rewrite rewrite 66 as as Now 6. e(d) e(d)(d) = diJJ(e(d) diff(e(d)(d)) 6. (d) = (d)) Now by by VL VL on on 1.2 1.2 with with e(d) e(d)(d) Now (d) 7. diJJ(e(d) diff(e(d)(d)) r e(d) e(d)(d) (which is is (diJJ(e(d) (diff(e(d)(d)) = e(d) e(d)(d)) --+_l_) 7. (d)) # (d) (which (d)) = (d)) �1-) F• f-1- Finish by by =*L =~L on on 7. 7. and and 6. 6. 0 [] Finish Implicit functions functions from f r o m relations. relations. A A common common way way to to define define functions functions is is implic implicImplicit itly in in terms terms of of relations. relations. Suppose Suppose R R is is a a relation relation on on A A x x B B and and we we know know that that for for itly every x x E A A there there is is aa unique unique yy in in B B such such that that R(x, R(x, y) y).. Then Then we we expect expect to to have have aa every x, Jf(x)). (x)) . How function function J f E CA A� -+ B B such such that that R( R(x, How do do we we specify specify this this function? function? To facilitate facilitate consideration consideration of of this this matter, matter, let let us us define define :J!y 3!y:A. P(y) to to mean mean there there To : A. P(y) is aa yy satisfying satisfying P P , , and and any any zz that that satisfies satisfies it it is is yy.. Thus Thus is Definition. 3!y": A. A. P(y) P(y) == = = :Jy 3y": A. A. P(y) & Vz Vz": A. A. (P(z) (P(z) =* =~ yy = zz in in A) A).. Definition. :J!y P(y) & We expect expect the the following following formula formula to to be We be true. true. Function Comprehension. Comprehension. "Ix Vx": A. =~ :JJ 3f :" A A � --+ B. B. "Ix Vx" Function A. 3!y" :J!y : B. B. R(x,y) R(x, y) =* A. A. R R (x, (x, J(x)) f (x)).. For many many instances instances of of types types A, B and and relation R we we can can prove prove this this formula formula by by For A, B relation R exhibiting r) for in N exhibiting a a specific specific function. function. For For example, example, if if we we define define Root(n, Root(n, r) for n, n, rr in N as as r 2 :::; < n n & &n n < < (r (r + + 1) 1) 22 then not only only can can we we prove prove Vx Vx": N. g. :J!r 3!r": N. N. Root(n, Root(n, r) r) but we then not r2 but we can also also define function root by primitive primitive recursion, recursion, namely namely can define aa function root by oot(O) == o0 root(O) root(suc(n)) = if if (root(n) (root(n)++ 1) then root(n) root(n) + + 11 else root(n).. else root(n) root(suc(n)) = 1) 22 _:::; nn then We know know that root(x)) eE N g --~ and Root Root (n, So perhaps We that )~ >. (x. (x. root(x)) �N N and (n, root(n))is root(n)) is true. true. So perhaps if there there are expressions for for defining prove the the conjecture. if are enough enough expressions defining functions, functions, we we can can prove conjecture. In set theory, functions are are usually usually defined defined as total relations, i.e., aa In set theory, functions as single-valued single-valued total relations, i.e., relation R on on A B is for all in A A there unique yy in in B B such relation R A xx B is aa function function iff iff for all xx in there is is aa unique such that that y) . The The relation relation R is aa subset subset of of A A xx B and this this R taken to to be the function. R(x, y). R is B, , and R is is taken be the function. R(x, Bour If the the underlying underlying logic Hilbert EE-operator) -operator) as If logic has has a a choice choice function function (or (or Hilbert as in in Bourbaki 1968b] or HOL (Gordon (Gordon and and Melham 1993] ), then then the of the the function function baki [[19685] or HOL Melham [[1993]), the value value of defined for the defined by by RR on on input input xx isis choice(y. choice(y. R(x, R(x, y)) y)) and and aa A >. form form for the function function isis )~(x. >.(x. choice(y. choice(y. R(x, R(x, y)). y)) . The choice choice operator operator would would not not only only prove prove the the implicit implicit function function conjecture, conjecture, but but itit The as well. well. That is would prove the the closely closely related related axiom axiom of of choice choice as would prove T h a t axiom axiom is � B). B). Vx" Vx : A. A. R(x, R(x, fJ(x)) A x i o m oof f C h o i c e . Vx" (x)) .. Axiom Choice. Vx : A. A. 3y" :Jy : B. B. R(x, R(x, y) y) =~ =* 3:JJf ": (A (A -~ We will We will see see in in section section 33 that that in in constructive constructive type type theory theory this this axiom axiom isis provable provable because the the theory theory has has enough enough expressions expressions for for functions. functions. because 718 718 R. Constable Set types types aand local set set theories theories 2.10. Set 2.10. n d local Another of of the the most most fundamental fundamental concepts concepts of of modern modern mathematics mathematics isis the the notion notion Another of set set or or class. class. Class Class theory theory arose arose out out of of Frege's Frege 's foundation of foundation for for mathematics mathematics in in Grundgesetze and and in in Principia Principia along along similar similar lines. lines. Even Even before 1900 Cantor Cantor was Grundgesetze before 1900 was creating aa rich rich naive naive set set theory theory which which was axiomatized in in 1908 1908 by by Zermelo Zermelo and and creating was axiomatized improved improved by by Skolem Skolem and and Fraenkel Fraenkel into into modern modern day day axiomatic axiomatic set set theories theories such such as as ZF ZF (Bernays [1958]) [1958]) and and BG BG (Ghdel (G6del [1931]) [1931]) and and Bourbaki's Bourbaki ' s set set theory theory ([1968b]). ([1968b]). (Bernays We could could formulate formulate aa full full blown blown axiomatic axiomatic set set theory theory based based on the type type Set. Set. We on the But type theory into which which ZF But the the type theory of of section section 33 is is an an alternative alternative into ZF can can be be encoded encoded (Aczel [1986]). [1986]). So So instead instead we we pursue pursue aa much much more more modest modest treatment of sets sets along along the the (Aczel treatment of lines of of Principia's Principia 's classes. classes. In In Principia, Principia, given given aa propositional propositional function function ri: whose whose lines range of of significance significance is is the the type type A, A , we we can can form form the the class class 2(r i: ( x) of of those those elements elements range A satisfying satisfying r . We write this this as as {x: {x : A[r A I (x)} . We call this this aa set set type or aa of We write We call type or of A class. two classes classes c~, a, fl (3 we we can can form the usual combinations of of union, union, aa U class. Given Given two form the usual combinations U (3fl,, intersection, universal class, A , and empty class, intersection, aa n M (3 fl,, complement, complement, a ~,, universal class, A, and empty class, r . The judgment associated with a a set set type what one one would The typing typing judgment associated with type is is what would expect. expect. Suppose A is EA Prop , then then Suppose A is a a type type and and P P E A -+ --+ Prop, H by setR setR [-I ~f- aa eE {x { x ':AA II P(x)} P(x)} by H H Ff-- aaEEAA f- P(a) P(a) gH Fm The rule for for using using an assumption about membership is is The rule an assumption about set set membership H, f-I,yy :" {x { x ': AA Ii P(x)} P(x ) } f~G G by by setL setL H, P(y) f-F- G H, yy": A, A, P(y) G As As with with the the other other rules, rules, we we can can choose choose to to name name the the assumption assumption P(y) P(y) by by using using the new uu.. In the justification justification by by setL setL new In Nuprl Nuprl there there is is the the option option to to "hide" "hide" the the proof proof of of P(y).. This hidden version version is is the default in in Nuprl. Nuprl. A A hypothesis hypothesis is is hidden hidden to to prevent prevent P(y) This hidden the default proof object object from This is necessary because because the the proof the from being being used used in in computations. computations. This is necessary the set set membership does not proof P (a) ; so the constructive membership rule, rule, setR, setR, does not keep keep track track of of the the proof P(a); so the constructive elimination elimination rule rule is is i-I, yy": {x" A I] P P(x) J FG by by /setL, IsetL, new new u u H, {x : A (x)}} ,, J f- G : A, [u H, i-I, yy'A, [u": P(y)] P(y)],, J J fJ- G. G. In In local local set set theories, theories, the the concept concept of of the the power power set, set, P(A) 7~(A) is is introduced introduced (c.f. (c.f. Bell Bell [1988], [1988], MacLane MacLane and and Moerdijk Moerdijk [1992]). [1992]). This This type type collects collects all all sets sets built built over over A A and and Prop. Prop. If If A A is is aa type, type, then then P(A) 7~(A) is is aa type. type. In In order order to to express express rules rules about about this this type, type, we we need need to to treat treat the the judgments judgments A AE E Type Type and and P P EE A A -+ ~ Prop Prop in in the the rules. rules. Thus Thus far far we we have have expressed expressed these these judgments judgments only only implicitly, not implicitly, not as as explicit explicit goals, goals, in in part part because because Type Type and and A A -+ --+ Prop Prop are are not not types types themselves, but themselves, but "large "large types." types." However, However, it it makes makes sense sense to to write write aa rule rule such such as as 719 719 Types /~ I~ {x {x:: A A II P(x)} P(x) } E e P(A) P(A) : fI [-I IF- A A E E Type Type fI fI I-~- P P E e A A --+ -+ Prop Prop We We can can also also imagine imagine the the rule rule r I 'I - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - , I H , Xx' 7: :P(A) ' ( A ) IF- 3P 3 P ' :AA --+ -~ Prop. Prop. (X (X = = {x { x ':AAI PI P ( x(x)} ) } in in P(A)) P(A)) . fI, 'I " I I J This introduces introduces the the large large type, type, (A (A --+ --~ Prop) Prop) into into the the type type position. position. Treating Treating this this This concept concept precisely precisely requires requires that that we we consider consider explicit explicit rules rules for for Type Type and and Prop, Prop, espe especially cially their their stratification stratification as as Typei Typei and and ProPi Propi.. We We defer defer these these ideas ideas until until section section 3.7. Let Let us us note note at at this this point point that that the the notion notion of of Prop Prop and and set set types types be be at at the the heart heart of of topos Essentially, the topos theory theory as as explained explained in in Bell Bell [1988]. [1988]. Essentially, the subobject subobject classifier, classifier, n ~ and and T : : 11 --+ --+ n f~,, of of topos topos theory theory is is an an (impredicative) notion of of Prop Prop and and the the subtype subtype T (impredicative) notion of propositions. The of true true propositions. The notion notion of of a a pullback pullback is is used used to to define define subtypes subtypes of of a a type type A "pulling back" A by by "pulling back" a a characteristic characteristic function function P P : :AA --+ -~ Prop Prop and and the the truth truth arrow arrow I P( x(x) T T : : 11 --+ -~ Prop Prop to to get get the the domain domain of of P P, , {x {x:: A AIR ) } .} . A A topos topos is is essentially essentially a a category products (n-ary) subobject classifier category with with Cartesian Cartesian products (n-cry) aa subobject classifier and and power power objects. objects. In In other other words, words, it it is is an an abstraction abstraction of of a a type type theory theory which which has has Prop Prop,, a a collection collection of of true true propositions, propositions, subtypes subtypes and and a a power power type, type, P ( A ) for for each each type. type. The The notion notion of of aa P(A) Grothendieck (c.f. Bell Moerdijk [1992]) Grothendieck topos topos (c.f. Bell [1988], [1988], MacLane MacLane and and Moerdijk [1992]) is is essentially essentially aa predicative concept. It ned in predicative version version of of this this concept. It can can be be defi defined in Martin-Lof Martin-Lhf type type theory theory and and in Nuprl, but beyond the in Nuprl, but that that is is beyond the scope scope of of these these notes. notes. (However, (However, see see section section 5.) 2.11. Quotient 2.11. Q u o t i e n t types types The The equality equality relation relation on on a a type, type, written written ss = = tt in in T T or or ss =T = T tt,, defines defines the the 's referential element element's referential nature. nature. The The semantic semantic models models we we use use in in section section 3.9 3.9 take take a a type type to to be be a a partial partial equivalence equivalence relation relation (per (per)) on on a a collection collection of of terms. terms. T, other Given Given a a type type T, other types types can can be be defined defined from from it it by by specifying specifying new new equality equality relations on relations on the the elements elements of of T T.. For For example, example, given given the the integers integers Z Z,, we we can can define define the the congruence to be congruence integers integers Z//mod Z//mod n n to be the the type type whose whose elements elements are are those those of of Z Z related related by by xx = iff nn divides = y y mod mod n n iff divides (x (x - y) y).. More : N. m More symbolically, symbolically, let let n n I[ m m mean mean that that n n divides divides m m,, i.e., i.e., 3k 3k:N. m = - k k ,* n n.. Then Then = y mod n iff n I (x y) . If rm(x, n) is the remainder when x is divided by n n,, xx = y mod n iff n [ ( x - y). If r m ( x , n) is the remainder when x is divided by then then clearly clearly x x = - y y mod rood n n iff i f f rrm( m ( xx,, nn) ) = -- rm(y, r m ( y , nn)) in in Z Z.. It It is is easy easy to to see see that that xx = - y y mod mod n n is is an an equivalence equivalence relation relation on on Z Z.. In In general, general, this this is is all all we we require require to type. If to form form a a quotient quotient type. If A A is is a a type type and and E E is is an an equivalence equivalence relation relation on on A A,, 720 720 R. Constable then / / E is then AAilE is aa new new type, type, the the quotient quotient of of A A by by E. E . The The equality equality rule rule isis xx == yy in in iff E(x, E(x, y) y) for for x, x, yy in in A. A . Here Here are are the the new new rules. rules. AAll / / EE iff is aa type type iff iff A A isis aa type type and and EE isis an an equivalence equivalence relation relation on on A A AAll / / EE is iI Ff- aa in in AAilE by quotient_member quotienLmember /~ liE by iI~f-a ianin H AA H, A//E, iI, xx :: All E, JJ Ff- b[x] b[x] in in BB by by quotientL quotientL f- b[x] b[x] in in BB iI, x: x : A, A, JJ FH, [-I, b[x/] in in B B iI, xx :: A, A, x' x' :: A, A, E(x, E(x, x'), x'), JJ Ff- b[x] b[x] == b[x'] For For PP to to be be aa propositional propositional function function on on aa type type A, A , we we require require that that when when aa == a' a' in in A A then then P(a) P (a) and and P(a') P(a') are are the the same same proposition. proposition. If If we we consider consider atomic atomic propositions propositions P(x) / / E , then / / E . . The equality of iff xx == tt in in AAilE, then aa == tt in in AAilE The rules rules for for equality of expressions expressions P (x) iff built from elements / / E will guarantee the nature of propositions over will guarantee the functional functional nature of propositions over built from elements of of AAilE A / / EE. . We 3.9 and literature on on Nuprl All We discuss discuss the the topic topic in in detail detail in in section section 3.9 and in in the the literature Nuprl Constable et al. [1986], [1986], Allen Allen [1987b]. [1987b]. Constable et al. The very important many subjects. We have have found it especially The quotient quotient type type is is very important in in many subjects. We found it especially natural in automata theory et al. [1998]), rational rational arithmetic arithmetic and of ( Constable et al. [1998]), and of natural in automata theory (Constable course, for congruences. For For congruence congruence integers we have course, for congruences. integers we have proved proved Fermat's Fermat ' s little little theorem this form: theorem in in this form: Theorem. {x : N lI prime(p)} prime(p)} .. Vx:g//mod Vx : Z/lmod p. p. (x x) (xpp = T h e o r e m . Vp Vp:: (x:N = x) Here mechanism suppresses the type equality when it can Here the the display display mechanism suppresses the type on on equality when it can be be immediately immediately inferred inferred from from the the type type of of the the equands. equands. Equivalence E q u i v a l e n c e classes. classes. It It is is noteworthy noteworthy that that quotient quotient types types offer offer aa computationally computationally tractable tractable way way of of treating treating topics topics normally normally expressed expressed in in terms terms of of equivalence equivalence classes. classes. For For example, example, if if we we want want to to study study the the algebraic algebraic properties properties of of Zllmod Z//mod n n it it is is customary customary to to form form the the set set of of equivalence equivalence classes classes of of Z g where where the the equivalence equivalence class class of of an an element element of these Z/mod nn.. The Zz is is [z] [z] = = {{ii :: Z g l Ii i = = zz mod mod n} n}.. The The set set of these classes classes is is denoted denoted g/mod The algebraic algebraic operations operations are are extended extended to to classes classes by by [z [Zl + [z,]t J + + [Z [z,]2 ] = = [z, + Zz,], 2], [Zl] [zl] * * [Z [z2] = [Z [Zll * * Zz2], etc. 2] = 2 ], etc. All All of of this this development development can can be be rephrased rephrased in in terms terms quotient quotient types. types. We We show show that that + + and a n d ,* are are well-defined well-defined on on Zllmod g//mod nn,, and and the the elements elements are are ordinary ordinary integers integers instead instead of of equivalence equivalence classes. classes. What What changes changes is is the the equality equality on on elements. elements. 2.12. 2.12. Theory Theory structure structure So So far far we we have have introduced introduced aa typed typed mathematical mathematical language language and and aa few few examples examples lists, Cartesian of of specific specific types types and and then then rules-for rules--for N, N, lists, Cartesian products, products, functions, functions, subsets, subsets, and and quotients. quotients. The The possibilities possibilities for for new new types types are are endless, endless, and and we we shall shall see see more more of of Types Types 721 721 them them in in sections sections 33 and and 4. 4. For For example, example, we we could could introduce introduce the the type type Set Set and and explore explore classical computational set theories. We introduce partial classical and and computational set theories. We can can introduce partial objects objects via via the the bar types types that that Constable Constable and and Smith Smith [1993] [1993] developed. developed. As As we we have have seen, seen, we we can can use use bar the Magic rule the Magic rule or or not not or or various various weaker weaker forms forms of of it. it. Some Some choices choices of of rules rules are are inconsistent, inconsistent, e.g. e.g. bar bar types types and and Magic Magic or or the the impredica impredicative Mendler [1988] products on tive f:::. /~ type type of of Mendler [1988] and and dependent dependent products on the the fixed fixed point point rule rule with with all the consistent all types. types. How How are are we we to to keep keep track track of of the consistent possibilities? possibilities? One One method method is is to to postulate postulate fixed fixed theories theories in in the the typed typed logic logic such such as as Heyting Heyting IZF (HA) (c.f. (HA + Arithmetic Arithmetic (HA) (c.f. Troelstra Troelstra [1973]) [1973]) or or Peano Peano Arithmetic Arithmetic (HA + Magic) Magic) or or IZF (c.f. Moerdijk [1995] (c.f. Beeson Beeson [1985] [1985],, Friedman Friedman and and Scedrov Scedrov [1983] [1983],, Joyal Joyal and and Moerdijk [1995],, Moerdijk Moerdijk and and Reyes Reyes [1991]) [1991]) or or Intuitionistic Intuitionistic Type Type Theory Theory (ITT) (ITT) or or Higher Higher Order Order Logic Logic (HOL) (HOL).. We We rely rely on on a a community community of of scholars scholars to to establish establish the the consistency consistency of of various various collections collections of axioms. axioms. Books Books like like Troelstra Troelstra [1973] study relationships relationships between between dozens dozens of of these these of [1973] study theories. The The space space of of them them is is very very large. large. theories. Another the "tree Another possibility possibility is is to to explore explore the "tree of of knowledge" knowledge" formed formed by by doing doing nitions and mathematics mathematics in in various various contexts contexts determined determined by by the the defi definitions and axioms axioms used used for any any result. result. We We can can think think of of definitions definitions and and axioms axioms as as establishing establishing contexts. contexts. for N.G. N.G. de de Bruijn Bruijn [1980] [1980] has has proposed proposed aa way way to to organize organize this this knowledge, knowledge, including including derivation derivation of of inconsistency inconsistency on on certain certain paths. paths. Essentially Essentially de de Bruijn Bruijn defined defined typed typed mathematical mathematical languages, languages, PAL, PAL, Aut-68, Aut-68, Aut AutQE, AutAut-II, which were used for for writing definitions and and axioms. axioms. 20 2~ He He proposed proposed aa QE, II , which were used writing definitions logical organizing definitions, definitions, axioms logical framework framework for for organizing axioms and and theorems theorems into into books. books. We We will will explore explore these these typed typed languages languages in in the the next next section. section. They They are are more more primitive primitive than than our our typed typed logic. logic. The The apparatus apparatus of of Automath Automath is is completely completely formal; formal; it it is is aa mechanism mechanism whose whose meaning meaning is is to to be be found found completely completely in in its its ability ability to to organize organize information information and and classify classify it content. Extending mathematics being it without without regard regard for for content. Extending this this attitude attitude to to the the mathematics being expressed expressed leads leads to to the the formalist formalist philosophy philosophy of of mathematics mathematics espoused espoused by by Hilbert Hilbert [1926]. with Principia [1926]. This This is is de de Bruijn's Bruijn's view view in in fact, fact, and and it it surely surely contrasts contrasts with Principia which which found meaning in truths written written into found its its meaning in the the logical logical truths into aa fixed fixed foundational foundational theory. theory. It will contrast Martin-Lof view, view, the It will contrast as as well well to to the the Martin-Lhf view, Girard's Girard's [1987] [1987] view, the views views of of Coquand Huet in Coq and expressed to Coquand and and Huet in Coq and my my own own view view (as (as expressed to aa large large extent extent in in Nuprl) Nuprl) in in which which the the logical logical framework framework is is organized organized to to express express computational computational meaning. meaning. It It is is noteworthy noteworthy that that the the three three influential influential philosophical philosophical schools-Formalism, schools--Formalism, Logicism, Logicism, and and Intuitionism, Intuitionism, can can be be characterized characterized rather rather sharply sharply in in this this setting setting (and (and coexist!). coexist!). An An Automath Automath book book is is a a sequence sequence of of lines. lines. A A line line has has four four parts parts as as indicated indicated in in Table Table 2. 2. Each Each line line introduces introduces aa unique unique identifier identifier which which is is either either aa primitive primitive notion, notion, PN, or PN, or a a block block opener opener or or is is defined. defined. The The category category part part provides provides the the grammatical grammatical category; category; type type is is aa built-in built-in category, category, defined defined types types like like nat nat are are another. another. The The lines lines form form two two structures, structures, one one the the linear linear order order and and the the other other aa rooted rooted tree. tree. 2° 2~"Automath is a language language which which we claim claim to be suitable for expressing expressing very large large parts of mathematics, in such a way that the correctness correctness of the mathematical contents contents is guaranteed as as as the rules of the grammar are obeyed." obeyed." de Bruijn [1980). [1980]. long as 722 722 R. Constable Constable indicator indicator identifier identifier definition definition category category PN 0 nat type PN nat type nn nat 0 nat real PN type real PN type 0 xX n real real - - Table 2: Sequence Table 2: Sequence of of lines lines The nodes nodes of of the the tree tree are are identifiers, identifiers, x x,, and and the the edges edges are are from from x x to to the the indicator indicator The identifier part. of line having of the the line having x x as as its its identifier part. The The complete complete context context of of x x is is the the list list of of the root. line uses indicators indicators from from x x back back to to the root. So So each each line uses as as its its indicator indicator the the last last block block opener opener in in its its context. context. When When the the definition definition and and category category components components are are included included with with x x,, the the result result is is what what de de Bruijn Bruijn calls calls the the tree tree of of knowledge. knowledge. Nuprl has has a a similar similar structure structure to to its its knowledge knowledge base, base, called called a a library. library. A A library library Nuprl consists consists of of lines. lines. Each Each one one is is uniquely uniquely named named by by an an identifier. identifier. These These can can include include the the equivalent equivalent of of block block openers, openers, called called theory theory delimiters delimiters (begin_thyname, (begin_thyname, end_thyname) end_thyname).. The The library library is is organized organized by by a a dependency dependency graph graph which which indicates indicates the the logical logical order order among delimiters) . Unlike among theories theories (the (the lines lines between between delimiters). Unlike in in Automath, Automath, the the theory theory structure structure is is a a directed directed acyclic acyclic graph graph (dag). (dag). Theories Theories can can also also be be linked linked to to aa file file system system or or a a database database which which provides provides additional additional "nonlogical" "nonlogical" structuring. structuring. The The Nuprl Nuprl 55 system system also also provides provides a a structured structured library library with with mechanisms mechanisms to to control control access to theories. collecting access to theories. There There are are two two modes modes of of accessing accessing information. information. One One is is by by collecting axioms, definitions, and theorems into axioms, definitions, and theorems into controlled controlled access access theories. theories. These These theories theories can can only use specific rules root. Each only use the the specific rules and and axioms axioms assembled assembled at at its its root. Each type type such such as as N N or or T is is organized organized into into a a small small theory theory consisting consisting of of its its rules. rules. 21 21 More More complex complex theories theories SS x T are built built by by collecting axioms. 22 22 We We will be specifying specifying certain certain important important theories theories are collecting axioms. will be later. One later. One of of them them is is Nuprl Nuprl 4, 4, the the fixed fixed logic logic in in the the Nuprl Nuprl 4.2 4.2 release. release. Another Another theory theory could Smith [1993]) could be be Nuprl Nuprl 4_bar, 4_bar, the the theory theory with with partial partial objects objects (Constable (Constable and and Smith [1993]) or uIZF, the or N NuIZF, the formulation formulation of of IZF IZF in in type type theory. theory. Another library we call free Another way way to to use use the the library we might might call free access. access. A A user user can can prove prove theorems rules whatsoever, Once aa theorem theorems using using any any rules whatsoever, even even inconsistent inconsistent collections. collections. Once theorem collection of is is proved, proved, the the system system can can define define its its rooLsystem, root_system, the the collection of all all rules rules and and definitions used and prove it. The ystem determines definitions used to to state state and prove it. The root...s root_system determines the the class class of of theories theories into into which which the the result result can can be be "planted." "planted." 2.13. Proofs 2.13. P r o o f s as as objects objects The The notion notion of of proof proof plays plays a a fundamental fundamental role role in in logic logic as as we we have have seen seen here. here. ' s proof proofs, and Hilbert Hilbert's proof theory theory is is a a study study of of proofs, and for for philosophical philosophical reasons reasons he he conceived conceived 21The 21The associated tactics are attached attached as well, well, see Hickey Hickey [1996b,1997]. [1996b,1997]. 22The 22The associated associated tactics can also also enforce enforce global global constraints on the theory such such as "decidable "decidable type checking." checking." 723 723 Types 3 Given of it it as as a a constructive constructive theory, theory, and and a a metatheory. metatheory. 223 Given the the central central role role of of proofs proofs of in all of mathematics, it it is not a a great great leap leap to begin thinking thinking about about proofs proofs as as in all of mathematics, is not to begin mathematical mathematical objects objects with with the the same same "reality" "reality" as as numbers. numbers. This This viewpoint viewpoint is is central central to intuitionistic and to intuitionistic and constructive constructive mathematics, mathematics, and and it it seems seems to to be be coherent coherent classically classically as as well. well. De De Bruijn Bruijn designed designed the the Automath Automath formalisms formalisms around around notion notion of of formal formal proofs proofs as objects, and ordinary ordinary objects objects such such as as functions functions could could depend on proofs. proofs. In In order order to to as objects, and depend on treat what what was was called called classical classical mathematics mathematics he he had had to to add add a a principle principle of of irrelevance irrelevance treat of proofs. 2244 However, ofproofs. However, to to bring bring proof proof expressions expressions fully fully into into the the mathematics mathematics as as objects objects means more more than than allowing allowing them them into into the the language. language. As As the the proof proof irrelevance irrelevance principle principle means shows, they they can can be be regarded regarded as as part part of of the the underlying underlying linguistic linguistic apparatus. apparatus. 225 To shows, 5 To make make proofs proofs explicit explicit objects objects with with a a referential referential character, character, we we must must define define equality equality on on nitional them (the (the kind kind of of equality equality called called book book equality equality in in Automath Automath as as opposed opposed to to defi definitional them equality equality which which holds holds for for all all terms terms whether whether referential referential or or not) not).. There objects. We There are are two two sources sources to to guide guide the the discovery discovery of of equality equality rules rules for for proof proof objects. We can turn turn to intuitionistic mathematics mathematics and and its its semantics semantics for for the the logical logical operators operators or or we can to intuitionistic we can look look to to proof proof theory and the the reduction reduction (or (or normalization normalization rules). rules). Neither Neither account account can theory and is classically conceived is definitive definitive for for classically conceived mathematics. mathematics. In In the the case case of of using using intuitionistic intuitionistic reasoning reasoning as as a a guide, guide, we we must must handle handle classical classical rules, rules, such such as as contradiction, contradiction, or or classical classical axioms like like the the law law of of excluded excluded middle middle "magic" "magic".. There There are are various various ways ways to to approach approach axioms this promising results results (Allen (Allen et al. [1990], this with with promising et al. [1990], Murthy Murthy [1991]' [1991], Girard Girard [1991]). [1991]). The The subject subject is is still still very very active. active. Another normalization theorems Another approach approach is is suggested suggested by by the the normalization theorems for for classical classical and and constructive deduction systems, constructive logics logics natural natural deduction systems, or or N-systems N-systems (due (due to to Prawitz Prawitz [1965]), [1965]), and and the the body body of of results results on on cut cut elimination elimination in in the the sequent sequent calculi, calculi, or or L-systems L-systems (arising (arising from from Gentzen Gentzen [1935]). [1935]). Unfortunately, Unfortunately, the the results results give give somewhat somewhat conflicting conflicting notions notions of of proof proof equality equality (c.f. (c.f. Zucker Zucker [1974,1977]' [1974,1977], Ungar Ungar [1992]) [1992]).. It It is is perhaps perhaps premature premature to to suggest suggest the the appropriate appropriate classical classical theory, theory, so so instead instead we we will will sketch sketch the the constructive constructive ideas details to ideas and and leave leave the the technical technical details to section section 33 where where we we will will explore explore carefully carefully Martin-Lars Martin-Lhf's interpretation interpretation in in which which the the computational computational content content of of a a proof proof is is taken taken as as the the object. object. Another Another prerequisite prerequisite to to treating treating proofs proofs as as objects objects is is that that we we understand understand the the domain domain of of significance, significance, the the type type of of assertions assertions about about proofs. proofs. This This is is another another point point that that is is not not entirely entirely clear. clear. For For instance, instance, the the views views of of Kreisel Kreisel [1981]' [1981], Scott Scott [1976], [1976], and and Tait Martin-Lof [1982,1983] Girard [1987]. Tait [1967,1983] [1967,1983] differ differ sharply sharply from from those those of of Martin-Lhf [1982,1983] and and Girard [1987]. One One of of the the key key points points is is whether whether we we understand understand a a proof proof p p as as a a proof proof of of a a proposition proposition P P,, p p proves proves P P,, or or whether whether provability provability is is a a relation relation on on proofs proofs so so that that Proves(p, Proves(p, P) P))) is is the the appropriate appropriate relationship. relationship. In In the the latter latter case case there there arises arises the the part of Hilbert's Hilbert's Program Programfor a formal foundation foundation of mathematics. 23That it had to be so was part Classical Classical parts parts of of mathematics mathematics were were to to be be considered considered as as ideal ideal elements elements ultimately ultimately justified justified by by constructive means. 24",,... proposition are 24 . . . we extend the language by proclaiming that proofs of one and the same proposition always definitionally equal. This extra rule was called 'proof irrelevance' .... irrelevance'...." 25This is quite different from taking them to be metamathematical metamathematicalobjects as is done in proof theory . . . a theory that theory.., that could be formalized in Automath. " 724 724 R. Constable Constable R. danger of of an an infinite infinite regress regress since since we we will will require require aa proof proof p~ p' of of (p (p Proves Proves P). P) . danger At some some level level itit seems seems that that provability provability must must be be aa basic basic judgment, judgment, like like the the typing typing At T. judgment tt EE T. judgment Ifwe we start start with with the the view view of ofthe the relationship relationship pp proves proves PP as as aa typing typing judgment, judgment, then then If we are are led led to to the the view view that that the the type type of of aa proof proof isis the the proposition proposition that that itit proves. proves. Thus Thus we propositions play play the the role role of of types types according according to to the the propositions_as_types propositions_as_types principle. principle. propositions This principle principle isis designed designed into into Automath Automath (but (but can can be be regarded regarded as as "linguistic"), "linguistic" ), This and itit is is the the core core of of both both Martin-Lhf Martin-LOf type type theory theory (Martin-Lhf (Martin-LOf [1982,1984,1983], [1982,1984,1983] ' and Nordstrom, Petersson Petersson and and Smith Smith [1990]) [1990]) and and Girard Girard type type theory theory (Stenlund (Stenlund [1972], [1972] ' Nordstrom, Constable et et al. al. [1986], [1986] , Girard, Girard, Taylor Taylor and and Lafont Lafont [1989]). [1989]). According According to to this this principle, principle, Constable proposition PP is is provable provable (constructivists (constructivists would would say iff there there is is aa proof proof pp whose whose aa proposition say true) true) iff type is is PP, , that that is is type iff for p, t-p f- p EE PP for some some p, f- PP iff Indeed, on on this this interpretation interpretation and and recognizing recognizing that that proof proof expressions expressions pp denote denote proofs, proofs, Indeed, f- PP by as just way of of writing writing we can see see the the sequent notation fI we can sequent notation H ~by p p as just another another way fI f- p E P . [-I~-pEP. The P by form can can be be considered is The /~ fI ~f- P by pp judgment judgment form considered implicit. implicit. Attention Attention is focused on P and the main concern concern is is that that there is some The f- pp EE PP focused on P , , and the main there is some inhabitant. inhabitant. T h e /fI ~ Fform is explicit, attention is is focused focused on on the the actual actual proof. rules could could all all be be form is explicit, and and attention proof. The The rules presented presented in in either either implicit implicit (logical) (logical) form form or or explicit explicit (type (type theoretic) theoretic) form. form. Consider Consider the is an the VL VL and and VR VR rules, rules, for for example. example. Here Here is an implicit implicit form. form. j aj Y . g[y]) H , fj' V: Vx x ' A: A. . P(x, P(x,)J~G by by VLU VL(f;a;y. g[y]) )J f- G H, Vx": A. P(x) P(x),, J J,, yy": P(a) P(a) fF- G G by by g[y] g[y] H, jf :"Vx fI f- A by HF-A by aa _ H f-F- Vx Vx": A. A. P(x) P(x) by by VR(x. VR(x. p[x]) p[x]) H H, H, xx": A A f-~- P(x) P(x) by by p[x] p[x] Here Here is is the the explicit explicit form form of of the the VL VL rule. rule. i-I, jf": Vx Vx": A. A. P(x), P(x), J J f-~-VLU VL(f;j aj a; y. y. g[y]) g[y]) EE G G fI, f-I, jf'Vx" A. P(x), P(x), J 2,, yy": P(a) P(a) f-F- g[y] g[y] EE G G fI, : Vx : A. fI fI f-F- aa EEAA We We will will discover discover in in section section 3.11 3.11 that that there there is is aa reasonable reasonable notion notion of of reduction reduction on on proof proof expressions expressions (which (which can can either either be be considered considered as as computation computation or or definitional definitional equality) equality) and and that that this this gives gives rise rise to to aa minimal minimal concept concept of of equality equality on on proofs proofs that that is is sufficient sufficient to to give give them them the the status status of of mathematical mathematical objects. objects. 2.14. Heyting's H e y t i n g ' s semantics semantics ' s interpretation Here Here is is Heyting Heyting's interpretation of of the the judgment judgment pp proves proves P P.. 11.. For atomic P we cannot base the explanation on propositional For atomic P we cannot base the explanation on propositional components components of of ' P because there aren t any. But it might depend on an analysis P because there aren't any. But it might depend on an analysis of of terms terms and and 725 725 Types their type type which which could could be be compound. compound. their We recognize recognize certain certain atomic atomic propositions, propositions, such such as as 00 = - 00 in in N N as as "atomically "atomically We true." That T h a t is, is, the the proofs proofs are are themselves themselves atomic, atomic, so so the the proposition proposition is is an an true." axiom. In In the the case case when when the the terms terms are are atomic atomic and and the the type type is is as as well, well, there there is is axiom. little left left to to analyze. analyze. But But other other atomic atomic propositions propositions can can be be reduced reduced to to these these little axioms by by computation computation on on terms, terms, say say 55 *900 = = 11 *900 in in N N.. axioms Some atomic atomic propositions propositions are are proved proved by by computation computation on on terms terms and and proofs. proofs. Some For example, example, suc(suc(suc(O))) suc(suc(suc(O))) = = suc(suc(suc(O))) suc(suc(suc(O))) in in N N is is proved proved by by thrice thrice For iterating the the inference inference rule rule suceq suc_eq iterating nn ==mm suc(n) = suc(m) = We might might take take the the object object suc_eq(suc_eq(suceq(zero_eq))) suc_eq(suc_eq(suc_eq(zero_eq))) as as aa proof proof expression expression We for this this equality. equality. On On the the other other hand, in such such aa case case we we can can just as well well consider consider for hand, in just as the proof proof to to be be a a computation computation procedure on the the terms terms whose whose result result is is some some the procedure on token indicating indicating success success of of the the procedure. procedure. token In general, general, the the proofs proofs of of atomic atomic propositions propositions depends depends on on an an analysis analysis of of the the In terms involved involved and and the the underlying underlying type type and and its its components. components. For For example, example, terms = bb in in AI A / /IEE might might involve involve a a proof proof the the proposition E(a, b).. aa = proposition E( a, b) So we we cannot cannot say say in in advance advance what what all all the the forms forms of of proof proof are are in in these these cases. cases. As As a a So general guide, guide, in in the the case case of of completely completely atomic atomic propositions propositions such such as as 00 = - 00 in in N N general in in which which the the terms terms and and type type are are atomic, atomic, we we speculate speculate that that the the proof proof is is atomic atomic as as 6 well. proofs we have aa special such as as axiom. axiom. 22~ well. For For these these atomic atomic proofs we might might have special symbol symbol such 2. proof of Q is is aa pair proves PP and and qq proves proves Q. of PP & &Q pair (/9, (p, q) q) where where pp proves Q. 2. AA proof proof of of P P and To be 33.. AA proof P vV QQ isis either either pp or or qq where where pp proves proves P and qq proves proves Q. Q . To be more explicit explicit we we say say it is aa pair where if if the designates P P then (tag, e) e) where the tag tag designates then ee is is more it is pair (tag, pp and Q,, then and ifif itit designates designates Q then ee is is q. q. 4. A A proof proof of P =~ Q is is a a procedure maps any proof pp of p),, a a 4. of P *Q procedure ff which which maps any proof of PP to to ff ((p) proof of of Q. proof Q. A proof 3x:: A A.. P[x] is aa pair where a a eE A and pp proves 55.. A proof of of 3x P[x] is pair (a, (a, p) p) where A and proves P[a]. P[a] . "Ix : A. P[x] P[x] is procedure ff taking taking any element aa of of A A to to aa proof 6. AA proof 6. proof of of Vx:A. is a a procedure any element proof f (a) of f(a) of P[a]. P[a] . Note, we we treat treat --,P as PP ==>_1_, *1.. , so so these these definitions definitions give give an an account account of of negation, negation, Note, P as but approaches, such but there there are are other other approaches, such as as Bishop Bishop [1967]. [1967]. We will will see see aa finer finer analysis analysis of this definition definition in the section section on type theory; theory; We of this in the on type there following following Martin-LSf Martin-Lof [1982] [1982] and and Tait Tait [1967,1983], [1967,1983], we will distinguish distinguish between between there we will canonical proof expressions )) canonical proof expressions and and non-canonical non-canonical ones ones such such as as add(suc(O); add(suc(O) ; suc(suc(O) suc(suc(O))) (which to aa canonical In this (which reduces reduces to canonical one one suc(suc(suc(O)))). suc(suc(suc(O)) ) ) . In this more more refined refined analysis analysis 26In Martin-LSf and in Nuprl atomic formulas are are reduced reduced to a token Martin-Lof type theory and Nuprl all proofs of atomic (axiom in Nuprl). Information that that might be needed from the proof is kept only at at the the metalevel. 726 726 R. Constable Constable we canonical proof we say say that that the the above above clauses clauses define define the the canonical canonical proofs, proofs, e.g. e.g. a a canonical proof of of P pair (p, L( '* R(x. (x, q)); proof of P & &Q Q is is a a pair (p, q) q/,, but but '* =~L(=~ R(x.(x, q/); p) P) is is a a noncanonical noncanonical proof of P P & &Q Q which "normalize" the which reduces reduces to to (p, (p, q) q} when when we we "normalize" the proof. proof. Although this this is is aa suggestive suggestive semantics semantics of of both proofs and and propositions, propositions, several several Although both proofs questions remain. questions remain. Given Given a a proposition proposition P, P, can can we we be be sure sure that that all all proofs proofs have have the the structure structure suggested suggested by by this this semantics? semantics? Suppose Suppose P P & &Q Q is is not not proved proved by by proving proving P P and decomposing an and proving proving Q Q but but instead instead by by a a case case analysis analysis or or by by decomposing an implication implication and and then existential statement, statement, etc.; then decomposing decomposing an an existential etc.; so so if if tt proves proves P P & &Q Q,, do do we we know know tt is aa pair? pair? is If objects, then relation on If proofs proofs are are going going to to be be objects, then what what is is the the right right equality equality relation on them? them? If tt proves proves P P&Q then is is tt at at least least equal equal to to aa pair pair (p, (p, q) q/?? What What is is the the right right equality equality &Q then If Q?? How on on propositions? propositions? If If P P - Q Q and and p p proves proves P P does does p p prove prove Q How can can we we make make sense sense structure of of Magic as proof object? object? It of Magic as a a proof It is is aa proof proof of of P P v V ..,p -~P yet yet it it has has no no structure of the the kind kind Heyting Heyting suggests. suggests. We We will will see see that that the the type type theories theories of of the the next next section section provide provide just just the the right right tools tools for for answering answering these these questions. questions. = 3.. Type T y p e theory theory 3 3.1. 3.1. Introduction Introduction Essential E s s e n t i a l features. features. In In this this section section II want want to to give give aa nontechnical nontechnical overview overview of of the the subject will discuss subject II am am calling calling type type theory. theory. II will discuss these these points: points: • 9 It It is is a a foundational foundational theory theory in in the the sense sense of of providing providing definitions definitions of of the the basic basic notions logic, mathematics, notions in in logic, mathematics, and and computer computer science science in in terms terms of of aa few few primitive primitive concepts. concepts. • 9 It It is is aa computational computational theory theory in in the the sense sense that that among among the the primitive primitive built-in built-in concepts concepts are are notions notions of of algorithm, algorithm, data data type, type, and and computation. computation. Moreover Moreover these notions are these notions are so so interwoven interwoven into into the the fabric fabric of of the the theory theory that that we we can can discuss discuss the the computational computational aspects aspects of of every every other other idea idea in in the the theory. theory. (The (The theory theory also also provides mathematics, as provides a a foundation foundation for for noncomputational noncomputational mathematics, as we we explain explain later.) later.) • 9 It It is is referential referential in in the the sense sense that that the the terms terms denote denote mathematical mathematical objects. objects. The The referential referential nature nature of of aa term term in in a a type type T T is is determined determined by by the the equality equality relation relation associated with T, relation is associated with T, written written s = tt in in T T.. The The equality equality relation is basic basic to to the the meaning meaning of of the the type. type. All All terms terms of of the the theory theory are are functional Junctional over over these these equalities. equalities. s= • 9 When When properly properly formalized formalized and and implemented, implemented, the the theory theory provides provides practical practical tools for expressing, performing, and reasoning about computation tools for expressing, performing, and reasoning about computation in in all all areas areas of of mathematics. mathematics. A three features A detailed detailed account account of of these these three features will will serve serve to to explain explain the the theory. theory. Under Understanding them standing them is is essential essential to to seeing seeing its its dynamics. dynamics. In In aa sense, sense, the the axioms axioms of of the the theory theory serve serve to to provide provide a a very very abstract abstract account account of of mathematical mathematical data, data, its its transformation transformation by by effective procedures, and effective procedures, and its its assembly assembly into into useful useful knowledge. knowledge. II summarized summarized my my ideas ideas on this this topic topic in in Constable Constable [1991]. [1991]. on Types Types 727 727 L a n g u a g e and a n d logic. logic. In In aa sense, sense, the the theory theory is is logic logic free. free. Unlike Unlike our our account account of of Language typed logic, logic, we we do do not not start start with with propositions propositions and and truth. truth. Instead Instead we we begin begin with with more more typed elementary parts parts of of language, language, in in particular, particular, with with aa theory theory of of computational computational equality equality elementary of terms terms (or (or expressions). expressions). In In Principia Principia these these elementary elementary ideas ideas are are considered considered as as part part of of the the meaning meaning of of propositions. propositions. We We separate separate them them more more clearly. clearly. We We examine examine the the of mechanism of of naming naming and and definition definition as as the the most most fundamental fundamental and and later later build build upon upon mechanism this an an account account of of propositions propositions and and truth. truth. this This analysis analysis of of language language draws draws on on the the insights insights of of Frege, Frege, Russell, Russell, Brouwer, Brouwer, This Wittgenstein, Church, Church, Curry, Curry, Markov, Markov, de de Bruijn, Bruijn, Kolmogorov, Kolmogorov, and and Martin-Lof, Martin-Lhf, and and Wittgenstein, it draws draws on on technical technical advances advances made made by by numerous numerous computer computer scientists scientists and and logicians. logicians. it We can can summarize summarize the the insights insights in in this this way. way. The The notion notion of of computability computability is is grounded grounded in in We rules for for processing processing language language (Church (Church [1940]' [1940], Curry Curry and and Feys Feys [1958] [1958],, Markov Markov [1949]) [1949]).. rules In particular, particular, they they can can bbee organized organized as as rules rules for for aa basic basic (type (type free) free) equality equality on on In 's theory expressions closely closely related related to to Frege Frege's theory of of identity identity in in [1903]. [1903]. The The rules rules explain explain expressions when two two expressions expressions will have the the same same reference if they they have have any any reference. reference. (We (We call call when will have reference if these computation computation rules, rules, but but they they could could also also be be considered considered simply simply as as general general rules rules of of these definitional equality equality as as in in Automath.) Automath.) De Bruijn showed showed that that to to fully fully understand understand the the definitional De Bruijn definitional rules, we we need need to to understand understand how how expressions expressions are are organized organized into into contexts contexts defi nitional rules, in tree of of knowledge knowledge as as we we discussed discussed in section 2.12. 2.12. in aa tree in section Frege not not only only realized realized the the nature nature of of identity identity rules, rules, but but he he explained explained that that the the Frege very notion notion of of an an object object (or (or mathematical mathematical object) object) depends depends on on rules rules for for equality equality of of very expressions which which are are intended intended to to denote denote objects. objects. The The equality equality rules rules of of aa theory theory expressions serve serve to to define define the the objects objects and and prepare prepare the the ground ground for for aa referential referential language, language, one one in in which the the expressions expressions can can be be said said to to denote denote objects. objects. which Frege also also believed believed that that the equality rules were not arbitrary but but expressed Frege the equality rules were not arbitrary expressed the the primitive truths about about abstract such as as numbers numbers and and classes. We build primitive truths abstract objects objects such classes. We build on on Brouwer's understanding of of the numbers N especially Brouwer's theme theme that that an an understanding the natural natural numbers N is is an an especially clear place to to build build as as possible with them. Here clear place to begin, begin, and and we we try try to as much much as possible with them. Here the insights insights of Brouwer [1975] van Stigt how to connect intuitions intuitions the of Brouwer [1975] (see (see van Stigt [1990]) [1990]) show show how to connect about number to equality of of expressions. expressions. Brouwer Brouwer shows shows that about number to the the rules rules for for equality that the the idea idea of natural natural number number and numbers are are meaningful meaningful because they arise arise from of and of of pairing pairing numbers because they from mental operations. operations. Moreover, Moreover, these these are abilities needed needed to the mental are the the same same abilities to manipulate manipulate the language of of expressions expressions (see [1988]). 227 language (see Chomsky Chomsky [1988]). 7 So and Brouwer (and unlike unlike formalists), formalists), we we understand understand type theory to to So like like Frege Frege and Brouwer (and type theory be referential, that that is, theory is about mathematical mathematical objects, be referential, is, the the theory is about objects, and and the the meaningful meaningful expressions expressions denote denote them. them. Following theory is created by by classifying Following Russell, Russell, we we believe believe that that aa referential referential theory is created classifying Not every is meaningful, meaningful, for for example, example, school school expressions every expression expression is expressions into into types. types. Not children is sometimes say say that % is not. not. We We sometimes that the the meaningful meaningful expressions expressions children know know that that 0/0 are are those those that that refer refer to to mathematical mathematical objects, objects, but but this this seems seems to to presuppose presuppose that that we we 27For Brouwer this language is required by an individual only because of the limits and and flaws in his or her her mental powers. powers. But for our our theory, theory, language language is essential to the communication among agents (human and artificial or otherwise) needed to establish public knowledge. 728 728 R. Constable know know what what such such objects objects are. are. So So we we prefer prefer to to say say that that the the task task of of type type theory theory is is to to provide the the means means to to say say when when an an expression expression is is meaningful. meaningful. This This is is done done by by classifying classifying provide expressions expressions into into types. types. Indeed Indeed to to define define a a type type is is to to say say what what expressions expressions are are of of that that type. type. This This process process also also serves serves to to define define mathematical mathematical objects. objects. 22s8 Martin-Lof suggested particular way Martin-LSf suggested a a particular way of of specifying specifying types types based based on on ideas ideas devel develFirst designate oped by oped by W. W. W. W. Tait Tait [1967,1983]. [1967,1983]. First designate the the standard standard irreducible irreducible names names for for elements belong to elements of of a a type, type, say say tl, t l , tt22 ,, ... ., . belong to T T.. Call Call these these canonical canonical values. values. Then Then based on on the the definition of evaluation, evaluation, extend extend the the membership membership relation relation to to all all t' t ~ such such based definition of that t' t ~ evaluates evaluates to to a a canonical canonical value value of of T; we say say that that membership membership is is extended extended by by T; we that pre-evaluation. pre-evaluation. Level [1908] observed L e v e l restrictions. r e s t r i c t i o n s . Russell Russell [1908] observed that that it it is is not not possible possible to to regard regard the the collection of collection of all all types types as as a a type type itself. itself. Let Let Type Type be be this this collection collection of of all all types. types. So So Type Type is not not an an element element of of Type. Type. Russell Russell suggested suggested schemes schemes for for layering layering or or stratifying stratifying these these is Set. The "inexhaustible "inexhaustible concepts" concepts" like like Type Type or or Proposition Proposition or or Set. The idea idea is is to to introduce introduce notions notions of of types types of of various various levels. levels. In In our our theory theory these these levels levels are are indicated indicated by by level level ypei . They will be indexes indexes such such as as T Typei. They will be defined defined later. later. Architecture A r c h i t e c t u r e of o f type t y p e theory. t h e o r y . What W h a t we we have have said said so so far far lays lays out out a a basic basic structure structure for linguistic material for the the theory. theory. We We start start with with a a class class of of terms. terms. This This is is the the linguistic material needed needed for communication. We for communication. We use use variables variables and and substitution substitution of of terms terms for for variables variables to to express x, y, s, tt be express relations relations between between terms. terms. Let Let x, y, zz be be variables variables and and s, be terms. terms. We We of variable denote of term denote the the substitution substitution of term ss for for all all free free occurrences occurrences of variable x x in in tt by by t[s/x] t[s/x]. . The details of The details of specifying specifying this this mechanism mechanism vary vary from from theory theory to to theory. theory. Our Our account account is is conventional conventional and and general. general. Substitution Substitution introduces introduces a a primitive primitive linguistic linguistic relationship relationship among among terms terms which which is is used used to to define define certain certain basic basic computational computational equalities equalities such such as as ap(>.(x.b); ap(A(x.b); a) a) = = bra/xl b[a/x].. There There are are other other relations relations expressed expressed on on terms terms which which serve serve to to define define computation. computation. We We write write these these as as evaluation evaluation relations relations tt evals_to evals_to t' t ~ also also written written tt .} $ t'. t ~. Some Some terms terms denote denote types, types, e. e. g. g. N N denotes denotes the the type type of of natural natural numbers. numbers. There There are are type build new Cartesian product type forming forming operations operations that that build new types types from from others, others, e. e. g. g. the the Cartesian product T1 x x T T22 of of T T1l and and T T2. Corresponding to to a a type type constructor constructor like like x • there there is is usually usually a a Tl 2 . Corresponding constructor on tl EE T1, Tb tt22 EE T T2 . By constructor on elements, elements, e. e. g. g. if if tl T22 then then pair(t pair(t1;1 ; tt2) T1 x • T2. By the the 2 ) EE Tl Tait condition above Tait pre-evaluation pre-evaluation condition above t't ~evals_to evals_to pair(tl pair(tl;; tt2) 2) tt'' Ee T1 T~ x • T T22 28The 2SThe interplay between expressions and objects has seemed confusing to readers of constructive type theory. In In my opinion this arises mainly from the fact that that computability considerations cause us to say more about the underlying language than is typical, but the same relationship exists in any formal account of mathematics. Types Types 729 729 Part of of defining defining aa type type is is defining defining equality equality among among its its numbers. numbers. This This is is written written Part as ss = - tt in in T. T. The The idea idea of of defining defining an an equality equality with with aa type type produces produces aa concept concept as like Bishop's Bishop's sets sets (see (see Bishop Bishop [1967], [1967], Bishop Bishop and and Bridges Bridges [1985]), [1985]), that that is is Bishop Bishop like [1967,p.63] said said ""... set is is defined defined by by describing describing what what must must be be done done to to construct construct an an [1967,p.63] . . . aa set element element of of the the set, set, and and what what must must be be done done to to show show that that two two elements elements are are equal." equal." The basic basic forms forms of of judgment judgment in in this this type type theory theory are are The t is is aa term term •9 t This is is aa simple simple context-free context-free condition condition on on strings strings of of symbols symbols that that can can be be checked checked This by aa parser. parser. We We stress stress this this by by calling calling these these readable readable expressions. expressions. by • 9T T is_a is_a type type We also also write write T T E E Type Type and and prefer prefer to to write write capital capital letters, letters, S, S, T, T, A, A, B B for for types. types. We This relationship relationship is is not not decidable decidable in in general general and and cannot cannot be be checked checked by by aa parser. parser. This There are are rules rules for for inferring inferring typehood. typehood. There tE ET T (type membership membership or or elementhood) elementhood) •9 t (type This judgement judgement is is undecidable undecidable in in general. general. This s= = tt in in T T (equality on on T) T) •9 s (equality This judgement judgement is is also also undecidable undecidable generally. generally. This Inference m e c h a n i s m . Since Since Post Post it it has has been been the the accepted accepted practice to define define the the Inference mechanism. practice to class of of formulas formulas and and the the notion notion of of proof proof inductively. inductively. Notice Notice our our definition of formula formula class definition of in section section 2.4, 2.4, also, also, for for example, example, a a Hilbert Hilbert style style p proof is a a sequence sequence of of closed closed formulas formulas roo/ is in F1 F 1,, . . . , , Fn Fn such such that that Fj Fi is is an an axiom axiom or or follows follows by by aa rule rule of of inference inference from from F Fj, Fkk for for j, F < i, i, k k < < ii.. A A typical inference rule rule is is expressed expressed in form of of hypotheses hypotheses above jj < typical inference in the the form above aa horizontal line with with the the conclusion conclusion below below as as in in modus modus poneus. poneus. horizontal line • . • A , AA~ => B B A, B B This presentation of that an an element element This definition definition of of aa proof proof includes includes aa specific specific presentation of evidence evidence that is in the class proofs. is in the class of of all all proofs. The above form of of aa rule rule can can be be used used to to present present any The above form any inductive inductive definition. definition. For For example, the the natural natural numbers numbers are are often often defined defined inductively inductively by by one with no no example, one rule rule with premise and another premise and another rule rule with with one. one. 0o EENN nn EENN suc(n ) E~ N suc(n) N This This definition definition of of 5I N isis one one of of the the most most basic basic inductive inductive definitions. definitions. It It is is aa pattern pattern for for all others, others, and and indeed, indeed, itit is is the the clarity clarity of of this this style style of of definition definition that that recommends recommends itit all for for foundational foundational work. work. Inductive definitions definitions are are also also prominent prominent in in set theory. The The article article of of Aczel Aczel [1986] [1986] Inductive set theory. Introduction to to Inductive Inductive Definitions" Definitions" surveys surveys the the methods methods and and results. results. He He "An "An Introduction bases his his account account on on sets of of rule rule instances instances of of the the form form x__ �X where X are are the the premises premises bases sets (I) where X and and xx the the conclusions. conclusions. A A set set Y Y isis called called (I)-closed -closed iff iff X X C_ �Y Y implies implies xx EE Y. Y . The The set set inductively inductively defined defined by by (I) is is the the intersection intersection of of all all subsets subsets Y Y of of A A which which are are C-closed. -closed. 7730 30 R. Constable 3.2. 3.2. Small S m a l l fragment fragment m arithmetic arithmetic We build small fragment fragment of of a a type type theory theory to to illustrate illustrate the the points points we we have have just just We build aa small made. The explanations explanations are are all all inductive. We let let 8 S and and T T be be metavariables metavariables for for made. The inductive. We ! ! types let, s,s, t,t, si, Si, titi ,, also s', t', t', s:, types and and let, also s', si, t: t i denote denote terms. terms. We We arrange arrange the the theory theory around around a a single single judgment, judgment, the the equality equality ss = = tt in in T T.. We We avoid avoid membership membership and and typehood typehood judgments judgments by by "folding "folding them them into into equality" equality" just just to to make account of make the the fragment fragment more more compact. compact. First First we we look look at at an an informal informal account of this this theory. theory. The intended meaning meaning of in T is that The intended of ss = -- tt in T is that T T is is a a type type and and ss and and tt are are equal equal elements it. Thus = tt in implies that elements of of it. Thus a a premise premise such such as as ss = in T T implies that T T is is a a type type and and that that and tt are are elements elements of of T T (thus (thus subsuming membership judgment) judgment).. 229 9 ss and subsuming membership N. If The The only only atomic atomic type type is is N. If 8 S and and T T are are types, types, then then so so is is (8 (S x T) T);; these these are are the only compound compound types. the only types. The canonical elements of of N N are are 00 and and suc(n) suc(n) where where n n is is an an element element of of N, N, The canonical elements canonical canonical or or not. not. The The canonical canonical elements elements of of (8 (S x T) T) are are pair(s; pair(s; t) t) where where ss is is of of type type 8 S and and tt of of type type T. T. The The expressions expressions 10J(p) lof(p) and and 20J(p) 2of(p) are are noncanonical. noncanonical. The The evaluation evaluation of of 10J(pair(s; lof(pair(s; t)) t)) is is ss and and of of 20J(pair(s; 2of(pair(s; t)) t)) is is tt.. The The inference inference mechanism mechanism must must generate generate the the evident evident judgments judgments of of the the form form ss = = tt in in T T according according to to the the above above semantics. semantics. This This is is easily easily done done as as an an inductive inductive definition. definition. The The rules rules are are all all given given as as clauses clauses in in this this definition definition of of the the usual usual style style (recall (recall Aczel Aczel [1977] [1977] for for example) example).. only atomic N. If We We start start with with terms terms and and their their evaluation. evaluation. The The only atomic terms terms are are 00 and and N. If ss and 20J(t) . Of and tt are are terms, terms, then then so so are are suc(t), suc(t), (s (s xx t) t),, pair(s; pair(s; t) t),, 10J(t), lof(t),2of(t). Of course, course, not not will not all will be meaning, e.g. all terms terms will be given given meaning, e.g. (0 (0 x N), N), suc(N) suc(N),, 10J(N) lof(N) will not be. be. Evaluation. E v a l u a t i o n . Let Let ss and and tt be be terms. terms. o0 evals_to evals_to 00 N N evals_to evals_to N N suc(t) suc(t) evals_to evals_to suc(t) suc(t) pair(s; pair(s; t) t) evals_to evals_to pair(s; pair(s; t) t) 10J (pair ( s; t)) 1of(pair(s; t)) evals_to evals_to ss 20J(pair(s; 2of(pair(s; t)) t)) evals_to evals_to tt Remark: Remark: s(N) s(N) evals_to evals_to s(N) s(N),, 10J(pair(N; lof(pair(N; 0)) 0)) evals_to evals_to N. N. So So evaluation evaluation applies applies to to meaningless formal relation, meaningless terms. terms. It It is is a a purely purely formal relation, an an effective effective calculation. calculation. Thus Thus the the base base of of this this theory theory includes includes a a formal formal notion notion of of effective effective computability computability (c.f. (c.f. Rogers Rogers [1967]) with various [1967]) compatible compatible with various formalizations formalizations of of that that notion, notion, but but not not restricted restricted necessarily necessarily to to them them (e.g. (e.g. Church's Church's thesis thesis is is not not assumed). assumed). Also Also note note that that evals_to evals_to is is idempotent; idempotent; if if tt evals_to evals_to t't' then then t' t' evals_to evals_to t' t' and and t' t' is is a a value. value. general g e n e r a l equality equality tt ll = in T tl = ttll = = tt22 in T tl = tt22 in in T T tt22 = = ta t3 in in T T - - tt22 in in T T tl tl evals_to evals_to t� t'1 t� t2 = tl in T tl = t3 in T t~ = = tt22 in in T T 29In the type theory of Martin-Lof [1982], a premise such as s = t in T Martin-Lhf [1982], T presupposes that that T T is a type and that that Ss EE T, T, t EE T. T. This must be known before before the judgment makes sense. 731 731 Types typehood t y p e h o o d and a n d equality equality 5 1N O0 ==0 iOn in tt = = t 't'i nin 5 1N suc(t) = = suc(t') suc(t') in in N 51 suc(t) ss = = s' s' in in S S tt ==t ' it'n Tin T pair(s; pair(s; tt)) == pair(s pair(s';' ; t') t ' ) iin n (S (S x T) T) The The inductive inductive nature nature of of the the type type N 51 and and of of the the theory theory in in general general is is apparent apparent from from its its presentation. presentation. That T h a t is, is, from from outside outside the the theory theory we we can can see see this this structure. structure. We We can use use induction induction principles principles from from the the informal informal mathematics mathematics (the (the metamathematics) metamathematics) can to say, say, for for example, every canonical canonical expression for aa number number is is either either 00 or or suc(n) suc(n). . to example, every expression for But But so so far far there there is is no no construct construct inside inside the the theory theory which which expresses expresses this this fact. fact. We We will will eventually eventually add add one one in in section section 3.3. 3.3. E x a m p l e s . Here Here are are examples examples oftrue of true judgments judgments that that we we can can make: make: suc(O) suc(O) = suc(O) suc(O) Examples. in N. N. This This tells tells us us that that 51 is a a type type and and suc(O) suc(O) an an element element of of it. it. Also Also pair(O; = in N is pair(O; suc(O)) suc(O)) = pair(O; suc(O)) suc(O)) in in (N (51 x x N) 51) which which tells tells us us that that (N (51 x N) 51) is is a a type type with with pair(O; pair(O; suc(O)) suc(O)) pair(O; aa member. a)) belongs belongs to member. Also Also loj(pair(O; 1of(pair(O; a)) to N 51 and and suc(loj(pair(O; suc(lof(pair(O; a))) a))) does does as as well well for for arbitrary arbitrary aa.. 0 Here is is a a derivation derivation that that suc(loj(pair(O; suc(lof(pair(O; suc(O)))) suc(O)))) = 2oj(pair(O; 2of(pair(O; suc(O))) suc(O))) in in N.3 51.30 Here 00 ==0 i0n in n N in N = 00 in in Nsuc(O) Nsuc(O) = suc(O) suc(O) = = suc(o) suc(o)in N 00 = pair(O; pair(O; suc(O)) suc(O)) = pair(O; pair(O; suc(O)) suc(O)) in in N 51 x N N loj(pair(O; lof(pair(O; suc(O))) suc(O))) = loj(pair(O; lof(pair(O; suc(O))) suc(O))) in in N N loj(pair(O; lof (pair(O; suc(O))) suc(O))) evals_to evals_to 00 2oj(pair(O; 2of(pair(O; suc(O)))= suc(O)))= 2oj(pair(O; 2of(pair(O; suc(O))) suc(O))) in in N N 2oj(pair(O; 2of(pair(O; suc(O))) suc(O))) evals_to evals_to suc(O) suc(O) loj(pair(O; lof(pair(O; suc(O))) suc(O))) = = 00 in in N N suc(loj(pair(O; suc(lof (pair(O; suc(O)))) suc(O)))) = suc(O) suc(O) in in N N 2oj(pair(O; 2of(pair(O; suc(O)) suc(O)))) = = suc(O) suc(O) in in N N suc(O) suc(O) = = 2oj(pair(O; 2of (pair(O; suc(O))) suc(O))) in in N 51 suc(loj(pair(O; suc(lof (pair(O; suc(O)))) suc(O)))) = = 2oj(pair(O; 2of(pair(O; suc(O))) suc(O))) in in N N Analyzing A n a l y z i n g the t h e fragment. f r a g m e n t . This This little little fragment fragment illustrates illustrates several several features features of of the the theory. theory. First, evaluation First, evaluation is is defined defined prior prior to to typing. typing. The The evals_to evals_to relation relation is is purely purely formal formal and and is is grounded grounded in in language language which which is is a a prerequisite prerequisite for for communicating communicating mathematics. mathematics. Computation Computation does does not not take take into into account account the the meaning meaning of of terms. terms. This This definition definition of of computability might be since we relies on computability might be limiting limiting since we can can imagine imagine a a notion notion that that relies on the the information information in in typehood, typehood, and and it it is is possible possible that that a a "semantic "semantic notion" notion" of of computation computation must be be explored explored in in addition, once the the types types are are laid laid down. down. 3311 Our Our approach approach to to must addition, once 30In 3~ type theory, we will write the derivations in the usual bottom-up style with the conclusion at the bottom, leaves at the top. 31 In IZF this is precisely the way computation is done, based on the information provided by a alin membership proof. 732 732 R. Constable computation is compatible with computation theory computation is compatible with the the view view taken taken in in computation theory (c.f. (c.f. Rogers Rogers [1967]). [1967]). Second, Second, the the semantics semantics of of even even this this simple simple theory theory fragment fragment shows shows that that the the concept concept of a a proposition proposition involves involves the the notion notion of of its its meaningfulness meaningfulness (or (or well-formedness). well-formedness). For For of example, what appears to be be aa simple example, what appears to simple proposition, proposition, tt = = tt in in T T,, expresses expresses the the judgments judgments that that T T is is a a type type and and that that tt belongs belongs to to this this type. type. These These judgments judgments are are part part of of understanding understanding the the judgment judgment of of truth. truth. To To stress stress this this point, point, notice notice that that by by postulating postulating 00 = = 00 in in N N we we are are saying saying that that N N is is a a type, type, that that 00 belongs belongs to to N N and and that that it it equals equals itself. itself. The The truth truth judgment judgment is is entirely trivial; entirely trivial; so so the the significance significance of of tt = = tt in in T T lies lies in in the the well-formedness well-formedness judgments judgments implicit in it. These judgments judgments are are normally normally left left implicit implicit in in accounts accounts of of logic. logic. implicit in it. These Notice Notice that that the the well-formedness well-formedness judgments judgments cannot cannot be be false. false. They They are are a a different different category of those about about truth. truth. To that 00 EE N category of judgment judgment from from those To say say that N is is to to define define zero, zero, and and to to say say N N is is a a type type is is to to define define N. N. We We see see this this from from the the rules rules since since there there are are no no separate "N is� separate rules rules of of the the form form "N is_a type" type" or or 00 is_a is_a N." N." Note, Note, because because tt = - tt whenever whenever tt is type, the is in in a a type, the judgment judgment tt = - tt in in T T happens happens to to be be true true exactly exactly when when it it is is well-formed. well-formed. Finally be clarified Finally the the points points about about tt = = tt in in T T might might be clarified by by contrasting contrasting it it with with sue suc in suc = - suc in O0.. This This judgment judgment is is meaningless meaningless in in our our semantics semantics because because 00 is is not not aa type. Likewise suc = sue in although N type, suc type. Likewise suc = suc in N N is is meaningless meaningless because because although N is is a a type, suc is is not not a a member member of of it. it. Similarly, Similarly, 00 = = sue suc in in N N is is meaningless meaningless since since sue suc is is not not aa semantics. None member member of of N N according according to to our our semantics. None of of these these expressions, expressions, which which read read like like propositions, is propositions, is false; false; they they are are just just senseless. senseless. So So we we cannot cannot understand, understand, with with respect respect to to our our semantics, semantics, what what it it would would mean mean for for them them to to be be false. false. Third, Third, notice notice that that the the semantics semantics of of the the theory theory were were given given inductively inductively (although (although informally), informally), and and the the proof proof rules rules were were designed designed to to directly directly express express this this inductive inductive definition. will be the full definition. This This feature feature will be true true for for the full theory theory as as well, well, although although the the basic basic judgments will involve both semantically semantically and judgments will involve variables variables and and will will be be more more complex complex both and proof theoretically. theoretically. proof Fourth, the semantic language. We Fourth, the semantic explanations explanations are are rooted rooted in in the the use use of of informal informal language. We speak of language is critical speak of of terms, terms, substitution substitution and and evaluation. evaluation. The The use use of language is critical to to ex expressing not treat treat terms terms as nor evaluation pressing computation. computation. We We do do not as mathematical mathematical objects objects nor evaluation as as aa mathematical mathematical relation. relation. To To do do this this would would be be to to conduct conduct metamathematics metamathematics about about the the system, system, and and that that metamathematics metamathematics would would then then be be based based on on some some prior prior informal informal language. language. When When we we consider consider implementing implementing the the theory, theory, it it is is the the informal informal language language which implement, translating translating it notation lying which we we implement, it to to a a programming programming notation lying necessarily necessarily outside outside of of the the theory. theory. Fifth, although the Fifth, although the theory theory is is grounded grounded in in language, language, it it refers refers to to abstract abstract objects. objects. This the equality (pair(O; sue(O))) This abstraction abstraction is is provided provided by by the equality rules. rules. So So while while 10f lof(pair(O; suc(O))) is is not canonical integer not aa canonical integer in in the the term term language, language, we we cannot cannot observe observe this this linguistic linguistic fact fact in in the the theory. theory. This This term term denotes denotes the the number number O0.. The The theory theory is is referential referential in in this this sense. sense. Sixth, Sixth, the the theory theory is is defined defined by by rules. rules. Although Although these these rules rules reflect reflect concepts concepts that that we meaningful, and we have have mastered mastered in in language, language, so so are are meaningful, and although although all all of of the the judgments judgments we evident, it Since the we assert assert are are evident, it is is the the rules rules that that define define the the theory. theory. Since the rules rules reflect reflect aa semantic the objects semantic philosophy, philosophy, we we can can see see in in them them answers answers to to basic basic questions questions about about the objects Types 733 733 of the the theory. theory. We We can can say say what what a a number number is, what 00 is, is, what what successor successor is. is. Since Since the the of is, what fragment fragment is is so so small, small, the the answers answers are are a a bit bit weak, weak, but but we we will will strengthen strengthen it it later. later. Seventh, the the theory theory is is open-ended. open-ended. We We expect expect to to extend extend this this theory theory to to formalize formalize Seventh, ever larger larger fragments fragments of of our our intuitions intuitions about about numbers, numbers, types, types, and and propositions. propositions. As As ever G6del showed, complete. So Ghdel showed, this this process process is is never never complete. So at at any any point point the the theory theory can can be be extended. extended. By By later later specifying specifying how how evaluation evaluation and and typing typing work, work, we we provide provide aa framework framework for for future future extensions extensions and and provide provide the the guarantees guarantees that that extensions extensions will will preserve preserve the the truths truths already already expressed. expressed. 3.3. 3.3. First F i r s t extensions extensions We could could extend extend the the theory theory by by adding further forms forms of of computation such as as a a We adding further computation such the evaluation term, prd, term, prd, for for predecessor predecessor along along with with the evaluation prd(suc(n)) prd(suc(n)) evals_to evals_to n. n. We We can can also also include include a a term term for for addition, addition, add(s; add(s; t) t) along along with with the the evaluation evaluation rules rules add(O; t) evals_to add(O; t) evals_to tt add(n; t) evals_to add(n; t) evals_to s' s' add(suc(n); t) t) evals_to evals_to suc(s suc(s')' ) add(suc(n); We We include, include, as as well, well, a a term term for for multiplication, multiplication, mult(s; mult(s; t) t) along along with with the the evaluation evaluation rule rule mult(O" t) evals_to 00 mult(O; t) evals_to ' mult(n; mult(n; t) t) evals_to evals_to m m add(m; add(m; t) t) evals_to evals_to aa mult(suc(n); t) evals_to mult(suc(n); t) evals_to aa These equalities. We These rules rules enable enable us us to to type type more more terms terms and and assert assert more more equalities. We can can easily easily prove, prove, for for instance, instance, that that add(suc(O); add(suc(O); suc(O)) suc(O)) = = mult(suc(O); mult(suc(O); add(suc(O); add(suc(O); suc(O))) suc(O))) in in N. N. But "theory" is is woefully woefully weak. It cannot cannot But this this "theory" weak. It internally express 9 internally express general general statements statements such such as as prd(suc(x)) prd(suc(x)) = - xx in in N l~l or or add(suc(x); ; y) y) = - suc(add(x; suc(add(x; y)) y)) for for any any x x because because there there is is no no notion notion of of variable, variable, add(suc(x) but but these these are are true true in in the the metalanguage. metalanguage. • • definition patterns 9 express express function function definition patterns such such as as the the primitive primitive recursions recursions which which were were used add, multiply multiply and used to to define define add, and for for which which we we know know general general truths. truths. • 9 express express the the inductive inductive nature nature of of N N and and its its consequences consequences for for the the uniqueness uniqueness of of functions functions defined defined by by primitive primitive recursion. recursion. Adding Adding capability capability to to define define new new functions functions and and state state their their "functionality" "functionality" takes takes us concrete theory one; from us from from a a concrete theory to to an an abstract abstract one; from specific specific equality equality judgments judgments to to functional functional judgments. judgments. These These functional functional judgments judgments are are the the essence essence of of the the theory, theory, and and they connecting to they provide provide the the basis basis for for connecting to the the propositional propositional functions functions of of typed typed logic. logic. So So we we add add them them next. next. 734 734 R. Constable Constable R. The simplest simplest new new construct construct to to incorporate incorporate isis one one for for constructing constructing any any object object The (primitive) by following following the the pattern pattern for for the the construction construction of of aa number. number. We We call call itit aa (primitive) by recursion combinator, combinator, R. R. ItIt captures captures the the pattern pattern of of definition definition of of prd, prd, add, add, mult mult given given recursion above. ItIt will will later later be be used used to to explain explain induction induction as as well. well. above. The defining defining property property of of RR isis its its rule rule of of computation computation and and its its respect respect for for equality. equality. The 32 The We present present the the computation computation rule rule using using substitution. substitution.32 The simplest simplest way way to to to to this this We bound variables variables (as (as in in the the lambda lambda calculus calculus or or as to to use use the the standard standard mechanism mechanism of of bound as in quantifier quantifier notation). notation). To To this this end end we we let let u, u, v, v, w, w, x, x, y,y, zz be be variables, variables, and and given given an an in exp of of the the theory, theory, we we let let u.exp u.exp or or u, u, v.exp v.exp or or u, u, v,x.exp v, x.exp or or generally generally expression exp expression U l , . . . , un .exp (also (also written written ~t.exp) u.exp) be be aa binding phrase. We that the the ui Ui are are Ux,...,un.exp binding phrase. We say say that binding occurrences occurrences of of variables variables whose whose scope scope isis exp. exp. The The occurrences occurrences of of ui Ui in in exp exp are are binding bound (by (by the the smallest smallest binding binding phrase phrase containing containing them). them). The The unbound unbound variables variables of of bound exp are are called called free, free, and and ifif xx isis aa free free variable variable of of ~.exp, u.exp , then then ~.exp[t/x] u.exp[t/x] denotes denotes the the exp every free free occurrence occurrence of of xx in in exp. exp o IfIf any any of of the the ui Ui occur occur free free in in substitution of of tt for substitution for every t, usual u.exp[t/x] ~.exp[t/x] produces phrase u'.exp' ~t'.exp' where where the t , then then as as usual produces aa new new binding binding phrase the binding binding 33 variables are prevent capture capture of variables of of t.t . 33 variables are renamed renamed to to prevent of free free variables b[t/v] evals_to evals_to cc b[t/v] R(0; v.b; u, i.h) evaZs evals_to R(O; t;t; v.b; u, v,v, i.h) -to cc R(n; t; v.b; u, v, v, i.h) ali] evals_to R(n; t; v.b; u, i.h) evals_to evals_to aa h[n/u, h[n/u, t/v, t/v, a/i] evals_to cc R(suc(n); t; v.b; u, v, v, i.h) R(suc(n); t; v.b; u, i.h) evals_to evals_to cc Here addition in Here is is a a typical typical example example of of R R used used to to define define addition in the the usual usual primitive primitive recursive recursive way. way. R(n; R(n; m; m; v.v; v.v; u, u, v, v, a.suc(a)) a.suc(a) ) We We see see that that R(O; m;; --R(0; m - ) ) evals_to evals_to m m , , i.e. i.e. 00 + +m m = =m m R(suc(n) R(suc(n);; m; m ; --- ) ) evals_to evals_to suc(R(n; suc(R(n; m; m ; --- ) ) )), , i.e. i.e. suc(n) suc(n) + +m m evals_to evals_to suc(n suc(n + + m) m) Once Once we we have have introduced introduced binding binding phrases phrases into into terms, terms, the the format format for for equality equality and and consequent consequent typing typing rules rules must must change. change. Consider Consider typing typing R. R. We We want want to to say say that that if if v.b v.b and and u, u, v, v, i.h i.h have have certain certain types, types, then then R R has has aa certain certain type. type. But But the the type type of of bb and and hh will will depend depend on on the the types types of of u, u, vv and and ii.. For For example, example, the the type type of of v.v v.v will will be be T T in in aa context context in in which which the the variable variable vv is is assumed assumed to to have have type type T T.. Let Let us us agree agree to to use use the the judgment judgment tt E ET T to to discuss discuss typing typing issues, issues, but but for for this this theory theory fragment fragment (as (as for for Nuprl Nuprl)) this this notation notation is is just just an an abbreviation abbreviation for for tt = = tt in in T T.. We We will will use use it it when when we we intend intend to to focus focus on on typing typing issues. issues. We We might might write write aa rule rule like like 32 32R R can can also also be be defined defined as as aa combinator combinator without without variables. variables. In In this this case case the the primitive primitive notion notion isis application application rather rather than than substitution. substitution. 33If 33If tI; ui isis aa free free variable variable of of tt then then itit is is captured capturedin in ii.exp[t/x] fi.exp[t/x] by by the the binding binding occurrence occurrence tlj ui.. 735 735 Types N vv EE Al A~ U u EEI N A 1 ii E EB B 22 nn EEI NN tt EEAA1 I bvb EEEE �I hh EE B B22 B22 R(n; t; v.b; u, v, v, i.h) R(n; t; v.b; u, i.h) E EB B22 The The premises premises uu EE N Al ii EE B N vv E E A1 B22 hh EEBB 22 reads reads ""hh has has type type B B22 under under the the assumption assumption that that u u has has type type N, N, u u has has type type Al A1 and and ii has has type type B B2." 2 ." For ease ease of of writing writing we we render render this this hypothetical hypothetical typing typing judgment judgment as as For uu ::NN, The syntax , vv ::AAI 1 , , ii ::BB 22 f~ h h E E B B2. syntax u u ::NN is is a a vvariant a r i a n t of of u u E E N N which which 2 . The stresses the typing stresses that that u u is is a a variable. variable. Now Now the typing of of R R can can be be written written nn EE N : AI f- bb EE B N,, vv:A1, : AI , ii:B2 : B2 f-F- hh EE B N tt EENN v v:A1FB22 u u :: N B22 R(n; t; v.b; v, i.i. h) R(n; t; v.b; u, u, v, h) E EB B22 This n, t,t, bb and possibly compound compound expressions This format format tells tells us us that t h a t n, and h h are are possibly expressions of of the the indicated v, u, u, ii as indicated types types with with v, as variables variables assumed assumed to to be be of of the the indicated indicated types. types. Following our our practice practice of of subsuming subsuming the the typing typing judgment judgment in in the the equality equality one, one, we we Following introduce introduce the the following following rule. rule. First First let let Principle_argument Principle_argument Aux_argument Aux_argument Base_equal ity Base_equality Induction_equality Induction_equality nn = = n' n' in in N N tt = = t' t ~ in in N 1N vv = in B2 = v' v ~in in A A1F= b' b~inB2 l f- bb = uu = = u' u ~in in N, IN, vv = = v' v ~in in AI, A1, ii = = ii '~in in B B22 fF- h h = = h' h' in in B B22 == --- == == Then Then the the rule rule is is Principle_argument Aux_argument Base_equality Induction_equality Principle_argument Aux_ar g u m e n t Base_equality Induction_equality R(n; t; v. v. b;b; u, v, e.e. h) R(n; t; u, v, h) = - R(n R(n';' ; tt';' ; vv'.'. b'; u',' , vv',' , ee'.' . h') h') in in B B22 b'; u Unit U n i t and a n d empty e m p t y ttypes. y p e s . We We have have already already seen seen a a need need for for a a type type with with exactly exactly one one element, element, called called a a unit unit type. type. We We take take 11 as as the the type type name name and and •9 as as the the element, element, and and adopt adopt the the rules: rules: .9=1 4.9 in l We We adopt adopt the the convention convention that that such such a a rule rule automatically automatically adds adds the the new new terms terms .9 and and 1 to to the the collection collection of of terms. terms. We We also also automatically automatically add add • 9 evals_to evals_to •9 1 evals_to 1 to indicate that to indicate that the the new new terms terms are are canonical canonical unless unless we we stipulate stipulate otherwise otherwise with with a a different different evaluation evaluation rule. rule. R. Constable 736 We will have reasons reasons later later for for wanting wanting the the "dual" "dual" of of the the unit unit type. type. This This is is the the We will have empty empty type, type, 0, 0, with with no no elements. elements. There There is is no no rule rule for for elements, elements, but but we we postulate postulate 0 0 is_a type is_a type from from which which we we have have that that we we 0 0 as as a a term term and and 0 0 evals_to evals_to 0 0 An handling 00 is An interesting interesting point point about about handling is to to decide decide what what we we mean mean by by assuming assuming xx EE O. 0. Does Does xx : :0OF -I- xx EE0 O make make sense? sense? Is Is this this a a sensible sensible judgment? judgment? We We seem seem to to be be saying saying that that if if we we assume assume belongs to O. We clearly know to 0 and and that that 0 0 is is type, type, then then x x indeed indeed belongs to 0. We clearly know xx belongs belongs to functionality vacuously vacuously since since there there are are no no closed closed terms terms t, with tt = = t' t' in in 0. It is is t, t't' with o. It functionality more interesting to more interesting to ask ask about about such such anomalies anomalies as as or x z : :0 0 F -I-z Ex1E 1 xx : :00F -I-x Ex NE N or or possible nonsense or even even the the possible nonsense xx : :O0~ I-N ENNE. N. What are we the design the theory? W h a t are we to to make make of of these these "boundary "boundary conditions" conditions" in in the design of of the theory? According to 0 ItAccording to our our semantics semantics and and Martin-Lors Martin-LSf's typing typing judgments, judgments, even even x x :"0 (suc = t, t't' in (suc = N N in in N) N) is is a a true true judgment judgment because because we we require require that that 0 0 is is a a type type and and for for t, in 0, if if tt = = t' t' in 0, then then suc suc E E N, N, N N E E N and sue suc = = N N in in N. N. Since Since anything anything is is true true for for 0, in 0, N and all t, t't' in 0, the true. all t, in 0, the judgment judgment is is true. This bizarre, but will be This conclusion conclusion is is somewhat somewhat bizarre, but we we will will see see later later that that there there will be other other types, I P( x()x}) } whose types, of of the the form form {{xx : : A AlP whose emptiness emptiness is is unknown. unknown. So So our our recourse recourse is is to to treat treat types types uniformly uniformly and and not not attempt a t t e m p t to to make make a a special special judgment judgment in in the the case case of of assumptions assumptions of of the the form form x x :: T T for for which which T T might might be be empty. empty. List list data data type almost as central to L i s t types. t y p e s . The The list type is is almost as central to computing computing as as the the natural natural numbers. numbers. We We presented presented this this type type in in the the logic logic as as well, well, and and we we follow follow that t h a t example example even though we special case even though we can can see see lists lists as as a a special case of of the the recursive recursive types types to to be be discussed discussed pleasing to later (section 4). later (section 4). The The rules rules are are more more compact compact and and pleasing to examine examine if if we we omit omit T and use the typing abbreviation of t E T for t = t in the typing context the typing context T and use the typing abbreviation of t E T for t = t in T T.. So So although we we will will write write a a rule rule like like 334 4 although aa EE A, A, 11 EE list(A) list(A) eons(a; cons(a; l) l) E list(A) list(A) Without W i t h o u t its its typing typing context, context, we we intend intend the the full full rule rule T T It- a a = = a' a' in in A A T T IF- 1l = = l' l' in in list(A) list(A) T T IF- eons(a; cons(a; l) l) = = eons(a cons(a';' ; ll') in list(A). list(A). ' ) in 34In 34In this section we use list(A) instead of A list to stress that we are developing a different theory than in theory than in section section 2. 2. 737 737 Types We also also introduce introduce a a form form of of primitive primitive recursion recursion on on lists, lists, the the combinator combinator L L whose whose We evaluation are: evaluation rule rule and and typing typing rules rules are: b[t/v] b[t/v] evals_to evals_to cc L(nil; p; v.b; h, t,t, v, v, i.g) L(nil; p; v.b; h, i.g) evals_to evals_to cc L(l, s, v.b, h, t,t, v,v, i.g) cdi] evals_to L(l, s, v.b, h, i.g) evals_to evals_to Cl c, g[a/h, g[a/h, lit, 1/t, s/v, ~1~, c,/i] evals_to C2 c2 L(cons(a; h, t,t, v,v, i.g) L(cons(a; l); l); s; s; v.b; v.b; h, i.g) evals_to evals_to C2 c2 Let b; h, h, t,t, v,v, e.e. g), and Let L[x; L[x; b, b, g] g] = = L(x; L(x; v. v. b; g), and HE in S HB === = V v = = Vi v ~ in S fF- bb = = b' b~ E E B, B, Hs A,, t =t t= t'~in Hs == == hh = hi h ~in inA i n llist(A), i s t ( A ) , vv = Vi v ~in in S, S, ii = i' i ~in i n BB~ fg =9g ~=i ng'Bin , B, C CAA == = = f~- a a = = a' a ~in in A, A, Css == = = f~ ss = = s' s ~in in S, S, and and C C Alist === CAli~t = fF- ll = = l' l' in in list(A), list(A), then then Hs C Hs HB nz Ca Ca~i,~ CA Cs Aliit L[cons(a; b; g] L[cons(a; l), l), b; g] = = L[cons(a'; L[cons(a'; I') l'),, b', b', g'] g'] in in list(A) list(A) L(nil; L(nil; v.b; v.b; h, h, t, t, v, v, i.g) i.g) = L(nil; L(nil; v.b'; v.b'; hi, h', t, t, V, v, i.g i.g')' ) in in list(A) list(A) Here Here are are typical typical generalizations generalizations of of the the functions functions add, add, mult, mult, exp exp to to N N list list to to illustrate L. For illustrate the the use use of of L. For the the list list (3, (3, 8, 8, 5, 5, 7, 7, 22)) the the operations operations behave behave as as follows. follows. Add Add addL addL is is (3 (3 + + (8 (8 + + (5 (5 + + (7 (7 + + ((22 + + 0)))) 0))))),) , multL multL is is 33 *9 88 *9 55 *9 h 7 9 22 *9 11,, expL expL22 is is (((((2)2 ) 2~)~)~)~)~. ((((( ) 7 ) 5 ) 8) 3 . a.add(h, a)) L(l; 0; addL(l) == = = L(1; 0; h, h, t, t, a.add(h, a)) addL(I) 1; h, m.mult(h, m)) multL(l) == = = L(l; L(1; 1; h, t, t, m.mult(h, m)) multL(I) ezpL(l)~ == = = L(I; n(1; k; k; h, h, t, t, e.exp(h, e.exp(h, e)). e)). expL(I)k The The induction induction rule rule for for lists lists is is expressed expressed using using L L as as follows. follows. Let Let Hs Hs == xz Ee list(A), list(A), Yy E e S, S, vv E e S S fF- j[nil/x, f[nil/z, v/y] v/y] = = bb in in B S and and let let HUst Htist == xx Ee list(A), A, tt Ee list(A), list(A), yy E e S, S, hh E e A, list(A), vv E e S, S, ii E e B B fF- j[cons(h; f[cons(h; t)/x, t)/z, v/y] v/y] = 9g in in B, B, then then Hs Htist xz EE list(A) v, i.g) list(A), , yy E e S S fF- j f = L(x; L(x; y; y; v.b; v.b; h, h, t, t, v, i.g) in in B B This This says says that that L L defines defines a a unique unique functional functional expression expression over over list(A) list(A) and and S S because because the the values values as as inductively inductively determined determined by by the the evaluation evaluation rule rule completely completely determine determine functions functions over over list(A) list(A). . 738 738 Constable R. Constable R. 3.4. FFunctions 3.4. unctions The judgment judgment xx -= xx in The in AA FI- bb -= bb in in BB defines defines aa function function from from AA to to BB whose whose rule rule isis given given by by the the expression expression b.b. We We know know this this from from the the functionality functionality constraint constraint in the the type type A, A , then then b[a/x] bra/xl == b[a'/x] bra' /x] in in implicit in in the the judgment, judgment, i.e. i.e. ifif aa == a'a' in implicit the expression in the type type B. B . Likewise Likewise ifif bl b1 isis an an expression in xx and and b'b' isis an an expression expression in in x' x' then then xx -= x' b' in two rules b' are in BB defines defines such such aa function. function. The The two rules b, b, b' are considered considered x' in in A A FI- bb -= b' equal in A. A . Also Also itit isis part part of of the the judgment judgment that that b[a/x] bra/xl -= b'[a'/x']. b'[a' /x'] . To To equal on on equal equal a,a, a'a' in extensional. this extent extent at at least least the the notion notion of of equality equality on on these these functions functions isis extensional. this Let us us look look at at patterns patterns of of functionality functionality that that involve functions as as arguments. arguments. The The Let involve functions N isis represented represented by by addition function function on on N addition add(loJ(z); 2of(z)) 2oJ(z)) EE NN N xx NN FI- add(lof(z); zz EE N We also know know that that We also l1 EE list(N) l i s t ( N ) FI- addL(l) addL(l) E e N. N. We the pattern pattern of of definition definition used multL, expL expL We know know that that the used to to form form addL, addL, multL, can extended to to any any binary binary function function Jf from to N using can be be extended from N N xx N N to N using fJLk Lk(1) = L(l; L(l; k; (h, a)). For any any specific we can can write write this function fJLk Lk(l), (l) = k; h, h, t,t, a.f a.f(h, a)). For specific fJ we this function (l) , but we would like to to express general fact as aa function saying: for for any but we would like express the the general fact as function of of fJ, , saying: any function from N to to N N and and any any kk in in N k; h, h, t,t, a.f(h,a)) a.J(h, a)) isis aa functional function from N N x x N N, , L(l; L(l; k; functional expression expression in in l,l, kk and and fJ. . In order to say this, this, we type for (N xx N) N) -+ -+ N the In order to say we need need aa type for fJ. . The The notation notation (N N is is the type used in section 2. We can add (A --+ -+ B) B) as type expression for A A and and B type used in section 2. We can add (A as a a type expression for B types. But we also need canonical canonical values for the the type, type, what should they they be? be? Can Can we types. But we also need values for what should we use use (x (x E EA A IF bb E e B) B) as as a a notation notation for for a a function function in in (A (A -+ --+ B) B ) ?? It notation; it similar to It would would be be acceptable acceptable to to use use just just that that notation; it is is even even similar to the the Bourbaki Bourbaki notation notation x x f-t ~-~ b(x b(x E E A, A, bb E E A) A) (see (see Bourbaki Sourbaki [1968a]). [1968a]). But But in in fact fact we we do do not not need need the type type information information to to define define the the evaluation evaluation relation relation nor nor to to describe describe the the typing typing rule. the rule. So So we we could could simply simply use use (x (x f-t ~ b) b).. Instead Instead we we adopt adopt the the lambda lambda notation notation 'x(x.b) A(x.b) more more familiar familiar in in computer computer science science as as we we did did in in sections sections 11 and and 2. 2. We We also also need need notation notation for for function function application. application. We We write write ap(f; ap(f; a) a) for for the the applica application tion of of function function J f to to argument argument a a,, but but often often display display this this as as J(a) f(a). . The The new new evaluation evaluation rules rules are: are: 'x(x.b) A(x.b) evals_to evals_to 'x(x.b) A(x.b) bra/xl b[a/x] evals_to evals_to c ap(,X(x.b); a) evals_to c The The typing typing rule rule is is xx ==x 'x' in A I-F bb ==b b' inA ' i nin BB ,X(x. .' II) in A(x. b) b ) == 'x(x A(x'./~) in (A (A -+ ~ B) B) 739 739 Types This rule rule generates generates the the type type (A --+ B) B) as as a a term. term. 33~ This (A -+ 5 3.5. 3.5. Duality D u a l i t y and a n d disjoint d i s j o i n t unions unions The called duals The types types 0 0 and and 11 are are called duals of of each each other other in in a a category category theory. theory. Here Here is is what means. The called terminal what this this means. The object object 11 is is called terminal (or (or final) final) because because for for every every type type A A,, there there is is a a unique unique map map iinn A A -+ -+ 1, 1, i.e. i.e. a a map map terminating terminating iinn 1, namely namely >.(x A ( x.. .•)). . The The object 0 0 is is initial initial since since for for every every type type A A,, there there is is a a unique unique map map initiating in 0, i.e. 0 object initiating in 0, i.e. -+ -+ A A,, namely namely >'(X.X) ~(x.x).36 . 36 The The duality duality concept concept is is that that the the arrows arrows of of the the types types are are reversed reversed in in the the definition. definition. 1 is is final final iff iff for for all all A A there there is is a a unique unique element element in in A A -+ -+ l1.. o 0 is is initial initial iff iff for for all all A A there there is is a a unique unique element element in in 0 0 -+ -+ A. A. We We will will examine examine another another useful useful duality duality next. next. be characterized The The type type A A x • B B can can be characterized in in terms terms of of functions. functions. In In category category theory theory this this is is done done with with a a diagram diagram c C V~/Pp � ff Z/ " ~ gg � A A ~f':-- A A •x B ~ B B which projection functions == which says says that that given given the the projection functions aa = - >.(x.loJ(x)) )~(x.lof(x)), , bb == - = >.(x.2oJ(x)) A(x.2of(x)) and A, 9g :: C B, there and any any functions functions J f :: C C -+ --+ A, C -+ --+ B, there is is exactly exactly one one map map p p denoted denoted (1, ( f , gg) ) E E CC- +-+ A •A x B such that J = a 0 p and 9 = b 0 Pi that is, for f o r zzEEC C J(z) f(z) = -- a((1, a ( ( f , gg) ) ((z)) z)) g(z) = = b((1, b((f, g)(z)). g)(z)). We We can can show show that that >.(z.pair(J(z) A(z.pair(f(z); i g(z))) g(z))) is is the the unique unique map map (1, (f, g) g).. In construction that In category category theory theory there there is is a a construction that is is dual dual to to the the product, product, called called co-product. diagram, so co-product. Duals Duals are are created created by by reversing reversing the the arrows arrows in in the the diagram, so for for a a dual dual we we claim claim this. this. C C f\ A ff ~jl :i pp "' r,,,, g g � A A ~� A A ++B B ~ Z BB Given inl EE A Given A, A , BB with with maps maps inl A -+ ~ A A + + B B , , inr inr E E B B -+ ~ A A + + B B and and maps maps Jf EE A A -+ -+ C C,, 9 g E E (B (B -+ -+ C) C) there there is is a a unique unique map map [1, [f, g] g] E EA A+ + B B -+ -+ C C such such that that [1, [ f , g] g ] o0iinl n l == f J and and [J, [ f , b] b ] 0o iinr n r == g . g. In In type type theory theory we we take take inl(a) inl(a),, inr(b) inr(b) to to be be canonical canonical values values with with evaluation evaluation 35Martin-L6f : A f-F-b E B since this means that 35Martin_LSf would only need the premise x x:A that A is a type. But in his system to prove x ::A fF-b E B requires requires proving A is..a is_a type type.. 36We 36We could also use A(X. ,k(x. aa)) for any aa EE A if there is one since under the assumption that that xx EE 00,, xx = - a a for any aa,, thus A(X. ,k(x. x) x) = = A(X. )~(x. aa)) in 00 � --+ A A.. 740 740 R. Constable Constable R. inl(a) inl(a) evals_to evals_to inl(a) inl(a) inr(b) evals_to evals_to inr(b). inr(b). inr(b) For AA and and BB types, types, AA ++ BB isis aa new new type type called called the the disjoint disjoint union union of of A A and and B. B. For But the the typing typing rules rules present present aa difficulty. difficulty. IfIf we we simply simply write write But in A A aa == da' in inl(a' ) in in AA ++ SB inl(a) == inl(a') inl(a) bb = =b b'~ in in BB inr(b) inr(b') in in A A ++ BB inr(b) == inr(b') then we we can can deduce deduce aa judgment judgment like like inl(O) inl(O) -= inl(O) inl(O) in in N N ++ suc(O) suc(O) which which does does then not make make sense sense because because N N ++ suc(O) suc(O) isis not not aa type. type. That That is, is, the the rules rules would would no no longer longer not propagate the invariant invariant that that ifif tt == tt in in TT then then TT is is aa type. type. propagate the is_a type, type , into into the the We could solve solve this this problem problem by by including including aa new new judgment, judgment, TT is_a We could theory. The The rules rules would would be be quite quite clear clear for the types built, namely: namely: theory. for the types already already built, N is_a type 1 is_a type 0 is_a type A is_a type type B is_a type A is_a B is_a type (A xx B) is_a type (A B) is_a type list(A) list(A) is_a is_a type type (A (A -+ -~ B) B) is_a is_a type type (A is_a type (A + + B) B) is_a type We We can can then then use use the the rules rules aa = - a' d in in A A B B is_a is_a type type inl(a) inl(a) = = inl(a inl(a')' ) in in A A+ + B B bb = - b' b~in in B B A A is. is_a type .a type inr(b) inr(b) = = inr(b inr(l/)' ) in in A A+ + B B We We will will see see in in section section 3.7 3.7 how how to to avoid avoid adding adding this this new new judgment judgment T T is_ is_ type type.. The gj is The map map [j, [f,g] is built built from from aa new new form form called called decide(d; decide(d; u.f(u); u.f(u); v.g(v)) v.g(v)) whose whose evaluation evaluation rules rules are are f(a) f (a) evals_to evals_to cc decide(inl(a); decide(inl(a); u.f(u) u.f (u);; v.g(v)) v.g(v)) evalS-to evals_to cc g(b) g(b) evals_to evals_to cc decide(inr(b) decide(inr(b);; u.f(u); u.f (u); v.g(v)) v.g(v)) evals_to evals_to cc The gj isis )..)~(x.decide(x; ( x.decide(x; u.f( u) ; v.g( v)) ) . ItIt isis easy The function function [j, [f,g] u.f(u); v.g(v))). easy to to see see that that [j, g] (inl(a)) =- ff(a) (a) and [f,g](inl(a)) and [j, (inr(b)) =- g(b) If, g] g](inr(b)) g(b).. Types 741 741 3.6. 3.6. Metamathematical M e t a m a t h e m a t i c a l properties p r o p e r t i e s of of the t h e type t y p e theory t h e o r y fragment fragment The theory with -* and The theory with base base types types 0, 11,, N and and type type constructors constructors x x,, list, list,--+ and + + is is sufficiently sufficiently complex complex that that it it is is worthwhile worthwhile analyzing analyzing its its properties. properties. First, First, it it is is based based on on a a simple simple inductive inductive model model of of computability computability and and typing typing that that is intuitively intuitively clear. clear. So So we we could could accept accept it it based based on on self-evidence. self-evidence. Indeed Indeed it it is is like like is PRA Church Church [1960] in that that regard-a regard--a manifestly manifestly correct correct theory theory baring baring mistakes mistakes of of [1960] in PRA formalization the intuitive this type theory leads formalization of of the intuitive ideas. ideas. Discussing Discussing this type evidence evidence for for the the theory leads us into philosophy and and Formal Formal Methods Methods studies of formalization formalization which are beyond beyond us into philosophy studies of which are the the scope scope of of the the work. work. Second, we Second, we can can prove prove various various properties properties of of the the formalism formalism by by syntactic syntactic means. means. For For instance: instance: Termination Termination of of Evaluation: Evaluation: If If ft- tt = = tt in in T T then then there there is is a a term term t' t' such such that that tt evals_to evals_to t' t' and and t' t' evals_to evals_to t' t'.. Subject Subject Reduction: Reduction: If If f~ tt = = tt in in T T and and tt evals_to evals_to t' t' then then ft- t' t ~= = t' t' in in T T.. Typehood: Typehood: If If ft- ttl1 = = tt22 in in T T then then T T is_a is_a type type,, and and ft- ttl1 = = t1 tl in in T T and and f~ tt22 = = tt22 in in T T.. Nontriviality: Nontriviality: There There is is no no term term tt such such that that ft- tt = = tt iinn O. 0. Consistency: Consistency: It It is is not not possible possible to to derive derive 00 = = suc(O) suc(O) in in N. IN. Third, Third, we we can can translate translate this this theory theory into into various various well-known well-known mathematical mathematical theories theories order, HA IZF set including including Heyting Heyting Arithmetic Arithmetic of of w w order, HA wW,, IZF set theory theory and and ZF ZF set set theory, theory, and and the the theories theories of of Feferman Feferman [1970,1975] [1970,1975].. There There are are also also categorical categorical models models of of this this ( Bell [1988]) . simple fragment using topoi simple fragment using topoi (Bell [1988]). / 3.7. 3.7. Inductive I n d u c t i v e type t y p e classes classes and a n d large large types types The belong to The types types defined defined so so far far belong to an an inductively inductively defined defined collection collection according according to is_a type last section. to the the scheme scheme for for T T is_a type in in the the last section. Let Let U1 U1 denote denote this this inductively inductively defi ned collection defined collection of of types; types; it it has has the the characteristic characteristic of of aa type type in in that that it it has has elements elements and and is is structured. structured. Evaluation Evaluation is is defined defined on on the the elements, elements, e.g. e.g. N 1N evals_to evals_to N, IN, (N (N x N) IN) evals_to evals_to (N (IN x N) N),, etc. etc. So So all all of of the the elements elements are are canonical canonical and and are are built built up up inductively themselves. N. It properties of inductively themselves. In In this this regard regard U1 U1 resembles resembles IN. It has has all all the the properties of aa type. type. We We want want to to make make U1 U1 a a type. type. So So we we add add rules rules for for its its elements elements in in terms terms of of equalities. equalities. For For example, example, there there are are rules rules 0 = = 0 in in U1 U1 and and A in1U1 A ==AA' ' i nin U 1U1 B B ==B B' 'inU A A xB B = = A A'' X x B B'' in in U1 U1 The have in in mind are these The equality equality rules rules we we have mind are these 1l ==l i n1 Uinl U1 N N ==NN i nin U IU1 0 O ==O0i ninUU1 t 742 742 R. Constable R. A==AA' B==B 'B' A ' i nin U 1U1 B (A a ' xx B') (A •x BB)) == ((A' B') inU1 in U1 in in Ux U1 list(A') in 0"1 U1 list(A) == list(a') list(a) in (A' --+ B') in (A --+ (A in U1 U1 -+ B') -+ BB)) == (A' B) == ((A' B') in in U1 U1 (A ++ B) (A a ' ++ B') This is is aa structural structural or or intensional intensional equality equality (used (used in in both both Nuprl Nuprl and and MartinMartin This Lof [1982]). [1982]) . ItIt turns turns out out that that this this equality is also also extensional since A A == BB in in U1 iff Lbf equality is extensional since U1 iff A implies implies aa EE B B and and conversely. conversely. This This is is the the only only type type so so far far whose whose elements elements are are EA aa E but itit does does not not include include all all types, types, in in particular particular U1 not in in U1 U1 according according to to our our types, but types, U1 isis not semantics. semantics. We have no no way way to to prove prove that that U1 to say say is not not in in U1. U1 • We We don't don 't even We have U1 is even have have aa way way to this. this. But But itit would would be be possible possible to to add add aa recursion recursion combinator combinator on on U1 U1 that that expressed expressed the idea idea that that U1 is the the least least type type closed closed under under these these operations. operations. The The combinator combinator the U1 is would have the the form form of of aa primitive primitive recursive definition would have recursive definition f(O, x) = f(o, = bo(x) b0( ) f(l, x) x) = (x) f(1, : b1 bl(X) = b f(N, x) f ( U , x ) = b2(x) 2 (x) f((A x) = h1 f((A x x B), B),x) h x(A, ( A ,B, B , ff(A, ( A , xx) ) , ,f f( B(B, , x )x)) ) ff (((A + B), x) = hh 44(A, (A + B),x) ( A ,B, B , ff ((A, A , xx), ) , ff( B(B, , x )x)) ) With this form recursion and and the corresponding induction rule we prove With this form of of reeursion the corresponding induction rule we could could prove 0, 1, 1, N, N, aa product, etc. that every every element element of of U1 that was either either 0, product, a a union, union, etc. U1 was Once regard types types as elements of then we can extend our Once we we can can regard as elements of a a type type like like U1 U1,, then we can extend our methods for building building objects, over N N or or by analysis over of Booleans, methods for objects, say say over by case ease analysis over aa type type of Booleans, say say iii ]~ etc. etc. to to building building types. types. Here Here are are two two examples, examples, taking taking iii ]~ as as an an abbreviation abbreviation of of 11 + (.) as tt and ( . ) as + 11 and and abbreviating abbreviating inl inl(.) as tt and inr inr(.) as f ff f .. Let A, T(ff) = B A(x.T(x)) is Let T(tt) T(tt) = A,T(ff) B,, then then )~(x.T(x)) is a a function function iii ]~ -+ --+ U1 U1.. If If we we build build aa generalization generalization of of iii ]~ to to n n distinct distinct values, values, say say iii" ]I~ = - ((1 ((1 + + 1) 1) + + . .. .. . + + 1) 1) n n times times defined IIi(suc(n)) = lIi(n) + with elements build defined by by iii ]~ = = 1, 1,]~(suc(n)) = ]~(n) 4- 11 with elements 1l bb ,, .. .. .., , n rib, then we we can can build b , then aa function selecting nn types, ) . function T(x) T(x)selecting types, T(i T(ib). b It It is is worth worth thinking thinking harder harder about about functions functions like like T T :: 11% ]~n -+ ~ U1 U1.• This This is is an an indexed indexed , T(n collection putting them collection of of types, types, {T(l { T ( l bb ),) , .· .· .·, T ( n bb))}}. . We We can can imagine imagine putting them together together to to form for instance form types types in in various various ways, ways, for instance by by products products or or unions unions or or functions functions T(l T(lb) T(nb) or b) Xx . . . x• T(n b ) or . . . T(n + ) T(l ) + or T(lb) + ' " + T(nb) or b b T(l T(lb) ~ ' ".-.-.+-+ T(n T(nb). b) . b) -+ • • • We We could could define define these these types types recursively, recursively, say say by by functions functions II H,, I: ~ and and e (9 if if we we could could have have inputs inputs like like this: this: m m in in N, N, T T in in B.n B,~ -+ ~ U1 U1,, II YIm(O)(T) T(ir~(1)) m (l)) m (O) (T) = T(i IIm(n)(T) = I I m ( n- l) 1)(T) T(im(suc(n))) II (n) (T) = II (T) xx T(i m (n m (suc(n))) m 743 743 Types where lan , kkb. where iira(k) selects the the k-th k-th constant constant of of ]~n, Likewise for for L: E and and 8 O.. However, However, m (k) selects b .3377 Likewise we are are unable unable to to type type these these functions functions H, E, 8 0 with with the the current current type type constructors. constructors. IT, L:, we We We could could type type them them with with the the new new ones ones we we are are trying trying to to define! define! In the the case case of of IT II and and L: E the the operations operations make make sense sense even even for for infinite infinite families families of of In types, E A types, say say indexed indexed by by T T E A -+ --+ U1 U1 for for any any type type A A.. We We can can think think of of IT H over over T EA that on input aa EE A (a) EE T(a) T E A -+ -~ U U11 as as functions functions f f such such that on input A,, we we have have f f(a) T(a).. For For L: EA the elements elements aa as E over over T T E A -+ --+ U1 U~ we we can can use use the as "tags" "tags" so so that that elements elements are are pairs pairs (a, t) where where tt EE T(a) (a, t) T(a). . These ideas ideas give give rise rise to to two two new new type type constructors, H and and L: E over over an an indexed indexed These constructors, IT family of of types types T T E E A A -+ --+ U1 U1.• We We write write the the new new constructors constructors as as IT(A; H(A;T) and family T) and E(A; T) T).. We We could could use use typing typing rules rules like like these these L:(A; p "1 A EA A E E U1 U1 T T E A - -+ + U1 U1 IT(A; n(A; T) T) E E U1 U~ L:(A; T) EE U1 E(A; T) U~ x E A k- f E T(x) )~(x.f) E H(A; T) F-aEA F-bET(a) pair(a; b) E E(A; T) The indicate that The dotted dotted lines lines forming forming the the box box indicate that this this is is an an exploratory exploratory rule rule which which will will pair(a; b) be be supplanted supplanted later. later. We We treat treat A(X.f) A(x.f) and and pair(a; b) just just as as before, before, so so we we are are not not elements to just new existing ones. adding new adding new elements to the the theory, theory, just new ways ways to to type type existing ones. With With IT II and and L: E and and using using induction induction over over N iN we we can can build build types types that that are are not not in A, f (suc(n)) = in this this U1 U~.• For For example, example, let let f(O) f(O) = = A,f(suc(n)) = A A x• f(n) f(n). . Then Then f f is is a a times. The function function N iN -+ --+ U1 U1 where where f(n) f(n) = = A A x • ..... . x • A A taken taken n n times. The actual actual function function is is A(n.R(n; A(n.R(n; A; A; u, u, t.A t.A x • t)) t)).. Now Now we we can can build build types types like like L:(N; E(N; A(n.R(n; A(n.R(n; A; A; u, u, t.A t.A x • t))) t))) and t.A x• t))) and IT(A; II(A; A(n.R(n; A(n.R(n; A; A; u, u,t.A t))) which which are are not not in in U1 U1.. We We could could imagine imagine trying trying to to enlarge enlarge the the inductive inductive type type class class U1 U~ by by adding adding these these operators operators to to the the inductive inductive definition. take up the next next section. definition. We We will will take up this this topic topic in in the section. Dependent D e p e n d e n t types. t y p e s . The The construction construction of of IT H and and L: 2 types types over over U U11 suggests suggests something something more more expressive. expressive. Instead Instead of of limiting limiting the the dependent dependent constructions constructions to to functions functions from from T EA can form type expression T E A -+ --+ U U1, we could could allow allow dependency dependency whenever whenever we we can form a a type expression 1 , we B[x] that B[x] that is is meaningful meaningful for for all all x x of of type type A A.. We We are are led led to to consider consider a a rule rule of of the the form form I-b A A E E U1 UI xx : :AA I-F B[x] B[x] E E U1 U~ fun(A; fun(A; x.B) x.B) E E U1 U1 prod(A; prod(A; x.B) x.B) E E U1 U1 37in(O) = inlm-l(inl(.)) and ira(n) = inlm-"(inr(.)). 744 744 R. Constable Constable We call call fun fun aa dependent dependent function function constructor and prod prod aa dependent dependent product.38 product. 38 We constructor and We We adopt adopt a a different different notation notation from from IT H and and I; E to to suggest suggest the the more more fundamental fundamental character character of of the the construction. construction. If If we we have have T T E EA A -+ ~ U1 U1,, then then IT(A; II(A; T) T) is is the the same same as as fun(A; x.T(x)) x.T(x)) and and I;(A; E(A; T) T) is is the same as as prod(A; prod(A; x.T(x)) x.T(x)).. But But now now we we can can iterate iterate fun(A; the same is, we the construction without the construction without going going beyond beyond U1 U1.• That That is, we postulate postulate that that U1 U1 is is closed closed under under dependent dependent functions functions and and products. products. This conception conception of of IT II and and I; E is is reminiscent reminiscent of of the the collection collection axiom axiom in in set set theory. theory. This relation on For example, in For example, in ZF ZF if if R(x, R(x, y) y) is is a a single-valued single-valued relation on sets, sets, then then we we can can form form to think ( y l I33x x E E A.R(x, A.R(x,y)}. Another way way to think of of collection collection is is to to have have a a function function y)} . Another {y ff :: A Set where Set and (x) IIxx EE A} A -+ ~ Set where A A E E Set and postulate postulate the the existence existence of of the the set set {J {f(x) A}.. The The similarity similarity between between collection collection and and these these rules rules is is that that we we can can consider consider B B in in fun(A; x.B) x.B) to to defi define function >.(x.B) A(x.B) from from A A into into U1 U1.. With With the the addition addition of of fun(A; ne aa function dependent intuitive model model becomes becomes more dependent types, types, the the intuitive more complex. complex. What What assurance assurance can can we consistent, e.g. we offer offer that that the the theory theory is is still still consistent, e.g. that that we we can can't' t derive derive 00 = -- 11 in in N N or or that that we we derive derive tt E E T T but but evaluation evaluation of of tt fails fails to to terminate? terminate? Can Can we we continue continue to to understand understand the the model model inductively? inductively? If If we we can can build build an an inductive inductive model model of of U1 U1 then then we we can be assured but of can be assured of of not not only only consistency consistency but of a a constructive constructive explanation. explanation. We We answer answer these questions these questions next. next. 3.8. 3.8. Universes Universes We We can can consider consider U1 U1 and and the the rules rules for for it it in in the the last last section section as as partial partial axiomatization axiomatization of of the the concept concept of of Type Type.. On On this this view, view, we we think think of of U1 U1 as as open-ended, open-ended, and and we we do do not not adapt adapt an an axiom axiom capturing capturing its its closed closed inductive inductive character, character, such such as as the the recursion recursion combinator for combinator for U1 U1 discussed discussed above. above. On other hand, hand, we think of On the the other we can can also also think of U1 U1 as as a a large large type type belonging belonging to to Type Type.. On construction on On this this view view the the axioms axioms for for U1 U1 reflect reflect the the rules rules of of type type construction on Type Type into into the the collection collection of of types. types. The The axioms axioms postulate postulate a a certain certain enrichment enrichment of of the the concept concept Type Type in in the the same same way way that that the the axiom axiom of of inaccessible inaccessible cardinals cardinals postulates postulates an an enrichment enrichment of of Set. Set. Similarly, Similarly, from from the the foundations foundations of of category category theory theory (Kreisel (Kreisel [1959]) [1959]),, Grothendieck's Grothendieck's concept concept of of a a universe universe is is a a way way of of modeling modeling large large categories categories (and (and is is equivalent equivalent to to inaccessible inaccessible cardinals). cardinals). If If we we take take the the view view that that U1 U1 is is a a universe universe (rather (rather than than Type), Type), then then it it makes makes sense sense to etc. To form U we extend U1 by adding , to form form larger larger universes, universes, say say U U2, then U U3, etc. To form U2 we extend U1 by adding 2 2 , then 3 this: U1 the the type type U1 U1 itself, itself, like like this: U1 = = U1 U1 in in U U2. 2• Martin-Lof Martin-Lhf and and Nuprl Nuprl axiomatize axiomatize a a universe universe hierarchy hierarchy indexed indexed by by natural natural num nummethod of doing this bers, bers, U Ui. The method of doing this is is to to add add U Uii = - U Uii to to UH1 Ui+l and and to to postulate postulate i • The cumulativity, cumulativity, that that any any type type A A in in U Uii belongs belongs to to all all Uj Uj for for ii < < jj .. So So the the universe universe rules rules are: are: A in U A= =A Ain Uii Ui =Ui in Ui+l A A ==AAi nin U Uj j for f o r i/ <<j .j. 8 Marlin-Lof calls this a dependent 33SMartin-Lhf dependentsum sum and writes writes E(x F~(xEE A)B A)B.. We think of it as as a generalization of of A A x• B B and and display display the the constructor constructor in in Nuprl Nuprl as as xx :: A A xx B B.. Types 745 745 It extend the It is is possible possible to to extend the universe universe hierarchy hierarchy further, further, say say indexed indexed by by ordinal ordinal numbers ord. ord. It It is is possible possible to to postulate postulate closure closure of of Type Type under under various various schemes schemes for for numbers generating such matters. generating larger larger universes; universes; Palmgren Palmgren [1991] [1991] considers considers such matters. Nuprl Nuprl has has been been designed designed to to facilitate facilitate index index free, free, or or "polymorphic" "polymorphic",, treatment treatment of of Ui.. Generally, Generally, the the user user simply universe as as Ui and and the the system system keeps keeps track Ui simply writes writes aa universe track of relative level numbers among them in of providing providing relative level numbers among them in terms terms of of expressions expressions called called level level expressions expressions which which allow allow forming forming ii + + 11 and and max( max(i,i, j) j).. The The theoretical theoretical basis basis for for this this is in in Allen Allen [1987b] [1987b] and and was was implemented implemented by by Howe Howe and and Jackson Jackson (see (see Jackson Jackson [1994c]). [1994c]). is 3.9. 3.9. Semantics: S e m a n t i c s : PER P E R models models The The principal principal mathematical mathematical method method that that we we have have used used to to prove prove the the soundness soundness of Nuprl Nuprl (and (and Martin-LM Martin-Lhf type type theory) theory) has has been been to to interpret interpret equality equality relations relations on on a a of type as as partial partial equivalence equivalence relations relations (("pers") over terms t e r m s- thereby thereby building building a a variety variety type "pers" ) over of term term model Stenlund [1972]). [1972]). We We use use aa method method pioneered pioneered by by Stuart Allen model (see (see Stenlund Stuart Allen of [1987a,1987b] 9 In [1987a,1987b] to to define define the the model model inductively. inductively. 339 In his his thesis thesis Allen Allen compares compares his his models models to to those those of of Aczel Aczel [1986] [1986],, Beeson, Beeson, and and Smith Smith [1984] [1984].. The The modeling modeling techniques techniques also membership relation relation is also borrow borrow from from Tait Tait [1967,1983] [1967,1983] in in that that the the membership is extended extended from from values values to to all all terms terms by by the the pre-evaluation pre-evaluation relation; relation; in in that that regard regard it it follows follows closely closely 0 Martin-Lhf's informal informal semantics.4 semantics. 4~ Martin-Lors ' s method method has Allen Allen's has been been remarkably remarkably potent potent in in all all of of our our work. work. Mendler Mendler [1988] [1988] used used the the technique technique to to model model the the recursive recursive types types defined defined in in Mendler, Mendler, Constable Constable and and Panangaden Panangaden [1986]), [1986]), and and Smith Smith [1984] [1984] used used it it to to model model our our bar bar types types for for partial partial objects objects Constable Constable and and Smith Smith [1993]. [1993]. Harper Harper [1992] [1992] gave gave one one of of the the most most accessible accessible accounts; heavily on Harper to accounts; II draw draw heavily on the the accounts accounts of of Allen, Allen, Mendler, Mendler, and and Harper to explain explain the method. method. the The is to x the the collection The first first step step is to fi fix collection of of terms. terms. The The next next step step is is to to equip equip the the terms terms Allen [1987b] with with an an evaluation evaluation relation, relation, written written now now as as tt ..$(. t' t'.. Allen [1987b] gives gives an an abstract abstract account account of of the the syntax syntax of of terms terms and and the the properties properties of of evaluation. evaluation. We We follow follow Mendler Mendler and less detail. and Harper Harper in in supplying supplying less detail. es El Assume that on Assume that on closed closed terms terms tt evaluation evaluation satisfi satisfies E1 and and E2: E2: i El. El. if if tt ..(. $ t' t' and and tt ..(. $ til t" then then t' t' = = tt",l, so so ..(. $ is is deterministic, deterministic, and and E2. E2. if if tt ..(. $ t' t' then then t' t' $..(. t', t', so so ..(. $ is is idempotent. idempotent. If call t't' aa (canonicaQ If tt ..(. $ t' t' then then we we call (canonical) value. value. Our intended to Our task task now now is is to to specify specify those those terms terms which which are are intended to be be expressions expressions for for mathematical and to mathematical objects objects and to specify specify those those terms terms which which are are expressions expressions for for types. types. We We and dependent carry out for the types carry this this out for the types built built from from N 51 using using products products and dependent functions. functions. We We distinguish distinguish these these as as two two tasks. tasks. The The first first one one is is to to consider consider membership, membership, and and the the 39In 39In his introduction Allen Allen says says "The "The principal principal content content of this thesis thesis is a careful careful development development of . . . a semantic with the intention of... semantic reinterpretation [of type theory] theory] with intention of making making the bulk bulk of type typetheoretic practice . . . independent constructive basis.. ..... . Moreover, practice.., independent of its original original type-theoretic and constructive Moreover, in the unfamiliar unfamiliar domain domain of intuitionistic type theory, theory, the reinterpretation can serve serve as a staff made of familiar familiar mathematical mathematical material." material." made 40We 4~ say that if t �$ t' and t' EE T, then t EE T. This This is the preevaluation preevaluationrelation relation of t' to tt.. 746 746 R. Constable R. second isis to to determine determine type type expressions. expressions. We We look look at at membership membership first first since since itit isis more more second basic. basic. According to to Martin-LSf, Martin-Lof, to to specify specify aa type type isis to to say say what what its its members members are, are, i.e., i.e., According which terms terms are are members, members, and and to to say say what what equality equality means means on on those those terms terms which which are are which members. Equality Equality will will be be an an equivalence equivalence relation, relation, E, E , on on some some collection collection of of terms. terms. members. Considered over over the the entire entire collection, collection, the the relation relation need need only only be be partial. partial. The The field field of of Considered the relation relation (those (those elements elements in in the the relation relation to to themselves, themselves, xEx) xEx) are are the the members members of of the the type. type. These These relations relations are are called called partial partial equivalence equivalence relations relations or or per per for for short. short. the The built-in built-in notion notion of of computation computation places places an an additional additional requirement requirement on on the the pers, pers, The namely, they must respect evaluation. evaluation. That is, if if tt $.J.. t't' and and tEr, tEr , then then t'Er. t' Er . We can namely, they must respect That is, We can Kleene equality equality on on terms, terms, tt ~_ � t' t' means that if if tt $.J.. ss or or say this succinctly by defi ning Kleene say this succinctly by defining means that s , then then tt $.J.. ss and and t't' $.J.. s, s , i.i. e.e. ifif either either term term has has aa value, value, then then both both have that same t't' $.J.. s, have that same value. So So we we require that tt _� t't' and and tEr tEr implies implies t'Er. t' Er . We We next introduce notation notation value. require that next introduce for this this notion notion of of type, type, starting starting with with the the idea idea that that types themselves are are mathematical mathematical for types themselves objects with with an an equality equality defi ned on on them. them. objects defined Let us see how how this this notion type membership for the natural numbers, numbers, Let us see notion of of type membership looks looks for the natural i. e. e. for for the the type type IN. N. Define relation Neq Neq on on terms i. Define the the relation terms inductively inductively by by Neq 0O 0o Neq aa Neq implies suc(a) Neq suc(b) geq bb implies suc(a) geq suc(b) a' implies aa Neq a' Neq Neq b' b' and and a a .J..$ a' a' and and bb .J.. $ b' b' implies Neq b. b. relation which minimal notation Neq Neq is is a a partial partial equivalence equivalence relation which determines determines a a minimal notation for for num numbers on recursion (lazily). bers on which which we we can can compute compute by by primitive primitive recursion (lazily). That That is, is, we we know know what what elements are zero, and nonzero numbers, numbers, we elements are zero, and for for nonzero we can can find find the the predecessor. predecessor. Next, Next, we we define define aa membership membership per per for for the the Cartesian Cartesian product product of of two two types types A A and and * denote B B with with a c~ and and f3 /? as as the the membership membership relations. relations. Let Let a c~* denote the the pre-evaluation pre-evaluation relation ab' and a', b.J.. b' . relation of of aa relation relation a ~,, that that is is aab a~b iff iff there there are are a', a', b' b' such such that that a' a'~b' and a.J.. aSa', bSb'. Define (pair(a; b) , pair(a'; b')) b')) Il aaa' Define a®f3 c ~ / ~ as as {{(pair(a; b),pair(a'; a~a' & & bf3b'} b~b'} **.. We We can can see see that that if if a a and and/3f3 are are value value respecting respecting pers, pers, then then so so is is a c~|® f3 . It It clearly clearly defines defines membership membership in in aa Cartesian Cartesian product product according according to to our our account account of of products. products. Finally, Finally, we we need need aa membership membership condition condition for for the the dependent dependent function function space space con constructor, structor, fun(A; fun(A; x. x. B) B).. This This is is aa bit bit more more complex complex because because for for each each element element aa of of A A,, B[a/x] B[a/x] is is aa type. type. SSoo we we need need to to consider consider aa family family of of membership membership relations relations indexed indexed by by aa type. type. The The members members of of the the function function type type will will be be lambda lambda terms, terms, .\(x. )~(x. b) b).. Let Let a c~ be be aa value value respecting respecting per per and and for for each each aa such such that that aaa a~a,, let let �(a) (I)(a) be be aa value value respecting respecting per. per. Define Define rra� IIc~(I) as as the the following following partial partial equivalence equivalence relation: relation: Ix'] } **.. { (.\(x. b) b),, .\(x'. b')) b'))II Va, W, a'. aaa' => bra/xl b[a/zl �(a) @(a)b'[a' b'[a'/x']} In In order order for for this this per per to to define define type type membership membership for for the the function function space, space, we we require require that that whenever whenever aaa' a~a',, then then �(a) (I)(a) = = �(a') (I)(a').. We We have have in in mind mind that that these these membership membership conditions conditions are are put put together together inductively. inductively. This This is is made made explicit explicit by by the the following following inductive inductive definition definition of of aa relation relation K K on on pers. pers. Types 747 747 K(Neq) K(Neq) K(ar if K(a) K ( a ) and and K(fl) if K(f3) K(a®f3) K(Ha(I)) if K(a) K ( a ) and and '(a) ~(a) = - cI>(a') (b(a') & & K(cI>(a)). K((~(a)). K(IIacI» if We can can prove prove inductively inductively that that all all the the pers pers in in K K are are value value respecting respecting and and all all We defi ne type membership. K define type membership. K provides provides a a per per semantics semantics for for the the small small type type theory theory based based on N, products products and Notice equality on N, and dependent dependent functions. functions. Notice equality on on pers pers is is extensional. extensional. T y p e expressions. expressions. The The inductively inductively defined defined set set K K determines determines a a collection collection of of mem memType bership bership pers pers which which represent represent types, types, but but it it does does not not relate relate these these to to the the terms terms used used N, N (N; x. x. decide(s; v. N to to name name types, types, e. e. g. g. terms terms such such as as N, N x • N N,, fun fun (N; decide(s; u. u. N; N; v. N x x N)) N)) and relationship next definition of and so so forth. forth. We We establish establish this this relationship next by by modifying modifying the the definition of K K to to include include names names for for types. types. Let Let M M be be the the following following inductively inductively defined defined binary binary relation. relation. N N M M Neq Neq A if AMa and BMf3 A xxBB M M aa®f3 | = cI> (a') and fun(A; x. B) fun(A; x. B) M M IIacI> Ha(b if if AMa A M a and and '(a) (b(a)=(~(a') and B[a/x]McI>(a) B[a/x]M@(a) This is is an an ordinary ordinary inductive inductive defi definition of a a binary binary relation. relation. Also, Also, it it is is easy easy to to This nition of Ma see see that that A AM a implies implies K K aa. . The The only only membership membership pers pers described described by by M M are are those those whose whose constituents constituents are are also also described described by by M M . . Moreover, Moreover, all all the the membership membership pers pers are are represented i. e. represented by by terms, terms, i. e. are are related related to to terms terms by by M M . . This This is is critical critical for for the the IIacI> Ha(I) pers pers because because it it guarantees guarantees that that cI> (I) is is represented represented by by aa term. term. Here Here are are three three critical critical facts facts about about M M.. Fact Fact 1 AMa A M a =} =~ K(a) K(a) Fact Fact 2 AMa A M a and and AMa' A M a ' =} =~ a a = = a' a' Fact Fact 3 AMa A M a and and A A � ~_ A' A' =} =~ A'Ma. A'Ma. These These facts facts can can be be proved proved by by M M induction. induction. Fact Fact 11 means means that that all all member member pers pers are are value value respecting, respecting, and and Fact Fact 33 means means that that the the type type names names are are value value respecting respecting as as well. well. Pers P e r s for for intensional intensional type type equality. equality. We We now now want want to to define define a a per per on on type type ex expressions type equality value respecting. pressions which which represents represents type equality and and is is value respecting. There There is is already already M a, Ma a' aa sensible sensible equality equality that that arises arises from from M M , , namely, namely, A A = -- A' A' if if A AM a, A' A'M ' and and aa = extensional equality. model the = a' a'.. This This is is an an extensional equality. We We want want to to model the structural structural equality equality of of section Here is section 3.7, 3.7, thus thus A A x x B B = - A' A' x x B' B' iff iff A A = - A' A' and and B B = - B' B'.. Here is the the appropriate appropriate definition of relation E terms. definition of aa binary binary relation E on on terms. NEN NEN A A x x BEA' BEA' X x B' B' if if AEA' A E A ' and and BEB' BEB' if xB)E fun(A'; x'. B') fun(A; fun(A; x B ) E fun(A'; x'. B') if AEA' A E A ' and and 30'. 3a AM A M t ~a and and A' A'M M aa and and Va, a'. a'. aaa' aaa' =} =v B[a/x]E B[a/x]E B'[a'/x'] B'[a'/x'] ' 748 748 R. Constable R. We now now summarize summarize the the approach approach described described above, above, starting starting from from EE and and following following We Harper ' s method method of of using using least least fixed fixed points points to to present present the the inductive inductive relations. relations. Harper's of pper semantics. Here Here is is aa summary summary of of the the per per semantics semantics along along the the SSummary u m m a r y of e r semantics. lines We define lines developed developed by by Harper Harper [1992]. [1992] . Let Let TT be be the the collection collection of of terms. terms. We define on on TT partial equivalence equivalence relation relation EE intended intended to to denote denote type type equality. If aEa aEa then then we we aa partial equality. If is aa type. type. If If aEd aEa' then then aa and and da' are are equal equal types. types. Let Let IEI lEI := {t: {t : TT ItEt}, I tEt} , say that that aa is say called the the field field of of E. E . Let Let TTIE the set set of of equivalence equivalence classes classes of of terms; terms; say say called / E bbee the [t]E == {x: {x : TT II xEt}. xEt} . Let Let PER PER denote denote the the set set of of all all partial partial equivalence equivalence relations relations on on [t]E T. T. Associated with with each each type type is is aa membership membership equality, equality, corresponding corresponding to to aa == bb in in A; A; Associated -+ thus for for each each aa EE II EE I,I , there there is is aa partial partial equivalence equivalence relation, relation, LL (a). (a) . SSoo LL EE TT/IEE -+ thus PER. PER. We require require of E and and each each LL (a) (a) that that they they respect respect evaluation, evaluation, i.e. alEa2 and and We of E i.e. if if alEa2 a� and a2 $t a~, a� , then then a~Ed a� Ea�2 .. Likewise Likewise for L((a) place of of E. E. el and a2 for L a ) i in n place a l t$ a~ Now consider consider how define E and L Now how we we might might define E and L mutually mutually recursively recursively to to build build aa model of of the the type For the the sake of simplicity, simplicity, we start with extensional model type theory. theory. For sake of we start with an an extensional define Ext Ext and mutually notion of of type equality, as it Ext notion type equality, as above. above. Call Call it Ext. . We We define and L L mutually recursively. recursively. y. ((xL(a)y) ((xL(a)y) {:} (xL(b)y)) iff "Ix, Vx, y. r (xL(b)y)) a Ext Ext b iff a2 )t iff sL(al •x a2)t tl, t2. t2 . S8 t~ pair(s1; pair(s l ; Ss2) t2 ) sL(al iff 3s ~81, & tt $t pair(tl pair(t1;; t2) l, S82, 2 , tl, 2) & & ss2L(a2)t2. & SlL(al)tl slL(a l )tl & & 2 L(a2 )t2 . ffL(fun(al;x, L(fun(al; x. aa2))f' x', b'. ff t$ A(X. iff 3x, 3x, b, b,x',b', A(x. b) b) & &! f '' t $ A(X'. A(x'. b') b') & 2 )) !' iff lx]L(a Vy, y' y' :: IExt(al) [Ext(al)ll .. (yL(al)Y' (yL(al)y' :::} =~ b[ b[y/x]L(a2[y/x])b'[y'/x']. Vy, Y 2 [ylx])b'[y'lx'] . This definition of This is is aa mutually mutually recursive recursive definition of Ext Ext and and L L,, and and it it reflects reflects our our intuitive intuitive understanding, understanding, but but the the definition definition is is not not aa standard standard positive positive (hence (hence monotone) monotone) induc inductive definition because tive definition because of of the the negative negative occurrence occurrence of of yy L L (a (al)y~ in the the clause clause defining defining l )y' in equal equal functions. functions. Allen Allen calls calls these these "half-positive" "half-positive" definitions; definitions; his his method method of of using using K K and and M M as as above above shows shows how how to to replace replace this this nonstandard nonstandard definition definition with with aa standard standard positive positive induction which induction which can can be be interpreted interpreted in in either either classical classical or or constructive constructive settings settings (for (for example, ZF or IZF or in aa theory theory of example, in in ZF or IZF or in of inductive inductive definitions, definitions, see see Troelstra Troelstra [1973] [1973] and and Feferman Feferman [1970]). [1970]). Definition. Definition. A A type type system system T is is aa pair pair (E, (E, L L /) where where E E is is aa value value respecting respecting per per (a) is on on T T and and for for each each aa E E lEI, IEI, L L(a) is aa value value respecting respecting per. per. Given Given type type systems systems ' (a) ; TT = (E, L (E', L') (a) = = (E, L)) and and TT'' == (E', L'),, define define T T I; _ TT'' iff iff E E I; _ E' Z' and and Va Va": lEI. [E[. L L(a) = L L'(a); ' ' that that is is T I; E__TT' iff iff T T' has has possibly possibly more more types, types, and and on on the the types types in in Tr it it has has the the same same equality. equality. Let collection of Let TS T S be be the the collection of all all type type systems systems over over T T with with evaluation evaluation t$.. It It is is easy easy to to see see that that T T SS under under I; _ is is aa complete complete partially partially ordered ordered set, set, aa cpo. cpo. The The relation relation Types Types 749 749 rT [.;;;_ r' ~" is is aa partial partial order order on on T T SS,, and and there there is is aa least least type type system system in in this this ordering, ordering, namely namely (¢>, (r ¢»r where where ¢>r is is the the empty empty set. set. A A non-empty non-empty subset subset D D of of TS T S is is directed directed iff every every pair pair of of elements elements in in D D,, say say r, T, rT'~,, has has an an upper upper bound bound in in D D.. Given Given any any iff directed set set D D of of type type systems, systems, say say rj Ti for for ii EE II,, it it has has aa least least upper upper bound bound f~ where where directed E /~ = = UEj UEi and and L L ~w ((a) a ) == Lj(a) L i ( a ) iiff any any Lj(a) Li(a)isis defined. defined. If If Lj(a) L i ( a ) iis s defined, defined, then then since since ri, Ti, rTjj for for ii =F ~ jj has has an an upper upper bound bound in in D D,, say say rk Tk,, we we know know that that L Lj(a) = Lj(a) Li(a) for for j (a) = aa EC Ej Ei U UE Ej, so the the type type system system ra Ta is is well well defined. defined. Also Also rw r~ is is least least since since if if rj Ti [.;;; E rT'~for for j , so all ii,, then then E Eii [.;;; E_E' E' for for all all ii,, so so UEj UEi [.;;; E E' E'.. all Theorem. e. T h e o r e m . For For any any cpo cpo D D with with order order [.;;; E,, if if F F EE D D -+ ~ D D and and F F is is monotone, monotone, i.i.e. xx [.;;; E yY =} =~ F(x) F(x) [.;;; E F(y) F(y) for for all all x, x, yy in in D D,, then then there there exists exists aa least least fixed fixed point point of of F F in in D D,, ii.e. an element element XXoo such such that that (i) (i) F(xo) F(xo) = = xo xo,, (ii) 5i) for for all all zz such such that that F(z) F(z) = - zz . e. an and XXoo [.;;; E_zz.. and We ne an We now now defi define an operation operation T T EE TS T S -+ ~ TS T S which which is is monotone monotone and and whose whose least least fixed fixed point, f ~,, is is aa type type system system which which models models our our rules. rules. point, D e f i n i t i o n . Let Let T((E, T((E, L») L)) = = (F*, (F*, M M } ) where where Definition. F = {{ ((N, N , NN) ) }} F U{(a b~) Il aEa' aEa ~& & bEb'} bEb ~} U{ (a xx b,b,aa'~ x• b') (a; x. (fun(a; x. b), b), fun(a'; fun(d; x'. x ~. b') b~)) II aEa' aEs & & uU{{ (fun Vy, y'. yL(a)y' Ix']} Vy, y'. yL(a)y' =} =~ b[y/x]Eb'[y' b[y/xlEb'[y'/x']} M (a) = M (a) = { Neq if if aa = = N N Neq L(a L(al)l ) I8i | L(a L(a2)if = a all x • a a22 2 ) if aa = II(L(al), )~(x. L(b))) = fun(al; b) & & fun(al ; x. x. b) I1(L(al) , A(X. L(b))) ifif aa = & Vy." Vy : lI L(ad b[y/x] eE IEI aell Ee lE IEII & L(al)l.l · b[y/x] lEI Theorem. T T is monotone in on TTSS. . Theorem. is monotone in F[.;;; on 3.10. s i n g ttype y p e ssystems y s t e m s tto o m o d e l ttype y p e ttheories heories 3.10. U Using model Allen ' s techniques techniques enable enable us us to to model model aa variety variety of of type type theories. theories. Let Let us us designate designate Allen's some models models for the theories theories discussed discussed earlier. earlier. We'll We'll fix fix the the terms terms and and evaluation evaluation some for the relation to to include include those those of of the the richest richest theory; theory; so so the the terms terms are: are: 0, 0, 11,, . ,. , IN, N, 0, 0, relation suc(t), suc(t) , prd(s), prd(s) , add(s; add(s; t), t) , mult(s; mult(s; t), t) , exp(s; exp(s; t), t) , R(n; R(n; t;t; v.b; v.b; u, u, v,v, i.h), i.h) , ss •x t,t , pair(s; pair(s; t), t) , prod(s; x.t), x.t) , fun(s; fun(s; x.t), x.t) , )~(x.t), A(X.t) , list(t), list(t) , (s.t), (s.t) , L(s; L(s; a; a; v.b; v.b; h, h, t,t, v,v, i.g), i.g) , ss ++ t,t , inl(s), inl(s) , prod(s; inr(t), and decide(p; decide(p; u.s; u.s; v.t). v.t) . inr(t) , and There isis also also the the evaluation evaluation relation relation ss evals_to evals_to tt which which we we abbreviate abbreviate as as ss $.t. t.t . We We There consider various various mappings mappings Tl T l :: TTS where 1I isis aa label label such such as as N, N, G, G, ML, ML, Nu, Nu , consider S ~-+ TTS S where etc. The The most most elementary elementary "theory" "theory" we we will will examine examine isis aa subtheory subtheory of of arithmetic arithmetic etc. involving only only equalities equalities over over IN N built built from from 0, O, suc(t), suc(t), prd(s), prd(s) , and and add(s; add(s; t). t) . This This isis involving modeled by by TTNN. . The The input input to to Wg TN isis any any pair pair (E, (E, LL)) and and the the output output isis (F, (F, M M) ) modeled where the the only only type type name name isis NN, , so so NFN, NFN, thus thus NN EE IFI, IFI , and and the the only only type type equality equality where 750 750 R. Constable Constable R. is is M M (N) (N) which which is is defined defined inductively. inductively. We We have have that that ss M M (N)t (N)t isis the the least least relation relation Neq such such that that Neq Neq tti iff (s $t 00 & & tt $t 00 vV 3s', 3s', t'. t'. ss $t suc(s') suc(s') &. &. tt $t suc(t')s'Neq suc(t')s'Neq t'). t') . ss Neq f f (s TN (E, p,(T N ) The map map TN takes The takes any any (E, L) L) to to this this (F, (F, M M) ) , , so so its its least least fixed fixed point, point, # ( T N ) is is just for successor We see (F, M M) .) . This This is is the the model model for successor arithmetic. arithmetic. We see that that in in this this model, model, just (F, N natural number number and N isis aa type, type, that that ss -= tt in in N N iff iff ss evaluates evaluates to to aa canonical canonical natural and tt evaluates to to the the same same canonical canonical number. number. evaluates The rules rules can can be be confirmed confirmed as as follows. follows. First, First, notice notice that that evaluation evaluation is is determinisdeterminis The tic and and idempotent idempotent on on the the terms. terms. As As we we observed, observed, the the general general equality equality rules rules hold hold in in tic any g ) is any type type system system (because (because M M ((N) is in in equivalence equivalence relation relation on on canonical canonical numbers). numbers) . 0, suc(O), suc(O) , suc(suc(O)),.., suc(suc(O)), . . . are are in in the the relation relation This follows follows by by showing showing inductively inductively that that 0, This M (N), (N) , i.e. i.e. in in the the field field of of the the relation. relation. The The fact fact that that M M (N) (N) respects respects evaluation evaluation M validates the last rule. validates the last equality equality rule. p,(TN) #(TN) ~F= 00 = -- 00 in in N N #(TN) suc(t) in in N. p,(TN) ~F= ss =-- tt iinn NN implies implies #(TN) p,(TN ) F=~ suc(s) suc(s) -= suc(t) N. The typing is also by induction Neq;j namely, namely, The typing rule rule for for successor successor is also confirmed confirmed by induction on on Neq if ss'' LL((N)t' if N ) t ' , , then suc(t') have then since since suc(s') suc( s') $t suc(s') suc( s') and and suc( t') $t suc(t'), suc(t') , then then we we have suc(s) M M (N)suc(t) (N)suc(t) as as required the typing typing rule. suc(s) required for for the rule. N , the model #(TN), and and the informal semantics semantics are essentially In the the case In case of of N, the model the informal are essentially the same. same. So So the theory fragment fragment for N can stand on own with with respect to the the the the theory for N can stand on its its own respect to N will will have have the the same same essential essential ingredient ingredient model. Even Even aa set theoretic semantics semantics for model. set theoretic for N ' s definition of an an inductive inductive characterization. instance, Frege that of characterization. For For instance, Frege's definition was was that J.L(TN ) , N == = = {x {x II "Ix. VX. (0 (0 E e X X & & Vy. Vy. (y (y E e X X * =~ s(y) s(y) Ee X)) X)) * =~ xx E e X} X} N where I 3u. (u where s(x) s(x) is is {z {zl:lu. ( u eEz z& & z - z{ u-} {U} = x ) }=. x) } . In In ZF ZF we we can can use use the the postu postulated infinite set, x} where lated infinite set, in in !f , , and and form form w == == {i {i :: in inf! II "Ix. Vx. natJike(x) nat_like(x) * =~ ii EE x} where natJike(x) both of nat_like(x) iff iff (0 (0 E e xx & & Vy. Vy. (y (y Ee xx * =~ suc(y) suc(y) Ee x)) x)) for for Suc(y) Suc(y) = = yy U U {y} {y}.. In In both of these definitions, nature of N is theory and ZF these definitions, the the inductive inductive nature of N is expressed. expressed. But But Frege's Frege's theory and ZF allow very general ways using this this inductive only used used it allow very general ways of of using inductive character. character. So So far far we we have have only it for for specifying specifying the the canonical canonical values. values. The model for The same same approach approach can can be be used used to to define define aa model for the the type type theory theory with with cartesian cartesian products. products. In In this this case case we we denote denote the the operator operator on on type type systems systems as as TN 2.. Given Given TN 2 ((Z, L)) L)) = = (F, (F, M M ) ,) , if if S S E e l[E[I and and T E e l[E[I then then S S x T E e IFI IF[,, and and M M (S (S x T) is is L L ((S) S ) IZI | L L ( T ) . For For this this system, system, TN 2 is is continuous, continuous, i.i. e. e. if if TO= = (¢J, (r ¢J) r and TN2(Ti) = -- Ti+I, then then #(TN 2) = -- Tw. and In In #(TN 2) all all the the rules rules for for the the fragment fragment of of section section 3.2 3.2 are are true. true. Again Again the the theory theory is is so so close close to to the the semantics semantics that that it it stands stands on on its its own. own. Notice Notice that that in in confirming confirming the the rule rule for for typing typing pairs, pairs, we we rely rely on on the the fact fact that that #(TN 2) is is aa fixed fixed point. point. #(TN 2) ~ ss = = s' s' in in S S and and #(TN 2) ~ (t (t = = t't' in in T ) iimply mply ( s' j t') #(TN 2) ~ pair( pair(s;Sj t) t) = = pair pair(s'; t') in in S S x T. TN2 ((E, T) TN2 (T;)2 Ti+l , p,(TN ) p,(TN22) F= p,(TN ) F= (T). 2 p,(TN ) Tw• E T E TN 2 p,(TN2 ) p,(TN2 ) F= T. T T) TN 2 TO 751 751 Types Types 7i Note, this Note, this fact fact would would not not be be true true in in any any fixed fixed Ti since since S S x • T T might might be be defined defined only only in in Ti+l. To To provide provide aa semantics semantics for for fun(A; fun(A; x. x. B) B) and and pTod(A; prod(A; x. x. B) B) we we use use the the map map TM TML L defined model is correct, we defined in in section section 3.9. 3.9. The The model is P,(TM #(TML). To prove prove the the rules rules correct, we recall recall L ) ' To the the meaning meaning of of sequents sequents such such as as x x EA A \F- B B type type and and x x EA A \t- ss = - tt in in T T.. p,(TMd implies p,(TM ~(TML) F ~ (x (X E A A \F-B B type) type)implies ~(TML) ~ fun(A; fun(A; x.B) x.B) type type L) F p,(TM \- bb E B) ~(TML) ~ (x (X e A A ~B) implies implies p,(TM #(TML) ~ >.(x.b) A(x.b) in in fun(A; fun(A; x.B) x.B) L) F L) F 7i+l . E E E E E E Modeling M o d e l i n g hypothetical h y p o t h e t i c a l judgments. j u d g m e n t s . The The meaning meaning of of x x EA A \F- bb E EB B is is that that A A is is type and and for for any any two two elements, a' of of A A,, B B[a/x] is aa type type and and B B[a/x] - B[a' B[a'/x] aa type elements, a, a, a' [a/x] is [a/x] = /x] ((i.e. i.e. B b[a'/x] B is is type type functional functional in in A), A), and and moreover, moreover, bra/xl b[a/x] EE B[a/x] B[a/x] and and bra/xl b[a/x] = = b[a'/x] in ne in B[a/x] B[a/x].. We We have have extended extended this this notion notion to to multiple multiple hypotheses hypotheses inductively inductively to to defi define xl E AI A1,... An \t- bb E B B.. This This definition can be be carried carried over over to to type type systems. systems. , . . . ,,xn xn E An definition can Xl E E E 3.11. 3.11. A A semantics s e m a n t i c s of of proofs The in section The discussion discussion of of proofs proofs as as objects objects and and Heyting Heyting semantics semantics in section 22 suggested suggested treating proofs objects and propositions as treating proofs as as objects and propositions as the the types types they they inhabit. inhabit. True True propo propositions are those inhabited by proofs. proofs. But But there there were were several several questions questions left left open open in in sitions are those inhabited by details of section 2.14 section 2.14 about about the the details of carrying carrying out out this this idea. idea. The The type type theory theory of of this this section section can can answer answer these these questions, questions, and and in in so so doing doing it it provides provides a a semantics semantics of of proofs. proofs. The The basic basic idea idea is is to to consider consider a a proposition proposition as as the the type type of of all all of of its its proofs proofs and and to to take take proof proof expressions expressions to to denote denote objects objects of of these these ' s semantics types. types. Based Based on on Heyting Heyting's semantics we we have have a a good good idea idea of of how how to to assign assign a a type type to to compound compound propositions propositions in in terms terms of of types types assigned assigned to to the the components. components. For For atomic atomic propositions propositions there there are are several several possibilities, possibilities, but but the the simple simple one one will will turn turn out out to to provide provide good semantics. consider only semantics. The The idea idea is is to to consider only those those atomic atomic propositions propositions which which can can aa good plausibly plausibly have have atomic atomic proofs proofs and and to to denote denote the the canonical canonical atomic atomic proofs proofs by by the the term term axiom. axiom. We We will will assign assign types types to to the the compound compound propositions propositions in in such such a a way way that that the the canonical will call canonical elements elements will will represent represent what what we we will call canonical canonical proofs. proofs. Moreover, Moreover, the the reduction reduction relation relation on on the the objects objects assigned assigned to to proof proof expressions expressions will will correspond correspond to to meaningful meaningful reductions reductions on on proofs. proofs. Proofs Proofs corresponding corresponding to to noncanonical noncanonical objects objects will will be canonical proofs. be called called non noncanonical proofs. The The correspondence correspondence will will guarantee guarantee that that noncanonical noncanonical proofs proofs p' p~ of of a a proposition proposition P P will will reduce reduce to to canonical canonical proofs proofs of of P P.. We We now now define define the the correspondence correspondence between between propositions propositions and and types types and and between between proofs proofs and and objects. objects. Sometimes Sometimes this this correspondence correspondence is is called called the the Curry-Howard Curry-Howard isomorphism. isomorphism. Curry-Howard C u r r y - H o w a r d isomorphism. i s o m o r p h i s m . For For the the sake sake of of this this definition, definition, if if P P is is a a proposition, proposition, we type, and we let let [P] [P] be be the the corresponding corresponding type, and if if p p is is a a proof proof expression, expression, we we let let [P] [p] be be the the corresponding corresponding element element of of [P] [P].. We We proceed proceed to to define define [[ ]] inductively inductively on on the the structure structure of proposition P from from section section 2.5. 2.5. of proposition P We consider consider only only atomic atomic propositions propositions of of the the form form aa = - bb in in A A.. The The type type 11.. We [a will have the atomic atomic proof Ia = = bb in in A] A] will have the proof object object axiom axiom if if the the proposition proposition is is 752 752 R. Constable R. axiomatically true. true. axiomatically for aa == bb in in A A evaluates evaluates to to aa canonical canonical proof proof built built If the the proof proof expression expression ee for If only from from equality equality rules, rules, then then we we arrange arrange that that ee $.j.. axiom. axiom . This This isis aa simple simple only form of of correspondence correspondence that that ignores ignores equality equality information. information. For For instance instance form e 2 )] $.j.. axiom. [transitivity(el ' e2)] axiom. [symmetry(3)] [symmetry(3)] $.j.. axiom axiom [transitivity(el, e' [e] [~] $.j.. e' [equality_intro(e)] $.j.. e'e' [equality_intro(e)] We also need need these these evaluation evaluation rules rules for for the the proof proof expressions expressions for for substitution substitution We also and type type equality. equality. and 2. 2. 3. 3. 4. 4. 5. 5. 6. 6. [p] ip] pi pi [p] $.j.. p' [p] $.j.. p' i [subst(p [eq(p; e pi e)] p .j.. ; H $ p' [~q(p; ~)] $.j.. p' [~ub~t(p; ~)1 [Q] and and [P Q] == [P] [P] xx [Q] iP & & Q] [[&R e2 )] -= pair([eli; pair([e l ]; [e2]), and [e2]), and ~ ( e(eI l , , e2)] 2of([e l ]/v)) . u, v. ~)1 e2 )] == [~](loy(i~ll)/u, [e2 ](lof([e l ])/u, 2oY([~11/~)). [&L(e i~L(~;l ; ~,,. = [P] [P] ++ [Q], [Q] , [P V V Q] [P Q] = [VRl(a)] == int([a]), inl([an , [URt(a)] [VRr(b)] == ~n~([b]), inr([bn , [UR~(b)] v. ee2)] [VL(d; u. u. el; e l i v. decide([d] ; u.[el]; u.[eIJ; v.[e2]). v.[e2 ]) . [YL(d; 2 )] == decide([d]; [P =} Q] = [P] -+ [Q] , IF =~ Q] = IF] --+ [Q], [=}R(x. e)] = i~R(~. ~)1 = >'(x.[e]) ~(~.M), , [g] [ap([J]; ~p]/y]. [P]/y] . [[~n(y; L(j; p; y. q)] p; y. q)] == M[ap([Y]; =} [P[x]]) = prod(A; x. [3x:: A. P[z]] = prod(A; x. [P[x]]),, A. P[x]] [3x [3R(a; [3R(a; p)] p)] = = pair(a; paid(a; [pn [P]),, [3L(p; (lof( [P])/u, 2of([P]/v)) [3L(p; u, u, v. v. g)] g)] = = [g] [g](lof([pl)/u, 2of([p]/v)). . [Vx:A. P[x]] = = fun(A; fun(A; x. x. [P[x]]) IF[x]]),, [Vx : A. P[x]] [VR(x. e)] = = >'(x.[e)) ~(x.[e]),, [ap([J] ; a)/y] [VL(j; a; y. y. e)] [VL(f; a; e)] = = [g] [g][ap([f]; a)/y].. Sequents S e q u e n t s to to typing t y p i n g jjudgments. u d g m e n t s . We We can can now now translate translate deductions deductions of of sequents sequents f/ Hn we [-I fFP P by by p p to to derivations derivations of of [f/] [/4] f~- [P] [/9] E E [P] IF].. Given G i v e n /f/ ~ = - X xll :"H H I ,, .. .. .., , xn xn :"Hn we take [f/] be x~ Xl EE HL where if type then then Hi take [/~] to to be H ~ , .. .. .., , Xn x~ E E H� H~ where if Hi is is aa type H~ = - Hi and and if if Hi is Hi ] . In this case is aa formula formula then then Hi H~ = = [[Hi]. In this case we we treat treat the the label label Xi xi as as aa variable. variable. Now to translate a deduction tree to a derivation tree we work Now to translate a deduction tree to a derivation tree we work up up from from the the leaves leaves translating translating sequents sequents as as prescribed prescribed and and changing changing the the rule rule names. names. The The proof proof system system was was designed designed in in that that we we need need not not change change the the variable variable names. names. Expressing E x p r e s s i n g well-formedness w e l l - f o r m e d n e s s of of formulas. formulas. The The introduction introduction of of U U1I combined combined with with the the propositions-as-types propositions-as-types interpretation interpretation allows allows us us to to express express the the pure pure proposi proposition tion of of typed typed logic logic more more generally, generally, and and we we can can solve solve the the small small difficulty difficulty of of insuring insuring that that A A+ + B B is is aa type type discussed discussed at at the the end end of of section section 3.5. 3.5. 753 753 Types According According to to the the propositions-as-types propositions-as-types principle, principle, Ul U1 represents represents the the type type __;; (small) (small) propositions, and and a a function function P P E E A A --+ --+ Ul U1 can can be be interpreted interpreted as as a a propositional propositional propositions, function. want to this logical function. When When we we want to stress stress this logical interpretation, interpretation, we we use use the the display display form form ProP the proposition Prop1l for for Ul U1 and and generally generally ProPi Propi for for Ui Ui,, and and we we call call ProPi Propi the proposition of of leveli leveli.. We propositions in We can can express express general general propositions in typed typed logic logic by by quantifying quantifying over over ProPi Propi and and Ui Ui.. Here Here are are some some examples examples from from section section 2. 2. 1. 1. VA, VA, B B": U U1. VP": A A --+ ~ PropNQ ProplVQ": B B --+ -~ ProP Prop1. l . VP l. Vx": A. A. Vy Vy": B. B. (P(x)&Q(y)) (P(x)&Q(y)) ¢:> r Vx Vx": A. A. P(x)&Vy P(x)&Vy": B. B. Q(y) Q(y).. Vx : A. �y : B. R(x, y)) . 2. VA, B B ':UU1I ..VR V R :' A A xxBB --+ -~ ProPl Prop~.. (�y (3y'B. Vx'A. R(x,y) =~ Vx Vx'A. 3y'B. R(x,y)). 2. VA, : B. Vx : A. R(x, y) � At this this level level of of generality, generality, we we need need to to express express the the well-formedness well-formedness of of typed typed At formulas in in the the logic rather than than as as preconditions preconditions on on the the formulas formulas as as we we did did in in formulas logic rather 2. This section 2. section This can can be be accomplished accomplished easily easily using using Ui Ui and and ProPi Propi.. We We incorporate incorporate into into the the rules rules the the conditions conditions necessary necessary for for well well formedness. formedness. For For example, example, in in the the rule rule fHP~� HF-P Q Q by by � ::~ _ fI, p P F'r- Q Q fI, We We need need to to know know that that P P and and Q Q are are propositions. propositions. We We express express this this by by additional additional well-formedness well-formedness subgoals. subgoals. A A complete complete rule rule might might be be v "1 fI / ~ F'r-- P P ~� Q Q by by �R ~ R at at ii H, p PF-Q fI, 'r- Q Pi fI f-I 'r~- P P E E Pro Propi Pi fI [-I 'rF- Q QE E Pro Propi _ t. I .t If the invariant can prove prove fI If we we maintain maintain the invariant that that whenever whenever we we can H 'r~ aa E E A A then then we we know A in aa Ui fI 'r-~ P know P in ProPi know A is is in Ui,, and and whenever whenever we we prove prove/~ P then then we we know P is is in Propi,, then then we we can can simplify simplify the the rule rule to to this this _ ,------, fI H t 'r-- -P P ~�Q Q by by H, p P t 'r-- QQ fI, fI [-I 'rf- P P E E ProPi Propi _ � = ~ at at i We add well-formedness We need need to to add well-formedness conditions conditions to to the the following following rules, rules, VR, VR, �R, =~R, VR, VR, Magic. Magic. We We already already presented presented �R; =vR; here here are are the the others. others. fI by VRI at ii VR VR H t'r-- PPVVQ Q by VRt at fI H F'r-- Pp fI H 'rt- Q Q EE ProPi Propi The The VRr VP~ case case is is similar. similar. VVR R Magic Magic fI [-I 'rF-Vx Vx": A. A. P(x) P(x) by by VR VR at at i-I, xx": A A 'rf- P(x) P(x) fI, fI 'r-~ AAEEUUi H i i fI /~ 'rF- P P V V op ~ P by by Magic Magic at at fI [-I 'r~P P EE ProPi Propi i 754 754 R. Constable 3.12. PProofs as pprograms 3.12. r o o f s as rograms The type type corresponding corresponding to to aa proposition proposition of of the the form form (Vx:A. (\Ix : A. 3y:B. 3y : B. S[x, S[x, y]) y]) isis The -+ yy: :BB •x ~S[x,y]]. [S[x, y]] . The The proof proof expressions, expressions, say say p, p , for for the function function space space xx: :AA --+ the this object object denotes denotes aa canonical canonical element element of of the the type. type. That That element element is is aa function function this b) where where for for each A, b[a/x] bra/xl eE y: y : BB •x IS[a, [S[a, y]] y]] and and if if lof 10](b[a/x]) and A(x. each aa eE A, (b[a/x]) e BB and -X(x. b) -+ B B and and let 2of(b[a/x]) 20](b[a/x]) eE [S[a, [S[a, lof(b[a/x])~. lo](b[a/x])] . So So the the function function ~(x. -X(x. lof(b)) 10](b)) e AA --+ let A -+ = )~(x. -X(x. lof(b)), 10](b)) , then then f] eE A B and and ~(x. A(X. 2of(b)) 20](b)) proves proves Vx:A. \lx : A. Six, S[x, f(x)]. ] (x) ] . -+ B f] = So we we can can see see that that the the process process of of proving proving the the "specification" "specification" Vx:A. \Ix : A. 3y:B. 3y : B. Six, S[x, y] y] So constructively creates creates aa program program f] for for solving solving the the programming programming task task given given by by the the constructively specification, and and itit simultaneously simultaneously produces produces the the verification verification ~(x. -X(x. 2of(b)) 20](b)) that that the the specification, program meets meets its its specification specification (c.f. (c.f. Constable Constable [1972], [1972], Bates Bates and and Constable Constable [1985] [1985] program and Kreitz Kreitz [n.d.]). [n.d.]). and E E Refinement style pprogramming. to R e f i n e m e n t style r o g r a m m i n g . This This style style of of programming programming provides provides a a way way to build the the program possible to to gradually refine build program and and its its justification justification hand-in-hand. hand-in-hand. It It is is possible gradually refine these two objects, filling filling only only as as much much detail detail as as necessary for clarity. for example, example, these two objects, necessary for clarity. So So for proof detail detail can can be be omitted omitted for for programming programming steps steps that obvious. The The extreme proof that are are obvious. extreme case we omit omit all all proof proof steps except those case of of "unbridled" "unbridled" programming programming arises arises when when we steps except those that come come automatically automatically as as part part of of the the programming, programming, e.g. e.g. certain certain "type "type checking checking that steps" and and the the over all logical logical structure structure of of the proof. steps" over all the proof. Explicit programming style. We can program program aa solution solution to : A. 3y : B. S[x, Explicit p r o g r a m m i n g style. We can to \Ix Vx:A. 3y:B. Six, y] y] directly by function f] E E A A -~ -+ B and then then proving \Ix : A. A. S[x, directly by writing writing a a function B and proving Vx: S[x, ](x)] f(x)].. Christine Paulin-Mohring studying how program information to Christine Paulin-Mohring [1989] [1989] is is studying how to to use use the the program information to help help drive drive the the derivation derivation of of the the proof. proof. 4. T Typed languages 4. y p e d programming programming languages 4.1. 4.1. Background Background Programming at Programming at its its "lowest "lowest level" level" involves involves communicating communicating with with specific specific digital digital ' s) . The hardware hardware in in "machine "machine language," language," sequences sequences of of bits bits (D's (O's and and ll's). The particular particular machine machine model model will will classify classify sequences sequences of of bits bits into into aa fixed fixed number number of of "types," "types," say say instructions, instructions, signals, signals, addresses, addresses, and and data; data; the the data data might might be be further further classified classified as as floating floating point point or or integer integer or or audio audio or or video, video, etc. etc. Programming Programming at at this this machine machine level level assembly language or or just just above above at at assembly language level level is is generally generally regarded regarded as as "untyped" "untyped" in in part part because because everything everything is is ultimately ultimately bits. bits. We We are are mainly mainly concerned concerned with with so-called so-called higher-level higher-level programming programming languages, languages, and and for for the the purpose purpose of of this this discussion, discussion, higher-level higher-level languages languages will will be be classified classified into into two two groups groups as as typed typed or or essentially essentially untyped. untyped. Two Two high high level level languages languages from from the the earliest earliest period Lisp. Fortran period are are still still "alive," "alive," Fortran Fortran and and Lisp. Fortran is is considered considered typed typed (though (though minimally) minimally) as as are are more more modern modern languages languages like like Pascal, Pascal, C C ++ ++ ,, ML, ML, and and Java. Java. Two Two of of Types Types 755 755 the the most most historically historically significant significant typed typed languages languages were were Algol Algol 68 68 and and Simula Simula 67. 67. Lisp Lisp is considered considered untyped untyped as as is is its its modern modern descendent descendent Scheme. Scheme. These These languages languages have have is notion of of run-time run-time typing typing in in which which data is tagged information during during aa notion data is tagged with with type type information execution. 68, ML, ML, and execution. Whereas Whereas Algol Algol 68, and Java, Java, for for example, example, are are statically statically typed typed in in that that data and and expressions expressions are are typed typed before before execution execution (at (at "compile "compile time" time" ).4 ).411 data One the computer One of of the the major major design design debates debates in in the computer science science community community over over the the years the value years has has been been about about the value of of rich rich static static typing, typing, represented represented by by Algol Algol 68 68 and and Simula, and Scheme. There Simula, and "untyped" "untyped" programming programming represented represented by by Lisp Lisp and and Scheme. There are are formal languages languages that that capture capture the the essence essence of of this this distinction. distinction. Lisp Lisp and and Scheme Scheme are are formal Barendregt [1981] represented represented by by the the untyped untyped lambda lambda calculus calculus of of Church Church [1960] [1960] (see (see Barendregt [1981],' Seldin [1972]) Stenlund [1972] [1972],' Hindley, Hindley, Lercher Lercher and and Seldin [1972]) on on which which they they were were modeled, modeled, Stenlund and and ML ML by by the the typed typed lambda lambda calculus calculus (see (see Barendregt Barendregt [1977], [1977], de de Bruijn Bruijn [1972]) [1972]).. We We have have seen seen the the untyped untyped lambda lambda calculus calculus in in section section 3.4. 3.4. Its Its terms terms are are variables, variables, abstractions, and applications denoted abstractions, and applications denoted respectively respectively Xi xi,, A(X. •(x. t) t),, and and ap(s; up(s; t) t) for for ss and and tt terms. terms. The calculus introduces introduces some The typed typed calculus some system system of of types types T T and and requires requires that that the the variables are are typed, typed, x x TT .. Usually Usually the types include include the the individuals, individuals, L~,, and and if if a, a,/~ are variables the types f3 are types, types, then then so so is is ((aa -+ --+ (3) ~).. The The untyped untyped lambda lambda calculus calculus can can express express the the full full range range of sequential sequential control and hence hence the the class class of of general general recursive recursive functions. functions. of control structures structures and For ap(x; x)))); ap(x; x)))) For example, example, the the Y Y combinator combinator A(f. A(f . ap(A(x. ap(A(x, ap(f; up(f; up(x; x)))); A(X. )~(x. ap(f; up(f; up(x; x)))) more A(X. xx)) ne recursive more commonly commonly written written A(f. A(f. JfA(x. xx)) A(X. A(x. xx) xx) is is used used to to defi define recursive functions. functions. We We have have that that Y(A(f. Y(A(f. F[J])) F[f])) = = F[Y(A(f. F[Y(A(f. F[J]))] Fir]))] so so that that Y Y "solves" "solves" the the recursive recursive definition definition J f - = F[J] F[f]. . In lambda calculus, In the the typed typed lambda calculus, Y Y is is not not typeable typeable because because the the self-application self-application A( x. ap( x; x)) A(x. up(x; x)) cannot cannot be be typed. typed. This This situation situation summarizes summarizes for for "typeless "typeless programming programming devotees" devotees" the the inherent inherent limitations limitations of of typed typed programming; programming; for for them them types types "get "get in in the the way." way." The The debate debate about about typed typed or or untyped untyped languages languages illustrates illustrates one one of of the the many many design design issues Other topics issues that that have have been been studied studied and and debated debated over over the the years. years. Other topics include: include: functional functional versus versus imperative, imperative, lazy lazy versus versus eager eager evaluation, evaluation, manual manual versus versus automatic automatic storage so forth. storage allocation, allocation, reflection reflection or or not, not, and and so forth. Many Many of of these these issues issues have have been been explored explored with with theoretical theoretical models, models, and and much much is about the design consequences. Indeed many is known known about the design consequences. Indeed many programming programming language language constructs setting of theories, e.g. constructs arose arose first first in in the the setting of formal formal logical logical theories, e.g. the the lambda lambda calculus, calculus, as algebraic type type systems, systems, binding binding mechanisms, mechanisms, block block structure, structure, abstract abstract data data types types ((as algebraic structures) modules. Just structures) and and modules. Just as as assembling assembling aa good good formal formal theory theory is is high high art, art, so so is is assembling programming language. assembling aa good good programming language. Both Both are are formal formal systems systems which which can can be be processed computers. But processed by by computers. But there there is is at at least least one one major major difference. difference. Good programming Good programming languages languages are are widely widely used, used, perhaps perhaps by by tens tens of of thousands thousands of of people their life times. Most Most logical people over over their life times. logical theories theories are are never never implemented, implemented, and and the the 2 best best of of those those that that are are might might be be used used by by less less than than one one hundred hundred people people over over aa lifetime.4 lifetime. 42 41 A compiler 41A compilertranslates high-level high-level language language programs programs into into another another language, language, typically typically a lower lowerlevel level language language such such as assembly assembly code code or native code code (machine (machine language). language). 42We 42We hope hope that the fact fact that Nuprl Nuprl contains contains a programming programming language language and that proofs proofs are will attract a significant significant audience. audience. executable will 756 756 R. Constable Constable believe that that this fact has has aa major consequence for for "theory "theory designers," designers," namely they II believe this fact major consequence namely they must must learn learn about about programming programming language language evolution. evolution. We We see see from from aa history history of of programming programming languages languages what what ideas ideas "work" "work",, what what com combinations of used. As binations of features features are are most most expressive, expressive, what what constructs constructs are are heavily heavily used. As with with the the evolution evolution of of natural natural languages, languages, the the speakers speakers exert exert aa force force to to mold mold the the language language to purpose. One to its its purpose. One of of the the lessons lessons of of programming programming language language history history is is that that types types are are ' s type component. We critical. critical. A A language language's type system system is is its its most most important important component. We also also know know that modularity modularity mechanisms are critical, critical, but this too is defined by the the type type system. system. that mechanisms are but this too is defined by The trend is The evolutionary evolutionary trend is toward toward ever ever richer richer type type systems-from systems--from the the fixed fixed types types of Fortran Fortran to to the the polymorphic polymorphic recursive recursive types types of of ML ML and and the the classes classes of of Java. Java. One One of might might argue argue that that this this development development must must eventually eventually subsume subsume the the type type systems systems of of the the mathematical true, and discussion of will mathematical theories. theories. II believe believe this this is is true, and our our discussion of type type systems systems will reveal reveal why. why. Role the role types in in programming Role of of types t y p e s in in programming. p r o g r a m m i n g . Let Let us us examine examine the role of of types programming (see (see the the excellent excellent article article by by Hoare Hoare [1972] [1972] as as well) well).. Fortran Fortran used used variable variable names names k, l,I, m, m, nn to beginning beginning with with ii,, j, j, k, to denote denote integers integers (fixed (fixed point point numbers), numbers), the the other other letters letters indicated indicated reals reals (floating (floating point point numbers). numbers). This This type type distinction distinction facilitated facilitated connection to to mathematical mathematical practice practice where where the the same same conventions conventions were were used, used, and and it it connection provided information information to to the the compiler compiler about about how how to to translate translate expressions expressions into into assem assemprovided bly language which also bly language which also made made the the distinction distinction between between fixed fixed and and floating floating numbers. numbers. Another Another important important type type in in Fortran Fortran and and Algol Algol was was the the array. array. Arrays Arrays represent represent sequences, matrices, matrices, tensors, tensors, etc. etc. A sequences, A typical typical specification specification (or (or declaration) declaration) of of this this type type dimensional array might might be be real real array[n, array[n, m] m],, a a two two dimensional array (matrix) (matrix) of of reals. reals. The The declaration declaration provides link to to important provides aa link important mathematical mathematical types types such such as as sequences sequences or or matrices, matrices, and and provides to the compiler on provides information information to the compiler on how how much much memory memory needs needs to to be be allocated allocated for for this data. this data. The links to The record record type type (or (or Algol Algol structure) structure) also also provides provides links to mathematical mathematical types types al :: and compiler. A and provides provides information information for for the the compiler. A typical typical record record syntax syntax is is record( record(a1 T ;1 . . . ;;an an ::Tn) Tn) where ai are called field Tt;... where T Tii are are types types and and ai are identifiers identifiers called field selectors. selectors. This This type type corresponds corresponds to to aa cartesian cartesian product product T T1l x ... .. . x• Tn Tn,, and and if if tt is is an an expression expression of of this this record record type, type, then then t.ai t.ai indicates indicates the the i-th i-th component, component, which which has has type type Ii Ti.. We We discuss discuss the the field field selectors selectors in in Section Section 4.4. 4.4. In In this this case case the the type type declaration declaration also also introduces introduces new new identifiers identifiers (or (or names) names) into into the the language. language. This This was was aa convenience convenience not not systematically systematically used used in in mathematics. mathematics. But But it bound it also also led led to to some some confusion confusion about about the the status status of of these these names names ai ai;; are are they they bound variables bound, what what is is their variables or or free? free? And And if if bound, their scope? scope? Here Here aa small small "convenience" "convenience" leads naming in leads to to interesting interesting new new questions questions about about scope scope and and naming in formal formal languages. languages. Algol , . . . , Tn) . This Algol 68 68 introduced introduced aa union union type, type, union(Tl union(Tt,...,Tn). This was was an an obvious obvious attempt link to mathematical types, attempt to to link to mathematical types, but but it it created created problems problems for for efficient efficient language language translation since the compiler might translation since the compiler might have have to to reserve reserve storage storage based based on on the the type type T Tii needing the needing the most most memory. memory. This This type type also also brought brought language language designers designers face face to to face face with set theory." in the with the the problems problems of of aa "computable "computable set theory." A A programmer programmer given given data data tt in the type B, C) type union(A, union(A, B, C) will will need need to to know know which which type type it it is is in. in. So So there there must must be be an an operation, like will decide operation, like decide(t) decide(t) which which will decide what what type type tt belongs belongs to. to. This This operation operation Types 757 757 is not not available available as as aa computable computable operation operation in in set set theory, theory, so so new new mathematics mathematics had had to to is be 68 was was rich rich in in aa "computable "computable mathematics mathematics of of types," types," and and its its be worked worked out. out. Algol Algol 68 reference reference manual manual isis aa type type theory theory which which inspired inspired both both logician logician and and computer computer scientist scientist alike. alike. Pascal the the union union type type was was considered considered to to be be aa variant of the the record record type. type. The The In variant of In Pascal record(xl :A1; : AI; x2:A2[x~]) which is is thought thought of of as as simplest such such structure structure is is essentially essentially record(x1 simplest X2 : A2 [xd) which aa union AI . This This is is aa restricted restricted version version of of our our union indexed indexed by by the the (necessarily (necessarily finite) finite) type type A1. dependent product product type type prod(A~; prod(AI; x. x. A2[x]) A2 [X]) from Section 3.7. 3.7. The The Pascal Pascal conception conception dependent from Section reveals both the the computational computational way to treat treat unions, unions, namely namely use use disjoint reveals both way to disjoint unions, unions, and and reveals the the implementation implementation strategy strategy (borrowed (borrowed from set theory)--use theory)-use elements elements from from reveals from set types are are type A1 Al as as tags tags on on the the data data to to keep keep track track of of the the disjunct. disjunct. So So ifif the the tag tag types aa type the booleans, booleans, ~, B, and and AI(i) A l (i) -= if then SS else else TT fifi then then prod(]~; prod{B; i.i. Al(i)) A l (i)) isis the the Algol Algol the if ii then 68 [i]).43 68 union(S, union(S, T) T) and and the the Pascal Pascal variant variant record record record(i record(i :]~; : B; x: x : A1 Adi]) . 43 Algol 60 60 and and Algol Algol 68 68 considered considered the the notion notion of of higher higher order order functions. functions. Algol Algol 68 68 Algol essentially had the the idea the type type fun(x:A)B fun (x : A)B as function from A to to essentially had idea of of the as the the type type of of function from A technology was up to to the the task task of functions B. But the the implementation implementation technology was not not up of returning returning functions B . But as values. values. This the community community to to implement implement it it correctly correctly as done in as This type type challenged challenged the as done in Scheme and ML closures. Scheme and ML with with closures. The function function space concept fun(x:A)B fun (x : A)B does does not not mean mean the the same same thing as the the The space concept thing as corresponding notion, A A -+ the constructive case. In corresponding mathematical mathematical notion, -~ B B even even in in the constructive case. In computational mathematics the elements elements ff of A -+ -+ B B are functions; that computational mathematics the of A are total total functions; that of A, A, f(a) f (a) converges converges to to aa value value bb in in B Whereas, the is, every element is, on on every element aa of B.. Whereas, the elements r1jJ of might diverge or abort of fu A)B are are partial partial functions, functions, that is, 1jJ(a) elements funn (x (x:: A)B that is, r might diverge or abort without returning This is without returning a a value. value. This is a a major major difference difference between between programming programming types types and and mathematical mathematical types. types. There There are are two two reactions reactions to to the the difference. difference. It It is is possible possible to to give give total total function function : A)B and semantics semantics to to fun(x fun(x:A)B and claim claim that that current current implementations implementations are are just just approxima approximations to logic with tions to the the idea. idea. The The full full concept concept emerges emerges in in a a programming programming logic with termination termination rules hand, one rules (Dijkstra (Dijkstra [1968]). [1968]). On On the the other other hand, one can can regard regard the the partial partial function function space space as a a new new mathematical mathematical construct construct and and try try to to work work out out axioms axioms and and models models for for it it as (Scott (Scott [1976], [1976], Plotkin Plotkin [1977]). [1977]). Both Both approaches approaches have have been been pursued. pursued. (x : A)B to Notice Notice that that it it is is aa simple simple manner manner to to extend extend fun fun(x:A)B to dependent dependent function function types types by by allowing allowing B B to to depend depend on on x x.. This This type type is is then then closely closely related related to to fun fun (A; (A; x. x. B) B) of of Section Section 3. 3. A A more more modern modern addition addition to to the the type type structure structure of of programming programming languages languages is is the the module module or or object object (or (or ADT ADT or or package package or or unit). unit). This This concept concept can can be be traced traced to to Simula Simula 67 and and is is well well developed developed in in Modula Modula and and SML. SML. Among Among the the interesting interesting experimental experimental 67 languages languages for for modules modules were were Russell Russell at at Cornell Cornell (Demers (Demers and and Donahue Donahue [1980] [1980],, Boehm Boehm et et al. al. [1986]) [1986]),, CLU CLU at at MIT MIT (Liskov (Liskov and and Guttag Guttag [1986]), [1986]), and and Modula Modula at at DEC. DEC. The The basic basic idea idea is is that that aa module module is is aa type, type, say say D D,, and and aa collection collection of of operations operations fi fi 43The 43The actual Pascal syntax is is very very baroque, and the so so called called free free unions unions are a well well known known place place for for "breaking" "breaking" the type discipline discipline since since the user user must must keep keep track of the dependency. dependency. Note Note the if_then_else_fi if_then_else_finotation is the Algol Algol 68 way of "bracketing" "bracketing" the conditional conditional with delimiters delimiters if, Ii fi.. 758 758 R. Constable Constable R. This isis the the type type of of aa structure structure in in algebra algebra (Bourbaki (Bourbaki on D D and and auxiliary auxiliary types. types. This on [1968a]) and and model model theory theory (Chang (Chang and and Keisler Keisler [1990]). [1990]). For For example, example, we we might might have have [1968a]) (D, (D, f,j, g,g, e} e) where where the the signature signature of of the the module module isis list list of of types types of of the the components, components, e.g. D D EE Type, Type, fj :: DD xx DD -4 -+ D, D, gg :: DD -4 -+ ~, 1m, ee E E D. D . AA group group would would have have signature signature e.g. G n v :: GG -4 G EE Type, Type, oopp :: GG xx GG -4 -+ G, G, iinv -+ G, G, ee EE G, G , and and then then there there would would be be axioms axioms saying that that op op isis associative, associative, inv inv isis an an inverse inverse and and ee an an identity. identity. saying The module module concept concept corresponds corresponds exactly exactly to to dependent dependent types types over over Type. Type. In The In Section 22 we we would would denote denote the the type type of of groups groups (signature) (signature) as as Section G 4 G) G :Type : Type xx op: op : (G (G xx G G--+ G) xx inv: inv : (G (G -4 -+ G) G) xx ee ::GG Except for the fact that the the function function types types in in the the programming programming type type are are partial partial Except for the fact that and Type Type has has less less mathematical mathematical structure, structure, the the algebraic algebraic concept concept and and the the programprogram and ming one one are are similar. similar. ming We will will see see that that the the notion notion of of subtype and inheritance inheritance that that is so critical modern We subtype and is so critical to to modern programming be nicely nicely captured theory. This This leads leads to to aa programming practice practice can can be captured in in our our type type theory. object-oriented programming programming (c.f. (c.f. mathematical treatment of the central concepts concepts in mathematical treatment of the central in object-oriented Meyer [1988]). [1988]). Meyer Looking over the types above we these uses. Looking over the types described described above we discern discern these uses. 1. Types 1. Types relate relate data data in in the the machine machine to to standard standard mathematical mathematical concepts. concepts. 2. Types express the domain of significance of a programming problem 2. Types express the domain of significance of a programming problem and and impose impose constraints to be constraints on on the the data data for for it it to be "meaningful" "meaningful" in in the the sense sense that that the the computer computer will not "crash" "crash" (attempt (attempt to to execute execute aa meaningless meaningless instruction) and the the data data will not instruction) and will not not fail to represent mathematical objects.44 objects. 44 Usually Usually these these constraints constraints will fail to represent mathematical can can be be rapidly rapidly checked checked to to provide provide some some level level of of assurance assurance that that aa program program is is sensible. sensible. 3. 3. Types Types provide provide aa notation notation for for structuring structuring aa solution solution by by decomposing decomposing aa task task into into components components (modules) (modules) and and levels levels of of abstraction. abstraction. 4. "debugging" ) aa computa 4. Types Types provide provide an an interface interface language language for for analyzing analyzing (("debugging") computation. tion. 5. 5. Type Type information information can can be be used used to to increase increase the the performance performance of of the the compiled compiled code. code. There There is is aa direct direct historical historical link link from from Russell Russell and and Church Church to to languages languages like like Algol Algol and and Lisp. Lisp. Also Also we we are are seeing seeing aa close close correspondence correspondence between between mathematical mathematical types types and and data data types: types: Cartesian Cartesian products products correspond correspond to to record record types, types, unions unions to to disjoint disjoint unions unions (or (or variant variant record record types), types), function function spaces spaces to to procedure procedure types, types, inductive inductive types types to to recursive recursive data data types, types, algebraic algebraic structures structures to to modules modules (and (and superstructures superstructures correspond correspond to to subtypes) subtypes).. The The integers integers are are included included in in some some programming programming languages languages as as the the data data type type "bignums" "bignums",, and and real real numbers numbers are are (badly) (badly) approximated approximated by by "floating "floating point point numbers" numbers".. In In aa sense sense the the system system of of data data types types provides provides aa computational computational type type theory theory capable capable of of organizing organizing and and unifying unifying programming programming problems problems and and solutions solutions in in 44Crashing 44Crashingcan can mean meanaa complete completefailure failureto to respond respondor or an an unwanted unwantedresponse responsefrom fromthe the operating operating system "bus error") "segmentation fault" ). system (("bus error") or or from fromthe the hardware hardware (("segmentation fault"). Types 759 759 the same same way way that that type type theory theory organizes organizes and and unifies unifies computational computational (also (also constructive constructive the and and intuitionistic) intuitionistic) mathemataical mathemataical problems problems and and solutions. solutions. The The continuing continuing (rapid) (rapid) evolution of of programming programming languages languages will will probably probably lead lead to to data data type type theories theories that that evolution subsume mathematical mathematical type type theories. theories. There There may may be be new new data data types types appropriate appropriate for for subsume expressing expressing the the problems problems of of interaction interaction as as well well as as those those of of "functional "functional action" action" which which now dominate. now dominate. Although Although the the similarities similarities between between types types and and data data types types just just enumerated enumerated is is com compelling and think it pelling and interesting, interesting, II think it is is also also important important to to understand understand the the differences. differences. These These differences differences challenge challenge us us to to find find logical logical foundations foundations for for new new types. types. 4.2. 4.2. Type Type E E type t y p e and a n d domain d o m a i n theory theory Given Given that that programming programming types types are are not not the the same same as as mathematical mathematical ones, ones, might might it it be allow aa type all types, types, precisely theory was be sensible sensible to to allow type of of all precisely the the notion notion that that type type theory was created in accordance with the the vicious One fact know created to to disallow disallow in accordance with vicious circle circle principle? principle? One fact we we know from the work of of Meyer and Reinhold Reinhold [1986] [1986] and Howe [1991,1989,1987,1996b] [1991,1989,1987,1996b] is is from the work Meyer and and Howe that adding the typing typing rule rule Type Type E E Type Type to the simply simply typed typed lambda allows to the lambda calculus calculus allows that adding the new typed among new terms terms to to be be typed among which which are are applications applications that that fail fail to to terminate. terminate. No No such such terms typed without without this this new the other other hand, this rule would not terms can can be be typed new rule. rule. On On the hand, this rule would not cause the type "collapse" in cause the type system system to to "collapse" in the the sense sense that that every every term term could could be be typed typed or or every every term term belongs belongs to to every every type type (as (as would would happen happen if if we we added added the the rule rule T T1l = = T T22 for for Indeed, we any any two two types types Tl T1 and and T T2). we know know that that such such aa type type system system has has aa nontrivial nontrivial 2 ) . Indeed, mathematical (Cardelli [1994] mathematical model model (Cardelli [1994],, Meyer Meyer [1988]) [1988]).. The The discovery discovery of of interesting interesting mathematical mathematical models models for for programming programming language language types programming language semantics. It types is is aa flourishing flourishing topic topic in in the the field field of of programming language semantics. It theory pioneered has led directly the rich rich subject has led directly to to the subject of of domain domain theory pioneered by by Dana Dana Scott Scott [1970a,1970b,1972,1976] [1970a,1970b,1972,1976] led led early early on on by by Gordon Gordon Plotkin Plotkin [1975]. [1975]. (The (The results results of of Plotkin Plotkin [1981], [1981], Abramsky Abramsky [1993] [1993], Reynolds Reynolds [1981], [1981], Cardelli Cardelli [1994], [1994], Mitchell Mitchell [1996], [1996], ' Gunter Gunter [1994] [1994],, Egli Egli and and Constable Constable [1976] [1976],, and and Abadi Abadi and and Cardelli Cardelli [1996] [1996] are are quite quite relevant relevant to to the the work work discussed discussed here.) here.) One One of of the the major major early early discoveries discoveries of of domain domain theory theory is is that that there there are are referential referential or or "denotational" "denotational" mathematical mathematical models models of of partial partial function function spaces, spaces, in in particular, particular, of of the the untyped untyped lambda lambda calculus calculus in in which which function function equality equality is is extensional extensional (see (see Scott Scott [1976]) [1976]).. The as been The challenge challenge for for domain domain theory theory hhas been to to relate relate these these models models to to the the standard standard mathematical mathematical types types and and type type theories. theories. This This remains remains an an active active area area of of research research with with especially especially promising promising recent recent results results in in analysis analysis (Edalat (Edalat [1994]). [1994]). Let Let us us call call types types which which allow allow diverging diverging elements elements partial partial types. types. Given Given that that there there is is aa consistent consistent theory theory of of partial partial types types allowing allowing Type Type EE Type Type and and that that this this rule rule drastically simplifies the drastically simplifies the theory, theory, we we proceed proceed to to explore explore it. it. One One view view of of this this theory theory is is that that it it speaks speaks about about aa domain. domain. Another Another is is that that it it is is aa "partial theory" which "partial type type theory" which will will require require refinement refinement as as more more constraints constraints are are added, added, such such as as totality totality restrictions. restrictions. But But until until we we require require totality, totality, the the vicious vicious circle circle principle principle has has no no force force since since its its consequence consequence is is merely merely aa nonwell nonwell founded founded concept concept (nontermi (nonterminating term) nating term).. This This approach approach to to type type theory theory permits permits aa great great deal deal of of freedom-partial freedommpartial 760 760 R. Constable Constable objects }, objects are are allowed, allowed, illogical illogical comprehension comprehension is is possible, possible, e.g. e.g. {x ( x : : Type Type II xx E Xx}, negative definitions are negative recursive recursive definitions are allowed allowed (see (see Section Section 4.3) 4.3),, and and concepts concepts need need not not be be referential required. It referential since since equality equality relations relations are are not not required. It will will be be left left to to the the programming programming logics these "unruly" logics to to impose impose more more logical logical order order on on these "unruly" types. types. One this theory products taken One of of the the first first benefits benefits of of this theory is is that that dependent dependent products taken over over Type Type provide module is provide a a notion notion of of module. module. The The signature signature (or (or type) type) of of a a module is M M :: Type Type x• F(M) F(M) M.. By where where F(M) F(M) is is a a type type built built from from M M such such as as M M x • M M -+ --+ M By iterating iterating this this construct construct we we get get the the general general structure structure of of a a module module (XO) x0:: Type Type x • Xl Xl:: TI Tl(X0) Xo x • . . . X• Xn (xo, . . . , Xn-l ) ' x n : : Tn Tn(xo,...,xn-1). 4.3. 4.3. Recursive R e c u r s i v e types types As As we we have have seen, seen, inductive inductive definitions definitions and and principles principles of of inductive inductive reasoning reasoning lie lie at at the and logic. the heart heart of of computational computational mathematics mathematics and logic. The The inductive inductive definition definition of of the the natural lists, and mind. The natural numbers, numbers, lists, and formulas formulas come come immediately immediately to to mind. The elements elements intro introduced inductively inductively can can be be represented represented in in computer computer memory memory by by linked linked data data structures structures duced A, say ), constructed constructed from from pointers. pointers. For For example, example, a a list list of of elements elements of of type type A, say ((aa Il ,, . ... ., , an an), would would be be represented represented by by a2 I] t-+ an I[ t-+ -Jr---+I[a2 -~ "-- � ~ ~ nil nil I[alal I[ t-+ I[an where where the the arrows arrows are are pointers pointers (data (data of of type type address address or or in in Algol Algol 68 68 terminology, terminology, references thus of type ref(A) references to to A A objects, objects, thus of type ref (A)).) . A A seminal seminal discussion discussion of of these these methods methods can can be be found found in in C.A.R. C.A.R. Hoare Hoare's's article article Notes Notes on on Data Data Structuring Structuring [1972] [1972]..45 4~ One the most One of of the most decisive decisive uses uses of of types types in in programming programming languages languages is is in in defining defining recursive recursive data data types types at at the the same same level level of of abstraction abstraction used used in in mathematics. mathematics. This This innovation innovation was was pioneered pioneered by by Lisp Lisp and and its its treatment treatment of of lists lists without without explicit explicit mention mention of pointers. The of pointers. The pointer pointer representation representation is is managed managed by by the the run-time run-time system system of of pro procollector is gramming language, language, and called aa garbage gramming and a a program program called garbage collector is used used to to dynamically dynamically manage manage the the allocation allocation and and deallocation deallocation of of memory memory for for lists lists and and other other inductive inductive structures. structures. In In programming programming these these inductive inductive types types are are called called recursive recursive types types or or recursive recursive data structures with recursive data structures by by analogy analogy with recursive programs. programs. They They include include circular circular data data structures, structures, unfounded unfounded lists lists (or (or streams) streams) and and other other "nonwell-founded" "nonwell-founded" recursive recursive data data ,, 6 that 4 The that would would not not be be considered considered as as properly properly "inductive. "inductive. ''46 The definition definition of of such such a a 45The small book Structured Structured Programming, Programming, Dahl, Dijkstra and Hoare [1972], [1972], is one of the gems of computer science. science. All of computer All three three articles articles are are closely closely related related to to the the subject subject of of this this section. section. 46Perhaps the reason for the popularity of the term "recursive data type" comes from Hoare's evocative evocative analogy: analogy: "There "There are are certain certain close close analogies analogies between between the the methods methods used used for for structuring structuring data . . . a discriminated union data and the methods for structuring a program which processes that that data data.., corresponds corresponds to to aa conditional conditional ..... . arrays arrays to to for ]or statements statements ..... . sequence sequence structure structure ..... . to to unfounded unfounded looping . . . The looping ..... The question question naturally naturally arises arises whether whether the the analogy analogy can can be be to to aa data data structure structure corresponding to recursive procedures." Types Types 761 761 type isis disarmingly disarmingly simple simple to to paraphrase paraphrase Hoare: Hoare: "write "write the the name name of of the the type type being being type defined inside inside its its own own definition." definition." In In his his notation notation we we write write defined type TT == F[T] F[T] type where FIX] type definition definition in in X. X . If If we we use use ++ for for disjoint disjoint union union and and 11 for for the the unit unit where F[X] isis aa type type and and xx for for cartesian cartesian product, product, then then here here are are the the definitions definitions for for natural natural numbers numbers type and lists lists over over aa type type A. A. and type NN := 11 ++N type N list LL := 1l +(A +(A xx L). L). list We will will use use aa more more compact compact notation, notation, writing writing aa single single term term with with aa binding binding construct. construct. We Our notations A xx L) L) where where N N and and LL notations for for these these types types are are #(N. J-t(N. 11 ++ N), N) , #(L. J-t(L. 11 ++ A Our are bound F[T] isis aa type type expression expression in in T, T , then then #(T. J-t(T. FIT]) F[T]) bound variables. variables. In In general, general, ifif FIT] are denotes type used above to giving the denotes the the recursive recursive type used above to illustrate illustrate Hoare's Hoare's notation. notation. In In giving the rules for for recursive recursive types, will use A --+ -+ B B and for the the programming rules types, we we will use A and xx:: AA -+ -~ B[x] B[x] for programming type fun so the the elements elements are functions. type f u n ((x x : :AA)B; ) B ; so are partial partial functions. J-t (x. F[x]) E Type redype_def 11.. H Ig F- # (x. F[x]) e Type rec_type_def H, xx ::Type Type I-F- F[x] H, Fix] E E Type Type 2. (x. Fix]) F[x]) redype_member 2. H g It- tt E e J-t # (x. rec_type_member H g It- tt E e F F [J-t(x. [#(x. F[x])] F[x])] 3. redype_elim 3. fI /~ IF- J-t # (t; (t; f, f, y. y. g[f, g[f , y]) y]) E e G G rec_type_elim fI, H, x x:: Type, Type, f f ::x z -+ --+ G, G, yy:: F[x] Fix] It- g[J, g[f , y] y] E e G G E J-t(x. F[x]) fI It R F- t e #(x. F[x]) The The term term J-t # (t; (t; f, f, y. y. g[J, g[f, y]) y]) is is called called aa recursion recursion combinator. combinator. It It is is the the recursive recursive program program associated associated with with the the recursive recursive definition. definition. The The evaluation evaluation rule rule is is 9g [-X z; f, y. g[J, [~ ((z. z . J-t , ((z; f, y. g[f, y])) y ] ) ) // ff,, t/y] t/y] .\.$ aa J-#t (t; (t; f, g[J, y]) y]) .\.$ aa f, y. y. g[f, The The operational operational intuition intuition behind behind these these rules rules is is this. this. A A recursive recursive type type type type T T = = F[T] F[T] is is well well formed formed exactly exactly when when its its "body" "body" F[T] F[T] is is aa type type under under the the assumption assumption that that T T is is aa type. type. This This is is "writing "writing the the name name of of the the type type being being defined defined in in its its own own definition." definition." To To construct construct aa member member of of the the type, type, build build aa member member of of F[T] FIT],, and and if if this this construction construction requires requires an an element element of of T T,, then then apply apply the the construction construction recursively recursively (in (in the the implementation, implementation, use use aa pointer pointer to to T T and and build build recursively). recursively). The The process process may may not not terminate terminate unless unless there there is is aa "base "base case" case" which which does does not not mention mention T T,, as as in in the the left left disjunct disjunct of of 11 + +T T or or of of 11 + +A A xx T T.. A A definition definition like like J-t(X. #(X. X) X) is is empty empty because because no no element element can can be be created, created, likewise likewise for for J-t(X. #(X. X X + + X) X) or or J-t(T. #(T. T T xx T) T).. Note Note however however that that J-t(T. #(T. T T -+ --+ T) T) will will contain contain the the element element -X(x. A(x. x) x) by by this this application application of of rules rules I-t- -X(x. A(x. x) x) Ee J-t(T. #(T. T T -+ -+ T) T) TT": Type Type I-F--X(x. A(x. x) x) EE T T -+ --+ T T T. T. Type, Type, xx": T T I-F-xx EE T. T. 762 762 R. Constable R. Associated with with #J.l (x. (x. F[x]) F[x]) isis aa method method of of recursive recursive computation computation (as (as Hoare Hoare Associated suggested and and as as we we know know from from inductive inductive definitions definitions in in mathematics). mathematics). If Ifthe the recursive recursive suggested type isis "well-founded" "well-founded" then then this this procedure procedure will will terminate, terminate, otherwise otherwise itit might might not. not. type The recursive recursive procedure procedure isis the the following. following. Given Given tt EE #J.l (X. (X. Fix]), F[x]) , to to compute compute an an The , use a program 9 that computes on elements of F[x] . This element of of type type G, element use a program g that computes on elements of F[x]. This C procedure may decompose decompose tt into into components components tt'~ of of type type #J.l (x. (x. Fix]). F[x]) . In In this this case, case, procedure g9 may call the the procedure procedure recursively. recursively. To To specify specify this this we we note note that that ifif we we consider consider that that tt call belongs to to Fix], F[x] , then then component component tt'~ will will belong belong to to X. X . The The recursive recursive call call of of the the belongs procedure isis represented represented in in the the rule rule by by the the function function variable variable fj from from X X to to G. We see see procedure C . We from the the evaluation evaluation rule rule that that this this is is used used exactly exactly as as aa recursive recursive call. call. from This method method of of organizing organizing the the rules rules comes comes from from Constable Constable and and Mendler Mendler [1985] [1985] This and Mendler Mendler [1988]; [1988] ; itit can can be be made made more more expressive expressive using using the the subtyping subtyping relation relation and SS E_ parameterized recursions. !;,;; T T and and dependent dependent function function types types and and parameterized recursions. First, First, with with dependent types types we we get get dependent g) EE Gin] (x. F[X]) fl, uu ':#J.l (X. /~, FIX]) f~ #J.l (u; (u; j, f, y. y. g) C [u] [x]) , yy": F[X] F[X] f-t-- 9g Ee G[y] i-I, X :"Type, Type, f" G[x]), fl, X j : (x" (x : XX ~-+ C C [y] The form of of recursive type to to depend depend on on aa The parameterized parameterized form recursive type type allows allows the the defined defined type parameter of type A The syntax syntax is is #J.l (X. (X. F[x]) @a parameter of type A.. The f[x]) @a (X. F[X])@a F[X]) @a EE Type Type Ip. fl f-~- J.#l (X. l p. /~ 2p. 2p. 3p. 3p. p. 44p. fl, i-I, X X": A A -+ ~ Type Type ft- F[x] f[x] E E (A (A -+ ~ Type) Type) fl H t f- aaE EA A flt /~t E E J.l # (X. (X. F[x]) F[x]) @a @a fl @y] (a) /~ fF- tt E EF F [,\ [A (y. (y. J.#l (X. (X. F[X])) FIX]))@y] (a) fl l (a; t;t; j, H f~ J.#(a; f, u, u, y. y. g) g) E EC G l (X. F[X])@u) fI, X X :"(A (A -+ -~ Type) Type),, Vu Vu :. A. A. (X(u) (X (u) !;,;; E_ J.~(X. F[XI)~u) fl, f-~- g[j, g[.f, u, u, y] y] EE C G f-~ aaEEA A f-F- tt EE J.l(X. #(X. F[x])@a F[x])@a u. '\~ (~. r. J.~(u; l u; r; gg [~'\ (u. r; j, S, u, u, y. y. gg))) IS, ~lu, t/y] .~ J.#(a; l a; t;t; jf,, u, u, y. y. 9g) $ c _ We We can can combine combine the the parameterized parameterized form form and and the the dependent dependent form; form; such such rules rules are are 't use given given in in Constable Constable et et al. al. [1986] [1986] and and Mendler Mendler [1988], [19SS], but but we we won won't use this this level level of of complexity complexity here. here. The The parameterized parameterized recursive recursive types types can can be be used used to to define define mutually mutually recursive recursive types types since since we we can can think think of of J.l(X. #(X. F[x])@u F[x])@u as as aa family family of of simultaneously simultaneously recursively recursively defined defined types. types. With With the the propositions-as-types propositions-as-types principle principle and and restricting restricting the the recursive recursive types types to to be be well-founded, well-founded, we we get get recursively recursively defined defined relations. relations. These These have have been been exploited exploited well well in in the the Coq Coq theorem theorem prover prover (Coquand (Coquand and and Paulin-Mohring Paulin-Mohring [1990] [1990],, Coquand Coquand [1990], [1990], Paulin-Mohring Paulin-Mohring and and Werner Werner [1993]). [1993]). With With recursive recursive types types and and disjoint disjoint unions unions and and aa unit unit type type we we can can define define natural natural numbers numbers and and lists lists as as we we have have shown. shown. Using Using record record types types we we can can define define pairs pairs of of numbers numbers which which gives gives us us integers integers and and rational rational numbers. numbers. (Using (Using function function types types we we can can define define the the computable computable reals; reals; see see Bishop Bishop [1967], [1967], Chirimar Chirimar and and Howe Howe [1991], [1991], Forester Forester Types 763 Booleans can be defined [1993].) [1993].) Booleans can be defined as as 11 + + 1. So the the number number of of primitives primitives for for aa rich rich 1. So type will examine type theory theory can can be be reduced reduced to to a a very very small small set. set. We We will examine some some especially especially interesting reductions reductions in interesting in the the next next section. section. E x a m p l e defining defining primitive p r i m i t i v e recursion r e c u r s i o n on on N. N. To To illustrate illustrate the the workings workings of of the the Example recursion recursion combinator combinator p,O #(),, we we use use it it to to define define primitive primitive recursive recursive functions functions from from N N to to G.. Suppose Suppose 1 f is is defined defined primitive primitive recursively recursively on on p,(X. #(X. 1 + X) X) to to G G by by G 1(0) S(0) = =bb I(suc(u)) f(suc(u)) = h(n, h(n, I(n)). f(n)). Then Then the the corresponding corresponding combinator combinator is is p, # (u; (u; l, f, u. u. decide decide (u; (u; v. v. b; b; v. v. h(v, h(v, l(v)))) f(v)))) whose typing typing is is seen seen from from the the judgment. judgment. whose X: : Type, Type, I ]:: X X -t --+ G, G, u u:: 1 + X X f~- decide(u; decide(u; v. h(v, I(v))) f(v))) E E G. G. X v. b;b; v. v. h(v, Typing Typing a a fixed fixed point point combinator. c o m b i n a t o r . While While the the recursion recursion combinators combinators are are essential essential for inductive types, indeed they characterize characterize them, them, in in aa rich rich enough enough partial type for inductive types, indeed they partial type theory defined. The theory they they can can be be defined. The idea idea is is to to use use the the richness richness of of the the recursive recursive types types to to assign assign a a type type to to a a fixed fixed point point combinator, combinator, like like Y Y.. Recall Recall that that the the Y Y combinator combinator is is abbreviated >.(x. g(xx))>.(x. g(xx))) or >.(x. g(xx)) abbreviated >' ik (g. (g. ~(x. g(xx))~(x, g(xx))) or still still further further by by letting letting w w= = )~(x. g(xx)) and We show that Y and writing writing Y Y as as >.(g. A(g. ww) ww).. We show that Y has has type type (T (T -t -+ T) T) -t -~ T T for for any any type type T T p,(X. X by by using using the the auxiliary auxiliary recursive recursive type type S S == = - #(X. X -t --+ T) T).. Here Here is is the the derivation. derivation. The T, the T. The The type type of of 9 g will will be be T T -t -+ T, the type type of of w w is is S S -t ~ T. The "trick" "trick" is is to to type type ap(x; x) to to be be of of type type T T.. We We examine examine the the typing typing derivation derivation for for w w.. ap(x; x) fp,(X. X F- >.(x. ,~(x. g(xx)) g(xx)) E E #(X. X -t --+ T) T) by by p,Jllembership #_membership by )~(x. g(xx)) g(xx)) E ES S -t -~ T T by -t --+ R R f-~- >.(x. gg :: T T - -t + T, T , xx::SS fby t- g(xx) g(xx) E ET T by -t ~ L L by xx E ET T by ap ap f-~- xx f-~- xx EE S by S -t ~ T T by unroll unroll x x f-~-xx EE S by S by hyp h y p xx Once Once we we know know that that w wE ES S -t ~ T T and and w wE ES S,, then then ww ww E ET T and and g(ww) g(ww) E ET T.. One corollary of this typing is that Y(>.(x. x)) belongs to the empty type p,(X. One corollary of this typing is that Y(:k(x. x)) belongs to the empty type #(X. X) X) called called void, void, since since >.(x. A(x. x) x) E E void void -t --+ void. void. But But Y(>.(x. Y()~(x. x)) x)) is is aa diverging diverging term, term, so so it it is is not not aa value value belonging belonging to to void. void. Indeed, Indeed, we we can can easily easily show show that that there there are are no no values values of of type type void. void. Now including the Now we we can can use use Y Y to to define define any any partial partial recursive recursive function, function, including the recursion recursion combinators of type p,(X. F) -t G . In general, >. (x. p,(x; I, u . g[l, combinators of type #(X. F) ~ G. In general, ~ (x. #(x; f, u. g[f, u])) u])) is is just just Y (>. (J. >.( u. g[l, u]))) . The type of 1 is (p, (X. F) -t G) -t (p,(X. F) -t G) Y ()~ (f. ~(u. g[f, u]))). The type of f is (# (X. F) --+ G) ~ (#(X. F) ~ G),, and and we we observed observed that that g[l, g[f, u] u] E EG G can can be be derived derived from from this this typing typing of of I f .. gg:T--+ : T -t T T Applying Applying this this general general construction construction to to primitive primitive recursion recursion we we get get the the term term Y(>.(J. v. b;b;v. v. h(v, l(v)))))) , which Y()~(f. >.(n. ik(n. decide(u; decide(u;v, h(v,f(v)))))), which is is R R,, the the primitive primitive recursion recursion combinator, (with (with bb and and h h as as parameters). parameters). combinator, 764 764 R. Constable types. Constable Constable and and Mendler Mendler [1985] [1985] and and Mendler Mendler [1988] [1988] gave gave conditions conditions IInductive n d u c t i v e types. needed to to guarantee guarantee that that recursive recursive types types #(X. j.L(X. F) F) define define only only total total objects. objects. One One needed be aa monotone monotone operation operation on on types types in in the the sense sense that that such condition condition isis that that FF be such ZX E � YY =~ => FIX] F[X] E� FLY]. F [Y] . We We also also studied studied conditions conditions to to guarantee guarantee that that elements elements of of these types types are are functional. functional. The The result result is is aa set set of of rules rules used used in in Nuprl Nuprl for for inductive inductive these types (c.f. (c.f. Constable Constable et et al. al. [1986], [1986], Hickey Hickey [1996a]). [1996a]). types When FF is When is required required to to be be monotone, monotone, then then we we cannot cannot define define the the type type #(X. j.L(X. X X --+ -+ T) T) used used in in typing typing Y. Y . Indeed, Indeed, itit is is not not possible possible to to type type Y Y nor nor divergent divergent elements. elements. For For this reason reason the the #(x; this j.L(x; f,f, u. u. g) g) recursion recursion forms forms are are needed. needed. They They provide provide the the structural structural induction rules rules for for inductive inductive types. types. In In Nuprl Nuprl these these induction induction rules rules for for recursive types induction recursive types Y(>.. (J. b)) b)) are are can be be used used to to prove prove that that certain certain applications applications of of the the Y Y combinator, combinator, Y(A(f. can indeed total total objects objects (see (see Constable Constable et et al. al. [1986]). [1986]). So So we we get get the the advantages advantages of of general general indeed recursive programs programs without without losing losing the the logical logical structure structure of of type type theory. theory. recursive 4.4. 4.4. Dependent D e p e n d e n t records records aand n d vvery e r y ddependent e p e n d e n t ttypes ypes We are core type type system system that that will of the We are aiming aiming to to exhibit exhibit aa small small core will generate generate all all of the types we studied. The step in direction that that we take here of considerable considerable types we have have studied. The step in this this direction we take here is is of practical value--it value-it builds builds record spaces. practical record types types from from dependent dependent function function spaces. n } be == {( 1I ,, .. .. .., , n} Consider the the record type record(x : Al, . . . , Xn Nn == Consider record type record(x1l "A1,..., xn "An). Let Nn be : An) . Let element enumeration type-it can can simply simply be be 1 ++. .. .. . + an taken n times. Define Define an nn element enumeration type--it + 11 taken n times. to Type. Then the the essential the record B(i) -= A~ from Arm Type. Then essential structure structure of of the record Ai from Nm to aa function function B(i) : Nn -+ this type, type, ff(i) is given by the the dependent dependent function space ee'Nn is given by function space -~ B(i) B(i).. Given Given ff in in this (i) is the i-th component. We display form form for for record record selection selection if if we ne is the i-th component. We obtain obtain aa nice nice display we defi define f· Xi == ff(i). (i) . f. xi ---- This definition of properties. In This definition of records records has has nice nice subtyping subtyping properties. In aa standard standard record record calculus calculus a a record record type, type, rr~, is aa subytpe subytpe of of record record type type rr2, written rr~l � _ rr2, iff rrll 2 , iff 2 , written l , is has fields. So has additional additional fields. So a a colored colored point point is is aa subtype subtype of of aa point point or or aa group group type type is is aa subtype subtype of of monoid monoid type, type, etc. etc. Our Our definition definition provides provides this this subtyping subtyping directly directly from from the subtyping that if the subtyping relation relation on on function function spaces. spaces. Recall Recall that if A A1l � __E_A A2, B1l � E_ B B22 then then 2, B A Also if A22 -+ -~ Bl B1 � _ Al A~ -+ --+ B B2. if Nn Nn � _ N Arm, and n n � _ m m,, and and B B~l (i) (i) = B B2(i) for ii E EN Nnn 2 . Also m , and 2 (i) for then then i'Nm -+ Bl(i) E_ i'Nn ~ B2(i). : Nn -+ Notice Notice that that f f E E (i (i": N Arm -+ B BI(i)) is an an element element of of ii'Nn -+ B B2(i) simply by by the the 2 (i) simply l (i)) is m -+ polymorphic nature polymorphic nature of of functions functions (Le. (i.e. they they are are rules rules given given by by >.. A terms) terms).. Encoding records. The : Al x• A E n c o d i n g dependent d e p e n d e n t records. The dependent dependent product product types, types, x x:A1 A2[x] offer aa 2 [x] offer A l ; X2 form form of of dependent dependent record record as as mentioned mentioned above. above. The The general general form form is is record(x record(x1l ::A1; x2:: A [x l , . . . , Xn- l]) . Can A2[xl];... An[Xl,...,xn-1]). Can we we also also define define these these records records as as dependent dependent Xn : An 2 [xl ] ; . . . ;;xn: functions? functions? The The existing existing dependent dependent function function space space is is not not adequate adequate for for this this task, task, but but Jason Jason Hickey Hickey [1996a] [1996a] has has discovered discovered an an extension extension that that he he calls calls very very dependent dependent function function Types Types 765 765 spaces. B[x]).. spaces. The Thebasic basicnotation notationisis fun(A; fun(A;f,f,x.x. B[f, B[f, x]) xl) as as opposed opposed to to fun(A; fun(A; x. x. B[xl) The function Theidea ideaisisthat thatthe thetype type BB can can depend depend not not only only on on the the argument argument to to the the function sosothat values" of of g(a) EE B[a], B[a] , but but now now the the type type of of BB can can depend depend on on "previous "previous values" that g(a) g,g ,sosog(a) the elements elements g(a) EEB[g, B[g,a]. a] . To Tosee see how how the the idea idea works, works, let's let's use use itit to to define define the ofofxlXl"At (a~,a2) where Note A2 A2 "A~ -+ Type, Type , and and an an element element is is (al , a2 ) where : A 2 (XI) . Note : Al --+ : Al • xx2X2"A2(x~). ala l eE Al, Imagine that that A I ,a2a2 eE A2(al). A2 (al ) . The The encoding encoding isis based based on on N2 N2 == {1,2}. {I, 2} . Imagine B(1) say this this if if we we had had We could could say where a~ a l eE A~. A I . We andwe wewant want B(2) B(2) == A2(a~) A 2 (al) where B(l)==A~, AI ,and the element g such that g(1) E A1. So if we add g as a parameter to B we can say the element g such that g(l) E A I . So if we add g as a parameter to B we can say B(g, B(g, 1)1) == A1 Al S(g, B(g, 2)=A2(g(1)). 2) = A 2 (g(1)). This 2),, gg is is referenced referenced only only This particular particular definition definition makes makes sense sense because because at at B(g, 2) atat previous basis for for defi defining the simplest simplest Hickey takes takes this this as as the the basis ning the previous arguments. arguments. Hickey very on gg as as prerequisite prerequisite to to very dependent dependentfunction function space. space. He He requires requires aa well-ordering well-ordering on forming theory we we can can get get away away with with formingthe thetype type (see (see Hickey Hickey [1996a]). [1996al ) . In In aa partial partial type type theory less. generate an an ordering ordering on on values values via via less. AA particular particular computable computable function function gg will will generate its Big, x] x] in in forming forming the the type, type, itscomputation. computation. So So we we can can allow allow arbitrary arbitrary expressions expressions B[g, but satisfying the the constraints constraints of of B. B. The The but itit will will be be empty empty unless unless there there isis aa function function satisfying (viciously circular) circular) rules rules are: are: (viciously fun(A; f,f, x.x. B) B) EE Type Type 1.1 . HfI ~f- fun(A; fI Ff- AA EE Type Type [-I fI, xx ": A, A, ff ": ffun(A; f, x. x. B) B) f-~ B [I, un(A; f, B EE Type Type fI F-f- A(x. .\(x. b)E b) E fun(A; fun(A; f,x. f, x. B) B) 2.2. /~ fI, x" x : AA ~f- bb eE B[A(x. B[.\(x. b)/f] /~, b)/f] g(a) eE B[g/f, B[g/ f, a/x] fI ~f- g(a) 3.3. H a/x] by by ap ap over over fun(A; fun(A; f, f, x. x. B) B) fI Ff- gg EE fun(A; fun (A; f, f, x. x. B) B) fIFf-- aaEEAA H With this this type type we we can With can define define dependent dependent products products as as prod(A; x. B[xl) == x. ifif xx -= 11 then 2 ; f,f,x. prod(A;x. B[x]) = = fun(N fun(N2; then AA else else B[J(l)]). B[f(1)]). 4.5. A A vvery 4.5. e r y small small type type theory theory The previous previous reductions The reductions show show that that we we can can define define aa very very rich rich type type theory theory using using only three primitive type constructors and one primitive type, namely Type only three primitive type constructors and one primitive type, namely Type.. x. B) types: types: Type Type AA ++ B B fun(A; fun(A; f,f,x. B) Jl(X #(Z.. B) B) values: values: inl(a), inl(a), inr(b), inr(b), .\(x. A(x. b)b) forms: forms: decide(t; decide(t; u. u. a;a;v.v. b)b) ap(t; ap(t;a)a) This This language language can can be be seen seen as as aa combination combination of ofthe the ideas ideasfrom fromConstable Constableand andMendler Mendler [1 985], Mendler 1988] , Hickey 1996a] ; itit isis inin the [1985], Mendler [[19SS], Hickey [[1996a]; the style style of ofMendler's Mendler'sthesis thesisusing using 's textbook Hickey's ] considers Hickey's key key reduction. reduction. The The language language FPC FPC ininGunter Gunter's textbook[ 1992 [1992] considersthe the nondependent nondependent recursive recursive types types in in aa similar similarspirit. spirit. 766 766 Constable R. Constable 5.. Conclusion Conclusion 5 In the the main, main, this this article article is is aa snapshot snapshot of of three three subjects subjects recently recently come come into into In alignment. This This conclusion conclusion addresses addresses research research dynamics dynamics driving driving these these subjects. subjects. alignment. T y p e d logic. logic. Many Many standard standard topics topics in in logic logic must must be be reworked reworked for for typed typed logic. logic. We We Typed have already already seen seen that that its its deductive deductive machinery machinery is is different, different, so we need need to to ask ask about about have so we normalization results for for natural natural deduction deduction (as (as in in Prawitz Prawitz [1965]) [1965]) or or cut cut elimination elimination normalization results for numerous numerous variants variants of of the the sequent sequent calculus calculus (with (with structural structural rules rules or or without, without, for tableau style style or or bottom bottom up, up, etc.) etc.) What What properties properties of of the the normal normal syntax syntax of of proofs proofs tableau reflect their their deeper deeper semantic semantic content? content? What What symmetries symmetries of of the the sequent sequent calculus calculus reflect reveal properties properties of of evidence? evidence? reveal The emergence emergence of of automated automated deduction deduction systems systems has has introduced introduced new new issues issues and and The questions. For For example, example, the the notion notion of of aa tactic-tree tactic-tree proof proof(Allen et al. al. [1990]) [1990]) illustrated illustrated (Allen et questions. here here is is aa novel novel structure, structure, and and its its use use in in refinement refinement logics logics (Bates (Bates [1979], [1979], Bates Bates and and Constable [1985]) [1985]) raises raises questions, questions, such such as, as, how how is is soundness and type type correctness correctness Constable soundness and of the the metalevel metalevel programming programming language language for for tactics tactics related related to to the the soundness soundness of of the the of logic? logic? The traditional traditional questions questions about about the the relative relative "power" "power" of of logical logical theories theories can can be be The posed for for typed typed logics, logics, and and the the various various translation translation results results such such as as the the Kolmogorov Kolmogorov posed and Godel translations and GSdel translations are are being being studied studied (Troelstra (Troelstra and and Schwichtenberg Schwichtenberg [1996]). [1996]). Chet Chet Murthy Murthy [1990,1992] [1990,1992] discovered discovered remarkable remarkable results results relating relating these these translations translations to to Plotkin's's CPS CPS translations, translations, and and he he proved proved Friedman's Friedman's [1978] [1978] theorem theorem for for aa fragment fragment Plotkin of Nuprl as part part of of this work (see (see also also Palmgren [1995a]). These These results have been been of Nuprl as this work Palmgren [1995a]). results have applied in in interesting interesting ways ways in in program program extraction extraction by by Murthy Murthy [1992] [1992] and and Berger Berger applied and Schwichtenberg Schwichtenberg [1996]. [1996]. Friedman "reverse mathematics" can be be and Friedman's's program program of of "reverse mathematics" can elaborated in in this this context well, and and now now programming programming logics can be considered in in elaborated context as as well, logics can be considered aa more more uniform manner (Kozen (Kozen [1977], [1977], Kozen and Tiuryn Wiuryn [1990]). uniform manner Kozen and [1990]). The logic has emerged in of logic and computer computer The subject subject of of applied applied logic has emerged in the the intersection intersection of logic and science. This This includes includes the the study science. study of of specification specification languages languages such such as as Z Z (Spivey (Spivey [1989]), [1989]), aa main main topic topic in methods. The languages of typed logic HOL, in formal formal methods. The languages of typed logic (say (say in in Coq, Coq, HOL, Nuprl, PVS) provide provide alternative alternative specification languages which which seem seem to have Nuprl, and and PVS) specification languages to have advantages automation. These typed logics logics can accommodate special special advantages over over ZZ in in automation. These rich rich typed can accommodate languages those needed needed in in temporal temporal logic and for languages such such as as those logic and for hybrid hybrid systems systems (Nerode (Nerode and Shore and Ho Ho [1994]). and Shore [1994], [1994], Henzinger Henzinger and [1994]). The field of automated automated deduction flourishing part part of of applied applied logic. logic. Presently, Presently, The field of deduction is is aa flourishing specialized tools tools such such as as model specialized checkers (c.f. model checkers (d. Clarke, Clarke, Long Long and and McMillan McMillan [1989], [1989], checkers (c.f. (d. Milner, Milner, Tofte Tofte and and Burch et et al. al. [1991], [1991] ' Henzinger Henzinger and and Ho Ho [1994]), [1994]), type Burch type checkers Harper [1991]), [1991]), and and arithmetic arithmetic decision decision procedures procedures are are already already used used by by industry industry in in Harper production. are also production. Integrated Integrated systems systems like like Coq, Coq, HOL, HOL, Nuprl, Nuprl, and and PVS PVS are also valuable valuable to to industry. 4~ The logic-based industrial systems has wealth of industry.47 The deployment deployment of of logic-based industrial systems has led led to to aa wealth of research problems and and challenges (Kreitz, Hayden and Hickey [n.d.]). For For example, research problems challenges (Kreitz, Hayden and Hickey [n.d.]). example, 47The late IBM Fellow, Fellow, Harlam Harlam Mills, Mills, said in December December 1984, 1984, "It is the kind of research research that 47The can change the course course of industrial history." Types 767 767 it is is becoming becoming imperative imperative to to share share libraries libraries of of mathematics mathematics between between provers. Howe's it provers. Howe's work work [1996a] [1996a] with with HOL HOL libraries libraries in in Nuprl Nuprl is is one one of of the the first first examples examples of of how how this this can can be done. done. Practical Practical deployment deployment relies relies on on several several years years of of investigating investigating the the underlying underlying be semantic semantic issues issues involved involved in in translating translating between between theories theories (Howe (Howe [1996b,1991]). [1996b,1991]). The to share The need need to share results results between between provers provers is is only only one one example example of of aa more more general general need to to build build more more open theorem proving proving systems. systems. These These systems systems should should be be able able to to need open theorem interface with with several several text text and and proof proof editors, editors, with with other other provers, with programming interface provers, with programming languages languages to to evaluate evaluate computable computable terms, terms, and and with with metalanguages metalanguages for for managing managing proof We discuss proof planning planning and and generation. generation. Nuprl Nuprl Version Version 55 is is one one such such system. system. We discuss these these problems in in aa wider wider context context in in Collaborative Collaborative Mathematics Mathematics Environments (Chew et et problems Environments (Chew al. [1996]). [1996]). al. Type theory. The The research research agenda agenda in type theory theory is is strongly strongly tied tied to logic as as this this T y p e theory. in type to logic article illustrates, providing new semantics. semantics. In In addition, addition, there there are are strong strong ties ties to to pure pure article illustrates, providing aa new (Gallier [1993]). and applied mathematics mathematics (Gallier [1993]). Indeed, Indeed, Martin-Loftype Martin-Lhf type theory theory arose arose as as an an and applied attempt foundational account the practice practice of of constructive attempt to to find find aa foundational account of of the constructive mathematics, mathematics, especially the style style of Bridges [1985], [1985], Mines, Mines, especially in in the of Bishop Bishop (Bishop (Bishop [1967], [1967], Bishop Bishop and and Bridges Richman and and Ruitenburg Ruitenburg [1988]). [1988]). This constructive mathematics mathematics is Richman This constructive is more more similar similar the practice of computational computational mathematics mathematics than to Intuitionistic mathematics to the practice of than to Intuitionistic mathematics to book can can be read in that that its its results results are are consistent consistent classically. classically. Indeed, Indeed, Bishop's Bishop 's book in be read as classical analysis or as as computational computational or or Intuitionistic Intuitionistic mathematics. as a a piece piece of of classical analysis or mathematics. Nuprl, in in fact, as an to provide foundation for Nuprl, fact, arose arose as an attempt attempt to provide a a foundation for computer computer science science numerical analysis, analysis, computer computer algebra, algebra, the the theory theory of algorithms and numerical of algorithms and computability. computability. It It was based based on on programming programming concepts concepts (Constable Constable and [1984]) was (Constable [1972], [1972], Constable and Zlatin Zlatin [1984]) influenced by Algol68 and in 1978 and and influenced by Algol68 and Simula, Simula, but but we we recognized recognized in 1978 the the power power of of Martin-LM semantics to this activity, Martin-Lhf semantics to organize organize this activity, and and in in Constable Constable and and Zlatin Zlatin [1984] [1984] used used his his semantics semantics to to improve improve our our earlier earlier design. design. As As computational computational mathematics mathematics has has gained gained importance, importance, more more work work has has been been done done to to systematize systematize it. it. For For example, example, the the algebra algebra underlying underlying aa computer computer algebra algebra system system AXIOM (Jenks (Jenks and such such as as AXIOM and Sutor Sutor [1992]) [1992]) is is constructive: constructive: consider consider the the definition definition of of an provides aa function, which will will divide an integral integral domain; domain; it it provides function, div div,, which divide aa * cc by by cc =F r oo.. In In general, claim that object "exists" "exists" is general, in in computer computer algebra, algebra, to to claim that an an object is to to give give an an algorithm algorithm to to construct construct it. it. A A current current active active area area of of research research is is expressing expressing the the concepts concepts of of computer computer algebra algebra in in constructive constructive type type theory. theory. It It is is especially especially promising promising that that the the work orderly account work provides provides an an orderly account of of the the types types and and domains domains used used in in algebra algebra systems systems - - for for example, example, compare compare AXIOM AXIOM (Jenks (Jenks and and Sutor Sutor [1992]) [1992]) or or Weyl Weyl (Zippel (Zippel [1993]) [1993]) to to ' s account Jackson Jackson's account in in Nuprl Nuprl [1994b,1994a]. [1994b,1994a]. Peter Peter Aczel Aczel is is considering considering Galois Galois theory theory in in LEGO LEGO (Pollack (Pollack [1995]), [1995]), and and more more work work of of this this sort sort will will be be done. done. Another Another important important topic topic in in the the same same vein vein is is the the use use of of type type theory theory to to organize organize the the foundations foundations of of numerical numerical mathematics mathematics by by Boehm Boehm et et al. al. [1986] [1986],, Chirimar Chirimar and and Howe Howe [1991]. [1991]. It It will will be be interesting interesting to to see see whether whether floating floating point point numbers numbers could could be be incorporated incorporated into into aa rigorous rigorous theory, theory, perhaps perhaps even even arranging arranging that that the the notion notion of of aa constructive number as which was constructive real real number as aa sequence sequence of of approximations approximations each each of of which was aa "floating number. It intriguing to "floating point" point" number. It is is intriguing to imagine imagine that that this this work work might might extend extend to to 768 768 R. Constable aa computational computational treatment treatment of of nonstandard nonstandard analysis analysis (see (see Nelson Nelson [1968], [1968], Wattenberg Wattenberg [1988]). [1988]). This This is is potentially potentially interesting interesting because because it it is is now now realized realized since since the the work work of of Loeb accounts of more Loeb that that nonstandard nonstandard accounts of probability probability applications applications can can be be significantly significantly more intuitive than their their classical intuitive than classical counterparts. counterparts. Category Category theory theory can can be be seen seen as as an an abstract abstract organization organization of of type type theory, theory, and and just just as as type type theory theory provides provides an an alternative alternative and and more more general general foundation foundation for for mathematics mathematics than too, category than set set theory, theory, so so too, category theory theory provides provides such such aa foundation. foundation. The The category category possible to called an an elementary elementary topos topos generalizes generalizes set set theory. theory. 448 It is is possible to develop develop aa 8 It called predicative os theory Martin-Lof type predicative version version of of top topos theory (the (the Grothendieck Grothendieck topos) topos) in in Martin-Lhf type theory (Palmgren [1995a]). Likewise, Likewise, category category theory theory can can provide provide models of type type theory (Palmgren [1995a]). models of theory (Crole [1993], theory (Crole [1993], Seely Seely [1987]). [1987]). The The categorical categorical models models allow allow new new kinds kinds of of constructive theorems for predicate calculus constructive completeness completeness theorems for the the Intuitionistic Intuitionistic predicate calculus Palmgren Palmgren [1995a] [1995a],, and and from from these these it it is is possible possible to to give give aa uniform uniform computational computational interpretation interpretation to to nonstandard nonstandard analysis analysis (Palmgren (Palmgren [1995a]). [1995a]). Typed T y p e d programming p r o g r a m m i n g languages. languages. The The research research agenda agenda in in programming programming languages languages is is the the most most fast-paced fast-paced of of the the three; three; like like everything everything in in computer computer science science it it is is driven driven by by curiosity, curiosity, by by technology, technology, and and by by market market forces. forces. Research Research is is put put to to use use before before the the "ink is "ink is dry." dry." Each Each small small result result seems seems to to explode explode into into an an industry. industry. Needs Needs for for secure secure mobile code code will will now influence as mobile now be be a a major major influence as code code reuse reuse and and modularity modularity were were before. before. Language Language research research depends depends on on aa deeper deeper understanding understanding of of the the design design space space and and on on range of of semantic semantic tools tools to to rapidly rapidly validate validate experimental experimental designs. designs. Our Our approach approach of of aa range "partial types" is attempts to this knowledge, theory, "partial types" is one one of of many many attempts to provide provide this knowledge, domain domain theory, and semantics (c.f. others (see Crary and theories theories of of operational operational semantics (c.f. Plotkin Plotkin [1981]) [1981]) are are others (see also also Crary [1998]). [1998]). Acknowledgments. A c k n o w l e d g m e n t s . II want want to to thank thank Kate Kate Ricks Ricks for for preparing preparing this this manuscript manuscript and Allen for helping with and Stuart Stuart Allen for helpful helpful comments comments on on earlier earlier drafts drafts and and for for helping with aa new new account 1987 thesis account of of his his 1987 thesis work. work. 6. Appendix 6. Appendix 6.1. 6.1. Cantor's C a n t o r ' s Theorem. Theorem. tion tion 2.9. 2.9. Here Here is is aa Nuprl Nuprl proof proof of of Cantor's Cantor's theorem theorem from from Sec Sec- *T 9T cantor cantor I3 diff : A -+ A : A . ..., (diff x = x» ~- VA V A :: UU. . ((3dill:A-+ A.. Vx Vx:A. -~(diffx=x)) => (Ve : A -+ d : A -+ : A . ..., (e x (Ve:A -+ A A -+ -+ A A.. 3 3d:A -+ A A.. Vx Vx:A. -~(e x ==dd» )) I BY veo THENW ut o BY Uni UnivCD THENW A Auto I I 11.. AA :: UU 48 48"The "The startling aspect of topos theory is that it unifies unifies two seemingly seeminglywholly whollydistinct mathe mathematical subjects: hand, topology subjects: on the one hand, topology and algebraic algebraic geometry, geometry, and on the other hand, hand, logic logic theory." MacLane MacLaneand Moerdijk Moerdijk [1992,p.l) [1992,p.1] and set theory." Types Types 769 769 Hdiff:A -~ A A.. 'v'x Vx:A. -~ (diff (dill x x= = x) x) 22.. 3 diff : A --+ : A . ..., 3. e: e: A A ---+ + A -A+ A--+ A 3. ~- 3 Hd:A -+ A A.. 'v' Vx:A. -~(e d) r d : A --+ x : A . ..., (e xx == d) I[ B Y DD2 2 BY I[ dill:: A A --+ --~ A A 22.. diff Vx:A. -~(diff ) 33.. 'v'x : A . ..., (diff xx == xx) + A A ---+ +A A 44.. ee:: AA ---+ I[ BY With Aa.diff (e a a a) a) 1] (D (D 0) O) THENW THENW Auto Auto BY With r[ A a . diff (e I[ ~- 'v'x Vx:A. -~(e = ((Aa.diff (e a a a) a))) r : A . ..., (e xx = A a . diff (e )) Il BY D D 0 0 THENW THENW Auto Auto BY I[ x :: AA 55.. x ~- ..., -~(e x = = ((Aa.diff (e a a a) a))) r (e x A a . diff (e )) Il BY D D 0 0 THENW THENW Auto Auto BY Il 6.. e e xx= = ((Aa.diff ) )) A a . diff ((ee aaaa) 6 ~- False False r Il BY With x x] (D 3) THENW Auto Auto BY With r[ee x xl (D 3) THENW Il 3. A - +AA --+ A 3 . ee :: AA- +--+ 4. x :: AA 4. x 5.. e e xx= = (Aa.diff ) )) 5 ( A a . diff ((ee aaa a) 6.. -1(diff ) = ) 6 ..., (diff ((ee xxx x) =ee xxxx) Il B Y DD6 6 BY Il ~diff ((ee xxx x) ) = - ee xxxx r diff Il BY RW RW (AddrC (AddrC [3; THENN Auto BY [3 ; I] 1] (HypC (HypC 5) 5) )) 00 THENW Auto II r diff (e x x) == (Aa.diff ( A a . diff (e (e aa a)) a) ) xx ~diff (e x x) lI BY THEN Auto Auto BY Reduce Reduce 00 THEN *C cantor_end ******************************** 6.2. SStamps Here is is aa complete complete Nuprl Nuprl proof proof for for aa simple simple arithmetic arithmetic 6.2. t a m p s pproblem. roblem. Here problem. We show any number number greater greater than than or or equal equal to to 88 can can be be written written as as a a problem. We show that that any sum of of 3's 3 ' s and and 5's. 5's. We We call call this this the the "stamps" "stamps" problem. problem. When When Sam Sam Buss Buss saw saw this this sum we discussed discussed aa generalization generalization which which is is included included in in Section Section 6.3. 6.3. Christoph Christoph theorem theorem we 770 770 R. R. Constable Constable Kreitz proved handwritten notes. notes. It Kreitz proved the the generalization generalization following following Sam's Sam's handwritten It is is interesting interesting that Nuprl Nuprl caught caught aa missing missing case case in in this this proof. proof. The The arguments arguments seem seem sufficiently sufficiently that self-contained self-contained that that we we present present them them without without further further comment. comment. F- 'v' V ii: :{{8 8 . .. . .} .. } . 3m, 3m,n n ::lN V. . 3 3 , *mm + 5+, n5=*i n = i I1I BY BY D D0 0 THENA THENA Auto· Auto. 1I 1i.. ii:: {8 . . .} {8...} I- 3 m,n: N . 3 Bm,n:/V. 3 ,*mm + 5+, 5n =*in = i 1I BY BY NSubsetlnd NSubsetInd 11 THEN Auto Auto.· 1I THEN 1l\ \ 1l l1.. ii::Z Z 11 22.. 00< i O] 01 THENA BY BY Decide Decide r[n THENA Auto Auto.· 1l\ \ 6. n n >>O 0 116. 1II1 BY DTerm DTerm rrm m + + 2] 0 THENM THENM DTerm DTerm rrn n - 11]1 0 0 THEN THEN Auto Auto.· 1i BY 21 0 \ \ 6 6.. ..., -~ (n (n > > 0) O) 1I BY m -- 3] 31 0 n+ 21 0 BY DTerm DTerm r[m 0 THENM THENM DTerm DTerm r[n + 2] 0 THEN THEN Auto Auto.· 1I II'- O 0 � < m --3 3 1I BY BY Suplnf SupInf THEN THEN Auto Auto Types 6.3. GGeneralized 6.3. e n e r a l i z e d sstamps t a m p s pproblem roblem Lemmata Nurpl Library. L e m m a t a ffrom r o m tthe h e SStandard tandard N urpl L ibrary. 'v' a , b : N N. . 00 << aa,*bb *T muLbounds_la 9T mul_bounds_ia ~f- kia,b: f- kia,b" 'v' a , b : N N+ + .. 0O<< aa,* bb *T mul_bounds_lb 9T mul_bounds_lb F*T muLpreserves_lt N ++. . aa< 9T m u l _ p r e s e r v e s _ l t ~f- k'v'i aa ,, bb':ZZ, . k'v'n i n :: N ~ nn**a a < n<*nb * b *T mUl_preserves_le N, . aa < � bb = => � nn**bb 9T m u l _ p r e s e r v e s _ l e ~f- k'v'iaa ,, bb-: ZZ, . k'v'n i n :' N ~ nn**a a < *T multiply_functionality_wrt_le 9T mult iply_funct ionality_wrt_le f- kiil,i2,jl,j2"N, 'v' i l , i2 , j l , j 2 : N . il i1 <_ � jl j l ~=> i2 i2 _� j2 j2 ~=> iih � jj ll **jj22 Fl * ii2 2 _< nn< n ~f- k'v'iaa::N N , . k'v'n l n::N N ++. . 00 _� aarrem e m n n A/\ aarrem emn< 'v'a: Z . kin'Z 'v'n : Z --0o. . aa == (a (a -~ + nn)) ,*nn+ a+rae mrem ~f- kia:Z, n n ± jj ii == 4i ==j j VVi =i-=j -j == i *T 9T rrem_bounds_l em_bounds_l *T 9T ddi i vv__rem_sum rem_sum *A pm_equal 9A pm_equal b b II a a a==b b == 3 B cc':ZZ .. a . c* c *T divisor_bound fa : NN N +. +. a I b => 9T divisor_bound ~ 'v' kia" . . 'v'b kib:: N a lb ~ a a _< � bb *A 9A ddivides ivides Newly Lemmata. N e w l y IIntroduced n t r o d u c e d Notions N o t i o n s and and L emmata. STAMPS STAMPS *T a= 9T ppm_equal_nat m_equal_nat k iaa::N N + +.. a = ± 4- 11 => ~ a a = = 11 f-~ 'v' *T fa ,b , c : Z . a 9T ddivisor_oCsub ivisor_of_sub ~ 'v' kia,b,c:Z, a II b b => ~ a a II cc => ~ a a II b b --c c ,b: Z *T 9T divisor_oCsub_self divisor_of_sub_self fF- 'v'a kia,b" Z .. a a II b b => ~ a a II b b -a a a *A 9A even even a is is even even == == 2 2 II a a a *A 9A oodd dd a iis s odd odd == == 2 2 II a a ++ 11 *T ,m: Z . m => odd f-~ 'v'b 9T oodd_mul dd_mul kib,m'Z, m **bbi sis o dodd d ~ b b iis sod d *T feven V 9T odd_or_even odd_or_even ~ 'v'z kiz:: Z Z .. z z is is even V z z is is odd odd *A aa and 9A stampproperty stampproperty and b b are a r e useful u s e f u l stamp stamp values values m :: N == 'v' k iii :: {{a a ++bb. .. ..}..} . 3 Bnn ,, m / V .. ii ==nn, a*+am+, bm * b Proof P r o o f of o f the t h e 'Induction' ' I n d u c t i o n ' Step. Step. *T 9T sstamp_pre t a m p _ p r e f~ 'v'a V a ,,bb': NN + + .. a a < (('v' k iii: '{ {(a ( a++ b) b ) .. .. ((2 2 ,*aa+ b+)b) - }. }. 3 B nn,, m m :: NN . . ii ==nn * a*+am + * bm)* b) => a a and and b b are are useful useful stamp stamp values values BY simple_prover BY simple_prover II 1l .. aa:: N N + + 22.. bb :: NN + + 33.. aa < 771 771 772 772 R. Constable Constable R. ~- 3 Bn,m:2V. In , m : N . ii ==nn* a* +am+, bm * b BY Cases Cases [[r[ii <<22,*a +ab+] b 1 ;;[i > 22 ,*aa++b ]b]1 ] ..... BY ri > ... 1l\ \ 1[ 66.. ii << 22**aa+ b+ b BY aIlE allE 4 4 r[i] 1I BY i 1 ..... ... II \\ > 22 ,*aa+ + bb 66.. ii > BY Assert r[ (a ( a ++bb) ) + + ((ii --bb) ) rem r e m aa E 6 {{ ((aa ++bb) ) ....((2 2 ,*aa+ b+)b) - }] }1 BY Assert THENL [Id; [Id; aIlE allE 4 4 r[ (a (a + + b) b) + + (ib) rem rem a a]] 1] I[ THENL (i - b) 1l\ \ ~- (a (a+b) + (i-b) r e m aa E 6 {{ ((aa ++bb) ) ....((2 2 **aa+ + b )b) - }- } + b) + (i - b) rem I[ IBY FLemmaOn FLemma0n ''rem_bounds_l' b 1] ;; r[a]] THEN SupInf SupInf ..... ... 1I BY rem_bounds_ 1 ' [[[i r i -- b a1 ] THEN II \\ ( a ++bb) ) + + ((ii --bb) ) rem r a m aa E 6 {{ ((aa ++bb) ) ....((2 2 **aa+ b+)b) - }- } 77.. xx": (a Bn,m:/V. ( a ++bb) ) + + ((ii --bb) ) rem r e m aa==n ,na*+am ,+bm * b 88.. 3 n ,m : N . (a BY thin 7 7 BY thin THEN Repeat Repeat existentialE existentialE I[ THEN THEN exI exI r[((i-b) + a a - 1) +n] «i - b) + 1) + n1 II THEN 1i\ \ 4.. ii :: {{ (a ( a ++bb) ) ....((2 2 **aa+ b+)b) - }- } -+ --+ (n (n:/V X m m:/V X ((iffin,a+m,b)) 1I 4 :N X :N X i = n * a + m * b» 6.. 2 2 ,*aa+ + bb � < ii 1I 6 n:: N /V 1[ 77.. n 118. 8 . mm:: NN 1[ 9 9.. (a ( a ++bb) ) + + (i ( i --bb) ) rem r e m aa==n ,na*+am ,+bm * b ~- 0 0 < < « ((i-b) ~- a a -- 1I)) + + nn i - b) + II IY AAs s s-;ert e r t r[22 ,* aa < < ii --bb] 1 1I BBY THENL [[ SupInfW SupInfWf; Assert r[2 <_ (i - b) b) + -? a a 1] THEN THEN SupInfWf SupInfWf ]] I[ THENL f ; Assert 2 � (i \\ 7.. nn:: NN 7 8.. m: 8 m : /V N 9 9.. (a ( a ++bb) ) ++ ((ii --bb) ) rrem e m aa= =n ,na*+am ,+bm * b m :: NN . . ii == « (((i-b) ) ,*a a + m+,m b*b I- B 3m ( i - b) §+ aa-- 1) 1 ) ++ nn) BY BY exI exI r[m] m1 II (((i-b) ) ,*a a + m+,m b*b I- ii == « (i - b) .'-+ aa-- 1) 1) ++nn) BY SubstInConcl ) ,*a a + m+,m b*b BY SubstInConcl r[(((i-b) « ( i - b) §+ aa-- i) 1) ++nn) « ( i - b ) §+ aa)) ,*aa--a )a) + * b 1 .... .... == (((i-b) + nn**a a + m+,mb ] THEN RevHypSubst 99 0 0 ... THEN RevHypSubst ... ; r a1 ] THEN THEN FLemma0n FLemmaOn ''div_rem_sum' di v_rem_ sum' [[ [ r ii -- b] b 1 ;[a]] THEN SupInf THEN SupInf .... .... - M ain T heorem. Main Theorem. *TStampThm *T StampThm ~I- V'v'aa,,bb:: NN +. + . aa< (a (a and are useful useful stamp stamp values values {:::: => a = 1 VV ((aa ==22 A/\ bbiis s o dodd) d) V (a (a == 33 A/\ bb == 4) 4) v vV ((aa ==33A b/\= Sb)=) 5» Types 773 773 References References M. ABADI L. CARDELLI M. ABADI AND AND L. 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Logic, Logic, 7, 7, pp. 1-112. Errata Errata ibid ibid 7, 7, p.156. p.156. in: Col/oque [1977] [1977] Formalization Formalization of of classical classical mathematics mathematics in in Automath, Automath, in: CoUoque International International de Internationaux du Logique, Logique, Colloques Colloques Internationaux du Centre Centre National National de de la la Recherche Recherche Scientifique, Scientifique, CNRS, Paris, CNRS, Paris, pp. pp. 135-145. 135-145. N aame m e Index Index N Beckmann, A., 210, 227, 227, 254, 254, 333 Beeson, M. J., 340, 340, 357, 357, 366, 366, 401, 411, 421, 441,458, 721, 774 429, 436, 441, 458, 462, 463, 721, Beklemishev, L. D., 486, 487, 489, 490, 492528, 541 496, 528, 718, 719, 719, 741, 741,774 Bell, J. L., 694, 718, 774 Bellantoni, S., 607, 630 Bellin, G., 73, 74 Belnap, N. D., 643, 643, 679 Bennett, C., 99 Bennett, J. H., 98, 99, 143 Benthem, J. van, 546 Berardi, S., 463, 774 488, 519, 521, 521,522, 530, 542 Berarducci, A., 488, 522, 530, U.,. , 434, 461, 461,463, 766, 774 Berger, V 463, 766, 484, 542 Bernardi, C., 484, Bernays, P., 31, 117, 145, 145, 339, 339, 476, 481, Bernays, 31, 76, 117, 554, 555, 555, 560, 560, 586, 633, 718, 506, 543, 554, 774 Berry, G. G., 113 Beth, E. W., 36, 58, 59, 74, 484, 545, 545, 554, 63O 630 Bezem, M., 436, 463, 465 Bibel, W., 780 Bishop, E., 356, 356, 401, 725, 725, 729, 729, 762, 762, 767, 767, 774 Bj~rner, 691,774 Bj0rner, N., 691, 774 Blair, H. A., 653, 653, 680 253, 300, 300, 333 Blankertz, B., 253, Blass, A., 74 Bledsoe, W. W., 690, 690, 711, 774, 775 711,774, Boehm, H.-J., 757, 757, 767, 774 462,464 Boffa, M., 462, 464 Bogdan, R. JJ.,. , 784 R., 640, 640, 661, 679 Bol, R., 661,679 Bonet, M. L., 600, 605, 616, 617, 630 600, 605, Boolos, G., 113, 113, 122, 122, 143, 143, 475-477, 485, 485,487, 487, 532, 533, 542 488, 490, 494, 495, 532, Boone, W. W., 78 Boppana, R., R., 616, 616, 630 630 Borel, F., F., 371 Borger, BSrger, E., 472, 473, 544, 633, 635 Bourbaki, N., 714, 714, 717, 718, 738, 758, 717, 718, 758, 774 Boyer, R. S., 690, 774, 775 690, 711, 711,774, Aagaard, M., 783 Aagaard, Abadi, M., 759, 759, 773 759, 773 Abramsky, S., 74, 759, W.,. , 31, 76, 126, 126, 143, 174, 189, Ackermann, W 143, 174, 203, 242, 553, 633 143, 207, 333, 334, 385, 392, Aczel, P. H. G., 143, 462, 718, 718, 729, 729, 730, 730, 745, 399, 421, 457, 462, 767, 773 Addison, J., 385 Aglianb, P., 542 Agliano, Aitken, W., 773 Ajtai, M., 604, 605, 607, 617, Ajtai, 617, 618, 618, 621,629 621, 629 Ali, M., 779 691, 692, 692, 709, 709, 720, 720, 723, 723, 745, Allen, S. F., F., 691, 768, 773, 776 747-749, 766, 768, Alon, N., 616, 630 Alur, R., R., 774, 774, 783 Anger, F. D., 779 Apt, K. R., 640, 661,679 640, 648, 649, 649, 661, 679 Arai, T., 203, 203, 204, 204, 267 550, 630 Arora, S., 550, Art~mov, Artemov, S. N., 485, 485, 487, 487, 489, 489, 490, 490, 497-499, 532, 534, 534, 535, 535, 540, 541 143, 190, Avigad, J., 80, 123, 123, 126, 126, 143, 190, 204, 340, 372, 399, 400 Avron, A., 485, 541 Baaz, M M.,. , 554, 568, 573, 587, 628, 630 193, 196, 196, 206, 268, 385, 391, Bachmann, H., 193, 392, 400 691,773 Backhouse, R. C., 691, 773 Bakker, J. de, 784 Bar-Hillel, Y., 144, Bar-Hillel, 144, 467 Barbanera, F., F., 459, 462 Barendregt, H. P., P., 429, 429, 462, 755, 755, 773 Barwise, J., 57, 57, 58, 74, 75, 77, 143, 143, 147, 206, 217, 229, 267, 267, 269, 269, 280, 280, 281, 205, 206, 217, 229, 291, 333, 333, 382, 382, 400, 283-285, 291, 400, 401, 404, 404, 405, 462, 462, 469, 472, 545, 405, 469, 472, 545, 635, 635, 682, 682, 690, 773 Basin, D., 709, 773 L., 754, 766, 773 Bates, J. L., 621,630, Beame, P., 599, 604, 607, 618, 618, 621, 630, 635 787 787 788 788 N a m e Index Index Name Boyle, JJ.,. , 77, 786 Braffort, P., 782 729, 767, 767, 774 Bridges, D., 729, Bromley, H. M., 776 151,228, Brouwer, L. E. J., 65, 151, 228, 366-368, 402, 686, 727, 727, 774, 774, 780, 780, 785 408, 686, Brown, A., 774 773, 775 Broy, M., 773, 727, 755, Bruijn, N. G. de, 688, 690, 721-723, 727, 774 153, 164, 164, 189, 189, 190, 190, 199, 199, 201, Buchholz, W., 153, 204, 242, 242, 249, 249, 253, 253, 254, 254, 266, 266, 300, 203, 204, 385, 392, 392, 401, 418, 463, 332-334, 385, 401, 402, 418, 662, 680 689, 775, Bundy, A., 689, 775, 780 766, 775 Burch, J. R., 766, Burnside, W., 78 Burr, W., 386, 386, 401 Buss, S. R., 46, 74, 80, 88, 92, 97, 99, 101107, 108, 108, 110, 117, 126, 126, 127, 105, 107, 110, 114, 114, 117, 127, 138, 143, 143, 144, 144, 190, 190, 204, 204, 364, 130-135, 138, 461, 463, 463, 481, 488, 488, 541, 401, 407, 434, 461, 571, 586, 587, 589, 599, 600, 604, 554, 571, 605, 618, 618, 619, 619, 622-627, 629-631, 634, 686, 769, 775 Butts, R. E., 472 Cannonito, F. B., 78 Cantini, A., 377, 377, 401 126, 157, 157, 185, 185, 196, 196, 213, 237, 300, Cantor, G., G., 126, 716, 718, 718, 768 Carbone, A., 542 Carbone, Carboni, Carboni, A., 463 Cardelli, L., 759, 773, 773, 775 Carlson, T., 493-495, 542 Carson, D. F., 23, 78 Carson, Cartwright, Cartwright, R., 774 Cavedon, L., 668, 680 Cellucci, C., 463 Chaitin, Chaitin, G. J., 113, 144 Chan, T., 690, 711,775 711, 775 Chang, C. C., 758, 775 Chang, C.-L., C.-L., 24, 25, 64, 74, 552, 567, 628, 631 Chang, E., 774 Chang, Chellas, B. F., 478, 542 Chew, L. P., 685, 767, 775 Chirimar, Chirimar, J., J., 762, 767, 775 Chomsky, N., 727, 775 Church, Church, A., A., 146, 146, 358, 358, 391,415,416, 391, 415, 416, 419,420, 419, 420, 422, 422, 433, 433, 434, 434, 437, 437, 440, 440, 442, 442, 476, 476, 686, 686, 688, 688, 694, 694, 695, 695, 711, 711, 714, 714, 727, 727, 730, 730, 741, 741, 755, 758, 774, 775, 777 777 Cichon, Cichon, E. E. A., A., 153, 153, 190-193, 190-193, 199, 199, 201, 201, 204, 204, 249, 266, 266, 333 333 249, Clark, Clark, K. K. L., 640, 640, 648, 648, 650, 650, 661, 661, 669, 669, 673, 673, 680 Clarke, E., 468, 468, 766, 766, 775 Clarke, Cleaveland, Cleaveland, W. R., R., 776 776 Clegg, M., 604, 604, 631 Clocksin, W. F., 64, 75 Clote, P., 96, 144, 144, 145, 145, 206, 206, 403, 403, 542, 542, 543, 571,605,625, 546, 571, 605, 625, 630-633, 781 Cobham, Cobham, A., 103, 103, 104, 104, 108, 108, 144 144 Cohen, Cohen, L. J., 334, 468, 468, 781 ColSn, M., 774 Colon, 628, 631 Compton, K. J., 628, Constable, R. L., 189, 189, 204, 204, 394, 394, 401, 685, 685, 688, 688, 690-692, 695, 695, 709, 709, 711, 711, 720-722, 724, 724, 726, 726, 745, 745, 754, 754, 759, 759, 762, 762, 764-767, 764-767, 773, 775-777, 782 Cook, S. 108, 144, 144, 340, 364, 365, S. A., 74, 101, 101, 108, 401, 434, 461, 463, 463, 550, 550, 552, 552, 592, 592, 594, 595, 616, 624-626, 629, 631, 633, 686 631,633, Cook, W., 604, 605, 632 Coppo, M., 774 Coppa, Coquand, C., 400, 400, 401 401 Coquand, T., T., 398, 398, 401, 401, 463, 463, 691, 691, 692, 692, 721, 721, 762, 776 Coullard, C. R., 605, 632 R., 604, 604, 605, 632 Craig, W., 56-59, 75, 117, 134, 541, 551, 552, 117, 134, 541,551,552, 612, 612, 613, 613, 632, 632, 634 634 Crary, K., 768, 776 Cremer, J. J. F., F., 776 776 Cresswell, M. J., 478, 478, 543 543 CraIe, Crole, R. L., 768, 776 Crossley, J. N., N., 206, Crossley, J. 206, 335, 335, 405, 405, 464, 464, 469, 469, 471, 471, 777, 780 C 595, 636 ubarjan, A. (~ubarjan, A. A., A., 595, 636 Curry, H. 47, 68, 76, 500, 500, 546, 546, 727, 727, 751, H. B., 47, 68, 76, 751, 774, 776 Cutiand, N. Cutland, N. J., J., 164, 164, 204 204 Dahl, O.-J., 760, 776 Dalen, D. van, 64, 405, 64, 66, 66, 68, 68, 77, 77, 366, 366, 402, 402, 405, 423, 424, 424, 431, 431, 433, 433, 436, 436, 411, 412, 412, 421, 421, 423, 448, 450, 450, 461, 461, 462, 462, 464-466, 471, 472, 472, 448, 464-466, 471, 498, 542, 542, 780, 780, 785 498, Damnjanovic, Z., Z., 460, 460, 463 463 18, 20, 20, 21, 21, 75 75 Davis, M., 18, J. W., 778 778 Dawson, J. A., 164, 164, 220 220 De Morgan, A., J. C. E., E., 405 Dekker, J. A. J., J., 757, 757, 776 Demers, A. 776 N., 358, 358, 401 401 Dershowitz, N., A., 144, 144, 635 635 Di Prisco, Prisco, C. A., Name N a m e Index Index Dijkstra, Dijkstra, E. W., 757, 760, 776, 777 Dill, D. L., 775 Diller, J., 339, 352, 401, 401, 434, 339, 340, 340, 352, 434, 458, 458, 459, 459, 464 Dimitracopoulos, C., 88, 98, 144, 146 Doets, K., 640, 648, 649, 680 Donahue, J., 757, 757, 776 776 Dorn, G. J. W., 773 773 Dowd, M., 593, 594, 601, 619, 632 601,619, Drabent, Drabent, W., 672, 672, 680 680 Dragalin, 434, 445, 461, 464, 577, Dragalin, A. G., 421, 421,434, 445,461,464, 632 405, 777, 777, 780 Dummett, M. A. E., 335, 335,405, 780 Dunn, J. M., 679 679 692, 777 777 Dybjer, P., 692, Dzhaparidze, K., see K. Dzhaparidze, G. K., see Japaridze, Japaridze, G. G. K. Edalat, 759, 777 Edalat, A., 759, 777 Eder, E., 552, 552, 554, 554, 564, 564, 600, 600, 628, 628, 632 632 Edmonds, J., 604, 631 Eggerz, P., 441,464 441, 464 Egli, H., H., 759, 759, 777 Ehrenfeucht, A., 586, 586, 632 632 Eichenlaub, C. D., 690, 690, 775, 775, 776 776 Eklof, P. C., 9, 9, 75 75 Emden, M. H. H. van, 640, 640, 680 680 Epstein, G., 679 679 Esakia, L. L., 539, 539, 542 542 Etchemendy, J., 690, 690, 773 773 Fairtlough, M. V. H., H., 80, 80, 153, 153, 164, 164, 190, 190, 204, 204, 242 242 Farmer, 554, 571, 632, 691, 777 Farmer, W. M., M., 554, 571,632, 691,777 Feferman, 32, 58, 75, 80, Feferman, S., 32, 58, 75, 80, 113, 113, 114, 114, 121, 121, 123, 123, 144, 144, 150, 150, 187, 187, 204, 204, 267, 267, 268, 268, 271, 271, 273, 273, 278, 278, 279, 279, 333, 333, 340, 340, 351, 351, 355, 355, 361, 361, 364, 364, 366, 366, 377, 377, 380, 380, 382-386, 382-386, 392, 392, 399, 399, 401-403, 429, 429, 463, 463, 464, 495, 495, 503, 503, 505, 543, 741, 748, 777, 543, 673, 673, 680, 680, 741,748, 777, 778 778 Fenstad, J. E., 402, 402, 404, 404, 405, 405, 472, 472, 635, 635, 680 680 Fermat, P., 720 Fermat, P., 720 Ferrante, 556, 632 Ferrante, J., J., 556, 632 Ferreira, F., 377, 402, 460, 464 377, 402,460, 464 Feys, R., 727, 727, 776 776 Fitting, 643, 651, 652, 680 Fitting, M., M., 643, 651,652, 680 Fontet, Fontet, M., M., 681 681 Forester, Forester, M. M. B., B., 762, 762, 777 777 Fourman, M. P., 448, 448, 464 464 Fraenkel, Fraenkel, A. A. A., 577, 577, 582, 582, 586, 586, 688, 688, 718, 718, 774 774 Frege, G., 3-5, 553, 3-5, 9, 9, 30, 30, 31, 31, 75, 75, 403, 403, 549, 549, 553, 554, 554, 570, 570, 590-607, 590-607, 617-626, 617-626, 630, 630, 631, 631, 633, 633, 634, 634, 685-687, 685-687, 689, 689, 694, 694, 695, 695, 714, 714, 718, 718, 727, 727, 750, 750, 777, 777, 779, 779, 780 780 Freiman, C. V., 775 463 Freyd, P. J., 421, 421,463 Friedman, H. 204, 205, H. M., 153, 153, 190, 190, 204, 205, 268, 279, 333, 334, 372, 372, 376, 377, 279, 333, 376, 377, 385, 419, 421, 385, 386, 386, 399, 399, 402, 402, 419, 421, 422, 422, 513, 571, 571, 581, 458, 461, 464, 465, 513, 581, 632, 766, 777 632, 721, 721,766, 777 Friedrich, W., 340, 366, 402 Frolov, I. T., T., 680 789 789 266, 266, 382, 445, 445, 628, Gabbay, D., 543 Gaifman, R., 88, 98, 98, 144 H., 88, Gallaire, H., 680 Gallier, J. H., 394, 397, 397, 398, 398, 402, 402, 465, 465, 767, H., 394, 767, 777 Galois, E., 767 767 Gandy, R. 0 O.,. , 286, 286, 636 Gavrilenko, Gavrilenko; Yu. V., 459, 459, 465, 465, 577, 577, 632 632 37, 44, 47, 66, 75, Gentzen, G., 10, 16, 36, 37, 123, 126, 144, 167, 169, 187, 190, 190, 205, 123, 126, 144, 167, 169, 187, 230, 231, 231, 233, 233, 240, 240, 241, 241, 338, 338, 342, 342, 361, 380, 382, 382, 384-386, 392, 392, 398, 461, 461, 552554, 564, 564, 571-574, 600, 600, 632, 632, 686, 686, 688, 723, 777 Gerber, H., H., 391, 402 391,402 Gertz, E., 74 74 Geuvers, H., 690, Geuvers, J. J. H., 690, 782 782 Ginsberg, Ginsberg, M. M. L., 643, 643, 680 680 47, 57, 68-70, 75, 75, 80, Girard, J.-Y., 47, 57, 68-70, 80, 126, 126, 144, 144, 153, 153, 164, 164, 190, 190, 193, 193, 205, 205, 249, 249, 334, 334, 340, 340, 393-400, 393-400, 402, 402, 459, 459, 465, 465, 554, 554, 632, 632, 657, 657, 658, 658, 675, 675, 679, 679, 680, 680, 689, 689, 690, 690, 692, 692, 695, 695, 721, 723, 724, 721,723, 724, 778, 778, 779, 779, 782 782 Gleit, Z., 484, 484, 543 543 Glivenko, I., 552 Glivenko, V. V. I., 552 Godel, K., 30, GSdel, K., 30, 66, 66, 75, 75, 80, 80, 92-94, 92-94, 100, 100, 102, 102, 108, 18-123, 126, 108, 112-116, 112-116, 1118-123, 126, 128, 128, 137, 137, 138, 138, 142, 142, 143, 143, 145-147, 145-147, 190, 190, 193, 193, 210, 210, 241, 241, 333, 333, 337-343, 337-343, 346, 346, 347, 347, 350, 350, 362, 362, 365, 365, 366, 366, 399, 399, 401-405, 401-405, 458, 458, 461, 461, 465, 465, 476, 481, 484, 476, 481, 484, 488, 488, 497-500, 497-500, 502, 502, 505, 505, 506, 506, 508, 508, 509, 509, 543, 543, 560, 560, 578, 578, 580, 580, 581, 581, 583, 586, 627, 628, 632, 686-689, 583, 586, 627, 628, 631, 631, 632, 686-689, 699, 733, 766, 778, 779, 699, 718, 718, 733, 766, 778, 779, 784 784 Goerdt, 605, 633 Goerdt, A., A., 605, 633 Goldfarb, W., 76, 76, 484, 484, 543 543 Good, 690, 778 Good, D., D., 690, 778 Goodman, N. 465 Goodman, N. D., D., 441, 441,465 Goodstein, R. R. L., L., 191, 191, 192 192 Gordon, 709, 717, Gordon, M., M., 690, 690, 691, 691,709, 717, 778 778 Gottlob, G., 402 402 Grayson, 434, 441 , 457, 465 Grayson, R. R. J., J., 422, 422, 434, 441,457, 465 Griffin, T. T. G., 708, 708, 709, 709, 778 778 790 790 Name Name Index Index Griffith, J. E., 775 Griffor, E., E., 400, 403 Griibner, GrSbner, W., 604, 631 Groote, Groote, J. F., 465 Grothendieck, A., 695, 719, 719, 744, 744, 768 Grundy, J., 779 779 Grzegorczyk, A., 174, 189, 204, 205, 207, 460, 205,207, 553, 553, 633 633 Guaspari, D., 495, 495, 496, 496, 507, 507, 528, 528, 543, 543, 544, 544, 546 Guenthener, F., 543 Gunter, Gunter, C. A., 765, 765, 775, 775, 778 778 Gunter, E., E., 759, 759, 778 Gutierrez, Guti6rrez, C., 74 757, 778, 778, 781 Guttag, J., 691, 691,757, Guttman, 691, 777 Guttman, J. D., 691,777 Hajek, 145, 147, 147, 190, 190, 205, Hs P., 80, 143, 143, 145, 205, 333, 372, 403, 528, 543, 403, 497, 497, 507, 507, 521, 521, 528, 543, 563, 574, 589, 619, 633 Hajas, 601, 603, 633 HajSs, G., 601,603, Haken, A., 605, 611, 616, 633 611,616, Hallnas, Halln~, L., 655, 680 Hardy, G. H., 152, 153, 158, 158, 186, 186, 190, 152, 153, 190, 205, 205, 242, 249 Harel, D., 498, 543 Harnik, V., 434, 460, 465 Harper, R., 745, 745, 748, 748, 766, 766, 776, 776, 778, 778, 782 782 Harrington, 146, 191, 205, 372, Harrington, 1., L., 122, 122, 126, 126, 146, 191, 205, 372, 376 376 Harrison, Harrison, J., J., 779 779 Harrop, R., 66, 66, 75 75 Hartmanis, Hartmanis, J., 686, 686, 779 779 Hartung, V., 386, 401 Hastad, 618, 633 Hs J., 618, 633 Hayden, M., 766, 766, 780 780 Heijenoort, Heijenoort, J. van, van, 75, 75, 77, 77, 403, 403, 777-779 777-779 Heine, E., E., 371 371 Helmink, L., 1., 692, 692, 779 779 Henkin, 1., 75, 145, L., 30, 30, 75, 145, 506 506 25, 75 Henschen, 1., L., 25, 75 Henson, C. C. W., 628, 631 Henson, W., 628, 631 Henzinger, T. T. A., 766, 766, 774, 774, 779, 779, 783 783 54, 55, 55, 59, 59, 61, 62, 75, Herbrand, J., 48-52, 48-52, 54, 61, 62, 75, 132, 132, 137, 137, 145, 145, 338, 338, 372, 372, 382, 382, 384-386, 384-386, 555, 573, 577, 589, 628, 632, 632, 634, 555, 573, 574, 574, 577, 589, 628, 634, 640, 645, 653, 640, 643, 643, 645, 653, 654, 654, 659, 659, 679, 679, 686, 686, 688 688 Heyting, A., 65, 367, 401, 403, 404, Heyting, A., 65, 147, 147, 341, 341,367, 401,403, 404, 408, 441, 441, 446, 448, 448, 450, 450, 457, 457, 465, 465, 467, 488, 721, 724, 488, 544, 544, 545, 545, 636, 636, 701, 701, 721, 724, 726, 726, 741, 751, 774, 780 741,751,774, 780 Hickey, 691, 692, Hickey, J. J., 691, 692, 722, 722, 764-766, 779, 780, 780, 783 783 Higman, G., 461 31, 37, Hilbert, D., 3, 3, 10, 10, 18, 18, 21, 21, 27, 27, 29, 29, 31, 37, 76, 76, 117, 145, 210, 338, 145, 210, 338, 339, 339, 345, 345, 403, 403, 476, 481, 506, 506, 543, 543, 552-555, 564, 564, 573, 573, 574, 585, 590, 590, 600, 600, 601, 601, 603, 603, 630, 630, 633, 633, 685, 686, 688, 688, 701, 701, 704, 704, 717, 717, 721-723, 729, 779 779 Hilpinen, R., 680 Hinata, S., 339 Hindley, 359, 403, 546, 755, Hindley, J. R., 76, 359, 403, 464, 464, 546, 774, 779 Hintikka, J., 36, 76, 472 Hirschberg, Hirschberg, D., 782 Ho, P.-H., 766, 766, 779 Hoare, C. A. R., 756, 756, 760-762, 776, 778, 778, 779 Hodges, W., 403, 403, 681 Hodgson, B. R., 100, 100, 106, 106, 145 Honsell, F., 778 Hoogland, E., 541 Hoogland, E., Hopcroft, J. E., 779 25, 26, 63, 64, 75, 636, Horn, A., 25, 636, 640 Horning, J., 691, 778 691,778 Hiisli, HSsli, B., 675, 680 Howard, 126, 145, Howard, W. A., A., 47, 47, 68, 68, 76, 76, 126, 145, 164, 164, 193, 196, 196, 199, 199, 205, 205, 206, 206, 338-340, 338-340, 351, 360, 360, 366, 366, 367, 367, 369, 369, 370, 370, 373, 373, 385, 385, 386, 386, 391, 392, 403, 436, 391,392,403, 436, 465, 465, 500, 500, 751 751 Howe, Howe, D. J., 691, 691, 745, 745, 759, 759, 762, 762, 767, 767, 773, 773, 775, 775, 776, 776, 779 779 Huet, G., 471,691,692, 471, 691, 692, 721,773, 721, 773, 776, 776, 779 779 Hughes, E., 478, Hughes, G. E., 478, 543 543 Hwang, L. J., 775 775 Hyland, J. M. E., E., 451, 451, 457, 457, 461, 461, 465, 465, 466, 466, 636, 636, 780 780 Iemhoff, R., 541 541 Igarashi, S., 690, 690, 780 780 Ignatiev, K. N., N., 494, 494, 529, 529, 543 543 Ignjatovic, 103, 110, 138, 144, 144, 629, Ignjatovid, A., 101, 101, 103, 110, 138, 629, 633 633 Impagliazzo, R., 599, 599, 604, 604, 607, 630, 607, 618, 618, 621, 621,630, 631, 633, 635 631,633, 635 Ito, Ito, T., 776 776 Jackson, 745, 767, Jackson, P. P. B., B., 709, 709, 711, 711,745, 767, 776, 776, 780 780 Jagadeesan, R., 74 74 Jager, 268, 291, 384-386, 402, 402, 472, Jiiger, G., G., 268, 291, 334, 334, 384-386, 472, 473, 473, 640, 640, 655, 655, 672, 672, 675, 675, 680 680 Jankov, Jankov, V. V. A., 466 466 Japaridze (Dzhaparidze) (Dzhaparidze),, G. K., 486, 486, 487, 487, 489, 512, 522, 489, 494, 494, 495, 495, 503, 503, 512, 522, 528-530, 528-530, Name Name Index Index 535, 535, 539-543 539-543 Jeffrey, R. 533, 542 R. C., 533, 542 Jenks, R. R. D., 767, 767, 780 780 Jeroslow, R. G., 495 Johannsen, J., 633 Johnson, D. SS.,. , 133, 133, 145 145 Johnson, S. D., 690, 775, 775, 776 Johnstone, P. T., 451, 457, 465 451,457, Joja, A., 469 Jonasson, A., 74 Jongh, D. H. J. de, 460, 460, 466, 466, 476, 476, 488, 488, 496, 496, 514, 515, 521, 528, 542, 544, 546 514, 521,528, Joosten, Joosten, J., J., 541 541 Jouannaud, Jouannaud, J.-P., 358, 401 Joyal, A., 466, 721, 780 721,780 Jumelet, Jumelet, M., 488, 544 Kabakov, F. A., 466 466 Kadota, N., N., 197, 197, 205 205 Kahle, R., 268, 334 Kanger, SS.,. , 36, 36, 76 76 Kapur, Kapur, A., 774 774 Kapur, D., 783 Kaufmann, M., 773 773 Kaye, 80, 137, Kaye, R. R. W., W., 80, 137, 145 145 Keisler, Keisler, H. J., 462, 462, 469, 469, 472, 472, 758, 758, 775 775 Kenny, A., 780 100, 106, 106, 145 145 Kent, C. F., 100, 205 Ketonen, J., J., 126, 126, 145, 145, 154, 154, 191, 191,205 Khakhanyan, V. 439, 458, 458, 466 V. Kh., 439, 466 Kino, A., 145, 146, 333, 333, 334, 334, 401-404, 145, 146, 401-404, 466, 466, 468, 468, 473, 473, 777, 777, 781 781 Kipnis, M. M., 466 Kirby, L. A. SS.,. , 84, 134, 136, 136, 137, 137, 146, 146, 153, 84, 134, 153, 191, 192, 191, 192, 205 Kleene, 150, 151, 153, 228, 228, Kleene, S. C., C., 9, 9, 31, 31, 76, 76, 150, 151, 153, 339, 356, 357, 362, 363, 339, 356, 357, 362, 363, 366, 366, 368-370, 368-370, 377, 383, 408, 377, 383, 384, 384, 386, 386, 390, 390, 391, 391, 403, 403, 408, 409, 420-422, 428, 428, 431, 433, 409, 411, 411, 412, 412, 420-422, 431, 433, 434, 449, 459, 459, 467, 467, 680, 746, 778 434, 449, 680, 701, 701,746, 778 Kleine Biining, 472, 473 Brining, H., 472, 473 Knoblock, Knoblock, T. T. B., B., 776 776 E., 780 Knuth, D. E., Kobayashi, S., 467 467 Kohlenbach, U. 340, 341, 356, 372, 372, 375U. W., W., 340, 341,356, 375377, 377, 403, 403, 407, 407, 436, 436, 467 467 Kolmogorov, 727, 766 Kolmogorov, A., A., 65, 65, 66, 66, 408, 408, 727, 766 Kondo, KondS, M., 385 Konig, J., 36, 366, 366, 371, 373, 439 KSnig, 371,373, Kowalski, Kowalski, R. R. A., A., 64, 64, 76, 76, 640, 640, 680 680 Kozen, D., 468, 468, 690, 690, 766, 766, 780 780 Kracht, M., 546 791 791 Krajicek, 108, 131-133, 143-145, Krajf~ek, J., 54, 76, 108, 206, 206, 403, 403, 542, 542, 543, 543, 546, 546, 554, 554, 564, 564, 569569571, 571, 593-595, 593-595, 598-601, 598-601, 605, 605, 607, 607, 612612619, 621, 621, 622, 624-627, 629-634 54, 76, 189, 205, 205, 249, Kreisel, G., 54, 76, 175, 175, 189, 249, 339, 339, 340, 353, 355, 357, 353, 355, 357, 361, 361, 366, 366, 369-371, 383, 404, 421, 433, 434, 467, 503, 505, 544, 549, 549, 554, 554, 564, 571-573, 587, 543, 544, 628-630, 628-630, 634, 634, 723, 723, 744, 744, 780 780 Kreitz, C., 754, 754, 766, 766, 770, 770, 780 780 Krentel, M. W., W., 131, 131, 145 145 Kripke, S., 66, 478, 480-482, 487, 488, 488, 490, 66, 478, 480-482, 487, 490, 494, 495, 514, 529, 530, 530, 534, 534, 535, 535, 539, 494, 495, 514, 529, 539, 541, 680, 700, 541,680, 700, 780 Krol', M. D., 421, 467 421,467 Kropf, T., 783 783 Kumar, R., R., 783 Kunen, K., 462, 652, 654, 462, 469, 469, 472, 472, 643, 643, 651, 651,652, 654, 658, 661, 668, 669, 658, 659, 659, 661,668, 669, 680 680 Kurtz, S. A., 468 Kushner, B., 470 470 Lacombe, D., 774, 774, 784 784 Lafont, Lafont, Y., 75, 75, 397, 397, 398, 398, 402, 402, 459, 459, 465, 465, 675, 675, 680, 680, 689, 689, 692, 692, 695, 695, 724, 724, 778 778 Lambek, 468, 694, Lambek, J., J., 421, 421,468, 694, 781 781 Lang, Lang, B., B., 779 779 Lassez, JJ.-L., .-1., 643, 643, 681 681 Liiuchli, L~iuchli, H., 460, 460, 468, 468, 700, 700, 781 781 Laudet, Laudet, M., 774, 774, 784 784 Lautemann, C., 74 Lee, R. 25, 64, 567, 628, 628, 631 631 R. C.-T., 25, 64, 74, 74, 552, 552, 567, Leeser, M., 783 Leeuwen, J. van, 401, 404, 679, 679, 780 401,404, 780 Leibniz, G. W., 686, 687 Leitsch, A., 402, 402, 628, 628, 630 630 Leivant, D., 190, 190, 205, 205, 460, 460, 463, 463, 488, 488, 686, 686, 781 781 Lercher, B., 755, 755, 779 Lercher, Lessan, H., 137, 137, 145 145 Levy, L~vy, A., 215, 215, 295, 295, 505, 505, 544 544 Lewis, Lewis, C. C. 1., I., 544 544 Lifschitz, V., V., 422, 461, 468 422, 437-439, 437-439, 457, 457, 461,468 Lincoln, P., 74, 76 Lindstrom, 145, 495, LindstrSm, P., 122, 122, 145, 495, 508, 508, 511, 511, 513, 513, 544 544 Lipton, J., 460, 460, 468 468 Lipton, R. R. J., 99, 99, 145 145 Liskov, Liskov, B., B., 757, 757, 781 781 Liu, C. 1., L., 781 781 Lloyd, W., 640, 661, 668, Lloyd, J. J. W., 640, 648, 648, 649, 649, 661, 668, 680, 680, 681 Lob, H., 75, LSb, M. M. H., 75, 117, 117, 118, 118, 122, 122, 145, 145, 189, 189, 205, 205, 481, 484, 486, 534, 538 481,484, 486, 491, 491, 496, 496, 531, 531,534, 538 792 792 Name Index Index Name Loeb, P., P., 768 768 Loeb, Lolli, G., G., 634 634 Lolli, London, R., R., 690, 690, 780 780 London, Long, D. D. E., E., 766, 766, 775 775 Long, Longo, G., G., 634 634 Longo, Lopez-Escobar, E. E. G. G. K., K., 17, 17, 57, 57, 76 76 Lopez-Escobar, Lorenz, K., K., 402 402 Lorenz, Los, J., J., 334, 334, 468, 468, 781 781 Log, Loveland, D. D. W., 24, 24, 25, 25, 64, 64, 76 76 Loveland, L6wenheim, L., L., 216 216 LSwenheim, Luckham, D., D., 23, 23, 24, 24, 76, 690, 780 780 Luckham, 76, 690, Luckhardt, H., 340, 340, 351, 351, 370, 370, 404, 404, 407, 407, 628, 628, Luckhardt, 634 634 Lund, C., C., 630 630 Lund, Luo, Z., Z., 692, 692, 781 781 Luo, Lusk, E., E., 77, 77, 786 786 Lusk, Lyndon, R. R. C., C., 57, 57, 76, 76, 78 78 Lyndon, Maass, W., 340, 340, 399, 399, 404 404 Maass, Maciel, Maciel, A., A., 74 74 Macintyre, A., A., 545 545 Macintyre, MacKenzie, D., D., 691,781 691, 781 MacKenzie, MacLane, 718, 719, 719, 768, 768, 781 781 MacLane, S., 694, 694, 718, Magari, R., 484-486, 484-486, 542 542 Magari, R., Maher, M. J., J., 643, 643, 681 681 Maher, M. Mahlo, P., 267 267 Mahlo, P., Main, M., 463 463 Main, M., Makkai, M., 460, 465 Makkai, M., 460, 465 Maksimova, 1. L., 461,468 461, 468 Maksimova, L. L., Mal'cev, A. I., I., 649, 649, 681 681 Mal'cev, A. Manna, Z., Manna, Z., 774 774 Marcja, Marcja, A., A., 634 634 Marcus, Marcus, R. R. B., B., 773 773 Markov, A. A., 347, 355, 355, 356, 356, 416, Markov, A. A., 347, 416, 417, 417, 439, 439, 441, 688, 727, 441,688, 727, 781 781 Marques, A., 460, Marques, A., 460, 464 464 Martelli, M., 672, 680 Martelli, M., 672, 680 Martin-Lor, 359, 394, 394, 398, 398, 400, 400, 401, 403, Martin-LSf, P., P., 359, 401,403, 441, 459, 404, 404, 441, 459, 468, 468, 635, 635, 688, 688, 691, 691, 692, 692, 695, 719, 721, 721, 723-725, 695, 698, 698, 700, 700, 719, 723-725, 727, 727, 728, 728, 730, 730, 736, 736, 739, 739, 742, 742, 744-747, 744-747, 767, 767, 768, 768, 773, 773, 776, 776, 779, 779, 781-784 781-784 Martini, Martini, S., S., 459, 459, 462 462 Mathias, 472 Mathias, A. A. R. R. D., D., 464, 464, 467-469, 467-469, 471, 471,472 Matijacevic, V., 113, Matijacevi~, Yu. Yu. V., 113, 781 781 McAllester, McAllester, D. D. A., A., 689, 689, 781 781 McAloon, McAloon, K., K., 399, 399, 402, 402, 462, 462, 464 464 McCarthy, McCarthy, J., J., 686, 686, 782 782 McCarty, McCarty, D. D. C., C., 458, 458, 468 468 McCune, McCune, W., W., 779 779 McGee, McGee, V., V., 532, 532, 542 542 McKinsey, McKinsey, J. J. C. C. C. C.,, 500, 500, 544 544 McMillan, K. McMillan, K. L., L., 766, 766, 775 775 Medvedev, Medvedev, Yu. T., T., 461,469 461, 469 Mehlhorn, K., 681 681 Melham, T., T., 691,717, 691, 717, 778 778 Mellish, C. S., 64, 64, 75 75 Melton, M., 463 463 Mendelson, E., 9, 9, 76, 76, 114, 114, 145 145 Mendler, N. P., 692, 692, 745, 745, 762, 762, 764, 764, 765, 765, 776, 776, 782 782 Mendler, P. F., F., 721,745, 721, 745, 762, 762, 764, 764, 765, 765, 782 Metakides, G., 402, 402, 404 404 Meyer auf auf der Heide, F., F., 636 636 Meyer, A. R., 759, 759, 776, 776, 782 782 Meyer, B., 758, 758, 759, 759, 782 782 Mikhajlov, A. I., 1., 464, 464, 466 466 Miller, D., 679, 679, 681 681 Mills, H., 766 766 690, 691,709, 691, 709, 766, 766, 778, 778, 782 782 Milner, R., 690, Mines, R., 767, 782 Minker, J., 680 Mints, G. E., 123, 123, 146, 146, 189, 205, 265, 265, 266, 189, 205, 459, 469, 635, 635, 776 Mislove, A., 463 394, 397, 404, 404, 468, 468, 759, Mitchell, J. C., 76, 394, 759, 775, 782 Miyatake, T., 572, 572, 634 Moerdijk, 1., 768, I., 466, 466, 694, 694, 718, 718, 719, 719, 721, 721, 768, 780-782 780-782 Moisil, G. C., 469 Moisil, 469 636 Monien, B., 636 488, 493, Montagna, F., F., 485, 485, 488, 493, 496, 496, 497, 497, 521, 532, 542-544, 589, 633 528, 532, Moore, G. H., 778 778 Moore, J. S., 690, 774 690, 711, 711,774 Moschovakis, J. R., 407, 428, 434, 407, 421, 421,428, 434, 469 469 Moschovakis, Y. N., 269, 269, 270, 270, 334, 334, 468, 468, 652, 653, 676, 681 Mostowski, A., 82, 83, 147, 147, 212, 212, 495, 495, 503, 545 545 Motwani, R., 630 630 Muller, G. H., 204, 204, 777 777 Mulvey, C. J., 464 402, 612, 634 Mundici, D., 402, Munkres, J. R., 9, 9, 76 76 Murthy, Murthy, C., 723, 723, 766, 766, 782 782 Mycielski, Mycielski, J., 586, 586, 632 Mycroft, A., 643, 643, 681 681 Myers, J. P., 775 775 Myhill, J., 145, 145, 146, 146, 333, 333, 334, 334, 401-404, 401-404, 419, 419, 421, 466, 468, 421,466, 468, 469, 469, 473, 473, 777, 777, 781 781 Nagel, Nagel, E., 467 467 Nagornyi, N. M., 470 470 Nahm, W., 458, 464 W., 340, 340, 352, 352, 401, 401,458, 464 Name Name Index Index Naumov, P., 776 Nederpelt, R. P., 690, 690, 782 Nelson, D., 469 469 98, 99, 99, 114, 118, 137, Nelson, E., 81, 98, 114, 118, 137, 146, 583, 583, 634 634 Nelson, G., 690, 782 Nelson, R. J., 768, 768, 782 782 Nepomnjascii, Nepomnja~ii, V. A., 99, 146 Nerode, A., 766, 781, 782 766, 781,782 Neumann, J. von, 591, 632, 686 591,632, 686 Niiniluoto, I., 1., 784 784 Nivat, M., 779 779 774, 784 Nolin, L., 774, 784 Nordstrom, B., 691, 724, 782 691,724, 782 Oberschelp, W., 544 544 Odifreddi, P., 401, 402, 404, 776 401,402, O'Donnell, M. J., 460, 460, 468, 774, 775 O'Leary, J., 712, 712, 783 783 Ong, C.-H. 1., L., 465 Oosten, J. van, 407, 457, 460, 407, 434, 434, 438-441, 438-441,457, 460, 469 Oppen, D., 690, 690, 782 782 Orevkov, V. 577, 629, 629, 634 V. P., 577, 634 Orey, S., 503, 503, 507, 507, 531, 543, 545 531,543, Overbeek, R., 77, 77, 786 786 Owre, 783 Owre, S., S., 691, 691,783 Pacholski, L., 205, 205, 545 545 Padoa, Padoa, A., 74 74 Palmgren, E., 268, 268, 334, 334, 692, 692, 745, 745, 766, 766, 768, 768, 783 783 Panangaden, Panangaden, P., P., 745, 745, 776, 776, 782 782 Papadimitriou, 133, 145 Papadimitriou, C. H., 133, Parikh, R., 74, 74, 87, 87, 98, 98, 99, 99, 102, 102, 112, 112, 146, 146, 472, 472, 496, 496, 544, 544, 545, 545, 552, 552, 554, 554, 568, 568, 571, 571, 577, 577, 628, 629, 635, 776 Paris, J. J. B., 84, 98, 98, 114, Paris, B., 80, 80, 84, 114, 122, 122, 126, 126, 134, 134, 136-139, 136-139, 143, 143, 146, 146, 147, 147, 153, 153, 190-192, 205, 621, 629, 635 205, 545, 545, 621,629, 635 Parsons, 175, Parsons, C., 84, 84, 111, 111, 123, 123, 134, 134, 136, 136, 146, 146, 175, 189, 189, 206, 206, 339, 339, 340, 340, 362, 362, 404, 404, 405 405 Paterson, S., 61, 61, 77 Paterson, M. M. S., 77 Paulin-Mohring, 692, 754, 776, Paulin-Mohring, C., C., 691, 691,692, 754, 762, 762, 776, 783 783 Paulson, L. C., 691, 783 691,783 46, 77, 77, 80, 80, 84, 84, 94, 94, 95, 95, 97, Peano, G., G., 31, 31, 46, 97, 101, 122, 122, 126, 175, 199, 199, 126, 144-147, 153, 153, 175, 204-206, 231, 231, 242, 261, 340, 341, 242, 247, 247, 261, 340, 341, 359, 359, 360, 360, 366, 366, 399, 399, 487, 487, 492, 492, 494, 494, 495, 495, 542-546, 542-546, 571-573, 571-573, 582, 582, 628, 628, 721 721 Peter, Peter, R., R., 189, 189, 206, 206, 242 242 Petersson, 691, 724, Petersson, K., K., 691, 724, 782 782 793 793 Petkov, P. P., 147, 147, 544-546, 636 Pfeiffer, H., 334, 781 334, 468, 468, 781 Pfenning, F., 679, 681 679,681 Phoa, W., 470 Pianigiani, 528, 544 Pianigiani, D., 528, Pingali, K., 775 Pitassi, T., 599, 599, 603, 603, 605, 605, 607, 607, 616-618, 616-618, 621, 621, 630, 635 630, 631, 631,635 Pitt, D. H., 783 Pitts, A. M., 441, 441,451,457, Pitts, 451, 457, 465, 465, 470, 470, 783 Platek, R., 290, 334 Plisko, V. E., 459, 459, 460, 470 Plotkin, G., 471, 757, 759, 471,757, 759, 766, 768, 773, 773, 778, 783 Podewski, K.-P., 334, 468, 781 Pohlers, W., 80, 80, 126, 126, 146, 146, 153, 153, 164, 164, 199, 199, 204, 204, 210, 210, 227, 227, 253, 253, 254, 254, 266-268, 266-268, 270, 270, 300, 300, 333, 463 333, 334, 334, 392, 392, 401, 401,463 Poigne, A., 783 783 Poll, E., 692, 783 Pollack, R., 691, 767, 783 691,767, Pollett, C., 74, 74, 143 143 Ponse, A., 546 Post, E. 1., L., 729 729 Powell, W., 445 445 Prawitz, Prawitz, D., D., 47, 47, 77, 77, 144, 144, 204, 204, 398, 398, 404, 404, 554, 554, 635, 635, 723, 723, 766, 766, 784 784 Presburger, M., 628 628 Pudlak, 131-133, Pudl~k, P., P., 54, 54, 76, 76, 80, 80, 98, 98, 108, 108, 118, 118, 131-133, 138, 143, 143, 145, 145, 146, 146, 190, 190, 205, 205, 497, 543, 562, 562, 563, 563, 568-571, 568-571, 573, 573, 574, 574, 581, 581, 582, 582, 584, 584, 585, 585, 589, 589, 590, 590, 593, 593, 594, 594, 599-601, 599-601, 605, 605, 607, 607, 614, 614, 616-619, 616-619, 621, 621, 624-626, 624-626, 628, 631, 633-635 628, 630, 630, 631,633-635 Putnam, 20, 21, 75, 495 Putnam, H., H., 18, 18, 20, 21, 75, 495 Quine, W., 694, 694, 784 784 Rabin, Rabin, M. M. 0., O., 628, 628, 635 635 Rackoff, C. W., 556, 556, 632 Rackoff, C. 632 Rajan, Rajan, S., 783 783 Ramsey, F., 122, 122, 145, 145, 191, 205, 619, 619, 635 Ramsey, F., 191,205, 635 Ratajczyk, 190, 206 Ratajczyk, Z., Z., 190, 206 328, 333, Rathjen, M., M., 273, 273, 291, 291, 304, 304, 328, 333, 335, 335, 400, 403 Raz, 605, 616, Raz, R., R., 605, 616, 617, 617, 630 630 Razborov, 134, 146, 613, 616, 616, 629, 629, Razborov, A. A. A., A., 134, 146, 613, 631, 635, 636 631,635, 636 Reckhow, R. A., 550, 550, 552, 592, 594, 595, 552,592,594, 595, 631, 631, 636 Reinhold, Reinhold, M. M. B., B., 759, 759, 782 782 Remmel, J. B., 630, 631, 634, 781 Remmel, J. B., 206, 206, 471, 471,630, 631,634, 781 794 794 Name Index Index Name Renardel de de Lavalette, Lavalette, G. G. R., R., 425, 425, 429, 429, 441, 441, Renardel 470, 546 546 470, Reps, T., T., 708, 708, 784 784 Reps, Reyes, G. G. E., E., 721,782 721, 782 Reyes, Reynolds, J. J. C., C., 393, 393, 394, 394, 404, 404, 692, 692, 759, 759, 784 784 Reynolds, Rezus, A., A., 692, 692, 784 784 Rezus, Richman, F., F., 457, 457, 463, 463, 469, 469, 767, 767, 782 782 Richman, Richter, M. M. M., M., 472, 472, 473, 473, 544 544 Richter, Ricks, K., K., 768 768 Ricks, Riggle, M., M., 774 774 Riggle, Riis, S., S., 629, 629, 636 636 Riis, Rijke, M. M. de, de, 514, 514, 545, 545, 546 546 Rijke, Robbin, J., J., 189, 189, 206 206 Robbin, Robinet, B., B., 404, 404, 784 784 Robinet, Robinson, A., A., 59, 59, 77, 77, 400, 400, 484 484 Robinson, Robinson, E., E., 451,466, 451, 466, 470 470 Robinson, Robinson, G., G., 23, 23, 63, 63, 77, 77, 78 78 Robinson, 18, 22, 22, 24, 24, 59, 59, 61, 61, 64, J. A., Robinson, J. Robinson, A., 18, 64, 77, 77, 648, 681 648, 681 Robinson, R. R. M., M., 46, 46, 82, 82, 83, 147, 503, 503, 507, 507, Robinson, 83, 147, 513, 533, 545, 560, 579, 586 513, 533, 545, 560, 579, 586 Robinson, T. T., 421,470 421, 470 Robinson, T. T., Rodriguez, R. V., 779 779 Rodriguez, R. V., Rogers, 464, 467-469, 467-469, 471, 472, 730, Rogers, H., H., 464, 471,472, 730, 732, 732, 784 784 Rootselaar, B. van, Rootselaar, B. van, 467 467 Rose, Rose, G. G. F., F., 459, 459, 470 470 Rose, E., 189, Rose, H. H. E., 189, 206, 206, 404 404 Rosenfeld, J. L., 1., 775 Rosenfeld, J. 775 Rosolini, G., 451, 466, 470 Rosolini, G., 451,466, 470 Rosser, 120, 146, Rosser, J. J. B., B., 120, 146, 358, 358, 484, 484, 495, 495, 496, 496, 542-545, 774 542-545, 581, 581,774 Roth, Roth, K.F., K.F., 628, 628, 634 634 Rudich, Rudich, S., S., 134, 134, 146 146 Ruitenburg, Ruitenburg, W., W., 767, 767, 782 782 Rushby, M., 691, 783 Rushby, J. J. M., 691,783 Russell, Russell, B., B., 31, 31, 77, 77, 600, 600, 634, 634, 636, 636, 685, 685, 687, 687, 694, 694, 698, 698, 714, 714, 727, 727, 728, 728, 758, 758, 784, 784, 785 785 Rydeheard, Rydeheard, D. D. E., E., 783 783 Sambin, 476, 484, 542, 545 Sambin, G., G., 401, 401,476, 484, 541, 541,542, 545 Saracino, Saracino, D., D., 400 400 Sasaki, Sasaki, J. J. T., T., 776 776 Scarpellini, Scarpellini, B., B., 470 470 Scedrov, 422, 434, 463, Scedrov, A., A., 76, 76, 421, 421,422, 434, 458, 458, 461, 461,463, 465, 721, 777 465, 470, 470, 471, 471,721,777 Schinzel, Schinzel, B., B., 544 544 Schlipf, Schlipf, J. J. S., S., 382, 382, 400, 400, 672, 672, 682 682 Schliiter, Schlfiter, A., A., 267, 267, 304, 304, 335 335 Schmer!, Schmerl, U., U., 199, 199, 206 206 Schmidt, Schmidt, D., D., 154, 154, 206, 206, 463 463 Schmidt, Schmidt, H., H., 462 462 Schmitt, Schmitt, P., P., 780 780 Schroeder-Heister, Schroeder-Heister, P., P., 655, 680, 681 681 Schiitte, Schiitte, K., K., 36, 77, 77, 80, 126, 146, 146, 164, 169, 187, 206, 221, 221, 268, 335, 380, 383, 385, 392, 404, 462, 462, 657, 658, 681 Schiitzenberger, Schiitzenberger, M., 774, 784 784 Schwichtenberg, Schwichtenberg, H., 144, 164, 175, 175, 189, 190, 193, 206, 361, 361, 404, 434, 461, 463, 500, 546, 680, 766, 774, 785 Scott, Scott, D. D. S., S., 204, 204, 428, 428, 445, 445, 448, 448, 464, 464, 471, 471, 509, 545, 723, 757, 759, 777, 784 Scott, P. J., J., 421,468, 421, 468, 471,629, 471, 629, 634, 694, 781 471 Scowcroft, P., 471 784 Seely, R. A. G., 768, 784 Seisenberger, M., 434, 461,463 461, 463 Seldin, J. P., 76, 359, 359, 403, 464, 546, 755, 774, 779 779 Selman, A. L., 775 775 Setzer, A., 334, 692, 784 633 Sgall, J., 631, 631,633 Shanin, N. A., 422, 471 691,783, Shankar, N., 76, 691, 783, 784 Shapiro, S., 468 Shavrukov, V. Y., 485, 485, 486, 495, 495, 496, 496, 521, 522, 545, 546 190, 205, 266, 334 Sheard, M., 190, 205, 266, 461,468 Shekhtman, V. B., 461, 468 404, 640, 640, 643, 643, 649, 649, 654, Shepherdson, J. C., 404, 661,681,778 661, 681, 778 Shoenemann, R., 634 634 340, 342, 342,404 Shoenfield, J. R., 340, 404 Shore, R. A., 471, 471, 766, 766, 782 782 Shostak, 784 Shostak, R. E., 690, 690, 711, 711,784 Sieg, W., 111, 111, 143, 143, 146, 146, 178, 178, 190, 190, 204, 204, 206, Sieg, 333, �, 339, m, 372, �, 382, �, 386, w 401,, �, 402, � 405,, m, 463 463 25, 74, 75, 77, 78, 691, 691,784 Siekmann, J., 25, 784 143, 207, 207, 333, 333, 334 Simmons, H., 143, 204, 291, 291,371,399, 402, 405, Simpson, S. G., 204, 371, 399, 402, 635 635 Sipma, Sipma, H. B., 774 774 Sitharam, M., 629, 629, 636 636 59, 216, 216, 219, 219, 242, 242, 247, 247, Skolem, T., 49-51, 59, 333, 346, 346, 355, 355, 377, 377, 378, 378, 386, 386, 497, 497, 577, 577, 333, 630, 630, 686, 686, 718 718 Skowron, Skowron, A., A., 632 632 Skvortsov, 468 Skvortsov, D. D. P., P., 461, 461,468 Skyrms, Skyrms, B., B., 144, 144, 204 204 Slagle, Slagle, J. J. R., R., 23, 23, 77 77 Slisenko, Slisenko, A. A. 0., 0., 636 636 Smith, Smith, B. B. C., C., 745, 745, 785 785 Smith, J. J. M., M., 401, 401,461,471,691,724, 782 Smith, 461, 471, 691, 724, 782 Smith, S. S. F., F., 721, 721,722, 745, 776 776 Smith, 722, 745, Name Name Index Index Smorynski, 147, 476, 477, 477, 484, Smoryfiski, C., 114, 114, 122, 122, 147, 487, 492, 494, 495, 504, 545 Smullyan, R. M., 99, 114, 147, 147, 554, 636 Solovay, R. M., 98, 118, 147, 118, 126, 126, 140, 140, 145, 145, 147, 154, 191, 205, 205, 476, 476, 481-483, 485-489, 154, 191, 492, 514, 522, 522, 532, 492, 495, 495, 496, 496, 514, 532, 534-536, 534-536, 540, 543-546, 557, 562, 589, 589, 590, 539, 540, 557, 562, 636, 636, 778 778 143, 147, 147, 175, Sommer, R., 126, 126, 143, 175, 190, 190, 199, 204, 204, 206 206 Specker, E., 144 144 Spector, 393, Spector, C., C., 286, 286, 340, 340, 349, 349, 350, 350, 366-371, 366-371,393, 395, 396, 400, 405 Spivey, J. M., 766, 766, 785 785 Srivas, M., 783 Staal, J. F., 467 Staples, Staples, J., 429, 458, 459, 471 SHirk, St~irk, R. F., 74, 655, 658, 664, 672, 681 Statman, Statman, R., 586, 586, 587, 587, 595, 595, 628, 628, 636 636 Stearns, Stearns, R., 686, 686, 779 779 Stein, Stein, M., M., 459, 459, 471 471 Stenlund, S., 724, 724, 745, 745, 755, 785 Stern, J., 206 206 Stigt, W. P. van, 727, 727, 785 785 Stockmeyer, L. J., 100, 100, 106, 106, 147 Strahm, Strahm, T., T., 268, 268, 334, 334, 335, 335, 423, 423, 471 471 Strannegard, C., 485, 541, 545 Strannegs C., 485, 494, 494, 531, 531,541,545 Streicher, T., 461, 471 461,471 Sudan, Sudan, M., M., 630 630 Suppes, P., P., 467, Suppes, 467, 469, 469, 780 780 Suslin, M. 1., I., 384 384 Sutor, R. S., 767, 767, 780 780 Swaen, 459, 471 471 Swaen, M. M. D. G., G., 459, Sweedler, 471 Sweedler, M. M. E., 471 Szabo, M. E., 75, 75, 144, 144, 778 778 Szegedy, M., 630 630 Tait, W. W., 16, 16, 17, 17, 77, 126, 126, 153, 153, 164, 164, 165, 175, 175, 206, 206, 220, 220, 232, 232, 254, 254, 269, 269, 271, 271, 282, 282, 295, 315, 338, 295, 315, 338, 339, 339, 359-361, 359-361, 370, 370, 377, 377, 381, 391, 381, 391, 397, 397, 405, 405, 459, 459, 472, 472, 655, 655, 673, 673, 674, 674, 682, 682, 723, 723, 725, 725, 728, 728, 745, 745, 785 785 16, 44, Takeuti, G., G., 16, 44, 54, 54, 57, 57, 76, 76, 77, 77, 80, 80, 97, 97, 108, 131-133, 138, 145, 147, 108, 123, 123, 126, 126, 131-133, 138, 145, 147, 187, 189, 187, 189, 206, 206, 267, 267, 335, 335, 398, 398, 405, 405, 571, 571, 574, 600, 625, 634, 636, 657, 682 574, 600, 601, 601,625,634, 636, 657, 682 Tarski, A., 82, 82, 83, 147, 467, 83, 147, 467, 500, 500, 503, 503, 533, 533, 544, 582, 636, 544, 545, 545, 560, 560, 562, 562, 581, 581, 582, 636, 686, 686, 699, 699, 785 785 Tatsuta, 461, 467, 472 Tatsuta, M., 461,467, 472 459, 465, P., 75, Taylor, P., 75, 397, 397, 398, 398, 402, 402, 459, 465, 675, 675, 680, 680, 689, 689, 692, 692, 695, 695, 724, 724, 778 778 Teitelbaum, T., 708, 708, 784 784 795 795 Tennenbaum, S., 533, 533, 540 Tharp, 458, 472 Tharp, L. H., 458, Thayer, F. J., 691, 777 691,777 Thiele, H., 462 Thomas, Thomas, W., 544 Thompson, S., 691, 785 691,785 Tiuryn, J., 766, 766, 780 Tofte, M., 766, 766, 782 Tompkins, R. R., 472 Toran, Tors J., 74 Troelstra, A. S., 64, 66, 68, 68, 74, 77, 77, 80, 80, 339342, 349, 350, 366, 402, 342, 349, 350, 352, 352, 357, 357, 366, 402, 403, 403, 405, 408, 412, 421, 423, 424, 431, 408, 411, 412, 424, 431, 434, 436, 436, 445, 448, 450, 450, 461, 461, 465433, 434, 467, 472, 500, 467, 471, 471, 472, 500, 541, 541, 546, 546, 721, 721, 748, 748, 766, 766, 780, 780, 785 785 Trybulec, A., 691, 785 691,785 Tseitin, G. S., 20, 459, 472, 472, 595, 595, 628, 628, 636 20, 77, 77, 459, 636 Thcker, Tucker, J. V., 190, 190, 207 207 Thran, 631, 632 Turin, Gy., 604, 604, 605, 605, 631,632 Thring, Turing, A., 87, 99, 103, 103, 104, 104, 106, 106, 146, 146, 161, 354, 354, 380, 380, 495, 495, 546, 546, 579, 579, 588, 588, 619, 619, 626, 626, 686, 686, 688 688 Tychonoff, A., 8 Ullman, Ullman, J. J. D., D., 779 779 Ungar, Ungar, A. A. M., M., 723, 723, 785 785 Uribe, J., J., 776 776 Uribe, T. E., 774 774 Urquhart, 340, 364, 364, 365, 365, 401, Urquhart, A., 101, 101, 144, 144, 340, 401, 434, 461, 463, 603, 607, 630, 635 461,463, Ursini, Ursini, A., A., 542 542 Valentini, S., 484, 484, 545 545 Van Gelder, A., 672, 672, 682 682 Vardanyan, 532, 534, Vardanyan, V. V. A., A., 532, 534, 540, 540, 546 546 Varpahovskij, Varpahovskij, F. 1., L., 473 Vaught, R. 1., 554, 637 L., 554, 637 Vavasis, Vavasis, S., S., 775 775 Veblen, 0 O.,. , 214, 214, 304, 304, 310, 310, 383, 383, 392 392 Veltman, Veltman, F., F., 514, 514, 515, 515, 544 544 Venema, Y., y., 546 546 Verbrugge, 542, 546 Verbrugge, R., R., 488, 488, 514, 514, 542, 546 Vesley, 146, 333, 401Vesley, R. R. E., E., 145, 145, 146, 333, 334, 334, 368, 368, 401404, 428, 434, 404, 428, 434, 466-468, 466-468, 471, 471, 473, 473, 777, 777, 781 781 Visser, A., 480, 485, 487-491, 494, 495, 495, 505, 480, 485, 487-491,494, 505, 513-515, 521, 521, 522, 513-515, 522, 528, 528, 530, 530, 541, 541, 544, 544, 546 546 Vliet, J. van, 784 784 Voorbraak, Voorbraak, F., F., 496, 496, 546 546 Vopenka, Vop~nka, P., P., 589 589 Voronkov, A., 473 Voronkov, A., 473 796 796 Name Name Index Index Vrijer, R. C. D., 690, 690, 782 Wadsworth, C., 690, 690, 691, 709, 778 691,709, Wainer, S. S., 74, 80, 143, 143, 164, 164, 175, 175, 189, 189, 190, 193, 203-207, 242, 333, 334 Wansing, H., 546 546 Wattenberg, F., 768, 768, 785 785 Wegman, M. N., 61, 61, 77 77 Wehmeier, K. F., 460, 461, 473 461,473 Weiermann, A., 190, 190, 199, 199, 201, 204, 207, 207, 241, 201,204, 241, 247, 249, 253, 266, 300, 333, 335 Weingartner, P., 773 Weispfennig, V. B., 400 400 Werner, B., 691, 762, 783 691,762, 783 Westerstahl, D., 144, 144, 204 Weyl, H., 345, 366, 405 Weyrauch, R., 690, 690, 785 685, 687, 687, 694, Whitehead, A. N., 31, 31, 77, 77, 685, 694, 785 785 Widgerson, A., 134, 134, 636 Wilkie, A. J., 80, 137-139, 143, 80, 114, 114, 118, 118, 137-139, 143, 147, 147, 583, 621, 626, 629, 635 621,626, Wilmers, G. M., 635 Wing, J. M., 691, 778 691,778 Wirsing, M., 779 779 Wittgenstein, 1., L., 688, 688, 694, 727, 727, 785 690, 785 Wolfram, S., 690, Woods, A., 599, 607, 630, 634 607, 618, 621, 621,630, Wos, L., 23, 25, 63, 75, 77, 78, 690, 690, 786 Wrathall, C., 99, 100, 100, 106, 106, 147 Wright, J. von, von, 779 779 Wrightson, G., 25, 25, 74, 75, 77, 78, 691, 784 691,784 Yannakakis, M., 133, 133, 145 Yao, A. C.-C., 618, 618, 636, 637 Zach, R., 628, 630 Zambella, D., 108, 132, 147, 522, 546 108, 132, 147, 485, 522, Zermelo, E., 577, 586, 688, 577, 582, 582, 586, 688, 718 718 Zippel, R., 767, 767, 775, 775, 786 786 Zlatin, D. R., 767, 776 Zucker, J. 1., I., 190, 190, 207, 207, 723, 723, 786 786 Subject Index Index Subject IE! IE1,, IIIn IIIn,, 46, 46, 84 84 LE1,, LIIn LIIn,, 84 LE! HA*), 341, 409, 409, 412, 412, 488, Heyting's (HA, (HA, HA* ) , 341, 721 higher-order Heyting Heyting (HAH), (HAH), 446 446 higher-order ( I D 1 ) ), , 269, inductive definition (ID! 269, 387, 387, 388 (I-HA), 430 intensional finite-type (I-HAl, finite-type (HAW (HAw)),, 429, 429, 741 intuitionistic finite-type see Heyting's intuitionistic first-order, see intuitionistic second-order (HAS), 441 (Aut-II~)), iterated comprehension (((II~-CA), ( ITt -CA) , (Aut-IIl ) ), 276, 277 I D abstraction, 423 operator, 301 acceptable operator, definition, 388, 388, 391, 391, accessibility inductive definition, 392 accessibility relation, 478 accessible part, part, 271 271 accessible Ackermann-Peter function, 242 active formula, 42 active sequent, 34 additive components, 213 additive connective, 71-73, 733 additive normal form, 213 212, 308 additively indecomposable, 212, 479, 516, see also also consequence adequate, 479, 516, 519, 519, see a n d structure and admissibility predicate, 283 (Adl)-(Ad3), axioms (Ad 1) - (Ad3) , 283 admissible, 217, 217, 304 admissible, believes believes to to be, be, 218 218 Algol, 755 analysis, see see also also arithmetic, 366, analysis, 366, 395 see ordinal analysis analysis, ordinal, see ancestor, 12, 32 direct, direct, 12, 12, 32 32 immediate, immediate, 12 12 anchored, anchored, 43, 43, 46, 46, 110, 110, 112 112 antecedent, antecedent, 10 10 anti-idempotency, anti-idem potency, 86 anti symmetry, 86 antisymmetry, 86 application, application, 715 715 arithmetic, arithmetic, see see also also bounded bounded arithmetic, arithmetic, number number theory theory and a n d Peano Peano arithmetic arithmetic bounded bounded theories theories I�o IA0,, 84, 84, 98 98 S~,' T� T~,, 46, 99, 102 52 ( P A 22 )),, 271,366 classical second-order (PA 271, 366 Dialectica theories HA HA # # , PA# P A # , 352 ' H-"A# # , P-A#, 364 fiA extensional extensional finite-type finite-type (E-HA), (E-HA), 430 430 fragments A fragments of of P PA BE! BE1,, BIIn BIIn,, 84 84 p}/ 797 797 798 798 Subject S u b j e c t Index Index associativity, 85 associativity, 85 atomic atomic formula, formula, 26, 26, 642, 642, 703 703 atomic atomic set set term, term, 295 295 attribute attribute grammar, grammar, 708 708 Automath, 721 Automath, 721 Automath Automath book, book, 721 721 autonomous autonomous iteration, iteration, 385 385 autonomous autonomous ordinal, ordinal, 383, 383, 385 385 autonomously autonomously inaccessible, inaccessible, 267 267 auxiliary auxiliary formula, formula, 12 12 axiom, axiom, 751 751 Axiom f3, 286, Axiom ~, 286, 288 288 Axiom Axiom of of Choice, Choice, 347, 347, 352, 352, 367, 367, 379, 379, 717, 717, see see also also dependent dependent choice choice AC, 271, 432 AC, 271,432 Extended, Extended, 424 424 quantifier-free, 426 426 quantifier-free, Rule Rule (ACR), (ACR), 432 432 axiom axiom schema, schema, 591 591 axiomatiable, axiomatiable, 117 117 axiomatization, axiomatization, 29, 29, 551 551 Bachmann Bachmann hierarchy, hierarchy, 385 385 Bachmann-Howard 196, 268, 268, Bachmann-Howard ordinal, ordinal, 193, 193, 196, 385, 391, 392 385, 391,392 bar, bar, 367 367 Bar 272, 273, Bar Induction Induction (BI), (BI), 272, 273, 366-368, 366-368, 428 428 bar bar recursion, recursion, 366, 366, 368 368 Bar Bar Rule, Rule, 273, 273, 383 383 bar bar theorem, theorem, 368 368 BASIC axioms, BASIC axioms, 101 101 Basic Basic Set Set Theory, Theory, BST BST,, 280 280 Beth's Beth's Definability Definability Theorem, Theorem, 58 58 bimodal , 494, 539 bimodal logic, logic, 491 491,494, 539 type, type, 492 492 binary binary tree, tree, 371 371 binding binding occurrence, occurrence, 734 734 binding binding phrase, phrase, 714 714 binumerate, binumerate, 504 504 block block opener, opener, 721 721 body, body, 649 649 Boolean circuit, 595 Boolean circuit, 595 bounded bounded depth, depth, 607 607 monotone, monotone, 615 615 Boolean Boolean function, function, 33 Boolean Boolean proposition, proposition, 702 702 bootstrapping, 85, 89 bootstrapping, 85, 89 bound 27, 69, bound variable, variable, 27, 69, 704, 704, 734 734 bounded 139, 364 bounded arithmetic, arithmetic, 97, 97, 139, 364 S� S~,, T� T~,, 99, 99, 102 102 lAo IA0,, 84, 84, 98 98 and and propositional propositional logic, logic, 619 619 bounded bounded consistency, consistency, 138 138 bounded bounded formula, formula, 83, 83, 170, 170, 437, 437, 439 439 bounded bounded proof, proof, 138 138 bounded bounded quantifier, quantifier, 82, 82, 109 109 bounded bounded set, set, 211 211 bounded bounded theory, theory, 82, 82, 97 97 Boundedness Boundedness Lemma, Lemma, 225 225 Boundedness Boundedness Theorem, Theorem, 224, 224, 303 303 Bounding Bounding Lemma, Lemma, 170, 170, 172 172 bounding bounding term, term, 88 88 Brouwer-Heyting-Kolmogorov Brouwer-Heyting-Kolmogorov (BHK) (BHK) interinterpretation, 65, 408 pretation, 65, 408 Calculus Constructions, 398 Calculus of of Constructions, 398 calculus calculus of of problems, problems, Medvedev's, Medvedev's, 461 461 call call by by name, name, 715 715 call call by by value, value, 715 715 cancellation cancellation law, law, 85, 85, 86 86 cancellation cancellation of of double double negations, negations, 438 438 canonical canonical proof, proof, 725 725 canonical canonical realizer, realizer, 415 415 canonical 16, 119 canonical term, term, 1116, 119 canonical canonical value, value, 745 745 canonically canonically false false proposition, proposition, 697 697 canonically canonically true true proposition, proposition, 694, 694, 697 697 Cantor 126, 157, 157, 185, Cantor normal normal form, form, 126, 185, 196, 196, 213 213 Cantorian Cantorian closed, closed, 300 300 cardinal, cardinal, 211 211 cardinal cardinal term, term, 308 308 cardinality, cardinality, 211 211 cartesian cartesian product, product, 343, 343, 696, 696, 756 756 category category theory, theory, 768 768 cedent, cedent, 10 10 characteristic characteristic set, set, 254, 254, 296 296 Characterization Characterization Theorem, Theorem, 416 416 choice, choice, see see axiom axiom of of choice choice and a n d dependent dependent choice choice Church's Thesis, 415 Church's Thesis, 415 CT, CT, 433, 433, 437 437 Extended, Extended, 416, 416, 438 438 Weak Weak Extended, Extended, 440 440 with with Uniqueness, Uniqueness, 437 437 Church-Kleene Church-Kleene ordinal ordinal WfK wcK ,, 217, 217, 222, 222, see see also constructive ordinal also constructive ordinal Church-Rosser Church-Rosser property, property, 358 358 circuit, Boolean circuit circuit, see see Boolean circuit Clark's Clark's equational equational theory theory (CET), (CET), 648 648 standard standard model, model, 648 648 class, class, 718 718 class class terms, terms, 280 280 Classification Classification Theorem, Theorem, 202, 202, 490 490 clause, 19, 598 clause, 19, 598 ground, ground, 62 62 Horn, Horn, see see Horn Horn clause clause 799 799 Subject S u b j e c t Index Index mixed, mixed, 19 19 negative, negative, 19 19 positive, positive, 19 19 program, program, 649 649 clause, clause, unit, unit, 24 24 clique clique problem, problem, 615 615 closed, closed, 69, 69, 212, 212, 269, 269, 282, 282, 642 642 closed closed recursive recursive term, term, 498 498 closed closed under under aa rule, rule, 272 272 closed closed under under substitution, substitution, 33 33 closure closure ordinal, ordinal, 269 269 club, club, 212 212 co-product co-product type, type, 739 739 cointerpretable, 503, 528 528 cointerpretable, 503, collapsibly less, « collapsibly less, <<,, 244 244 collapsing collapsing function, function, 242, 242, 243, 243, 304 304 Collapsing 309, 310 Collapsing Theorem, Theorem, 195, 195, 309, 310 collection, 12, 216, collection, 84, 84, 1112, 216, 321, 321, 327, 327, see see also also replacement replacement � -Collection, 280 E-Collection, 280 combinator, 429, 715 715 combinator, 69, 69, 344, 344, 360, 360, 423, 423, 429, combinatory combinatory completeness, completeness, 344 344 commutativity, commutativity, 85 85 compactness compactness propositional, propositional, 88 Comparisons Comparisons Lemma, Lemma, 160 160 complement, complement, 19 19 complementary complementary relation, relation, 641 641 complete, complete, 44 implicationally, implicationally, 591 591 complete complete context, context, 722 722 complete complete induction, induction, 86 86 complete complete justification, justification, 708 708 complete complete partial partial order, order, 748 748 completed 650 completed definition, definition, 650 completeness completeness combinatory, combinatory, 344 344 cut-free, cut-free, 33 33 first-order, 30 first-order, 30 implicational, implicational, 66 infinitary, infinitary, 165 165 interpretability 519 interpretability logic, logic, 518, 518, 519 modal modal logic, logic, 478-480 478-480 paramodulation, 63 63 paramodulation, propositional, 66 propositional, provability provability logic, logic, see see arithmetic arithmetic complete completeness ness resolution, 20, 62 resolution, 20, 62 SLDNF, SLDNF, 671 671 w-Completeness w-Completeness Theorem, Theorem, 224 224 completion, completion, 650 650 Clark, Clark, 650 650 partial, partial, 672 672 composed composed set set term, term, 295 295 composition, composition, 96, 96, 103 103 compound compound proposition, proposition, 697 697 comprehension, 717, see comprehension, 289, 289, 291, 291,717, see also also arith arithmetic, arithmetical comprehension, metic, arithmetical comprehension, and and recursive recursive comprehension comprehension axiom CA) , 271, 366, 379, axiom ((CA), 271,366, 379, 446 446 full, full, 442, 442, 458 458 rule CR) , 279 rule ((CR), 279 ITt -Comprehension Theorem, II~-Comprehension Theorem, 287 287 computable computable real, real, 762 762 computation computation formula, formula, 183, 183, 185, 185, 186 186 computational complexity, complexity, cc cc ((.)) ,, 245 245 computational concatenation, 124, 425 concatenation, 124, 425 Condensation Condensation Lemma, Lemma, 216 216 conditional conditional function, function, 103 103 confluent, confluent, 358 358 consequence consequence 2-adequate, 2-adequate, 651 651 4-adequate, 4-adequate, 651 651 lower, lower, 651 651 upper, upper, 651 651 conservative, conservative, 506 506 Conservative Conservative Class, Class, 419, 419, 428 428 consistent, consistent, 30, 30, 138, 138, 506, 506, 530, 530, see see also also bounded bounded consistency consistency and a n d free-cut free-cut free free consistency consistency w-consistent, 487, 494, w-consistent, 119, 119,487, 494, see see also also weakly weakly w -consistent w-consistent constant constant objects objects functor, functor, 450 450 constant constant symbol, symbol, 26 26 constructible constructible hierarchy, hierarchy, 215, 215, 267 267 constructive 391, see constructive ordinal, ordinal, 391, see also also ChurchChurchKleene Kleene ordinal ordinal WfK W~K Continuity Continuity Generalized, 427 Generalized, 427 Weak, Weak, 434 434 continuous, continuous, 212 212 continuous continuous cut cut elimination, elimination, 265 265 contraction 1 , 73 contraction rule, rule, 111, 73 :3 -contraction, 53 3-contraction, 53 propositional, propositional, 53 53 controlled controlled access access theories, theories, 722 722 controlling 314, 321 controlling operator, operator, 314, 321 cotolerance, cotolerance, 503, 503, 528, 528, 530 530 countable countable tree-ordinal, tree-ordinal, 154 154 counterwitness, counterwitness, 504 504 counting, counting, length length bounded, bounded, 93 93 course course of of values, values, 223 223 critical critical function, function, 383 383 critical critical ordinal, ordinal, 214 214 - 800 800 Subject S u b j e c t Index Index critical critical successor, successor, 516 516 cryptograph� cryptography, 617 617 Curry-Howard Curry-Howard isomorphism, isomorphism, 47, 47, 68, 68, 500, 500, 751, 751, 752, 752, see see also also formulas formulas as as types types currying, currying, 343 343 cut 36, 73, 109, 166, cut elimination, elimination, 36, 73, 109, 166, 231, 231, 251, 251, 299, 299, 574 574 continuous, continuous, 265 265 partial, partial, 657 657 semantic, semantic, 658 658 Cut Cut Elimination Elimination Theorem, Theorem, 16, 16, 37, 37, 65, 65, 109, 109, 169, 169, 233, 233, 235 235 165, 656 cut formula, 12, 165, cut 12 cut free free proof, proof, 12 cut cut in in aa model, model, 562 562 inductive, inductive, 98, 98, 118, 118, 140, 140, 562 562 cut cut rank, rank, see see rank rank Cut 168 Cut Reduction Reduction Lemma, Lemma, 168 cut rule, 11, 231, 554, 656, 656, 708 cut rule, 11,231,554, 708 cutting 604, 616 cutting plane plane proofs, proofs, 604, 616 D-interpretation, D-interpretation, 346, 346, see see also also Dialectica Dialectica ininterpretation terpretation D-interpreted, 348 D-interpreted, 348 dag -like proof, dag-like proof, see see proof proof database, database, 23, 23, 63 63 deduction rule, deduction rule, 600 600 deduction 6, 477, 477, 600 deduction theorem, theorem, 6, 600 definable definable function, function, see see also also provably provably recursive recursive and a n d provably provably total total Qi-definable, Qi-definable, 131 131 E� -definable, 102 S~-definable, 102 El -definable, 87 $1-definable, 87 ,,),-definable E? -DEF(")')') ), ~'-definable ((S~ ), 172 172 E -definable, 281 S-definable, 281 relative -function, 284 relative E S-function, 284 definable definable predicate predicate A� -definable, 102 A~-definable, 102 Ao -definable, 86 Ao-definable, 86 A-definable, 281 A-definable, 281 define, define, 504 504 definedness 715, 728, 728, 745, definedness U ($),) , 411, 411,715, 745, 749 749 defining axiom, 86-88, 102, 284 defining axiom, 86-88, 102, 281, 281,284 definition by cases, 423 definition by cases, 423 definition definition form, form, 650 650 denotational denotational semantics, semantics, 759 759 dependency dependency graph, graph, 722 722 dependent dependent choice choice (DC), (DC), 369, 369, 379 379 dependent dependent function, function, 744 744 dependent dependent product, product, 744 744 dependent dependent record, record, 764 764 dependent type, type, 743 dependent 743 depth depth E-depth, E-depth, 53 53 formula, formula, 37, 37, 569, 569, 599 599 Kripke Kripke model, model, 478 478 proof, proof, 569, 569, 599 599 term, term, 567 567 derivative, derivative, 214 214 derived derived model, model, 480 480 descendent, descendent, 12, 12, 32 32 direct, direct, 12, 12, 32 32 immediate, immediate, 12 12 descendent descendent recursive, recursive, 266, 266, see see also also recursive recursive descent descent functional, functional, 159 159 Detachment Detachment Lemma, Lemma, 255 255 Diagonal Diagonal Lemma, Lemma, 119 119 diagonalizable diagonalizable algebra, algebra, see see Magari Magari algebra algebra diagonalization, diagonalization, 119, 119, 716 716 diagram, diagram, 219 219 Dialectica Dialectica interpretation, interpretation, 338-340, 338-340, 346, 346, 458, 458, see see also also D-interpretation D-interpretation Diller-Nahm Diller-Nahm interpretation, interpretation, 458 458 directed, directed, 749 749 directed directed acyclic acyclic graph, graph, 13 13 discharge, discharge, 47 47 disjoint sets, 613, disjoint NP AfT~ sets, 613, 617 617 disjoint disjoint union, union, 739, 739, 740, 740, 757 757 disjunction disjunction property, property, 66 66 DP, DP, 419, 419, 432 432 distinguished, distinguished, 222 222 distributivity, distributivity, 85 85 divides, divides, 89 89 division, division, 89 89 domain, domain, 27, 27, 501 501 domain domain closure closure axiom axiom (DCA), (DCA), 649 649 domain domain theory, theory, 759 759 domain domain type, type, 715 715 double-negative double-negative translation, translation, see see negative negative translation translation downward downward persistency, persistency, 301 301 dual, dual, 739 739 duality, duality, 739 739 effective, effective, 453 453 canonically, canonically, 453 453 eigenvariable, eigenvariable, 32, 32, 48, 48, 110 110 Elementary Elementary Analysis Analysis (EL), (EL), 425, 425, 439 439 elementary elementary function, function, 164 164 elementary elementary inductive inductive definitions, definitions, 270 270 elementary elementary topos, topos, 768 768 Elimination 236, 258 Elimination Lemma, Lemma, 236, 258 elimination elimination rule, rule, 48 48 Elimination Elimination Theorem, Theorem, 237 237 Embedding Embedding Theorem, Theorem, 179 179 SSubject u b j e c t IIndex ndex empty type, type, 735, 735, 736 736 empty endsequent, 11 11 endsequent, enumerating function, function, 212 212 enumerating envelope, 651 651 envelope, i-calculus, 31,554 31, 554 e-calculus, i-number, 157, 157, 202, 202, 213, 213, 392 392 e-number, iO , 157, 213 213 e0,157, equality axiom, axiom, 29, 29, 32, 32, 63, 63, 233, 341 , 648, 656 656 equality 233, 341,648, equality formula, formula, 642 642 equality equality functional, functional, 430 430 equality Equality 319 Equality Theorem, Theorem, 319 equivalence class, class, 720 720 equivalence equivalence of realizabilities, realizabilities, 410 410 equivalence of essential reflexivity, reflexivity, 493 493 essential essentially reflexive, reflexive, 505 505 essentially essentially II~, II� , E~, �� , 279 279 essentially evaluates to to (evals_to), (evals_to), 730 730 evaluates exchange rule, exchange rule, 1111 excluded middle, law of, of, 64, 64, 342 342 excluded middle, law 3-fre� 413, 413, 416, 3-free, 416, 440 440 expansion expansion V-expansion, 51 V-expansion, 51 strong -expansion, 51 strong V V-expansion, 51 Explicit Definability Explicit Definability (ED), (ED), 432 432 Explicit Definability Definability for for Numbers 419 Explicit Numbers (EDN), (EDN), 419 explicit definition, 58 58 explicit definition, exponentiation, 91, 98, 98, 139, 139, 156, 156, 169, 169, 193 193 exponentiation, 91, ordinal, 213 213 ordinal, exportation, 301 301 exportation, expression, 295 expression, 295 Extended Extended Axiom Axiom of of Choice Choice (EAC), (EAC), 424 424 Extended (ECR), 419, 419, 439, Extended Church's Church's Rule Rule (ECR), 439, 440 440 Extended (ECT), 416, 416, 438 Extended Church's Church's Thesis Thesis (ECT), 438 extended extended Frege, Frege, see see extension extension Frege Frege system system extended resolution, 599 extended resolution, 599 extension, 20, 645 extension, 20, 645 extension system, 592 extension Frege Frege system, 592 extension extension variable, variable, 20 20 extensional, extensional, 121 121 Extensional Extensional Continuous Continuous F\mctionals Functionals (ECF), (ECF), 433 433 extensional extensional equality, equality, 350 350 extensionality, extensionality, 216, 216, 295 295 EXT, EXT, 434, 434, 446, 446, 458 458 extraction extraction function, function, Ext E x t , , 91 91 factoring, factoring, 61 61 faithfully faithfully interpretable, interpretable, 485, 485, 503, 503, 528 528 Fan Fan Functional, Functional, 433 433 Fan Fan Rule, Rule, 434, 434, 436 436 Fan Fan Theorem, Theorem, 366, 366, 428 428 fast-growing 158, 195 fast-growing hierarchy, hierarchy, 152, 152, 158, 195 801 801 feasibly interpretable, interpretable, 514 Feferman Feferman provability, provability, 495 495 Fermat's Fermat's little little theorem, theorem, 720 final, final, 739 739 finite subtheory, 501 finite type, type, 343 finite finite symbols, 429 429 finite type type symbols, First Incompleteness Incompleteness Theorem, Theorem, 120 120 First first-order logic, logic, 29, 341 269, 399 399 point, 269, fixed point, fixed point fixed point combinator, combinator, Y, Y, 763, 763, 764 764 fixed point theorem, 484, 484, 521 fixed-point fixed-point term, 308 forcing forcing relation, relation, 478 478 formal proof, 2 formal formula formula first-order, 26, 31,642 31, 642 propositional, 3, 701 formulas as types, 47, 68-70, 459, 500, formulas 500, 751 Fortran, 754 Foundation axiom, 216, 216, 217 Foundation Lemma, 323 Foundation Theorem, 324 fragment, fragment, 80 80 515, see also Kripke frame, Veltman Veltman frame, 515, frame frame and and Visser Visser frame frame free free access, access, 722 722 free 16, 43, 112 free cut, cut, 16, 43, 46, 46, 112 free cut elimination, 42, 109, 109, 1111, 1 1 , 112 Free-cut Elimination Theorem, 16, 44, 46, 47, 65, 65, 109 109 free cut cut free, free, 16, 16, 43 43 free free-cut free-cut free free consistency, consistency, 138 138 variable, 27, 27, 69, 69, 703, 703, 734 734 free variable, FV, FV, 409 409 free free variable variable normal normal form, form, 33 33 free-cut free-cut elimination, elimination, 16 16 freely freely substitutable, substitutable, 27 27 Frege Frege proof, proof, 591 591 Frege Frege rule, rule, 591 591 Frege Frege system, system, 5-10, 5-10, 591, 591, see see also also extension extension Frege substitution Frege Frege and and substitution Frege bounded bounded depth, depth, 599 599 Full Type Structure, 430 function, function, 715, 715, 738 738 function function comprehension, comprehension, 717 717 function function symbol, symbol, 26 26 function function type, type, 429, 429, 714 714 functional, functional, 343 343 functional model, 356 functional functional of of finite finite type, type, 356 356 fa Fo,, 214, 214, 383 383 802 802 Subject S u b j e c t Index Index Generalized Continuity (GC) Generalized (GC),, 427, 427, 439 Rule Rule (GCR), (GCR), 428 428 Gentzen's Gentzen's Hauptsatz, Hauptsatz, see see cut cut elimination elimination Gentzen's Theorem, Gentzen's Theorem, 240 240 Girard 393, see Girard interpretation, interpretation, 393, see also also polymor polymorphism phism globally globally essentially essentially reflexive, reflexive, 505 505 goal, goal, 649 649 empty, empty, 649 649 form, 661 form, 661 Giidel-Bernays GSdel-Bernays (GB) (GB) set set theory, theory, 589 589 Giidel GSdel {3 ~ function, function, 94 94 Giidel Lemma, 119 GSdel Diagonal Diagonal Lemma, 119 Giidel GSdel Fixpoint Fixpoint Lemma, Lemma, 119 119 Giidel 14 GSdel number, number, 92, 92, 1114 good good representation, representation, 261 261 Goodstein 191 Goodstein sequence, sequence, 191 Groebner Groebner proof proof system, system, 604 604 Grothendieck Grothendieck topos, topos, 719 719 ground ground clause, clause, 62 62 ground ground literal, literal, 62 62 ground ground resolution, resolution, 62 62 group, group, 758 758 Grzegorczyk Grzegorczyk hierarchy, hierarchy, 174 174 Hajas HajSs calculus, calculus, 601 601 Hardy Hardy functions, functions, 158, 158, 249 249 Hardy 152, 158, Hardy hierarchy, hierarchy, 152, 158, 242 242 Harrop Harrop formula, formula, 66 66 head, head, 649, 649, 714 714 height, 486 height, 168, 168, 178, 178, 486 Kripke model, 478 Kripke model, 478 Herbrand Herbrand disjunction, disjunction, 574 574 Herbrand Herbrand function, function, 50 50 Herbrand Herbrand proof, proof, 52 52 Herbrand Herbrand structure, structure, 645 645 Herbrand Herbrand universe, universe, 643 643 Herbrand Herbrand variant, variant, 574 574 Herbrand's Herbrand's Theorem, Theorem, 48-56 48-56 Herbrandization, Herbrandization, 50 50 Hereditarily Hereditarily Effective Effective Operations Operations (HEO), (HEO), 357, 357, 431 431 hereditarily recursive (HRE), hereditarily extensional extensional recursive (HRE), 390 390 hereditarily hereditarily majorize, majorize, see see majorize majorize Hereditarily Hereditarily Recursive Recursive Operations Operations (HRO), (HRO), 357, 357, 430 430 Heyting Heyting arithmetic, arithmetic, see see arithmetic arithmetic Heyting's semantics, 724, 751 Heyting's semantics, 724, 751 Hierarchy Hierarchy Theorem, Theorem, 173 173 higher-order higher-order logic, logic, intuitionistic, intuitionistic, 445 445 Hilbert-Bernays-Liib Hilbert-Bernays-LSb Derivability Derivability Conditions, Conditions, 117 117 Hilbert Hilbert style style system, system, 29, 29, 553, 553, 729 729 Hilbert's Hilbert's program, program, 338, 338, 339 339 homomorphism, homomorphism, 606 606 honest, honest, 160 160 Honesty Honesty Theorem, Theorem, 163 163 Horn Horn clause, clause, 25, 25, 63-64 63-64 Howard ordinal, see Howard ordinal, see Bachmann-Howard Bachmann-Howard ordiordinal nal hyperarithmetic set, hyperarithmetic set, 380 380 Hyperarithmetical Quantifier Theorem, 229 Hyperarithmetical Quantifier Theorem, 229 hyperjump, hyperjump, 270 270 hyperresolution, hyperresolution, 22 22 hypothetical hypothetical judgement, judgement, 699 699 idempotency, idempotency, 415 415 identity identity axiom, axiom, 656, 656, see see also also equality equality axiom axiom identity-free identity-free derivation, derivation, 657 657 ILM ILM frame, frame, see see Veltman Veltman frame frame implication implication logical, 28, 32 logical, 28, 32 tautological, tautological, 44 implication ally complete, implicationally complete, 591 591 implicit implicit definition, definition, 58 58 impredicative impredicative polymorphism, polymorphism, 393 393 impredicative impredicative systems, systems, 266 266 table table of, of, 332 332 incompleteness, 18-122, 241 incompleteness, 1118-122, 241 Independence 418, Independence of of Premise Premise (IP), (IP), 347, 347, 352, 352,418, 432 432 induced induced model, model, 502 502 induction, 232, 272 induction, 232, 272 O -induction, 154 ~-induction, 154 transfinite, 1 1 , 380, 417 transfinite, 187, 187, 200, 200, 210, 210, 2211,380, 417 TI 286 T I ,, 224, 224, 238, 238, 261, 261,286 type type 11 (Res) ( R e s ) , , 378 378 induction induction axiom, axiom, 46, 46, 178 178 IND, 10 IND, 83, 83, 1110 length length (LIND), (LIND), 101, 101, 110 110 polynomial polynomial (PIND), (PIND), 101, 101, 110 110 Induction Induction Lemma, Lemma, 234 234 induction induction rule, rule, 45, 45, 176, 176, 200, 200, 735, 735, 737 737 IND, 46, 10 IND, 46, 1110 PIND, 46, 10 PIND, 46, 1110 inductive inductive cut, cut, see see cut cut in in aa model model inductive 269, 285, 285, 387, 399, inductive definition, definition, 269, 387, 391, 391,399, see see also also arithmetic arithmetic inductive inductive extension, extension, 677 677 inductive inductive norm, norm, 269 269 inductive inductive set, set, 276 276 inductive inductive type, type, 764 764 inductively definable, definable, 269 inductively 269 infinitary 165, 361 infinitary logic, logic, 17, 17, 165, 361 infinite infinite height, height, 486 486 infinity, infinity, axiom axiom of, of, 217, 217, 280 280 Subject S u b j e c t Index Index inhabited, inhabited, 705 705 initial, initial, 739 739 initial sequent, sequent, 32 32 initial input argument, argument, 665 665 input input resolution, resolution, 24 24 input input input variables, variables, 665 665 instance, 648, 703 instance, 49, 49, 591, 591,648, 703 intensional, 14, 1116, 16, 121 intensional, 113, 113, 1114, 121 in Q, 118 in Q, 118 Intensional Intensional Continuous Continuous Functionals Functionals (ICF), (ICF), 433 433 intensional intensional equality, equality, 350 350 interactive interactive proof, proof, 550 550 interpolability, interpolability, 529 529 interpolant, 56, 612 interpolant, 56, 612 Interpolation Interpolation Theorem, Theorem, 56-58, 56-58, 612 612 interpretability interpretability logic, logic, 514 514 interpretable, interpretable, 503 503 interpretation, 502, see interpretation, 502, see also also arithmetic arithmetic realrealization ization cointerpretation, 503, 528 cointerpretation, 503, 528 faithful, faithful, 503, 503, 528 528 feasible, 514 feasible, 514 weak, weak, 503, 503, 528 528 interpretation interpretation (structure) (structure),, 27 27 introduction introduction rule, rule, 47 47 intuitionistic 341, 411 intuitionistic logic, logic, 64-70, 64-70, 341,411 natural natural deduction, deduction, 48 48 Inversion Inversion Lemma, Lemma, 167, 167, 256, 256, 302 302 Inversion Inversion Theorem, Theorem, 13 13 irreducible, irreducible, 358 358 irrelevance irrelevance of of proofs, proofs, 723 723 isomorphic isomorphic !1-sets, 12-sets, 451 451 iterated iterated admissibility, admissibility, 291 291 /tAd, ItAd, 291 291 RltAd, RltAd, 292 292 iterated iterated closure, closure, 304 304 iterated 276 iterated comprehension, comprehension, 276 iterated 486, 490 iterated consistency, consistency, 486, 490 iterated iterated hyperjumps, hyperjumps, 270 270 iterated iterated inductive inductive definitions, definitions, 203, 203, 270, 270, 271, 271, 273, 273, 392, 392, see see also also arithmetic arithmetic iterated iterated reflection, reflection, 495 495 Java, Java, 754 754 judgment, judgment, 729 729 jump, 239, 239, 380 jump, 380 jump hierarchy, jump hierarchy, 276 276 justification, justification, 707, 707, 710 710 Kleene Kleene basis basis operator, operator, 377, 377, 384 384 Kleene Kleene basis basis theorem, theorem, 384 384 803 803 Kleene Kleene equality, equality, 746 746 Kleene's Kleene's 0 O,, 150, 150, 153 153 Kleene-Brouwer Kleene-Brouwer ordering, ordering, 228 228 Kond6-Addision theorem, theorem, 385 385 Kondo-Addision Konig's KSnig's Lemma, Lemma, 439, 439, see see also also Weak Weak Konig's KSnig's Lemma Lemma KPT KPT Witnessing Witnessing Theorem, Theorem, 131 131 Kreisel's 587 Kreisel's conjecture, conjecture, 571, 571,587 Kripke Kripke frame, frame, 478, 478, 535 535 Kripke Kripke model, model, 478, 478, 515, 515, 535 535 Kripke Kripke semantics, semantics, 700 700 Kripke-Platek Kripke-Platek set set theory theory KP K P --, , 216 216 KPw K P w -- , , 217 217 KP, KP, 217, 217, 280 280 KPw, KP w, 217, 217, 280 280 KPl, KP1, 283 283 KPi, KPi, 289 289 KP KP ,B ~,, 290 290 KPI KP1 with with iterated iterated admissibility, admissibility, 292 292 Lowenheim-Skolem L6wenheim-Skolem Theorem, Theorem, 216 216 labeled labeled assertion, assertion, 705 705 >.-abstraction, A-abstraction, 68 68 >.A-calculus, -calculus, 68, 68, 755, 755, 759 759 lambda 696, 709, lambda notation, notation, 696, 709, 714 714 >.-term, )~-term, 68 68 language, language, 44 least 282, 387 least fixed fixed point, point, 282, 387 least least number number principle, principle, see see minimization minimization axaxiom iom leftmost leftmost branch, branch, 384 384 Leivant's Leivant's Principle, Principle, 488 488 length, 92, 103, 103, 425 length, 92, 425 length length induction, induction, see see induction induction axiom axiom length minimization, length minimization, see see minimization minimization axiom axiom see also size, proof length, length, proof, proof, 13, 13, 564, 564, see also size, proof level, level, see see type type level level Levy Levy hierarchy, hierarchy, 215, 215, 295 295 Lifschitz Lifschitz topos, topos, 457 457 limit limit ordinal, ordinal, 281 281 Li m , 211 Lira, 211 limited limited iteration iteration on on notation, notation, 104 104 limited limited recursion recursion on on notation, notation, 365 365 line, line, 721 721 linear linear arithmetic arithmetic (SupInf), (SupInf), 711 711 linear linear bounded bounded automata, automata, 99 99 linear linear implication, implication, 72 72 linear linear logic, logic, 70-74 70-74 MALL, 73 MALL, 73 linear 270, 286 linear order, order, LaO /.0(.),, 270, 286 linear linear proof, proof, 551 551 linear linear resolution, resolution, 24 24 804 804 Subject Index Index Subject linear space, 99 linear time hierarchy, 99 linked list, 760 686, 754 754 Lisp, 686, list, 713 list type, 736 18, 598, 598, 642 literal, 18, ground, 62 negative, 19, 642 positive, 19, 642 LSb's Theorem, 122 Liib's local predicativity, 253 local reflection principle, 490 reflexive, 505 locally essentially reflexive, logic of proofs, 497 logic program, 649 allowed, 668 definite, 649, 649, 667 definite, general, 649 normal, 649 411-412,746 logic with partial terms (LPT), 41 1-412, 746 logical argument, 665 logical axiom, 5, 111, 176, 656 1 , 17, 176, logical consequence, 647 logical framework, framework, 685 logical implication, 28, 32 logical rule, 32, 656 Magari algebra, 485 magic rule, 707, 707, 709 part of an inference, 255, main part 255, 299, 299, see also also principal formula majorizable, 434, 434, 436 majorization hierarchy, 160 majorization properties, 159 majorize, 373, 434 354, 356, 356, Markov's principle 347, 352, 352, 354, principle (M), 347, 416 maximality, 460 131, see see also maximization axiom, 131, also minimization axiom meta-predicative, 268 metatheory, 488 Midsequent Theorem, 574 minimal logic, 48 minimization axiom, axiom, 86, 95 length (LMIN), 101,110 length 101, 110 MIN, 83, 110 minimization operator operator (#), (IL), 266, 377, 378 Mizar, Mizar, 691 ML, 754 modal logic completeness, completeness, 478 completeness completeness theorem, theorem, 480 480 modal modal operators operators D,O, 0 , 0 , 477 I-q,, 477 [J D,, !:::,. A,, 491 0 DR ,, 496 DR [> ~>,, 514 514 E., I:;i E + ,, 528 I:n, » >>,, 529 529 0 O,, 529 529 D,V,3,539 0 , 'v' , 3 , 539 modal modal propositional propositional logic, logic, 477 477 modal modal systems systems K,L,K4,S, K , L , K 4 , S , 477, 477, 478 478 $4, 481, 481,497 S4, 497 A,D, 487 487 A,D, CS,CSM, 492, 492, 493 493 CS,CSM, LP, LP, 497 497 IL,ILM, IL,ILM, 514 514 TOL,TLR,ELH, T O L , T L R , E L H , 529 529 Lq,S5, Lq,S5, 539 539 Sq, Sq, 539 539 QL,QS, QL,QS, 540 540 modality, modality, 73 73 modally modally expressible, expressible, 490 490 mode, mode, 665 665 mode mode assignment, assignment, 665 665 model, 647 model, 28, 28, 501, 501,647 modified modified realizability, realizability, see see realizability realizability Modula, 757 Modula, 757 module, 757, 760 module, 757, 760 Modulus Uniform Continuity 433 Modulus of of Uniform Continuity (MUC), (MUC), 433 modus modus ponens, ponens, 5, 5, 706, 706, 729 729 monotone monotone operator, operator, 269 269 monotonic, monotonic, 282 282 monotonicity monotonicity axiom, axiom, 493 493 Monotonicity Monotonicity Lemma, Lemma, 225 225 most most general general proof, proof, 568 568 move, move, 524, 524, 525 525 multiplicative multiplicative connective, connective, 71-73, 71-73, 733 733 multiply 189 multiply recursive, recursive, 189 N-interpretation, 342, 342, see see also also negative negative transtransN-interpretation, lation lation natural deduction, deduction, 47-48, 47-48, 69, 69, 600 600 natural natural numbers, 711 711 natural numbers, natural proofs, proofs, 134 134 natural ND-interpreted, 348 348 ND-interpreted, necessitation, 477, 477, 498 498 necessitation, negation as as failure, failure, 661 661 negation negative clause, clause, 19 19 negative negative formula, formula, 437, 437, 439 439 negative negative occurrence, 15 15 negative occurrence, 805 805 Subject S u b j e c t Index Index negative negative translation, translation, 66, 66, 67, 67, 338, 338, 341, 341, 342, 342, 355, 355, 370, 370, 392, 392, 766 766 neighbourhood 426 neighbourhood function, function, 426 no-counterexample 54, 340, no-counterexample interpretation, interpretation, 54, 340, 355, 355, 362 362 node, 478 node, 221, 221,478 non-logical symbols, 81 non-logical symbols, 81 non-schematic non-schematic theory, theory, 117 117 norm, norm, 242 242 oo-norm, c~-norm, 216 216 norm norm function, function, 201 201 normal, normal, 498, 498, 499 499 normal form, form, 358 358 normal normal normal function, function, 212 212 normal normal modal modal logic, logic, 477 477 normalizable, normalizable, 358 358 normalization, normalization, 17 17 normalizing, normalizing, 358 358 Nullstellensatz, Nullstellensatz, 603 603 Number Number Theory, Theory, NT, NT, 232, 232, see see also also arithmetic arithmetic second-order, second-order, NT NT2,271 2 , 271 numeral, numeral, 81, 81, 116, 116, 119, 119, 220, 220, 409 409 numeral wise representability, numeralwise representability, 113 113 numerate, numerate, 504 504 Nuprl, Nuprl, 722 722 object, object, 757 757 object object assignment, assignment, 28 28 object-oriented object-oriented programming, programming, 758 758 occurs occurs check, check, 60 60 w-consistent, w-consistent, see see consistent consistent !1 gt function, function, 447 447 !1 451 gt functionset, functionset, 451 !1 powerset, 451 powerset, 451 !1 gt predicate, predicate, 447 447 !1 product, product, 447 447 w w provability, provability, 487, 487, 494 494 !1 relation, relation, 447 447 !1 set, 446 gt set, 446 one-way one-way function, function, 617 617 ontological ontological axiom, axiom, 216, 216, 217 217 Operations Operations Hereditarily Hereditarily Effective, Effective, 431 431 Hereditarily Hereditarily Recursive, Recursive, 430 430 operator, operator, 300 300 operator controlled derivable, operator controlled derivable, 301 301 operator operator controlled controlled derivation, derivation, 253, 253, 254, 254, 300 300 optimal optimal propositional propositional proof proof system, system, 626 626 oracle, oracle, 106 106 order 222, 288 order type, type, otyp, otyp, 212, 212, 221, 221,222, 288 ordered ordered pair, pair, 696 696 ordinal, 210, 280, see ordinal, 210, 280, see also also tree tree ordinal ordinal 229, 230 ordinal ordinal analysis, analysis, 229, 230 IIl n~-,-, 229 229 for for set set theories, theories, 321-331 321-331 of of NT, NT, 240 240 profound, profound, 263 263 K-, 219 ~-, 219 IIg-, II ~ 247 247 ordinal ordinal arithmetic, arithmetic, 156, 156, 193 193 ordinal ordinal notation, notation, 495, 495, see see also also tree-ordinal tree-ordinal ordinal ordinal of of aa formula formula IIHl~,, 229 HIE, , 229 IFI nO ', 260 [F[Ho 260 2 ordinal ordinal of of aa theory theory IIA IIAxll ~, ' 216 216 x ll oo IIA IIAxll ~,, 216 216 x II F IIA I Axll n2 , 217, 219 x II K , IIA IIAxll~, IIAxII~, x II Ei" IIIAxlln~, IIA x l 1 n' , 228 IIAxlln;, 228 n8 IIA IIAxll, x l l , :228 IIA [[Ax[[ ~cK, 229 CK , 229 x l 1 E71 ]Ell table table of of impredicative impredicative theories, theories, 332 332 ordinal operator, operator, 300 300 ordinal ordinal ordinal sum, sum, 212 212 ordinal ordinal term, term, 308 308 ordinal ordinal terms, terms, 308 308 Orey Orey sentence, sentence, 531 531 Orey Orey set, set, 531 531 output output argument, argument, 665 665 output output variables, variables, 665 665 pairing, 423, 429, 429, 445 pairing, 70, 70, 423, 445 pairing 177, 216, pairing axioms, axioms, 177, 216, 279 279 parameter parameter variable, variable, 33 33 parameters, parameters, par(-) par(-),, 258, 258, 300 300 paramodulation, paramodulation, 63 63 parentheses, parentheses, omitting, omitting, 5, 5, 26 26 Parikh Parikh provability, provability, 495 495 Parikh's Parikh's Theorem, Theorem, 87, 87, 112 112 partial partial combinatory combinatory algebra, algebra, 424 424 partial partial continuous continuous application, application, 426 426 Partial Partial Continuous Continuous Operations Operations (PCO), (PCO), 426 426 partial (per), 719, 719, 745, partial equivalence equivalence relation relation (per), 745, 746, 746, 748 748 partial recursive, partial recursive, 172 172 in in an an ordinal, ordinal, 217 217 Partial Partial Recursive Recursive Operations Operations (PRO), (PRO), 424 424 partial partial type, type, 759 759 Pascal, Pascal, 754 754 path, path, 221 221 Peano 175, 231,352, 231, 352, 721, Peano arithmetic, arithmetic, 84, 84, 175, 721, see see also also arithmetic arithmetic persistence, persistence, 170 170 � -Persistency, 280 E-Persistency, 280 806 806 S u b j e c t Index Index Subject persistency downwards, 301 downwards, upwards, 301 Hi-completeness, II I -completeness, 494 pinning down, 267 pointer, 760 491,495 polymodal logic, 491, 495 393, 715, 715, 745 polymorphic, 393, polymorphic A-calculus, F, 394 polymorphism, 393 polynomial calculus, 604 polynomial growth rate, 98, 100 133, 134 Polynomial Local Search (PLS), 133, PLS function, 133 polynomial size tree (pst) proof, 564 polynomial time, 103, 103, 104, 104, 106 polynomial time hierarchy, 105-108 polynomially equivalent, 552 polynomially numerates, 578 polynomially simulates, 552 positive clause, 19 positive formula, 643 positive occurrence, 15, 282 positive resolution, 22 power type, 445 predecessor, 89, 423, 733 n-predecessor, 154 immediate n-predecessor, 154 predicate provability logic, 531 predicative, 268 Predicative Elimination Lemma, 237, 302 predicative polymorphism, 394, 398 predicativity, 267 prenexification, 51 E-preservativity, 488 b-preservativity, prime powers, 90 prime primes, 90 primitive notion, PN, 721 primitive recursion, 82, 96, 733 175, 189, 189, 219, 363, primitive recursive, 175, 363, 364 primitive recursive arithmetic, arithmetic, see see arithmetic primitive recursive function, 82, 96 defining equations, equations, 82 primitive recursive predicate, 96 PRWO, 264 primitive recursive well ordering, PRWO, principal principal formula, 12, 46, 110, 110, 112, 112, see see also also main main part part of an inference principal term, 308 probabilistically checkable checkable proofs, 550 product product topology, 9, 373 product product type, type, 429, 739 profound, 263 clause, 649 649 program clause, program program rules, rules, 656 656 stratified, 660 programs as deductive deductive systems, 655 programs as theories, 655 progressive, progressive, 187 187 Prog, Prog, 225, 225, 238, 238, 286 projection, 70, 96, 103 PROLOG, PROLOG, 64, 64, 668 668 proof, 550 length, length, see see length, length, proof sequence-like sequence-like (dag-like), (dag-like), 13, 13, 551 551 tree-like, 13, 550 proof proof by contradiction, 707 proof proof equality, equality, 723 723 proof expression, 708 116, 263, 263, 476, 476, 498, 499 proof predicate, 116, proof system associated associated to to theory, theory, 624 624 cutting plane, 604 cutting extension extension Frege, 592 Frege, 5-10, 591 Frege, 5-10, 591 bounded bounded depth, depth, 599 599 Groebner, Groebner, 604 604 Haj6s HajSs calculus, 601 Hilbert Hilbert style, style, 29, 29, 553 553 Nullstellensatz, Nullstellensatz, 603 603 polynomial polynomial calculus, calculus, 604 604 propositional, propositional, 550 550 optimal, 626 optimal, 626 quantified, quantified, 600 600 resolution, 18-26, 59-64, resolution, 18-26, 59-64, 598-599, 598-599, see see also also resolution resolution substitution Frege, substitution Frege, 591 591 proof theoretic proof theoretic ordinal, ordinal, 228, 228, see see also also ordinal ordinal of of aa theory theory proofs 679, 754 754 proofs as as programs, programs, 679, proposition, proposition, 694 694 category 694, 695 category Prop P r o p , , 694, 695 propositional propositional function, function, 695 695 propositional propositional logic, logic, see see Frege Frege system, system, proof proof system, quantified quantified propositional propositional logic system, logic and resolution resolution and and bounded bounded arithmetic, arithmetic, 619 619 and propositional rule, rule, 11,710 11, 710 propositional propositional theory, 484, 485 propositions as types, types, 724, 724, 752 752 propositions as proto-effective, 453 453 proto-effective, canonically, 453 453 canonically, provability logic, logic, 476, 476, 487, 487, 489, 489, 491,492 491, 492 provability provability predicate, predicate, 116 116 provability provably recursive, recursive, 87, 87, 173, 173, 199, 199, 202, 202, 248, 248, provably Subject S u b j e c t Index Index 353, 354, 364, 370, 498, see 353,354, 364, 370,498, see also also definable definable function function in 189, 253, in P P AA, , 189, 253, 362 362 in P RovREC(T» , 173 173 in T T ((PROvREc(T)), provably 498, 587, 587, see provably total, total, 498, see also also provably provably recursive recursive Prover-Adversary 596 Prover-Adversary game, game, 596 pullback, 719 pullback, 719 pure pure proof, proof, 701 701 pure proposition, 700, 700, 701 pure pure propositional propositional function, function, 700 700 pure type, 343 pure type, 343 pure typed typed function, function, 701 701 pure Q, R (theories (theories of of arithmetic) arithmetic),, 82-83, 82-83, 507, 507, Q, R 513, 513, 560, 560, 579 579 quantified propositional propositional logic, logic, 600 600 quantified quantifier quantifier exchange exchange property, property, 100 100 quantifier quantifier rule, rule, 32, 32, 109, 109, 710 710 Quantifier Quantifier Theorem, Theorem, 286, 286, 287 287 quantifier quantifier theorem, theorem, hyperarithmetical, hyperarithmetical, 229 229 quasi tautology, 49, quasitautology, 49, 52 52 quotient quotient type, type, 719, 719, 720 720 ramified ramified analysis, analysis, 383, 383, 385 385 ramified ramified set set theory, theory, 294 294 Ramsey's Ramsey's theorem, theorem, 619 619 random random restriction, restriction, 607 607 range 715 range type, type, 715 rank, 297, 361, 525, 642, rank, 168, 168, 178, 178, 221, 221,297, 361,525, 642, 656 656 realistic, realistic, 485 485 realizability, 66, 407-462 realizability, 66, 407-462 abstract (;1;:), 424 abstract (r_), 424 extensional extensional (re,rne,rnet) (re,rne,rnet),, 439, 439, 440 440 function function (rf), (rf), 427, 427, 428 428 function function with with truth truth (rft), (rft), 427, 427, 428 428 Lifschitz Lifschitz (rIn, (rln, rlf), rlf), 437 437 modified 432, 434 modified (mr), (mr), 429, 429, 431, 431,432, 434 function function (mrf), (turf), 434 434 numerical 443, 457 457 numerical (mrn) (turn),, 434, 434, 443, with with truth truth (mrt), (tort), 431 431 naming naming conventions, conventions, 422 422 numerical 410, 413, 413, 418, 418, 442, numerical (rn), (rn), 408, 408, 410, 442, 444, 444, 446, 446, 455 455 numerical numerical with with truth truth (rnt), (rnt), 413, 413, 442, 442, 457 457 q, 422 q, 421, 421,422 sset et theory, theory, 458 458 realization, realization, see see arithmetic arithmetic realization realization realizational realizational instance, instance, 532 532 record record type, type, 756, 756, 764 764 � -Recursion Theorem, E-Recursion Theorem, 281 281 807 807 recursion, recursion, see see bar bar recursion, recursion, limited limited recur recursion, primitive primitive recursion, transfinite re sion, recursion, transfinite recursion cursion recursion recursion operator, operator, 425 425 recursive, recursive, 172 172 ,-recursive REC(,),» ),, 172, 7-recursive ((REC(~/) 172, see see also also descendescendent dent recursive recursive recursive recursive comprehension comprehension (RCA), (RCA), 371 371 recursive recursive type, type, 760 760 recursively recursively inaccessible, inaccessible, 289 289 recursively 228, 304 recursively regular, regular, 228, 304 recursor, 232, 344, recursor, 232, 344, 345, 345, 348, 348, 349, 349, 360, 360, 362, 362, 364, , 763 364, 378, 378, 387, 387, 429, 429, 734, 734, 737, 737, 761 761,763 redex, 358 358 redex, reduced reduced sequence, sequence, 222 222 reduces, reduces, 358 358 reduces reduces in in one one step, step, 358 358 reducibility reducibility candidate, candidate, 397 397 reducible, 222, 358, reducible, 222, 358, 359 359 Reduction 256, 302 Reduction Lemma, Lemma, 235, 235, 256, 302 refinement refinement logic, logic, 704 704 �-Reflection, E-Reflection, 280 280 reflection reflection principle, principle, 217, 217, 218, 218, 280, 280, 281, 281, 490, 490, 624 624 iterated, iterated, 495 495 reflexive, reflexive, 505 505 reflexivity, reflexivity, 86 86 reflexivity reflexivity axiom, axiom, 494 494 regular regular axiom axiom system, system, 248 248 regular regular counterwitness, counterwitness, 504 504 regular 11 regular ordinal, ordinal, 2211 regular regular ordinals ordinals (Reg) (Reg),, 304 304 topological topological closure closure Reg Reg,, 304 304 regular regular term, term, 308 308 regular regular witness, witness, 504 504 relation relation symbol, symbol, 26 26 relative relative translation, translation, 501 501 relativization, 18, 216 relativization, 1118, 216 Relativized -Recursion Theorem, Relativized � E-Recursion Theorem, 284 284 relativizing relativizing formula, formula, 501 501 remainder, remainder, 89 89 �-Replacement, E-Replacement, 280 280 replacement, 109, 1110, 10, 112, replacement, 84, 84, 94, 94, 109, 112, 135, 135, 412, 412, 445, 445, 447, 447, see see also also collection collection and a n d strong strong replacement replacement resolution, 18-26, 59-64, 598-599 resolution, 18-26, 59-64, 598-599 ground, ground, 62 62 hyper-, hyper-, 22 22 input, input, 24 24 linear, linear, 24 24 negative, negative, 23 23 positive, positive, 22 22 808 808 Subject S u b j e c t Index Index positive unit, unit, 25 25 positive R-resolution, 61 61 R-resolution, semantic, 23 23 semantic, set of of support, support, 23 23 set SLD, 26, 640, 640, 661 SLD, SLDNF, 640, 640, 661 661 SLDNF, unit, 24 resolution proof, 20 20 resolution resolution refutation, refutation, 19, 19, 598 598 resolution resolution rule, rule, 19, 19, 598 598 resolution 19, 61, 61,664 resolvent, 19, 664 restricted arithmetic (Arith), 711 restricted quantifiers, 215 371,766 reverse mathematics, 371 , 766 rewrite system, 358 (RP),, 457 Richman's Principle (RP) see Q, R Robinson arithmetic, see root, 478 Rosser ordering, 496 Rosser Rosser provability, 120, 120, 495, 496 121,496 Rosser sentence, 121, 496 Rosser's Theorem, 120 run time typing, 755 satisfiable, 4, 19, 28 satisfied, 28 satisfy, 28, 61 115, 117, 117, 552, 554 schematic theory, 115, Scheme, 755 scope, 704, 734 221,222, search tree, 221, 222, 228 Second Incompleteness Theorem, 121, 121, 137, 476, 583 formalized, 506 order logic, 271 second order self-realizing, 415 self-reference, 118 self-referential, see see Diagonal Diagonal Lemma Lemma semantic semantic resolution, 23 semantic semantic tableau, tableau, 36 Semantical Main Lemma, 223 Semantical semantics, 27 semi-formal calculus, calculus, 231,234, 231 , 234, 298 semiformula, 31 semiterm, 31 sentence, sentence, 27 sentential sentential rule, rule, 317 317 separated, separated, 453 453 canonically, canonically, 453 453 A-Separation, fl. -Separation, 280 280 separation, separation, 216, 216, 321 321 Separation Separation axiom, axiom, 216 216 sequence, sequence, 713 713 sequence sequence coding, coding, 91-94 91-94 sequence-like sequence-like proof, proof, see see proof proof sequent, sequent, 10, 10, 705 705 empty, empty, 10 10 initial, initial, 11 11 upper, upper, lower, lower, 11 11 sequent 600 sequent calculus, calculus, 10, 10, 31, 31,600 LJ, LJ, 64 64 LK, LK, 32 32 PK, PK, 11 11 sequential sequential theory, theory, 560, 560, 562 562 set set existence existence axioms, axioms, 216 216 set set of of support support resolution, resolution, 23 23 set set terms, terms, 295 295 set set theory, theory, 718 718 set set type, type, 718 718 Shanin's Shanin's algorithm, algorithm, 422 422 sharply sharply bounded bounded quantifier, quantifier, 82 82 side side formulas, formulas, 12 12 signature, signature, 758 758 simple simple contradiction, contradiction, 596 596 Simula, Simula, 755 755 simulate, simulate, 624 624 simultaneous simultaneous inductive inductive definition definition (SID), (SID), 676 676 size size proof, 142, 551, proof, 142, 551, see see also also length, length, proof proof term, term, 567 567 skeleton, 42, 114, skeleton, 42, 114, 568 568 Skolem Skolem function, function, 50 50 Skolem functional, Skolem functional, 377, 377, 378, 378, 386 386 Skolemization, Skolemization, 50, 50, 346 346 slash I ), 420-421 slash (([), 420-421 Aczel, 421 Aczel, 421 SLD, SLD, SLDNF, SLDNF, see see resolution, resolution, completeness, completeness, and a n d soundness soundness slow-growing hierarchy, 152, 157, 157, 194 194 slow-growing hierarchy, 152, slow-growing operator, G, 152, 156 156 G , 152, slow-growing operator, smash function function (( ##) ),, 81, 81, 99, 99, 100 100 smash social proof, proof, 22 social Solovay function, function, 482 482 Solovay sorting, 393 393 sorting, sound, 480 480 sound, soundness soundness first-order, 30, 30, 33 33 first-order, HAW , 432 432 HA~, HA HA'I ,, 438 438 HA* , 414 414 HA*, strong, 414, 414, 417, 417, 420 420 strong, weak, 414, 414, 417 417 weak, implicational, 6, 6, 13 13 implicational, intuitionistic many-sorted, many-sorted, 448, 448, 449 449 intuitionistic modal logic, logic, 478 478 modal Subject S u b j e c t Index Index propositional, 6, 6, 13 13 propositional, resolution, 19 19 resolution, SLDNF, 669 669 SLDNF, space representable, representable, 161 161 space sparse set, set, 626 626 sparse species, 392 species, Spector(-Howard) interpretation, 367 Spector(-Howard) Spector-Gandy Theorem, Theorem, 286 Spector-Gandy spectrum II~-spectrum, 228 rrt -spectrum, 228 20-spectrum, 246 LY -spectrum, 246 H~-spectrum, 247 rrg speed up, 497 speed square root, 90 square 215, 295 stage in constructible hierarchy, 215, stg, 295 stg, stage of an inductive inductive definition, 269, 269, 281 stage standard interpretation, 295 starting function, 242 static typing, 755 stratification, 728 stratified program rules, 660 411,447 strict, 411, 447 strong fragment, 81 strong inference, inference, 111, 1 , 32 strong interpretation, 502 109, 110 strong replacement, 96, 109, strongly critical, 214, 308 SC,, 214 SC strongly strongly critical critical components, components, SC, SC, 305 305 strongly normalizable, 358 strongly normalizing, 358 strongly positive, 388 structural rule, 11,301,317, structural 11, 301, 317, 708, 710 structure, 27 adequate, 650 equational, 648 four-valued, 644 free term, term, 670 Herbrand, Herbrand, 645 lower three-valued, 644 two-valued, 644 644 upper upper three-valued, three-valued, 644 structured structured tree-ordinal, tree-ordinal, see see tree-ordinal subformula, subformula, 704 subformula subformula property, property, 13, 13, 111,573 111, 573 subobject subobject classifier, classifier, 719 substitution, substitution, 5, 5, 27, 27, 59, 59, 116, 116, 341, 341, 567, 567, 648, 648, 728, 734 closed closed under, under, 33 empty, empty, 648 variable variable renaming, renaming, 59 809 809 substitution substitution Frege Frege system, system, 591 591 substitution substitution operator, operator, 232 232 substitution substitution rule, rule, 591 591 subsume, subsume, 22 22 subsumption, subsumption, 22 22 subtheory, subtheory, 501 501 subtraction, subtraction, 89, 89, 349 349 subtree subtree ordering, ordering, 154, 154, 193 193 subtype, subtype, 693 693 succedent, succedent, 10 10 successor, successor, 96, 96, 103, 103, 220, 220, 232, 232, 344, 344, 360, 360, 409, 409, 423, 423, 429, 429, 516 516 successor 304 successor ordinal, ordinal, 211, 211,304 superarithmetic superarithmetic theory, theory, 504 504 superexponentiation, superexponentiation, 37, 37, 81, 81, 138, 138, 139 139 support, support, set set of, of, 23 23 supremum, supremum, 211 211 surjection, surjection, 445 445 Suslin Suslin quantifier, quantifier, 384 384 switching switching lemma, lemma, 618 618 symmetric symmetric sum, sum, 213 213 Syntactical Syntactical Main Main Lemma, Lemma, 223 223 system system F, F, 394 394 T -predicate, Kleene's, T-predicate, Kleene's, 409 409 tableaux tableaux proof, proof, 704 704 tactic, tactic, 709 709 tactic tree tactic tree proof, proof, 766 766 tactical, tactical, 709 709 tail, 125, 714 tail, 125, 714 tail tail model, model, 480, 480, 490 490 Tait 165, 220, 220, 232 Tait calculus, calculus, 16-18, 16-18, 165, 232 Takeuti's Takeuti's conjecture, conjecture, 398 398 Tarski's Tarski's conditions, conditions, 560 560 tautological tautological implication, implication, 44 tautology, tautology, 4, 4, 505 505 Tautology Lemma, 233 Tautology Lemma, 233 tautology rule, rule, 317 317 tautology term, 26, 31, 220, 642, 642, 703 703 term, 26, 31,220, A-calculus, 68 68 A-calculus, term model, 357, 357, 358 358 term model, terminal, 739 739 terminal, tertium non non datur, datur, see see excluded excluded middle, middle, law law tertium of of theory, 29, 29, 501 501 theory, theory delimiters, delimiters, 722 722 theory theory of of implication, implication, 600 600 theory thin, 708 708 thin, thread, 221 221 thread, three-valued closure closure ordinal, ordinal, 653 653 three-valued TOL model, model, 530 530 TOL tolerance, 503, 503, 528-530 528-530 tolerance, topos, 421,441,451,452, 421, 441, 451, 452, 457, 457, 461,719 461, 719 topos, 810 810 topos theory, theory, 719 719 topos transfer, 525 525 transfer, transfinite induction, induction, ssee ee induction induction transfinite transfinite recursion, recursion, 211,281 211, 281 transfinite transitive, 210 210 transitive, transitivity, 86 86 transitivity, translation, 501 501 translation, tree, 221 221 tree, tree of of knowledge, knowledge, 722 722 tree tree relation, relation, 222 222 tree proof tree-like proof, proof, see tree-like see proof tree-ordinal, 154, 154, 191,386 191, 386 tree-ordinal, finite type type theory theory (OR"{), ( OR'j'), 386, 386, 387 387 finite structured, 154, 198 198 structured, 154, trichotomy, 86 86 trichotomy, truth, 28, 501 501 truth, 28, truth assignment, assignment, 3, 3, 702 702 truth truth complexity, complexity, 219 truth 219 t e , 224, tc, 224, 297 297 truth definition, definition, 137, truth 137, 139, 139, 142, 142, 220 220 truth provability logic, logic, 487 truth provability 487 truth value, value, 694 truth 694 contradictory, contradictory, 643 643 false, 643 643 false, true, 643 true, 643 undefined, 643 undefined, 643 type, 68, 68, 342, 342, 692, 692, 703 703 type, of aa term, term, 343, 343, 429 429 of type assumption, type assumption, 703 703 type level, level, 343, 343, 452 type 452 type type structure, structure, 343 343 type type system, system, 748 748 type type theory, theory, 726, 726, 767 767 typed typed A-calculus, A-calculus, 755 755 typed typed propositional propositional formula, formula, 703 703 typing typing context, context, 703 703 typing judgment, 698, 698, 735 typing judgment, 735 unbounded unbounded quantifier, quantifier, 82 82 unbounded unbounded set, set, 211 211 uncountable uncountable cardinal, cardinal, 304 304 unification, 55, 59, 59, 567, unification, 55, 567, 648 648 unification unification algorithm, algorithm, 60-61 60-61 Unification Unification Theorem, Theorem, 60 60 unifier, unifier, 59, 59, 567, 567, 648 648 most most general, general, 60, 60, 567, 567, 648 648 uniform, uniform, 452 452 canonically, canonically, 452 452 Uniform Uniform Continuity Continuity Modulus Modulus of, of, 433 433 Uniformity Uniformity Principle Principle (UP), (UP), 442, 442, 453 453 Uniformity Uniformity Rule Rule (UR), (UR), 443 443 union union axiom, axiom, 216, 216, 279 279 Subject Index Index Subject union type, 756 unique unique factorization, factorization, 90 90 unit unit clause, 24 unit unit resolution, resolution, 24 24 unit unit type, type, 700, 700, 735 735 universal closure, universal closure, 32 32 universe, universe, 27, 27, 394, 394, 398-400, 398-400, 744 744 universe rules, 744 unpairing, unpairing, 429 429 unrestricted unrestricted quantifiers, quantifiers, 215 215 unsecured 277, 287, unsecured sequences, sequences, 230, 230, 277, 287, 290 290 untyped A-calculus, A-calculus, 755, 755, 759 759 untyped unwinding, 338 unwinding, 338 upward upward persistency, persistency, 301 301 valid, 28, 28, 32, 32, 448, 448, 478, 478, 535, 535, 647, 702 valid, 647, 702 valid element, 701 valid formula, 4 valid inference, 115 variable, 3, 3, 26, 26, 702 variable, 702 free and and bound, 734 free bound, 31,703, 31, 703, 704, 704, 734 variant, 648 term, 42 term, 42 Veblen Veblen function, function, 214 214 Veblen hierarchy, hierarchy, 383 Veblen 383 Veblen 214 Veblen normal normal form, form, 214 Veltman frame, Veltman frame, 515 515 very very dependent, dependent, 765 765 very function, 764 very dependent dependent function, 764 very type, 764 very dependent dependent type, 764 very very weak weak fragment, fragment, 81 81 Visser frame, frame, 530 530 Visser Weak Continuity Continuity (WC), (WC), 434 434 Weak Weak Weak Extended Extended Church's Church's Thesis Thesis (WECT), (WECT), 440 440 weak weak fragment, fragment, 81 81 weak weak inference, inference, 11 11 Weak Weak Konig's KSnig's Lemma Lemma (WKL), (WKL), 371, 371,374 374 Weakening Weakening Lemma, Lemma, 167 167 weakening weakening rule, rule, 11, 11, 73 73 weakly -consistent, 119 weakly w w-consistent, 119 weakly weakly compact compact cardinal, cardinal, 331 331 weakly weakly inaccessible, inaccessible, 304 304 weakly weakly interpretable, interpretable, 503, 503, 528 528 weakly weakly introduced, introduced, 43 43 weakly weakly positive, positive, 388 388 well 222 well founded, founded, 221, 221,222 Wf(-<) 286 Wf (-<),, 272, 272,286 well -<) , 274, well ordered, ordered, WO( W0(-,:), 274, 286 286 well-specified, 485 485 well-specified, witness, witness, 52 52 witness witness comparison, comparison, 496 496 Subject Index witness witness predicate, predicate, 123, 123, 127 127 Witnessing 131, 255 Witnessing Lemma, Lemma, 123, 123, 128, 128, 131,255 witnessing witnessing substitution, substitution, 52 52 world, world, 478 478 Zermelo-Fraenkel (ZF) set theory, theory, 589 589 Zermelo-Fraenkel (ZF) set 811 811 This Page Intentionally Left Blank

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