Logic Colloquium 1982: Proceedings

LOGIC COLLOQUIUM '82 Proceedings of the Colloquium held in Florence 23-28 August, 1982

Edited by

G. LOLL1 Dipartimento di lnformatica Univer.sitadi Torino Torino Italy

G. LONG0 Dipartimenio di lnformatica Universita di Pisa

Pisn Itnly

and

A. MARCJA Dipartimento di Matematica Lihern Universita degli Stud di Trento Trento Italy

1984

NORTH-HOLLAN'D AMSTERDAM 0 NEW YORK. OXFORD

QELSEVIER SCIENCE PUBLISHERS B . V . , 1984 All rights reserved. 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 the prior permission of the copyright owner.

ISBN: 0 444 86876 3

Published by: Elsevier Science Publishers B . V P.O. Box 1991 1000 BZ Amsterdam The Netherlands Sole distributors for the U . S .A . and Canada. Elsevier Science Publishing Company, Inq; 52 Vanderbilt Avenue New York, N . Y . 10017 U.S.A.

Library of Congress Cataloging in Publication Data

Logic Colloquium (1982 : Florence, Italy) Logic colloquium '82. (Studies in logic and the foundations of mathematics ; v. 112)

Bibliography: p. 1. Model theory--Congresses. matics)--Congresses. 3. Lambda I. Lolli, Gabriele, 1942111. Marcja, A. (Annalisa) IV.

.

@9*7*L64 1982 511' .8 ISBN 0-444-86876-3 ( U . S . )

2. Categories (Mathecalculus--Congresses. 11. Longo, Giuseppe. Title. V. Series.

84-1630

PRINTED IN T H E NETHERLANDS

V

PREFACE

The Logic Colloquium ‘82 has been held in Florence (Italy), from 23 to 28 August, 1982. The date had been so chosen in order to allow a combined participation to the Warsaw ICM. Notwithstanding the postponement of the latter, more than 200 logicians were able to attend the Logic Colloquium and the Summer Meeting of the ASL. The organization and the program of the conference were undertaken by a Committee consisting of E. Casari (Florence), J.E. Fenstad (Oslo), G . Lou (Turin), G . Longo (Pisa), A. Marcja (Trento), and D. van Dalen (Utrecht). The organization has been made possible by the financial support of many institutions: a substantial help came through the Comitato per la Matematica of the Italian Consiglio Nazionale delle Ricerche; further contributions were made accessible from the University of Florence, the city of Florence and the Regione Toscana, and the Ente Provinciale per il Turismo di Firenze. The conference was sponsored by the International Union of History and Philosophy of Science, Division of Logic, Methodology and Philosophy of Science. The main topics chosen for the Colloquium were: Model Theory, with particular emphasis on models of Peano arithmetic (a small if not exactly pertinent contribution to the celebrations for the 50th anniversary of Peano’s death); Categorical Logic; Lambda-calculus. Much care and effort have been put by the invited speakers into investigating and further stretching the overlapping and crossbreeding of these areas. We regret that the present volume does not faithfully reflect the balance of the three topics at the Logic Colloquium, since some of the speakers in categorical logic and in lambda-calculus were unable to submit the written text of their lectures (while the editors were guilty against the readers of too much waiting). We hope that their contributions will however be made accessible in other ways to all interested logicians. We are grateful to Elsevier Science Publishers B.V. for inserting this volume in their glorious series in Logic and the Foundations of Mathematics. November 1983

G . Lolli (Torino) G . Longo (Pisa) A. Marja (Trento)

LOGIC COLLOQUIUM '82 G. Lolli, G.Long0 and A. Marqa (editors) @Elsevier Science Publishers B. V. (North-Holland), 1984

1

LECTURES ON NONSTANDARD MODELS OF ARITHMETIC

Commemorating Guiseppe Peano C. Smoryikki Department of Mathematics The Ohio State University Columbus, OH 43210

USA

Contents § 0.

1. 2. 3.

4. 5. 6. 7. 8. 9. 10.

Preface The Beginnings (The 1950s and Earlier) The R6le of the Infinite Integer (The 1960s. I) Extensions of Models (The 196Os, 11) Saturation Properties (The 1970s. I) Recursively Saturated Models of Rich Theories (A Digression) The Arithmetised Completeness Theorem (The 197Os, 11) Powerful Arithmetisations (The 1970s, 111) Diversity (The 1980s) A Dead End Summer Reading List

0. PREFACE Guiseppe Peano died in 1932; nonstandard models of arithmetic were born the following year. Thus, Peano never studied models of arithmetic and it may seem odd to commemorate him with a series of lectures on such a topic. However, one of the more active areas in the study of what has come to be called Peano arithmetic, o r P A , is the study of its nonstandard models and this subject seems thus appropriate. Besides, with the subject beginning just after his death, we might say it picked up where he left off. It is my intention subject. so.

to

give a partly historical account of the development of the

Since these are commemorative lectures, it seems most appropriate to do

Moreover, there are now two collections of papers edited by Kenneth McAloon

e t aZ. with expositions of recent work, while there is nothing about the pioneering efforts of the 1950s and 1960s.

Progress was much greater in this period

than most people realise.

I will depart from the historical line in two important respects.

First, much

early work concerned only "strong" nonstandard models of arithmetic rather than arbitrary models of P A .

A nonstandard model was called "strong" if it elemen-

tarily extended the standard model No= (w;+,.,',O)

and was of particular interest

only because of a lack of familiarity with the strength of P A :

One needed to

know, say, that all integers (standard and nonstandard) had certain coding prop-

c. S M O R ~ S K I erties that ordinary integers had and, unaware that such properties were provable outright in P A , one simply assumed them to hold by assuming the model to be strong. A second departure from the historical line must occur when several independent trends develop simultaneously. This first happened around 1960, when nonstandard model theory reached its first level of maturity. Because of time and space limitations, I shall have to assume the reader familiar with a great deal of what can be done within PA.

This means that, with respect to,

say, the hypothesis of strength mentioned above, I can discuss the full results available without the irrelevant hypothesis-- even though the full result came a decade later.

It also means that some important developments, which preceded

formalisation, will look more trivial than they should. My notation will be fairly standard-- at least by my standards-- and I hope everything will be fairly self-explanatory. 1.

THE BEGINNINGS (THE 1950s AND EARLIER)

It was in a paper published in 1929 that Thoralf Skolem first suggested the possibility that the standard numbers might not be alone, that there might be poor In SkoZem 1 9 3 3 , he was able to

imitations satisfying the same first-order laws.

psove this relative to any finite set of axioms. This, of course, could have been established by appeal to Gb'del's Incompleteness and Completeness Theorems; or, it could have been done by the now-common appeal to the Compactness Theorem.

The

former approach would seem to have little bearing on the larger problem of obtaining nonstandard models satisfying a l l the true sentences of arithmetic; and neither of these abstract approaches would very likely have been to Skolem's taste. In any event, in the following year in SkoZem 1934 he published a proof of the existence of strong nonstandard models of arithmetic, i.e. structures 1'4 =

(M;+,-,',O)

not isomorphic to No but, nonetheless, making true the same sentences

as N o . Skolem's original papers were in German.

He republished the proof in English in

SkoZem 1955 in the proceedings of a conference-- the same proceedings in which bos published his theorem on ultraproducts. Although this was merely a repetition

of his earlier proof, it is this later paper to which most people refer.

Perhaps

this is because this paper is in English, a much easier language than German for many of us; perhaps it is simply that this paper, appearing in a slim little volume that has been reprinted, is in more private libraries and hence more accessible; or perhaps it is the juxtaposition with bos' paper that strikes one's fancy-- for, Skolem's construction of a nonstandard model is something of an ultrapower construction.

Sketch of Skolem's construction:

Given No

=

(u;+,*,',O),

let F consist of

3

Lectures on Nonstandard Models of Arithmetic all functions F:w

+

w

definable in No.

... of F it is

From an enumeration FO,F1,

not difficult to construct, by diagonalisation, a function G:w all F ,F. i J

E

+

w

such that, for

F,

< F.G(x) eventually F G(x) i J or eventually F.G(x) > F . G ( r ) .

or

eventually F.G(x) = F.G(x) J

J

Using G, one can define an equivalence relation :by F. 1

5

F. iff eventually F.G(x) = F.G(r) J

J

and shew the structure F/: to be an elementary extension of N 0 '

It should be noted that, although Skolem's construction resembles the ultrapower construction so much that one feels like calling it such, an important element is missing.

Skolem's use of the diagonalising G in place of the now-usual ultrafil-

ter relies heavily on the countability of the arithmetic language; his method does not yield the existence of nonstandard models when, say, a continuum of predicates naming all sets of natural numbers is added to the language. For this latter, one must use one of the standard abstract existence theorems of logic. It is also worth mentioning that Skolem's goal in constructing nonstandard models was philosophical: He aimed to shew that first-order logic could not characterise the number series; he did not care to start a new subject.

Until the 1960s, this

was generally the case-- nonstandard models of arithmetic were either objects of philosophical interest or tools, not objects of mathematical interest in their own right.

The major counterexample to this was an observation made by Leon Hen-

kin in his paper Henkin 1950 on the Completeness Theorem for Type Theory. announced the order type of a nonstandard model of arithmetic to be w where 8 is a dense linear order.

( Exercise: If, in particular,

+

He

(w*+u)8,

0 is countable,

it must be the order type of the rationals. A related, but trickier, exercise due (I believe) to Klaus Potthoff is this: not have the order type of the reals.

Shew, in the uncountable case, 0 can

)

The important, i.e. useful, fact about the order type of a model M = (M;+,*,*,O) of PA is that M begins with w and then follows this with the nonstandard or infi-

nite integers. The significance of the infinite size of the nonstandard integers is manifold: They code various paths to infinity and, particularly in Nonstandard Analysis, simulste limit processes; they code infinite sets; and they code nonprincipal ultrafilters in the algebra of definable sets of natural numbers. all this, their potential usefulness is clear.

With

In the early 1950s, however, not

all of this was known: The first application of infinite integers depended merely on their size. In RyZZ-Nardzewski 1952Czesaaw Ryll-Nardzewski essentially proved the following theorem: 1.1.

Theorem.

PA is not finitely axiomatisable.

c. S M O R ~ S K I

4

Ryll-Nardzewski actually proved something a bit stronger: If T is any finitely axiomatised theory in a language extending that of PA and T is true in some expansion of No to accommodate the extended language, then T does not prove some instance of induction in this language. This result is still rather weak.

As I

mentioned in the Preface, many early results were proven in weak form because it was necessary to assume certain arithmetic truths held in a nonstandard model and, through lack of familiarity with the power of PA, the best guarantee of this assumption was that the nonstandard model was a strong one.

Dropping this assump-

tion, the final result along these lines is the following: 1.2.

Theorem.

PA is essentially unbounded, i.e. no consistent extension T of PA

can be given by axioms of a fixed bounded complexity. This theorem was first published by Michael Rabin in Rubin 1962 and is thus often referred to as Rabin's Theorem, although he explicitly announced it to have been known to others-- most probably including Solomon Feferman, Georg Kreisel, Dana Scott, Stanley Tennenbaum, and Hao Wang.

A s the reader might guess from the long

list of names, the road from Theorem 1.1 to Theorem 1.2 was a long one.

Indeed,

it cap now be recognised to have been one of the two major themes in nonstandard model theory in the 1950s, if one may refer to the then-primitive development as "nonstandard model theory".

Today, we have many proofs of Theorems 1.1 and 1.2,

among which is a modification of Ryll-Nardzewski's proof of Theorem 1.1. Let M be an arbitrary model of T ? P A and let MI be an elem-

Proof of 1.2:

0

entary extension of M which contains at least one nonstandard integer a which is 0

infinite relative to M ~ i.e. , a > b for every element b

E

J M ~ J .Suppose n is a

finite positive integer and U is a set of C -sentences true in Mo. struct a model Mz

b

U but such that M2

y

The construction is very simple: Let IN

We will con-

PA. 2

I

be the closure of IN

I

{ a } under all M 2 is automatic-

0 "

total functions definable in Ml by parameter-free Zn-formulae. ally a Cn-elementary substructure of M I , whence it satisfies U.

The reason that M 2 is not a model of PA is also simple: There is a Cn+I-definable function F which can schematically be proven in PA to eventually dominate each C definable function, i.e. for each parameter-free C -formula natural number

I such

@VoVl

-

there is a

that

where Ji defines F. ( F is obtained by a simple diagonalisation on the uniform C enumeration of Zn-definable partial functions. ) In Mz, F is not total; in fact,

-

Fa is not defined.

To establish this last claim, suppose Fa exists, i.e. M 2 k 3 V 1V 2$'av102, where I )' is Il

and 3v2$'defines F .

Let b be the image of a under F in M2 and let c witness

5

Lectures on Nonstandard Models of Arithmetic this fact, i.e. M2 Fa = b in MI.

I=

$ 'abc.

function G such that b

=

Since $ ' E

M2

k $'&

E

calling that a > d for every d E lMo 1 , function H.

nn,

1M21, there is some d G(a,d) in Ml. Letting Hvo

Now, since b

=> Ml

IW 0 I

E

k I)'&?

, whence

and some C -definable

supv
,V ) , and re0 1 we have b < Ha in MI for some 1 -definable =

G(V

I- 0

But this readily yields a contradiction: For, there is a finite x

such that P.4l-V

vo

> HV ) and a is quite infinite, whence b

> :(FVO

0

=

Fa > Ha in

&ED I wish to emphasise here the r61e played by the size of a relative to Mo: The integer a is so large that a Zn+l-function eventualzy dominating all C -functions already dominates them at a.

Thus, the mere closure under Z -functions, which

yields a C -elementary substructure, does not yield closure under the provably total majorising function.

Through such an application, the mere size of an in-

finite integer presents itself as a useful tool, one which could still be exploited unaided in the 1950s in the work of Feferman, Scott, and Tennenbaum, and even as recently as 1975 in a paper of Alex Wilkie. As I remarked just before proving Theorem 1.2, the passage from Theorem 1.1 to Theorem 1.2 was one of two main themes in the study of nonstandard models.

The

other, which nowadays one can see to partially merge with this one in the work of Feferman, Scott, and Tennenbaum, is a couple of decades older and more technical. This theme is the complexity of models and goes back to the second volume of Hilbert and Bernays, specifically to a result Paul Bernays proved and included in the volume: 1.3.

Theorem.

Let

0 be

any sentence of a given language.

If $I is consistent,

i.e. if $I has a model, then $I has an arithmetically definable model, i.e. a model whose domain is an arithmetically definable set of natural numbers and whose primitive relations are also arithmetically definable. Actually, the entire satisfaction relation for the model can be taken to be arithmetically definable-- the satisfaction relation for the language of @ is simply a new predicate describable by a new axiom @ and one can apply the result to $ A $ . The fact that the description could be given by a single axiom @ was not immediately recognised.

This was the decade* in which Stephen Kleene (KZeene 1952.4)

(and later William Craig and Robert Vaught) proved one could finitely axiomatise r.e. theories by the addition of new predicates.

Thus, when the early researchers

became interested in extending Bernays' result to r.e. theories, the direct reduction to the finite case by finitising the theory in telrms of a satisfaction relation did not occur to them and they proved the result anew. Bernays had proved Theorem 1.3 by arithmetically analysing the proof o f Godel's

*In fact, Robinson's finitely axiomatised Q was not published until 1953.

c. S M O R Y ~ K I

6

Completeness Theorem for the predicate calculus.

The new result was established

(by Gisbert Hasenjsger, Stephen Kleene, and Hao Wang) by doing the same to modifications of Henkin's proof. We will discuss this later in 96, where we will see that this final completion of Bernays' Theorem is a useful tool. This realisation came, however, around 1970; in the 1950s the question was simply one of the complexity of models. The upper bound on the complexity of a model of an r.e. theory is readily established by inspection: If T is a consistent r.e. theory, then the arithmetic encoding of the proof of the Completeness Theorem shews T to have a model on an initial segment of the set of natural numbers and the definition of satisfaction for this model (and consequently each of its primitive relations) to be A2.

[ Recursion theoretically, this means the model is recursive in

-

0'.

Later

improvements were made by Joseph Shoenfield and, ultimately, by Carl Jockusch and Robert Soare; cf. ShoenfieZd 1960 and Jockusch and Soare 2972A8B. ) For lower bounds on the complexities of the interpretations, one can again ask two questions: How complicated must the primitive relations be; and, how complicated must the satisfaction relation be? Well, one r.e. theory's satisfaction relation is another finitely axiomatised theory's primitive relation and, globally, the questions conflate.

For specific theories (like P A ) or fixed languages (e.g.

that of one binary relation symbol), however, the equivalence is nontrivial and one usually asks for large laver bounds on the complexities of the primitive relations. In the 1950s, Kreisel and Andrzej Mostowski alternated in a series of papers, the

main outcomes of which were the existence of r.e. and finitely axiomatised theories with no recursive models.

Mostowski even shewed that, whereas the set

of sentences true in all models is r.e., that of all sentences true in all recur-

sive models is not even arithmetically definable. In 1960, Vaught offered yet another generalisation. In the late 1950s, the problem of the difficulty of constructing nonstandard models of PA, i.e. the complexity problem for models of P A , was considered by Feferman, Scott, and Tennenbaum, who published a short series of abstracts. Among other things, they reconsidered Skolem's ultrapower-like construction and shewed, e.g., that no homomorphic image of the Xn-functions could model PA.

This was

largely an application of size, 6 Za Ryll-Nardzewski (or, at least, my exposition of the latte&

proof, above).

However, their work led Tennenbaum to an important

discovery: Infinite integers not only have infinite size, but they also code infinite sets.

Looking at the encoding yields:

1.4. Theorem (Tennenbaum's Theorem). Let M = (w;+,X,',O) be a nonstandard model of PA and let 4m be any formula of the language of arithmetic (with parameters

Lectures on Nonstandard Models of Arithmetic from M allowed).

X

=

Then In fact, X is recursive in each of

is recursive in i , x .

( Notes: i. Note that

M k G }

{ x E w :

+,x.

denotes the successor function of the model; 0 the zero element. is recursive in

isince X I

=

x

iI

, where 1 is the unit of M.

Without

loss of generality, we can assume 0 to coincide with 0 and use the simpler notation. and x,was not

In the first published proof in Ehrenfeuckt and KreiseZ

1966 it is noted that X is recursive in

credits the observation on

+.

to Feferman.

X

Kenneth McAloon, in M c A l o o n 1 9 8 2 ,

)

First, we recall that an integer x can serve as a code for a

Proof of 1 . 4 :

finite set of integers.

There are many ways in which this can be done, my least For an integer X, let Dz

favourite being one expecially suited for this proof. consist of all 9 ' s such that the (y +l)-th where p o , p l ,

+

The extra flourish, that X is recursive in each of

ii.

stated by Tennenbaum.

prime divides x :

... is the sequence 2 , 3 , ... of primes.

y

E

iff p (z,

Dx

Y

A simple induction on v 0 proves the Aussonderungsaxiom, PA

1 vVo3VIv

V2(

V 2 E

Du 1

E V2

VoA@V2

),

for any formula @ v 2 . Let a be an infinite integer in M, let $ be given, and let E w: M 1. Applying Aussonderung to @ and a, we obtain b E lMl such

6

X = { x

that X = { x

E

w: M

13: E

iff

This shews X to be r.e. in shews o

-

X to be r.e. in

Recursivity in

X

3 1.

But then

M I G I iff E 3 c E w ( ~ k rX * c = E ) - 3 , E w ( M k C i... ic b ) ( p , times).

iff

EX

f.

Since w - X corresponds to

i , whence

X is recursive in

is a similar affair:

the same proof

l@,

f.

From the equivalence between z E X and the

representability of b as a p -fold sum of c's, we get the equivalence with the X

representability of Zb as a p -fold product: x X b some d , where e = 2

E

X iff d

.

x

...

x

d

e , for

&ED

Tennenbaum's Theorem has some immediate corollaries:

1.5. Then

Corollary. +,X

Let M = (w;i,x,',O)

be a strong nonstandard model of arithmetic.

are not arithmetical.

This can, of course, be hierarchically refined. 1.6.

Corollary.

If M = ( w ; f , ~ , ~ , O )

The main such refinement is:

is a nonstandard model of P A , then +,x are

not recursive.

Proof:

It suffices to choose @ so that

This is a simple recursion theoretic trick:

x

E

o: M k

61

is not recursive.

If $,J, define two provably disjoint

effectively inseparable r.e. sets, say A,B, respectively, then A 5 X =

c. S M O R ~ S K I

8 { J:

E

61

o: M

and B n X = @, whence X is not recursive.

I have one more important corollary to cite.

BED

Before giving this, however, I must

remark that the proof of Corollary 1.6 did not depend on the full power of Tennenbaum's Theorem. fic formula

It really only depended on establishing Theorem 1.4 for one speci-

@.For this, the proof needs:

i. some minimal arithmetic-- say,

Robinson's Q; ii. the totalities of the exponential and prime enumerating functions; and iii. two instances of induction to establish Aussonderung for

0,I$.

If we

conjoin these to obtain a single axiom 9, Corollary 1.6 can be rewritten as

1.6'.

Corollary. If M

= (w;+,x,',O)

b

$ is nonstandard, then

+,X

are not recur-

sive. Conjoining with ii, any axiom forcing the model to be nonstandard yields a sentence having no recursive models.

In fact, with ii, we can easily prove Mostowski's

result:

1.7.

Corollary. The set of sentences valid in all recursive models is not

arithmetical.

Proof:

The only recursive model of $ is the standard one.

Hence, if fJ is any

arithmetic sentence, $ 4 0 is valid in all recursive models iff 0 is true in the standard one.

Since arithmetical truth is not arithmetical, the Corollary follows. QED

The sentence ii, consisting mainly of two odd instances of induction conjoined with a few natural axioms is a bit odd and one can ask for an aesthetically pleasing finite theory to which Corollary 1.6 can be applied.

It is not hard to see that

(over &) Z1 -induction suffices to establish the totalities of the exponential and prime enumerating functions and C

-

1

and Il -Aussonderung. 1

One can also ask for the weakest theory which can be used in place of $.

Z -1nduc1 tion is rather strong and one can use weaker theories, as shewn in McAZoon 1982.

Tennenbaum's Theorem can also be generalised in another direction, as we shall see in 55, below. I might also add that the question of the complexity of a model has of late resumed an interest; cf., e.g., Marker 1982. A

2.

THE ROLE OF THE INFINITE INTEGER (THE 1960s I)

The year 1960 was something of a watershed in the development of nonstandard model theory. Prior to 1960 most papers on nonstandard models of arithmetic were aimed at metamathematical targets, the nonstandard models themselves not being of central interest.

Then, suddenly, this changed: In the early part of the decade any num-

ber of papers appeared in which the models themselves were studied. There were papers on the additive group of integers (negative ones added) of a nonstandard

9

Lectures on Nonstandard Models of Arithmetic model, on the nonstandard model as a semi-ring, on the algebra of definable sets in a nonstandard model, and on the forms of embeddability on nonstandard models into one another. The variety is such that a straight historical discussion of the period is pointless.

Instead, I shall outline a few central themes.

First, let me hark back to the r8les played by the infinite integers. While mere size considerations remained important (cf. for example Ehrenfeucht and KreiseZ 1 9 6 6 ) , greater sophistication arose.

The following theorem relating the infinite

to the finite integers is a cornerstone of Abraham Robinson's Nonstandard Analysis; in the proper language, it is the lemma upon which the equivalence of the standard and nonstandard definitions of the limits of sequences depends.

2.1.

Theorem. Let M be a strong nonstandard model of arithmetic, i.e. a proper

elementary extension of No. i. ii.

Proof: <=.

Let

3 infinite a( M infinite a( M

$V

have only v free.

6)

b vv0 3vl

iff N~ iff N~

b Q;; )

+ xuo vvl

> v o @vl

> vo $vl.

Because i and ii are duals, I shall only prove i. Observe: No

b

V V o 3v1 > v 0 @vl =>

M

k

Vv0 3Vl >

Vo

@v1. Picking u

infinite, the right-hand-side of this implication produces an infinite v

=>.

Let a be infinite such that M

MI=

a > ;*&

->

M1=3 v

Since cc was arbitrary, we conclude N 0

>

b 6.Let x E o and 3: @vl => No b 3v1

0

1'

observe

>

3: @ v l .

V V 0 3v1 > V 0 @ v l *

QED

This theorem is slightly disappointing in that it assumes M to eZementariZy extend

No.

There are hierarchical variants which the reader can work out for himself and

also something called Overspill to handle the non-elementary case.

I shall shortly

discuss the latter; but, before doing s o , I would like to remark that the present theorem, however much it assumes, is rather useful.

It is, for example, the key

to one of the corollaries of the main theorem of Jeff Paris and Leo Harrington: Paris and Harrington proved the independence of various true 112 sentences @ = V V 0 3 v 1 ~ 0 V 1 , with ii, E

the least y such that No tions.

A,.

The (recursive) function-- call it G-- taking x to

I= wT

eventually dominates all provably recursive func-

These two facts are equivalent, the latter trivially implying the former.

One can, with some effort, prove directly the majorisation result and thereby conclude the underivability of $; conversely, and more easily, one can appeal to Theorem 2.1.11 and the model theoretic construction underlying the independence proof:

Most expositions of the Paris-Harrington Theorem demonstrate the underiv-

ability of $ by starting with an elementary extension M of No in which @ is true. For each infinite integer a

E

/MI and its image b = Ga, they shew the existence of

an initial segment I 5 IWI such that i. a the inherited functions. Since

E

I and b t! I,and ii. I I= PA when given

I is a model of

P A , it is closed under all provably

recursive functions. For such a function F, from the facts that Fa

E

I,Ga t! I,

c. S M O R Y ~ K I

10

and I is an initial segment of M, it follows that Fa < Ga. Theorem 2.l.ii yields: N o

b

Bvo vvl > v o (FVl >

Since a was arbitrary,

GOl).

I have omitted a few things here, such as an absoluteness of the provably recursive functions, and I have used much stronger conditions on M than are necessary for the above argument; but I have illustrated the use of the Theorem. The reader familiar with Ramsey’s Theorem may wish to use the other half of Theorem 2.1 to

( Similarly, the reader familiar

reduce the finite form to the infinite form.

with the work of Paris and Harrington may wish

to

reduce their variant to the

infinite Ramsey Theorem. The lazy reader may wish to wait for 56, below.

)

In the non-elementary case there is, as I said, a replacement for Theorem 2.1: 2.2.

(Weak Overspill).

v

5

0 1

Let

$v0V2

( M I= & )

iff

(Strong Overspill).

Let

x ii. $71

Let M

Theorem (Overspill Principle). i.

E

w

defines a function F : I M ( Vz

w

E

I= PA

have only

be nonstandard and b

Vo,VI

3 infinite a ( M @v0VIv2

+

E

(MI.

free. Then

I=

vv0 <

ii $ v o b ) .

have only vo,V1,v2 free and suppose

IM(. Then 3 infinite a

( Fx is infinite ) iff

v

c < a (Fc is infinite ).

To prove this we first need a small lemma: Lemma. Let M

2.3.

I= PA

be nonstandard. Then: w is not parametrically definable

in M.

Proof: Let b

E

(MI and @ v ~ with u ~ only vO,vl free be given and suppose w

lMl: M I= @I 1 . Obviously, M I= $Ob and M k V v o ( $vob -+ $(vo’,b) ), whence M Vv 4V 5 and w = (MI, contrary to the nonstandard hypothesis. a

=

E

0

&ED

0

( The reader who has not yet solved the exercise in 51 on the order type w + (w*

+ w)8,

namely that of shewing 8 not to have the order type of the reals will

now get a hint:

Let a

E

Proof of OverspiZZ:

Assume

Vx

E

Vx

w M

infinite a ii.

E

I=

w

(M

@ccb

IMI be nonstandard and consider

m:

z

E

w

1. )

i.

The right-to-left implication is trivial.

1.

Let

JrvoV1

be the formula V v 2 <

Vo~V2Vl.

Clearly

I@. Since w is not parametrically definable in M, there must be some

E

IM( such that M

I=

Vv0 <

a

Again the one direction is trivial.

To establish the other, let F

denote the function defined and apply Weak Overspill to the formula

FVO

> vo:

For

all finite I, Fx is infinite, whence Fx > x, Thus, for a few infinite a ’ s following w, Fa > a , i.e. Fa is infinite.

&ED

The Overspill Principle is, in its weak form, apparently due independently to Rabin and Abraham Robinson and, in its strong form, to Robinson.

Weak Overspill

is used a great deal in the study of nonstandard models of arithmetic, as we shall

Lectures on Nonstandard Models of Arithmetic

11

shortly see.

For now, however, a simple application will suffice: Suppose a E 1 1 1

is infinite.

Then there is an infinite b

finite, so is ZX; whence 2" < a .

E

IMI such that 2b < a .

For, if x is

By Overspill there is also an infinite b such

b < a.

that 2

In its strong form, Overspill is extremely useful in Nonstandard Analysis.

Applied

in that language, with F replaced by l/F, it becomes the Infinitesimal Prolongation Theorem:

... is an internal sequence of infinitesimals, then for a few

If r o , r l ,

infinite integers a , r

remains infinitesimal. One can consult, e.g. Martin Davis'

book for examples of the usefulness of this principle.

In the study of nonstandard

models of arithmetic, however, Strong Overspill has not made much of an appearance until recently, and then only tangentially as an insight behind the Indicator Theory of Laurie Kirby and Jeff Paris.

Even here, unless there is some recent

work I am unaware o f , there are no results which cannot as easily be proven by other means.

( Hence,

I shall not discuss indicators in the sequel. )

Both Overspill and the preceding Theorem 2.1 are manifestations of the r81e of an infinite integer as a path to infinity: An infinite integer behaves somehow like the integers it encounters in its path.

In an ultrapower, an infinite integer is

literally such a path; in the more general case, my words are mystical but of some heuristic value.

In any event, I view these last two theorems as sophisticated

views of the sizes of infinite integers. in $ 1 was their coding power.

The other property of integers we used

Tennenbaum's Theorem was a crude mixture of this

coding power and size. We now take a more sophisticated look at this. 2.4.

Definitions. Let M be a model of arithmetic. A set X c w is standard on M

if there are a formula @ J ~ with u ~ only u O , V l free and an element b

E IMI such that x E w: M I= I@ 1. The collection of standard sets of M is called the standard system of M, written SSyIMl. A number a E IMI codes X E SSy(MI iff X =

X

=

I

.c E w:

MI=;

E

D;; 3 .

Here, by Dx I mean the finite set canonically indexed by

X.

We already used the

prime decomposition of numbers as a means of encoding finite sets.

Generally,

however, it is more convenient (i.e. I prefer) to use the following: Do

=

{

1;

D

=

,..., xn- I

x0

1,

+ ... + ZXn-I ... > xn-I'

if x = 2"O

x* >

and

Before discussing standard sets and their codes, I should warn the reader that the terminology presented here is not universally used.

Following the deplorable

logical tradition of referring to sets of natural numbers as real numbers, the standard sets are often called the reazs of the model, SSyfMI being thus denoted

RM.

Moreover, for reasons soon to be evident, SSyfMI = R

Scott set.

M is often called a

(Well, actually SSy(MI is proven to be a Scott set and then called

the Scott set of the model.

)

c. S M O R Y ~ S K I

12

Tennenbaum's Theorem depended on the fact that in a nonstandard model every standard set possessed a code.

2.5.

A bit more is true:

Let M I = PA be nonstandard, X

Lemma.

E

SSyfMl.

X has arbitrarily small

infinite codes.

Proof:

Let X be defined through $vv,F, a an arbitrary infinite integer, c

infinite such that 2c < a , and F defined by FV

=

Then

{

1,

@vF

0,

+vF.

C-2

d

c

=

~v

+ zC-'

< 2'

<

a

i=l

is a code for X. The lemma yields, in the nonstandard case, a characterisation of standard sets: They are precisely those which have codes. The culmination of the work of Feferman, Scott, and Tennenbaum was to have been a joint paper of Scott and Tennenbaum. Their manuscript was, unfortunately, lost; there was, however, a paper by Scott in which a characterisation of standard systems was given.

x

x is completion

2.6. Definition. Let be a class of sets of natural numbers. c l o s ed, or c-ctosed, if it satisfies: i.

x is an algebra of sets, i.e. x is closed under the boolean operations

ii.

x is closed under relative recursiveness, i.e. imply Y

iii.

x

E

X

E

X

satisfies a weak form of Kgnig's Lemma:

If X

E

x and Y

X

x codes an infin-

ite binary tree (in the strong sense that the elements of X code finite sequences of 0 ' s and 1 ' s ) .

then some Y

E

x codes an infinite

path through X.

( A s remarked above, this terminology is not universal; the term "Scott set" is often applied. )

The choice of the term "completion closed" will be evident shortly.

The lost Scott-Tennenbaum paper included a proof of the following.

2.7.

Let M I=PA.

Theorem.

Then SSyM is c-closed.

Proof sketch (nonstandard case: for the standard case cf. 56): nonstandard and

i.

That

x = SSy(M/.

Let M be

x is closed under boolean operations is obvious as the formulae

defining the standard sets are closed under the logical connectives. ii.

That

x is closed under relative recursiveness follows from the charac-

13

Lectures on Nonstandard Models of Arithmetic

The details are routine,

terisation of relative recursiveness as being "A -in". 1 but grubby, and I omit this part of the proof.

Suppose X E

iii. This is the interesting part of the proof.

x codes an in-

finite binary tree. By K6nig's Lemma, there is an infinite path Yo through X; but Y

0

might not lie in

x.

Write X

3: E

=

to:

M k @

I= 38 ( $uz,lh(u) 0 lh(v) denotes the length of the sequence coded by V . to conclude M

the existence of Y

a,c

f

IMI, c infinite, such that M k @,,lh(a)

=

x codes an infinite path through X.

} for some

)

=

Then Y

E

o, where

{ (a)3::

=

w )

3: E

E

QED He actually proved

To state this, recall that a set

sentable in a theory T if there is a formula $G 0 with only X = { Z E O :

and b and use

Apply Overspill to obtain

c.

In Scott 2962, Scott proved a converse in the countable case. a relatively strong converse.

I$

for each y

Vo

Xs

w is repre-

free such that

?'!-&I T C l & l .

w - x = i 3 : E w :

The collection of sets representable in T we denote R e p i T l .

With all this we can

now state Scott's Theorem: 2.8.

Theorem (Characterisation of Standard Systems).

of sets of natural numbers. i. ii. iii. Proof:

Let

x be a countable family

The following are equivalent:

x is c-closed x R e p @ ) for some complete T Z PA in the language of arithmetic =

X

= S S ~ ( M for )

some M

PA. Implication ii => iii follows from

We have already proven iii => i.

the existence of a minimum model M

T

to prove i => ii.

of T: SSy(MT) = R e p ( T ) .

Thus, it suffices

For the sake of convenience, I will only shew i=> iii. This

requires a lemma:

2.9.

Lemma.

Let

x be c-closed and let X

E

sistent theory T (in a recursive language). is also coded in

x.

x code a set of axioms for some conThen some consistent completion of P

roof: Let, for some axiomatisation of T , E

x.

x= I

'$':

is an axiom of T

A l s o , let I $ ~ , @ ~be , .a. .recursive enumeration of all sentences in the lan-

guage of T .

A completion of

T is essentially an infinite path through the binary

tree

more exactly, it is an infinite such path consistent with 2'. is Ill in T, not recursive in T, and the fact that T

E

Consistency with P

x does n o t

guarantee

x to

c. S M O R Y ~ ~ S K I

14

include the tree of paths consistent with T .

eo

el

;!:$I,..., those paths $o ',I$]

However, it does include the set of

which have no proofs of their inconsistency with T

.~

of code at most k .

x+

=

I

/x\

where I) =

'

yclhkc) predicate calculus of The

+ set X

V].

+

coding an infinite path

+ through X .

E

Let T' have axioms I) for x X

leave to the reader to shew, a consistent, complete theory. it is recursive in its axiomatisation and T' = Before returning to the proof of Theorem 2.8,

'$':

x

x.

Choose Y f Y. T' is, as I Since T' is complete,

is an infinite binary tree recursive in X, whence X

T'b $

1

E

E

x.

&ED

let me pause to remark that this

lemma not only explains the terminology adopted, but it also provides a very Not only will we use it immediately,

useful property of completion-closed algebras. bur also we will apply it later.

Proof vf 2.8, continued:

x be a countable c-closed family of sets of ... . We construct a modelM with

Let

natural numbers with an enumeration Xo,X1, SSyIM) = Let C =

x in stages by a Henkin argument.

{ Co, Cl,...

be a new set of constants and let L

,...,cn-Ij.

arithmetic augmented by { go

S t a g e 0. S t a g e 2n

E

x

Let T0 in Lo be any completion in of PA. i1. Let TZni3 be TZn u { E % : x E Xn U{

Note that, if T2n

S t a g e 2n

be the language of

f

i2.

x,

then T2n+l

Let T2ni2

E

1 ;

f

n

x.

be any completion in LniI

%

:

x ?!

Xn

n

of TZnil

such that TZnil

x.

Let T = U Tn, M the term model of T (i.e. the model generated by all definable

n functions, using the constants of C).

Since M

b T 2 PA, M

is a model of arith-

metic. The fact that

x ESSy@l) is clear:

Xn = {

X E 0:

TI

E

% 1

= { X E W:

; E % 1

Mk

n The converse follows also by construction: Let

b

E

IMI is explicitly definable from some c. MI= v u ( $ W E

c-+

$*Gi ...c. v

for some $*.

E

SSy(M).

n @U0VI

,...,-ci

"0

)

"k-1 Thus, we can represent X f SSy(M) as

be any formula. Since any

k-I

, we

have

1.

Lectures on Nonstandard Models of Arithmetic

for all m > 2i

Tm

E

x.

T~

[ z E w:

=

+ 2, i

Thus X

=

I-

L

max{iO,

x.

E

...ci k - l 0 -

xci

15

1

...,ik-l 1 .

Thus, X is recursive in Tm, where

@ED

( Remark: The proof that i => ii is trickier. One does not have new codes to make elements of

x standard.

Thus, one alternates p a r t i a l z y completing the

theory-- first the C -sentences, then I: -sentences, etc.-- and representing 1 2 by using independent formulae of increasing complexity. ) elements of

x

Theorem 2.8 can be generalised. For example, one can replace PA by any consistent r.e. extension To (using Craig's observation that an r.e. theory has a recursive axiomatisation to handle stage 0).

A more important generalisation due indepen-

dently to Don Jensen and Andrzej Ehrenfeucht (Jensen and Ekrenfeuckt 1 9 7 6 ) and David Guaspari (Guaspari 1 9 7 9 ) is the following: 2.10.

Theorem.

Let P be a consistent completion of PA in a language extending

that of arithmetic by a countable set of individual constants. Let able c-closed family of sets. i. Rep(!?) 5 ii.

x be a count-

The following are equivalent:

X

There is a model M

b T with SSy(M) =

x.

The proof of Theorem 2.10 is a bit mare delicate. The extra constants offer no real problem-- they can be handled finitely many at a time. one only has R e p ( T )

c

x and not T x. E

The problem is that

However, for each finite n,m the Il

consequences of T referring only to the first rn constants is a set in

x.

-

Thus,

in the construction underlying the proof that i=> ii, one simply handles progressively larger chunks of T as well as progressively larger sets of constants.

I omit the details. Theorem 2.10 is, of course, of interest in its own right, but it is also a lemma. The theories T to which one wishes to apply the result are the complete diagrams of models.

As apparently first noticed by Guaspari, this allows an iteration of

Theorem 2.8 which, by a direct limit argument, yields: 2.11.

Theorem.

Let

x be a family of at most ,ql

sets of natural numbers.

The

following are equivalent: i. ii. 2.12.

x is c-closed X=

S S ~ ( M )for some M

Corollary (Assuming CH).

I=

PA.

x 5 P(w)

is c-closed iff

x is the standard sys-

tem of a model of PA. 2.13.

Open Problem.

Can CH be eliminated as an assumption in Corollary 2.12?

The derivation of Theorem 2.11 from 2.10 is entirely routine and I omit it. is time to move on to another topic.

It

We will not leave standard systems for very

c. S M O R Y ~ K I

16

long:

they are of extreme usefulness.

EXTENSIONS OF MODELS (THE 1960s 11)

3.

Till now I have largely considered single nonstandard models. Another development of the 1960s was the study of the embeddability of one nonstandard model into another, i.e. of generalising the r8le of N o as the standard model to allow nonstandard models to be standard relative to other models.

There is a big

difference in the relation between a nonstandard model M and an extension N and the relation between N (NI

-

a

IN1 -

E

0

and its nonstandard extensions: The new integers of

[MI could fail to be larger than all the integers of /MI. Thus we say

]MI is M-infinite if a > b for all b

E

IMI, and a is M-finite otherwise.

There are three possible types of extensions of models (relative to the crude criterion of the existence of M-finite and M-infinite integers): 3.1.

Definition. Let M _ C N be models of PA. i.N is an end extension of M, or M is an initial segment of N , E J N J - J M Jis M-infinite. N is a c o f i n u l extension of M, written M c N, if every a

written M c N, if every a ii.

E

IN1 -

IMI is M-finite. iii.

N is a mixed extensionofM if IN1 - \MI contains both M-finite and

M-infinite integers. All three possibilities are realised: 3.2.

Theorem.

Let M be a nonstandard model of PA. M has elementary mixed

extensions. 3.3.

Theorem. Let M

3.4.

Theorem.

b PA.

M has proper elementary end extensions.

Let M be a nonstandard model of PA. M has proper elementary

cofinal extensions. Theorem 3 . 2 is a triviality: Two new constants and compactness or any proper ultrapower provide the extension.

Theorem 3 . 3 is nontrivial.

It was first proven

by R. MacDowell and Ernst Specker in MacLbwelZ and Specker 1961.

I postpone a

proof of this Theorem until 57, when I will give a stronger result. section I will also give some related results for the countable case. is from Rubin 2962 and is relatively easy:

In the next Theorem 3 . 4

One doesn't have to construct the

cofinal extension directly; one merely extracts it from a mixed extension. The modern descendent of Rabin's extraction is the following.

3.5.

Theorem (Splitting Theorem).

Let

MzN

be models of PA. There is a unique

model k f f k PA such that M

Cc

?8* Ce N.

Moreover, the cofinal portion of the extension is elementary.

17

Lectures on Nonstandard Models of Arithmetic The Splitting Theorem is due to Haim Gaifman (Gaifian 1 9 7 1 ) and was the end product of a development begun by Rabin and partially furthered by A. Adler and Grigorii Chudnovskil.

It is a fundamental result in that it shews the crude

order theoretic trichotomy given by Definition 3.1 to be model theoretically significant:

Cofinal and end extensions are the, so-to-speak, building blocks of

all extensions and it suffices to study these two extreme cases.

Moreover, it

points to a fundamental difference in the r6les of the two kinds of extensions:

3.6.

Corollary.

Cofinal extensions are elementary.

The reader can easily deduce 3.6 from 3.5. If M

elementary case of 3.5:

5

Another easy exercise is given by the

N, there is a unique model

4f such that M Sc1'8~

The full Theorem 3.5 is nontrivial-- it is equivalent to the formalisa-

4 e N. bility within PA of the solution to Hilbert's 10th Problem.

Because of this, the

result is language-dependent; if one adds a few new predicates to the language, L the result can fail. If we let PA denote the extension of PA to a language L L L containing that of arithmetic-- assuming full induction in L-- and let A o , z I , etc. denote the quantifier classes of the extended language, the Splitting Theorem becomes 3.7.

L L Let M E N be models of PA and assume the extension is A -elemen0 Then there is a unique fif such that M
Theorem.

tary.

Of 3.5 and 3.7 I will only prove 3.5. 3.8.

Lemma.

Let M 5 N be models of PA.

,...,anb1

shewn, be given and let a. i.

If

M ii.

is a Zl-formula,

)I

If @ is a

r[

1

-formula,

$zo...&-l=>

If

M

Let $ n ~ ~ . . . v ~with - ~ , free variables as

E IMI.

- b $Fo...zn-l==> N b $ a o . ..an-l

N iii.

To do this I need a lemma.

c$

M

is provably A,

k $zo..

$zo...-an- 1

in P A ,

iff N

b $Zo..

Parts ii and iii are immediate consequences of part i, which follows from the provable equivalence in PA of all C -formulae with existential ones and the per1 sistence of existential formulae under extensions.

Proof

Of

3.5:

obvious:

Let M 5 N.

We want M

=

{ a

E

3b

=

a

E

INI: INI:

Moreover, it is easy to see that

Q

A -function. 1

Ifif I

fif INI(a < b )

7

N.

The choice of

fif

is

is M-finite }.

!ffis a

i.e. it contains 0 and is closed under segment of N,

E

structure for the arithmetical language,

I,+,*.

In fact, since

fif

is an initial

is closed under any PA-provably polynomially bounded total

c.S M O R Y ~ S K I

18

One A -function that can be proven total and polynomially bounded in PA is GEdel's 7 @-function: There is a A -definable term @ for which the following are satisfied: 1

for each n there is a polynomial &(VO,...,un-l)

i.

1 V U ~ . . . U ~ -3~V n 1 B(uo,vl) 5 V o

PA

ii. PA

c vvO...un-l

iii. PA

&(u,,

$vo...u

3 V ~ . . . U ~ - $~ u o...u

PA

iv.

<

-

18fl is

,..., R ( v , z ) ) ,..., ~ ( u , n - I ) )

@ ( V o ,B ( U 2 ,u o ) ) .

First, expand the language to make

8 is a provably A 1-function,

( N , @ ) and, by the bound ii,

vi )

=

(B(v,O)

< ul

aU2vVo

Now let us prove the Theorem in earnest. Since

3 u@

such that

B(V,,i)

u $ (B(v,o)

t-f

n-1 n- 1

v. PA 1 v v o < U 1 3 U 2 $ u o U 2 + I omit the routine, but boring, proofs.

primitive.

...,u ~ - ~ nc\ ) ( i
8

( M , B ) is still a substructure of

closed under

8. We shew (M,B) to be an

elementary substructure of (Mef,@) by induction on the length of the prefix of a prenex formula,

@ = QOVO* * * q - l u n - l J, 9 where we suppress mention of the free variables and J, is quantifier-free. By iii and iv, we can assume the quantifiers alternate in kind. We call $ 3 if and V

Qo

is 3

otherwise.

Basis: Suppose @

E

Let a

b

(M,8)

E

1~8~ and 1a

< b

1Ml.

E

b v v o < Wo => (N,B)I= v 8) b 6=' (4f,B) I= G,

V v o W o => W,B)

=> ( N ,

6) b

For n = 1 , the subcase $

The case n = 0 is trivial. Q l , say $ = ' f V o $ v o .

(!@,

(M,8) b $ =>

Formally: We prove by induction on n that

since J, is quantifier-free. Since, moreover, a

E

Igfl was

$.

is trivial.

E

Observe -

v0 < b Wo,

by 3.8

arbitrary, we have:

(M"f,B) b v ~ o ~ o . Induction s t e p :

If n > I, there are again two cases:

Case 1.

$

E

3n. This is trivial.

Case 2.

$

E

Vn.

Using v above,

(M,B)

Write $

=

Vvo3

I= V U o 3 U l J , = >

ulJ, and let a

(M,f3)

I=

3VlVV 0

E

<

Igfl and a

< b

E

IMI.

~J,(Uo,B(Vl,~O)).

But the right-hand side of this is basically 3n, whence Case 1 applies to yield (Mef ,B) b ~ U l V U 0 < ~ J , ( U , , B ( U l , V O ) ) => (4 f.B) 3VlJ,(Z,B(Vl,Z))

I=

and, since a

E

=> (4f,B) I= a u1 $Civ arbitrary, (4 ,@) -/- v U o 3 U l

lffl was

8'

$VoVl.

( N.B.

The essen-

t i a l ~- n e w of the formula in question was based on contraction of quantifiers, which 2n

!ffdepends on the bound

i-- or a different n-tupling function.

) &ED

The interested reader is referred to MZ&k 1982B for an axiomatic analysis of the proof of the Splitting Theorem (in the form 3.7) and to Motohashi A&B for additional observations on this result. I also refer to my survey Smoryn'ski 1 9 8 1 A

Lectures on Nonstandard Models of Arithmetic

19

for more on cofinal extensions; there is much of interest in them, but I do not have the space to devote to them here. I have given more space to the Splitting Theorem and given a more detailed proof of it than many expositors would because it is rather more important than it appears.

It does, as I said above, establish the significance of the crude order-

theoretic trichotomy of extensions into cofinal, mixed, and end extensions, and it justifies focussing one's attention on the two extremes. Moreover, as first observed by Wilkie, it has its applications.

(Cf. Theorem 4 . 8 , below, or, for a

simpler application in the style of Wilkie, Smoryriski 1 9 8 1 A . )

I shall close this

section with an amusing application. The Tarskian attempt of the 1950s to turn model theory into algebra resulted in a number of preservation theorems-- theorems characterising those sentences preserved under various kinds of restrictions, extensions, and homomorphic images. With the emergence of the notion of an end extension came also the problem of preservation under end extensions. This was settled in Feferman and XreiseZ 1 9 6 6 by appeal to a many-sorted interpolation theorem.

Straight model theory also

suffices to prove the characterisation, and, in the arithmetic case, the result is

an easy consequence of the Splitting Theorem: 3.9.

Theorem. Let L be a language extending that of arithmetic. Let

I@~...V

n- 1

be an arithmetical formula. The following are equivalent: i. ii.

Proof:

for all models M c

Mb

&o...Fn-l

PAL

c Qi

1

L

and a.

Nel= Go..., n-1

=> for some ?-formula

-

N of PA

,...,an-l

IMI,

E

JI,

JI.

The proof that ii => i is routine.

i => ii. Suppose Qi is upward preserved under all end extensions. By Theorem L 3 . 7 , it follows that @ is preserved under all A -elementary extensions. Letting 0 L L' extend L by a stock of primitives sufficient to make all A -formulae equivalent 0

to quantifier-free ones, we see that @ is preserved under all extensions M 5 N of models of P A L ' . By a standard result of model theory, @ is existential in PAL' , L whence 1; in PA &ED

.

4.

SATURATION PROPERTIES (THE 1970s I)

The model theoretic notions of types and of saturated and special models crystallised in the 1960s.

Although a crude mixture of recursion theory and model theory

can be found already in the late 1930s with Bernays' arithmetisation of the completeness theorem and one can find further discussion of the complexity of models in the 1950s and 1960s, the refined mixture of the two branches of logic-- namely, recursive saturation-- only arrived in the 1970s.

As one might predict on the

c. S M O R Y ~ S K I

20

basis of the fact that recursive saturation is a marriage of model theory and arithmetic definability, this notion is of great importance in the study of nonstandard models of arithmetic.

In fact, recursive saturation-- of a sort-- was

being applied in nonstandard model theory before it was realised that such a concept existed.

In the present section, I will discuss these early applications

and one or two minor issues related t o recursive saturation; in the next section,

I will digress to discuss recursive saturation in a more general, but arithmetically relevant, context. First, of course, come some definitions: 4.1.

Definitions. Let M be a model for some (not necessarily arithmetic) lan-

,...,

bmwl E 1Ml given parameters, and ~ U ~ . . . U ~ - ~ U ~ . .nim-1 .U a set of guage, bo formulae with only the free variables shewn. -

i. set T 5

M

7,

-

. ba~type - ~ over M if, for every finite The set ? ~ ~ . . . ~ ~ - ~ b ~ . .is

k

(M

$v o...v

S U ~ . . . U ~ - ~

-

b

-

n-1 o..*bm-l

1

tJET The type 7 ~ ~ . . . v ~ - j b ~ . . . b ~is - ~recursive if the set TU o...v

ii.

(or, rather, the set of its numerical codes rtJ1) iii.

-

-

The type ~ v ~ . . . ~ ~ - ~ b ~ . . .isb ~a -r-type, ~ for

(e.g. all formulae, Z -formulae, TI -formulae), if every

n

iv.

M b M k

B V o...v

-

The type ?v0.. .v n-l 5o...bm-l

n- 1

$

r

E 7

a set of formulae belongs to

r

is realised in M if

( ~ ~ o . . . v n ~ 1 5 0 . . . b ), i.e. for some

~o...Zn-lbo...Fm-l, for all

nim-1

is recursive

m-1

a.

,...,an-l IM\ E

E 7

V. M is recursively r-saturated if every r-type over M is realised in M; M is recursivezy saturated if M is recursively r-saturated for the set r of all formulae of the language of M.

N.B.

In the arithmetic language, we can restrict our attention to 1-types ‘ruz

with only one parameter because of the arithmetic ability to code finite sequences. We will be interested, however, in models of theories in languages not admitting such coding. One of the keys to the arithmetic importance of these notions is the following notion: 4.2.

Definition. Let

r

be a set of formulae. We say

r

a h i t s a truth definition

(more accurately: a satisfaction relation) in PA if there is a formula Tr (v such that, for all PA

where ( v 0 ,

$ V ~ . . . V ~ -E~

r

r

b@U~...V~-~

...,Unm1)

v ) 03

I

with free variables as shewn,

tf Tr (r$’,(vo,...,V r n-1 ) ) , is the code of the sequence V o ,

...,V n-1‘

The bearing of this notion on recursive saturation is the following lemma implicit in Robinson 1963 and very nearly explicit in Friedman 1 9 7 3 : 4.3.

Lemma (Robinson-Friedman Lemma).

Let M be a nonstandard model of PA and

21

Lectures on Nonstandard Models of Arithmetic suppose of

T,

M

r

admits a truth definition. Then: M is recursively r-saturated.

~ a recursive type over M. By the recursiveness Proof: Let b E IWI and T U be there is a formula Tau(u ) such that, for any formula @ u U o , @VV E T iff

0

0

Let, further, Trr be the truth definition for I-.

Tau('$').

From the fact that

M I=

T

is a type, we have, for each

3 u 1 V r@'

<

1 ( Tau('$uUi)

BY Overspill, for some infinite c

M

I= 3

Lc E 0 ,

+

Trr(r@',(U,E))

).

-+

Trr(r$l,(U,E))

).

]MI,

E

c ( Tau('@Uu;)

v1 V r @ ' <

lM1 witness this last, we see for each realises W E in M. Letting a

E

@UUo E T

that M

I=

a,i.e.

a &ED

The application Robinson made of this lemma is the arithmetic undefinability of arithmetic truth: 4.4.

Corollary.

The class of arithmetic formulae does not admit a truth defini-

tion in PA.

Proof sketch:

Then every non-

Suppose Tr(Uo,VI) gave a truth definition.

standard model of PA would be recursively saturated.

But we can easily construct

nonstandard models which are not recursively saturated:

(6 la

integer b in a nonstandard model N and let

Take, e.g., any infinite

Ryll-Nardzewski) M be the sub-

model of N generated from b by the parameter-free Skolem functions. does not realise the recursive type

x u D $7) OE

+

TUF

Clearly M

of formulae of the form

3 u o < V@V$.

A small remark:

&ED

Corollary 4.4 is not quite as strong as GEdel's result (usually

attributed to Tarski, who rediscovered and first published it) on the undefinability of the truth of sentences. The application Friedman made of the Robinson-Friedman Lemma is a bit deeper: 4.5.

Let M , N be countable nonstandard models of PA.

Theorem. i.

ii.

The following are equivalent: a.

M is embeddable in N

b.

SSy(M) 5 SSy(N) and

T h , (M) 5 f i x ( N )

The following are equivalent: a.

M is isomorphic to an initial segment of N

b.

SSy(M) = S S y ( N )

and

Thz (M) 5 T h z l ( B ) . 1

A few words of explanation are in order:

For

r

=

TI or E l

(the existential or,

respectively, Cl-formulae), T h r ( K ) is the set of sentences in

r

Matijaseviz's result, we could restrict our attention to !Z'k3

in both parts.

explicit reference to Th

true in K.

By The

in ii is, however, the form holding in expanded lan=1

guages as well as the one reflecting more directly the closure properties needed

c. S M O R Y ~ K I

22 in the proof.

In each equivalence, the implication a => b is fairly direct:

(1) The inclusion

of the standard system of M in that of N follows from i. the rigidness of w in the A, nature of the coding of standard sets, and

all nonstandard models, ii. iii. L e m a 3.8:

For a

(MI and z

E

u, MI=

E

3: E

W

iff N ) =

M

c

E

N are both nonstandard, one can choose a code for X in M.

That !7‘h3(M)

sions of the theories are similarly easy.

IY.

If X

equality of standard systems in ii.b follows from Lemma 2.5:

E

(2)

The S S y ( N ) and

The inclu-

5 T h 3 ( N ) is a familiar

fact of model theory; if the embedding is initial, Theorem 3 . 9 can be invoked. Indeed, ignoring the r6le of the standard systems, Theorem 4.5 can be viewed as a generalisation of the model-theoretic preservation theorems.

(I shall, in fact,

derive the hard part of Theorem 3.9 as a corollary to Theorem 4.5 below.) The proofs of the converse implications are facilitated by the following lemma: 4.6.

Lemma. Let K be nonstandard, a.

definition Try. The r-types

Let, for any a

Proof:

-

I= 6 O...an-l,

the parameters a,,,

x= Clearly X n u

SSy(K).

TUZo..

To see that

9l

IKI, and let T admit a truth

T

-a

.an- 1

-

a

E

is the r-type of a with respect to

K

-_

I= Trr(‘$’,(a,a0

,..., -an-I ) )

be a r-type over K, with X =

I

r$’:

}-

-

@ao..

E T

1

is realised, either repeat the proof of Lemma 4 . 3 or

T

-

T‘VU

D j ; + TrT(rI$l,(u,~o

E

be the set of all I$vVO...un-l

uVO...Vn-l

Let b be any infinite integer in IKl and consider

replace T by the recursive type r

E

over K realised in K are precisely those

T U~~...U,-~

-I < b: n- 1 SSy(K1 codes T ~ .

Conversely, let E

.. -

IK1,

E

i.e.

{ FI$~uo...u E

,...,an-,

O...an-,

.

coded in S S y ( K )

such that K

TU;

o...a

- -

b consisting of all formulae n- 1

,..., -an-l)),

where X = Db nu.

&ED

Proof of fieoran 4.5:

We have but to prove the implications b => a.

To do

this, we let M , N be given with convenient enumerations of their elements. We

,... of

construct such enumerations a O,al

model of N that the function mapping a A bit of notation:

r

we let T~

o.

.

i

\MI and bo,bl

For K a model of PA, co

denote the r-type of c o ,

.c

,... of an appropriate sub-

to bi is the desired embedding.

,...,em-,

E

IKI, and T either 3 or Zl,

...,cm- 1 over K.

m-1 i.b. => a. Let M,N be countable with SSy(M) Moreover, let a o , a l , enumeration b ,b Stczge

-

n.

...

5 SSy(N) and Th3 (M) 5 T h , (N). be the given enumeration of IMI. The construction of the

,,... proceeds by stages.

Suppose bo,...,b,-,

-

.an-l ‘to. ..bn-l

Tao.. LL

have already been chosen so that (*)

23

Lectures on Nonstandard Models of Arithmetic ( M ) 5 T h R (N) ) . The claim is that ~b~...b,_~ is a type over N and is in SSy(N).

(if n = 0,this reads: 3

anao...an-1

T =

By the Lemma, T is coded in S S y ( M ) , whence the inclusion of standard systems It thus suffices to shew T to be a type over N.

places T in S S y ( N ) . For any @ ,,

easy:

-

* *

9

T>

@k-l-E

.

M I= mbiZnC0. .an-l => i

I=

M

existence of b

Let b

E

3v

[m+ivC,..

I=

3v

1

.“,-l

z

=> N by the assumption (*).

But this is

-

3,

[ fi@;vb0.. .bn-l i

IN1 be any element of N realising

T.

( The

is, of course, guaranteed by the Lemma. )

I leave to the reader the verification that the map ai to bi constitutes the desired embedding. ii.b. => a.

This half of the proof is a bit more complicated in that we must

map back to guarantee the function constructed maps M onto an initial segment of N . such that, for each n,

A s in part i, we construct ao,...,an-l,b,,...,bn-l

c.

z1 h,. ..an-l

(*)

c ‘b0.. 1 .bn-l’

The w-stage construction alternates in direction.

Even-numbered stages.

Given a,,

...Jan-Z,boJ.. .,bn-l

satisfying (*), let a

be the first element of the given enumeration of I M ( not already listed and choose

bn exactly as in part i such that

Odd-numbered stages,

I1 ‘a,...a

c

5

(**)

- ‘b0...bn‘

satisfying (*), let bn be

Given a,,...Jan-I,bO,...Jbn-l

the first element of the given enumeration of IN1 such that br, < bk for some k

n-1.

(If no such bk exists, move on to the next stage.)

5

Choose any convenient

k for which bn < bk and fix it. We want to find an E ILY such that (**) holds. By the Lemma and the To this end, we look at the n -type T of b ,bo,...,bn-l. 2 assumed equality of standard systems, T E S S y ( M ) .

We thus need only shew that T

is a type over M . For any $o,.

N

.., ok I= En

<

E

T,

.

-

Kk A m$iKnEo.. bn-? i

=> N => M

b

3v < 3v <

Zk

[fi@iVbg.

Zk

(fi$iva,.

..%n-2 ) ..an-l ),

?.

since the sentence in question is TIl and since (*) yields

-

“1

Thus,

T

n- 1 Tbo...bn-l‘ is a type over M and, by the Lemma, it is realised in M by some an’

Again, I leave to the reader the routine task of verifying that the map ai to bi

c. S M O R Y ~ S K I

24

&ED

has the desired properties. Corollary. Let M be a countable nonstandard model of PA.

4.7.

Then: M is iso-

morphic to a proper initial seEment of itself. One proof proceeds by repeating the second of the above constructions-- being careful not to map M above a certain fixed element of itself. be had:

A slicker proof can

By the MacDowell-Specker Theorem (Theorem 3 . 3 , to be proven in § 7 ) , M has

a countable proper elementary end extension N which, by Theorem 4.5, can be initially embedded in M.

Composing the maps yields an isomorphism of M onto a proper

initial segment of itself. Theorem 4.5 is hardly the last word in back-and-forth constructions between nonstandard models of arithmetic. A fairly exhaustive list of basic embeddability results can be found (each with “constructive” and ”slick” proofs) in my survey

Smory6ski 1981E. 4.8.

Here let me cite merely the following theorem of WiZkie 1977B: Let M , N be countable nonstandard models of PA.

Theorem.

The following are

equivalent: i. M is arbitrarily highly initially embeddable in N , i.e. M is isomorphic to arbitrarily large initial segments of N ii. SSy(M) = SSy(N)

and T h

%

(M) -

= *% (N)

-

For a proof cf. WiZkie 1977B or ShroryGski 1981E.

There are, of course, many other such results deserving mention.

I like Wilkie’s

result because it has an interesting application to the construction of end extensions.

For that matter, Friedman’s Theorem has its applications to the study of

end extensions.

I cite one application of each.

First, there is the arithmetic case of the Feferman-Kreisel result: Theorem. Let Q, be a sentence of the language of arithmetic (plus a few

4.9.

constants if one wants a result about formulae).

The following are equivalent:

Q, is preserved under end extension: If M M I= @ => N I= Q, ii. for some $ E zlJ PA Q, ++ 9. i.

Proof:

CeN

are models of PA,

This is Theorem 3 . 9 , which we have already proven.

Theorem 4.5.ii, we assumed 4.9.ii=> 4.9.1.

In proving half of

Thus, I really intend only to use

4.5.11 to give a new proof of the more difficult implication i = > ii.

I prove the equivalent dual result: If I$ is preserved under restriction to initial segment, then Q, is equivalent to a I$-sentence. Let

I$

be given and T = PA

countable model of T .

+

{

TI E

111:

PA

1 Q, + TI

1.

Suppose M is an arbitrary

If we can shew MI= Q,, it will follow thatPAC I$++

TI~~...,,T%

25

Lectures on Nonstandard Models of Arithmetic for some IIl-sentences vl,...,v

E

T , whence

9 is

equivalent to a 17 -sentence. 1

Clearly it suffices to embedd M as an initial segment of some model N Now PA

T h z (M) is consistent; else PA f 9 7(UlA A5mm-1) for some 0 I’ 1 Th (M), contradicting the fact that the II -consequence I( U l ...fi ~

E

El

is true in M

I=

1

T. Now, the truth definition for X 1-formulae shews 7 !h ‘

+ 9 + Thz

Let T’ be any completion of PA

coded in SSy(M).

(which exists by the completion-closure of SSy(M)). model of T’ with S S y ( N )

=

embeddable in N, whence M

( Query:

I$.

...

+ 9+

...,um-1

I= PA +

SSy(M) (Theorem 2.10).

I=

5

(M) to be

(M) coded in SSy(M) 1 Finally, let N be a countable

By Theorem 4.5.ii M is initially

I$.

QED

Is the present proof model theoretically more pleasing than that of the

preceding section? ) As an application of Wilkie’s Theorem 4 . 8 , I cite the following application of

Wilkie 1 9 7 7 B : 4.10.

Theorem. Let M be a nonstandard model of PA and let T be any consistent

completion of PA. The following are equivalent: i. ii.

~N(M

c~NI=T)

R e p ( T ) C_ SSy(M)

and

Thz (M)

5 T.

1

The remarkable thing here is that M is not assumed to be countable--

Proof:

the countable case is nearly trivial.

Theorem 4 . 8 , in conjunction with the

Splitting Theorem and the Arithmetised Completeness Theorem (56, below), yields the uncountable case. i = > ii.

Left to the reader.

ii => i.

Countable ease.

Let M be given and countable. If Rep@) 5 SSy(M)

and T is complete, there is (by Theorem 2.10) a countable model N =

SSy(M).

k

T with SSy(N)

If T 33% (M), Theorem 4.5.ii applies and M is isomorphic to an init-

ial segment of N:

M

C

N

I= T.

General ease (countable and uncountable): Let M be given, T I Th plete theory with Rep(T) 5 SSy(M).

By the Arithmetised Completeness Theorem,

there is an end extension K of M modelling PA

+ TrI12,

i.e. the 17 -theorems of T. 2


By a Lgwenheim-Skolem argument, one can find a countable K

b

E

lKol is M-infinite, and ii. S S y ( K o ) 2 R e p ( T ) .

countable model Mo

(M) a com-

T with SSy(Mo) = S S y ( K o ) .

such that i. some

Apply Theorem 2.10 to obtain a

By Theorem 4.8, we can initially

in K above b , i.e. the image of Mo contains b . Composing the initial 0 0 embedding of Ma in K and the embedding of KO into K, we can assume M K. 0 0 -

embedd M

Apply the Splitting Theorem to find N

=

A$f

satisfying Mo

CcN

Ce

K.

Since M,N

c. S M O R Y ~ ~ S K I

26

are initial segments of K , either M the latter inclusion fails and M

C

C

N

N or N

C

M. As b

E

lMol 5 I N I , b

T.

E! IMI, &ED

Now for an abrupt change of topics: Note that we have thus far only explored the consequences of the saturation inherent in every nonstandard model of arithmetic-recursive r-saturation for

r

admitting a truth definition-- and not full recursive

saturation. Among countable recursively saturated models, the ability to realise full types allows the construction of elementary maps:

4.11.

Theorem. i.

Let M , N be countable recursively saturated models of PA.

The following are equivalent:

M is elementarily embeddable in N b. M is (properly) cofinally embeddable in N a.

C.

ii.

Th(M) = Z'h(N)

and SSy(M)

5 SSy(ilT)

The following are equivalent: b.

M is isomorphic to N M is isomorphic to a (proper) elementary initial segment of N

C.

Th(M)

a.

=

Th(N)

and SSy(M) = S S y ( N ) .

I refer the reader to my survey Smoryn'ski 1981B for detailed proofs and credits. The basic results-- namely, the criteria for elementary embeddability and isomorphism-- are exercises for the reader. a minor point:

I would even say "easy exercises", but for

In proving Theorem 4 . 5 , we relied on Lemma 4 . 6

made of the existence of a truth definition for

r

where use was

in shewing the r-types over M

realised in M were precisely those coded in S S y ( M ) .

This appeal to the truth def-

inition can be replaced by appeal to an appropriate recursive type-- which will be done in greater generality in the next section. Friedman's 1973 paper had a tremendous effect on the study of nonstandard models of arithmetic.

It was largely responsible for the renaissance of nonstandard

model theory in the 1970s.

Among those papers largely inspired by his paper were

WiZkie A , Jensen a d Ehrenfeucht 1 9 7 6 , Kirby and Paris 1 9 7 7 , WiZkie 197?B, Guasp a r i 1 9 7 9 , L i p s h i t z 1 9 7 9 , H&ek and Pudlak 1 9 8 0 , and Srnoryn'ski 1981B.

Moreover,

several of these have had sequels themselves. Perhaps the most interesting of these is the paper of Don Jensen and Andrzej Ehrenfeucht, which will be the point of departure of 55, below.

The marriage of recursive saturation with nonstandard model theory begat a number of additional developments which I cannot discuss here.

I refer the reader to

H6jek and B d Z a k 1 9 8 0 , KotZarski A , and Smoryriski 1982B for more advanced backand-forth arguments, and to K o t Z a r s k i , Krajewski and LachZan 1 9 8 1 , Lachlan 1 9 8 1 ,

Kaufman 1 9 7 7 , Kossak A , Shelah 1978B, Schmerl 1978 and Schmerl A for further interesting properties of recursively saturated models of arithmetic.

27

Lectures on Nonstandard Models of Arithmetic 5.

RECURSIVELY SATURATED MODELS OF RICH THEORIES (A DIGRESSION)

Donald Jensen wrote up some joint work with Andrzej Ehrenfeucht in 1973. When he died in 1974, the task of editing the manuscript fell to Harry Simmons. The final paper, Jensen and Ehrenfeucht 1 9 7 6 , appeared in 1976.

The notion of recursive

saturation, due to J.P. Ressayre and J. Schlipf, was only first mentioned in print in 1975.

It is historically false to say that the Jensen-Ehrenfeucht paper had

any results on recursive saturation.

However, the paper falls short of proving

many basic results on recursively saturated models only in the non-appearance of the words "recursively saturated".

The present section is a sort of elaboration

of ideas implicit in the Jensen-Ehrenfeucht paper.

I worked the details out in

December of 1979 during a visit to Heidelberg, but never published them because Julia Knight and Mark Nadel, in Knight and Nude2 1 9 8 2 A , went a bit further in a key direction.

However, I've never been particularly pleased with their exposi-

tion and its emphasis on the useless necessary and sufficient conditions; to my mind, it would have been more appropriate to emphasise the recognisable sufficient condition of richness.

Nonetheless, I refer the reader to their paper for more

general results on expandability. Our point of departure is the following pair of results of Jensen and Ehrenfeucht: 5.1.

Theorem. i. ii.

5.2.

Theorem.

Let M = (lMl;+,*,',O) be a nonstandard model of PA. Then: (lMl;+,<.a) is recursively saturated ([MI - (01;.,1) is recursively saturated. Let M , N be countable nonstandard models of PA.

(lNI;+,<,O) are isomorphic, then

so are

(IMl

-

If (IMI ;+,<,a)

and

{Ol;*,l) and ( I N 1 - (Ol;-,l); and

conversely, if the models of the latter pair are isomorphic, so are those of the

.

former

Closely related (and of some interest in the model theoretic assault on complexity theory) is the following result due (I believe) t o Wilkie. 5.3.

Theorem. i.

Let M be a nonstandard model of PA and a , b

and @,@ are the restrictions of ii.

E

(
+,'

to

IMI. c

E

IMI: c < a


if M is countable and Th(
is isomorphic to (
2975.

)

If one looks carefully at Jensen and Ehrenfeucht's proof of Theorem 5.2, one sees that the model M is needed only to invoke Theorem 5.1 and thereafter it is properties of recursively saturated models of Presburger-Skolem Arithmetic, PSA

=

Th(w;+,.,O),

and of Skolem's Multiplicative Arithmetic,

c. S M O R Y ~ ~ S K I

28

that are used.

Moreover, these properties are guaranteed by the richness of the

class of recursive types of the given theories. Without further ado, let me introduce the class of rich theories and the study of Following this, Theorems 5.2 and 5.3.ii will

their recursively saturated models.

( As for Theorems 5.1 and 5.3.i, let me note that

be recognised as trivialities.

the Robinson-Friedman Lemma applies:

The full truth definitions for the reducts

of 5.1 exist by the provable quantifier-eliminability within PA of PSA and SMA (Cf. Presburger 1 9 3 0 and Skolem 1931 for PSA, CegieZski 1 9 8 0 and Nude2 1 9 8 0 for SMA, and CegieZski, McAzoon and W i h e r s 1982 for both) and the Ao-nature in M of

all questions about
)

A theory T in a recursive (preferably finite) language L is

Definition.

}new, of formulae in one free variable

r i c h if there is a recursive sequence,

such that, for any disjoint finite sets X,Y of natural numbers, we have

+@jvl.

C R V cM\@ivn.

iEX 5.5.

JEY

The obvious examples are PA and ZF. Other examples, in increas-

Examples.

are:

ing order of the complexity of @ i.

of w independent sets.

The theory P

sequence of monadic predicates P,,,P,, "

A

This theory has a countable

... (no equality) and axioms

)

3v (/?(\PjVAh 1 P . V ZEX .icy J

for any disjoint finite sets X,Y of natural numbers.

Evidently, the choice

P,"'

will do. ii. Atoms.

The theory BA

-

an

_ _

where a c , a l , iii.

5v,

... are constants naming the designated atoms. PSA

=

Th(w;+).

3vo(v where p

=

Simply choose

pnvo).

denotes the n-th prime and mu the m-fold sum v iv.

SMA

$lnv:

=

T h ( w - {O};.).

3vo(v

with p , as above and V.

reads

of Boolean Algebras with Countably Many Distinguished

One chooses,

=

+

... + v .

Here one chooses

%,

v0

vm the m-fold product.

Skolem's Divisibility Arithmetic, SDA = T h ( w - {Ol; We want to define 3 v o ( v o is primeAvon+l I v h y v on+2 I v ) .

"v0 divides V I " .

To do this, we define a few auxiliary predicates: vo = I : Prim(vo):

"Vl(V0~Vl)

T V 0= I ~

V

(V V

IV

+ v l = v0V V 1

= 1)

I),

where

V 0 IV 1

Lectures on Nonstandard Models of Arithmetic

29

Primpow(v ) : 0 n-Chain(U ) : 0

n-Chain(u) uo

=

asserts that

1 and all divisors distinct, i.e. that v has at least n prime factors.

Primn(vo) : Primpow(Uo)A Primn(v) asserts u =

$nu:

where

.

1 (n+l)-Chain(vo).

for some prime V We now define ) p , Primn'l(ul)h Primn+2 ( v , ) 0

A

~

~

3vou1v2 (Prim(v A u

nated

V;

n-Chain(vo),,

~

'

~

~

u

1- ~

~

~

~

A

u

~

u

A

1

~

~

+ vi. The theory ODAG of Ordered Divisible Abelian Groups with a Desig+ + Positive Element, i.e. ODAG = Th(Q;+,<,O,l)= Th(R;+,<,O,l)= Th(R ;+,<,l,e), + Q,R,R denote the sets of rational numbers, real numbers, and positive real

numbers, respectively.

If we let

element, mt the rn-fold sum

t+

denote the constant designating the positive

... + t for any term t, and m the particular

instance mi, we can choose

$nu: \x/{ K 1.P+'v < 2m+1: o 5 m < 2n 1 . ( Explanation: Consider the iterated bisection of the "unit interval" based on 0 $nu asserts that u lies in the left half of the result of the n-th bisec-

and 1. tion.

) vii.

Any extension (by language or axioms) of a rich theory is rich.

Of

+.

particular interest is the theory RCF of Real Closed Fields, which contains ODAG

( For later reference, note that in RCF

-

uo < v l :

a u 2 ( l V 2 = O A U0

< is definable in terms of

+ v 22

o,+,.,=

by:

= V l ) .

Thus, the choice of $ U of vi can, for RCF, be translated into an existential

) formula in the language with primitives o,T,+,*,=. By point vii and the weakness of the theories cited, there are evidently a great many rich theories.

5.6.

Nonexamples.

There are also quite a few non-rich theories: Any theory T with a countable saturated model fails to be rich.

These include: i.

h'o-categorical a. b.

theories, e.g.

the theory of Equality on an Infinite Domain, Em = Th(w;=) the theory of the Order of the Rationals or Reals, OR = Th(Q;<) = Th(R;<)

ii.

sa-categorical theories for a > 0, e.g. a.

the theory of Algebraically Closed Fields of Characteristic 0, ACFO =

b.

UDAG iii. iv.

Th(C;+,*,O,l),where C is the complex plane

the theory of Uniquely Divisible (Torsion-Free) Abelian Groups, =

!i%(Q;+,O)

=

Th(R;+,O) = Th(R+;*,l)

w-stable theories occasional other theories, e.g. Th(w;<,O).

~

~

c. SMORYI~SKI

30

The reason a theory possessing a countable saturated model is not rich is simply that this model has only countably many types, while every model of a rich theory If T is rich, with

possesses at least continuum many types:

InEW witnessing

@nV

this richness, then, for each set XC w, the set

I

T (v) = X

$,v:

n

x 1

E

x I

n L

U{

is a type over every model of T. And clearly, if X # Y, then T ( v ) and -rY(V) are X

incompatible, whence distinct. The types

T (0)

X

If a

are the key to the mystery:

E

lMl realises

T (1))

X

in a model

M of T, then a c o d e s the set X in M. To emphasise this point, we will use a different notation: n E v : @nU. ( N.B. n is a subscript of the formula n

E

v,

@zv in an arithmetic lann E w: M k n E a 1 is coded

not a participant (unless @ u is uniformly defined as guage). by a .

) Thus, if a

E

lMl realises T XV , then X

k

The family of sets coded in a model M

=

T is analogous to the standard system of a

nonstandard model of arithmetic: 5.7.

Let T be a rich theory, { n

Definitions.

E V

InEw, a recursive sequence of

unary formulae witnessing the richness of T, and M a model of T. An element a

E

\MI codes a set X 5 w iff the condition cited above holds. A set X 5 w is s t a n dard on M (or, a r e a l of M) if X has a code in M. The standard system, SSy(M), of M is the collection of all standard sets of M. There is a serious ambiguity in this definition: not merely on M, but on the choice of for example, to construct a model M when we choose n choose n

E

E

E,

k Pw

in which SSy(M) =

v to be PZnU ; and SSy(M)

v to be P

The standard system of M depends,

i.e. of the formulae n = { X

5 w:

E

v.

XC w:

It is easy,

X is finite

X is cofinite

1,

I,

when we

2n+lV'

For recursively saturated models, however, there is no ambiguity:

5.8.

Let T be a rich theory and M a recursively saturated model of T.

Lemma.

The standard system of M is independent of the choice of

Proof: iff M

Let

Let X

given.

k

n

E~

Since

E,

and E ~ ,i.e. the sequences { n E v InEw and { n E~ v InEo, be 1 be standard with respect to El, i.e. let a E IMI satisfy n E X

E

5w

a,

-

TUU:

E.

for all n YZ Ez

U

E

++

w.

Consider the recursive set of formulae,

a:

YZ E

n

E

w

I.

witnesses the richness of T, this is a type over M.

we see that b codes X relative to

E

2'

The symmetric argument yields the converse: are standard with respect to

Letting b realise it,

All sets standard with respect to

E

1'

E

2

&ED

Since we are only interested in recursively saturated models of T; the lemma justifies our not mentioning

E

in "SSy(M)", i.e. we need not refer to "SS~(M,E)".

31

Lectures on Nonstandard Models of Arithmetic The significance of the standard systems of recursively saturated models of rich theories is the same as that of standard systems of (recursively saturated) nonstandard models of arithmetic. 5.9.

Let T be a rich theory and M a recursively saturated model of T.

Lemma.

The types realised in M are precisely those coded in S S y ( M ) .

Proof:

-

-

mbo...bm-l Suppose

Let M be a recursively saturated model of a rich theory T and let be a type over M . Let b

is coded in S S y ( M ) :

T

E

[ M I be such that, for $wv0...

(the set of formulae with free variables v , v o ,

M

!=

'$'

z. -Simulate - -

E

T

..

by the set

..., m-1 V

),

V m-1 @vbo...bm-l E T

&m+l iff

-

o...

@vz bm-l: @ E km+l I. This is easily seen to be a recursive type over M, whence realised in M ; but any

~ ' v b ~b,-*b: .

a

E

/MI

realising

T'

Conversely, if a.bo

{ r@l

E

%

-+

automatically realises

,...,bmml

E

T.

\ M I , the type of a over M in parameters bo

,...,bm-l

(i.e. the (m+l)-type of ~ , b ~ , . . . , b ~ -is ~ ) seen to be coded by appeal to the recursive type -

-

@abo...Emm-lt+

-r'vabo...bm-l:

r$l

E 0:

@ E

R",+l 1.

&ED

An immediate corollary is: Theorem. Let M ,M be countable recursively saturated models of a rich 1 2 theory T with equality. The following are equivalent:

5.10.

M1 is isomorphic to M z

i.

and SSy(MI) = S S y ( M z ) .

Th(Mz) = Th(M2)

ii.

I leave the proof to the reader. reduces to: S S y ( M l )

=

SSy(Mz).

I also note that, if T is complete, condition ii

( Exercise: Prove Theorems 5.2 and 5.3.ii. )

The key to the parenthetical exercise is that, if M is a countable nonstandard model of PA and Mo is its

+-

or --reduct, then SSy(M) = S S y ( M o ) .

In fact, the

standard systems of countable recursively saturated models of rich theories are precisely those of nonstandard models of arithmetic. 5.11.

Theorem.

Let M be a recursively saturated model of a rich theory.

Then:

SSy(M) is c-closed.

Proof: n

E

i. ly.

Let M be a recursively saturated model of a rich theory and let

v InEw witness this richness. S S y ( M ) is an a2gebra.

Let X,Y

Consider TTX(V):

Tx n Y ( V ) :

{ n YZ

E V

++

1E E

-

a:

E V t-t YZ E ; A n

E

SSy(M) with codes a,b

?' Ei W E

z:

Evidently, these are recursive types over M.

E

]MI, respective-

1 n

E W

1.

Any element of M realising

T

,

~

c. SMORYI~SKI

32

codes the complement of X; any element realising ii. X

E

T~~~

codes X n Y .

SSyiM) is ctosed under r e k t i v e recursiveness. To shew Y

SSy(M).

Let Y be recursive in

SSy((M), we first recall a convenient characterisation of

E

relative recursiveness: Y is recursive in X iff there is a partial recursive function f such that, for all n

Let b

n

E

Y

iff

n

L

Y

iff

w,

E

3 km ( D~ 5 X , , D ~ n x 3 k m ( Dk C X , , D m r l X

=

0 hf(k,m,n)

=

=

BAf(k,m,n)

=l).

0)

-

1M1 code X and f reduce Y to X and consider, i E FA/xz 7 i E 7 n E v : f(k,m,n) = T V ~ : iEDk i"Dm ~ { r ) c \i E /x\ I i E 7 + --, n E v : f ( k , m , n )

E

{m

One easily sees

TVF

to be an r.e. type over M.

recursive ones,

TVZ

is realised by some a

Let X

E

=

11.

iED,

iEDk

iii.

0 } U

E

Since r.e. types are equivalent to

lMI. Evidently, a codes Y.

SSy(M) code an infinite binary tree and let b

E

(MI code X.

To

conclude that an infinite path through X is coded in SSy(M), we will again realise an appropriate recursive type. tion:

To describe this type, we must recall some nota-

We assume an identification of w with the set of all finite sequences of

0 ' s and 1 ' s under which the usual operations and relations on finite sequences are

In particular, < > is the (number identified with) the empty sequence,

recursive.

(for i

=

0 , l ) is the sequence of length 1 with sole entry

catenation of the sequences m , n ,

TVEbe the set of all formulae of the forms

With this notation, let >

v,

E

i, m*n is the con-

and m _c n asserts that the sequence n extends m.

n

E

v

+

n*

E

vvn*<~> E v,

n

E

v

-+

n

E

6

for all n , and also all formulae of the form E

v),

for all m,n such that m

$n

-(m

us that

E

vAn

T V ~ ,which

and n

m.

Since X is infinite, Kb'nig's Lemma tells

asserts v to code an infinite path through X, is a type over M.

&ED

Recursive saturation does the rest. Shortly we shall have an interesting application of Theorem 5.11.

First, however,

we want to prove the converse and one of its applications. 5.12.

Theorem.

Let T be a consistent, complete rich theory and

c-closed family of sets. i. ii.

a countable

The following are equivalent:

TEX There is a recursively saturated model M

I=

5" with SSy(M) =

x.

The proof is sufficiently similar to that of Theorems 2 . 8 and 2.10 that we can safely skip it. Note that Theorem 5.12 refers only to complete theories.

In the incomplete case,

Lectures on Nonstandard Models of Arithmetic

33

one has: 5.13.

Corollary. Let T be a consistent rich theory and If some axiomatisation of T is coded in

family of sets.

sively saturated model with standard system 5.14.

Corollary.

Let T be a consistent r.e. rich theory and

x a countable c-

Then T has a recursively saturated model with standard

closed family of sets. system

x.

x a countable c-closed x, then T has a recur-

X.

( What about the uncountable analogue to Theorem

5.12?

An easy application of the

Arithmetised Completeness Theorem shews that uncountability is no greater obstacle in the rich case than in the arithmetic case:

If a complete rich theory T and a

x with T x are given, and if x = SSy(M) for some model M of arithSSy(M ) for some recursively saturated model M b T . In particumetic, then x 0 0 lar, Theorem 5.12 extends to the case where x is of cardinality ,q,. However, it x for simple 2'. The characterisation may be easier to construct M with SSy(M) c-closed set

E

=

=

trivially holds without restriction for P and Knight and Nude2 19824 shew it holds without restriction for PSA.

)

The immediate corollary to the existence and uniqueness theorems (5.10 and 5.12) and the independence of SSy(M) of the choice of

E

is expandability:

5.15. Theorem. Let T c T be complete rich theories with respective finite 0- 1 languages Lo 5 Ll with equality and let M be a countable recursively saturated

model of T o .

The following are equivalent:

TI E SSy(M) ii. M can be expanded to a recursively saturated model of T 1 , i.e. there i.

is a recursively saturated modelMl of T pretation on /MI of the primitives of L

Proof:

i => ii.

1

obtained from M by an appropriate inter-

l-Lo'

Let M be a countable recursively saturated model of T o .

Construct a recursively saturated model MI of Tl with SSy(Ml) = S S y ( M ) .

Letting

M

be the L -reduct of M and appealing to Lemma 5.8 inside M we have SSy(M ) = 0 0 1 1 0 SSy(Ml) = S S y ( M ) . By Theorem 5.10, M is isomorphic to Mo. Pulling the structure of MI back to M yields the expansion desired. ii => i.

&ED

By Lemma 5.9.

Refinements: Obviously, To need not be assumed complete as T h ( M ) is both rich and complete. If T is not assumed complete, but some axiomatisation of T is coded 1 1 in S S y ( M ) , it has a completion so coded and the existence o f an expansion to this completion yields that for T I .

In particular, a countable recursively saturated

model M of a rich theory T is expandable to a recursively saturated model of

!2%(M) U T l , for any recursive theory T I consistent with T h ( M ) . This latter version of expandability-- to recursive theories-- has nothing to do

c. S M O R Y ~ K I

34

with richness; it is a basic expandability result called respzendence and due independently to J.P. Ressayre and to Jon Barwise and John Schlipf.

The present

sharp form for rich theories was also proven in Knight and Nude2 1 9 8 2 4 , where a generalisation to arbitrary theories in finite languages in terms of "ideals" is given. Resplendence-- the expandability of countable recursively saturated models t o recursive theories-- has many applications to models of arithmetic, particularly to recursively saturated models of arithmetic. Many of these applications, however, are not as powerful as those obtainable by more direct methods; I refer the reader to the papers cited at the end of the previous section for such applications. What I really want to do now is digress even further from the subject of nonstandard models of arithmetic and prove an analogue to Tennenbaum's Theorem. Recall that Tennenbaum's Theorem (actually, Corollary 1.6)

told us that no non-

standard model of PA could have recursive operations of addition and multiplication.

By Theorem 5.1,

the

+-

and --reducts of a nonstandard model of ?A are

recursively saturated models of PSA and SMA, respectively; by Theorem 5.15, converse is countably true:

the

A countable recursively saturated model of ?SA

is the +-reduct (--reduct) of a nonstandard model of PA.

(SMA)

Hence, there are no

recursive, recursively saturated models of ?SA or SMA. What we shall see is that this is a better way of looking at Tennenbaum's Theorem-- it asserts the nonexistence of recursive, recursively saturated models of decent rich theories. Before proving the analogue of Tennenbaum's Theorem, we need two definitions. 5.16.

Definition.

Let I MI be M as identity. We say M

Let M be a model for a given recursive language L .

w and let equality, if it occurs in L , be interpreted in

is recursive7y presented if there is a recursive satisfaction definition for atomic formulae of L in M:

MI=

...,xn- 1

The relation, for xo,

Go...Zn-1 '

is a recursive relation of

r$Uo...V

' and

n- 1

E

w and formulae I$,

(X~,...,Z~-~) (the usual recursively

obtained n-tuple). If L is finite, this amounts to each primitive relation and function being recursively interpreted in

W.

The notion of recursive presentation is the appropriate version of model theoretic complexity. 5.17.

The notion of decency of richness is the following:

Definition. A rich theory T is e x h t e n t i a Z Z y r i c h if the recursive sequence

of formulae witnessing the richness of 7' can be chosen to be purely existential. Examples will follow shortly. First, let me state the main result:

Lectures on Nonstandard Models of Arithmetic 5.18.

Let M be a recursively saturated

Theorem (Tennenbaum's Theorem Revisited).

Then: M is not recursively presented.

model of an existentially rich theory. Proof:

Observe first that S S y ( M ) , being completion-closed, contains a nonLet X be such a set and

recursive set-- e.g. some consistent completion of PA. let k M

E

35

( M I = u code X.

I= $nzo., .Kr where

Then: n

avo..

E

X

iff M

n

E

3 m0

iff

is the formula n

.V

n

k

E

,...,m p -I

s.t.

v with$ quan'tzifier-

n

free. Evidently, X is r.e. in any presentation of M . Since SSy(M) is closed under complementation, T X is also r.e. in any presentation, whence X is recursive in such.

Since X is non-recursive, the representation is

&ED

not recursive. 5.19.

Examples.

The following theories are existentially rich and, hence, have

no recursively presented, recursively saturated models: i.

PSA; cf. 5.5.iii

ii. SMA; cf. 5.5.iv iii.

ODAG+; cf. 5.5.vi

iv.

RCF; cf. 5.5.vii.

RCF, at least two of +,.,<

I n fact, in any recursively saturated model of

must be nonrecursive; but any one of these can be

recursive. For further information on variants of Tennenbaum's Theorem for recursively saturated models, cf. Macintyre 1 9 8 1 .

Therein Angus Macintyre goes much further in

ruling out the recursive presentability of recursively saturated fields of various kinds.

He gives an analysis analogous to mine of Tennenbaum's Theorem and cites

Lipshitz' citation of Tennenbaum as another source. If I may add: Mark Nadel informs me he has also proven a generalisation of Tennenbaum's Theorem. Let me finish with a minor problem: 2.20.

Open Problem. Is SDA (cf. 5.5.v) existentially rich?

6. THE ARITHMETISED COMPLETENESS THEOREM (THE 1970s 11)

I have already made several references to the Arithmetised Completeness Theorem; indeed, I have twice referred to its applications. In the present section, I propose to consider it in detail. 6.1.

Theorem.

In its weakest form, it is simply the following:

SSy(N ) is c-closed. 0

Proof sketch:

Recall that S S y ( N o )

=

Def(No), the family of subsets of u

definable in N o , i.e. the class of arithmetical sets. The closure of the family of arithmetically definable sets under boolean operations is trivial; that under relative recursiveness follows from Post's Theorem. The crucial point is the existence of an arithmetically definable infinite path

c. S M O R Y ~ K I

36

through any arithmetically definable infinite binary tree.

By an argument analo-

gous to that in the proof of Theorem 5.11, it suffices to find an arithmetical To each infinite binary tree X we can

completion of any arithmetical theory:

associate recursively in X a theory T any completion of which recursively codes an infinite path Y through X. To complete a theory T with an arithmetically definable axiomatisation, simply

... of all sentences of the language and look

take a recursive enumeration @,,Q,,

for the left-most consistent path through the usual tree (cf. the illustration in T the proof of Lemma 2 . 9 , above). The consistency question is n2, whence the path T T is recursive in II?, i.e. it is A2. &ED

( Remark: The reduction is not absolutely necessary. Given an infinite binary tree X, one can search for the left-most infinite path through X:

Start at < >.

Given u, if there are infinitely many elements of X beyond U* list wise list u*.

other-

O*;

The path Y so generated is recursive in the question, "Are there

X

infinitely many elements greater than u?", which question is II2-- whence Y can be X chosen A3. The reduction reduces this degree. ) The Arithmetised Completeness Theorem generalises Theorem 6.1 in two ways.

First,

by adding witnessing constants, it produces a model, not merely a completion; second, it is formalised within P A , whence valid over any model thereof.

Before

going on to the full result, I should first like to give some sort of application of the result at hand.

It is not, perhaps, the most convincing of applications;

but it allows me to illustrate the relation between infinite integers and ultrafilters:

I shall translate the ultrafilter proof of Ramsey's Theorem into a

nonstandard one and use Theorem 6.1 to obtain sQme extra information. First, some background on Ramsey's Theorem:

6.2.

Definition.

Let X be a set of natural numbers and let 2.y be positive

integers. i.

(,") is the set of subsets of X of cardinality

X,

i.e. the set of

s-element subsets of X ii. A function ~:(jcx) colouring, of

(,XI

P; o r :

6.3.

+

y (

I

O , I , ...,y-

iii. A subset Y 5 X is P-homogeneous ( o r : Y monochromatic ) if P is constant on (s). Theorem (Ramsey's Theorem).

subset of w , and ) : ( P

Y 5 X.

=

+

I 1

) is a

homogeneous with respect t o

Let s,y be positive integers, X an infinite

y a partition.

There is an infinite P-homogeneous set

Moreover, Y can be chosen arithmetically definable

Proof:

By induction on x > 0.

it will remain relatively fixed.

p a r t i t i o n , o r y-

€1

X,P.

We shall denote x by n in the following since

Lectures on Nonstandard Models of Arithmetic

Basis.

For n

31

1 , this is merely the assertion that a finite union of finite

=

In this case, Y is clearly recursive in X,P.

sets is finite.

Induction step.

Without l o s s of generality, we may assume X

=

w.

Let

(w;+,*,',O,P). By Theorem 6.1, SSy(N 0 ,P) is c-closed. 0 Choose (M,P), by Theorem 2.10, to be any nonstandard model of T h ( N ,P) with 0 SSy(M,P) = SSy(NO,P). The canonical embedding of ( N ,P) into (M,P) is clearly P:(nyl)

y and ( N ,P)

-+

elementary.

=

0 Moreover, the restriction to w of any definable subset of or a rela-

tion on (M,P) is definable in (No,P). Let a

E

Using a, deviously define the rela-

\(M,P)I be any nonstandard integer.

tion R on wn+'

by

Rxo...xn-lx:

P(xo,...,xn~l,x)

=

P(xo,...,xn-l'a).

R is codable in (M,P), whence definable in (NO,P). Let @Vo...Vn-lV

Proof:

Suppose P ( X ~ , . . . , X ~ - ~ ,=~ )i < y .

(M,P) I=

Go....: n- I x

-

-

whence (No,P) b Go...Fnn-lx

(No,P)I= v v ( Since p(x0

&o...Fn-Iv

,...,

=

++

For any x E w, P(X0 Xn-,,X) = P(xo Xn-l,a) tf P(F0,. 'xn-l ,x) = T , P(xo X ~ - ~ ~ = X i.) Since x was arbitrary, P; o...xE-lv i whence (M,P) satisfies the same.

,...,

f-t

-

define R.

.. ,...,

i, (M,P)I=

Proof of 6.3, continued:

1

,...,

,,

QED

I$LZ~...X~-~U.

We enumerate a prehomogeneous set Z as follows:

... < znm1 be arbitrary natural numbers. n, such that for any subsequence Suppose we have generated z o < ... < zk-Il k < z i < ... < zi , we have (No,P) b @zi .,.zi By the Claim, for each z zi . i0 0 n -0 - n-1 n ... < z i , we have (M,P) c. ...zi a. From this it follows by Theorem 2.1.i n- 1 '0 n-1 z for all that there are infinitely many z ' s in w such that (N ,P) $yi ...z 0 o in-?

Let z o <

-

such subsequences.

Choose z k to be the least such.

After w steps, we have enumerated Z arithmetically in P,X.

=

zo,zl,

... 1

recursively in R-- hence

Now define a new partition Q: (

Z

)

+

y by

Q is well-defined and arithmetical in P,X. P(X~,...,X~-~,~). hypothesis, there is an infinite Q-homogeneous Y 5 Z. Let yo <

... < y,

P(yo, ...,y

and wo <

n)

= =

... < w

be elements of Y.

Q(Xo,

...,Wn-1 ) ,

Q(wo,

=

P(wo

Observe:

=

P(wO,...,wn-l,~

by Q-homogeneity

,...,wn-l,a), by

=

We claim Y is P-homogeneous:

P(yO,...,yn-Z,a), since the yi's are in Z Q ( y o , ...,y,- I ) , by definition

=

...,xn-l)

By induction

definition

),since the w.'s.are in Z .

Since Y is arithmetical in Q, it is arithmetical in P,X.

c. S M O R Y ~ K I

38 6.4.

Remarks.

As I mentioned earlier, this is a modification of a proof

i.

using ultrafilters (cf. Chang and Keisler's model theory text or Bell and Machover's logic text).

A more faithful representation of such a proof proceeds by

replacing the model constructed by appeal to Theorem 6.1 by any proper elementary This proof, which I leave to the reader as an exercise,

extension of ( N , P ( W ) , E ) . 0

does not give the extra information on the relative arithmetical definability of the homogeneous set. ii.

As first shewn in Jockusch 1 9 7 2 , this relative arithmetical definability

can also be established by analysing a direct proof of Ramsey's Theorem; in fact, this latter gives us explicit arithmetical bounds and the schematic provability of Ramsey's Theorem within PA.

This doesn't seem possible with the above proof or

via the more general Arithmetised Completeness Theorem.

We will see in 87, below,

however, that the above proof can be extended in the face of a yet stronger arithmetisation. To illustrate yet once more the usefulness of infinite integers, let me offer the following reduction of the Finite Ramsey Theorem to the Infinite Ramsey Theorem, as promised in

6.5;

Corollary

following the proof of 2.1.

82,

(Finite Ramsey Theorem).

Let x,y,W be positive integers.

There

is a positive integer z o > x such that, for all z > z u and all partitions P:(")

X

-f

y, there is a P-homogeneous set X 5 z ( = 10,

Proof:

Fix x,y,

...,2-1)

) of cardinality w .

and W and assume the resulting assertion to be false.

This

assertion has the form ~ v O v v >l v o $ ; its negation is thus V u u 3 vI > u o 7 $. Apply Theorem 2.l.i: Let M be a nonstandard elementary extension of Nu and let u

E

IWI be nonstandard and P:Ca) X

homogeneous set exists. partition on

(-), which

+

y any coded partition for which no w-element

A contradiction follows by noting that P restricts to a

has an infiniEe P-homogeneous set Y, any w-element

&ED

subset of which is also P-homogeneous.

I first learned the present reduction from Joram Hirschfeld; I find it a bit easier than the direct appeal to Kb'nig's Lemma. the Paris-Harrington Variant:

Both proofs also quickly yield

The Theorem remains true if the cardinality con-

dition on X (that X have cardinality W) is replaced by one that X be relatively large (i.e. card(X)

2 min(X) , x + l ) .

( Exercise: Carry out this proof. ) The

Finite Ramsey Theorem also has a reasonably direct proof with fairly explicit bounds; the Paris-Harrington Variant does not (cf. e.g. Paris and Hurrington 1577 or Ketonen and S o ~ o u a y1581 ) . Getting back on track, recall Theorem 1.3 and the ensuing discussion by which any consistent r.e. theory has, not only an arithmetically definable completion, but also an arithmetically definable (in fact, A,)

model.

construction can routinely be formalised within PA.

The fact is that the whole The problem is that, although

39

Lectures on Nonstandard Models of Arithmetic the construction is routine, it is a notational morass.

I turn now to the unin-

spiring discussion of syntactic preliminaries.

6.6. Notational Conventions. i. ProvL (v ,v ) is a A -representation of provaPC 0 1 1 bility within the predicate calculus PC based on the language L. That is, we assume, for

Ji,x in

PCL$ iff 3 2 E w P A L ProvpC(uOJ L rJi’)

a. b. ii.

the language L,

( PAlProvgC(;,‘jl’) A ProvpC(ul, L ‘Ji *

If $uvo is any formula withu free and

other free variables) such that PAC$vuo

L

formula Prov($)(v

‘U 0 ’ 1’ $(r$7,uo) }, by

$:

+

ProvpC(uO~V1*(rX”), L

‘-x-).

vo either free or vacuous (and no

”U is a sentence of L”, we define a

v 2 ) , supposedly representing proofs overPC from axioms

L(uO;ul Prov($) iii.

+

)

xT)

,u2) :

3 finite

Q5

I U : $UU

0

, r m ~ U;

I(ProvpC(ul L

+

)).

We also define :

Pr($)(uo;uz)

ulProv($)(VL 0

;v ,u ) , 1 2

suppressing mention of the language. There are two contexts in which we want to use this notation, the syntactic and Semantically, we choose a model M k P A and a parameter a

semantic.

define a theory T with axioms X =

1

$:

k $(rJi7,z) 1. The

M

IMI and M

extends T. Syntactically, one wants to choose a fixed numeral ; 0

and look at the theory T given by the axiomatisation X In this case, the set of theorems of T is

Ji:

contained in

E

set { J i :

T ~PI-($](F~;~J~’) 1.

I ) :

Ji: -

=

M

PA bPr($)(~~;~$~)

I= $J(~J~’,F~C,, 1. 1

and is

The free variable v o in $uu0 allows us

to handle everything unifbrmly.

6.7.

More Notation.

i.

Con($)vo

is the assertion -Pr($)(uo;y),

where A is

a convenient refutable sentence of the language L . ii.

If T is a theory with a more-or-less canonical representation @ of its

($1.

axiomatisation, we write Pry and ConT for Pr($)

and Con

assume such a canonical representation exists.

If T1 = T

If T is r.e., we

+ $,

and $ defines the

axiomatisation of T, we assume TI canonically given, relative to T, by $IUu0 :

$vvo

V

v =‘I)?.

Finally, we sometimes write PrT and ConT, when no canonical

representation is given, to denote the predicates used when some specific representation is in mind. iii.

If xuu

0

has u free, possibly

v0 free, and no other variable free, and

x satisfies PA C p u 0 “U is a sentence of L“, we x defines truth for some structure in the language I,.

if, for a given language L , let Mod(dVo Mod(&

0

assert that

+.

is the conjunction of the following: a.

v

Sentences ~ i ~ , X(~VJ,YJ,~ ~ i ~ ( ,uo)

t-t

xf$;

, u , ) ~ x ( ‘ $ ; ,vo) )

c. S M O R Y ~ K I

40 for

0

E

I

A , V,

1

+

V sentences J, ( x('7 $',V0) C f ix('-$',Vo) ) V ' I @ with ' only v free ( X ( ~ Q U J N ' , V , ) ++ qVx('I$"v0) ) where Q E c V ,2 1 and u" is a name for the object represented by 0. ( If one doesn't like u", rephrase this in terms of satisfaction of J, by V . ] b.

C.

With all this notation we can state the Arithmetised Completeness Theorem: 6.8.

Theorem (Arithmetised Completeness Theorem; Syntactic Form).

Let

@V0

be

an arithmetic formula with u free, v o free or vacuous, and no other free variables. Suppose, for some language L , P A F @VUo "V is a sentence of L". a formula p u g with the same free variables as +VVo such that

Then: There is

-+

i. PA ii. PA iii.

1pvo I-

+

"u is a sentence of L"

Con($)vo

+

Con

-+

PA

C+lV'0

.@v0 pv0 Mod(JI)vo. -+

The proof just consists of arithmetising the proof of the Completeness Theorem and is routine; nonetheless, or therefore, I shall not repeat it here.

Eventually,

the details will be published.

The Syntactic Form of the Arithmetised Completeness Theorem is rather powerful. I n proof theory (in the Orey Compactness Theorem-- cf. Fefemun 1960-- and in Smory6skC 1982C) this power is occasionally necessary; in model theory it-particularly the uniformity-- is generally not necessary. To state the semantic form, we need a definition:

6.9. Definition. Let M be a model of PA and N a structure for some primitive recursively encodable language L . We say that N is s t r o n g l y definable in M iff i.

IN(

=

IM[

ii. the satisfaction relation for N is parametrically definable in M, i.e. there is a formula Tr(uo;ul,t)Z) and some b E /MI such that for all $u0... V n- 1 and a . a E IN(, N !=Go a iff M l=Tr(F;r$',(zo n-1 n- 1

...

,...,

6.10.

,..., an-l)).

in $) b

M

I=

Let M ( , P A ,

Theorem (Arithmetised Completeness Theorem; Semantic Form).

+vva a formula always defining a set of sentences, and (if u E

Con($)),

1M(. Let

JI:

M

occurs nonvacuously

0

@(r$',F)1 axiomatise a theory

2'.

then there is a model N 1=T strongly definable in M.

I N / should be infinite, we can interpret equality in

If M

ConT (i.e.

Moreover, if

N by identity.

Theorem 6.10 is a fairly immediate corollary to Theorem 6.8 and I omit the details. We shall shortly see some good conditions on T guaranteeing M to recognise its consistency. First, let us note an immediate consequence of strong definability. 6.11.

Theorem.

definable in M.

Lst M be a nonstandard model of PA and suppose N is strongly Then:

N is recursively saturated.

Lectureson Nonstandard Models of Arithmetic

41

The proof has, more-or-less, already been given: The result is a variant of the Robinson-Friedman Lemma ( 4 . 3 ; but recall also 5.1 and 5.3). If the theory T contains PA, the connexion between M and a strongly definable N

k2'

6.12.

is much stronger: Lemma.

Let M,N be models of PA with N strongly definable in M.

There is a

canonically M-definable embedding of M onto an initial segment of N .

Proof: Define F:IMl -+ IN1 ( = IMI ) by recursion: F ( O ) = O N , F(S#) = M ( F a ) , where OM (ON), SM (S,) are the zero element and successor function of M N (respectively, N ) . The definability of SN and ON in M, together with recursion in M guarantee that F is well-defined and M-definable.

S

That F is a homomorphism, i.e. that it preserves atomic structure is proven as follows: F preserves the successor function by definition. Preservation of +,* follow by induction in M and the recursion equations in M and N .

+ 5) =

addition, weprove V V ( F ( V

Since

+

F(a

+ 0)

F(a

+ b')

=

Fa

=

FF) by induction on b Fa + 0 = Fa + FO N N N FV +N

+ b ) ' ) = S$(a + b ) = +NSgb = Fa +N F ( b ' ) .

=

F((a

=

Fa

vo

is preserved, so is <:

< vl

t+

E

E.G.

for

IMl :

SN(Fa +N Fb)

3v2(vo

+ v2'

=

vI). Hence F is one-

one, i.e. an embedding. Finally, to see that F maps onto an initial segment, we shew M

I=

VV <

N

by induction on b : note N

I=

V v ( v
z3v0(FV0

=

Fv)

For b = 0, the result is vacuous. For the induction step,

sNE

++

v cN Z v v

=

Z ).

&ED

Lemma 6.12 was multiply discovered: S l a t e r 1972, H&jek 1972, Sinoryn'ski 1973A&B, MeAloon 19784, and Guaspari 1979. Lema 6.12 has its applications; but these depend on our model M's recognising the consistency of a given theory 2'.

Gsdel's Second Incompleteness Theorem tells

us this cannot always be done-- for the obvious definitions of many T ' s .

A

theorem of Mostowski allows us to get around the gb'delian difficulty:

6.13. Theorem (Reflexiveness). Let L be a language extending that of arithmetic and let To be a finite subtheory of PAL axiomatised by ,$nn-l. Then,

PA

b Con , i.e.

PA

1 Con(+),

where +v is

TO

Wv

=

'$2.

. ..

The proof of Reflexiveness, that PAb proves the consistency of each of its finite subtheories, lies beyond the scope of the present lectures: It is a matter of proof theory, not model theory. However, I can remark that it is not particularly difficult-- one proves Cut-Elimination or Herbrand's Theorem for P?, proofs in P 8 are rendered direct.

A direct proof of

-I($~A

...

A$~-,)

by which all uses only

c. S M O R Y ~ S K I

42

L

formulae of bounded complexity and one can use the partial truth definition in PA to shew by induction that all formulae involved in the direct proof are true. Since

7 ( w 0 ~ . . .

A $ ~ - ~is)

n o t true, the given direct proof did not exist--

whence 6.13. 6.14.

Corollary.

Let M be a nonstandard model of PA.

There is a model N

PA,

with M c N and N strongly definable in M. Woof:

It suffices to note that M

I=

for some $ defining PA.

Con($)

The easy

Let $ v be the “canonical” definition of a set of 0 By Theorem 6.13, M Con($lF for each finite V5 V o .

exhibition of I$ is as follows: axioms of PA and ?vv *: z.

By Overspill, M

usual axioms of PA. 6.15.

Remark.

$ o V ~

-

Con($)a

E

(MI. But I$va defines the

Apply 6.10 and 6.12.

QED

A s I mentioned in the above proof, the formula $vvo exhibited is

With a little work, one gets a uniform definition

the easy one.

of PA for which PA kcon(?). Theorem:

for some infinite a

$0

of the axioms

The uniform method yields the Orey Compactness

For an r.e. theory T to be interpretable in P A , it is necessary and

sufficient that every finite subtheory of 7’ be so interpretable (whence its consistency provable in PA).

I refer the reader to F e f e r m a n 1960 and Grey 1961 for

more on this matter. A s one might gather from this Remark, Corollary 6.14 is a 1970s version of 1960s

This version is due to H6jek 1972 and S m o r y f i s k i 1982C (but cf. S m o r y f i s k i

work.

1973A&B) and was rediscovered by McAZoon 1978A and Gzcaspri 1979. McAloon actually offers the model theoretically nicest version, but I defer this till later.

Right now I want to give an application of the results already at hand.

The following is the uncountable generalisation of Theorem 5.12: 6.16.

Application.

x a c-closed x is the standard system of some model of PA, then

Let T be a consistent complete rich theory,

family of sets, and T

x.

E

If

x is the standard system of some recursively saturated model of 2’. In particular, there is a recursively saturated model of T with if x has cardinality at most s,, standard system x. Proof: Let M be a nonstandard model of PA with S S y ( M ) = x. Case 1 .

M

I= ConT

for some designation of T. By Mostowski’s Theorem (6.13)

we also have M (= ConpA for some encoding of PA. are consistent theories

so

Now, the argument that, if T1,TZ

is the theory of disjoint pairs of structures

modelling T I , T 2 , respectively, can be formalised within PA. Thus, M I= Con be a strongly definable model of produced by means of Theorem 6.10.

By Theorem 6.11,

is recursively saturated.

SSy() = SSy(A) = SSy(N).

By Lemma 6.12,

But, by Lemma 5.8,

we can regard N as an end extension

43

Lectureson Nonstandard Models of Arithmetic

.

of M, whence SSy(N) = SSy (M)

Thus, SSy ( A ) = SSy(M), as was to be proven.

M b ConT for any designation of T. I claim that some nonstandard

Case 2.

initial segment of M believes T to be- consistent. To find this, first apply the Lzwenheim-Skolem Theorem to obtain a countable nonstandard elementary submodel Mo

< M with T E SSy(Mo)

coded by, say, a E \ M o \ .

Expand the language of Mo to include a new predicate Tu defined by Tv: V Note that (M ,T) is merely a definitional expansion of Mo. 0

E

D;;.

Let L denote the

extension of the arithmetical language by the predicate Tu.

-

L

Now consider the theory T given by Th (M ) , PA , Con(Tu,V_
c

0

holds in ( N ,T) under an appropriate numerical interpretation of 0 is coded in S S y ( M o ) , whence it has a consistent completion TI

E

TuAV <

c allows us

Moreover,

TO

SSy(Mo).

Let (Ml,T) be a countable model of Tl with SSy(MI,T) = S S y ( M o ) . Friedman's embeddability criterion, observe that the bound

c.

Before applying

c in the formula

to replace this formula by a T-free formula @uZ, with new

parameter b , say, for which MI

b

Con

(@UZ)Z.

Now map MI isomorphically onto an initial segment of Mo and compose embeddings:

MI

ceM o <

M.

Apply the Splitting Theorem to get M

is a nonstandard initial segment of M, whence SSy($) Con(+vq,

i.e.

4fI=ConT for some definition

+UE

cc$f =

ceM.

S S y( M ) .

The model $f But $f

of the axioms of T, whence

&ED

Case 1 applies.

Application 6.16 is as much an application of the Splitting Theorem as it is one of the Arithmetised Completeness Theorem. There is a nice application of the Arithmetised Completeness Theorem which allows u s to avoid an appeal to the Splitting Theorem. This is Hamid Lesan's proof (Lesan 2 9 7 8 ) of Wilkie's Theorem 4.10

characterising the theories of end extensions of nonstandard models.

First, let me offer McAloon's statement of the Arithmetised Completeness Theorem: 6.17.

Theorem. Let M be a nonstandard model of PA and 2'2 PA a theory consistent

with the Infl-diagram of M, i.e. T is consistent with Th ( ( M ; u ) , E: 1 ~ 1 ) . If 'n+1 some axiomatisation of T is coded in S S y ( M ) , there is a strongly definable C elementary end extension N of M modelling T .

Proof sketch:

If T is consistent with the Xnfl-diagram of M, then, by a

hierarchical generalisation of Mostowski's Reflexiveness Theorem and the trick of Corollary 6 . 1 4 , M recognises this consistency. One now applies the Arithmetised Completeness Theorem to construct N modelling M and the I: -diagram of M. nf1 of course, satisfy more Zn+l-sentences, but on all I: -sentences, the two structures agree.

N may,

&ED

c. SMORYI~SKI

44

For full details, cf. McAZoon 1 9 7 8 4 . Now for Lesan's application: Let M be a nonstandard model of PA and T a consistent com-

6.18. Application.

plete extension of PA. ~

i.

I

~

The following are equivalent: (

M

C

~

N

~

T

)

R e p ( T ) E SSy(M).

ii.

The implication i => ii is immediate.

Proof:

To prove ii=> i, define a sequence M = M

C

M

C

... of models M

satisfying:

I= PA + nn+,(T) Mn ' x n Mn+l'

a. M

b. where n,+](T)

of T. This is done by noting TIn+z(T') + that

is the set of n,+]-theorerns

+ Th'n+1 ((M,;a) , . I I ) is consistent: If not, there is x E Cn+l such M, b x ~ ~ . . .while ~ ~ -nn+2(T) ~ , + PA 1 V V o . . . ~ m - l i po...V ,,,-I* But, v uo.. 1 pa.. E nn+l (T),whence Mn b B u g . . .urn-, l p o . . a PA

contra-

diction. Chqose N =

u

&ED

M,.

n The Arithmetised Completeness Theorem has other applications.

For some of these

cf. Smoryn'ski 1 9 7 3 A&B, Srnoryn'ski 1 9 7 7 , M c A t o o n 1 9 7 8 4 , and Srnoryn'ski 1982C.

7.

POWERFUL ARITHMETISATIONS (THE 1970s 111)

The Arithmetised Completeness Theorem is a fairly powerful tool.

As we saw, it

has many applications and it gives a variety of end extensions-- particularly, C -elementary ones.

But it does not seem to yield fully elementary end extensions.

Moreover, I see no way of obtaining, e.g., from it.

the fully formalised Ramsey Theorem

However, Ramsey's Theorem can be proven in PA and elementary end

extensions exist.

Moreover, these facts are intimately related.

The existence of an elementary end extension of any given nonstandard model M of PA was first proven in MacDowelZ and Specker 1 9 6 1 by an ultrapower construction:

Like Skolem before them, R. MacDowell and Ernst Specker took definable functions over M and made appropriate identifications by means of a carefully constructed ultrafilter on the algebra of definable sets. not a great deal of) care was needed:

In the countable case, some (but

Most ultrafilters would yield mixed

extensions and some will yield cofinal ones.

Also, in the countable case one can

simply appeal to a type-omitting argument (which uses Dirichlet's S c h u b f a e h p m h i p

--

i.e. the trivial case of Ramsey's Theorem).

The uncountable case requires a

great deal of uniformity because of the essentially countable nature of the construction.

45

Lectures on Nonstandard Models of Arithmetic In the early 1 9 7 0 s nonstandard modellists took a closer look at this uniformity.

Robert Phillips introduced the notion of a c o n s e r v a t i v e elementary end extension, his student Alan Cantor (in Cantor 1 9 7 2 ) proved No to have conservative proper elementary end extensions, Phillips (in PhiZZips 1 9 7 4 ) proved the existence in the general case, and Gaifman (in Gaifman 1 9 7 6 ) provided a nearly definitive treatment of the subject by introducing the notion of a d e f i n a b t e t y p e . 7.1.

Definition.

Let

Mc

N be models of PA.

We say N is a c o n s e r v a t i v e e x t e n -

sion of M if the restriction to 1M1 of every subset of IN1 parametrically definable in N is so definable in M. In the special case in which M is the standard model N o , an extension is conservative just in case its standard system consists of the arithmetic sets (or, for an extended language, those arithmetic in the new primitives).

If we generalise

from standardness to M-standardness, we can redefine conservativeness: Definitions. Let M 5 N be models of PA. A subset X 5 IMI is called an 7.1'. M-standard s e t of N, or more simply an M-set of N , if X is the restriction to lMl of a parametrically definable subset of N . The collection of M-sets of N is the M-standard s y s t e m , SSyM(N), of N . N is a c o n s e r v a t i v e e x t e n s i o n of M i f f SSYM(M) = SSYM(N).

We will not make full use of this conceptual breakthrough.

I merely wish to use

the notion of an M-set to explain conservativism. 7.2.

Theorem. i. ii.

Let M 5 N

be models of PA.

N is an end extension of M:

The following are equivalent:

M ce N

M and N have the same bounded (i.e. bounded in M) M-sets.

Moreover, each of these conditions implies: iii. lMI

7.3.

M and N have the same parametrically definable bounded subsets of

Corollary.

Let

MsN

be models of PA.

If the extension is conservative,

then it is an end extension. The corollary is immediate.

Proof of Theorem 7 . 2 :

i = > ii.

The proof is almost entirely analogous to

that slewing the standard systems of nonstandard models related by end extension to coincide:

If X

it has a code < 2c

E

SSyM(K), where K = M or N , is bounded by some c E (MI. I leave the details to the reader.

E

IMl , then

( This part is surprisingly easy and the reader might prefer to attempt the proof before reading further. ) Suppose M and N have the same bounded M-sets. Let b E I N ( - (MI and consider X = { a E (MI: N b a < % 1 . ii => i.

X

5 IMl

contains 0 and is closed under successor.

If b were M-finite, X would be

c. SMORYI~SKI

46

a bounded M-set of N , whence one of M--

whence X would be parametrically definable

in M.

But then induction in M would yield X = 1 M 1 , contradicting the boundedness

of X.

Hence, b is M-infinite. iii.

Since b

E

IN1 - IMI was arbitrary, M c

N.

&ED

Obvious.

I'm not sure whom to credit 7.2 and 7.3 to:

The proof of the elernentary case can

be found in Gaifman 2 9 7 6 , but, presumably, the result was also known to Phillips.

( As usual, I am separated from my library and am relying

on my shaky memory.

)

Theorem 7.2 seems not t o have been stated in print prior to 3noryfiski 1 9 8 1 A , even though it requires nothing new in the matter of proof.

Then again, the gain

of 7.2 over the elementary case of 7.3 is primarily heuristic:

It illustrates

nicely the difference in the r8les of cofinal and end extensions: The former add new definable sets; the latter can change truths.

( Remark: Using 7.2, one can see that the last paragraph of the proof of Lemma )

6.12 was unnecessary.

Anyway, for o u r present purposes, the importance of this digression is this: To construct an elementary end extension, it suffices to construct an elementary conservative extension.

I will present here Gaifman's method of doing this: Simply put, if M < K is given, and c E 1x1 - ] M I , there is a minimal extension of M in

K containing c-- namely, the model N generated over IMI

free Skolem functions.

U I c ] by the parameter-

Thus, it suffices to construct such an extension.

More-

over, because PA has built-in Skolem functions, constructing the model N amounts to constructing the type realised by c over M .

Guaranteeing the extension N will

be conservative is done by constructing a (locally) definab2.e type for c . For the sake of applications, it is handy to proceed in some generality:

7.4.

Definition. Let L be a countable language extending that of PA and let M be L Let Lf be the extension of L. by constants naming the various

a model of PA

.

elements of \MI. An L+-type (i.e. one naming all the elements of M )

Tv

over M is

a definubte type (perhaps better: a %ocalZy definabZe t y p e ) over M iff: F o r each L-formula

@ V ~ . . . V ~ -with ~

with only v o

u v O...Vn-Z

-

-

$vao.. .an-I E

only

v , v ~ , . . . , V ~ - ~ free,

,...,V n-1

TV

iff M

Thinking in terms of an element c

there is an L-formula

free such that, for all a.

,...,an-l

E

\MI,

I= u$Zo.. .Zn-3. E

IN1 realising

TU

in N

> M, TV

a

1M1 : N

is definable iff

(locally) each property of c is determined in M: N

k

iff MI=o$

$Cao...zn-2

o-*.a*-1. -

Switching roles of parameters and variables, if X X = { a

E

IMI : M

b

0

@

1

=

E

b

1 , then

and, if N is generated by M over c, the extksion is

readily seen to be conservative. So we see that the problem of producing elementary end extensions reduces to

47

Lectureson Nonstandard Models of Arithmetic producing definable types.

One more point:

We want the extension to be proper,

a

to include the formula V # for each a E /MI. Since T v is to be the type of an element c, TV is complete; since c will be M-infinite, it follows that

i.e.

TV

a

TV also includes v > for each a E IMI. Now, the one thing we know about M-infinite elements of elementary extensions is that they satisfy all properties

held by "almost all" elements of M and any property they satisfy are satisfied cofinally through M-- cf. e.g. Theorem 2.1.

-

Thus, in deciding the truth of

-

-

@ao.. .an-I ' the first question to ask is if M b B v o 3 v I > vo$vl~o.. .an-I. If - the answer is yes, we are free to put $Cao...an-I into the type TU as we are generating it.

There are only two problems:

avoiding clashes and being uniform.

But these obstacles are surmountable.

7.5.

Lemma.

Definable types exist.

Sketch of the construction: within PA

L

.

The idea is to ignore M and L

i

by working

Since M believes lMl to be w , a formal induction will work.

Since there are arithmetical pairing functions, it suffices to construct U $VV

0

with only

V,vo

V

$ 0 for free. To this end, let $ovvo,$IVvo,... be an enumeration of

L-formulae in these variables. The idea o f the proof is made clearer if we switch from formulae to sets and declare ourselves to be constructing a "definable ultrafilter" on the algebra of parametrically definable sets instead of constructing a type. $Vvo is taken to define, for each parameter

xa

=

c

b

E

(MI: M

a

E

Thus, a formula

[MI, a set

l='$E 1.

Since M believes itself to be standard and since we wish to work within P A , we

... of isuch. sets. ... .

think of '$ as defining a sequence Xo,XI,

sponding to the i-th formula ${, we denote by Xo,X?;, The crucial lemma is this:

The sequence corre-

. .

...

If X is a given unbounded definable set and Xi,X';,

is a given definable sequence of sets, then there is a definable unbounded.set

Y

5X

such that, for each n, Y is eventually inside X"

or disjoint from X i

,

i.e.

i is unbounded. Moreover, this is provable: . n i P A L L V nn(ltY n Xi is bounded" V "Y - Xn is bounded").

exactly one of Y

n X"

and Y - X

The provability in PAL follows from the fact that Y is defined via a recursion and the proof of the basic properties is inductive. So much for talking about the lemma! Let us now see how it is proven: Given X ; xi xi we define a sequetfceX.Yo ?YI 2 such that each Y n is unbounded

...

o,

and either Y c X: n-

and

or 'f,

n X; , if

=

@ by recursion:

this is unbounded

c. S M O R Y ~ K I

48

[ 1

if this is unbounded

Ynn X t + l ,

yn+l

=

Y

- x n+l,

otherwise.

L Although it is not immediately apparent, this recursion is admissible in PA

The desired set Y is simply the diagonal of this sequence:

I Yni: i

E

1, we have

w

segment { 0, ...,ynn

Y

ynn:

=

n

E

w }.

Writing Y

Note that, except for the initial

}, Y is either contained in X

i

or disjoint from it.

0 Now for the actual construction: Start with the enumerations X o , 0

,... and the set Z-l

I$,,$,

given by 0

and X o , X l ,

... to obtain Y

PAL

1Vn

("Z

n

= Zo

X

= w

bounded" v "Z

is 1

1

... .

Any definable set is of the form X = X i.e. define

TU

by:

OI$Vo:

(i.e. 1M1 in M).

Apply the lemma to Zm1

i

-X

is bounded").

Etc.

for some i , n ( n can be nonstandard when

Put X into the ultrafilter iff Z. n X" Z

If @Vv0is an L-formula, say

v u 1 3 v2 >

...; X1o , ...;...

satisfying

Then apply the lemma to Zo and X o , X l ,

interpreting the result in M).

.

=

0=

I$.VUo,

n

is unbounded,

then

Ul ( $ i ~ z A $ i V 2 ~ o ) ,

where Qi defines Z

i'

I fear I must leave the miserable verification that everything I claim holds to

QED

the energetic reader. Theorem. Let M ( = P A .

7.6.

Proof:

Then M has a proper elementary end extension.

QED

Immediate.

A few comments might be in order before we discuss applications: MacDowell and Specker's original construction was a sort of Skolem ultrapower-- one takes all parametrically definable functions d Zu Skolem, but carefully chooses one's ultrafilter, as in the proof of Theorem 7.5.

It turns out, after all the extra

work, that it is considerably harder to construct nonconservative elementary end extensions than conservative ones-

Kaufman's "rather classless model" (Kauftnan

2977)

provides an example of an uncountable model with no nonconservative exten-

sion.

For the countable case, cf. BZass 1 9 7 4 , PhiZZips 1974B, and Potthoff 1 9 7 6 .

( Exercise: There is a "natural" class of nonconservative elementary end extensions. Shew:

If M and iV are recursively saturated and N is an elementary

end extension of M, then the extension is not conservative. (Kotlarski)

)

As for applications, Gaifman, in Guifman 1 9 7 6 , shewed how the Ehrenfeucht-Mostowski Theorem on the existence of models generated by indiscernibles followed, without appeal to Ramsey's Theorem, from Lemma 7.5.

This is not too surprising:

The schematic derivability of Ramsey's Theorem in PAL is equivalent to the L In the last section, we saw existence of definable types over all models of PA

.

how the truth of Ramsey's Theorem followed from the mere existence of a conserva-

49

Lectures on Nonstandard Models of Arithmetic tive elementary end extension of No.

A minor variation of the proof (Exercise.)

shews that the schematic provability follows from the existence of such extensions of arbitrary nonstandard models.

For the converse, i.e. the construction of

definable types on the basis of the formalised Ramsey Theorem, cf. P i Z z a y 1981C.

( The schematic derivability of Ramsey's Theorem in PAL was established in Jockusch 1 9 7 2 .

Incidentally, Jockusch gives exact bounds on the arithmetic

complexities of the homogeneous sets in terms of those of the given partitions. I note that the proof via conservation yields explicit bounds, but confess I McAloon, in McAZoon 1978B,

haven't checked how close they come to Jockusch's. derives upper bounds model theoretically.

)

Another obvious application is Theorem 6.1:

SSy(No) is c-closed.

any proper conservative extension of No, SSy(M)

=

For, if M is

SSy(No) and we know from

Theorem 2.7 that SSy(M) is c-closed for nonstandard M. Other applications include generalisations to nonstandard models of anything we proved for No by appeal to appropriate infinite integers. The existence of

)I/-

1

like models, of course, follows as in ordinary model theory from the existence of elementary end extensions of countable models... and certain two-cardinal

I shall close this section with a delightful deriva-

theorems follow from this.

tion of a two-cardinal theorem.

7.7.

Let T be a theory in a language L possessing a unary predicate

Definition.

symbol U

.

If M is a model of T , we denote the interpretation of U in M by

For cardinal numbers of IMI and

7.8.

ff are

K

1. A,

K

A, respectively.

and

Application. Let T be a theory in a countable language L .

(Ho,n)-models for each n

Proof:

E

ff = {

w.

Then:

For any

K

2

A

O,...,n-I 1 .

If Ma is an (/!-'o,n)-model We can also expand M

of T, we can assume IM

model will satisfy Tn

=

T

+ PALi + v U(UV

By compactness, we get an (,~'o,h'o)-model: and add the axioms

c#

for all z

E W.

++

I

= w

to include the arithmetic :elations

and satisfy full induction in the augmented language, say L

n). Replace n in T

+.

In fact, the new

v <

by a new constant 7

Let M be a model of this new theory Tc.

/ ' Since M is an (,L)0,,50)-model, M expands to ( M ; F ) where F: 11'41

onto.

Suppose T has

.)I/,, T has a (K,A)-model.

Without loss of generality, we can assupe L disjoint from the

language of arithmetic. and

ff.

we say M is a (~,A)-modezof T if the cardinalities

+

ff is one-one and

Applying the Upward Lowenheim-Skolem Theorem toget an elementary extension

( N ; F ) of cardinality A, one also gets, by dropping F a (A,A)-model

Now iterate

K

of T,.

times the construction of elementary end extensions to get a

(K,A)-

model (Since U is interpreted by a bounded initial segment, no new elements occur at this time.) of T,.

Restricting this new model to the language L produces a

c. S M O R Y ~ S K I

so (K,X)-model of T.

&ED

The above proof is due to David Slater ( S l a t e r 1 9 7 2 ) .

The success of the argu-

ment may be a fluke; at least it is not clear how to obtain the more standard two-cardinal theorems by such an argument. 8.

DIVERSITY (THE 1980s)

It is still too early to attempt a discussion of the nonstandard model theory of the 1980s.

The only clear thing is the diversity the subject will display;

already one can discern several distinct directions: "hard" model theory, the relation to complexity theory, and the invasion of proof theory.

In addition to

these, there are, of course, continuations of programmes begun in the 1970s, when diversification began in earnest. It would exceed the scope (actually: the page limit) of this paper to go down any of these diverse paths any respectable distance.

In sections 1-7 I have presented

what I consider a minimal background and introduction to the study of nonstandard models of arithmetic.

I strongly recommend the reader next turn to the two

collections of expositions edited by Ken McAloon for further introduction to more recent trends.

Here, I wish merely to say a few unorganised words about some of

the things I've not discussed. My major omission has been the study of initial segments by Laurie Kirby and Jeff Paris (Kirby 2 9 7 7 , Kirby and Paris 1 9 7 7 , P a r i s 1 9 8 0 , and P a r i s 1 9 8 2 ; but cf. also P i Z z a y 2 9 8 2 B ) .

This study of combinatorially determined initial segments

modelling mere fragments of PA has had unexpected consequences-- most notably independence results in combinatorics (cf. e.g. P a r i s 1 9 7 8 , MeAZoon 1 9 7 9 , MeAZoon 1980A&B, Kirby 1 9 8 0 , Zbierski 1 9 8 0 , Tverskoy 1 9 8 0 , CZote 2 9 8 1 , von der Twer 1 9 8 2 ,

Friedman, MeAZoon and Simpson 1 9 8 2 , Ratajczyk 1 9 8 2 , Clote and M c A l o o n A , and

h d l a ' k A ) and in analysing the complexities of solutions to problems (cf. MeAZoon 2978B and CZote 1 9 8 1 ) .

Their work has even shed light on the study of elementary

end extensions (cf. Smoryiiski 1 9 8 1 C ) .

Fortunately, this development has been

amply documented and exposited, whence, though it is a major omission, it is not a serious one. A serious omission is the Kochen-Kripke notion of f u Z f i Z l a b i Z i t y .

In response to

Paris and Harrington's application of the work of Kirby and Paris to the problem of the combinatorial incompleteness of P A , Simon Kochen and Saul Kripke, in

Koehen and Kripke 1 9 8 2 , introduced an ultrapower-like construction useful for obtaining further such independence results.

Moreover, Kripke has derived from

this the ultrapower-free notion of fulfillability and he and Joseph Quinsey (the latter in his excellent thesis &u<nsey 1 9 8 0 ) demonstrated the general utility of fulfillability in the study of PA and its nonstandard models.

(I might note, for

example, that Quinsey has used fulfillability to detour the use of the Splitting

51

Lectures on Nonstandard Models of Arithmetic Theorem in establishing results in the uncountable case.)

When this material

becomes more widely exposited, it should find itself frequently applied. Very fashionable and used in the major omitted work of Kirby and Paris is their method of indicators.

Indicators are, as I announced back in section 2 , the

arithmetic manifestation of Strong Overspill.

Despite their fashionability and

an intuitive appeal, I am not much taken with indicators.

To my mind, i. they

fail as explanations of, say, the combinatorial incompleteness results because their whole machinery does not help construct the crucial indicators or prove them to be indicators, and ii. where they are successfully applied, they obscure more than they reveal.

The reader will have to either take my word for this or

dismiss my remark out of hand:

I have a lovely example, but am afraid to give it

because of the inevitable rumours about my personal vendettas; already I have heard it said that I "fight my battles fiercely" because I did not cite a modal logician in an algebra paper I wrote. Of course, I could be wrong about indicators in one area-- the study of models of subtheories of PA.

Back in the Preface I announced that the nonstandard pioneers

of the 1950s and 1960s proved theorems about strong models of arithmetic, i.e. elementary extensions of No, because they were not familiar with the extent to which such things could be formalised in PA.

In this complexity-conscious age,

the drift has been toward studying models of fragments ofPA (cf. e.g. Paris 1981,

Wilkie and Paris 1981, McAloon 1982, Gaifian and Lhitricopou1ou.s 1982, and

Paris and Dimitricopoulous 1 9 8 2 ) .

I am not familiar enough with this work to

determine if the use of indicators can be avoided. Another topic I've neglected that has assumed an increased interest is that of

nonstandard satisfafaction classes.

This notion originated with Abraham Robinson

(Robinson 1963) and was rejuvenated by S. Krajewski (Krajewski 1974A&B, m j e w s k i 1 9 7 6 ) and has begun to be energetically studied in Poland (cf. also KotZarski,

Krajewski and LachZan 1981, Lachlan 1981, and Ratajczyk 1 9 8 2 ) . I think I've said enough.

In the next section, I close by following a line that

does n o t go very far. 9.

A DEAD END

In contrast to initial segments, end segments of models of arithmetic have not been much studied.

In fact, they have not been studied at all.

Jensen and

Ekrenfeucht 1 9 7 6 speculates briefly on the idea that an end segment is somehow the "ultimate" arithmetic untouched by early irregularities and propose two problems (numbers 3 and 4 ) on end segments.

In February 1977 I solved these

problems in a simple and disappointing manner:

A nonstandard model of arithmetic

is uniquely determined by any of its end segments.

c. S M O R ~ S K I

52 9.1.

Theorem. Let M be a model of PA and suppose ]MI = I UE, where I is an

initial segment of M and E the end segment lMI recoverable from the structure E

-

I. Then: M is completely

(E;+,-).

=

The recovery is a variation of the construction of a field of quotients of an integral domain. The only subtle point is the means of singling out the allowable quotients. For, the formula, v0I1v1: 3 V Z ( V l = V0V2)’ only defines divisibility in the domain of quantification: For a , b

EbalIb

3 c

iff

E

E

E,

~(~bb=&).

For a proper definition of divisibility, one needs some intermediate notions: vo

((

a v (v

vl:

2 2

vo < v l :

svo

vo

((

v0I7J1: v

+ v2

=

vl)

I

vo <

v

=

B v

< u 1 (v2 - v 0 )

VlAVV2

I v (v + 0 1 1 0

I).

Note that < defines in E the restriction to E of the ordering of M, S the suc-

I

cessor function of M, and

the restriction to E of the divisibility relation of

M. With these, we can now “interpret“ M in E by:

M ( v o , v I ) : vl I v o (u

0’

v ) E (v

1

(Vo,V1)

(V,,V,)

2’

v ):

‘0’3

3

0 ( u 2 ,v 3 ) 0(v2,v3)

=

vlvZ

+ vlvz,~1v3) (vo”2,vIv3).

= (VOV3 =

The interpretation of M is: 9.2.

Lemma.

Let M

=

(M/Z;@/5,@/-)

via the map ( a , b ) / !

-+

{ (a,b) .-

E

E X E:

E

M(Z,F) 3 .

Then:

(IMl;+,*)

a / b , where ( )/E denotes the obvious quotient structure

modulo E. The proof of 9.2 is routine. From Lemma 9 . 2 , one quickly concludes Theorem 9.1.

Another corollary is the

following: 9.3.

Corollary. Let M ,M be models of PA with isomorphic end segments El.E2, 1 2 MI,M2 are isomorphic.

respectively. Then:

A closer look at the construction yields a few extra conclusions.

Combining the

interpretation of M in E with the coding properties of the standard systems of nonstandard models, one can prove the following: 9.4. Corollary. I plete Ill.

On the other hand:

The theory of nonstandard parts of nonstandard models is com-

Lectures on Nonstandard Models of Arithmetic 9.5.

53

Corollary. The theory of end segments of nonstandard models of PA is recur-

sively axiomatisable.

10. SUMMER READING LIST Before I begin, let me cite two books of particular value: Kenneth McAloon, Editor 1980

Mo&Zes de 11Arithm6tique, Aste'risque 73.

Chantal Berline, Kenneth McAloon and J.-P. Ressayre, Editors 1981

Model Theory and A r i t h m e t i c , Springer, Heidelberg.

These contain a number of expositions of more recent and advanced material than

I have covered in these lectures. As for the rest of the bibliography, a few words of explanation are in order:

I

have not listed abstracts unless the papers they abstracted did not appear. Unpublished papers that got absorbed into larger published works or which I doubted would ever get published or which I have not seen I have conservatively decided not to list.

But for this, I have been fairly liberal and have included

some papers cited in the body of my lectures which are not truly papers on models of arithmetic. There is no hope that the list is up-to-date, but it offers a decent introduction to the literature. Fred Abramson and Leo Harrington 1978

Models without indiscernibles, JSL 43, pp. 572-600.

A. Adler 1969

Extensions of nonstandard models of number theory, ZML 15, pp. 289-290.

Norman L. Alling 1965

Rings of continuous integer-valued functions and nonstandard arithmetic, Trans. AMS 118, pp. 498-525.

Jon Barwise and John Schlipf 1975

On recursively saturated models of arithmetic, in: D.H. Saracino and V.B. Weispfenning, eds., Mode2 Theory and Algebra; A Memorial !Tribute t o Abraham Robinson, Springer-Verlag, Heidelberg.

Chantal Berline 1981

Iddaux des anneaux de Pdano (d'apr'es CHERLIN), in: and Ressayre, cited above.

Berline, McAloon

Andreas Blass 1972

The intersection of nonstandard models of arithmetic, JSL 37, pp. 103106.

c. S M O R Y ~ K I

54 Andreas Blass (cont.) 1974

On certain types and models for arithmetic, J S L 39, pp. 151-162.

1977A

End extensions, conservative extensions, and the Rudin-Frolik ordering, Trans. AMS 225, pp. 325-340.

1977B

Amalgamation of nonstandard models of arithmetic, JSL 42, pp. 372-386.

1978

A model-theoretic view of some special ultrafilters, in: A. Macintyre, L. Pacholski and J. Paris, eds., Logic Colloquiwn '77, North-Holland, Amsterdam.

1980

Conservative extensions of models of arithmetic, Arch. f. math. Logik 20, pp. 85-94.

George Boolos 1973

A note on Beth's

theorem, Bull. Acad. Polon. Sci. 21, pp. 1-2.

Alan Cantor 1972

Minimal proper elementary extensions of N with respect to definable relations, Dissertation, Univ. S. Carolina.

Hans Georg Carstens 1972

uber die Kompliziertheit numerischer Modelle, Dissertation, Minster.

1975

Reducing hyperarithmetic sequences, Fund. Math. 89, pp. 5-11.

1976

0 A -Mengen, Arch. f. math. Logik 18, pp. 55-65. 2

Patrick Cegielski 1981

Theorie Elementaire de la multiplication des entiers naturels, in: Berline, McAloon and Ressayre, cited above.

Patrick Cegielski, Kenneth McAloon and George Wilmers 1982

Zoe'

Modsles re'cursivement saturgs de l'addition et de la multiplication des entiers naturels, in: D. van Dalen, D. Lascar and T.J. Smiley, eds., Logic Cozloquiwn '80, North-Holland, Amsterdam.

Chatzidakis

1981

La reprgsentation en termes de faisceaux des modiles de la th6orie glgmentaire de la multiplication des entiers naturels, in: Berline, McAloon and Ressayre, cited above.

Greg Cherlin 19758

Ideals of integers in nonstandard number fields, in: D.H. Saracino and V.B. Weispfenning, eds., Mode2 Theory and Algebra; A Memorial Tribute to Abrahmn Robinson, Springer-Verlag, Heidelberg.

1975B

Ideals in some nonstandard Dedekind rings, Logique et Analyse 18, pp. 349-406.

Lectures on Nonstandard Models of Arithmetic

55

Grigory Chudnovsky 1971

Certain arithmetical problems, (Russian), Kiev Univ. preprint.

Peter Clote 1980

Weak partition relations, finite games, and independence results in Peano arithmetic, in: L. Pacholski, J. Wierzejewski and A.J. Wilkie, eds., Model Theory of Algebra and Arithmetic, Springer-Verlag, Heidelberg.

1981

Anti-basis theorems and their relation to independence results in Peano arithmetic, in: Berline, McAloon and Ressayre, cited above.

Peter Clote and Kenneth McAloon A

Two further combinatorial theorems equivalent to the 1-consistency of Peano arithmetic,

Maryvonne Daguenet 1972

Un modsle non standard de l'arithmgtique, Compte Rendus 274, Serie A, pp. 685-688.

A. M. Dawes 1972

First-order hierarchies in general models and in models of Peano arithmetic, Dissertation, Toronto.

1977

End extensions which are models of a given theory, ZML 23, pp. 463-467.

S.D. Denisov 1972

Models of noncontradictory formulas and the Ershov hierarchy, (Russian), Alg. i Logika 11, pp. 648-655; English translation: Alg. and Logic 11 (1974), pp. 359-362.

Max Dickmann 1980

Types remarquables et extensions de modsles dans 1'arithm;tique Peano, I, Asterisque 73, pp. 59-117.

de

Lou van den Dries 1980

Some number theory and model theory of weak systems of arithmetic, in: L. Pacholski, J. Wierzejewski and A.J. Wilkie, eds., Mode2 Theory of AZgebra and Arithmetic, Springer-Verlag, Heidelberg.

Andrzej Ehrenfeucht 1973

Discernible elements in models for Peano arithmetic, JSL 38, pp. 291292.

Andrzej Ehrenfeucht and Georg Kreisel 1966

Strong models for arithmetic, Bull. Acad. Polon. Sci. 14, pp. 107-110.

Solomon Feferman 1958

Arithmetically definable models of formalized arithmetic, Notices AMS 5 , pp. 679-680.

c. SMORYE~SKI

56 Solomon Feferman (cont.) 1959

Models of arithmetic obtained from various sets of functions, Notices AMS 6, pp. 382-383.

1960

Arithmetization of metamathematics in a general setting, Fund. Math. 49, pp. 35-92.

Solomon Feferman and Georg Kreisel 1966

Persistent and invariant formulas relative to theories of higher order, B u l l . AMS 72, pp. 480-485.

Solomon Feferman, Georg Kreisel and Steven Orey 1962

1-consistency and faithful interpretations, Arch. f. math. Logik 6, pp. 52-63.

Solomon Feferman, Dana Scott and Stanley Tennenbaum 1959

Models of arithmetic through function rings, Notices AMS, pp. 173-174.

Harvey Friedman 1970

Iterated inductive definitions and $-AC,

in:

A . Kino, J. Myhill and

R. Vesley, eds., I n t u i t i o n i s m and Proof Theory, North-Holland, Amsterdam. 1973

Countable models of set theories, in: A.R.D. Mathias and H. Rogers, eds., Cambridge S m e r School i n Mathematical Logic, Springer-Verlag, Heidelberg.

1975

One hundred and two problems in mathematical logic, JSL 4 0 , pp. 113129.

Haim Gaifman 1967

Uniform extension operators for models and their applications, in: Crossley, ed., S e t s , Models and Recursion Theory, North-Holland, Amsterdam.

1970

On local arithmetical functions, and thier applications for constructing types of Peano's arithmetic, in: Y. Bar-Hillel, ed., Mathematical Logic and Foundations of S e t Tkeory, North-Holland, Amsterdam.

1972

A note on models and submodels of arithmetic, in: W. Hodges, ed., Conference i n Mathematical Logic-- London '70, Springer-Verlag, Heidelberg.

1976

Models and types of Peano's arithmetic, Annals math. Logic 9, pp. 223-

J.

306.

Haim Gaifman and Constantine Dimitracopoulous 1982

Fragments of Peano's arithmetic and the MRDP theorem, in: E. Engeler, H. Lzuchli and V. Strassen, eds., Logic and Algorithmic, L'Enseignement Mathgmatique, Geneva.

Lectures on Nonstandard Models of Arithmetic Robin 0. Gandy 1958

Note on a paper of Kemeny's, Math. Ann. 136, p. 466.

D.C. Goldrei, Angus Macintyre and Harry Simmons 1973

The forcing companions of number theories, Israel 3. Math. 1 4 , pp. 317337.

Andrzej Grzegorczyk 1971

An unfinitizability proof by means of restricted reduced power, Fund. Math. 73, p p . 37-49.

David Guaspari 1979

Partially conservative extensions of arithmetic, Trans. AMS 254, p p . 47-68.

Petr Hgjek 1971/ 1972

On interpretability in set theories I, 11, Comment. Math. Univ. Carol. 1 2 , pp. 73-79; 13, pp. 445-455.

1981A

Completion closed algebras and models of Peano arithmetic, Comment. Math. Univ. Carol. 22, pp. 585-594.

1981B

On interpretability in theories containing arithmetic 11, Comment. Math. Univ. Carol. 22, pp. 667-688.

Petr Hdjek and Pave1 Pudlsk 1980

Two orderings of the class of all countable models of Peano arithmetic,

in: L. Pacholski, J. Wierzejewski and A.J. Wilkie, eds., Model Theory of Algebra and A r i t h m e t i c , Springer-Verlag, Heidelberg. Gisbert Hasenjsger 1953

Eine Bemerkung zu Henkin's Beweis fur die Vollstzndigkeit des Prgdikatenkalkcls der ersten Stufe, JSL 18, pp. 42-48.

Kurt Hauschild 1963

ffber die Charakterisierbarkeit der Zahlenreihe in gewissen Nichtstandardmodellen der Arithrnetik, ZML. 9 , pp, 113-116.

1971

Nichtaxiomatisierbarkeit von Satzmengen durch Ausdrzcke spezieller Gestalt, Fund. Math. 72, pp. 245-253.

Leon Henkin 1950

Completeness in the theory of types, J S L 15, pp. 81-91.

G. Hensel and Hilary Putnam 1969

Normal models and the field

Z;, Fund. Math. 64, pp. 231-240.

Joram Hirschfeld 1972

Existentially complete and generic structures in arithmetic, Disserta-

c. S M O R Y ~ K I

58 Joram Hirschfeld (cont.) tion, Yale. 1974

Models of arithmetic and the semi-ring of recursive functions, in: A . Hurd and P. Loeb, eds., Vietoria Symposim on Nonstandard Analysis, Springer-Verlag, Heidelberg.

19758

Finite forcing and generic filters in arithmetic, in: D.H. Saracino and V.B. Weispfenning, eds., Model Theory and Algebra; A Memorial Tribute t o Abraham Robinson, Springer-Verlag, Heidelberg.

1975B

Models of arithmetic and recursive functions, Israel J. Math. 20, pp. 111-126.

Joram Hirschfeld and William Wheeler 1975

Forcing, Arithmetic and Division Rings, Springer-Verlag, Heidelberg.

Paul Howard 1972

A proof of a theorem of Tennenbaum, ZML 18, pp. 111-112.

Don Jensen and Andrzej Ehrenfeucht 1976

Some problem in elementary arithmetics, Fund. Math. 92, pp. 223-245.

R.G. Jeroslow 1971

Non-effectiveness in S. Orey's arithmetical compactness theorem, ZML 1 7 , pp. 285-289.

Carl G. Jockusch 1972

Ramsey's theorem and recursion theory, JSL 37, pp. 268-280.

1974

Il1 classes and boolean combinations of recursively enumerable sets,

0

JSL 39, pp. 95-96.

Carl G. Jockusch and Robert I. Soare 0

19728

II1 classes and degrees of theories, Trans. AMS 173, pp. 33-56.

1972B

Degrees of members of JIo classes, Pac. J. Math. 40, pp. 605-616. 1

Matt Kaufman 1977

A rather classless model, Proc. AMS 6 2 , pp. 330-333.

Jean Pierre Keller and Denis Richard 1978

Remarques sur les structures additives des modzles de l'arithmgtique, Compte Rendus 287, Serie A , pp. 101-104.

J.G. Kemeny 1958

Undecidable problems of elementary number theory, Math. Ann. 135, pp. 160-169.

Lectures on Nonstandard Models of Arithmetic

59

Jussi Ketonen and Robert M. Solovay 1981

Rapidly growing Ramsey functions, Annals of Math. 113, pp. 267-314.

Laurie Kirby 1977

Initial segments of models of arithmetic, Dissertation, Manchester.

1980

La mgthod des indicatrices et le th6orsme d'incomplgtude, Astdrisque 73, pp. 5-18.

1982

Flipping properties in arithmetic, JSL 47, pp. 416-422.

Laurie Kirby, Kenneth McAloon and Roman Murawski 1981

Indicators, recursive saturation and expandability, Fund. Math. 114, pp. 127-139.

Laurie Kirby and Jeff Paris 1977

Initial segments of models of Peano's axioms, in: A.H. Lachlan, M. Srebrny and A. Zarach, eds., S e t Theory and Hierarchy Theory V, Springer-Verlag, Heidelberg.

Steven C. Kleene 19528

Two papers on the predicate calculus, Memoirs AMS 10.

1952B

Introduction t o Metmathematics, van Nostrand, Princeton.

Julia Knight 1973

Complete types and the natural numbers, JSL 38, pp. 413-415.

1975

Types omitted in uncountable models of arithmetic, JSL 40, pp. 317-320.

19768

Omitting types in set theory and arithmetic, JSL 41, pp. 25-32.

1976B

Hanf numbers for omitting types over particular theories, JSL 41, pp. 583-588.

A

Additive structure in uncountable models for a fixed completion of P,

Julia Knight and Mark Nadel 19828

Expansions of models and Turing degrees, JSL 47, pp. 587-604.

1982B

Models of arithmetic and closed ideals, JSL 4 7 , pp. 833-840.

Simon Kochen and Saul Kripke 1982

Nonstandard models of Peano arithmetic, in: E. Engeler, H. Lguchli and V. Strassen, eds., Logic and AZgorithmic, L'Enseignement MathGmatique, Geneva; and: L'Enseignement Mathgmatique 28 ( 1 9 8 2 ) , pp. 211-231.

Roman Kossak 1980

An application of definable types of Peano's arithmetic, Bull. Acad. Polon. Sci. 28, pp. 213-217.

A

A certain class of models of Peano arithmetic,

c. SMORYI~SKI

60 Roman Kossak (cont.) B

L

C

A note on satisfaction classes,

-,q-elementary

equivalence of w -like models of PA, 1

Henryk Kotlarski 1980

On Skolem ultrapowers and their non-standard variant, ZML 26, pp. 227-

236.

A

On elementary recursively saturated cuts in models of PA, I-IV,

B

On cofinal extensions of models of arithmetic,

Henryk Kotlarski, S. Krajewski and Alisdair H. Lachlan 1981

A construction of satisfaction classes for nonstandard models, Canad. Math. Bull. 24, pp. 295-297.

S . Krajewski

19748

Predicative expansions of axiomatic theories, ZML 20, pp. 435-452.

1974B

Mutually inconsistent satisfaction classes, B u l l . Acad. Polon. Sci. 22, pp. 883-887.

1976

Nonstandard satisfaction classes, in: W. Marek, M. Srebrny and A . Zarach, eds., S e t Theory and. Hierarchy Theory, Springer-Verlag, Heidelberg.

Georg Kreisel 1950/ 1951

Note on arithmetic models for consistent formulae of the predicate calculus, Fund. Math. 37, pp. 265-285.

1953

Note on arithmetic models for consistent formulae of the predicate calculus, 11, in: Actes du XIemeCongrBsInternational de PhiZosophie, North-Holland, Amsterdam.

1968

Survey of proof theory, JSL 33, pp. 321-388.

Alisdair H. Lachlan 1981

Full satisfaction classes and recursive saturation, Canad. Math. Bull: 24, pp. 271-292.

D. Lascar 1980

Une indicatrice de type "Ramsey" pour l'arithmgtique de Peano et le formule de Paris-Harrington, Astgrisque 73, pp. 19-30.

Hamid Lesan 1978

Models of arithmetic, Dissertation, Manchester.

Leonard Lipshitz 1979

Diophantine correct models o f arithmetic, Proc. AMS 73, pp. 107-108.

61

Lectures on Nonstandard Models of Arithmetic Leonard Lipshitz and Mark Nadel 1978

The additive structure of models of arithmetic, Proc. AMS 68, pp. 331336.

R. MacDowell and Ernst Specker 1961

Modelle der Arithmetik, in: I n f i n i t i s t i c Methods, Pergamon, London.

Angus Macintyre 1980

Ramsey quantifiers in arithmetic, in: L. Pacholski, J. Wierzejewskj and A.J. Wilkie, eds., Model Theory of Algebra and Arithmetic, Springer-Verlag, Heidelberg.

1981

The complexity of types in field theory, in: M. Leman, J. Schmerl and R. Soare, eds., Logic Y e a r 1979-80, Springer-Verlag, Heidelberg.

1982

Residue fields of models of P, in: L.J. Cohen, J. Lo;, H. Pfeiffer and K. -P. Podewski, eds. , Logic, Methodo Zogy and Phi Zosophy of Science V I , North-Holland, Amsterdam.

Angus Macintyre and David Marker A

Degrees of recursively saturated models,

Angus Macintyre and Harry Simmons 1975

Algebraic properties of number theories, Israel J. Math. 22, pp. 7-27.

Attila Mate' A

Nondeterministic polynomial time computations and nonstandard models of arithmetic,

Larry Manevitz 1976

Internal end-extensions of Peano arithmetic and a problem of Gaifman, J. London Math. SOC. (2) 13, pp. 80-82.

Larry Manevitz and Jonathan Stavi 1980

0 operators and alternating sentences in arithmetic, JSL 45, pp. 1442 154.

A

Wiktor Marek and Marian Srebrny 1976

Urelements and extendability, in: W. Marek, M. Srebrny and A. Zarach, eds., Set Theory and Hierarchy Theory, Springer-Verlag, Heidelberg.

Kenneth McAloon 1977

Consistency statements and number theories, in: CoZZoque International de Logique, CNRS, Paris.

19788

Completeness theorems, incompleteness theorems and models of arithmetic, Trans. AMS 239, pp. 253-277.

1978B

Diagonal methods and strong cuts in models of arithmetic, in: A . Macintyre, L. Pacholski and J. Paris, eds., Logic Cozzoquim ' 7 7 , North-

M. Guillaume, ed.,

c. S M O R Y ~ K I

62 Kenneth McAloon (cont.) Holland, Amsterdam. 1979

Paris et the'or'eme d'incomplgtude (d'apres Forms combinatoires d'autres), in: Semimire Bourbaki V O Z . ' 1 9 7 7 / 1 9 7 8 E h o s e s 507-524, Springer-Verlag, Heidelberg.

1980A

La rapport entre la mgthode de Ggdel et la mgthode des indicatrices pour obtenir des rgsultats d'independence, Astgrisque 73, pp. 31-39.

1980B

Progressions transfinies de thgories axiomatiques, formes combinatoires du thgor'eme d'incomplgtude et fonctions recursives ?I croissance rapide, Astgrisque 73, pp. 41-58.

1982

On the complexity of models of arithmetic, JSI. 47, pp. 403-415.

Elliot Mendelson 1961

On non-standard models for number theory, in: Y. Bar-Hillel, ed., Essays on the Foundations o f Mathematics, Magnes Press, Jerusalem.

iarko Mijajlovi6 A

Submodels and definable points in models of Peano arithmetic,

George Mills 1977 1978

Extensions of models of Peano arithmetic, Dissertation, Berkeley. A model of Peano arithmetic with no elementary end extension, JSL 43, p p . 563-567.

1979

Substructure lattices of models of arithmetic, Annals Math. Logic 16, pp. 145-180.

Didier Misercque 1980

Sur le trellis des V 1-formules fermges de l'arithmgtique de Peano, Compte Rendus 290, Serie A, pp. 571-573.

1982

The nonhomogeneity of the E-tree-- answer to a problem raised by D. Jensen and A. Ehrenfeucht, Proc. AMS 84, pp. 573-575.

Josef Mlzek 1976

Twin prime problem in arithmetic without induction, Comment. Math. Univ. Carol. 17, pp. 543-555.

1978A

End extensions of countable structures and the induction schema, Comment. Math. Univ. Carol. 19, pp. 291-308.

1978B

A note on cofinal extensions and segments, Comment. Math. Univ. Carol. 19, p p . 727-742.

Carl Morgenstern 1982

On generalized quantifiers in arithmetic, JSL 47, p p . 187-190.

Lectures on Nonstandard Models of Arithmetic

63

Andrzej Mostowski 1952A

On models of axiomatic systems, Fund. Math. 39, pp. 133-158.

1952B

On direct products of theories, JSL 1 7 , pp. 1-31.

1957

On recursive models of formalized arithmetic, Bull. Acad. Polon. Sci. 5 , pp. 705-710.

Nobuyoshi Motohashi A

Preservation theorem and relativization theorem for cofinal extensions,

B

A normal form theorem for first order formulas and its application to

Gaifman's splitting theorem, Gert H. Msller 1961

Nicht-Standardmodelle der Zahlentheorie, Math. Zeitschr. 77, pp. 414438.

Roman Murawski 19761 1977

On expandability of models of Peano arithmetic I, 11, 111, Studia Logica 35, pp. 409-419; 35, pp. 421-431; 36, pp. 181-188; Correction: Studia Logica 36.

1978

Indicators, satisfaction classes and expandability, preprint, Poznan.

1979

Indicators and the structure of expansions, preprint, Poznan.

1980

Some remarks on the structure of expansions, ZML 26, pp. 537-546.

1981

Incompleteness of Z -definable theories via indicators, Func. et Approx. 11, pp. 57-63.

A

A simple remark on satisfaction classes, indiscernibles and recursive

0

saturation, Mark Nadel 1980

On a problem of MacDowell and Specker, JSL 45, pp. 612-622.

A

The completeness of Peano multiplication,

Steven Orey 1961

Relative interpretations, ZML 7 , pp. 156-153.

F. Pabion 1982

Saturated models of Peano arithmetic, JSL 47, pp. 625-637.

Jeff Paris 1972

On models of arithmetic, in: W. Hodges, ed., Conference in Mathematic a l Logic-- London '70, Springer-Verlag, Heidelberg.

1977

Models of arithmetic and the 1-3-1 199.

lattice, Fund. Math. 95, pp. 195-

c. SMORYI~SKI

64

Jeff Paris (cont.) 1978A

Note on an induction axiom, J S L 43, pp. 113-117.

1978B

Some independence results for Peano arithmetic, JSL 43, pp. 725-731.

1980

A hierearchy of cuts in models of arithmetic, in: L. Pacholski, J. Wierzejewski and A.J. Wilkie, eds., Model Theory of AZgebra and Arithmetic, Springer-Verlag, Heidelberg.

1981

Some conservation results for fragments of arithmetic, in: McAloon and Ressayre, cited above.

Berline,

Jeff Paris and Constantine Dimitracopoulous A

Truth definitions for A.

formulae,

Jeff Paris and Leo Harrington 1977

A mathematical incompleteness in Peano arithmetic, in: J. Barwise, ed., Handbook of Mathematical Logic, North-Holland, Amsterdam.

Jeff Paris and Laurie Kirby 1978

X

-

collection schemas in arithmetic, in: A. Macintyre, L. Pacholski

and J. Paris, eds., Logic Colloquium '77, North-Holland, Amsterdam. Jeff Paris and George Mills 1979

Closure properties of countable non-standard integers, Fund. Math. 103, pp. 205-215.

Robert G. Phillips 1971

On the structure of nonstandard models of arithmetic, Proc. AMS 27, pp. 359-363.

1972

Addition in nonstandard models of arithmetic, JSL 37, pp. 483-486.

19748

Omitting types in arithmetic and conservative extensions, in: A. Hurd and P. Loeh, eds., V i c t o r i a Symposium on Nonstandard Analysis, Springer-Verlag, Heidelberg.

1974B

A minimal extension that is not conservative, Mich. Math. J. 21, pp. 27-32.

Anand Pillay 1981A

Models of Peano arithmetic (a survey of basic results), in: McAloon and Ressayre, cited above.

Berline,

1981B

Cuts in models of arithmetic, in: above.

1981C

Partition properties and definable types in Peano arithmetic, in: Berline, McAloon and Ressayre, cited above.

Berline, McAloon and Ressayre, cited

Klaus Potthoff 1967

Untersuchungen Eber Nichtstandardmodelle, Dissertation, Hannover.

Lectureson Nonstandard Models of Arithmetic

65

Klaus Potthoff (cont.) 1969

Uber Nichtstandardmodelle der Arithmetik und der rationalen Zahlen, ZML 15, pp. 223-236.

1970

Ideale in Nichtstandardmodellen der ganzen Zahlen, ZML 16, pp. 321326.

1972

Ordnungseigenschaften von Nichtstandardmodellen, in: G. Asser, J. Flachsmeyer and W. Rinow, eds., Theory of S e t s and Topology; I n Honour of Felix Hausdorff, VEB, Berlin.

1976

A simple tree lemma and its application to a counterexample of Phillips, Arch. f. math. Logik 18, pp. 67-71.

1978

Orderings of types of countable arithmetic, ZML 24, pp. 97-108.

M. Presburger 1930

uber die Vollsthdigkeit eines gewissen Systems der Arithmetik ganzer Zahlen, in welchem die Addition als einzige Operation hervortritt, in: Comptes Rendus I Congrzs des Maththaticiens des Pays Slaves, Warsaw.

Pavel PudlCk A

Another combinatorial sentence independent of Peano's axioms,

Pavel Pudlhk and Antonin Sochor A

Models of the Alternative Set Theory,

C. Puritz 1971

Ultrafilters and standard functions in non-standard arithmetic, Proc. London Math. SOC. ( 3 ( 22, pp. 705-733.

Hilary Putnam 1965

Trial and error predicates and the solution to a problem of Mostowski, JSL 30, pp. 49-57.

Joseph Quinsey 1980

Some problems in logic, Dissertation, Oxford.

Michael 0. Rabin 1961

Non-standard models and the independence of the induction axiom. in: Y. Bar-Hillel, ed., Essays on t h e Foundations of bkzthernatics, Magnes Press, Jerusalem.

1962

Diophantine equations and non-standard models of arithmetic, in: E. Nagel, P. Suppes and A . Tarski, eds., Logic, Methodology and Philosophy of Science, Stanford.

2. Ratajczyk

1982

Satisfaction classes and combinatorial sentences independent from arithmetic, ZML 28, pp. 149-165.

c. S M O R Y ~ S K I

66 J.-P. Ressayre 1980

Types remarquable et extensions de modsles dans l'arithmgtique de Peano 11, (with appendix by M. Dickmann), Astgrisque 73, pp. 119-154.

Denis Richard 1977

On external properties of nonstandard models of arithmetic, Publ. Dep. Math. (Lyon) 14, no. 4, pp. 57-75.

1980

Saturation des modsles de Peano, Compte Rendus 290, Serie A, pp. 351353.

1981

De la structure additive 'a la saturation des modsles de Pgano et 'a une classification des sous-langages de l'Arithmectique, in: Berline, McAloon and Ressayre, cited above.

Abraham Robinson

C.

1963

On languages which are based on nonstandard arithmetic, Nagoya Math. J. 22, pp. 83-117.

1971

Infinite forcing in model theory, in: J.E. Fenstad, ed., Proceedings of t h e Second Scandinavian Logic Symposim, North-Holland, Amsterdam.

1973

Nonstandard arithmetic and generic arithmetic, in: P. Suppes, L. Henkin, G . C . Moisil and A. Joja, eds., Logic, Methodology and PhiZosophy o f Science I V , North-Holland, Amsterdam.

1961

Model theory and non-standard arithmetic, in: Pergamon, London.

Infinitistic Methods,

Ryll-Nardzewski 1952

The role of the axiom of induction in elementary arithmetic, Fund. Math. 39, pp. 239-263.

James Schmerl 1973

Peano models with many generic classes, Pac. J. Math. 46, pp. 523-536; Errata: Pac. J. Math 92 (1981), pp. 195-198.

1978

Extending models of arithmetic, Ann. Math. Logic 14, pp. 89-109.

1981

Recursively saturated rather classless models of Peano arithmetic, in:

M. Lerman, J. Schmerl and R. Soare, eds., Logic Year 1979-80, Springer-Verlag, Heidelberg. A

Recursively saturated models generated by indiscernibles,

James Schmerl and Steven G . Simpson 1982

On the role of Ramsey quantifiers in first order arithmetic, JSL 47, pp. 423-235.

Dana Scott 1961

On constructing models for arithmetic, in: Pergamon, London.

Infinitistic Methods,

Lectures on Nonstandard Models of Arithmetic

67

Dana Scott (cont.) 1962

Algebras of sets binumerable in complete extensions of arithmetic, in: J. Dekker, ed., Recursive Function IPheory, A M S , Providence.

Saharon Shelah 1978A

End extensions and numbers of countable models, JSL 43, pp. 550-562.

1978B

Models with second order properties 11. trees with no undefined branches, Ann. Math. Logic 14, pp. 73-87.

John Shepherdson 1964

A non-standard model for a free-variable fragment of number theory, Bull, Acad. Polon. Sci. 12, pp. 79-86.

1965

Non-standard models for fragments of number theory, in: L. Henkin, P. Suppes and A. Tarski, eds., The Theory of Models, North-Holland, Amsterdam.

Joseph Shoenfield 1960

Degrees of models, JSL 25, pp. 233-237.

Harry Simmons 1976

Each regular number structure is biregular, Israel J. Math. 23, pp. 347-352; Correction: Israel J. Math. 26 (1977), p. 95.

1977

Existentially closed models of basic number theory, in: R. Gandy and M. Hyland, eds., Logic CoZZoquizim ‘76, North-Holland, Amsterdam.

Steven G. Simpson 1974

Forcing and models of arithmetic, Proc. AMS 43, pp. 193-194.

Thoralf Skolem 1929

uber einige Grundlagenfragen der Mathematik, Skrifter Vitensk. i Oslo I, no. 4 , pp. 1-49.

1930

uber einige Satzfunktionen in der Arithmetik, Skrifter Vitensk. Oslo I, no. 7, pp. 1-28.

1933

uber die UnmEglichkeit einer vollstzndigen Charakterisierung der Zahlenreihe mittels eines endlichen Axiomensystems, Norsk Mat. Forenings Skrifter, Ser. 11, nr. 1-12, pp. 73-82.

1934

Uber die Nichtcharakterisierbarkeit der Zahlenreihe mittels endlich oder abzghlbar unendlich vielen Aussagen mit ausschliesslich Zahlvariablen, Fund. Math. 23, pp. 150-161.

1955

Peano’s axioms and models of arithmetic, in: Mathematical Interpretat i o n s of Formal Systems, North-Holland, Amsterdam.

David Slater 1972

Non-standard languages and thier applications, Dissertation, Rutgers.

c. S M O R Y ~ S K I

68 C. Smoryikki 1973A

Investigation of intuitionistic formal systems by means of Kripke models, Dissertation, Chicago Circle.

1973B

Applications of Kripke models, in: A. Troelstra, ed., Metamathematical Investigation of I n t u i t i o n i s t i c Arithmetic and Analysis, SpringerVerlag, Heidelberg.

1977

The incompleteness theorems, in: J. Barwise, ed., Handbook of Mathematical Logic, North-Holland, Amsterdam.

1981A

Cofinal extensions of nonstandard models of arithmetic, Notre Dame J. Formal Logic 22, pp. 133-144.

1981B

Recursively saturated nonstandard models of arithmetic, J S L 46, pp. 259-286; Addendum: J S L 47 (1982), pp. 493-494.

1981C

Elementary extensions of recursively saturated models of arithmetic, Notre Dame J. Formal Logic 22, pp. 193-203.

19828

A note on initial segment constructions in recursively saturated models of arithmetic, Notre Dame J. Formal Logic 23, pp. 393-408.

1982B

Back-and-forth inside a recursively saturated model of arithmetic, in: D. van Dalen, D. Lascar and T. Smiley, eds., Logic Colloquiwn '80, North-Holland, Amsterdam.

1982C

Nonstandard models and constructivity, in: A. Troelstra and D. van Dalen, eds., L.E.J. Brouwer Centenary Symposium, North-Holland, Amsterdam.

C. Smory6ski and Jonathan Stavi 1980

Cofinal extension preserves recursive saturation, in:

L. Pacholski,

J. wierzejewski and A. Wilkie, eds., Mode2 Theory of Algebra and

Arithmetic, Springer-Verlag, Heidelberg. Robert M. Solovay A S.

On interpretability in set theories,

F. Soprunov 19758

Strong nonstandard models of arithmetic, (Russian), Dokl. AN SSSR 220, pp. 293-296; English translation: Sov. Math. Doklady 16 (1975), pp. 80-84.

1975B

Initial segments of nonstandard arithmetics, (Russian), Dokl. AN SSSR 223, pp. 576-577; English translation: Sov. Math. Doklady 16 (1975). pp. 968-970.

1976

Countable nonstandard models of arithmetic, (Russian), in: D.A. Bocvar and V.N. Grisin, eds., Studies i n S e t Theory and Nonclassical Logics, (Russian), Izdat. "Nauka", Moscow; English translation: Selecta Math. Sovietica 1 (1981), pp. 303-315.

1979

Lattices of nonstandard arithmetic, (Russian), in: Studies in NoncZassica1 Logic and S e t Theory, (Russian), Izdat. "Nauka", Moscow.

Lectureson Nonstandard Models of Arithmetic

69

Ernst Specker 1957

Eine Verschzrfung des Unvollst&digkeitssatzes Bull. Acad. Polon. Sci. 5 , pp. 1041-1045.

der Zahlentheorie,

Teodor Stihi 1976

Forcing and Peano arithmetic, Dissertation.

1978

Induction and complete sequences of conditions in the sense of Cohen, Rev. Roum. Math. Pures et Appl. 23, pp. 129-133.

Stanley Tennenbaum 1959

Non-archimedean models for arithmetic, Notices AMS 6 , p. 270.

Akito Tsuboi 1982

On M-recursively saturated models of arithmetic, Tsukuba J. Math. 6 , pp. 305-318.

A.A. Tverskoy 1980

A sequence of combinatorial judgements which are independent of Peano

arithmetic, Vestnik Moskov. Univ. Matem. 35, no. 5, pp. 7-13; English translation: Mosc. Univ. Math. Bull. 35, no. 5 (1980), pp. 6-13. Tassilo von der Twer 1981

Some remarks on the mathematical incompleteness of Peano's arithmetic found by Paris and Harrington, in: R. Jensen and A. Prestel, eds., S e t Theory and Model Theoi-y, Springer-Verlag, Heidelberg.

Jouko Vyingnen 1979

A new incompleteness in arithmetic, (Finnish, English summary), Arkhimedes 31, pp. 30-37.

Hao Wang 1951A

Arithmetic translations of axiom systems, Trans. AMS 71, pp. 283-293.

1951B

Arithmetic models for formal systems, Methodos 3, pp. 217-232.

Alec J. Wilkie 1975

On models of arithmetic-- answers to two problems raised by H. Gaifman, J S L 40, pp. 41-47.

19778

On models of arithmetic with non-modular substructure lattices, Fund. Math. 95, pp- 223-238.

1977B

On the theories of end-extensions of models of arithmetic, in: A. Lachlan, M. Srebrny and A. Zarach, eds., S e t Theory and Model Theory V, Springer-Verlag, Heidelberg.

1978

Some results and problems on weak systems of arithmetic, in: A. Macintyre, L. Pacholski and J. Paris, eds., Logic ColZoquhm '77, North-Holland, Amsterdam.

c. S M O R Y ~ ~ S K I

70

Alec J . Wilkie (cont.) 1980

Applications of complexity theory to 1 -definability problems in

0

arithmetic, in: L. Pacholski, J. Wierzejewski and A. Wilkie, eds., Model Theory of Algebra and A r i t h e t i c , Springer-Verlag, Heidelberg. 1981

On discretely ordered rings in which every definable ideal is principal, in: Berline, McAloon and Ressayre, cited above.

1982

On core structures for Peano arithmetic, in: D. van Dalen, D. Lascar and T. Smiley, eds., Logic Colloquiwn '80, North-Holland, Amsterdam.

A

On the arithmetical parts of strong theories, unpublished.

Alec 3. Wilkie and Jeff Paris 1981

Models of arithmetic and the rudimentary sets, Bull. SOC. Math. Belg. Ser. B 33, pp. 157-169.

George Wilmers 1980

Minimally saturated models, in: L. Pacholski, 3. Wierzejewski and A. Wilkie, eds., Model Theory of Algebra and Arithmetic, SpringerVerlag, Heidelberg.

George Zahn 1971

On the additive structure of non-standard models of arithmetic, Dissertation, Univ. S. Carolina.

Pawel Zbierski 1980

Indicators and incompleteness of Peano arithmetic, Acta. Cient. Venezolana 31, pp. 487-495.

After typing the above, I had a chat with David Marker, who gave me the following additional items: Julia Knight, Alisdair Lachlan and Robert I. Soare A

Two theorems on degrees of models of true arithmetic,

Henryk Kotlarski A

Automorphisms of recursively saturated models of PA,

David Marker 1982

Degrees of models of true arithmetic, in: loquiwn ' 8 1 , North-Holland, Amsterdam.

Jeff Paris and Alec J. Wilkie A

A -sets and induction,

0

George Wilmers A

Bounded existential induction,

J . Stern, ed., Logic Coz-

LOGIC COLLOQUIUM '82 G. Lolli, G. Long0 and A . Marcia (editors] 0 Elsevier Science Publishers B. V. (North-Holland), I984

71

STRONG SATURATION A N D THE FOUNDATIONS OF STABILITY THEOPY

John T. B a l d w i n Department o f Kathematics, S t a t i s t i c s and Computer Science U n i v e r s i t y o f I l l i n o i s a t Chicago

The n o t i o n o f n o n - f o r k i n g i s one o f t h e p r i n c i p a l t o o l s o f s t a b i l i t y t h e o r y . Since b e i n g i n t r o d u c e d b y Shelah L197E1,

t h e r e have been a number o f e o u i v a l e n t

d e f i n i t i o n s o f t h e concept: i n t e r m s o f t h e fundamental o r d e r i n Lascar-Poizat w i t h v a r i a n t s stemming from an i d e a o f Z i e g l e r i n Balduin-198x and and i n t e r m s o f r a n k i n Shelah-197E and Lascar C19761.

Harnik-Harrington-19?2

We

b e g i n h e r e b y g i v i n g an o p e r a t i o n a l d e f i n i t i o n o f n o n - f o r k i n g as a r e l a t i o n s a t i s f y i n g c e r t a i n axioms.

Then a l l t h e o t h e r d e f i n i t i o n s can be viewed as p r o o f s we w i l l show i n

o f t h e e x i s t e n c e o f a r e l a t i o n s a t i s f y i n g t h e axioms.

Poreover,

t h e s p i r i t o f Lascar ( c f .

t h a t c e r t a i n o f these

a l s o Harnik-Harrington-19?2)

axioms a r e c a t e g o r i c a l and t h u s c o n c l u d e t h e i d e n t i t y o f t h e v a r i o u s n o t i o n s .

Our

axioms a r e a n o t h e r approach t o t h e problem o f a s s i g n i n g a d i m e n s i o n t o a mathematical s t r u c t u r e .

A key consequence o f s t a b i l i t y t h e o r y i s t h e a b i l i t y t o

d e f i n e such a s t r u c t u r e t h e o r y f o r a wide c l a s s o f s t r u c t u r e s .

A f t e r d e s c r i b i n g t h e axioms we want t h e n o n - f o r k i n g r e l a t i o n t o obey we g i v e another p r o o f o f t h e e x i s t e n c e o f such a r e l a t i o n .

T h i s p r o o f i s a v a r i - a n t on

t h a t i n L a s c a r - P o i z a t w i t h some o f t h e more c o m p l i c a t e d symmetry arguments s i m p l i f i e d by t h e use o f s t r o n g l y s a t u r a t e d models ( d i s c u s s e d i n 4 ) .

T h i s paper

i s p r i m a r i l y i n t e n d e d t o i n c u l c a t e t h e v i e w o f n o n - f o r k i n g i n d i c a t e d above. v i e w p o i n t i s e x p l o r e d i n g r e a t e r d e t a i l i n Paldwin-1982. i d e a s i n n o v e l ways.

However,

This

we d e v e l o p two

I n s e c t i o n 2 we suggest s y s t e m a t i c usage o f t h e t e r m i n o l o g y :

t h e t y p e p a l m o s t b e a r s a c e r t a i n r e l a t i o n (e.g. b e a r s t h a t r e l a t i o n t o e v e r y model

definable over) t o a set A i f p

M c o n t a i n i n g A.

I n section

4 we e x p l o r e a new

and s i m p l e r d e f i n i t i o n o f S h e l a h ' s concept o f an FaX ( T ) - s a t u r a t e d model. m a t e r i a l h e r e i s a p e r m u t a t i o n o f a subset o f B a l d w i n - 1 9 8 ~ .

The

Details o f proofs

o c c u r o n l y where t h e y a r e new; o t h e r w i s e we r e f e r t o e a r l i e r work.

J.T. BALDWIN

72

Our n o t a t i o n i s t h a t g e n e r a l l y used i n model t h e o r y and p a r t i c u l a r y i n s t a b i l i t y theory. model

C

A l l models and s e t s d i s c u s s e d a r e c o n t a i n e d i n an e x t r e m e l y s a t u r a t e d

( t h e monster model).

types w i t h a circumflex,

e.g.

We c a l l a t y p e o v e r

C

a g l o b a l t y p e and w r i t e g l o b a l

6.

1. ABSTRACT DEPENDENCE RELATIONS

I n t h e 1 9 7 0 ' s Van d e r Waerden

and Whitney

generalized the notions o f linear

independence i n v e c t o r spaces and a l g e b r a i c independence i n a l g e b r a i c a l l y c l o s e d f i e l d s t o o b t a i n t h e n o t i o n o f a dependence system.

VAN DER WAERDEN'S A X I O M S

T h i s r e l a t i o n i s most e a s i l y d e s c r i b e d i n t e r m s o f a

i)a depends on (a).

int

dependin

on a s e t X.

(Reflexivity)

ii)I f a depends on X and X

CY

t h e n a depends on Y.

(Monotonicfty)

iii)I f a depends on X and each x f X depends o n Y t h e n a depends on Y. ( T r a n s i t i v i t y o f dependence) iv)

I f a depends o n X U b u t a does n o t depend on X t h e n b depends

on X U €a).

(Exchange axiom)

v) I f a depends on X t h e n a depends o n a f i n i t e subset o f X.

(Finite

character)

These axioms a r e n o t so c a t e g o r i c a l as we m i g h t n a i v e l y t h i n k .

[ X I t o denote t h e subspace g e n e r a t e d b y X, dim(CA U E l ) = dimCAI f a i l s f o r a l g e b r a i c a l l y closed f i e l d . i m p o r t a n t "hidden axiom":

For example,

using

t h e law i n v e c t o r spaces:

+ dim(B)

Moreover,

- dim(A 1 B) i t t u r n s o u t t h a t t h e r e i s an

i f X i s an independent s e t ( i n a v e c t o r space o r f i e l d )

13

Strong Saturation and the Foundations of Stability Theory then,

up t o isomorphism,

t h i s independence.

t h e r e i s o n l y one way t o add a neu element x p r e s e r v i n g

Finally,

t h e most i m p o r t a n t consequence o f t h e s e axioms,

that

a l l bases f o r a space have t h e same c a r ' d i n a l i t y t u r n s o u t n o t t o r e q u i r e t h e f u l l s t r e n g t h o f axiom iii). We d e s c r i b e here,

n o t a g e n e r a l i z a t i o n o f t h e Van d e r

Waerden axioms as i t i s ueaker i n some senses and s t r o n g e r i n o t h e r s b u t r a t h e r another d e f i n i t i o n o f independence which s p e c i a l i z e s t o t h e v e c t o r space and f i e l d cases.

We d i f f e r from t h e t r a d i t i o n a l a l g e b r a i c approach i n s e v e r a l ways.

The a l g e b r a i c

approach c o n s i d e r s t h e r e l a t i o n : t h e p o i n t a depends o n t h e s e t B.

This r e l a t i o n

can be t r i v i a l l y extended t o : a depends o n E o v e r C. B i f some n o n - t r i v i a l

(e.g.

I n f i e l d s a depends on

p o l y n o m i a l i s s a t i s f i e d b y a and elements o f B;

a depends on

B o v e r C i f some n o n - t r i v i a l p o l y n o m i a l u i t h c o e f f i c i e n t s from C i s s a t i s f i e d b y a and some e l e m e n t s o f 8.) We u o r k d i r e c t l y u i t h a t r i p l e , (and u l t i m a t e l y ) b y a s e t A. B.

a,P,C.

We r e p l a c e t h e element a b y a seguence 3

Further,

we r e p l a c e t h e sequence 3 b y i t s t y p e o v e r

we f o r m u l a t e o u r n o t i o n as a r e l a t i o n between a t y p e p o v e r a s e t B and

Thus,

Since ue a r e u o r k i n g i n t h e monster model u h i c h i s s a t u r a t e d model t h e r e

a s e t A.

i s a c t u a l l y l i t t l e d i f f e r e n c e b e t u e e n d e a l i n g u i t h a complete type, t h e p a i r (F,B).

For,

t(2;B),

and

a l l r e a l i z a t i o n s o f t h e t y p e c a n b e mapped t o c b y an

automorphism u h i c h f i x e s B.

However,

i)

d e a l i n g u i t h t y p e s has t h r e e advantages.

I t a l l o w s us t o d e a l u n i f o r m l y u i t h f i n i t e sequences g i v i n g no p r e f e r e n c e t o

ii) I t a l l o u s us t o compare t u o p a i r s (T;B)

singletons. t(F;B),

t(a;E).

p a i r (F;B)

Thus,

and (a;E)

by considering

iii) I t makes i t c o n v e n i e n t t o speak o f a p p r o x i m a t i o n s t o t h e

b y d i s c u s s i n g s u b t y p e s o f t(F;B).

i n o u r d i s c u s s i o n h e r e ue u i l l f o r m a l i z e t h e n o t i o n

over A b y d e f i n i n g :

t(B;B)

i s f r e e o v e r A.

B does n o t depend on

I n t h e remainder o f t h i s s e c t i o n ue

l i s t t h e axioms ue u a n t t h i s r e l a t i o n t o s a t i s f y .

Much o f t h e m a t e r i a l i n t h e

l a t t e r p a r t o f Shelah-1978 can b e s i m p l i f i e d b y s y s t e m a t i c use o f t h e s e axioms.

B

J.T. BALDWIN

14 1.1 FREE EXTENSIOhlS.

A n o t i o n o f freeness,

1i s

a c o l l e c t i o n o f p a i r s (p,A).

Each p i s a ( n o t n e c e s s a r i l y complete) t y p e which d e t e r m i n e s i t s domain: B. (p,A)

€

I we

w i t h a p o i n t c r e a l i z i n g p and w r i t e t(c;B)

1.2 ISOMORPHISM CONVENTIOF!. isomorphism,

1 A.

say p i s f r e e o v e r A and w r i t e p

i.e.

Alternatively

, we

If

may d e a l

f A.

We r e q u i r e any n o t i o n o f f r e e n e s s t o be c l o s e d under

i f f i s an isomorphism and (p,A)

€

F then

(f(p),f(A))

€

L,

1.3 MONOTONICITY A X I O M S . M.I

M2.

1.4

Ep

If a

If A

C A

1-

and p i s f r e e o v e r A t h e n q fs f r e e o v e r A.

2

and p i s f r e e i s over A1 t h e n p i s f r e e o v e r A

T R A N S I T I V I T Y A X I O M FOP INDEPENDENCE.

Let C

B

C dom

p.

2'

I f p i s f r e e over B

and p J B i s f r e e o v e r C t h e n p i s f r e e o v e r C. Note t h a t t h i s axiom is m i t e d i f f e r e n t from t h e t r a n s i t i v i t y o f dependence e n u c i a t e d as iii)i n t h e Van d e r Uaerden t i s t .

I n particular,

we w i l l e s t a b l i s h

t h i s axiom f o r o u r n o t i o n o f dependence i n t h i s paper w h i l e t h e a t t e m p t t o approximate t r a n s i t i v i t y o f dependence l e a d s t o t h e s u b j e c t o f r e g u l a r t y p e s . However,

i t a150 easy t o o b t a i n t h e analogue o f t h e exchange p r o p e r t y ,

symmetry axiom.

the

We m e n t i o n i t h e r e a l t h o u g h ue do n o t t r e a t i t e x p l i c i t l y l a t e r

i n t h e paper s i n c e s u r p r i s i n g l y i t can be d e r i v e d from t h e c a t e g o r i c i t y o f a s m a l l e r s e t o f axioms ( c f . s e c t i o n 6 ) .

1.5 SYMMETRY A X I O M .

I f t(B;

B U

f6>)

i s f r e e o v e r B t h e n tt6;B

o v e r 8.

1.6

EXISTENCE AND EXTENSION A X I O M S .

El.

t(6;A)

i s f r e e o v e r A f o r any

6 and A.

U fB))

i s free

15

Strong Saturation and the Foundations of Stability Theory

E2.

I f p i s f r e e o v e r A and domp

CB

t h e r e i s a p1 € S(P) which extends

p and such t h a t p1 i s f r e e o v e r A.

As u s u a l we d e n o t e b y S(B)

1.7 DEFINITION.

Let

t h e c o l l e c t i o n o f a l l complete t y p e s o v e r F.

be a n o t i o n o f f r e e n e s s .

A type p i s s t a t i o n a r y over A i f

i f ql, q 2 € S t B ) e x t e n d p and a r e f r e e o v e r A t h e n q1 =

f o r e v e r y 8 c o n t a i n i n g A, 42'

Me say p i s s t a t i o n a r y i f p i s s t a t i o n a r y o v e r domp. The f o l l o w i n g axiom expressed a h i d d e n proper'ty o f v e c t o r spaces; t h e r e i s o n l y one way (up t o isomorphism) t o f r e e l y e x t e n d an independent s e t .

As n o t e d i n 5.14

t h i s axiom imposes a bound on t h e number o f such e x t e n s i o n s and t h i s bound s u f f i c e s f o r o u r purposes.

1.8 BOUNDEDNESS OR NON-TRIVIALITY A X I O M .

Every t y p e o v e r a model i s s t a t i o n a r y .

2. DEFINABILITY AND "ALMOST"

I n t h i s s e c t i o n we d i s c u s s t h e r e l a t i o n between such n o t i o n s as p i s d e f i n a b l e over A and p i s d e f i n a b l e o v e r e v e r y model c o n t a i n i n g A.

We do n o t assume T i s

stable i n t h i s section.

2.1

DEFINITION.

there i s a

i)The t y p e p € S ( B ) i s o v e r A i f f o r each f o r m u l a +(X;6) f4(F;5)

w i t h 5 € A and such t h a t

€ p

4 <-> 3'4.

ii) The t y p e p € S ( B ) i s d e f i n a b l e o v e r A i f f o r each +(F;6) € p t h e r e i s a f o r m u l a d+(g;S)

( o v e r A) such t h a t

+(F;6)

€ p iff

b dbt6;i).

iii) The t y p e p € S ( B )

i s f i n i t e l y s a t i s f i a b l e i n A i f f o r each

i s an 3 € A such t h a t

+(3;6).

+tx;6)

€ p there

J.T. BALDWIN

76

Now we i n t r o d u c e a n a t u r a l a p p r o x i m a t i o n t o each o f t h e s e n o t i o n s . s e c t i o n we f o c u s on almost d e f i n a b i l i t y .

In this

The r o l e o f a l m o s t s a t i s f i a b i l i t y

becomes c l e a r i n s e c t i o n 5 .

2.2

DEFINITION.

p i s over M .

A,

The t y p e p € S t E ) i s almost o v e r A i f f o r each model M c o n t a i n i n g S i m i l a r l y p i s almost d e f i n a b l e over A

o r almost s a t i s f i a b l e i n

A

i f p i s d e f i n a b l e o v e r o r f i n i t e l y s a t i s f i a b l e i n e v e r y model c o n t a i n i n g A.

Note,

t h a t f o r m a l l y p i s a l m o s t o v e r A means t h a t f o r each

M t h e r e i s a f o r m u l a #b(Z;ii)

with

4 <->

f + and t h e

compactness arpument shows t h a t t h e f o r m u l a

++

+b does

+(%;6)

f p and f o r each

may depend on M.

An easy

not depend On M.

We w i l l r e a u i r e t h e f o l l o w i n g n o t i o n l a t e r .

2.3 DEFINITION.

Then p i s s t r o n g l y d e f i n a b l e o v e r B i f

L e t p be a t y p e o v e r A.

t h e r e i s an e x t e n s i o n p ' o f p t o a complete t y p e o v e r A U @ such t h a t p ' i s d e f i n a b l e o v e r 8.

The f o l l o w i n g n o t i o n s a r e from Shelah-1978.

2.4 DEFINITION.

The t y p e p i s d e f i n a b l e almost o v e r A i f f o r each f o r m u l a

t h e r e i s a f o r m u l a d 4 ( 9 ) which i s a l m o s t o v e r A such t h a t f o r any

F b(3i;C)

4(li;9) € p iff

d+(F).

2.5

DEFINITION.

( R € FE(A))

i) The r e l a t i o n R(X;Y)

i s a f i n i t e eauivalence r e l a t i o n over A

i f R i s a n e q u i v a l e n c e r e l a t i o n which h a s o n l y f i n i t e l y many c l a s s e s

and which i s d e f i n a b l e w i t h p a r a m e t e r s f r o m A .

ii)The f o r m u l a + ( X I depends o n t h e f i n i t e e q u i v a l e n c e r e l a t i o n R i f t h e s o l u t i o n set o f

4 is

a u n i o n o f c l a s s e s o f R.

I1

Strong Saturation and the Foundations of Stability Theory Now we can t i e t h e s e i d e a s t o g e t h e r .

2.6 THEOREM.

Let

p

be a g l o b a l t y p e and A a s e t .

The f o l l o u i n g a r e e q u i v a l e n t .

i)p i s d e f i n a b l e almost o v e r A. ii)p i s almost d e f i n a b l e o v e r A.

iii)p has a bounded number o f c o n j u g a t e s o v e r A. ( I f t h e n o t i o n o f c o n j u g a t e i s u n f a m i l i a r see 4 . 3 below.) PROOF.

Clearly,

i)-> ii). I f i f ) h o l d s t h e n f o r any P c o n t a i n i n g A say w i t h

= 4, e v e r y c o n j u g a t e o f p i s d e f i n a b l e o v e r M. conjugates.

Cow,

suppose iii)h o l d s .

number o f c o n j u g a t e s .

Then,

Thus

f o r each

T h i s i m p l i e s as i n Shelah-1978

depends on a f i n i t e e a u i v a l e n c e r e l a t i o n o v e r A. t h e r e a r e lii

< k, e l e m e n t s o f

M such t h a t :

p

has a t most 4

b(F;y),

IAl

IT'

d 4 has a bounded

( c h a p t e r 111) t h a t d 4

Thus f o r any

M containing A

k b ( F ; E ) <-> V<E(F,iii):i

< k>.

3. FREE EXTENSIONS O F TYPES OVER M O D E L S I N STAELE THEORIES.

Now,

a s t a b l e t h e o r y can b e d e f i n e d as one f o r which i f p i s o v e r A t h e n p i s

d e f i n a b l e o v e r A. by H a r n i k .

3.1

We u i l l use a p a r t i c u l a r y s t r o n g form o f t h i s r e s u l t e n u n c i a t e d

H e n c e f o r t h ue d e a l o n l y w i t h s t a b l e t h e o r i e s .

LEMMA ( C H a r n i k - H a r r i n g t o n

19821.

I f T i s stable,

f o r every formula

4

and each

p € S ( A ) t h e r e e x i s t L - f o r m u l a s e(!?;Z) and M 5 ; Z ) such t h a t :

i)

For some 3, e(X;3)

ii)For e v e r y 6 € A,

I=

€ p.

i f e(%;a) € p t h e n

[bt!?;6)

€ p iff

At6; B) 1.

Now we w i l l d e f i n e a f r e e e x t e n s i o n o f a t y p e o v e r a model and show t h a t t y p e s o v e r models a r e s t a t i o n a r y .

We a l s o p r o v i d e some o t h e r i m p o r t a n t

c h a r a c t e r i z a t i o n s o f f r e e e x t e n s i o n s of t y p e s o v e r models.

In the following

s e c t i o n we i n t r o d u c e an i m p o r t a n t concept and t h e n r e t u r n t o t h e t a s k o f d e f i n i n g

J.T. BALDWIN

78 f r e e extensions o f a r b i t r a r y types.

I f p i s a type over a model, M,

then we w i l l choose t h e f r e e extension o f p t o be

one which i s d e f i n a b l e over M .

Such an extension e x i s t s and i n f a c t i t has

several other n i c e p r o p e r t i e s which we record nou.

f o r some

ZI

ii)(THE FUNDAMENTAL ORDER) Then p

,

over A i f

b(ic;5) 8 p.

f A,

2 q,

and q i n S(B).

Let p be i n S(A)

i f every formula which i s represented i n p i s represented i n

-

5

q and q

5

n a t u r a l l y extend t h i s n o t i o n t o p

2

q.

p

i)The formula bCii;l) i s represented i n t h e type

3.2 DEFINITION.

We w r i t e p

q if p

p.

When C q

i s represented i n p i s represented i n o.

domp

fl

domq we can

i f every formula i n L ( C ) )

which

We denote t h e equivalence

c l a s s o f q by Cql.

3.3 D E F I N I T I O N . Let M be a model o f T and an h e i r o f p € S(M) i f q extends p and p coheir o f p on A i f t(A;M

3.4 THEOREM.

M be contained

-,,,

q.

i n A.

Let a r e a l i z e p.

Then q i n S(A)

is

Then t(B;A)

is a

U 3 ) i s an h e i r o f t(3,M).

Let M be a model o f a s t a b l e theory T,

p f S(M)

and M

4 C,

then p has

a unique d i s t i n g u i s h e d extension t o a complete type over C which i s i t s - h e i r , coheir,

and d e f i n a b l e extension.

The proof o f t h i s can be found i n Lascar-Poizat o r i n any o f t h e various p r e p r i n t s o f Baldwin-198x.

4.

STRONGLY SATURATED MODELS

We uant nou t o d e f i n e t h e f r e e extension o f a type p uhose domain may n o t be a

79

Strong Saturation and the Foundations of Stability Theory For t h i s our v a r i o u s n o t i o n s o f a l m o s t w i l l be u s e f u l .

model.

e x p l o r e one f u r t h e r concept.

Eut f i r s t ,

we must

T h i s n o t i o n was i n t r o d u c e d b y Shelah as

F a X - s a t u r a t i o n and p l a y s an i m p o r t a n t r o l e i n t h e c o u n t i n g o f models o f s t a b l e theories.

4.1

However,

The model M i s s t r o n g l y - X - s a t u r a t e d

DEFINITION.

X, any

<

t h i s i s i t s f i r s t use a t such a b a s i c p o i n t i n t h e t h e o r y .

i f f o r each s e t A w i t h ( A ( <

X-type o v e r A which i s f i n i t e l y s a t i s f i e d i n M i s r e a l i z e d i n M.

I f M i s a X - s a t u r a t e d model o f a s t a b l e t h e o r y and X > IT1 t h e n M i s

4.2 LEMMA.

strongly-X-saturated. PROOF.

Suppose

lA( <

such t h a t t(e;A)

X and Let A

< X and

g u a r a n t e e s t h a t t(e;A) L e t d d e f i n e p = t(3;M) o f d.

A

0

4 N 5 M.

< X.

i f fk d+(m,fi)

.

-

L e t e be an element

Ao.

For any f i n i t e sequence 2 from A1

U 2) i n t o an e q u i v a l e n t t y p e q o v e r a subset N

Given t h i s c o n v e r s i o n t h e X - s a t u r a t i o n o f M

i s r e a l i z e d i n M. By 3.1,

o v e r M.

o n l y I T ( p a r a m e t e r s a r e used i n t h e range

L e t N be a submodel o f M w i t h A.

and w i t h I N 1

and A1 = A

i s f i n i t e l y s a t i s f i a b l e i n M.

we w i l l show how t o c o n v e r t t(e;A of M with IN(

M = A.

and a l l t h e s e p a r a m e t e r s c o n t a i n e d i n N

Cow f o r any m € M, 3 E A

- Ao,

6

and

from Ao,

kb(m,Z,fi)

So t h e r e q u i r e d t r a n s l a t i o n t y p e i s Cd+(9;x,p)(x,fi):+(x;a,S)

PTo I. see t h i s t y p e i s c o n s i s t e n t n o t e t h a t f o r any

4

f i n i t e s a t i s f i a b i l i t y t h a t t h e r e i s an e ' E M such t h a t

k4(e,3,6)

I= 4(e',3,6)

€

i m p l i e s by and t h u s

k

d+(e',fi).

4.3 DEFINITION. write p

€ S(M),

PROOF.

CM

and p,a

E S(M).

We say p i s c o n j u g a t e t o q o v e r A and

q i f t h e r e i s an automorphism o f M which f i x e s A and maps p t o q.

4.4 THEOREM. p,q

Let A

Let

IfM i s IMI-saturated,

i f p -A q then p

e

r e a l i z e p and

aA

a

X

>

(TI,

(MI

= X, and ( A 1 < X t h e n f o r any

q.

r e a l i z e q.

Choose f o r each

4,

f o r m u l a s e ( i i ; i ) and

J.T. BALDWIN

80

'(g;r) t

t o d e f i n e t(F;M)

IT1

Cb(X;FI)

U 2 ) = t(B';A

atn;6)

a).

Let

with

a(X;6') Thus a

€ p and

611 f q .

<-> U1';6')1

-Cb(X';B')

CB

and I B I

5 IAI

f o r each e chosen

€ p and f o r every I€ M,

4M

such t h a t

be an automorphism o f M f i x i n g A and t a k i n g E t o

(E

<-> A(iii;a

Now suppose -Cb(F;ii)

q.

M with A

Now b y s t r o n g s a t u r a t i o n choose 8'

p.

U

€ B such t h a t

Now f o r each 0 and each e, A depending on

8'.

'.

<-> u~1;6)1 €

6

4

That is,

c o n t a i n a l l parameters necessary t o d e f i n e p.

according t o 3.1 t h e r e i s a

t(B;A

Let P

i n the manner o f 5.1.

b,

if

etX;6)

Since p * A q,

€ p then

a(R;a

6)

€

t h e r e e x i s t 6 ' and I'

€ p c o n t r a r y t o our choice o f

e and

p = q as r e q u i r e d .

Lemma 4.2 i s i n Shelah-1978 ( u s i n g h i s d e f i n i t i o n ) and Theorem 4.4 i s i m p l i c i t i n Lascar-Poizat

5.

.

FREE EXTENSIONS OF ARBITRARY TYPES

Now we g i v e

our d e f i n i t i o n o f a f r e e e x t e n s i o n o f an a r b i t r a r y type.

5.1

DEFINITION.

M

T with A

Let

4 M,

b

f3

€

S(4)

Then

6

does n o t f o r k over A i f f f o r every

i s d e f i n a b l e over M.

f o r k over A i f f o r some

b f StL)

with p

Let p be a type over 8.

4 6,

f3 does n o t f o r k over

Then p does not A.

Note t h a t t h e f o l l o w i n g two lemmas a r e immediate from t h e d e f i n i t i o n o f d e f i n a b l e almost over and t h e d e f i n i t i o n o f n o n f o r k i n g .

5.2 LEMMA.

For any g l o b a l type

0, b

does n o t f o r k over A i f and o n l y i f

is

s t r o n g l y d e f i n a b l e almost over A.

5.3 LEMMA.

The n o n - f o r k i n g r e l a t i o n s a t i s f i e s t h e f o l l o w i n g axioms. M.I

If

q5

p and p does n o t f o r k over A t h e n q does n o t f o r k over A.

81

Strong Saturation and the Foundations of Stability Theory I f A1

M2.

2 A2

and p does n o t f o r k o v e r A 1 t h e n p does n o t f o r k o v e r

A2'

I f p d.n.f.

E2.

o v e r P and domp

p and such t h a t p,

5

B t h e r e i s a p,

€ S(B)

which e x t e n d s

o v e r A.

d.n.f.

I t i s easy t o e s t a b l i s h t h e f o l l o w i n g r e s u l t b y compactness.

I t corresponds t o

t h e e x i s t e n c e o f a bound i n t h e L a s c a r - P o i z a t t r e a t m e n t .

5.4 THE EXTENSION LEMMA. t h a t f o r every M w i t h A

5.5 THEOREM.

Let p € S(A).

E M,

6

There e x i s t s a

i s the heir o f

p

SCL) w i t h p L b such

€

plM.

Ifp i s a t y p e o v e r A t h e n p does n o t f o r k o v e r A.

Now u s i n g s t r o n g l y s a t u r a t e d models we show t h e s i g n i f i c a n c e o f almost satisfaction.

5.6 LEMMA.

Let A

C@

and p € S ( E 9 ;

p d.n.f.

o v e r A i f f p i s almost s a t i s f i e d i n

A.

PROOF.

I f p d.n.f.

over A then p

5

p f o r some g l o b a l t y p e b which i s d e f i n a b l e

o v e r M f o r each M c o n t a i n i n g A (and t h u s b y 3.4 converse,

f i n i t e l y s a t i s f i e d i n M).

Choose b y t h e E x t e n s i o n Lemma a

suppose p i s almost s a t i s f i e d i n A.

model M which i s I B I + - s a t u r a t e d and such t h a t t(M;B)

d.n.f.

5.7 DEFINITION.

over A.

d.n.f.

f i n i t e l y s a t i s f i e d i n M and t h u s r e a l i z e d i n M b y some F.

-C t(F;B)

For t h e

Then p i s

Then b y m o n o t o n i c i t y , p

o v e r A as r e q u i r e d .

For any A and any p € S(A),

upper bound o f W = CCql: p

5a

t h e bound o f p, p ( p ) i s t h e l e a s t

and q i s a t y p e o v e r a model).

Now we r e g a i n t h e d e f i n i t i o n from L a s c a r - P o i z a t .

We show t h a t f o r k i n g can be

82

J.T. BALDWLN

described i n terms o f t h e bound.

5.8 LEMMA.

5 6,

Let A

p € S(B)

and suppose p extends t o

L e t M be any model c o n t a i n i n g P and plA

f o r k over A.

p

5q

€ SCL) u h i c h does n o t

€ S(M).

&

Then Cql

Cpl M I .

p

Since

N i s s t r o n g l y (PI'-saturated

Choose N such t h a t

PROOF.

d.n.f.

5 l M i s r e a l i z e d i n N by some t.

over A,

be an automorphism o f

E uhich

f i x e s A and maps

extension o f q t o a g l o b a l type. suppose b(X;g)

Since t(P;N)

That is,

4(Y,g) i s

Then,

t o E.

represented i n a

Let

a

Let

4 denote t h e d e f i n a b l e

r e a l i z e a and l e t a

Now

MI.

T h i s i m p l i e s t h a t f o r some

m

€ M,

f o r some iii' € a M, +(t;iii').

4, hence i n 4, hence i n

uith A

over A.

d.n.f.

= (a 6) I (a M) = t(E;a

i s almost s a t i s f i a b l e i n A,

For any A and p € E(B)

5.9 COROLLARY.

a

a t(a;M)

is represented i n [PI.

€ F(A)

+(E;iii).

and t(M;N)

4 P.

q.

I f q = p ( A t h e n p f o r k s over A

i f f D(p) < D(p1A). PROOF.

p and 4 r e a l i z e p(p) and P ( p ( A ) .

Let

ue see Cp1

= C41.

b

and

4

over A t h e n by 5.8

d.n.f.

I f p f o r k s over A f o r every $ extending p t o

c o n t a i n i n g A such t h a t

Cpl < CPIMI.

If

SCL) t h e r e

i s an M

p i s n o t t h e d e f i n a b l e e x t e n s i o n o f P l M so by 3.4

But by C o r o l l a r y 5.8,

CplMl 5 CQI.

We nou can deduce t h e

t r a n s i t i v i t y axiom w i t h o u t e f f o r t .

5.10 COROLLARY. d.n.f.

5.11

Let A

5

over A t h e n p d.n.f.

DEFINITION.

i)

B

4

I f p d.n.f.

C and l e t p € S ( C ) .

over A.

X (T) i s t h e l e a s t i n f i n i t e c a r d i n a l such t h a t f o r each

f i n i t e sequence 3 and ascending sequence o f s e t s
t(Z;Ait,)

over B and p l B

< X

(T)>,

f o r some i

does n o t f o r k over Ai.

i i ) k T ) i s t h e l e a s t i n f i n i t e c a r d i n a l such t h a t f o r every type p t h e r e i s a set A

5 domp

uith

IAl < k T )

and p d.n.f.

over A.

83

Strong Saturation and the Foundations of Stability Theory I t i s r a t h e r easy t o show:

5.12

LEMMA.

k T )

< X

5 k(T)

(T)

Now since p f o r k s over plA, represented i n B(plA);

5.13 THEOREM.

+.

Cb(p)l c o n t a i n s a formula + ( i i ; p ) which i s n o t

thus i t i s immediate t h a t :

For every T,

x

(TI

5

IT!+.

Now ue can e a s i l y conclude t h e f o l l o w i n g r e s u l t .

5.14 LEMMA.

For any t y p e p, p has a t most 2’

contradictory nonforking

extensions. Choose A

PROOF.

-C R.

2 domp

such t h a t p d.n.f.

Now plA has a t most

ZIT’

over A.

Choose M w i t h

IMI 5 IT1

and P

extensions t o t y p e s over models and they are a l l

s t a t i o n a r y so plA has a t most 21T1 p a i r w i s e c o n t r a d i c t o r y n o n f o r k i n g extensions t o g l o b a l types.

@y t r a n s i t i v i t y o f independence and m o n o t o n i c i t y t h i s bound extends

t o p.

6. THE C A T E G O R I C I T Y O F THE DEPENDENCE A X I O M S

Although t h e r e a r e a number o f ways o f d e f i n i n g independence i n s t a b l e t h e o r i e s we can now see t h a t i f they meet a f e w l i m i t e d c r i t e r i a they are e q u i v a l e n t .

THEOREM.

axioms,

Suppose T admits a dependence r e l a t i o n s a t i s f y i n g t h e m o n o t o n i c i t y t h e isomorphism convention,

and each type has o n l y a bounded number o f

p a i r w i s e c o n t r a d i c t o r y f r e e extensions. type

fs i s f r e e over A i f f

PROOF.

Then T i s s t a b l e .

i s d e f i n a b l e almost over A.

L e t A be an a r b i t r a r y subset o f a model o f T. (T)

Moreover,

Then

IS(A))

the global

J.T. BALDWIN

84

i s a t most 2XCr'

x A and t h u s T i s s t a b l e .

For,

each p i n S ( A )

i s f r e e over a

s e t o f power Less t h a n X ( T ) and each t y p e o v e r a s e t o f power X ( T I has a t most A f r e e extensions.

Now suppose t h e g l o b a l t y p e b i s f r e e over A. so

Then a l l c o n j u g a t e s o f b a r e f r e e

b has a t most A c o n j u g a t e s and t h e r e f o r e b y 2.6 i s d e f i n a b l e almost o v e r

A.

REFERENCES

J.

Baldwin,

T.

ms.

C19821

J.T.

Baldwin,

V.

The e l e m e n t a r y t h e o r y o f a b s t r a c t dependence r e l a t i o n s ,

book on s t a b l e t h e o r i e s ,

H a r n i k and L. H a r r i n g t o n ,

vol.23

no.1,(1976),

Shelah,

45 pages i n ms.

"Rank and d e f i n a b i l i t y i n s u p e r s t a b l e t h e o r i e s " ,

D. Lascar and B.

S.

i n preparation.

Fundamentals o f Forking, t o appear Proceedings o f

Model Theory Year i n Jerusalem, D. Lascar,

56 pages i n

I s r a e l J. Math.,

53-87. Poizat,

"An i n t r o d u c t i o n t o f o r k i n g " ,

J.S.L.,

44(1979),

C l a s s i f i c a t i o n Theory and t h e Number o f Non-isomorphic

Models,

330-351. North

Holland(1978). 8. L. Van d e r Waerden, Moderne Algebra,

2nd e d i t i o n ,

Springer-Verlag

(Berlin),

1937. H. Whitney,

57(1935),

On t h e a b s t r a c t p r o p e r t i e s o f l i n e a r dependence,

509-533.

Amer. J . o f Math.,

LOGIC COLLOQUIUM '82 G. Lolli, G. Long0 and A . Marcjg (editors) 0 Elsevier Science Publishers B. V. (North-Hollnnd), 1984

85

UNDECIDABILITY OF RATIONAL FUNCTION FIELDS IN NONZERO CHARACTERISTIC Gregory L. Cherlin* Rutgers University INTRODUCTION The main result to be proved here is the undecidability of the rational function field F(t) over any infinite perfect field F of characteristic p > 0 . Actually I was aiming at something that I would consider a good deal more interesting: the undecidability of fields of formal power series in characteristic p. (I am grateful to Jan Denef for pointing out that my approach was well suited to the case of the rational function field.) The decision problem for power series fields of characteristic p (in the language of fields) is very much open, and nothing in the present paper refutes the possibility that their theories are model complete in a natural language. The plan of the paper is as follows. It is convenient to begin with a discussion of the undecidability of the theory of any infinite field F of nonzero characteristic with respect to certain weak monadic second order logics. 'Ihis is of interest because this theory will be interpretable in the theory of the field F(t) OK F((t)), in (unfortunately) an extended language continuing a predicate for the constant subfield F, as well as the valuation v on F((t)) (or its restriction to F(t)). All of this happens to be quite simple, but a small encoding trick is needed involving the Artin-Schreier polynomial 2 - x . present paper.)

(This is the new point of the

Turning to the case of rational function fields, we can show that the use of the valuation and the predicate for the constant subfield is inessential, though its elimination requires artificial devices. Thus we obtain the main theorem. Naturally we will be unable to resist a concluding discussion of the case of power series fields. For earlier discussion see [ 3 , 5 I . The most clear-cut statement we can make at present is the following. If K = F((t)) is a power series field, let the A-K-E language for K be the language of valued fields enriched by both a predicate for the constant subfield F and the "cross-section" function THEOREM.

x:

2

-+

K defined by

X(n) = tn.

Let F be a perfect field of characteristic p.

Then F((t))

is

undecidable in the A-K-E language. ( ' ) Of course, this formulation of the state of affairs suppresses a lot of relevant information. See SS2.4 for details. I mention some open problems on definability in power series fields in S5. *This report was prepared while the author was visiting Simon Fraser University. The research was supported in part by NSF Grant MCS 8 1 0 2 3 8 3 , and in part by the Alexanderyon-Humboldt Foundation, which funded a visit to the University of Bonn. I am grateful for the hospitality shown by the Seminar for Logic and Foundational Research at Bonn.

G.L. CHERLIN

86

Field theory in weak monadic second order logic

S1.

Notation 1. 2.

F is an infinite field of characteristic p > 0. Q, is the collection of finite subsets of F.

3.

Q+

is the collection of all finite subgroups of the additive group of

For greater clarity we may sometimes write 4.

Tho(F) by Q,

or or

Th+(F) is the theory of Q+ respectively.

QO(F), Q+(F)

F

rather than

Q,,

F. Q+.

in the language of fields extended

More precisely, if we introduce the two-sorted structures: OF = 0

then

Tho(F), Th+(F)

are

Th('F),

Th(+F)

in the usual sense.

We will prove the undecidability of Tho IF) model of arithmetic in a natural way, and I will mention another treatment of

F

Th+(F).

Tho (F)

encodes a

encodes T h O ( F ) . At the end

suggested by Macintyre.

0

The undecidability of If

Th+(F)

and

Th+(F)

Th (F) :

contains a transcendental element u, then the set

u

N

= {l,u,u

2

,...

with the operations u"@un = and urn 43 un = urn provides a natural model of arithmetic, which can be described using Q, and the parameter u. (The introduction of such auxiliary parameters is of course completely harmless.) To define uN we introduce the formula "y Pow x" defined as: or 3 X 6 Q [ I ,f X and x F X and 'da E X (ax f y => ax 6 X I " . y @

Oar

" y = 1 or x This means that

.

We define @ on uN using multiplication in F, and we define is in x from i+) using the set theoretic power of Q,. (One can do the same thing in

any semigroup having an element of infinite order.)

To encode a model of arithmetic in the general case, when F may be algebraic over the prime field, we introduce finite cardinals as equivalence classes of certain sets, and we then derive the arithmetic operations from cardinal arithmetic: Given z € F and X,Y,Z finite subsets of F, let " ( z , Z ) : X + Y" the following relation R(a,b) induces a bijection between X and (1)

a

+

mean that Y:

bz € 2 .

-

Let " X Y" mean that z,Z can be found so that (2.2): X Y. This relation is Q,,-definable, but it is simply the equicardinality relation, for if X,Y are +

finite sets we can find z € F so that all the elements a + bz (a E XI b € y) are distinct, and if X,Y are of the same cardinality we can then select Z satisfying (2.Z): X Y.

-

87

Undecidability of Rational Function Fields in Nonzero Characteristic

Let an F-integer be an equivalence class of finite subsets of F with respect to the equicardinality relation. The ordering on F-integers is induced by settheoretic inclusion (X X' Y) and addition and multiplication may be derived from union and encoded Cartesian product as follows. Let " 2 X + Y" mean that Z is the disjoint union of the sets X ' X and Y ' * Y. This induces the relation C = 5 + q on F-integers. For multiplication, let z: Z XXY mean that Z = {a + bz: a C X, b E Y} and that those elements a + bz are all distinct. Then the relation 32: 2 XxY induces multiplication of F-integers.

-

<

-

-

-

The undecidability of

-

Th+(F)

We will see that the quantifier Q,

can be interpreted in the quantifier Q+,

via the following very natural encoding procedure. Associate to finite additive subgroups A,B set S(A,B,c) given by:

5

(F,

+)

and a parameter c

F

the

{x c A: ( x + c ) - ' C B).

(2)

We need only show that an arbitrary set suitable A,B C Q+(F) Let A

and

be the additive subgroup of

that the elements { (a+c)-': a E A}

QO(F) occurs as S(A,B,c)

for

F generated by

{ (aX1-l : a E S}.

S

and choose c c F

so

are linearly independent over

finitely many elements c will serve). generated by

S €

c E F.

Then

Let

B

F (all but P be the additive subgroup of F

S(A,B,c) =

S.

0

Thus the undecidability of Ih+(F) follows from that of Th (F). Macintyre observes that it is also a consequence of a result of W. Baur I21 and Martyanov [ 1 3 1 , or rather of the proof of that result. They showed that the theory of a vector space (e.g. over Fp) equipped with five distinguished subspaces (as predicates) is undecidable Th+(F) encodes the theory of a finite vector space equipped with n distinguished subspaces (for any fixed n), which for suitable n is also undecidable. The proof of Baur and Martyanov used the word problem for l-generated groups. If we replace this by the word problem for finite semigroups, we get the result in the form needed here. I have recorded this last observation as a point of interest, but we will need the flexibility afforded by the reduction of Q, to Q+ later on.

Other quantifiers Let us define a weak second order of finite subsets of F which is of F is contained in an element Q, and Q+, but it is natural to

quantifier over the set F as a collection Q unbounded in the sense that any finite subset of Q. The only examples needed here will be ask whether every infinite field has an

undecidable theory with respect to any weak second order quantifier. Using an idea of hret [ 7 ] one can show: Proposition 1.

Let

ThQ(F)

F

be a p.a.c.

field which is not separably algebraically closed.

is undecidable for any weak second order quantifier Q on F.

Then

G.L. CHERLIN

88

x

Let Q* be the collection of finite subfields of the algebraic closure F P of the prime field. Then ThQ* (k ) is undecidable. P Nonetheless it seems quite plausible that a weak second Order quantifier with a 2.

-

decidable theory could be constructed on F

P'

though I do not see how to carry

this out. 52. Undecidability of

for

K/F

K =

F(t) or

F( (t)).

is perfect, infinite, and of characteristic p. K will be a field lying between F ( t ) and F ( ( t ) ) . In the present section K will be considered exclusively as a valued field with respect to the valuation v induced by the natural valuation on F((t)). Let K/F denote the structure obtained from K by adjoining a predicate for the constant subfield F of K. We will show that Th(K/F) is undecidable. This suggests that if it should happen that Th(K) is decidable, it is unlikely that this could be proved by a structural analysis of saturated models. F

We proceed by interpreting 'I~I+(F) into Th(K/F), in the terminology of S 1 . This means that we will encode quantification over finite additive subgroups of F into the language of K. We will make use of the function T(X) additive polynomial, and the choice of algebraic considerations. To motivate reader might want to determine the set Associate to x C K the set that the collection:

xp-x. We could actually use any was motivated only by irrelevant the following encoding procedure, the z [Kl for K = F ( (t)) before continuing. =

z

S(x) = {y C F: 32 z ( z ) = xy).

Our claim will be

{S(x): x F K , v(x) < O } is exactly Q + ( F ) .

Hence Q+(F)

Since z ( z +z ) = z ( z l ) 1

2

+

r ( z 2 ) , S(x) will be an additive subgroup of

v(x) < 0, the finiteness of write

u = Cu tn

with

can be encoded into K/F.

u

S(x)

is seen as follows.

If u = z ( z )

F F, then for any negative integer

F.

When

and we

i we have the

relation, whose verification we leave to the reader:

= 0. ipn u = xy, where x is fixed and condition on y: p(y) = 0. To Assuming xi f 0, this relation (If ipn ,t Z

take

u

The sum is finite because

i

is negative.)

If

y F F varies, then ( * ) imposes a polynomial see this raise ( 1 ) to a high power of p. is nontrivial, and hence is satisfied by only

finitely many elements y. We have seen that {S(x): v(x) < 0)

is contained in Q + ( F ) .

For the converse,

if A is a finite additive subgroup of F, we seek an element x of the particular form

89

Undecidability of Rational Function Fields in Nonzero Characteristic

such that S(x) = A. Here n is chosen so that pn According to ( * ) , the elements a F S(x) satisfy: n

is the cardinality of

A.

n-i

1 xiap

= 0.

i=O Conversely any solution of ( 2 ) will lie in S(x). n l/Pi u = 1 uitmpl and C ui = 0 then u C rIF(t)l

(In Other words, if

5

T[KI.)

i=O

Our claim accordingly takes on the following form. = x. € F elements, we seek coefficients c n-i 1

pn

vanishes precisely on A.

...,an, and

al,

A.

...,c

Determine co,

This is easily done.

If A F Q+(F)

Let

a,,

...,a

so that the corresponding polynomial

is not identically zero.

has

so that the polynomial

be a basis for P(a)

vanishes at

(This just requires us to take a

nontrivial solution of a homogeneous linear system.) Now the form of

P

shows that its set of zeroes is an additive subgroup

of F, hence includes A. Since the degree of P is pn, P vanishes only on A. (Notice also for use in S3 that P must have only simple zeroes, so co f 0.) Thus the element of

x

of the form ( 2 ) with

step we used the hypothesis that F

x. = c

~ encodes - ~

A.

In this

is perfect.

S3. Rational function fields

With minor modifications the method of S2 can be used to prove the undecidability of the rational function field F(t) in the pure language of fields. Here F is infinite, perfect, and of characteristic p. Oddly enough the analogous result has not yet been proved in characteristic zero, with F = C providing the outstanding test problem. (R. Robinson managed to prove the undecidability of F(t) for F formally real [ 1 5 1 . ) Of course if F is finite then F(t) is undecidable for much the same reason that Q is (see 1161 for an elegant version of this argument, due originally to, Ershov and Penzin using a method of J. Robinson). This case can certainly not be handled by the method used here. On the other hand one might also want to include imperfect fields, and it appears quite likely that the present approach can be extended to cover this case - but not without considerable effort, involving a m r e precise analysis of the effect

G.L. CHERLIN

90

of the encoding in SZ in the imperfect case, and some study of particular elliptic curves (see below). We may begin the proof of the undecidability of F(t) in the language of fields by a discussion of the role of the valuation and the predicate for the constant subfield in the argument given in S2. With a change of notation, define S(x) for x C F(t) to be the set:

The point established in S2 was that: {S(x) fl F: v(x) < 0) = Q+(F). The role of the valuation is marginal here. It is used to exclude choices of the parameter x which encode infinite sets, but this can be done in other ways. An examination of the end of ~2 shows that the relevant parameters x may be taken to satisfy: 3y Thus

{S(X)

x =

n F:

yp + t-l.

3y x =

+ t-'}

= Q+(F).

The real problem is to eliminate the use of the predicate for F. If F is definable in F(t), as is often the case, there is of course no problem. In general one can define an infinite subset C F in F(t) using a nonrational curve, following an idea used by R. Mbinson. Given such a set C, the arguments of SSl-2 show that finite subsets of C) is interpretable in F(t). a,b,c E F(t), define S(a,b,c) to be the set:

QO(C) (quantificationover

More explicitly, given elements

{x C F(t): x C S(a), (x+c)-l t S(b)}. Evidently

{S(a,b,c)

n

F: a,b,c E Fft), 3x a = xp + t-'} 1

{S(a,b,c) n C: a,b,c C F(t), 3x a =

2

+ t- 1

is QO(F), and hence

is Qo(C). 0

Thus we need only find an F(t)-definable subset C 5 F such that 'Ih (C) is undecidable. For this we will use a nonrational Curve E defined over F, more specifically an elliptic curve with defining equation p(x,y) = 0 with p quadratic in x (further normalization is superfluous). Let E = {(x,y) C E: x,y t F(t)}, which is the same as EF since E is P(t) nonrational. Let C = {y t F(t): 3x (x,y) C EFft)}. If F is quadratically closed then C = F and hence the undecidability of F(t) follows already from SS1-2. If F happens to be algebraic over F , so that in particular F has P at most one quadratic extension F', then F is still definable in F(t), for similar reasons: F = { y C F(t): 3x € F'(t) (x,y) E El. Since F'(t) is interpretable in F(t), our claim follows. Thus we may concern ourselves exclusively with the case in which F contains an element transcendental over F We may then select the curve E so that it P'

91

Undecidability of Rational Function Fields in Nonzero Characteristic contains a point of infinite order over F. Let E = E C F(t) X F(t) equipped Fwith the usual group operation +, definable over F(t). (Take + to be a partial function, since we work only with the affine points on E.) Since E is undecidable, as noted already in S 1 .

has a point of infinite order, T h o ( € )

Of course a slight gap remains to be filled in, since E F (t) Introduce

.

C = { x F F(t): 3y(x,y) C E or Let C' = {x+yz: x,y,z € C}.

is not a subset of

(y,x) F El.

Given X F C'

and

z C C

let E(X.2)

be

{ (x,y) F E: x+yz C X I .

Using

E(x,z)

we can encode QO(E)

in Qo(C').

Hence Th

0

( )

is interpreted

in Th (F(t)) , and the proof is complete. S4.

Power series fields

In characteristic zero most questions about the model theory of power series are settled by the work of Ax-Kochen and Ershov on unramified Hensel fields in characteristic 0 . In particular the theory of F ( (t)) is Turing-equivalent to the theory of F and this is true even in the highly extended language which includes the language of valued fields together with both a predicate for the constant field and the cross-section defined by x (n) = tn (x: Z + F( (t)) "inverts" the valuation). It was known (but more or less forgotten) that this is certainly not the case in characteristic p, and that even power series over a finite field become undecidable if the cross-section is allowed. In 131 this is attributed to Ax (followed independently by 8 . Jacob). A proof is given in [ 3 1 , and since the proof contains the only really convincing evidence that the case of characteristic p is different from the characteristic 0 cases at the level of the language I will give it in detail (or rather a simple variant ~ of fields, which is noticeably more general).

As a preliminary remark, observe that the valuation is already definable in the language of fields. Indeed if p,n are relatively prime then {x: 3y yn = ( 1 + txn)I is just o = {x: V(X) 1. 01. Of course v canimmediately be reconstructed from 0

.

Thus for our purposes the languages of fields and of valued fields are the same language. It seems to me that the problem of decidability in this language of power series fields F((t)) of characteristic p is best thought of (for the present) as containing three separate cases (always assuming F is perfect): 1. 2.

3.

F

F F

infinite, non-kaplansky (see below). infinite, kaplansky. finite.

For the definition of a kaplansky field, recall that an additive polynomial n i P(x) is one of the form 1 ciXp , the usual example being xp-x. I will call a i =O field F a kaplansky field if P[Fl = F for every additive polynomial in FIxl. This is usually called Kaplansky's residual condition - which seems unnecessarily wordy in OUT context - and it has been studied by Delon [61. who showed that

G.L. CHERLIN

92

it is equivalent to the nonexistence of an extension of degree divisible by

p.

Although there is no sound basis for any prediction at the moment, it would not be terribly surprising if F((t)) should be undecidable for all non-kaplansky fields, but decidable for decidable kaplansky fields. (This is idle speculation.) I will prove below: Theorem. Let F be a (perfect) non-kaplansky field. Then F((t)) undecidable in the language of fields with cross-section.

is

Let m e emphasize that I am just reworking an old result. I find the first part of the proof interesting because it does not use the cross-section. Accordingly we will work in the language of (valued) fields until further notice. Notation. If

x = Exiti E F((t)), let

= c [i mod pn]

x.tJ. jzi mod Pn i

x

is definable from x (and the parameter t).

X

Lemma 1. -

Of course the

ti mod pnl

only values of

i

that come into consideration are i = O , l , ...,pn- 1.

Let yi = x [i mod pnl

.

men

, have

yo,...,y

the following two properties:

pn-1

+... + Ypn-l

(1)

x = y + y1

(2)

vi t-iyi c Kpn.

Conversely (1,2) force yi = x Ii mod pnl

.

m e lemma follows easily.

To digress briefly, notice that we have just defined a "pairing function" (actually, a p-tupling function). Hence it is difficult, if not imposible, to conceive of a quantifier-elimination procedure, since there is no such thing as a "single" quantifitr

.

Lemma 2. Let P(x) E F[Xl be an additive polynomial. Then the following predicate Po(y) on F( (t)) is definable in the language of fields, using the parameter

t:

"yo E PIFI" (This means y = E yiti with yi E F, and yo E P[Fl.) Indeed, the definition of

Po(y)

is simply:

93

Undecidability of Rational Function Fields in Nonzero Characteristic where pn is the degree of P. In one direction, if y' € PIF((t))l then clearly y;, C P[Fl, so (*) implies Po(y). In the other direction, if P o ( y )

, and let v(z) = k-p". If v(z) > 0 then easily [O mod pnl z 6 P[F((t))l, so let y' = z in this case. If v(z) '< 0 find z 1 6 P[F((t))]

holds let z = y

v(z-z 1 ) > v(z) and let

with

property Po(z).) treated,

OK

z' =

Replacing z

by

.

(If v(z) = 0, we use the [O mJd Pnl z', we may conclude by the first case

(2-2,)

by induction.

At this point we introduce the cross-section to get the effect of f u l l secondorder quantification over the value group 2 in a non-kaplansky field. For x C F((t)) let Z(x) be the set:

where

is the predicate corresponding to an additive polynomial P with

Po

P[F] # F. This Z ( x ) is an arbitrary subset of on N , the undecidability follows. S5.

N

.

Since we already have

+

A few open problems

Although we speak in term of decidability and undecidability, we are really just studying the structure of the definable subsets of F( (t)) or F(t). The hard questions concern definability in the pure language of fields, but there are also open problems in enriched languages. 5.1

The language of fields

1.

Is the constant subfield definable?

This for me is a key question. My investigations have produced nothing worth going into here. 2.

Is the predicate 'n is a power of p" definable?

This is a predicate on the value group. One can certainly conceive of this happening, although there is no concrete evidence for it. 2A. Is the theory of

F((t)) undecidable in the language of fields enriched by the above predicate?

This is extremely likely. Van den Dries points out that Presburger arithmetic enriched by the "pk"-predicate more to work with. 3.

decidable, but we of course have considerably

Is the predicate "the leading coefficient of x is 1" definable?

This means x = iL&x)xiti field is finite, using

= 1.) This is trivially so if the base with x v (XI = 1 for a F F*. q

G.L. CHERLIN

94 3A.

If F is infinite (and perfect), is F((t)) undecidable in the language of fields extended by the above predicate?

This also seems extremely likely. 4.

Are there any nontrivial definable predicates on than those of the type catalogued in S4?

This, of course, is the real question. its accompanying comment. 5.2

F((t)), other

The reader may formulate question 4A. and

Ehriched languages

Let L(S), L(F) be the languages of valued fields extended by unary predicates S,F respectively, which will be interpreted as follows in power series fields F((t)) (for F perfect, infinite).

s = Itn: n t

x 1

F = constant subfield It turns out that in this context these languages are equivalent - more precisely L(S) has the same O-definable relations as L(F,t) - and strong: they amount to the language of F enriched by quantification over countable subsets. The method used is a somewhat awkward elaboration of the coding in S2. no light on weaker languages.

It casts

5.3 Wel-completeness

Consider the language of fields extended by predicates interpreted in F( (t)) to mean:

where

R

(and x = (*)

R*(x,y,z,

...

)

varies over some extension of the language of fields xiti,...).

The following is conceivable:

If F is model-complete in the extended language then is model-complete.

This is true in any case in characteristic zero, and it does not contradict anything known in characteristic p. Of course such a statement would not be obtainable from a model-theoretic trick. It is essentially a strong statement about algebraic geometry over F( (t)), in a rather vague form.

Undecidability of Rational Function Fields in Nonzero Characteristic

95

References Kochen, S., "Diophantine problems over local fields I, 11," Amer. J. Math. 187 (19651, 605-630 and 631-648. [2] Baur, W., 'Undecidability of the theory of abelian groups with a subgroup,". PAMS 55 (1976). 125-18. [31 Becker, J., Denef, J., Lipschitz , L., "Further remarks on the elementary theory of formal power series rings," in Model Theory of Algebra and Arithmetic, Pacholski et. al. eds., LNM 834, Springer-Verlag, NY 1980, pp. 1-9. [4] Cohen, P., "Decision problems for real and p-adic fields," Comm. Pure Appl. Math. 22 (19691, 131-153. [5] Delon, F., "Hensel fields in equal characteristic p > 0," in Model Theory of Algebra and Arithmetic, Pacholski et. al. eds., LNM 834, Springer-Verlag, NY 1980, pp. 108-116. [6] Delon, F., "Quelques propri6tgs des corps valugs en thgorie des modGles," Thise d'Etat, Univ. Paris 7, 1982. 171 Duret, J.-L., "Les corps faiblement alggbriquement clos non separablement clos out la propriit; d'indipendence,' in Model Theory of Algebra and Arithmetic, Pacholski et. al. eds., LNM 834, Springer-Verlag, NY 1980, pp. 136-157. (81 Ershov, Yu., "On elementary theories of local fields," Alg. Log. 4 (1965), [l] Ax, J.,

5-30. [91 Ershov, Yu., "Cn the elementary theory of maximal normed fields, "Doklady 165 (1965), 1390-1393. [lo] Greenberg, M., "Rational points in henselian discrete valuation rings," Publ. I.H.E.S. 31 (1966), 59-64. [ll] I. Kaplansky, "Maximal fields with valuation I, 11," Duke Math. J. 9 (1942). 303-321 and 12 (1945), 243-248. [12] Kochen, S . , "The model theory of local fields," in Logic Conference, Kiel 1974, LNM 499, Springer-Verlag, NY 1975, pp. 384-425. 1131 Martyanov, V., "The theory of abelian groups with predicates that

distinguish subgroups and with endomorphism operations (Russian)", Alg. Log. 14 (19751, 536-542. [141 P. Ribenboim, Th6orie des valuations, Presses Univ. Montreal, 1964. [15] Robinson, R., "The undecidability of pure trans. ext. of real fields" Zeitschr. f. math. Logik und Grundl. d. Math. 10 (1964), 275-282. [16] Rumely, R., "Undecidability and definability for the theory of global fields," TAMS (1980), 195-217. [17] Schilling, 0 . . The theory of valuations, AMS Math. Survey, 1980. [181 Serre, J.-P., Corps LOcaux, Hermann, [19] M. Ziegler, "Die elementare Theorie der henselchen K&per," Thesis, K6ln 1972.

1. I have since found more precise information, described without proof in S5.2.

LOGIC COLLOQUIUM '82 G. Lolli, G. Long0 and A. Marcia (editors) 0 Elsevier Science Publishers B. V. (North-Holland), 1,984

91

Remarks on T a r s k i ' s problem concerning

Q

, +,

,

*

exp )

Lou van den Dries

S choo 1 of Mathema tics The I n s t i t u t e f o r Advanced Study Princeton, New J e r s e y 08540 U.S.A.

.INTRODUCTION

,+ , * ) s t r u c t u r e @, +, * , exp)

In h i s monograph on t h e elementary theory of t h e s t r u c t u r e T a r s k i asked whether h i s r e s u l t s could be extended t o t h e ([T,

exp(x) = e x

p. 451). ( I n s t e a d of

,

b u t t h i s makes l i t t l e d i f f e r e n c e s i n c e

as t h e unique f u n c t i o n of t h e form

Tarski suggested the f u n c t i o n exp

is d e f i n a b l e i n

x b> f (ax)

t h e axioms mentioned i n [T, p. 57, n o t e 201 f o r from adequate, see e.g.

(R

(R

f (x) = 2'

,+ , * , f)

which is i t s own d e r i v a t i v e ;

Th@

,+ , , f)

are far

[D-W].)

Before we d i s c u s s T a r s k i ' s question, l e t u s b r i e f l y review some a s p e c t s of

his work on

I, +,

(1) D e c i d a b i l i t y of ( 2 ) Tha,

+ , .) =

a)

Th@,

and see what use has been made of i t :

+ , -) ,

t h e o r y of r e a l closed f i e l d s

(3) Elimination of q u a n t i f i e r s f o r

@t

(4) P r o p e r t i e s of d e f i n a b l e s u b s e t s of (5) P r o p e r t i e s of d e f i n a b l e f u n c t i o n s

,

, < , 0 , 1 , + , .) , lRn ,

.

These a s p e c t s are c l o s e l y r e l a t e d i n T a r s k i ' s work, b u t i t makes sense t o d i s c u s s them s e p a r a t e l y .

(1) is a n i c e r e s u l t i n i t s own r i g h t and q u i t e u s e f u l

i n many t h e o r e t i c a l d e c i d a b i l i t y questions, b u t has otherwise n o t been important

i n s e t t l i n g open problems, a s f a r as I know.

(2) i s sometimes u s e f u l i n proving

p r o p e r t i e s of r e a l closed f i e l d s : i n c e r t a i n cases t h e only known proof c o n s i s t s of f i r s t e s t a b l i s h i n g t h e p r o p e r t y f o r t h e f i e l d of reals by transcendental methods and then invoking (2).

(This i s c a l l e d T a r s k i ' s p r i n c i p l e . )

(2) and (3)

combined g i v e a t r i v i a l and improved s o l u t i o n of H i l b e r t ' s 17th problem,

,

L. VAN DEN DRIES

98

and some important g e n e r a l i z a t i o n s , due t o A. Robinson. The c e n t r a l r e s u l t i n T a r s k i ' s work seems t o me (3) a s I hope t o i n d i c a t e in

t h e d i s c u s s i o n of (4) and (5) below.

(Also, (1) and (2) are easy consequences Concerning (4): t h e s i n g l e most

of T a r s k i ' s method of e s t a b l i s h i n g ( 3 ) . )

f r u i t f u l f a c t i s t h e s o - c a l l e d Tarski-Seidenberg theorem: t h e image of a s e m i -

i s a semialgebraic

a l g e b r a i c s u b s e t of lRm under a semialgebraic map lRm ->Elu s u b s e t of

.

Rn

C l e a r l y t h i s i s t h e same a s t h e e x i s t e n c e of a q u a n t i f i e r

elimination f o r the structure than (3). (R

Q

,< ,

+ , *)

( r ) r ER,

which is s l i g h t l y weaker

(Semialgebraic = q u a n t i f i e r f r e e d e f i n a b l e with parameters i n

,< , + ,

.).)Another important p r o p e r t y of semialgebraic sets i s t h a t they

have only f i n i t e l y many connected components, and t h a t each component i s a l s o semialgebraic'

;

see [ K ] f o r a n i c e use of t h i s r e s u l t .

The b a s i c f a c t about (5) i s t h a t a continuous semialgebraic f u n c t i o n Rn ->R

t h i s follows

is bounded i n a b s o l u t e value by a polynomial function:

e a s i l y from ( 3 ) ; an important a p p l i c a t i o n occurs i n [ H 8 r , p. 2761.

A simple proof, due t o

r e s i s t giving one o t h e r b e a u t i f u l a p p l i c a t i o n :

K. McKenna, t h a t t h e inverse of a b i j e c t i v e polynomial map

a polynomial mal.

From complex a n a l y s i s we know t h a t

c l e a r l y the r e a l valued f u n c t i o n

z

+>

I cannot

p

-1

p:En

->

En

is holomorphic, and

i s continuous and semi-

Ip-'(z)l

a l g e b r a i c , hence bounded by a ( r e a l ) polynomial f u n c t i o n ( i d e n t i f y i n g

E2n). Therefore, by L i o u v i l l e , How t o extend a l l t h i s t o

p Q

-1

i s a polynomial map.

, + , , exp

is also

En

with

Q.E.D.

)?

I t seems t o me t h a t concentrating most a t t e n t i o n on t h e analogue of (l), t h a t i s , d e c i d a b i l i t y of the elementary theory, i s a waste of time:

consider

f o r example t h e perplexing problem of deciding t h e statements p(e, e

with

p EZ[X1,X2,X3,

...I

f r e e p a r t of the theory.)

.

e

e

, ee ,

...) = 0

,

(And t h i s i s j u s t a t i n y p a r t of the q u a n t i f i e r

A n a t u r a l 'exponential'

f i e l d ' does n o t seem l i k e l y (but see [vdD

11

analogue of ' r e a l closed

[D-W]),so I d o n ' t expect

This p r o p e r t y i s n o t an obvious consequence of T a r s k i ' s work; see a l s o the end of t h i s Introduction.

Remarks o n Tarski's Problem Concerning (W,f, . , exp)

99

an a t t r a c t i v e analogue of (2) f o r our exponential s t r u c t u r e . More p l a u s i b l e problems a r i s e i n the attempt t o extend (3),

(4) and (5).

To e x p l a i n t h i s l e t us go back t o t h e r e s u l t t h a t each semialgebraic s e t has only f i n i t e l y connected components, each semialgebraic.

This follows from

2 C o l l i n s [C] i n which a new d e c i s i o n method for the r e a l s is constructed, much more time e f f i c i e n t than T a r s k i ' s .

us h e r e :

But t h i s e f f i c i e n c y aspect does n o t concern

we a r e i n t e r e s t e d i n C o l l i n s ' key geometric idea, which he c a l l s

" c y l i n d r i c a l decomposition"; i t i s p a r t l y an a l t e r n a t i v e to, p a r t l y a considerable sharpening of t h e

notion

of q u a n t i f i e r elimination.

what a c y l i n d r i c a l decomposition of a s e t

X cRn

I n (3.6) we s h a l l define

is.

For the moment, we

i s the d i s j o i n t union of f i n i t e l y many

only mention t h a t such a s e t

X

a c e l l being a s u b s e t of IRn

m bmeomorphic t o a space R

,

m

,< n

.

The following c o n s i d e r a t i o n s i n d i c a t e t h a t a l t e r n a t i v e s t o (naive) quant i f i e r e l i m i n a t i o n a r e q u i t e welcome i n the s i t u a t i o n we a r e facing.

FAILURE OF 'NAIVE ' QUANTIFIER ELIMINATION The example below shows t h a t the elementary theory of

QR

,< ,

+ , , exp

(r)r

) does n o t admit e l i m i n a t i o n of q u a n t i f i e r s .

In

f a c t , much more i s t r u e :

Let

Proposition. functions,

Fi:R

"i

->

(Fi)i R

.

I

be any family of

Then the s t r u c t u r e (R,

admits q u a n t i f i e r e l i m i n a t i o n i f and only i f each (Note: Semialgebraic = d e f i n a b l e i n (p

,< ,

Fi

( r ) r ER,

(total) r e a l <,

analytic

(r)r ER,+,

-,

(Fi)i EI)

i s semialgebraic.

+,

)

, by

Tarski.)

The key t o t h i s n e g a t i v e r e s u l t is t h e following example which we s h a l l t r e a t before proving the proposition. Example. (Osgood, see [A, p. 1331)

.

f(x,y) = y.exp(x/y) {(x,y,z)ly in

@,<,

>

Then i t s graph

Define G(f)

O A 3 t ( z = y*exp(t)A t y = x))

* 9

exp

f a XR>

->R

by

i s the s u b s e t

of lR3

which i s obviously d e f i n a b l e

1.

The r e d u l t is a c t u a l l y due t o H. Whitney. See [& p. 1101 f o r an e l e g a n t proof.

L. VAN DEN DRIES

100 Claim. -

There a r e no ( r e a l ) a n a l y t i c functions

an open b a l l i n IR3

,

0

centered a t

such t h a t

boolean a l g e b r a generated by the s u b s e t s Proof. and

The c r u c i a l f a c t s about

i s n o t an "algebraic" function.

f

i = 1,

...,k ,

must vanish on

+0

Fi

nU

f o r each

d.3)

(Fi = 01 of

U

,

i

G(f)

and

nu

,

,

Fi:U ->IR

for all

Pd(x,y,f(x,y))

= 0

A t l e a s t one of the c E G(f)

Fi

nU

,

,

with

would c o n t a i n a whole neighborhood of

..., Pd a homogeneous polynomial of n U . Then f o r a l l 0 < X < 1 we have

+

d

.

XPl(x,y,z)

Take

d

+

with

2 XP2(x,y,z)

Pd

,

(x,y) E R x l ? '

for a l l

# 0

+

,

and

... ,

hence

and we see t h a t f

would be an a l g e b r a i c

o

We model the proof of the p r o p o s i t i o n on the argument j u s t given. f a n x I?->

IR be a n a l y t i c and d e f i n e

f (XI,.

..,xrrcl)

G(f)

i s quantifier free definable i n

= xrrclF(xl/xrrtl,.

.

. , X ~ / X ~ + ~ )

@

.

Let

1R by

If we assume t h a t i t s graph

, < , (r)r ER, + , * ,

P(x,f(x)) = 0

variables with

s u i t a b l e value

X >0

analytic function

for the

FXSZn ->IR

n+lst

,

s a t i s f i e s i d e n t i c a l l y an equation polynomial i n

n+l

variables.

a l g e b r a i c function, hence

.

3 , We assume here t h a t converges on U

U

F

for a l l

x EXn X R ?

0

.

,

(Fi)i EI)

then w e d e r i v e e x a c t l y as before t h a t t h e r e is a nonzero real polynomial n+2

F1

say

and

= Po(x,y,z)

Pd(x,y,z) = 0

->

,

+ P1 + P2 +

(x,y,z) E G(f)

Let

0 = F1(Xx,Xy,Xz)

F SZn

.

f(Xx,Xy) = M(x,y)(X>O,y>O)

(otherwise t h e r e would be

F1 = P 0

(Xx,Xy,XZ) E G(f)

function.

U

belongs t o t h e

Suppose t h e f u n c t i o n s

i s i d e n t i c a l l y zero.

G(f)

3 c E R ). Write degree

are:

nU

,

have the property we want t o r e f u t e ; we may of course assume

t h a t none of t h e

Fi(c)

f

G(f)

> 01 ,

{Fi

...,Fk :U ->a

Fl,

P

in

Substituting a

v a r i a b l e , we derive from t h i s t h a t the

given by (x,

,...,xn> &>

Q(x F (x)) = 0

' A

,

x Enn

xF(xl/X

,

The l e m below shows t h a t then

Q

,...,

xn / A )

a nonzero r e a l FX i s a semi-

i s a l s o semialgebraic.

i s taken so s m a l l t h a t t h i s Taylor s e r i e s of

F1

Remarks on Tarski's Problem Concerning (lR,+, . , exp)

kuima.

If a continuous function g d ->IR

equation Q(x,g(x))

'= 0

,x

101

satisfies identically an

E Rn , where Q is a nonzero real polynomial in

n+l variables, then g i s semialgebraic. From the results on pp. 106-110 of [El, it follows that Rn can

Proof.

be partitioned into semialgebraic subsets Ao,A1, connected such that if x E .A 1

-< i 5 m

and x E Ai

functions pil(x) < Pij:Ai ->lR

,

,

then Q(x,Y)

...,Am

of x

... Am

vanishes identically, while if

the real roots of Q(x,Y)

... < Pik(i)(x)

with A1,

.

are given by continuous

(Obviously, these functions

are semialgebraic.) By continuity and connectedness, g must

coincide on each Ai (i

> 0) with one of the functions Pij

,

hence g

... U Am is semialgebraic. Since every p .A is the limit of points in A1 u ... U Am , the value of g at p is semialgebraically determined by its values on A1 u ... u Am . It follows that g is semirestricted to A1

algebraic.

u

0

The proposition forces us to look for new ways of solving (the realistic part of) Tarski's problem. A line of attack which seems quite promising to Define a k-manifold, k EN

En , n 2 k

,

to be a real analytic submanifold M of some k equipped with an analytic isomorphism h : M " - I R We introduce

.

an IR-algebra

classes ~

goes roughly as follows:

,

of k-manifolds, and for each manifold

for each k a class Jf(k)

M E &@k)

me

(

strutted in stages.

~

&(M)

of

(real) analytic functions on M'.

for1 each M (N.B.,

introduce a l l of M ( M )

,

The

the algebra d ( M ,) are con-

as soon as we have M

,

we do not necessarily

at once.)

At stage 0 we introduce all semialgebraic k-manifolds and all semialgebraic (analytic) functions on them.

To obtain new manifolds, and functions

on them, as well as new functions on the manifolds already available, we use several constructions of which the following three are the most important.

Let

M

9 k-manifold already available.

102

L. VAN DEN DRIES

rf

(i)

are a l r e a d y a v a i l a b l e , then we introduce &he

f,g:M ->R

k-mani f o 1d graph(f)

t h e k+l-manifold

def (-m,f)M ,

(f,g)M

the k+l-manifolds If

(ii) f

0

fo,

...,f d :M ->R

+ fl Y +...+ ,

graph(f)

d fdY

(f,g)M

,

def

(x,f(x)) : x E MI

{(x,y) : f ( x ) (f,m)M

,

<

&

y

<

,

(if

g(x)]

f

<

g

fl M

),

&

defined s i m i l a r l y .

(-m,m)M,

a r e a v a i l a b l e , then w e introduce t h e f u n c t i o n

0" M XR (-m,f)M

,

,

=

(f,m)M

and a l s o i t s r e s t r i c t i o n s t o

(where

a r e as i n (i)), as w e l l

f,g

a s a l l those a n a l y t i c f u n c t i o n s which a r e a l g e b r a i c over t h e 'polynomial' functions so obtained. (iii) $:M

X1,

.

=IR k

...,\

be t h e coordinate functions on

M

defined by t h e c h a r t

Then we introduce as new functions on

M

those a n a l y t i c

f:M ->R

functions

which s a t i s f y a system of d i f f e r e n t i a l equations -a=f

where each

Fi

Fi(X1,.

..,%,f) ,

,...,k

i = 1

is a f u n c t i o n a l r e a d y a v a i l a b l e , e.g.,

,

v i a constructions ( i )

and ( i i ) . Now t h e d e l i c a t e p a r t is t o show t h a t t h e z e r o s e t of each f u n c t i o n i n

,

i d ( M )

,

M

as w e l l a s i t s complement, i s a d i s j o i n t union of

.

U ./fci) This can only be proved i f the coni l k s t r u c t i o n i s done i n e x a c t l y the r i g h t o r d e r ( t h e ' s t a g e s ' have t o be indexed

f i n i t e l y many manifolds i n

by a s u i t a b l y chosen w e l l o r d e r i n g ) , and i f t h e c o r r e c t induction hypotheses are selected.

(The i n d u c t i o n hypothesis on the "asymptotic behavior of

z e r o s e t s " seems t o be t h e c r u c i a l problem, b u t I have an i d e a as t o what i t should be.

Since p r e c i s e statements are complicated and I expect t o come back

t o i t l a t e r , I ' l l leave i t a t these vague i n d i c a t i o n s . ) Remarks

(1) All f u n c t i o n s b u i l t up from real constants, v a r i a b l e s , and t h e o p e r a t i o n s

+ , * , and

exp

a r e e v e n t u a l l y introduced v i a c o n s t r u c t i o n ( i i i ) , b u t i t i s

Remarks on Tarski's Problem Concerning

(R,+, , exp)

103

'

i n t e r e s t i n g t h a t i t i s the system of d i f f e r e n t i a l equations which is c r u c i a l t o prove t h e r e q u i r e d r e s u l t s f o r t h e s e functions (and t h e i r z e r o s e t s ) , and n o t the d e f i n i n g expression i n terms of

+,

*

, and

exp

.4 )

I f t h e program w e just sketched works out, then one o b t a i n s a Tarski-

(2)

Seidenberg theorem f o r a very l a r g e c l a s s of sets and functions, including a l l those d e f i n a b l e i n o u r exponential s t r u c t u r e , as w e l l as the r e s u l t t h a t each of

these s e t s i s a d i s j o i n t union of f i n i t e l y many k-manifolds.

Quite a p a r t from

i t s i n t e r e s t f o r T a r s k i ' s problem, i t would be very d e s i r a b l e to o b t a i n r e s u l t s

i n t h i s d i r e c t i o n , f o r example, i n connection with H i l b e r t ' s 1 6 t h problem, see [Hovl, [ P I . References [vdD2] and [Hov] g i v e p a r t i a l r e s u l t s i n t h e s p i r i t of t h e pro-

(3)

gram sketched above, and l e d m e t o t h e p r e s e n t formulation. I n t h i s paper w e s h a l l c a r r y o u t a more r e l a x e d i n v e s t i g a t i o n which has t h e advantage of introducing the n o t i o n of c y l i n d r i c a l decomposition i n the most

9 be

n a t u r a l and p a i n l e s s way:

of

lRm

9.

any expansion of

.

@I,< A s) ubset

i s c a l l e d d e f i n a b l e i f it i s d e f i n a b l e by a formula i n the language of

- t o be introduced i n t h e @ , < , + , ') - we s h a l l

Under a convenient assumption on T h ( 9 )

! next s e c t i o n , and obviously s a t i s f i e d i f &= inter alia: (*)

is piecewise continuous, c . f .

Each d e f i n a b l e f u n c t i o n IR ->W

(**) Each d e f i n a b l e s u b s e t of IRm

(2.2).

is the d i s j o i n t union of f i n i t e l y many

c e l l s , each of which i s a l s o d e f i n a b l e . See (3.11).

(Cell = space

k

homeomorphic t o an R .) (It is of some i n t e r e s t t h a t t h i s g i v e s a proof, based on q u i t e g e n e r a l

p r i n c i p l e s , of Whitney's " f i n i t e number of components" r e s u l t on semialgebraic

sets.)

I f we assume moreover that

2 is

an expansion of

(R

, < , +) ,

4 , The treatment i n [vdDL] could have been s i m p l i f i e d and generalized cons i d e r a b l y i f t h i s p o i n t had been c l e a r e r t o me.

then

L. VAN DEN DRIES

104

(*) and (**) can be strengthened i n the sense t h a t i n (*) the function

,

f o r each

n

piecewise

c e l l s as

Cn-submanifolds of

,

Cn

and i n (*),

.

lRm

one can take, given any

n

f

is,

,

the

In an appendix we s h a l l indicate the proof

of t h i s and make some remarks on the connection with the important notion of Whitney s t r a t i f i c a t i o n .

GENERAL CONVENTIONS

d= (A, ...)

In t h i s a r t i c l e we consider L-structures a f i r s t - o r d e r language with equality.

L

is said to define the set

{al,.

..,an)

An L-fomula

1 &F

E An

$

...,vn,vn+,,)

= $(vl,

1 &p

{a E A

i s given and

x

X C Am

,

Y

C

...,xm, yl,. ..,Y,)

A function

f:X

->

E

1

If

denotes the s e t

is called

is

X

if

i n that their

Y) of

.

A*

(A-)definable i f i t s graph

(A-)definable.

vary over elements of R = {0,1,2,,

the number of elements of

A"

in the usual way with the subset

..,xm) E X, (yl,. ..,Y),

(xl,.

A1-

A(X C Am)

k, a,m,n

We l e t

X

is f i n i t e ;

..]

otherwise

-.

and w r i t e

#X =

#X

for

Theories of f i n i t e type

(1.1) L e t

(A,<)

be a nonempty dense l i n e a r order without endpoints.

-- ,

notational purposes we adjoin two 'endpoints'

-- < a < + = subset that

#(x,A)

2

we allow parameters.

we i d e n t i f y X X Y

An

G(f) = {(x,f(x)) Ix E X I C Awl

51.

then

.

An

generalize definable subsets of

An

defining formula a r e LA-fomulas, i.e.,

{(xl,

,

An

..,an) ]

, ( i n d) , n

$(x,a)] ; these s e t s a r e s p e c i a l cases of A-definable s e t s :

A-definable subsets of

If

E

@,

..,vn),

$(vl,.

$(al,.

and a s e t of t h i s form i s c a l l e d a definable subset of

s= ...) ,

and

a

for a l l

(a,b)

&&!

and

b

a

EA

.

Let

(r E Ala < r < b3 of

2=

A

A ,

u

+m

.

E-m,+=J

where

--
a r e uniquely detemined by the interval.

c l e a r l y i n f i n i t e and

(A,<)-def inable.

,

For

and p u t ~n i n t e r v a l i s a


<+m

.

An i n t e r v a l i s

N ote

Remarks on Tarski‘s Problem Concerning (R, +, . , exp)

A subset

(1.2) Definition.

of

X

105

i s c a l l e d of f i n i t e type i f

A

is the

X

union of a f i n i t e s e t and f i n i t e l y many i n t e r v a l s . Example: X

If

(A,<)

then a s e t

= @,<)

has only f i n i t e l y many connected components.

(1.3) D e f i n i t i o n . bd(X) = {a

E

X

For

C

we d e f i n e t h e boundarx of

A

Note t h a t f o r

(A,<) = @,<)

m. (1)

bde)

(2)

if a

0

-OD

,

... < am .

awl

Let

...,am] ,

=

X

i = 1,

..,m

...,m I

A

m is

{-1, + l j

al

<

... < am

each i n t e r v a l

,

X

then, p u t t i n g

,

(ai,ai+l)

N

A

be of f i n i t e type and

m of

i = 0

,...,m ,

. {al,

...,am]

i t s boundary,

i s t h e sequence 7 = < T1,T2,

X

X

...,

T

:

+1 i f

(ai,ai+l)

C

X

-1 i f

(ai,ai+l)

C

A\X

:

ai E X

+1 i f

a f i n i t e sequence

.

where

o r d i s j o i n t from

X

C

Then t h e

i = 0,.

where f o r each

and f o r each

Then:

L e f t t o the reader.

(1.5) D e f i n i t i o n .

<

as

.

i s o f f i n i t e type.

e i t h e r i s p a r t of Proof. -

X

is f i n i t e ;

bd(X) = (al, =

.

XI

t h i s gives t h e u s u a l n o t i o n of boundary of

X C A

Suppose

t o be t h e set

X

c o n t a i n s both a p o i n t

and a p o i n t o u t s i d e

a s u b s e t of t h e t o p o l o g i c a l space IR (1.4)

a

Alevery i n t e r v a l containing

from X

al

i s of f i n i t e type i f and only i f

X CR

7=

<

T1,.

..,72 ,

i s c a l l e d t h e l e n g t h of

7

.

a l l whose terms

‘Ti are i n

~

~

~

L. VAN DEN DRIES

106

(1.6) In the remainder of t h i s section we assume t h a t the language a binary predicate symbol

<

J?f= (A,<,

and t h a t the L-structure

expansion of the dense l i n e a r ordering without endpoints Definition.

We c a l l an L-theory

T

A-definable subset of

d= (A,<, ...)

We c a l l

A

of f i n i t e type i f

(This condition implies t h a t each

contains

...) a

.

(A,<)

(extending the theory of dense l i n e a r

order without endpoints) of f i n i t e type i f each model property t h a t each

L

d= (A,<, ...)

has the

i s of f i n i t e type.

m(&)

is of f i n i t e type.

A-definable subset of

i s of f i n i t e

A

type, but the converse implication i s probably not valid.) (1.7) (bd

=.

For each

$) (vl,

... , v ~ + ~such )

Proof.

(1.8)

for a l l

that for a l l

&f= (A,< ,...)

If

then there i s

a

...,v,+~) E

there is a formula

An :

Proof. 3 vn+l $(vl,

4

With respect t o

...,vn+,)


i s of f i n i t e type and

k = k EN

.

a E A”

m

3 x 3y(x

$(vl,

Clear.

-.

L-formula,

L-formula

such t h a t

,...,v ~ + ~i s) an { O , l , ...,k 3 U {=I

$(vl

# $(a,A) E

the ‘ i n f i n i t a r y formula’

T h ( d )

i s equivalent t o the formula

A V V ~ + ~<( X vn+l

< y ->

$(v,,

we can use a simple ...,v ~ + ~ ) ) ) and ,

model theoretic compactness argument t o e s t a b l i s h the existence of a number

k

as required. (1.9) Proposition. an L-formula. subsets

X1,

each

and

i

Proof. #(bd $(x,A))

Let

d= (A,< ,...)

be of f i n i t e type and

Then there i s a f i n i t e p a r t i t i o n of

...,)(M, x

E

and there a r e d i s t i n c t types Xi

the s e t

There is

-< k ,

k EN

#(x,A)

i s of type

such t h a t f o r each

by (1.4), (1.7) and (1.8).

An

#(vl

,...,v ~ + ~ )

i n t o d i s j o i n t definable

T(l), T(i) x E An

...,

T(M)

such t h a t f o r

.

we have

Now each s e t

$(x,A)

i s of

Remarks on Tarski's Problem Concerning (IR,f , . , exp) f i n i t e type, and t h e type of

$(x,A)

5

i s moreover of length

107

f i n i t e l y many types a c t u a l l y occur among t h e types of t h e sets Let

T(l),,..,T(M)

f o r which

be these types and d e f i n e

$(x,A)

has type

.

T(i)

,

2k+l

so only

,

$(x,A)

as t h e s e t of a l l

Xi

x

E

x

E

An.

An

0

Remarks (1)

E

x

For

let

Xi

T(i)

have length

bd b(x,A) = ( f i , l ( x ) , . . . , f i , m ( i ) ( x ) ~ the functions

fi,

:xi ->

+ 1,

2m(i)

where

<...

fi,l(x)

Then


d.

are d e f i n a b l e i n

A

and l e t

(2)

The p r o p o s i t i o n w i l l be used l a t e r i n an i n d u c t i v e argument (see (3.10).

02.

Piecewise c o n t i n u i t y of d e f i n a b l e f u n c t i o n s

9i s

Throughout t h i s s e c t i o n

t h a t each IR-definable s u b s e t of

R

an expansion of

@,<)

has only f i n i t e l y many connected components.

9 i s of

( This assumption seems t o be somewhat weaker than:

m.

(2.1) f

under

n

f

,

f-'(b)

.

E

b

Then

is

f(J)

,

so t h e r e i s

J

i s c o n s t a n t on

f

f i n i t e type.) IR-definable, then

.

I

J

of

whose image

I

i s the union of the f i n i t e l y many s e t s

J

IR-definable s e t

Then

f : I ->R

Assume f i r s t t h a t there i s a s u b i n t e r v a l

is f i n i t e .

hence t h e J

i s an i n t e r v a l and

i s continuous i n a t l e a s t one p o i n t of Proof, -

J

I cIR

If

with the property

b

such t h a t

n f-'(b) J'

,

J

n f-'(b)

is infinite,

must c o n t a i n a s u b i n t e r v a l

hence

f

J'

of

i s continuous a t each p o i n t of

I n the remainder of t h i s proof we s h a l l assume t h a t each s u b i n t e r v a l of has i n f i n i t e image under of closed segments f[an,bn]

f

.

[an,bn] c I

with

0

< bn-an <

l/n

,n2

Jn of length

1

,

<

l/n

such t h a t

.

[al,bl]

and

length

<1

J1

n [an,bn] n= 1 we take an i n t e r v a l of

i s continuous a t the unique p o i n t i n a r e obtained a s follows:

contained i n the

for

J1

IR-definable i n f i n i t e s e t

i s chosen a s a closed segment contained i n the

It is

.

rn

f

I

I n d u c t i v e l y we s h a l l f i n d a descending sequence

i s contained i n an (open) i n t e r v a l

c l e a r t h a t then

J'.

f(1)

,

and

[al,bl]

IR-definable i n f i n i t e s e t

L. VAN DEN DRIES

108

,

f-l(J,)

Jnfl C f [ a n , b n ]

< b&l-an+l

0

(2.2)

Corollary.

< If

,

obviously

Proof.

.

I

f[a,b]

->

(a,b)

on which

g

,

al

<

a t which

a

,


then

... < a m ,

a o = a ,

.

(ai,ai+l)

f

f

i s c o n s t a n t o r s t r i c t l y monotone.


a

,

and assume t h a t

f

,

c

by:

[c,d]

E

g ( y ) = minCx

<

... < a

of

If

,

e i t h e r c o n s t a n t on (ai,ai+l)

Proof.

.

,

d

,

[c,d]

so

g

->R

f:(a,b)

a. = a (ai,ai+l)

I n particular

,

a

m

i s continuous on

[c',d']

R-definable,

,

[c',d']

under

0

then t h e r e a r e

such t h a t f o r each

l i m f(x) x1a

and

(a,b)

exist i n

l i m f(x) xt b

i

<m

is

f

cannot c o n t a i n a whole s u b i n t e r v a l o f

,...,amml) ,

4

.

which have no neighborhood on which

i s c o n s t a n t , o r continuous and s t r i c t l y monotone, i s

{al

By

o r continuous and s t r i c t l y monotone on

The s e t of p o i n t s i n

must be a f i n i t e s e t

.

is s t r i c t l y monotone on

is = b

and we

[ a , b ] : f ( x ) = y)

i s s t r i c t l y monotone on t h e image of

is

f

is n o t c o n s t a n t

f ( g y ) = y ), and t h i s image c o n t a i n s an i n t e r v a l .

(since

(a,b)

IR - d e f i n a b l e then

is

i s i n j e c t i v e , and by t h e p r e v i o u s c o r o l l a r y g [c',d']

f

i s n o t continuous i s

f

f : I ->R

c o n t a i n s a segment

[a,b]

(2.4) Corollary.

f

EI

Then

and t h e r e f o r e

1<

<

a.

i s a n i n t e r v a l and

I CIR

a,b

a subsegment

aO < a

R -definable,

is

Hence, i t must be a f i n i t e s e t . 0

Take

g:[c,d]

definition

g

R

By the previous c o r o l l a r y we may as w e l l assume t h a t

continuous.

define

(Jn+l)

a - d e f i n a b l e , b u t i t cannot c o n t a i n a whole s u b i n t e r v a l of

e. If

[a,b]

-1

[an+l,bn+l]Cf

i s continuous on each s u b i n t e r v a l

t h e r e i s a s u b i n t e r v a l of

on

we choose a n i n t e r v a l

then we choose

that is, there are

f

by t h e p r e v i o u s lemma.

(2.3)

->

The s e t of p o i n t s i n

Proof.

;

l/n+l

f:(a,b)

such t h a t

[an,b ]

Given

.

l/n+l

i s piecewise continuous,

a = b m

<

of l e n g t h

with

.

< bl-al < 1

0

with

(a,b)

,

with

a = a0 < a l <

by t h e p r e v i o u s lemma.

It i s e a s y t o s e e t h a t on each s u b i n t e r v a l (ai,ai+l) c o n s t a n t , o r continuous and s t r i c t l y monotone.

a-definable,

...
the function

f

and So i t

= b

.

is either

Remarks on Tarski's Problem Concerning (lR,+, . , exp) 13.

Cylindrical

Decomposition

-

(3.1) This s e c t i o n c o n t a i n s the main r e s u l t , Theorem (3.7). Recall t h a t W = R

ventions and d e f i n i t i o n s . a l l an e x p a n s i o n 9

u

(3.2) Given a nonempty d e f i n a b l e s u b s e t

of

X

the s e t of d e f i n a b l e continuous f u n c t i o n s

f(x)

<

g(x)

.

(--,m]

for a l l

For

E

x

f,g

and

-m

G(f) = { ( x , f ( x ) ) : x E X ] CRW1


we write

(f,g)X

I t i s h e l p f u l t o view

parametrized by

X

(f,g)X

.

i s a l s o homeomorphic t o IR

,

and

(3.3) By i n d u c t i o n w e d e f i n e f o r each (il,

...,i n )

X .)

namely n = 1: F(0) F(1)

no= (01

{a] with

of

E F(il, F(0)

F(1) .)

,

f

F(il,

and

f 6 C(X) and i f and

we l e t

E

f,g

,

t(X)

f(x)
,

then

G(f)

.

...,in) of

nonempty de-

F 0 has onl'y one member,

a ER

W

definable i n

9.

whose ( f i n i t e ) endpoints

G(f)

with

F(il,. f E C(X)

.. , i n ) ,

...,i n , l ) t o c o n s i s t of a l l sets ...,i n ) , f , g E 8(X) , f < g . (Note: and

if

9.

More generally, given the c o l l e c t i o n

X

f )

and each f i n i t e sequence

0

0

c o n s i s t s of a l l i n t e r v a l s of

F(il,

>

g

k+l ( f , g ) X i s homeomorphic t o R

2

be

.

c o n s i s t s of a l l

are definable i n

define

C(X)

.

f

c o n s i s t s of a l l graphs

(or

k X i s homeomorphic t o R

there is o n l y t h e empty sequence

n = 0:

we l e t

and we p u t


If

((x,y)lx E X

n

refers

are i d e n t i f i e d with the

m

of zeros and ones a c o l l e c t i o n

f i n a b l e s u b s e t s of

, f

,

20

m

as t h e family of i n t e r v a l s

Note t h a t i f k

,

Wm

be t h e graph of

f o r t h e set

Rm and Rn

.9.

E ?(X) we w r i t e

X . mere

(So the

The term ' d e f i n a b l e ' a s

f:X ->W

corresponding c o n s t a n t f u n c t i o n s defined on

f

(1.6).

and functions between s u b s e t s of

33"

t o d e f i n a b i l i t y by an L - f o m l a i n the s t r u c t u r e

u

We f i x once and f o r

which i s of f i n i t e type, c.f.

@,<)

applied t o s u b s e t s of

F i r s t some con-

.

(-m,m)

r e s u l t s of Sections 1 and 2 a r e a v a i l a b l e t o us.)

?(X) = C(X)

109

X

we define

E

F(il,

(f,g)X

F(il,.

...,i n )

,

.. , i n , O ) and we

where

t h i s g e n e r a l i z e s the d e f i n i t i o n

to

L.VAN DEN DRIES

110 Clearly, disjoint.

k = il

where only i f let

Also, i f

+

X

,

n

f o r given

E

.

E

X

F(n)

.

.,in) a r e mutually

k i s homeomorphic t o R

X CRn

dim(X) = k

collections

2"

a c o l l e c t i o n of d e f i n a b l e s u b s e t s of

(3.4) For

then

.

F(il,.

dim(X) = n

Note:

such t h a t

Rn

n .)

...,i n )

,

We

F ( ~ ) is

so

.


dim(X)

F(il,

,

i f and

( A l l t h i s i s i m e d i a t e by induction on

be the union of the

F(n)

collections

,

We p u t

i s open i n Rn

X

.., i n )

F(il,.

... + in .

2"

the

we s h a l l d e f i n e a d e f i n a b l e homeo-

morphism %:X

If

E

.

where

,

< n-1

dim@) = dim(hX)

h(X)

h(X)

By induction t h i s shows t h a t i f

In c a s e Y = h(X)

i s a graph

X

X = (f,g)y

Suppose


G(f)

$:x ->

and l e t

implies

where

Y

where

E

,

< n-1

dim(Y)

be t h e s e t

(3.5) For each s e t X p a r t i t i o n s of Dec(X)

X

n = 0

we have

Let

n

20

X E F(n)

T:G(f)

X

,

C(Y)

Y

X


F(n-l)

(y,r)

,

we put

+>

y

f , g E e(Y)

, gh;' . Clearly %

E ?(hY)

. . Then

,

and we

l i f t s t o a homeo-

.

we s h a l l d e f i n e a c o l l e c t i o n F(n)

.

DeC(X)

of f i n i t e

We c a l l a member of

. 0

X =W is

.

%'

We g e t

$(y),r)

F(n)

f

i s definably

so by r e c u r s i o n we may assume t h a t

, gh;l)h(y) +>

,

F(n-l)

E

X

{XI

,

since ;

F(O) =

we let

1x3

ho),

and the only

be t h e only member of

.

Dec(X)

X

E

i n t o s e t s a l s o belonging t o

( f i n i t e ) p a r t i t i o n of

of

in

a decomposition of

For

Let

(%'

hx:X Z h(X) : ( y , r )

morphism

f

then

FOCI

Y be t h e p r o j e c t i o n map

Y

h(X)

,

dim X = k

belonging t o

h :Y Z h(Y) has a l r e a d y been defined. let

.

F(n'l)

then one can r e p e a t t h e same c o n s t r u c t i o n with

homeomorphic t o an open s u b s e t of JRk

dim(X)

E

h(X)

and suppose and

f E C(X)

l i f t s t o one of

->

X

G(f)

Dec(X)

. ,

has been defined f o r a l l

Then each f i n i t e p a r t i t i o n -1 namely (TT-'(A1), ...,Tr (Ad)]

i s the p r o j e c t i o n on t h e f i r s t

n

coordinates.

X

E

F(n)

g ={A1, ,

. ...,Ad]

where

Let us c a l l t h i s

Remarks o n Tarski's Problem Concerning (R, +, . , exp)

'lifted' partition

gf.

Then we define Dec(G(f)) for f,g E ?(X)

The definition of Dec((f,g)X) complicated. Let again

9 E DecO() .

Then

=

B={Al,..., Ad

111

(gf 1 9E Dec(X)3

,

f

4 g

,

.

is a bit more

be a decomposition of X

,

i.e.,

9 induces a partition

E(f(Al,g(A

,...,(flAd,gIA )Ad 1

A1

of

(f,g)x

,

which we denote by

A second kind of partition of

,...,fm E c"(x)

fo,fl

partition of

with

(f,g)X

and the sets G(fi)

gfTg . (f,g)X

f = fo < fl <

is obtained as follows: let

... 4 fm = g

consisting of the sets

,

0< i< m

.

on X

(fi,fi+l)X

,

.

'ken we have the

i = O,..,,m-1

We denote this partition by

and call it a proper decomposition of

.

(f,g)X

,

(foIfl~...Ifm)X

("Proper" because it is not

induced by a decomposition of X .)

g =(A1, ...,Ad] E Dec(X) and that for each Ai a is given. Then -9/1 u ... u gd proper decomposition giof (f(Ai,glAi)Ai is a partition of (f,g)X which we denote by . We define Dec((f,g)X) as consisting of all partitions <3; [ sl,. ..,2d) > . We call 9 the base of the decomposition <9;( gl,. ..,3 d 1 > . Suppose now that

(3.6) Let us say that a decomposition A

is the union of a subcollection of

in

9 is contained in

of lRn

9,

partitions a set A CRn

if

in other words, if each set

A or disjoint from A

.

The following terminology will not be used in the remainder of this section, but it might be helpful in comparing with results in [ C ] . decomposition is a decomposition of lRn some

n

.

,

that is, a member of

A cylindrical

Dee@) ,

A cylindrical decomposition which partitions a set A CIRn

called a cylindrical decomposition of A

.

is also

With this terminology the next

theorem states that for any definable sets A1,AZ, cylindrical decomposition.

for

...,Am clRn

there is a c o m n

L.VAN DEN DRIES

112

n-1 X CIR

( a ) For each d e f i n a b l e s e t

X i n t o d e f i n a b l e s e t s on each of which

f i n i t e p a r t i t i o n of

(b) Given any d e f i n a b l e s u b s e t s p a r t i t i o n i n g each of

0

X =

or

...,Am

A1,

...,Am .

Al,

The proof i s by i n d u c t i o n on (since

f:X ->B

and d e f i n a b l e

.

n

of

Rn

t h e r e is a

i s continuous.

f

9€ DecQRn)

there i s

n = 1 statement (a) holds t r i v i a l l y

For

i s a s i n g l e t o n ) , and (b) i s an easy e x e r c i s e i n boolean

X

a l g e b r a u s i n g the f a c t t h a t each

Ai CIR

i s of f i n i t e type.

The induction

s t e p s take p l a c e i n t h e proof of t h e following t h r e e lenrmas.

m.

(3.8) all

for

n

.


N

Suppose

>

1 and statements (a) and (b) of (3.7) hold f o r

Then statement (a) holds f o r

Proof. -

Let

n = N-1

we may r e s t r i c t t o t h e c a s e t h a t

x

be d e f i n a b l e and

cIRN-l

Suppose f i r s t t h a t

dim X

< N-1

homeomorphism h:X 2' X '

where

XI

.

.

n = N f:X X

-> E

R

definable.

Using (b)

.

F(N-l)

Then, by ( 3 . 4 ) , we have a d e f i n a b l e

E F(N'2)

.

The f u n c t i o n

fh-':X'

->a

is d e f i n a b l e , so by the i n d u c t i o n hypothesis t h e r e i s a f i n i t e p a r t i t i o n of i n t o d e f i n a b l e s e t s on each o f which partition via o f which

f

w e g e t a f i n i t e p a r t i t i o n of

h

X

i n t o d e f i n a b l e s e t s on each

The set

Y = {x

dim X = N-1

E Xlf is

.

use (b) f o r

n = N-1

Y

.

X\Y

continuous a t

x)

contained i n

Y

,

so

which a r e contained i n

i s dense i n

Y

t o g e t a decomposition g o f lRN-'

The s e t s i n

9which

a r e contained i n

X

arguments used i n t h e c a s e

dim X

< N-1

and a r e open must be

< N-1

The sets i n

,

9

and so the

apply.

To prove t h e claim we t a k e any nonempty open s e t i s continuous i n a t l e a s t one p o i n t of

.

which p a r t i t i o n s both X

but n o t open a r e of dimension

X

i s definable, we can

i s continuous o n each of those sets.

f

.

Then X i s open i n

I f we accept t h i s claim f o r a m m n t then, s i n c e

and

I f we l i f t t h i s

is continuous.

W e a r e l e f t with the case Claim. -

is continuous.

fh-'

X'

U

U C X

.

To show t h a t

w e use e x a c t l y t h e same arguments

f

Remarks on Tarski's Problem Concerning (lR,+, . , exp) The r o l e of the decreasing segments

a s i n the proof of (2.1).

113

i s now

[an,bn]

U

of course taken over by a decreasing sequence of closed b a l l s contained i n

their diameter tending t o

11 (xl ,...,%-1) 11

by the norm b a l l s are

@ere we l e t the distance on rPN-' = max (lxll

,...,IX+~~) ,

be defined

i n order t h a t the

R-definable.)

m.

(3.9)

.

0

,

Assume t h a t

N

Then any two decompositions

>1

n = N-1

and statement (b) of ( 3 . 7 ) holds f o r

gland g2of

EN have a COrrrmOn refinement,

t h a t is, there is a decomposition of RN p a r t i t i o n i n g each of the sets i n

2B1 u s 2 .

Proof.

Note t h a t RN =

(-=,-& N

s t r u c t u r e of decompositions of R t h a t @) holds f o r a conmmn base

where

n

= N-1

see (3.5).

.

.,Bd]

qi

n-1

is a decomposition of B

Bij

,

fh

,

and

,

g* p a r t i t i o n s

in

(-=,")

,

,

are definable and continuous.

we have:

Bi Bi

= (-"Jfl/...lfpl-)Bi

, gCr:,Bi - 3 3 R

Fix a set

(proper) decompositions of

By the assumption t h a t (b) holds f o r that

gland g2have

.

can be partitioned i n t o f i n i t e l y many definable sets any

This gives us the

So we can use the assumption

to reduce t o the case t h a t

9 and say gli,32ia r e

fh

,

.

X =lRwl

9 ,say

g=[B1,.

where the

where

B

Now c l e a r l y

Bi

such that, given

is

< $(x)

either for a l l

x

E

Bij:

fh(x)

o r for a l l

x

E

Bij:

fX(x) = g,(x)

or for a l l

x

E

Bij:

fh(x)

n = N-1

each of the sets

Bij

there i s

> g,,(x)

g*E DecCpN-')

. Now it should

,

,

.

such

b e c l e a r how to

.

L.VAN DEN DRIES

114

the

gp )

f X and

g* (and using r e s t r i c t i o n s s e t i n glU g2. [I

RN w i t h base

c o n s t r u c t a decomposition of

which p a r t i t i o n s each

of

Note t h a t Lemma (3.9) i m d i a t e l y extends t o a f i n i t e c o l l e c t i o n of de-

.

N

compositions of lR

It i s t h i s s t r o n g e r form t h a t we s h a l l use i n t h e

f i n a l s t e p which follows now.

e. Suppose

(3.10)

statement (b) holds f o r Let

Proof. consider

N-1

Ai

Formula f o r

>

B c n(Ai)

f(B,l)

gB % (--I

n = N

Then statement (b) holds f o r

...,Am

A1,

of IRN

be given.

nRN ->lRN-'

where

n = N

and

.

Fix an Ai

and

i s t h e p r o j e c t i o n on

We a r e going t o apply (1.9) t o t h e d e f i n i n g We a l s o use

These c o n s i d e r a t i o n s give us a decomposition

partitioning

there is

n Ai

.

t a k i n g i n t o account Remark (1) following (1.9).

,...,f(B,k):B

n-l(B)

statement (a) of (3.7) holds f o r

rr(Ai)

t h e hypothesis of t h e lemma. . o f lRN-'

,

d e f i n a b l e sets

coordinates.

,

1

n = N-1

a s l y i n g over

Ai

the f i r s t

N

n(Ai)

and such t h a t f o r each s e t

B E

si

with

k = k(B) and t h e r e a r e d e f i n a b l e continuous f u n c t i o n s with

->lR

f(B,1)

<

... < f(B,k)

on

B

such t h a t

is p a r t i t i o n e d by t h e decomposition f(B, 1)

1.. . I

f(B,k) 1 9

n - l ( ~ ) = (-a,m)B

of

.

It i s

il-l(B)

c l e a r t h a t t h i s g i v e s us a decomposition

B of

WN

which p a r t i t i o n s

9; ,...,

Ai

.

p a r t i t i o n s each of

Now a c o m n refinement

A1

,...,Am .

n il(Ai)

=@

>

@ E Dec@RN)

of

u

This completes t h e proof of Theorem (3.7). (3.11) Corollary.

Each d e f i n a b l e s u b s e t of

Rn has only f i n i t e l y many connected

components, and each component i s a l s o d e f i n a b l e .

Remarks on Tarski's Problem Concerning (IR,+, . , exp)

Proof.

By (3.7) each definable subset

decomposition

9of

.

lRn

A

of

Rn

is partitioned by a

Now each of the f i n i t e l y many s e t s i n

(definable) c e l l , hence connected.

Each component of

of f i n i t e l y many c e l l s belonging t o

9,

115

9i s

a

i s therefore a union

A

and i s therefore definable.

( 3 . 1 2 ) Remarks (1) Theorem ( 3 . 7 ) was derived under the assumption t h a t

9i s

arguments i n ( 3 . 2 ) expansion

2 of

of f i n i t e type.

-

f i n i t e type

T h ( 2 ) ). Conversely, a weak form o f ( 3 . 7 )

(which i s r e a l l y an assumption on implies t h a t

9i s of

To be precise:

a l l of the definitions and

( 3 . 6 ) make sense and go through without change f o r any

@,<)

,

whether of f i n i t e type or not.

Now we have the

following f a c t .

If

2 is

finable subset titioning

A

,

an expansion of A

of lRn

then

,

@,<)

n

9i s of

2

1

,

with the property t h a t f o r each dethere i s a decomposition of IRn

f i n i t e type.

(Hint:

par-

derive f i r s t (1.9) f o r

d= 9-1 (2) The assumption that

9i s

with names f o r r e a l numbers.

of f i n i t e type remains v a l i d upon expanding

9

Therefore, Theorem ( 3 . 7 ) also applies t o

R-definable s e t s and functions where of course the notion of decomposition of Rn

i s s u i t a b l y relativized.

(3) The condition t h a t concrete case.

9i s

of f i n i t e type might be d i f f i c u l t t o prove i n a

The point of t h i s paper i s j u s t t o show what more i s true i f

t h i s condition holds.

For 'exponentiation' one may t r y to use t h i s knowledge

the other way around:

since I expect t h a t

QR,

<,+ ,

, exp)

is of f i n i t e

type, b u t don't see any model theoretic technique t o prove t h i s , I prefer t o s t a y inside W

and use the analytic machinery available there t o e s t a b l i s h

d i r e c t l y the conclusion of Theorem ( 3 . 7 ) . method t o achieve t h i s .

The introduction sketches a (tentative)

L. VAN DEN DRIES

116

(4)

9i s

of f i n i t e type i f and only i f each i n f i n i t e R ' - d e f i n a b l e s u b s e t

,

of R'

where

...) =9, has nonempty i n t e r i o r

9' =

@I,<,

(i.e.,

contains

an i n t e r v a l of IR' ). (Exercise.) The second p a r t of t h i s equivalence a l s o makes s e n s e f o r o t h e r topological-algebraic structures. analogue of t h i s second h a l f f o r

Now Macintyre [Mac] e s t a b l i s h e d t h e p - a d i c a l l y closed f i e l d s .

t o use this f a c t t o prove an analogue of Theorem (3.7) numbers

It would be n i c e

f o r t h e f i e l d of p-adic

5 .

(5) The r e l a t i o n between ' c y l i n d r i c a l decomposition' and ' q u a n t i f i e r e l i m i n a t i o n ' i s the following:

i f each q u a n t i f i e r f r e e d e f i n a b l e s u b s e t of

t i f i e r f r e e d e f i n a b l e c y l i n d r i c a l decomposition, then t i f i e r elimination.

Rm has a quan-

T h ( 2)

This is how C o l l i n s [C] proves t h a t

QR

admits quan-

,< ,0 ,

1

, + , .)

has q u a n t i f i e r e l i m i n a t i o n .

(6) Let

9be

an expansion of

@

.

,< ,+ , -)

P o s t u l a t i n g t h e f i n i t e type

c o n d i t i o n i s of course a r a t h e r d r a s t i c way of avoiding t h e G'ddel phenomena t h a t would appear i f

N were definable. I n t h i s connection I would l i k e t o know t h e

answer t o the following question: i n f i n i t e d i s c r e t e s u b s e t of IR

i s the s e t N

d e f i n a b l e whenever some

is definable?

APPENDIX

Throughout t h i s appendix we assume t h a t

2 is

an expansion of

@,

<,

i s of f i n i t e type, i.e.,

with the p r o p e r t y t h a t each B-definable s u b s e t of B has only f i n i t e l y many connected components. I\

Recall t h e n o t a t i o n IR = IR

(A.l)

w.

R-definable.

I

Suppose Then both 4

s i m i l a r l y ) e x i s t i n IR

f'(a-)

.

U(-OP,OD]

i s an i n t e r v a l , = lim

hf 0

a

E

I

f(a+h)-f(a) h

+)

, and

and

f:I

f'(a+)

->a

is

(defined

Remarks on Tarski's Problem Concerning (lR,+, . , exp)

Proof.

W e s h a l l j u s t treat the case of

f'(a+)

.

117

(The case of

f'(a-)

i s handled s i m i l a r l y . ) 4 = l i m i n f f(a+h)-f(a) hL0 h

Suppose

r

Choose a r a t i o n a l h

>0

that

such t h a t

t h e s e t of a l l

h

contradiction.

0

w.

(A.2)

2

rh

.

>

0

such t h a t

Suppose

i s an i n t e r v a l ,

I

increasing, and i t s inverse

Proof.

the maps

Suppose f ' (a')

continuous on a l l of

Proof.

with

all

I

on (A.4)

J

,

(Note:

a EI

.

Then

f

f(1)

l/f'(a+)

is strictly

has the

if

f(a) = b

.

the c o n t i n u i t y assumption cannot

,

a

.

Then

+>

f '(a-)

E P and an i n t e r v a l

J

I

hence

x E J

g

If

I

for a certain

around

a

a

E

I

a E I

.

g'(x+)

. .

Then

such t h a t

Then t h e continuous function

has t h e property t h a t

>0 ,

g'(x-)

g : J --3P

< 0 for

would be both s t r i c t l y increasing and s t r i c t l y decreasing

by the previous lemma.

m.

.

for all

f ' ( a + ) = f'(a-)

f'(a-)

for all

is continuous, and

i s continuously d i f f e r e n t i a b l e on I

f

>

f'(x-)

f : I ->R

a r e w e l l defined, r e a l valued, and

f'(a+)

g(x) = f(x)-cx

,

i s continuous and

defined on the i n t e r v a l

It s u f f i c e s t o show t h a t

>c>

x

fdl

f : I ->lR

for a l l

I i s an i n t e r v a l ,

Suppose the contrary, say c

such

0

a I->

there is

>0

< r h is IR-definable,

i s defined and equal t o

f")'(b+)

Lemma.

f'(x+)

>0

L e f t t o the reader.

be omitted.) (A.3)

b u t a l s o a r b i t r a r i l y small h

f(a+h)-f(a)

i s defined ( i n IR ), and

property t h a t

Then t h e r e are a r b i t r a r i l y s m a l l

This s i t u a t i o n is incompatible with the f a c t t h a t

A

f'(a+)

,

< rh

f (a+h)-f(a)

f(a+h)-f(a)

.

a< r
with

.

L = l i m sup f(a+h)-f(a) hi0 h

Contradiction.

i s an i n t e r v a l and

t h e r e are only f i n i t e l y many x

E

I

0

f:I --3R

such t h a t

+ f'(x )

i s R-definable, =

&m

.

then

L. VAN DEN DRIES

118 Proof. {x

E

1

I

Suppose f o r example t h a t t h e R-definable s e t

f'(x+) =

is infinite.

m]

So t h i s s e t c o n t a i n s a whole i n t e r v a l ,

and f o r the sake of d e r i v i n g a c o n t r a d i c t i o n , we may a s w e l l assume t h a t f'(x+) = that

for a l l

.

I

By ( 2 . 2 ) we may f u r t h e r r e s t r i c t t o t h e case

i s continuous, which implies t h a t

f

whence

E

x

2

f'(x-)

0

for a l l

EI

x

f

,

I

i s s t r i c t l y i n c r e a s i n g on

.

A f t e r f u r t h e r shrinking t h e i n t e r v a l , we may a l s o assume t h a t w e are i n one of two c a s e s :

(1) f ' ( x - )

= a

for all

E

x

I

i s finite for a l l

(2) f ' ( x - )

I

continuous on

.

b

,

whence

I

,

x C> f'(x-)

and

is

(This uses again ( 2 . 2 ) . )

I n c a s e (1) the i n v e r s e for a l l

E

x

f-'

of

f

satisfies

(f-')'(b+)

is . constant, c o n t r a d i c t i n g i t s i n j e c t i v e n e s s .

f-'

case (2) we can apply t h e same argument a s i n t h e proof of Lemma ( A . 3 )

Then a.

<

Proposition. f

... <

am

Proof. that

functions

Suppose

f : (a,b)

->

R

i s &definable,

a

<

,

a.

,

= a

a = b m

such t h a t

f

f

.

b

i s continuously d i f f e r e n t i a b l e

.

(aiJai+l)

By ( 2.2) we may a s w e l l assume t h a t f'(a+)

and

are finite for a l l

f'(a-)

a b> f'(a+)

and

a

b> f'(a-)

i s continuous, and by

f a

E

I

.

But then t h e

a r e R-definable,

hence

0

piecewise continuous, and t h e r e s u l t follows i m e d i a t e l y from ( A . 3 ) .

(A.6)

to get

i s piecewise continuously d i f f e r e n t i a b l e , t h a t i s , t h e r e a r e

on each s u b i n t e r v a l

(A.4)

In

0

a contradiction.

(A.5)

=0

= (f-')'(b-)

Corollary.

i s piecewise

t o i n c r e a s e with

Proof.

Suppose Cn

,

f : ( a , b ) ->R

f o r each

n

6N

.

i s IR-definable,

a

(The number of 'pieces


'

.

Then

i s allowed

n .)

This follows from P r o p o s i t i o n (A.5) by i n d u c t i o n on

n

.

0

Remarks on Tarski's Problem Concerning (IR,+, . , exp)

(A.7)

119

Let us a l s o mention t h e following i n t e r e s t i n g consequence of the R

hypothesis t h a t each IR-definable s u b s e t of

i s of f i n i t e type; we s h a l l

n o t use i t i n t h e r e s t of t h i s appendix. Proposition.

T h ( 9 ) has d e f i n a b l e Skolem functions.

equivalence r e l a t i o n on a d e f i n a b l e s u b s e t of

Also each d e f i n a b l e

has a d e f i n a b l e s e t of

Rn

representatives. and (4.1)] f o r the theory

The proof i s s i m i l a r t o t h e one given i n [vdD3,(1.2) of r e a l closed f i e l d s .

(A.8)

The c o n s i d e r a t i o n s of ?-submanifold

of

XIn

,

X

'defining'

ha)

?(X)

= {f:X

FM(ilJ

..., i n )

(ilJ...,in)

->1

be a c l a s s of

dim(X)

.

C?


,

Put F

De\(X)

So t h e

X E F$)

for

=

u

b u t we r e q u i r e t h e

.

FM(ilJ...,in)

hO[) E F$-')

then

M C -isomorphism.

g)

C'-submanifolds

the d e f i n i t i o n i s

{0,1] ;

in

and similar t o t h e one i n (3.3),

n

is a

Note t h a t

and the homeomorphism

?-analogue

as i n (3.5),

of (3.4) holds.

and we can s t a t e the

of meorem (3.7).

?-analogue

(~.9) Theorem. n z l

f is a d e f i n a b l e

F u r t h e r we l e t

and

We now d e f i n e following

I

we p u t

functions t o be

X E F$)

h:X

For a d e f i n a b l e

f o r each sequence

by i n d u c t i o n on

if

53 can now be r e f i n e d as follows.

of lRn

.

M C -function]

.

M E IN

F i x a number

suppose

s= , < , + , ...I

i s of f i n i t e type.

. m:

(a) For each d e f i n a b l e s e t f i n i t e p a r t i t i o n of

X

X CIRn-'

into

and d e f i n a b l e

E F$-"

?-manifolds

f:X

->

R

there is a

on each of which

f

$-function. (b) Given any d e f i n a b l e s u b s e t s

A1,

p a r t i t i o n i n g each of

.

AIJ...,Am

...,A

m

of

Rn

t h e r e is

9E

DecM(Rn)

L. VAN DEN DRIES

120

The proof is along the lines of the proof of (3.7)

and w e leave i t to the reader

to supply the e x t r a work needed i n the

of ( 3 . 8 ) .

(A.6),

is

induction on

3

(A.lO)

M

(?-version

(Hint:

use

and the f a c t t h a t a function on an open subset of lRm

i f f i t s p a r t i a l derivatives up to order

M

e x i s t and a r e continuous.)

This d i f f e r e n t i a b l e analogue of Theorem (3.7) could be useful i n a proof

of the following conjecture: L e t the expansion $(vl,

...,v-)

an

2 of

%-fornula.

@

,< ,+ , . )

be of f i n i t e type and

Then the family of =-definable

contains only f i n i t e l y many homeomorphism types.

sets

CIn p a r t i c u l a r , there i s a

uniform bound on the number of connected components of

Xa

g a ranges

(A.ll) Mather [Mat,p. 2181 gives a beautiful proof f o r the case of semialgebraic s e t s , i.e.,

for

2=

@

,< ,+ ,

-

). The main tools i n h i s proof

a r e the Tarski-Seidenberg theorem and the existence of a f i n i t e Whitney s t r a t i f i c a t i o n f o r each semialgebraic s e t .

In our more general s i t u a t i o n , the

analogue of the Tarski-Seidenberg theorem i s automatically true since we work i n the category of a l l =-definable

sets.

But t o give a s e t a Whitney

s t r a t i f i c a t i o n means t o p a r t i t i o n the s e t i n t o d i f f e r e n t i a b l e manifolds such t h a t a c e r t a i n technical condition holds, and t h i s technical condition is unfortunately not generally s a t i s f i e d by the

$-decomposition

of Theorem (A.9).

When we look a t Wall's proof i n [W] t h a t semialgebraic s e t s have f i n i t e Whitney s t r a t i f i c a t i o n s , i t seems nevertheless quite plausible t h a t t h i s proof e s s e n t i a l l y goes through i n our general context, mainly because Theorem (A.9) i s available. It i s remarkable t h a t Mather obtains h i s homeomorphisms by integration of a semialgebraic vector f i e l d , an operation which leaves the semialgebraic context. This i s one more reason t o t r y t o extend the structure suggested i n the Introduction.

4,< ,+

, *)

as

Remarks on Tarski's Problem Concerning (IR, +, ' , exp)

121

REFERENCES G. E. Collins, Quantifier Elimination for Real Closed Fields by Cylindrical Algebraic Decomposition, Automata Theory and Formal Language, 2nd G.I. Conf. , Kaiserslautern, pp. 134-183, Berlin, Springer-Verlag, 1975. L. van den Dries, Exponential Rings, Exponential Polynomials, and Exponential Functions, to appear in Pacific Journal of Mathematics. L. van den Dries, Analytic Hardy Fields and Exponential Curves in the Real Plane, to appear in Am. Journal of Math. L. van den Dries, Algebraic Theories with definable Skolem Functions, submitted. B. Dahn and H. Wolter, On the theory of exponential fields, preprint. L. Hormander, Linear Partial Differential Operators, Springer-Verlag, Berlin, 1963. A. G. Hovanskii, On a class of systems of transcendental equations, Soviet Math. Dokl. 22 (1980) 762-765. P. J. Kahn, Counting Types of Rigid Frameworks, Inv. Math. 55 (1979) 297-308. S.

kojasiewicz, Ensembles Semi-Analytiques, mimeographed notes,

IHES, 1965.

A. Macintyre, On definable subsets of p-adic fields, JSL 41 (1976) 605-610. J. Mather, Stratifications and Mappings, in: Dynamical Systems, ed. by M. Peixoto, pp. 195-232, Academic Press, 1973. C. Pugh, Hilbert's 16th problem: Limit cycles of polynomial vector fields in the plane, in: Dynamical Systems Warwick 1974, pp. 55-57, SLN 468, 1975.

-

A. Tarski, A Decision Method for Elementary Algebra and Geometry, 2nd ed., revised, Berkeley and Los Angeles, 1951. C. T. Wall, Regular Stratifications, in: Dynamical Systems 1974, pp. 332-344, SLN 468, 1975.

- Warwick

LOGIC COLLOQUIUM '82 G. Lolli, G. Long0 and A . Marqa (editors) 0 Elsevier Science Publishers B. V. (North-Holland), 1984

123

SOUS GROUPES D'AUTOMORPHISMES D'UNE STRUCTURE SATUREE 'Vanid LASCAR

C.N.R.S.

Paris

ABSTRACT. If Misasaturated structureand A c M , letAut (M) denote the A group of automorphisms leaving A pointwise fixed. We study conditions which are related to the following one : for all finite subsets of M, A and B

,

the subgroup of Aut(M)

generated by

(M). We investigate also the AutA(M) uAutB(M) is precisely Aut A nB class of subgroups of Aut(M) containing a group AutA(M), for some finite A cM. 1.-

Soit M une structure saturge de cardinalit6 non dgnombrable. Si A est un sous-

ensemble de M, on notera Aut (M) le groupe des automorphismes de M laissant A A point par point fixe. On va s'intgresser darts cet article aux sous-groupes de Aut(M)

qui contiennent un groupe de la forme Aut (M), pour un ensemble A fini. A Je pr6fSre attendre la fin de l'article pour dgvoiler les vraies raisons de cet int6rSt. Pour l'instant, et pour faciliter l'exposition des rssultats, on va 6tu-

dier la condition suivante : (0) Pour tous sous ensembles finis de M, A et B, on a

<

AutA(M) u Autg(M)

(Si xcAut(M),

> = AutA

B(M).

c x > d6signe le sous groupe de Aut(M)

engendrg par x).

PremiBres remarques.(aussi valables pour les conditions ( 1 ) et (2) ci-dessous) : cette condition porte en fait sur la thgorie de M plut8t que sur M : si M' est une autre structure saturse, 616mentairement gquivalente

M, alors la condition

(0) est vgrifige pour M si et seulement si elle l'est pour M'. I1 est d'autre

part clair que l'on a toujours

AutAnB(M).

Maintenant, un 616ment de Aut (M) laisse nscessairement fixe tous les points A d6finissables sur A ; pr6cisons : D6finition. On dit que a c M est dgfinissable sur A s'il existe une formule @(v

)

D. LASCAR

124

1 une v a r i a b l e l i b r e e t I p a r a m s t r e s d a n s A t e l l e que a s o i t l e s e u l p o i n t d e M satisfaisant $(a). S i a v d r i f i e $ ( a ) , e t que l ' e n s e m b l e { C E M ; M k $ ( c ) } e s t f i n i , on d i t que

a e s t a l g d b r i q u e s u r A. L e lemme s u i v a n t e s t a l o r s Q v i d e n t :

C e l a m o n t r e q u ' e n f a i t , il v a u t mieux se r e s t r e i n d r e d a n s l a c o n d i t i o n (0) aux e n s e m b l e s A e t B q u i s o n t d g f i n i s s a b l e m e n t c l o s ( i . e . s u r A, a l o r s

a E

s i a est d d f i n i s s a b l e

A). Cela donne :

( I ) pour t o u t A e t B c M q u i s o n t l a c l 8 t u r e d d f i n i s s a b l e d'ensembles f i n i s ,

O n v a a u s s i c o n s i d d r e r une c o n d i t i o n e n c o r e p l u s f a i b l e .

(2) pour tous A e t B c M q u i s o n t la c l 8 t u r e a l g d b r i q u e d'ensembles f i n i s < AutA(M) u AutB(M) >

= AutA

B(M).

11.- Doqnons un exemple oii ces c o n d i t i o n s n e s o n t p a s r e m p l i e s : l a t h d o r i e T d e M e s t l a t h Q o r i e d ' u n e r e l a t i o n d ' d q u i v a l e n c e R a y a n t une i n f i n i t 6 d e classes, chacune d t a n t i n f i n i e . Soient A = {a), B

=

I1 e s t c l a i r que A e t B s o n t

{b}, avec a f b e t M C R ( a , b ) .

t o u s deux a l g d b r i q u e m e n t c l o s , e t que

& { f EAut(M) ; M k R ( a , f ( a ) ) } .

Montrons q u ' e n f a i t



= {f rAut(M)

; M +R(a,f(a))}.

En e f f e t , s o i t f E A u t ( M ) t e l que M C R ( a , f ( a ) )

; supposons d ' a b o r d que f ( b ) # a . I1

e x i s t e a l o r s g r Aut (M) t e l q u e g ( b ) = f ( b ) . Donc g-' A -1 o f E . f =gog B

0

f

E

AutB(M) e t

S i f ( b ) = a , on c h o i s i t un p o i n t c , d i f f d r e n t d e a e t d e b, mais t e l que

125

Sous Groupes d'Automorphismes d'une Structure Saturke

M t=R(a,c).

I1 existe des automorphismes g1 et g2 tels que g (b) =c, g (c)=a, 1

2

gl(b) # a et gg(b) #a. D'aprZs ce que l'on vient de voir, g1 et g2 sont dans et il suffit de remarquer que g-' 1

o

g;lof(b)

=b.

Autrement dit, bien que la condition ( I ) ne soit pas remplie, il suffit d'ajouter les classes modulo R pour qu'elle le soit : considQrons la thgorie T' dans un langage compos6 de deux prgdicats unaires U et V, d'un prddicat binaire R et une fonction unaire f ; T ' assurera que :

-

U et V sont deux ensembles disjoints dont la reunion est le modele tout

entier.

- R est une relation d'gquivalence sur les QlQments de U, avec une infinit6 de classes, chaque classe Qtant infinie.

-

f est une application surjective de U sur V telle que, pour tout a,b dans

U, f(a) =f(b)

si et seulement si M kR(a,b).

Un modele de T ' n'est rien d'autre qu'un modsle de T dans lequel on a rajoutd des reergsentants pour les classes modulo R (les 61Cments de V) l'application f devant Stre consid6rge comme l'application canonique. La th6orie T' est

essen-

tiellement &gale 2 la thgorie T, en ce sens qu'elles sont mutuellement interprgtables l'une dans l'autre et que la cat6gorie de leurs modeles sont gquivalentes. Mais dans T', si a E U , {a} n'est pas dgfinissablement clos. On doit considerer A' ={a,f(a)}

et si on reprend les hypotheses de tout 2 l'heure, B ' ={b,f(b)},

avec f(a) =f(b).

On a alors bien cAutA,(M) uAutg,(M) > = Autlf(a)l(M)

=AutA,nB,(M).

111.- C'est Shelah, dans 141, qui a introduit le premier des Ql6ments pour repr6senter les classes d'6quivalence modulo une relation dgfinissable, et il leur a donne le nom d'Ql6ments imaginaires. Ainsi il semble que si une thQorie satisfait la condition (l), ces Q16ments imaginaires ne soient pas nQcessaires : ils sont d6j2 en quelque sorte dans le modhle. Poizat dans C3l gtudie une condition de ce genre, qu'il appelle"6limination des imaginaires". Ce que nous aimerions montrer, s o u s l'hypothese ( I ) ,

c'est que :

Pour tout n E w , R(x,;) il

relation d'gquivalence sur

#,

ddfinissable, et

a

E

Mn,

existe un ensemble B c M , fini tel que

(autrement dit, B

et la classe de

a modulo R

sont dgfinissables l'un sur l'autre

Mais ceci n'est pas tout-2-fait vrai : prendre pour T la thQorie d'un ensemble

D. LASCAR

126 infini et

En fait Poizat s'est heurt6 a la mdme difficult6 et a d 2 introduire une autre condition : "l'glimination faible des imaginaires". Pour nous, c'est 11 que la condition (2) interviendra :

En fait, il est facile de voir, par compacit6, que n et k ne d6pendent que de

,Wk)

R, et que l'on peut aussi supposer qu'il existe une formule $(;

b.

sont exactement les suites satisfaisant

telle que les

$(;,b).

-

-

Esquisse de d6monstration.- L'id6e est de trouver deux suites a l et a2 telles que {f cAut(M)

; M f=R(a,f(a)))

=

al

et M !=R(:,;~)

a2

AR(~,;~).

Mais ceci n'est pas toujours possible : prendre la th6orie d'une fonction 2 2 ff x [ s ( x ) # X A s (x) # X A s (x) =XI et pour

unaire s avec comme axiome

2

R(x,y) = C x = y v x = s ( y ) v x = s (y)].

Mais c'est presque vrai :

Pour A c M , notons Aut f (M) (le groupe des automorphismes A-forts) le sous A groupe engendr6par u {AutN(M) ; AcNxM}. I1 faut penser 1 ce groupe comme 6tant l'ensemble des automorphismes laissant la cleture alggbrique de A fixe. C'est pr6cis6ment le cas si T est stable et si l'on a rajoutg les 616ments imaginaires. Dans le cas gdngral, il faut consid6rer la cl6ture alg6brique dans un sens plus Btendu. Si a est un 6lgment imaginaire (i.e. la classe

a € Mn modulo une relation

d'gquivalence dgfinissable sur Mn), on peut aussi d6finir Aut (M) et Aut f (M). De mdme on peut dire que 1'616ment f3 (r6el ou imaginaire) est dgfinissable sur a si Auta(M) cAut (M), et que B est algdbrique sur a si

B

n'a qu'un nombre fini de

conjugu6s par les 616ments de Aut (M). Cela est dquivalent B dire que Aut f,(M)

c

Autg(M).

Maintenant, les r6sultats de C23, (th6orSme 614) nous permettent d'affirmer

127

Sous Groupes d'Automorphismes d'une Structure Saturk qu'il existe

al

et

a2 dans Mn

t e l l e s que M k = R ( a , a l )

et

nR(a,a2)

< A u t f- (M) u Aut f- (M) > = Aut fa(M) a2

al s i a e s t l a c l a s s e de

Soient

a modulo

R.

a l o r s A l e t A2 l e s c l B t u r e s a l g e b r i q u e s de

= < A u t f- (M) u Aut f - (a) > al a2

Aut f,(M)

I1 s u i t donc que

c1

c

est dzfinissable sur A n A 2 , 1

CURULLAIRE 2.- A w e c

b&

...,bk

l e s conjugu6s de

Le.4 rnCme.4 hypofh2hen que

a2.

On a a l o r s


7

lui-mdme algebrique sur a,

6

de A 1 n A 2 . par Aut (M).

e t l a mdme chose e s t v r a i e pour une s u i t e f i n i e prendre pour b l , b 2 ,

et

I1 s u f f i t a l o r s de

Le Th@ohZme I , 2 e x h i e un eMne.mble

B td que Aut fa(M) c_Autg(M) sAuta(M).

1V.- On va maintenant donner quelques exemples. a) Les corps algsbriquement c l o s . C ' e s t un f a i t assez b i e n connu que (2) e s t sat i s f a i t e dans ce cas. On va en donner une preuve c o l o r e e de s t a b i l i t d .

K. u K 1

Soient K, K 1 , K 2 , K ' des corps algebriquement c l o s , avec K 1 n K 2 = K , c K ' , K e t K de degre de transcendance f i n i au-dessus de K. I1 s ' a g i t de 1

2

montrer que G =

2

<

AutK (K')uAut (K') > e s t :gal 1 K2

21 AutK(K').

C ' e s t f a c i l e B f a i r e s i K1 e t K2 s o n t alggbriquement d i s j o i n t s au-dessus de K ( l e s f a n a t i q u e s de l a t h e o r i e des modgles pourront d i r e que K l e t K2 sont a l o r s

K-indgpendants, dans l e sens du forking, e t appliquer un r e s u l t a t general). Sinon on n o t e r a par K3 l a c l 6 t u r e algebrique du corps engendrg par K u K 2 , e t par n 1 1' "2. n + n - p l e s degrds de transcendance r e s p e c t i f s de K l , K e t K au-dessus de K. 1 2 2 3 La d6monstration s e f e r a par recurrence sur p , l e c a s p = O Q t a n t d6j5 rSgl6. Soient Cl B = (B(I),@(2),..

...,

a ( n ) ) une base de transcendance de K I ' 1 .,B(n ) ) une base de transcendance de K2 c h o i s i e de t e l l e s o r t e 2 que B ( l ) , B ( 2 ) , ...,B( n - p ) s o i e n t algebriquement independants au-dessus de K I e t 2 donc que B(n2 - p + I ) , B(n ) s o i e n t algebriques au-dessus de 2 K1 (B( I ) , ,B(n2 - P ) )

-

=

(a(l),a(2),

...,

.. .

.

D. LASCAR

128 S o i t maintenant

it= ( B ' ( l ) ,

...,B'

(n,))

dans K'

tel

que t(B'/K3)

e s t une

e x t e n s i o n non b i f u r q u a n t e d e t(B/K ) . 1 P a r h y p o t h b s e , s i a e s t a l g g b r i q u e s i m u l t a n d m e n t au-dessus a l o r s a t K . P u i s q u e t ( K , uK2/K(B'(1) (par symdtrie),

... ,B'

B f , n2 - p

(n2 - P I ) ,

alors a

K(B'(1)

E

d e K 1 e t de K2,

e s t h d r i t i e r de t ( K 1 uK2/K)

s i a e s t a l g d b r i q u e s i m u l t a n d m e n t au-dessus

B'(n2 - p ) ) e t K2(Bi ( l ) , montre que l e s

,...,O'(n2 - p ) )

de K 1 ( B ' ( l ) ,

,. .., B ' ( n 2

-p)).

..., Ceci

< i < n 2 ne s o n t p a s a l g d b r i q u e s s u r K 2 ( ~ i , ~ ; , . . . , ~ ~ 2 - P ) .

S o i e n t f un K -autornorphisme e n v o y a n t sur i'e t K' = f (K2). P u i s q u e 1 2 Aut , ( K ' ) = f Aut ( K ' ) f - ' , Aut c G . D ' a u t r e p a r t , K e t K ' s o n t K -inddpendants 2 2 1 K2 Kg donc K n K ; c K 1 , donc K n K ' = K . Mais o n v i e n t d e v o i r q u e l e d e g r d d e t r a n s c e n 2 2 2 d a n c e du c o r p s engendrd p a r K2 e t K; e s t s u p d r i e u r 1 2n - p : o n p e u t donc a p p l i 2 quer 1'hypothSse de rdcurrence.

";

b) La d d m o n s t r a t i o n p r d c d d e n t e se g c n d r a l i s e e t l ' o n o b t i e n t :

c ) On l a i s s e r a au l e c t e u r l e s o i n de m o n t r e r que l e s o r d r e s d e n s e s ( a v e c ou s a n s extrdmitds) s a t i s f o n t la condition ( 1 ) .

d) Les c o r p s r d e l s c l o s n e s a t i s f o n t n i ( 1 ) n i ( 2 )

: s i M e s t un c o r p s r d e l c l o s

s a t u r d , on p e u t e n e x t r a i r e deux p o i n t s a e t b , a l g d b r i q u e m e n t i n d d p e n d a n t s , t e l s que p o u r t o u t e n t i e r n, n < a < b < 2 a . I1 e s t a l o r s c l a i r que t o u t dldment d e < Aut (M) u Aut

b

(M)> l a i s s e l ' e n s e m b l e {c E M ; il e x i s t e un e n t i e r n t e l

que

0 5 c s n a } globalement f i x e , c o n t r e d i s a n t l a c o n d i t i o n (2).

e ) On r e m a r q u e r a h u s s i q u e d a n s l ' e x e m p l e c ) l a c o n d i t i o n d e f i n i t u d e p o u r l e s ensembles A e t B est n d c e s s a i r e : prenons A = ta. ; i a o < b < a l < . . . < an < b n <...

E

u], B = {bi ; i E u}, a v e c

; a l o r s A n B = @ , mais t o u t Q l e m e n t d e

< A u t (M) uAutB(M) > l a i s s e l ' e n s e m b l e { x t M ; il e x i s t e n t e l que c < a A ment f i x e .

1

globale-

On p o u r r a i t v o i r q u ' i l e n e s t d e m6me pour les c o r p s a l g d b r i q u e m e n t c l o s : s o i t K un c o r p s a l g c b r i q u e r n e n t c l o s , d e d e g r d

d e transcendance f i n i , i n c l u s dans

K ' un a u t r e c o r p s a l g d b r i q u e m e n t c l o s , non ddnornbrable. On c o n s i d b r e d e s Q l d m e n t s

de K ' ,

..., a l g d b r i q u e m e n t i n d c p e n d a n t s e t K1 e t K ( a l , a 2 , ...,an, ...) e t d e K ( b , a l + b a 2 , a 2 + b a 3 , . . . , a

b , a l , a 2 , ...,a n ,

algdbriques de

On m o n t r e r a a l o r s q u e K 1 n K = K , mais q u e s i f 2

E

K2 l e s c l S t u r e s

+ban,...). n- 1 < A u t (K') u Aut (K') > , a l o r s K1 K2

129

Sous Groupes d'Automorphismes d'une Structure Saturee f(K 1) est de degr6 de transcendance fini au-dessus de K 1 '

V.- La question qu'il est maintenant nature1 de se poser est de savoir si l'adjonction des Qldments imaginaires suffit toujours 1 rendre ( 1 )

ou ( 2 ) vraie. Mais en

regardant d'un peu prEs le cas des corps rsels clos, on voit tout de suite que la amen6 1 considerer les deux conditions suivantes :

r6ponse est non. On est donc (3)

Pour tout ensemble fini A et B, = {frAut(M)

; f laisse fixe tout dl6ment ima-

ginaire dgfinissable sur A et sur B}. (4)

P o u r tout ensemble fini A et B,

<

If tAut(M)

;

f

laisse fixe tout Clement imaginaire algebrique sur A

ou bien f lai sse fixe to IX BlBment imaginaire algebrique sur B} >

{ft Aut(M)

=

; f laisse fixe tout G16ment imaginaire alg6brique

sur A et sur B}.

On va s'interesser 1 un problsme plus g6ndral : disons que le s o u s groupe Aut(H)

G de

est 1 support fini s'il existe un ensemble fini A (un support de G ) tel que

AutA(M) c G . L'idde (fausse) est que si G est 21 support fini, il existe des dldments imaginaires {ai ; it:I} tels que G est pr6cisGment l'ensemble des automorphismes laissant les ai fixes. Dans ce cas, chacun des a. doit ltre definissable sur tout support de G . Le thdorkne suivant nous dit que ceci devient vrai si on 6largit encore la notion d'imaginaires :

G = {gcAut(M)

;

M t=R(a,g(a))}.

Par exemple, si la thdorie T est 5 -cat6gorique, la formule R ne peut Ztre que dgfinissable et il en d6coule alors :

COROLLAIRE 4.-

leb thZohieA

$ -cat&gohiqued d a t i d d o n t La condition ( 3 1 .

Dsmonstration du theorhe 3 . - Considerons la relation d'gquivalence sur Mn :

E(G,~) si

et seulement s'il existe gr G tel que g(X) =;.

D. LASCAR

130

On remarque d'abord que si f cAut(M)

alors f E G : en effet,

et si E(a,f(a)),

il existe g E G tel que f(a) =g(a),

-

M +E(a,b), Aut-(M) b

c

alors Aut-(M) b

et donc g-'f ~Aut-(M)c G. Ensuite que si a est conjugue de Aut-(M) par un element de G, donc a

G.

- -

-

-

Maintenant supposons que pour quatre suites c, c', c l , ci de t(C

, :'/a)

=

t(cl

,:;/a).

8,on

11 existe alors un eldment g E G tel que g(C)

=

ait

c1 et

- -

~(c,:')

est vrai si et seulement si E(c~,c;) l'est : g(C') =c', et par consequent I ceci montre que la satisfaction de E ( c , c ' ) ne depend que de t(c,:'/a).. Soit donc

P l'ensemble des 2n-types sur

c,C' EM"

4'

--

(E(c,c')

a

tel que, t(c,c'/a)

<=>

E

P)

.

On a alors

Malheureusement, cette dernisre formule peut contenir

---

tient cependant une formule @(x,y,z),

a comme paramstre. On ob-

disjonction infinie de conjonctions infinies

de formules finitaires sans paramstres, et telle que

Supposons que F; E Mn r6alise le mdme type pur que

a

---

; alors @(x,y,b)

dsfinit

aussi une relation d'equivalence sur Mn. Voyons sa signification ; soit f E Aut(M)

tel que f(b)

=a.

Alors p o u r tout

E

'

Mn,

donc si et seulement s'il existe g c G tel que gf ( C ) = f ( C ' ) En conclusion, M h(c)

=;'.

k @ ( C , C ' , c ) si et seulernent s'il existe h e f-lGf tel que

En particulier, si E(a,b)

Posons R(;,;')

=[t(;)

=

tG')

=

---

est vrai, alors M l=@(x,y,a)

---

t(i) A Y ;(~(x,y,x)

est une relation d'gquivalence et sa satisfaction par

--

<=>

- --

<--@(x',y,x~))~

c,c'

---

@(x,y,b). ;

~(i,i')

ne depend que du type

pur de c,c' ; elle peut donc s'exprimer s o u s la forme requise par le thGorSme. I1 nous suffira donc de rnontrer que pour tout

6E

M", E ( a , b ) si et seulement si

R(a,i).

131

Sous Groupes d'Automorphismes d'une Structure Saturbe I ) Supposons v d r i f i d e E(g,G), i . e . M M

x --I=

V

V

@(b,y,b)).

---

---

I=

---

O(a,b,a).

( @ ( x , y , a ) < = = >@ ( x , y , b ) ) e t donc que M I= Mais d ' a u t r e p a r t , il e s t c l a i r que E(G,;)

On a vu a l o r s que

---

V 9 (@(,b,y,a)<= > -s i e t seulement s i E(a,y)

Donc

e t on v o i t donc que M I = R ( a , b )

2 ) Rdciproquement, s i on suppose M F R ( a , b ) , a l o r s M

---

---

O(b,y,b). Mais puisque M C @ ( b , b , b )

, on

---

V f (O(a,y,a)

---

<=>

v o i t que M k O ( a , b , a ) .

VI.- MSme dans l e c a s des c o r p s algdbriquement c l o s , il e x i s t e des groupes fini:-ment supportds q u i ne s o n t pas l'ensemble des f i x a t e u r s d'un c e r t a i n nombre d'dldments. Par exemple, G={gEAut(M) ; g ( a ) e s t a l g d b r i q u e s u r a}, oil a e s t un 616ment t r a n s c e n d a n t . Cependant, c ' e s t l a v e r s i o n f a i s a n t p o r t e r l e s automorphismes f o r t s q u i e s t v r a i e dans un c a d r e beaucoup p l u s gdngral. Disons que GcAut(M) e s t

2 support presque f i n i s ' i l e x i s t e un ensemble f i n i A t e l que Aut f (M)cG. A

La ddmonstration e s t a s s e z technique e t u t i l i s e l e s p r o p r i d t d s du rang U (voir [ I ] )

e t l e thdorsme de l a base canonique de Shelah ( v o i r C 4 1 ) . Avant de

l ' a b o r d e r , montrons :

COROLLAIRE 6 . - La c o n d i t i o n ( 4 ) at v h a i e pout Led theohied b u p m t a b L ~de mng

dini. Preuve du c o r o l l a i r e . -

S o i e n t A e t B f i n i s . Posons G = < { f e Aut(M) ; f l a i s s e f i x e

t o u t Qldment imaginaire a l g e b r i q u e s u r A ou b i e n f l a i s s e f i x e t o u t dldment imagin a i r e a l g d b r i q u e sur B} > e t

G' ={fEAut(M) ; . f l a i s s e f i x e t o u t dldment imaginaire algdbrique

s u r A e t s u r B).

I1 e s t c l a i r que G c G ' , que Aut f (M) c G e t que Aut fB(M) c G ; l e groupe G a A donc un support presque f i n i e t on p e u t l u i a p p l i q u e r l e t h d o r h e 5 : s o i t a 1'616ment i m a g i n a i r e q u i en ddcoule. C e t dlgment d o i t 6 t r e algdbrique s u r A : sinon s o i t N ddnombrable, AcN-(M t e l que a e t N s o n t A-independants ; a n ' e s t pas algd-

D. LASCAR

132

brique s u r N e t s i a' r d a l i s e s u r N u {a} une e x t e n s i o n non b i f u r q u a n t e de t ( a / N ) ,

a ' n ' e s t pas a l g d b r i q u e s u r a, e t i l e x i s t e g~ Aut (M)cAut fA(M) c G t e l que N

g(a)

= a ' , c o n t r e d i s a n t l a conclusion du thdorsme 5. D e mBme, a e s t a l g 6 b r i q u e

s u r B. f l a i s s e f i x e a e t t o u s les Qldments i m a g i n a i r e s a l g 6 b r i q u e s

S i donc f E G ' ,

s u r a. Puisque T e s t s t a b l e , c e l a s u f f i t pour a f f i r m e r que f E A u t f (M) ( v o i r

r21) e t donc que f

E

G.

Passons 1 l a preuve du th6orSme 5 : s o i t a un Gldment i m a g i n a i r e t e l que Aut fa(M) c G , avec U ( a / ' $ )

minimum (dvidemment, il f a u t d t e n d r e l e rang U, avec

s e s p r o p r i d t d s , aux t y p e s d'616ments i m a g i n a i r e s ; l e l e c t e u r q u i n ' e s t pas convaincu que c e l a peut s e f a i r e devra e n r i c h i r l a t h d o r i e T comme il a dtd f a i t au second paragraphe). Choisissons g E G t e l que s i

B=g(a), alors U(B/a)

e s t maximum

( i c i on u t i l i s e l ' h y p o t h s s e que T e s t de rang f i n i ) . I1 f a u t montrer que U(B/a) = O . Comme on l ' a d d j 2 vu, Aut f (M) c G .

B

S o i t N un sous modSle 616mentaire de M contenant a e t t e l que U(B/N) =U(B/a). On s a i t que l e type de 6 s u r N admet une base canonique, q u i e s t un dldment imagia l g d b r i q u e s u r a. S o i t maintenant E M t e l que t ( B / N u t B } ) e s t une ex1 1 t e n s i o n non b i f u r q u a n t e de t(B/N). I1 e x i s t e a l o r s hEAutN(M) c G t e l que h(B)

naire a

=al,

(M) c G . D ' a u t r e p a r t , 6 e t

ce q u i montre que Aut f

B, s o n t indgpendants au-dessus

61 de ao, e t p a r un r d s u l t a t de C21

< A u t f (M) uAut f

B

:

(M) > ZAut f

Bl

(M) 0

e t p a r consdquent

Aut f

(M) C G

.

aO

D'aprSs l a minimalit6 de U ( a / B ) , e s t a l g d b r i q u e s u r a, U(a /'$) = U ( a / $ )

on v o i t que U ( a / ' $ ) > U ( a / ' $ ) , mais comme

e t a e s t a u s s i algdbrique s u r a .

c1

( i c i on

u t i l i s e l e f a i t que l e rang U e s t f i n i ) . Donc, e n f a i t Aut fa (M) =Aut fa(M). 0

Rappelons que g

E

G e t que g ( a ) =

B. Posons a1 = g-'(B1).

Alors

U(a,/a) =U(B,/B), e t p a r l ' h y p o t h s s e de maximalit6 de U(B/a),

U(Bl/B)

U(B/a) =U(B,/a) =U(B1/N U

{ B } ) . Par conszquent, U ( B l / B ) =U(B1/N u {B}),

l e type de B, au-dessus de Nu{B} ne b i f u r q u e pas au-dessus de B. O r

c1

et

qui est sa

base canonique d o i t B t r e a l g d b r i q u e sur 6. Toujours parce que U(ao/(b)= U ( B / c d ) , d o i t a u s s i d t r e a l g d b r i q u e s u r ao, donc s u r a.

VI1.- Concluons : on a baptist les c l a s s e s d ' d q u i v a l e n c e modulo une r e l a t i o n d d f i -

133

Sous Groupes d'Automorphisrnes d'une Structure Saturte

n i s s a b l e "Bldments i m a g i n a i r e s " .

Mais, e n f a i t , i l s n e s o n t g u b r e p l u s i m a g i n a i r e s

que l e s a u t r e s p u i s q u ' i l s u f f i t d ' e n r i c h i r l a t h 6 o r i e T e n une t h 6 o r i e T ' p o u r l e s r e n d r e "rCels", e t c e t e n r i c h i s s e m e n t e s t v r a i m e n t i n n e s s e n t i e l : l a c a t B g o r i e C(T) des modales de T, a y a n t pour morphismes l e s i n j e c t i o n s d l B m e n t a i r e s n e change p a s . S i l ' o n n e s ' i n t d r e s s e q u ' l c e t t e c a t d g o r i e ( p a r exemple s i on essaie d e c l a s s i f i e r

ou de dBnombrer l e s m o d P l e s ) , T e t T ' s o n t s t r i c t e m e n t e q u i v a l e n t s . La c l a s s e d e s sous g r o u p e s B s u p p o r t f i n i e s t i n v a r i a n t e p a r " b i - i n t e r p r s t a b i -

1 i t B " : s i M e t M' s o n t des s t r u c t u r e s s a t u r g e s i n t e r p r e t a b l e s l ' u n e dans l ' a u t r e , e t s i GcAut(M) e s t 5 s u p p o r t

a l o r s Aut(M) e s t canoniquement isomorphe 1 Aut(M'), f i n i , il e n e s t d e mSme d e s o n image d a n s Aut(M').

C ' e s t d ' a i l l e u r s l a mSme chose

pour l e s g r o u p e s d e l a forme AutA(M), 02 A e s t un ensemble f i n i d ' b l h e n t s imagin a i r e s . Mais bvidemment, c e l a n e v e u t p a s d i r e que l a n o t i o n "Otre 1 s u p p o r t f i n i " e s t d d f i n i s s a b l e 1 p a r t i r d e l a s e u l e s t r u c t u r e d e groupe d e Aut(M), n i mOme de l a c a t d g o r i e a b s t r a i t e C(T) d a n s l a q u e l l e Aut(M) e s t p l o n g 6 . Cependant, l e thdorbme 7.12deC21 montre que l ' o n e n e s t p a s l o i n . Consid6rons l e s f o n c t e u r s d e C(T) d a n s Ens, l a c a t e g o r i e d e s ensembles, e t d i s o n s qu'un t e l f o n c t e u r F e s t c o n t i n u s i p o u r t o u t e f a m i l l e (Mi ; i c I) f i l t r a n -

-

(Thdorbme 7 . 1 2 )

: s i F : C(T) + Ens e s t c o n t i n u e t a < F(M), a l o r s

{ g < A u t ( M ) , F ( g ) ( a ) = a } e s t un sous g r o u p e 1 s u p p o r t p r e s q u e f i n i .

- Si

GcAut(M) e s t un groupe I s u p p o r t f i n i , a l o r s i l e x i s t e F : C(T) + Ens, c o n t i -

nu e t a

E

F(M) t e l que G = {g

E

Aut(M), F ( g ) ( a ) = a ) .

En e f f e t , s o i t R l a r e l a t i o n d ' d q u i v a l e n c e s u r

M"

dont l'existence e s t a f f i r -

mBe p a r l e thBorbme 3 . La f o r m u l e q u i l a d d f i n i t d B f i n i t une r e l a t i o n d ' d q u i v a lence sur N n ,

s i N e s t n ' i m p o r t e q u e l modale d e T. S o i t F l e f o n c t e u r q u i B N f a i t

correspondre Nn/R,

-

et 1 y : N

+

N'

l ' a p p l i c a t i o n q u i 1 c/RcNn/R f a i t correspondre

y(c)/R ; F e s t a l o r s un f o n c t e u r c o n t i n u , e t l e thBor2me 3 a f f i r m e que G={gcAut(M) ; F ( g ) ( a / R ) } = a / R . Quels s o n t donc c e s g r o u p e s G t e l s q u ' i l e x i s t e un f o n c t e u r c o n t i n u F e t acF(M) a v e c G = { g c A u t ( M ) ; F ( g ) ( a ) = a } ? E s t - c e q u e c e s o n t l e s g r o u p e s

I sup-

p o r t p r e s q u e f i n i ? Remarquez q u e s i A c M e s t a l g e b r i q u e s u r un ensemble f i n i , a l o r s Aut (M) p o s s s d e c e t t e p r o p r i d t 6 e t n ' e s t p a s n d c e s s a i r e m e n t 5 s u p p o r t f i n i . A D ' a u t r e p a r t , s u p p o s o n s , p o u r B l i m i n e r c e problame q u e les g r o u p e s B s u p p o r t presque f i n i s o n t t o u s G-finies).

I s u p p o r t f i n i (comme c ' e s t l e cas p o u r l e s t h e o r i e s

S i , de p l u s T e s t $ o - c a t d g o r i q u e

oii T e s t s u p e r s t a b l e d e r a n g f i n i ,

a l o r s on a une c a r a c t d r i s a t i o n d e s g r o u p e s d e b forme A u t (M), a v e c A ensemble A

D. LASCAR

134

fini d'Q18ments imaginaires de M. Dans les autres cas, il est nature1 de consid& rer des dlements hyperimaginaires, qui sont les classes d'gquivalence modulo une relation dgfinissable par une formule infinitaire, c o m e dans le theorbme 3 . Le malheur c'est qu'onne peut paslesrajouter s o u s peine de perdre la compacitg, mais on peut se poser la question de savoir s'il y a une raison de pr6fCrer les imaginaires aux hyperimaginaires, autrement dit, si les groupes de la forme Aut (M) sont caractgrisables dans la catggorie C(T). A Une dernibre remarque : munissons Aut(M) simple. Alors les s o u s groupes 1 support

de la topologie de la convergence

fini sont exactement ceux qui sont au

voisinage de CII, et forrnent d'ailleurs un systsme fondamental de voisinage. Donc, dans le cas oii T est G-fini, on r6cuplre la structure topologique de Aut(M)

1 partir de la catggorie C(T). 0 0 0

c11

D. LASCAR : Ranks and definibability in superstable theories, Israel J. of Math, vol. 2 3 , NO1, 1976.

[21

D. LASCAR : On the category of models of a complete theory, J. of Symbolic

Logic, vol. 4 7 , N02, June 1982.

131 B. POIZAT : Une th6orie de Galois imaginaire, 1 paraitre. 141

S. SHELAH : Classification theory, North Holland, Amsterdam, 1978.

Daniel LASCAR

C.N.R.S.

U.E.R. de Mathgmatiques UNIVERSITE PARIS VII 75221, PARIS, Cedex 0 5 , FRANCE.

LOGIC COLLO@UIUM '82 G. Lo& G. Long0 and A . Marcia (editors) 0Elsevier Science Publishers B. V. (North-Holland), I984

135

ALGEBRAIC LOGIC AND DIAGONAL PHENOMENA Roberto Magari Dipartimento di Matematica Universiti di Siena Italia

1. The algebra of logic (and perhaps even modern logic itself) begins with Boole,

but a study of this subject as it is now understood may be said to begin with P.R. Halmos (1955), who extends and sistematically uses the duality theory for hemimorphisms which we owe in a less general form to the work of Jonsson and Tarski. With the exception of certain developments and speculations, this theory largely anticipates Kripke's theory, and in general terms constitutes the main algebraico-topological scheme into which one frequently falls when moving from syntax to semantics and viceversa. In what currently seems to be its most general form, the theory may be presented thus : First of all an extremely elementary reminder of the "Galois connections". Let A,B be sets, SAxB. We can define:

A

=A

,XAX and, it follows that and come out inversely monotonic, X A = putting K = 1 , = , K and >L are Moore's operators. This trivial situatior is fundamental in logic (validity),in Galois theory, in algebraic geometry, in Stone's theorem etc., and is perhaps the most general way of treating the sintax-semantics transition. It is easy to find conditions for be topological, compact etc. and for K be algebraic etc. Now let A,B be Boolean algebras, S and T the respective dual spaces, and let us think A, (B) as thealgebra of the clopen sets of S,(T). Let us define a ), between AxB and TxS, putting: (p,q) A (y,x) iff: if x i p then y r q. It is easy to see that the closed sets in 1 are the "quasi-boolean" relations, such, that is to say, that: i) For every p t A ii) For every y E T

R-I

R

(p) is a closed set (of T) (y) is a closed set (of S )

and we also see that: (1.1) The images and counter-images of closed sets in quasi-boolean relations

are closed. The closed sets in K seems less manageable, but it is obvious that however one takes R C TxS, the R combines at every p < A the elements of a filter. In these iq:(p,q) i X f is bijective from the closed sets in K , the g defined by (gX)p=

n

R. MAGARI

136

given closed sets to the "quasi-hemimorfism" from A to B, that is to say the functions f from A to the closed sets of T where: f0 = 0 (p,qc A ) f(p+q) = fp+fq . , Given that 4 = g-1, , :\. = c . g-', K = g' K. g-', the operators A , , ?I give us the ordinary Halmos duality (for hemimorfisms and boolean relations) extended to "quasi hemimorfisms" on the one hand and to the "quasi-boolean" relations on the other.Substituting * for A , /.. , one moves a q.e. to the dual relation and vice versa with the simple formulae:

- -

-

R'p

= I3-I

(P

(p)

yf*x iff, for every p:-'A,if x

h

a-

p then y c

x,

A)

fp

2. Diagonal operators. The customary transition to Lindenbaum's algebra of a given theory does not usually allow an adequate treatment of diagonal phenomena. (For a first approach, barring errors, see Magari 1975 page 366 nr.8 problem 5). Bearing in mind that the ordinary predicate "Theor" ( " T I ' ) is substitutable in respect of the Lindenbaum relation, (if I- p w q then I- '? 6')we can, first and foremost, introduce into Lindenbaum's algebra of Peano's arithme;p j = [ 'i'($)i ,as has already been tic an operator : with the formula: : done in the works of Kent and Simmons. It easily follows: t l = 1 :(pq) = :p tq S,'P 2-P :(p -t 4)6 r p 4 cq. The novel element lies in the introduction, for each abstract polynomial f(x,yl,y2, ,yn) with x restricted only by r , of a new polynomial g(yl,y2...yn) and the identity: dY1,Y2 Yn) = f(dY1.Y2' 'Y,). Yl.'.' ,Yn). We hereby obtain a variety of algebras of which Lindenbaum's algebra of-arithmetic is obviously one. Godel's first theorem states (for these algebras) (and therefore for a vast class of theories):

?('a)),

<

...

,...,

...

2.1 If A is non trivial algebra of the above mentioned variety (not for every p is :p = 1) then Card A > 2.

Dem. In A there exists a p where p = Y c p . It cannot be that p = l otherwise : p = 1, b :p = 0, p = 0. If it were that p = 1 it would be that Z O F 1 and A would be trivial. The second theorem stetes that: 2.2 If A is not degenerate (0 # 1) then

Dem. Let us take a p where p = V Z p We have v p = ';p c Y p = rtp

T O

>

0.

137

Algebraic Logic and Diagonal Phenomena ;IP I. i Y'P c o = cp. ,,p = 7 p = t'p but as we have already noted, pfl. Lbb's theorem becomes: 2.3 If

<

:p

p then p=l.

:p ,< p. There exists in A a q where q= :q -4p, and we find: : --:p eq= > ( pq-p) < q but since :q g :.-q is also =qlp from which rq+p = 1 q = l :q= 1

Dem. Let

p = l

3. Fixed Point.

A theorem which we owe to C . BERNARD1 (1975/76) and generalized by G. SAMBIN (1976) allows a great simplification of the study. First of all let us call diagonalizable algebra any system:

il= T l = 1 r (pq) = :p t q C ( ? P - ? P ) < ?P G = \': V given

(20) (2.1) (22)

.

(C3)

,in which is a boolean algebra

'(p,q,t A) (Lab's theorem formalized)) What we have here are algebras in which:

(p,q t. A)

( 7 is a "hemimorfism")

and it is clear that we also have: (24)

(n5)

(c6) (77) (:8) (79) (710) (711)

5

is monotonic (increasing) (p. v q ) a fiP * v c q 611P( G P :,(p.q).< ".P. " q ':la r p r; (p. Y r p)= T p <7g:np= G 1 if p \< 2: p then p=O G

(p,q C- A; n

>

0)

We also see that under ( 611, ( 7 2), the ( 6 3) is equivalent to the conjunction of (56),(ell). The theorem states that: 3.1

Let

f(X,yl ,y2,...,y, ) be a polynomial of the variety of diagonalizable algebras. It follow that there exists one and only one polynomial

R. MAGARI

138

..,

g(yl,y2....,yn), such that g(Y19Y2,. Yn)=f(g(Yl,Y2,...,Yn),Yl,Y2,...Yn) and identity. There exists an effective procedure for the construction of g (indeed for the I.D.A. "Intuitionistic diagonalizable algebras") An extension to M.D.A. (modal diagonalizable algebras), that is to say systems in which is a D.A. and k a closure where Zk = k T = ii is the work of R.Magari (1978). Fixed point algebras of Smorynski. I can only briefly mention the very important paper of Smorynski which introduces and studies the "Fixed point algebras", 4.1 (Definition). A fixed point algebra is a pair (B,A) of boolean algebras such that: (i) B is a subalgebra of AA and contains the constants. (ii) B is closed for composition. (iii) Each x ~ has B a fixed point in A. 4.2 (Definition). AnFPA is closed if x x 4 (x+a), hx<(a.x)dB ( a € A , x t B ) The foundamental theorem says that for every finite closed FPA there exists a diagonal operator 3 on A such that every A & B is a T -polynomial.

Status of P.A. We diagonalizable algebras may be seen as an abstraction of Lindenbaum's P algebra of Peanian arithmetic. Up to what point is it an appropriate abstraction? A foundamental result is that of R. Solovay (1976) 5.1 P is generic in the variety of diagonalizable algebras.

Montagna reinforces the result as follows: 5.2 Free diagonalizable algebra on a denumerable infinity of generators in subalgebra of P.

The unclear problem of how much of the complex structure of peanian arithmetic is reproduced in diagonalizable algebra is still open and should give rise to new research on a large scale. With reference to this there is a simple but important observation of Claudio Bernardi and Franco Montagna. In Peanian arithmetic if f is a recursive function, and f represent it, there exists a proposition t such that:

I-

+ ($3)

-

T(l(@))

This has the consequence that in diagonalizable algebra associated with P.A. for every polynomial f, there exists a p where: 5.3 Tf(p) = T'p a property which is not valid in all D.A. The study of D.A. with this property has only just begun.

139

Algebraic Logic and Diagonal Phenomena 6. Authology in D.A.

Any diagonalizable algebra A , can easily be considered as a Lindenbaum algebra for example of a suitable theory T; what then is the status of the theory of first order of D.A.? It is clear that, in a easily definable sense, this theory contains propositions of T and propositions of an appropriate metatheory of T. It must therefore be possible to partially define on the basis of such theory an operator which translates the ordinary " ". To clarify this with an example: if T is P.A. then the ordinary theorem of Lob:

'c,

6.1 If ?($) 3 p then t- p belongs to a metatheory of T. In

it transcribes:

6.2

If t p 6 p then p=l In P.A. its "formalization" is valid, and this is:

6.3

'i('ir3 ()--t p'

)

.--$

'i(9)

That is to say that this is a proposition of P.A. In A we have the element 6.4

T ( t p 4 p) y ? p or rather, in t we have the assertion:

6.5 6.5!

t p

C (

+ p) +'Cpp=l

T ( C p +p)

,C

or in other words:

F p

The applications of to the espression 6.3 gives us 6.1. In the same way, the application of the analog of which we want to define to 6.5 will give us 6.2. We also have an inverse operator, the analog of the so-called "formalization". T o define this more clearly, a convenient point of departure is a 't of the first order with identity, S , likewise furnished with a denumerable infinity of propositional variables. Let u s now establish a bijection of the set of the individuals variables to the set of the propositional variables, and for convenience sake substitute * for both the established bijection and for its opposite. Let us likewise establish a propositional variable, P. We many now define a cf frgm the set of the terms of ?; to the set of open propositions with the formulas: 'fo=Po

A

(7P0)

$ 1 =-lye

q- x

= x*

y9t = 1 T t $(t,+t,) = 4tl v cf t2 p t = t e 1 To return to the example, the application of The 9' then defined by 7' (t, t,) = 4 (t, t2) analogously gives us the open formulae.

*

-

'f to the term

6.4 gives us 6.2

140

R. MAGARI

Its application to 6.5 for example gives us 6.2. Similarly an inverse, )' , is identified. The principla theorem in question is: 6.6 tl -7- t2 is an identity of 2 if an only if <'(t12t2) is valid. The demonstration requires complicated algebraico-topological instruments. T h e y must be zxtended to all the formulae. An example of trivial corollary is:

6.7 The set of the valid formulae in

2 is decidable

It must be noted that 2 , on the other hand, is undecidable (F.Montagna). Finally one must note that the operator y ("formalization") is in its turn of the type S

.

7. Logical Methods However insignificant the distinction presumably may be, practice has shown that certain advantages are to be gained by switching between the algebraic terminolo gy used here and the more precisely "logical" terminology in which ? is considered as a modal operator; Kripke's models are used in the place of duality for hemimorphisms, sequent calculus is used and so on. For these methods see Smorinski (1982) and Sarnbin (1982). 8. Algebraic properties.

Roughly speaking the variety U of the topological algebras ( C . A . ) and the variety W of the diagonalizable algebras (D.A.) are subvarieties of the variety V of the M.D.A. (Consider the algebras in which K is identical, consider the algebras ) in which 1=0

....

Simple algebraic properties of W,U,V,

I recall that: Proposition 8.1 W is ideal. Proposition 8.2 W as a good theory of ideals in the sense of Ursini the 1-ideals and the 0-ideals are precisely the boolean filters closed under 2 ( t -filters) and the boolean ideals closed under Is- ( 'C-ideals). Proposition 8.3 The semisimple algebras of W are precisely the "trivial" algebras (the D.A.s in which 6 1 = 0 ) and form a subvariety. The unique (up to isomorphisms) simple D.A. is the two-elements trivial algebras. Proposition 8.4 Every proper K -ideal is extendible to a maximal 5-ideal and every maximal G-ideal is a maximal boolean ideal. Proposition 8.5 The radical (intersection of maximal 6 -ideals) of a D.A. is the boolean ideal generated by 61. For U we have: Lemma 8.6 U admit a good theory of ideals and is ideal. Proof. Obvious. The 1-ideals are precisely the

"

L -filters" i.e. the boolean

141

Algebraic Logic and Diagonal Phenomena filters closed for L and the 0-ideals are precisely the "k-ideals" i.e. the boolean ideals closed for k.

Lemma 8.7 The k-ideal generated by a non empty M G A (where A is a C . A . ) is the boolean ideal generated by k(M) (Of course the k- ideal generated by 0 is i0.r) Proof. It is sufficient to prove that the boolean ideal k(M) generated by k(M) kqi for a convenient ncrr),qi&M.So is a k-ideal. Let p G k(M), then p < icn k2 qi = i t kqi. kp kifn kqi =

-

iFn

Lemma 8.8 Every proper k-ideal of a C . A . k-ideal.

is extendible to a maximal (proper)

Proof. Obvious via Zorn's Lemma. Theorem 8.9.The radical (meet of maximal k-ideals) of C . A . A is 2p:k rkp=li. Proof. Let be kvkp=l, J a k-ideal and J* the k-ideal generated by JUIpl. If J* = A then l=j+kp for a convenient j ; J and so: kp b'kp kYkp kj

3 6

< =

f.j

j kj 1

S o J=A. It follows that p is the radical. Let be k *kp=l. Then the k-ideal generated by r k p is proper and so there exists a maximal k-ideal J to which kp belongs. So p f J and p do not belongs to the radical. It is easy to give C . A . s in which the radical is not trivial, for example the four-elements algebra O,la, a in which kO=O, kl=l, k v a = l , ka=a. In fact we have:

Theorem 8.10 The semisimple algebras of U from a subvariety and are precisely the monadic algebras ( " " ) . Proof. Of course a C.A. A is semisimple iff its radical is LOj, i.e. iff: (*) if p # 0 then k Y kp # 1 (pc-A). Now we have to prove that ( * ) is equivalent (in the C . A . s ) to: (**) k kp = pk't Suppose ( * I and pfO. Then kp f 0, kp # 1, k V k p = r'kp # 1. So ( * ) implies ( * * ) . If ( * ) fails then there exists a q for which

("")

See P.R.Halmos (1955) A monadic algebra is a topological algebra in which every open element is closed, i.e. in which the closure operator is a "quantifier".

R. MAGARI

142

kvkq

>

rkq.

Putting p=k Y kq. kq we have p > 0 and k v kp = k Vk(k r kq. kq) = kv(k v kq. kq) = k( fl k r kq + v kq) = = k r k r kq+kykq), V k y kq + k v k q = l contrary to ( * * ) . So ( * * ) implies ( " 1 and the theorem follows. Corollary 8.11 The simply algebras of U are precisely the simple monadic algebras i.e. the C.A.s in which kp = 1 for every pfO. Proof. Obvious. Remark. If follows that a maximal k-ideal is not in general a maximal boolean ideal. Now we can easy study the situation for V. If A is a M.D.A. put hp = kp + .T p ( p easy to verify, and we have:

A). h is a closure operator, as it is

Lemma 8.12 V admit a good theory of ideals (see A.Ursini 1972) The 0-ideals are precisely the boolean ideals closed for h (h-ideals) and the 1-ideals are precisely the boolean filters closed for yhV. Proof. V is substantially a variety of groups with operators so it admits a good theory of ideals. It is easy to verify the other propositions. Now of course, referring to h, all that we say before for C.A.s is valid also for M.D.A.s. In particular: Corollary 8.13. V is ideal. Proof. The M.D.A. < A , + , . , 7 ,O,l,k,7 > has the same congruences of and the result follows. Remark. Let u s remark also that if A is M.D.A. then Gl belongs to the radical, because = h Y c l=kYSl+ 6 P' F 1 >/ h v h G l = h V ( k G 1 + $ 1) VFl+Kl

=

1

9. Meaning, Truth and D.A.S.

It is possible a non Tarskian analysis of theconcept of truth. Let us take as an example Peano Arithmetic. It is clear that there is point in introducing a concept of truth for its propositions only if we consider it applied, which happens if we use it, for example, as a metatheory of formal systems. Wishing now to introduce a concept of truth, it seems plausible to introduce it only for formulas that we wish to consider "meaningful". Which subset Po of the set of arithmetical formulas, do we wish to consider to be the set of meaningful propositions? We can propose to include in such a set exactly those propositions recognised by the theory itself, as it were, as verifiable or falsifiable, i.e. the p for which one of the formulas: k P jr6) I-TP 3 +6-& is verified, where T is the ordinary predicate "Theor". I shall not provide all the details; developping the idea and using a simplification due to A. Ursini one arrives at proposing as a set Vo of the "true and

Algebraic Logic and Diagonal Phenomena

143

meaningful" propositions the deductive closure of the set: L p : not I- q p and I- i p i ? ( r ~ 3 ) This deductive turns out to be in the arithmetical hierarchy. With usual techniques it is not difficult to "express" Vo with a suitable predicate bo and the following conditions p i Vo; Vo( $' ) c Vo turn out to be equivalent, thus, this concept of truth is not subject to Tarski's limitation Of course there are still some limitation of Godel's type:

z2

a) by Feferman's lemma there is a p with: (analogous of Godel's first theorem) Ip H 7 Go ( 3) ..-

b) 7 i O ( . O #

Vo

(analogous of Gddel's second theorem)

. -

# 0 ' ) are meaningless, so in a way every But both the p in a) and ,VO('O discrepancy cases which leads us to distinguish between theory and metatheory. From an abstract point of view, forgotting all possible "phylosophical" meaning, it is interesting the construction of Vo from T, ? and this construction can be iterated obtaining for P.A.:

(Aldo Ursini 1976). The results obtained from myself (R.Magari 1975) and A.Ursini can be extended to every theory which "expresses" a predicate "Theor" and the natural ambient for this study are the D.A. Let A be a boolean algebra, S its dual space (think A as the algebra of the clopen sets of S), e a diagonal operator on A, Z = Y C Y , < the inverse of the dual relation of ;J Of course the analogous of T is now i.1; and the anal2 gous of i. is E Now theanalogous of V is the boolean filters, F generated by: M = i p i A : p # O a n z fp rvp;=ipkA:p#Oand 6 p g p i . as in the following example: It is possible that F be improper (

.

.

<

9.1 (ex.) Let be S= i a , b $ , 4 = 0. Now A= g ( S ) and F;{aj = F i b $ = 0. So a,b h M and O C F . We will speak in this cases of semantical inconsistency of A. Now: 9.2 A is sem. consistent iff for all p,qr A, p=O,q#O, p>,T;p,q)(Tq we have Pq f 0 . 9.3 A is sem. cons. iff 4 is upper filtered. Now for investigate on F we can take theclosed set of S , C = n F = A M = = n : p : P + ~ , p ) Tp5. It is useful the lemma 9.4 < S, 4 > is inductive. Now we have an useful classification. 9.5 (i) There are in < S , d > at least two terminal distinct points not associated in 4 In this case C = 0. This is the only case of semantic inconsistency. 9.5 (ii) There is only one terminal point, a, and a {a: in this case C=ia] d A. 9.5 (iii) There is only one terminal point a and (a a: in this case C= {a$ f A. 9.5 (iv) The number of terminal points is > 1 but they are a l l associated in 4 in this case C is the set of terminal points and no subset of C is in A.

.

R.MAGARI

144

Now, M sing the techniques of 56 we can "formalise" F putting: <:(Yp.-~~)v Tqik:(>qi. is.) = : o + 9.6l+p=C.p+

v

a=Q

...where

This is possible, of course, if the convenient families have suprema and ( .;-(p.m qi ) + infima. Of course if j = ) i r we have jp = 7 p + ( .,' 7 qi + <- ( >'qi))) = l... The search for validity of: 9.7 If p~ F then ip c F and the search for compatibility of F with the congruence ry associated with F goes at further distinction in the case (ii) of 9.5:

xi

9.5 ii.1 There exists in S a b < a such taht, for every point x < S - (a,bj , x < b . 9.5.ii.2 Otherwise. We can proof that 9.7 is valid for all except 9.5 (ii,l) with b p. For compatibility with we have: if (i) (semantic inconsistency): Yes, but T=@,k is the id. 11 (ii.1) N O : a c P + a, a FO " (ii.2) Yes, but F is trivial: PO = fi1 = 1 " (iii or iv) (semantic consistency and Lu-consistency: now we will call these algebras "regular") : Yes.

-

4

+-

10. Progressions of Boolean Algebras with hemimorphisms.

7's

For hemimorphisms 7 with ~T we can generalise the previous results. We can study also thefinal product of iteration of the procedure and we find the following possibilities: (i) The final algebra A, is trivial (0=1): In this case 'c = 3 -el and A, is in the case 9.5(i) (ii) A has two elements. is the total relation; Form a logical point of view this means (iii)C=S, that C 1=1 and for p f l c p = 0.

LOGIC COLLOQLIIUM '82 G. Lolli, G. Long0 and A . Marcia (editors) 0 Elsevier Science Publishers B. V. (North-Holland), 1984

145

ON LOGICAL SENTENCES IN PA

Saharon Shelah Department of Mathematics The Hebrew University, Jerusalem, Israel Department of Mathematics Ohio S t a t e University, Columbus, Ohio, USA I n s t i t u t e ofAdvanced Studies The Hebrew University, Jerusalem, Israel Department of Mathematics University of California, Berkeley, Calif

.,

USA

Contents 5

1 . A representation of PH

5

2. On al-comprehension axiom 1 [We suggest a solution t o ? / n i - C A o

g 3 . A true 5

IT:

= Paris-Harrington/PAl

sentence in PA, n o t provable in PA.

4. On theories with incomparable consistency strength. [We show how t o produce such reasonable theories. We also draw the reader's attention t o reasonable examples where theconsistency strength areequal b u t t h e r e i s no interpretation].

ii 1 . A representation o f PH

We give in t h i s section a representation of Paris Harrington [PHI r e s u l t s , in a way which will be helpful 1.1. Definition: An

F

E

1)

M9"

2)

M

L,

later.

(L,n)-model i s a sequence

i s an L-model

I' Me

9"

5

n > such t h a t :

except t h a t functions a r e p a r t i a l (so M9" # 0 ) .

i s a submodel of M,+1

F

M = <M : 9"

M

e+ 1 ( f o r a . + l

5

n)

b u t f o r every function symbol

i s a t o t a l function (with range 5

I am very grateful t o Leon Henkin f o r saving the manuscript and t o Annalisa MarcJa f o r taking care of i t s typing. The author would l i k e t o t h a n k the NSF and the United States-Israel Binational Science Foundation f o r p a r t i a l l y supporting t h i s research.

S. SHELAH

146 1.1 .A. N o t a t i o n : Let

dp(cp)

-M =<Ma:l

a

5

5

-

n>,

1

v and 1 $ 1

be t h e q u a n t i f i e r depth o f

@(X)

1.2. D e f i n i t i o n : F o r a f o r m u l a

E

L

only,and o n l y atomic terms used,w.l.o.g.) and an (L,n)-model

(D

o f l e n g t h > dp(v)

5

M-ri~.il = <M,:i.

n>.

i s t h e l e n g t h o f J,

.

( w i t h n e g a t i o n i n f r o n t o f atomic formulas

a

and a sequence (i.e.

n

t

dp(cp))

a(;)

M,,

from

= t(i),

we d e f i n e when

by i n d u c t i o n on t h e q u a n t i f i e r d e p t h o f cp :

fiI=cp[a] if

fi

: a.

M. = <Mi,,

atomic

: as u s u a l ( n o t e t h a t o n l y a t o m i c terms were used, and

t h e l e n g t h of

fi

i s > 0 = dep(cp),

hence we can compute

t h e terms and check t h e s a t i s f a c t i o n o f t h e r e l a t i o n in

1.3.

Claim: 1 )

a)

h a s a dp(+)-model

IJJ

I f a sentence

b) u, has an n-model

2)

J,

3) I f

fi

M,)

$

has an n-model t h e n

( i n f a c t an m-model whenever

5

m

5

n)

satisfying

has a model i f f i t has an n-model f o r e v e r y $

dp($)

has no n-model t h e n t h e r e i s a p r o o f o f

n i J,

. o f length

I+ln

2I'I

( t h e l e n g t h o f a p r o o f i s t h e number o f symbols i n i t ) .

4) If

J,

has an n-model t h e n t h e r e i s no c u t

free proof o f 1 $

o f length

5

n.

Remark. We have n o t t r i e d t o m i n i m i z e t h e f u n c t i o n and numbers n o r we s h a l l do i t elsewhere i n t h i s a r t i c l e .

Proof: 1) a) I f of

J,

i s an n-model o f

u, t h e n

fi[oydp(J,)li s

a

dp($)-model

(check t h e d e f i n i t i o n ) .

2 ) Suppose

i s an n-model o f J , . We d e f i n e by i n d u c t i o n on

a.

5

n,

A,

5 N,

On Logical Sentences in PA

147

such t h a t :

a ) l e t A, in

L

b) i f

At

appears in J , ) i s defined, 3 y (p(y,x) i s a subformula of

then f o r some b b

every non logical symbol

consist of a l l individual constants (w.1.o.g.

E

N2+1,

So clearly

IA,,

(we can forget

A

i s theunionof

and

E

ice’lsn’i= coCb,al;

“IA,I

a.

a

ji,

we demand t h a t there i s such

with a l l such b ’ s .

s Cnr.of individual constants in ~i I

+

”

,[Q,,tll I= ( 3 Y ) m ( Y i ) ,

c A e’

as the multiplication by

5

IJ,~

and

i s more than needed). As

models are non empty

Now l e t MI = Ne ? A

e’

and we can prove t h a t f o r every subformula

e(;)

of

JI, a c A R ’

n-a.

i C e s n l I= e [ a l

dp(e(i)) :

t

iff

i C e , n l I=

eG1

Me = M

, so

(just like Tarski Vaught c r i t e r i o n ) .

2) If

M

i s a model of

5

then l e t ( f o r a l l a. )

J,,

n> i s an n-model of n If f o r every n <Me : R

<Me : .t

J,. 5

n > i s an n-model of

<Me : R < w > which i s an w-model of J,

.

Easily

3 ) Also immediate.

=

X ; ~ - ~ , ~

[ $ ] ( m i ) dp(’)

n-model of

J,,

X ,... ; , 3x nl , . . . , x n

and

e

J,,

Me

by compactness there i s

i s a model of

n

[There i s a quite short proof (of length <

3xi ,...,

easily

says t h a t

dn-1

e

( ~ i I l * l) ,

Now

J,

where m,, = 141, m

< { x im : m

e i s quantifier f r e e .

showing

5

m i ’ i 5 j};

e can be refuted

j

J,

.

1- 3 xX ... xm,-1 o i+i 5

by a

=

n>

i s an

truth table of

lJ,T 1.

size 2 1 ~ 1 1

4 ) Just l i k e the proof t h a t every model s a t i s f i e s any provable sentence.

* * * * *

148

S.

Let

PA

SHELAH

be Peano a r i t h m e t i c , and

o f t h e i n d u c t i o n scheme o f l e n g t h

PAk

are included ( b u t except t h e instances o f

k

5

be Peano a r i t h m e t i c when o n l y i n s t a n c e s

t h e i n d u c t i o n scheme t h e r e a r e o n l y f i n i t e l y many axioms, which a r e included,hence i s finite).

PAk

Let

PAPL

be l i k e

PA, b u t wetake o n l y t h e i n s t a n c e s o f t h e

i n d u c t i o n h y p o t h e s i s w i t h no parameters, and and

PAk.

I t i s known

(Friedman t h e s i s , I t h i n k )

s i s t e n t . It i s c l e a r t h a t the consistency o f n

2

k,

m o n o t o n i c i t y we can t a k e o n l y

= <Mra..

.Q

5

n> where

If

rQ+l> r i ,

r Q> 1,

i.e. there i s

t h e axioms o f if

a r e equi-con-

n t k

,

PAFL has an n-model" ( b y

k-!=

r, < rl < r 2 <

... c

r

.

n

PA minus i n d u c t i o n h o l d s . So we want t h a t

fik ( 3 x ) c p ( x )

x < r o such t h a t

PAPL

i s equivalent t o " f o r every

t i o n and m u l t i p l i c a t i o n r e s t r i c t e d t o t h i s , and where

f o r any f o r m u l a cp(x),

PA,

PL

PApL? I t h i n k t h a t t h e obvious c h o i c e i s k r = {0,1,2, ..., r - 1 1 w i t h t h e u s u a l a d d i -

i s the set

M

that

PA

.

n = k)

What i s t h e most n a t u r a l n-model o f

R

PA

has an n-model "and t o " f o r e v e r y

PAk

i s the intersection o f

PA[L

cplxl

then A

R!=(3x)Cu(x)~(Vy)((~(y) + x s y ) l ,

(Vy)(cp(y)

+

x sy)

( s i s definable,

letting

r o= 2

o r can be added as a r e l a t i o n ) . T h i s suggests d e f i n i n g

..., r n )

Fw(rl,

= m i n { x < rl : <M

if F$ ( r 1 ,

. . .,r

r

: Q = O , n>i=cp(x) Q

F (rl,r3,..., rn+l) = Fcp(r2,r3,...,r cp

nti

otherwise Let value o f


is

+

n 0,

t

...

r be homogeneous f o r FC (where m > 2n). I f t h e c o n s t a n t m F i on i n c r e a s i n g sequences f r o m {r,, ...,rml i s 1 , t h e n n e c e s s a r i l y

ro<

<

..., r m )

m, so i f

m

: a. < m-nt2 > i s s t r i c t l y d e c r e a s i n g ( i n Q ) ; hence

i s l a r g e enough compare t o

r,,

t h e constant value o f

hence t h e i n s t a n c e s o f i n d u c t i o n h o l d i n t h e n-model

We can work w i t h a l l t h e f o r m u l a s o f l e n g t h s k,

F " ( r ,. . . ,rn t 2 ) = cp

So i f

t



0

if

F (rl ,...,rntl) 'i

1

say =

(Pi

: II = 0, n > . rQ (is i,,)and d e f i n e

Fcp.(rz 1

F&

<M

,...,rn+z 1 f o r every

i

5

i,

otherwise

i s indiscernible f o r

F" and l a r g e enough <Mr cp

:isn>

i

149

On Logical Sentences in PA

PL k ' (Paris Harrington

i s an n-model o f PH

So

PA

partition

theorem)

i s enough t o p r o v e t h e c o n s i -

stency o f PA.

2. On t h e

Ti-comprehension axiom.

Simpson and Schmerl CSSl f o l l o w i n g M a c i n t y r e proved t h a t t h e c o n s i s t e n c y s t r e n g t h of PA a u g m e n t e d by t h e Ramsey q u a n t i f i e r ,

PA(QMM), and o f

1

nl-CAD

(see 2.2) a r e t h e same, i n f a c t t h e y a r e b i i n t e r p r e t a b l e . M a c i n t y r e ' s aim was t o s t r e n g t h e n P A so as t o e l i m i n a t e t h e well-known incompleteness f o r f i n i t e combina-

[MI and ISSI, i . e . p r e s e n t a

t o r i c s . Here we suggest an answer t o a q u e s t i o n f r o m finitary

combinatorial p r i n c i p l e capturing

IT:

a:-CA,,

based on end-homogeneity

i n s t e a d o f homogeneity (see t h e book CEHMRI). T h i s p r i n c i p l e was c o n s t r u c t e d from t h e p r o o f and n o t as u s u a l by m i n i t u a r i z i n g i n f i n i t a r y ones. I t was c l e a r t h a t i t s i n f i n i t a r y analogy i s " p a r a m e t r i z e d Galvin-Prikry"

so we have i t s c o n s i s t e n c y s t r e n g t h e q u i v a l e n t t o t h a t o f

ni-CA,,

.

T h i s n a t u r a l l y h i n t s t h a t t h e y a r e a c t u a l l y e q u i v a l e n t . Simpson takes i t on h i m s e l f t o prove t h i s and succeeds. T h i s work was done i n summer 1979, a f t e r h e a r i n g on t h e r e s u l t s o f Friedman I F 1 1 and j u s t a f t e r Simpson has e x p l a i n e d us [MI and [SSI (and b e f o r e Friedman and Simpson s t a r t e d i n v e s t i g a t i n g t h e q u a s i - o r d e r o f t r e e s ) , and we would l i k e t o thank Simpson f o r t h e c o n v e r s a t i o n . 2.1. N o t a t i o n : F o r a s e t

A

o f n a t u r a l numbers, we c a l l

i f i t i s a f u n c t i o n f r o m i n c r e a s i n g sequences f r o m

A

o f power k ) For B

2.2.

A,,

c

f, A

A

into

..., Am E n, ,...,A,-,,D)

=

H

Remark: n:-CA,

f

i s c o n s t a n t on

k

( o r subsets o f

B.

H a hereditary function f o r

H(p,A,

H(p,A,

2.2A.

o f length

A

as above

i s f-homogeneous i f

As

a k-colouring o f

{O,l}.

N o t a t i o n : We c a l l

variable).

A

f

0

,...,Am) f o r every

belongs t o Am

n

for

p < n,

m,

and i f i t i s z e r o t h e n

{O,ll, (i.e.

if

H

i s monotonic i n t h e l a s t

i s always l i k e t h i s we would n o t r e q u i r e h e r e d i t a r i t y e x p l i c i t e l y . i s a t h e o r y "speaking" on n a t u r a l numbers and r e a l s ( i . e

150

S.

SHELAH

sets o f n a t u r a l numbers) s a t i s f y i n g : i f

$(Y,x)

i s a formula, p o s s i b l y w i t h r e a l

I x : (3Y) $ ( Y , x ) l

parameters b u t a l l q u a n t i f i c a t i o n a r e o n n a t u r a l numbers, t h e n

( x v a r i e s on n a t u r a l numbers,

i s a real

Y

on r e a l s ) .

D e f i n i t i o n : We d e f i n e a combinatorial p r i n c i p l e which corresponds t o t h e e x i -

2.3.

stence o f "end homogeneous sequence" i n Erdos-Rado terminology: : For any f u n c t i o n

CPeh(n,k,e)

whose values are k - c o l o u r i n g o f

-homo for

when m

Am/p

c

1

r

... < ik

p < Min Ar-,,

5 &,

Am

is

i, <

5

-

and f u n c t i o n

n,

n such t h a t ( a ) (A,(

subsets o f

F, k-place, defined on subsets on

2

m s a

<

H(p,A,

,

or

Am

each

( c )
,...,Am_,,Am/p)

= 0

ii : ( p , i ) n A,

H, t h e r e are A,

and ( b )

Min A,

i

5

CPCh (n,k)

n = {O,..,,n-ll

subsets o f

there a r e A, 2 A, Am i s

i s H-end-homo,

,...,Am-l,Ar)

= H(p,A,

2.5.

Claim:

...IAt

i.e.

'k

)-

where

k, a.

f o r every

F d e f i n e d on sequences o f

whose range a r e k - c o l o u r i n g o f

A, 5 n such t h a t (a)

...,Am- 1 )-homogeneous

F(A,,

A,?

2

F(A i,,...,A.

p}.

means: f o r every f u n c t i o n

... 2 Ak,

2

and

e ) holds.

CPeh (n, k,

Definition:

2.4.

2 A,

is

Remark: So t h e combinatorial statement we are i n t e r e s t e d i n i s : f o r some n

n,

lAkl

and ( c ) 1

n 2

and f u n c t i o n

Min A k

H

and ( b ) each

i s H-end-homo.

The statements

V k

Ya 3 n

CPeh

(n,k,a)

Yk

3 n

CPLh

(n,k)

a r e equivalent. Proof: Immediate. 2.6.

Claim:

Suppose

PA

+ Vk3n

CPIeh (n,k)

i s c o n s i s t e n t . Then

is consis-

T:-CA,

tent. of

Proof: So t h e r e i s a non standard model non-standard

k

~ eand, choose

(? kcp&

n

E

PA

+

Yk 3 n CPBh

(n,k).

Choose a

such t h a t

(ny3k)

Now we s h a l l d e f i n e

F,

so l e t

shall define a (3k)-colouring o f n,

be a sequence o f subsets o f F(A).

n,

and we

151

On Logical Sentences in PA

r s n,

F o r t h i s f o r e v e r y n a t u r a l numbers

r = {0,1,2,3,..

universe

t o r ) ,individual

c o n s t a n t s 0,l

p l a c e r e l a t i o n ) . The language

and r e l a t i o n s
m,

a sentence o f l e n g t h

L depends on ,(A)

involving only

m, t m 2 . Now we d e f i n e

be t h e model w i t h

<

&(a))

(i.e. A, i s a one

only. many ways, choose one so t h a t

i < m2, i s n o t t o o l a r g e compare t o

,...,i 3 k - 1 ) :

F(A) (io,il

G ( a ) (io ,...,i2k-1 ) =

Ai,

(a.

a.

We can code t h e sentences o f t h i s language i n

if

ar =IfTrlJ\l

let

.,r-ll, f u n c t i o n s : a d d i t i o n and m u l t i p l i c a t i o n ( r e s t r i c t e d

i t i s zero

G(h) (ik ,...,i3k-1 )

and one o t h e r w i s e , where

G(1) L-sentece

i s the f i r s t (iO,...,izk-,) and

J,


Cil, ...,ori

[A1,Oci 0

CAI> k k- 1

1

We now w i l l d e f i n e a f u n c t i o n i.e.

we w i l l d e f i n e i t i s zero i f

whenever

p s i, which i s

H(p, A,

i*,io ,...,im E A +,.


H.

,...,A m+ 1)

... <m'i

..., Ai [ h ] > m

B

which i s an

*

:

e(-,

p codesa f i r s t o r d e r f o r m u l a

< p < B < i,
i, o r

m = dp(e) t 8

+

-, A,

,...,A,-,),

t "8 d e f i n e a t r e e and f o r some

x , y < i

D the i n t e r v a l

and

8

A+,,

X,Y

(x,y)

t * f o r every

x < Y

5

has a member below

t*

( i n the tree)" o t h e r w i s e t h e v a l u e i s one

A

(so i f As

m+ 1

i s empty t h e v a l u e i s zero, n o t e

C P I e h (n,3k)

such t h a t ( f o r each

holds, m)

Am

H

i s hereditary).

A, 2 A, 2 ... 2 Ak, subsets o f n , ...,Am- 1)-homogeneous, and

there are is

F(A,,

i s H-end-homo. Let f o r n

a.

a.

< k

= Min A,

Now we d e f i n e a model

. c%

which w i l l be a model o f

a;-CAo:

t h e n a t u r a l numbers

152 of

S. SHELAH

63 a r e

(remember

k

i s not standard).

A d d i t i o n , m u l t i p l i c a t i o n 0, 1, < The f a m i l y o f r e a l s

QR = {B 5

a r e i n h e r i t e d from

c.

( = s e t s o f n a t u r a l numbers) i s t h e f a m i l y

aN:

f o r some s t a n d a r d

il,

i s f i r s t order definable

B

( w i t h parameters) i n t h e model

(a3,,+,

X,

Al,...,Ail)}

ni-CA,.

The o n l y n o n - t r i v i a l p a r t i s So we have t o p r o v e t h a t

tP

(FO) where

aN:

E

63

i=

( 3 Y ) j,(Y,p,i)}

i s a f i n i t e sequence of s e t s f r o m

. I t i s w e l l known t h a t w.1.o.g.

e(-,-;

a tree

p,

i)

@I

G3 t j,CY,P,q,A1

9 . .

As

(F1) i.e. (F2)

u s i n g a parameter Am(*J+l

is

F(A,,

( V X9Y) [ x

we can r e p l a c e f o r some

Y v a r i e s on

61,.

by

m(*),

i s f i r s t order definable i n

Y

...,Am ( * ) A

~

)-homogeneous E A m( * ) + I

(aN,A, ,...,Am(*)+1 )

a E A m( * ) + I

there i s

Am(*)+17 Y an element below By t h e H-end-homo,

>

f o r some

in

(@"A1,

...,

cBN .

q(*)

Am(*)+1

The model

...,A

"
Y

A X

t h e branch has a member i n each i n t e r v a l o f

(remembering K6nig

order,

Y i s an i n f i n i t e branch i n

..Am]

Then f o r some t r u e l y f i n i t e

)

,

€aN.Suppose

t r u e l y f i n i t e m, q

m(*)

J,

"says" t h a t

(whose o r d e r i s i n c l u d e d i n t h e n a t u r a l o r d e r o f & N ) .

By t h e d e f i n i t i o n o f

A

j,

€aR a,, f i r s t

<

y

A

(X < 2 <

y)]

s a t i s f i e s " f o r every

e(-,-,p,q,A,

we can r e p l a c e

lemma) t h e s e t

Y)

Hence

m(*)+i'

i n the interval

Am(*)+i t* ( i n t h e t r e e

( 3 2 E

-t

t* such t h a t f o r e v e r y

,

(F2)

i t i s easy t o see t h a t

m(*)+l

x < y < a (x,y)

there i s

,...,Am)". by

m+l,

hence c l e a r l y

153

On Logical Sentences in PA

ip

€ f i N:

{p

E

a k ( 3 Y ) *(Y,

aN:

f o r every a

P, ill = E

y

...,Am )

BR a s

which belongs t o 2.7.

t* ( i n t h e t r e e

(x,y)

B(-,-,p,q,

l.

required.

m: For every n a t u r a l number

3 n CPQh ( n ,

in t h e i n t e r v a l

Am+1

E

t h e r e i s an element below Al,

t* such t h a t f o r every

there i s

A,+,

x < y < a , X E Am + l ,

k we can prove i n PA(Q

MM

) t h e statement

k).

Proof: F i r s t we prove t h a t t h e conclusion i s t r u e , i . e . t r u e i n t h e universe

6?)if

i t i s a model of say second o r d e r Peano a r i t h m e t i c .

Then f o r every n , t h e r e i s a p a i r

Suppose t h a t the conclusion f a i l s f o r k.

(Fn, H ) which forms acounterexample t o CP'

of functions

eh

n

We now d e f i n e by induction on i For

5

k i n f i n i t e sets

so t h a t R i + ,

Ai,

5 Ai.

i = 0 t h e r e i s no problem. Let A, = B, = t h e set of a l l natural numbers.

So suppose we have defined

A

for j s i

j

and we s h a l l define

By t h e i n f i n i t e Ramsey theorem we can get an i n f i n i t e ( a ) f o r every y Ai n y )

Now

(n,k).

z

<

in

A:,

f

j

for j = i+l.

Af 5 Ai such t h a t : F ( A , n y , A, n y , ...,

i s t h e function

Y¶Z

A

Z

restricted t o y.

f

does not depend on

z, i . e . i f

y

z,

< z,,

E

A

then

YSZ

SO l e t

f

= f

Y

for y

Y,Z

( b ) For

Y,

<

<

A' i '

z

Y, < y 2 i n

f

= YlZ,

f

YJ,

.

1

fy,ry, = fy,ry0 [Simply apply Ramsey theorem t o t h e natural t h r e e place c o l o u r i n g , f i r s t f o r ( a ) then f o r ( b ) l So

f

=

U If

Y

rz

are i n A i l

: z < y

i s a k-colouring ( o f t h e natural num-

b e r s ) . By t h e i n f i n i t e Ramsey theorem t h e r e i s i n

&3,

an i n f i n i t e s e t A:

A:,

which i s f-homogenous. Now we s h a l l deal with

H.

Again t h e r e is an i n f i n i t e

A: 5 A f

(A;

of course) such t h a t :

(c) i f H Z ( p , A, n y ,

y

<

z

. .., Ai

are i n

A:,

ny,

9)

< z

a r e in

95 Ai

n

y,

p

does not depend on

<

y

t h e n t h e value of

z.

Moreover (d) i f

x

<

y

A;,

9 5 Ai

nx,

p

<

x then t h e value of

BR,

S. SHELAH

154

..., Ai

HZ(p, A, n y.

For every p

9)

n y,

63,

E

and i n f i n i t e s e t

zero i f f f o r a r b i t r a r i l y large q HZ(p, A,

..., Ai

y,

n

<

. 9):

z in A i / q

i t is

the value of

i s zero. Clearly a f i n i t e change i n

HZ, and "for a r b i t r a r i l y large q

"for every large enough q "

z )

we define T ( p ,

5 A:

f o r every y

E ( % ~

n y, D n y - q )

n o t change the value of

(and on

does not depend on y

3

does

can be replaced by

"

(see 2 . 2 ) .

Now we want t o apply the Galvin Prikry theorem t o T, more exactly t o a parametrized version of i t

( p as the parameter). Simply i t e r a t e the usual Galvin-

Prikry theorem on the natural numbers, and take the diagonal intersection. What we get we call

.

A;

Again remembering 2.2, and the conclusion of the Galvin-Prikry theorem, we can

so t h a t f o r every p,

find Aitl 5 A:, Ai/q

the value of

i f f o r some q > p ,

..., A 1.

HZ(p,A, n y ,

n Y, Ai+,nY-q)

f o r every y

i s zero,

<

z in

then 9 = P

will serve. If

..., Ak

A,, n

A,,

9

Aktl

are defined, choose y

y > contradicts thechoice of

Fz, HZ

<

z in Aktl

and < A , n y,

.

However we want t o prove t h i s in PA(QMM) (and not in ZFC o r i n second order number theory). The proof i s the same, replacing "a s e t of natural numbers" by "a definable s e t of natural numbers". Why i s < < F , H > : n < W > definable? We canchoose for each n , a minimal n n pair < F n , H n > (by a simple enough coding). We can apply the i n f i n i t e Ramsey theoMM

rem as Macintyre [MI proves theparallel statement holds (from PA(Q ), of course the proof depends on the formula defining the colouring and the i n f i n i t e s e t ) . Wore exactly he proves that i f

,...,xn,

~(x,

-

p, z)

A

i s i n f i n i t e and definable,

i s such that

then there i s a definable i n f i n i t e

V x1

B c A and co

<

,...,xn

( 3 2 < c ) ~ ( x ,..., , xn,p,z)

c such that

We are l e f t with the "parametrized Galvin-Prikry". Let a < * b mean:
<

a , b code f i n i t e increasing sequences, ,

e ( a ) > respectively, and ca(m) : m < a ( a ) > i s a proper i n i t i a l seg-

ment of < b ( m ) : m

<

e(b)>.

155

On Logical Sentences in PA We d e f i n e

A:

A:

(after

has been d e f i n e d )

by d e f i n i n g by i n d u c t i o n on

I, kk, 6, such t h a t k,

(a)

<

... <

k

2- 1

L o , C1,

and

..., sI-l

{0,11

E

> < * a < b and f o r e v e r y p < k f o r some q, (QMMa,b) C < k o , ..., k f i - l y, z i f q < y < z, y E A?, z E A? t h e n HZ(p, A, n y, A . n y.

(b) f o r every

...,

1

Ia(m) : m < z ( a ) > / q ) =

1

cPl.

c ~ =- 0~ . Now we can c a r r y t h e d e f i n i t i o n (as i n [ M I n o t i n g t h e f o r m u l a s we use have (c) i f compatible w i t h

(a) t (b),

bounded complexity).So we f i n i s h t h e p r o o f o f 2 . 7 .

5 3. A t r u e

ny-sentence o f P A n o t p r o v a b l e i n PA

I n summer '80 Friedman and H a r r i n g t o n o f f e r e d h o t l y t h e i r view t h a t i t i s one o f t h e main problems o f contemporary l o g i c t o f i n d mathematical sentences as ment i o n e d i n t h e t y t l e , as w e l l as t o f i n d n a t u r a l t h e o r i e s w i t h incomparable c o n s i stency s t r e n g t h . The " t e c h n i c a l d i f f i c u l t y " i s i m m a t e r i a l ; i n f a c t t h e easiness o f t h e p r o o f may i n d i c a t e t h e profoundness and n a t u r a l i t y o f t h e sentence. Now an answer t o such q u e s t i o n i s n a t u r a l l y more open t o debate t h a n t h e usual mathematic a l problem.

A s t h e a u t o r d i d n o t want t o go i n t o such d i s c u s s i o n , and H a r r i n g t o n wanted a s o l u t i o n , an agreement was reached: i f t h e a u t h o r c o u l d f i n d a s o l u t i o n which H a r r i n g t o n would t h i n k i s O.K.,

he would w r i t e i t up, d i s c u s s i t and p u b l i s h i t .

The c o n t e n t o f t h i s s e c t i o n was done i n s p r i n g '81, H a r r i n g t o n O.K.ed i t , as w e l l as I 4 ( w h i c h was done i n summer '80) b u t was t o o l a z y t o f u l f i l l h i s promise. 3.1. C o n t e x t :

N be a n a t u r a l number

1) Let

r

= < r : I < I(;)> 9"

a finitesequence

o f n a t u r a l numbers.-

2) L e t

K = K i = {(A,

<,

R)

: A

6 R

a subset o f

N,

<

a l i n e a r order o f

A

a sequence o f r e l a t i o n s o v e r an

A

,

,

r -place

Ik e ( R ) = I(;)}.

3 ) Members o f

K

a r e denoted by

A, B

,

letting

power o f t h e s e t o f t h e s e t o f elements o f 4) I n (4a)

K

A =

/ ~ ,l

so

i s the

llAll

A.

we d e f i n e

A <en B

( B an end e x t e n s i o n o f

A) i f A

i s a submodel o f

B

and

S. SHELAH

156 x

IBI

E

,

- 1.4

(4b) A < B

y

satisfying

implies

y < x.

of

A'

into G

IIb/l 5 IlAll

A',

A <

en B over

i n t o @(N)

Let

+

1,

A < B and f o r any en t h e r e i s an embedding

.

A

f r o m @(N)

3.2. E ( N , r , k , n ) - p r i n c i p l e : F

PI

( B an u n i v e r s a l end e x t e n s i o n o f A) i f

A'

5) A function

1 ) Domain:

E

F

is

be a k - p l a c e f u n c t i o n s a t i s f y i n g

i s d e f i n e d on i n c r e a s i n g sequences o f l e n g t h

2) Choice F u n c t i o n :

..., A k )

F(A,,

...

3 ) Isomorphism I n v a r i a n c y : i f A 1 < i s an isomorphism

g

from A k

< Ak

onto

k

K

from

.

IA,[

E

if / G ( A ) I z f ( 1 A I ) .

f-small

B

k

E

K,

B1 <

mapping

... < B k

At

onto

E

Bil

K , and F

then

there and

g

commute, i .e. F(B~.

..., B k ) =

4 ) Weak H e r e d i t a r i t y : F o r e v e r y such t h a t : i f

f

B

-1

=

0)

then

Then - there

A,

+)

A2 <

<

i s a submodel o f

B~

F(A,

...,

g(F(Al,

,...,A k )

= F(Bl

...

. k t h e r e i s an x -small f u n c t i o n

< Ak,

f(Bil-l)

Ai,l

n Ail

5 Bil

(stipulating

,..., B k ) .

i s an i n c r e a s i n g sequence < A :~ il < n > on which

F

depends on

the f i r s t structure only. 3.3.

Fact

F to

F'

if

N > 22 k+n+'(r(i)+l),

F a s i n 3.2 and

N' >

a k - p l a c e f u n c t i o n s a t i s f y i n g 1 ) - 4 ) o f 3.2 f o r

N,

t h e n we can e x t e n d

N ' , i n one and o n l y one

way. __ P r o o f . By 3.2 ( 4 ) ( 3 ) (as i n t h e p r o o f o f 1.3 ( 1 ) ( b ) ) . 3.4.

Claim: I n

PA+PH

$* = ( V

r,

we can p r o v e (the

k, il)

r,

(N,

k , n ) - p r i n c i p l e h o l d s f o r e.g.

,k+n+ f C r ( i ) + l l

).

N = 2 __ P r o o f : D e f i n e by i n d u c t i o n on

m

e,

i s s m a l l e r enough t h a n

putation). Applying

PH

and i n d i s c e r n i b l e f o r F ' ( A ~, 1

e,

a model

Am<*A

e

we g e t a s e t (F'

as i n 3.2

..., A .

'k

) = F'(A. Ji

.

e

c

K

w i t h universe

il,

so t h a t i f

( j u s t t a k e c a r e o f 3.1 ( 4 ) ( b ) , easy com-

C

o f n a t u r a l numbers

for

,

A

N'

..., Ajk)

(V x < y ~ C ) C 2 2 ~ < y l ,

l a r g e enough)

,

and

ICI

>

4 Min C

157

On Logical Sentences in PA (we use an e q u i v a l e n t v a r i a n t o f P.H.).

: i


F

C>,

E

As i n 5 1

depends on t h e f i r s t v a r i a b l e o n l y . Now as i n t h e p r o o f o f 1.3

( 1 ) ( b ) we c o l l a p s e t h e s o l u t i o n below 3.5.

Claim: I n

PA+$*

N

.

we can p r o v e t h e c o n s i s t e n c y o f PA (hence

Proof. We can b u i l d a non-standard model o f dard,

N

l a r g e enough, and

..., Ak)

F(A,,

mal code f o r which

1

( v y)

e a s i l y f o r subsequences o f

r

PA,+$*,

<4> d e f i n e

F :

i s d e f i n e d as f o l l o w s :

let


{3Xcp(X.y)

=

..., A k > / = +

M, choose

cp(x,y)

3X[cp(X,y)

A

(v

z

< X)

and t h e n

( i f t h e r e i s one).

A,,

cp(x,y)

non s t a n -

be a f o r m u l a w i t h m i n i -

ic p ( Z , j ) l

(by the lexicographic order o f

E

k, n

"induction fails,i.e.

and t h e n t a k e minimal x

PAP$*).

1''

<Max

y.

y o , yl,

The r e s t i s as i n I 1

...>

)

the only

a d d i t i o n a l p o i n t i s why a r e a d d i t i o n and m u l t i p l i c a t i o n d e f i n a b l e ? T h i s i s by 3.1 ( 4 ) ( b ) .

5

4. On c o n s i s t e n c y s t r e n g t h

Let extending CON(T)

T PA

denote h e r e a ( r e c u r s i v e ) t h e o r y ( w i t h f i n i t e - s o r t , f i n i t e - l a n g u a g e ) b u t i t may a l s o "speak" on r e a l s and even a r b i t r a r y s e t s .

be t h e sentence ( i n

D e f i n i t i o n : We say

PA

T, scs T,

language) s a y i n g

T

i s consistent.

( t h e consistency strength o f

equal) than t h e consistency strength o f

T2)

if

PA

Let

t- CON(T,)

T,

i s smaller (or -f

CON(T,):

I t was observed t h a t e s s e n t i a l l y " l a r g e c a r d i n a l axioms a r e l i n e a r l y ordered"

(though i n some cases t h i s "has n o t y e t been proved").More e x a c t l y i t seems t h a t a l l s e t t h e o r i e s which has been c o n s i d e r e d so f a r , a r e l i n e a r l y o r d e r e d by Solovay

( I t h i n k ) has

found

T's

which a r e

5

-incomparable,

s

cs b u t t h e y were

.

cs " p a r a d o x i a l " (i .e. have s e l f - r e f e r e n t i a l sentences). We s h a l l t r y t o g e t more r e a sonable ones.

* * * * * * * Let

PA+

be

PA t C O N ( P A ) .

We work i n s i d e

P A . A model w i l l mean one which

i s definable. Let

T,,

T,

be c o n s i s t e n t t h e o r i e s ( i n o u r " u n i v e r s e " which s a t i s f i e s PA).

s. SHELAH

158

T,

" s a y i n g " t h e r e i s a model o f

(think o f

PA+,

PA

+

CFMSI, o r o f course As T,

+

ZFC,

T, " s a y i n g " t h e r e i s a model o f

ATR, see Friedman, McAloon and Simpson

T,tCON(TI)

hence t h e r e i s MI

( 1 ) q Q ( n ) says

n

i s c o n s i s t e n t , hence ( b y Godel incompleteness has a model M,

M,

E

f ''$2(n)''

I=

such t h a t

(n

. By

t h e r e q u i r e m e n t on T,

i s a non-standard i n t e g e r )

where

i s a n a t u r a l number and t h e r e i s a p r o o f o f s i z e

n

PA

ZFC+large c a r d i n a l s ) .

T, +CON(T,) + iCON(T, +CONTI)

iCON(T,),

n

of

t 1 CON ( Te)

PA

f ( 2 ) +,(n)

says

$

Q

(n)

but

f ( 3 ) T~ = PA + ( 3 n ) r$,(n) i s consistent with

i s t h e f i r s t such number.

n

As we have assumed t h a t

with

TI,

o r use

or

i s consistent,

theorem) M,

I rrl-CAo;

T, says t h a t

has a model, c l e a r l y

+-,~,,(2~")1 f

+ (Vm) l$,(m)

PA + 3 n$,(n)

( a s even

PA+

PA

i s consistent

.

PA+)

By a theorem o f Friedman (based on a n a l y z i n g Godel incompletness theorem) f

( 4 ) Tb = PA + ( 3 n ) C$,(n)

'

( 5 ) PA+

+

Ta, Tb

are

s

cs

-incomparable.

L e t us p r o v e e.q.

*

Ta 1- CON(Ta)

rBecause f o r any model nition

11

PA

We s h a l l p r o v e t h a t csTb

217

+.

i s consistent with

Ta

$,(2

f

o f a mode:

phism from

No

Clearly

N,

of

N o of PA

(i.e.

PA++Ta, No

b e i n g a model o f

1: "N,

i n t o a p r o p e r i n i t i a l segment o f N,

satisfies

of bounded f o r m u l a s and

Ta -PA,

has a d e f i -

i s a model o f P A " ) and an isomorN,.

as end e x t e n s i o n s p r e s e r v e t h e s a t i s f a c t i o n

f

$,(x),

PA+

$,(x)

a r e such f o r m u l a s . 1

Note a l s o t h a t (6)

PA

+

.

Ta I- 7CON(Tb)

[Otherwise t h e r e i s a model a d e f i n i t i o n o f a model segment o f No I= T

a' formulas,

N, No

(i.e. f "$,(n)

and

f

N, o f No

No

of

PA+Ta+CON(Tb)

Tb a n d a n i s o m o r p h i s m g of

hence i n

No

No

there i s

onto a proper i n i t i a l

s a t i s f i e s t h e sentences s a y i n g t h o s e t h i n g s ) . AS 211 f and -I $,(2 ) " f o r some n, b u t as $, $1 a r e bounded f commute w i t h e x p o n e n t i a t i o n , c l e a r l y N, I. "$,(g(n)) and

159

On Logical Sentences in PA 1 $,(22g(n))".

So

f 2m N, I= Tb hence f o r some m, N , I= "$,(m) and $ , ( 2 ) " . f f and $,(g(n))",hence by q 2 ' s definition g ( n ) , m have t o be

But

f

N , k "$,(m)

equal. B u t

N, k "$,(2

2m

)

and 7 $ 1 ( 2 ' g ( m ) ) " hence

g(n),

rn

should be unequal,

contraddictionl. By ( 5 ) and ( 6 ) clearly

PA+

+

C O N ( T a ) I$

(as PA+ + T~

CON(Tb)

is

consistent (by ( 3 ) ) ; t h i s implies

PA I f CON(Ta)

(7)

+

So Ta $ csTb.

CON(Tb)

. Tb $ csTa

i s t o t a l l y analogous. n f NOW (Woodin suggests) wecan replace the sentences ( 3n ) r $ , ( n ) + - 1 $ ~ ( 2 ' ) I n f ( 3 n ) r $ , ( n ) + $ J 2 ' ) I by inequalities of the indicator functions corresponding t o

and

The proof of

f , , f,

T I and T, ( i . e . the function T , , T,).

t o the consistency of

exhibiting the

E.g. ( V n ) [ i f

f,(n)

ITsentences :

corresponding

i s defined then

so i s

f , ( f , ( f , ( n ) ) 1 and i t s negation. So i f we accept those functions as "mathematical"

( n o t j u s t reasonable methamathematical ) we get mathematical theories of incomparable consistency strength. (Originally we have used three t h e o r i e s ) . We a r r i v e t o the dangerous question o f which PA

function: on

see Paris and Harrington [ P H I ,

T's

have matheratical indicator

on many theories ( l i k e ZFC+large

cardinal) see Friedman rFl1 on ATR, see Friedman McAloon and Simpson rFMSl on 1

IT,-CA,

see 5 2. Alternatively f o r an indicator function

f ( f * ( n ) + l ) , and use

$,

$3

where

$,

f

f*

define

= "the f i r s t

by

f*(O) = 0

n f o r which f ( n ) i s

:Q

f*(n+l)=

mod 4".

+ * * * * * * * Notice the following two phenomena

( A ) For any two natural s e t theories, not only they are

5

cs

-comparable, b u t one

i s interpretable in the other ( o r expected t o be s o ) . ( 6 ) Similarly, f o r any theories, e.q. undecidab l i t y r e s u l t s are gotten by i n t e r -

pretation. Friedman rFr21 proved a theorem saying ( A ) i s really true. Concerning ( B ) however, in CSh1, (under C H )

the monadic theory of the re 1 order i s proven undecidable

without the usual interpretation. I n Gurevich and Shelah

CGSl1 t h i s

i s explained i t i s a Boolean-valued i n t e r p r e t a t i o n , and by CGS21 the usual i n t e r pretation i s impossible. Now we can t r a n s l a t e i t t o ( A ) : l i s t the reasonable axioms f o r the monadic theory of the real order (considered as a two-sort model).

S. SHELAH

160

REFERENCES

CEHMRl P. Erdos, A. H a j n a l , A. Mate and R. Rado, C o m b i n a t o r i a l s e t t h e o r y , N o r t h H o l l a n d P u b l . Co.

CF11

H. Friedman, On t h e necessary use o f A b s t r a c t s e t t h e o r y , Advances i n Mathem a t i c s 41 (1981), 209-280.

TF21

H. Friedman, T r a n s l a t a b i l i t y and r e l a t i v e c o n s i s t e n c y .

C FMS 1 H. Friedman, K. McAloon and S.G. Simpson, A f i n i t e c o m b i n a t o r i a l p r i n c i p l e which i s e q u i v a l e n t t o t h e I - c o n s i s t e n c y o f p r e d i c a t i v e a n a l y s i s . CGSll

Y. Gurevich and S. Shelah, The monadic t h e o r y and t h e n e x t w o r l d . I s r a e l J.

Math. CGS21 Y. Gurevich and S. Shelah, A r i t h m e t i c cannot be i n t e r p r a t e d i n monadic theory o f 8. [MI

A. M a c i n t y r e , Ramsey q u a n t i f i e r s i n a r i t h m e t i c , Proc. o f a L o g i c Symp. .(Karpacz 1979) ed. L. P a c h o l s k i and A. W i l k i e , S p r i n g e r V e r l a g L e c t u r e Notes i n Mathematics.

[PHI

J . P a r i s and L. H a r r i n q- t o n ,- A mathematical incompleteness i n Peano a r i t h m e t i c , Handbook o f Mathematical L o g i c , ed. Barwise, N o r t h - H o l l a n d Publ. Co.., 1977, 1133-1142.

CSh 1

S. Shelah, The monadic t h e o r y o f o r d e r , Annals of Math. 102 (1975), 379-419.

CSSl

S.G. Simpson and J.Schmer1, On t h e r o l e o f Ramsey q u a n t i f i e r s i n f i r s t o r d e r a r i t h m e t i c , J. Symb. L o g i c .

LOGIC COLLOQUIUM '82 G. Lalli, G. Long0 and A . 'Marcia (editors) 0Elsevier Science Publishers B. V. (North-Holland), 1984

161

CONTINUOUS TRUTH I Non-constructive Objects Michael P . Fourman Department of Mathematics Department of Pure Mathematics Columbia University Uni vers i t y of Sydney New York, N . Y . 10027 N.S.W. 2006 U.S.A. Australia

W e g i v e a general theory of the l o g i c of p o t e n t i a l l y i n f i n i t e o b j e c t s , derived from a theory of meaning f o r statements concerning these o b j e c t s . The paper has two main p a r t s which may be read independently but a r e intended t o complement each o t h e r . The f i r s t p a r t i s e s s e n t i a l l y philosophical. In i t , we d i s c u s s the theory of meaning. We b e l i e v e t h a t even t h e s t a u n c h e s t r e a l i s t must view p o t e n t i a l i n f i n i t i e s o p e r a t i o n a l l y . The second p a r t i s formal. In i t , we consider t h e i n t e r p r e t a t i o n of l o g i c i n t h e gros topos of sheaves over t h e category of separable l o c a l e s equipped with t h e open cover topology. We show t h a t general p r i n c i p l e s of c o n t i n u i t y , l o c a l choice and l o c a l compactness hold f o r t h e s e models. We conclude with a b r i e f discussion of the philosophical s i g n i f i c a n c e of our formal r e s u l t s . They allow us t o reconc!le our explanation of meaning w i t h the "equivalence thesis , t h a t 'snow i s white i s t r u e ' i f f snow is white.

PROLEGOMENON Classical mathematics i s based on a p l a t o n i c view of mathematical o b j e c t s . The meanings of mathematical statements a r e determined t r u t h - f u n c t i o n a l l y . T h i s Fregean explanation of meaning j u s t i f i e s c l a s s i c a l l o g i c . The d e f i c i e n c i e s of such a view a r e amply discussed by Dummett C19781. A c o n s t r u c t i v e mathematician r e j e c t s t h e completed i n f i n i t i e s of classiGa1 mathematics. For h i m , t h e objects of mathematics a r e e s s e n t i a l l y f i n i t e . The meaning

of q u a n t i f i c a t i o n over i n f i n i t e domains is given o p e r a t i o n a l l y i n terms of a theory of c o n s t r u c t i o n s . T h e r e s u l t i n g l o g i c includes Heyting's p r e d i c a t e c a l culus and o t h e r p r i n c i p l e s ( e . g . choice p r i n c i p l e s ) .

As Dummett has s t r e s s e d , one t a s k of any philosophy of mathematics i s t o explain the a p p l i c a b i l i t y of mathematics. The p o t e n t i a l i n f i n i t i e s of experience exceed t h e f i n i t e o b j e c t s of t h e s t r i c t c o n s t r u c t i v i s t . They demanda mathematics of inf i n i t e objects. Naive a b s t r a c t i o n leads t o the i d e a l i n f i n i t e o b j e c t s of c l a s s i c a l mathematics. This i d e a l i s a t i o n has enjoyed remarkable success. However, the meaning of statements .of c l a s s i c a l mathematics remains problematic. Brouwer C19811 introduced t o mathematics p o t e n t i a l l y i n f i n i t e o b j e c t s such a s f r e e choice sequences. Consideration of t h e s e j u s t i f i e d , f o r Brouwer, i n t u i t i o n i s t i c l o g i c , including various choice and continuity princip2e.s. W e s h a l l consider a general notion of non-constructive o b j e c t . For us, t o present such a notion i s t o give a theory of meaning f o r statements involving non-constructive o b j e c t s .

Our non-constructive o b j e c t s a r e not t h e p l a t o n i c ideal o b j e c t s of c l a s s i c a l mathematics nor t h e f i n i t a r y o b j e c t s of pure constructivism. They a r e p o t e n t i a l l y

M.P. FOURMAN

162

i n f i n i t e o b j e c t s r e l a t e d t o t h e l a w l e s s sequences o f K r e i s e l 119681and t o Brouwer's f r e e - c h o i c e sequences ( T r o e l s t r a 119771). The meanin s o f s t a t e m e n t s about t h e s e o b j e c t s cannot be g i v e n i n terms o f t r u t h c o n d i t i o n s ?as f o r c l a s s i c a l P l a t o n i s t mathematics) o r i n terms of c o n s t r u c t i o n s ( a s f o r n a i v e c o n s t r u c t i v i s m ) . The essence o f t h e s e n o n - c o n s t r u c t i v e o b j e c t s l i e s i n t h e i r i n f i n i t e c h a r a c t e r . They a r e n o t , i n g e n e r a l , t o t a l l y grasped. They a r e g i v e n i n terms o f p a r t i a l d a t a which may l a t e r be r e f i n e d . Meaning f o r statements a b o u t n o n - c o n s t r u c t i v e o b j e c t s i s g i v e n b y s a y i n g what d a t a j u s t i f i e s a g i v e n a s s e r t i o n .

To d e s c r i b e a p a r t i c u l a r n o t i o n o f n o n - c o n s t r u c t i v e o b j e c t i s t o d e s c r i b e t h e t y p e o f d a t a on which i t i s based. We c o n s i d e r v a r i o u s such n o t i o n s . Each c o n c e p t i o n o f d a t a g i v e s an e x p l a n a t i o n o f meaning w h i c h extends t h e range o f meaningful statements and may b e viewed as i n t r o d u c i n g new o b j e c t s i n t h a t i t a s c r i b e s meani n g t o new forms o f q u a n t i f i c a t i o n . I n f a c t f o r each t y p e o f d a t a we i n t r o d u c e a c o n c r e t e r e p r e s e n t a t i o n o f t h e n o n - c o n s t r u c t i v e o b j e c t s based on i t . Such a p r o j e c t i s n o t n o v e l : B e t h 119471 i n t r o d u c e d h i s models t o p r o v i d e j u s t such an e x p l a n a t i o n o f meaning f o r c h o i c e sequences. Our models g e n e r a l i s e Beth's. Dumnett 119771 makes a l e n g t h y c r i t i q u e o f t h e view t h a t t h e i n t e n d e d meanings o f o f t h e l o g i c a l c o n s t a n t s a r e f a i t h f u l l y r e p r e s e n t e d on B e t h t r e e s . Since o u r models g e n e r a l i s e B e t h ' s t h e y appear prima f a c i e t o be s u s c e p t i b l e t o t h e same c r i t i c i s m s . However, Dummett's remarks on t h e (non)-consonance o f t h e i n t e n d e d meanings o f t h e c o n n e c t i v e s w i t h t h e i r i n t e r p r e t a t i o n i n B e t h t r e e s a r e d i r e c t e d a t a d i f f e r e n t problem f r o m t h e one we address. Dummett appears t o have o i e r l o o k e d t h e p o s s i b i l i t y o f s e p a r a t i n g t h e problem o f e x p l a i n i n g t h e c o n s t r u c t i v e meaning o f statements c o n c e r n i n g l a w l i k e o b j e c t s f r o m t h a t o f e x p l a i n i n g t h e i n t u i t i o n i s t i c meaning o f statements c o n c e r n i n g c h o i c e sequences. Although we know o f no s a t i s f a c t o r y e x p l a n a t i o n o f c o n s t r u c t i v e t r u t h ( i n p a r t i c u l a r , we agree w i t h Dummett t h a t B e t h models do n o t g i v e one), such a s e p a r a t i o n appears n a t u r a l . I t i s p o s s i b l e t o c o n c e i v e o f c o n s t r u c t i v e t r u t h i n d e p e n d e n t l y o f c h o i c e sequences. Given such a c o n c e p t i o n , Beth models p r o v i d e an account o f t h e i n t r o d u c t i o n o f n o n - l a w l i k e o b j e c t s . I t i s t h i s t y p e o f account we have g e n e r a l i s e d . By way o f example we now c o n s i d e r two n o t i o n s o f d a t a c l o s e l y r e l a t e d t o Beth models. They b o t h a r i s e f r o m t h e same i n f o r m a l p i c t u r e . The Imagine r e c e i v i n g f r o m Mars an i n f i n i t e sequence a o f n a t u r a l numbers. p i c t u r e i s o f a t i c k e r - t a p e which produces an i n d e f i n i t e l y c o n t i n u e d f i n i t e i n i t i a l segment a o f t h e sequence CL. (We w r i t e CL E a t o mean t h a t a i s an i n i t i a l segment o f a . ) We want t o examine t h e consequences o f t r e a t i n g such undetermined sequences s e r i o u s l y as sequences. ( L a t e r we s h a l l i n t r o d u c e more i n t e r e s t i n g examples )

.

A n a i v e view o f t h i s example c o n s i d e r s t h e stages b y which i n f o r m a t i o n a r i s e s : a t any stage, t h e p o s s i b l e f u t u r e d a t a i s r e p r e s e n t e d b y t h e c o l l e c t i o n N (where n E N) must appear. N o t o n l y do we n o t y e t know which o f t h e s e p o s s i b i l i t i e s w i l l occur, i t i s n o t y e t determined which w i l l o c c u r . On t h e b a s i s o f t h i s d a t a we may cons t r u c t many sequences. The s i m p l e s t o f these, a i s g i v e n b y t r a n s c r i b i n g t h e d a t a as i t a r r i v e s . Thus on t h e b a s i s o f d a t a b 5 a, we a r e j u s t i f i e d i n a s s e r t i n g t h a t a i s an i n i t i a l segment o f a. We w r i t e t h i s b i t CL E a. (We o r d e r sequences by s e t t i n g b 5 a i f a i s an i n i t i a l seqment o f b s i n c e t h e n b a l l o w s fewer p o s s i b i l i t i e s f o r CL.)Another sequence B i s g i v e n by f i r s t w r i t i n g down a f i x e d f i n i t e sequence b and t h e n c o n t i n u i n g w i t h t h e incoming data. S c h e m a t i c a l l y , a i l 6 t b*a where * denotes c o n c a t e n a t i o n , and hence " o b v i o u s l y " , f o r any c 5 a, we have c i k D E b*a. We want t o make a l l such "obvious" assumptions d b o u t t h e n a t u r e of j u s t i f i c a t i o n e x p 1 i c i t ; s o we g i v e i t s two b a s i c s t r u c t u r a l p r o p e r t i e s . J u s t i f i c a t i o n should be persistent

alC $ a*blt 0

Continuous Truth I and inductive

a* < n > l t $ f o r a l l n alk @

163

E

N

P e r s i s t e n c e r e f l e c t s t h e i d e a t h a t knowledge, once j u s t i f i e d , i s secure. The i n d u c t i v e c l a u s e comes f r o m r e f l e c t i o n on t h e i n f i n i t e c h a r a c t e r o f a. Given a E a, t h e c o l l e c t i o n { a * I n c N I covers a l l p o s s i b i l i t i e s f o r f u t u r e data. I n general, i f we s t i p u l a t e b l k $ f o r b E B 5 N
ample, any monotone f u n c t i o n g: N
Then a monotone map N
+

P w i l l r e p r e s e n t a non-constructive

We do n o t w i s h t o i n t r o d u c e t e c h n i c a l i t i e s h e r e . L a t e r we s h a l l g i v e a d e f i n i t i o n o f all- 4 , f o r f i r s t - o r d e r $ , b y i n d u c t i o n on t h e s t r u c t u r e o f 9. F o r t h e moment we j u s t remark t h a t such a d e f i n i t i o n o f a l l $ can b e g i v e n and t h a t t h e i n t e r p r e t a t i o n s o f t h e c o n n e c t i v e s a r e c o m p l e t e l y determined, i n t h e c o n t e x t o f o u r requirements on j u s t i f i c a t i o n , b y r e q u i r i n g t h a t t h e r u l e s o f p o s i t i v e l o g i c be v a l i d . I n o u r p r e s e n t case t h i s would amount t o g i v i n g B e t h ' s semantics f o r i n t u i t i o n i s t i c l o g i c w i t h a s l i g h t l y m o d i f i e d n o t i o n o f " b a r " . B e t h ' s semantics a r e w e l l known t o be e q u i v a l e n t t o t h e t o p o l o g i c a l i n t e r p r e t a t i o n o v e r B a i r e space e x p l o i t e d b y S c o t t [ I 9 6 8 1 and Moschovakis 119731;our m o d i f i c a t i o n r e p l a c e s B a i r e space b y f o r m a l B a i r e space ( Fourman and Grayson C19823). We now r e t u r n t o o u r p i c t u r e o f t h e t i c k e r - t a p e . What we have done i s t o g i v e a r e p r e s e n t a t i o n o f t h e subjective e x p e r i e n c e o f r e c e i v i n g word f r o m Mars, a n ext e r n a l view o f how t h e w o r l d w i l l l o o k when d a t a a appears on t h e t i c k e r - t a p e . T h i s view i s dependent on t h e c o n t i n g e n c i e s o f what d a t a i s a v a i l a b l e . B u t mathem a t i c s s h o u l d be t i m e l e s s and a b s o l u t e . T h i s r e q u i r e m e n t appears t o exclude cons i d e r a t i o n o f p o t e n t i a l l y i n f i n i t e o b j e c t s . We now a t t e m p t t o r e s o l v e t h i s contradiction.

-

P i c t u r e a room w i t h a t i c k e r - t a p e , A and v a r i o u s sequences i n progress, a,p,y d e s c r i b e d above f o r example. Now suppose t h a t on t h e t a p e we have t h e ' f i n i t e sequence b. We have d u l y n o t e d t h a t a E b, 8 E b*b, y E g ( b ) . Consider now ano t h e r room A ' w i t h a t i c k e r - t a p e which, as y e t , i s b l a n k and t h r e e non-constructive objects defined by c l t a' E b*c

CIF8 '

E

b*b*c

c ~ F Y g(b*c) ' . I n t h i s room on t h e b a s i s . o f no d a t a we can a l r e a d y n o t e t h a t a' E b, 8 ' E b*b, g ( b ) . Furthermore d a t a b*c a r r i v i n g i n room A w i l l always have t h e same conI n f a c t the sequences f o r a . 8 , ~as d a t a c a r r i v i n g i n room A ' has f o r a ' , B ' , y ' . mathematics (and l o g i c ) o f t h e two rooms, A w i t h d a t a b and A ' w i t h no data, s h o u l d b e t h e same. We want t o add t h i s t o o u r f o r m a l t r e a t m e n t . T h i s i s done by r e g a r d i n g incoming d a t a n o t as changing t h e w o r l d b u t r a t h e r as e f f e c t i n g a t r a n s f o r m a t i o n w h i c h changes o u r view o f t h e w o r l d . We c o n s i d e r n o t a p a r t i c u l a r t i c k e r - t a p e b u t r a t h e r t h e uses w h i c h c o u l d b e made o f such an i n d e t e r m i n a t e sequence t o generate n o n - c o n s t r u c t i v e o b j e c t s . Data j u s t becomes a way o f t r a n s y' 6

M.P. FOURMAN

164

forming one such process i n t o a n o t h e r , g e n e r a l l y l e s s f r e e : i t s r e s t r i c t i o n . We give a general d e f i n i t i o n of t h i s transformation a s follows: c l k $(sib) i f f b*clk $(6) (Where 6 i s a non-constructive o b j e c t given by s t i p u l a t i n g what d a t a j u s t i f i e s $ ( 6 ) f o r various $.) For example, a l b = a ' = a; Olb = a ' ; y l b = y ' .

This change of viewpoint amount s f o r m a l l y t o a change in our r e p r e s e n t a t i o n of d a t a . Formerly we considered the p a r t i a l l y ordered set o r t r e e N < N as representing various possible s t a t e s of information. Incoming data changes t h e world i n t h a t i t places us i n a new s t a t e . Now we consider N"' a s a c o l l e c t i o n of transformations which a c t t o change our view of t h e world. Formally i t i s convenient t o r e p r e s e n t t h e data a s t h e monoid of f i n i t e sequences under concatenation; i f g: N C N P r e p r e s e n t s a non-constructive o b j e c t y then y l b i s represented by gob where b : a b*a a c t s by l e f t concatenation. The notion of j u s t i f i c a t i o n i s t o be s t a b l e under such a change of perspective: a I b $ ' ] b i f f b*alk $ (where I b i s applied t o t h e non-constructive parameters of $ . ) -f

We now consider examples of a more general type of non-constructive o b j e c t intend-

ed t o r e p r e s e n t p o t e n t i a l i n f i n i t i e s of experience. We base our d e s c r i p t i o n , f o r t h e sake of e x p o s i t i o n , on a view o f c l a s s i c a l experimental physics which we asc r i b e t o t h e nineteenth century. B r i e f l y i t runs a s follows: Physics i s based on measurement. Experiments determine values o f parameters a t o a c e r t a i n degree of p r e c i s i o n . Generally some e r r o r i s i n e v i t a b l e b u t i t may i n p r i n c i p l e be made a r b i t r a r i l y small ( t h i s i s t h e assumption which leads us t o l a b e l t h i s a s a nineteenth century noti o n ) . Now, we refuse t o admit t h e c l a s s i c a l a s c r i p t i o n of a c t u a l values t o these parameters. A t f i r s t consideration t h i s may appear c h u r l i s h . There i s an apparent d i f f e r e n c e between a sequence determined only by t h e f r e e w i l l of a Martian and a physical value. W e leave a s i d e the question o f whether this i s an actual d i f f e r ence because t h i s question misses t h e p o i n t . The point i s t o a s k , "How can we assign meaning t o statements concerning such q u a n t i t i e s , i n p a r t i c u l a r how should we understand q u a n t i f i c a t i o n over such q u a n t i t i e s ? " Our r e f u s a l amounts t o denying the coherence of any explanation based on the assumption t h a t every sentence has a determinate truth value, e i t h e r true o r f a l s e . We r e f e r t o Dummett f o r e l a b o r a t i o n of t h i s point. The p o s s i b l e r e s u l t s of experiments a r e concrete by experiment t h a t a E U. These possible U form assumes t h a t a r b i t r a r y refinement of our methods presented by saying t h a t the V i < U representing cover U

.

however. I n general we may f i n d a poset IP. The c o n c e i t which i s i n p r i n c i p l e possible i s rea c e r t a i n degree .of refinement

For example, measurement of a q u a n t i t y c l a s s i c a l l y represented by a real parameter could be represented by taking f o r P t h e poset of r a t i o n a l open i n t e r v a l s , with t h e s t i p u l a t i o n t h a t f o r each E > 0 an open i n t e r v a l U i s covered by t h e c o l l e c t i o n of a l l s u b i n t e r v a l s of length 5 E , a l s o t h a t t h e c o l l e c t i o n of a l l proper s u b i n t e r v a l s of U covers U. In general then we consider a poset IP of p o s s i b l e outcomes f o r an experiment. We a s k , a s a technical convenience, t h a t i f p and q re resent a p r i o r i compatible r e s u l t s ( i . e . i f t h e r e i s an r with r 5 p and r 5 qp then we can consider the outcome which c o n s i s t s j u s t of g e t t i n g t h e s e two r e s u l t s ( i . e . we have an infinium p A q E P). We a l s o consider no information as a possible r e s u l t ( i . e . Ip has a

Continuous Truth I t o p element T ) . The p o s e t demand t h a t t h i s b e

IP

i s equipped w i t h a n o t i o n o f c o v e r i n g f a m i l y .

r e f Zective

stabZe

165

We

i p } covers p I f U covers p and q

monotone

5

p t h e n Iq

w Iw

A

E

U l covers q

I f V 2 U covers p t h e n V covers p.

The n o t i o n o f a c o v e r i n g f a m i l y i s c r u c i a l t o o u r e x p l a n a t i o n o f meaning f o r i n complete o b j e c t s . I t f o r m a l i s e s t h e sense i n which t h e y a r e p o t e n t i a l l y i n f i n i t e . We a v o i d t h e metaphor o f W r i g h t C19811 which r e p r e s e n t s such a c o v e r i n g f a m i l y as embodying t h e r e c o g n i t i o n t h a t t h e s t a t e o f i n f o r m a t i o n i s capable o f e f f e c t i v e enlargement t o one o f t y p e a* because i t seems t o l e a v e open t o us t h e c h o i c e o f n o t p e r f o r m i n g t h i s enlargement. The i d e a we have i s t o i n t r o d u c e c o n s i d e r a t i o n o f a p a r t i c u l a r t y p e o f i n c o m p l e t e o b j e c t b y s p e c i f y i n g t h e t y p e o f d a t a which generates i t . T h i s s p e c i f i c a t i o n i n c l u d e s a n o t i o n o f c o v e r i n g f a m i l y . D i f f e r ences o v e r w h i c h i s t h e p r o p e r c o l l e c t i o n o f c o v e r i n g f a m i l i e s do n o t a f f e c t t h e b a s i c c o n c e p t i o n b u t m e r e l y l e a d t o d i f f e r e n t types o f data. We a r e n o t as mathematicians o r l o g i c i a n s i n t e r e s t e d i n t h e r e s u l t o f a p a r t i c u l a r experiment. Rather, we a r e i n t e r e s t e d i n t h o s e p r o p e r t i e s which would remain i n v a r i a n t no m a t t e r what t h e outcome o r methodology o f a p a r t i c u l a r experiment. It i s n o t t h e r e s u l t b u t t h e uses t o which t h e r e s u l t m i g h t be p u t i n d e f i n i n g mathem a t i c a l q u a n t i t i e s which i n t e r e s t us. Were t h e temperature s c a l e n o n - l i n e a r , o r t h e t i m e s c a l e g i v e n by t h e unequal t i m e o f t h e sun, p h y s i c s would be d i f f e r e n t ( i t was). B u t mathematics and l o g i c s h o u l d be immune t o such v a g a r i e s . Our s o l u t i o n i s s i m i l a r t o t h a t we employed i n g i v i n g an o b j e c t i v e view of. open data. The p o s s i b i l i t y we envisage i s t h a t o f a change o f s c a l e which i n some sense r e f i n e s o u r p o s s i b i l i t i e s f o r measurement. The measurements o f t h e o l d cont e x t s h o u l d be meaningful i n t h e new one b u t t h e new one may a f f o r d f i n e r d i s t i n c t i o n s . To d e s c r i b e such a change o f s c a l e i s t o say which new o b s e r v a t i o n s q E Q a r e t o b e viewed as r e f i n i n g an o l d o b s e r v a t i o n p E IP. We w r i t e t h i s r e l a t i o n q 5 f * ( p ) and demand t h a t i t be

monotone

p q

muZtip Zicative

pi I i

continuous

{q I q

f*(p) q 5 f*(p') q 5 f*(p A p ' )

5

E

p'

5

I

r

covers p

f*(pi)

some i

E

5

f*(p)

11 covers r

.

The m o t i v a t i o n f o r t h e f i r s t two i s c l e a r . C o n t i n u i t y may b e viewed as t h e r e quirement t h a t a p r e v i o u s c o n v i c t i o n t h a t a c e r t a i n f a m i l y covers, cannot be overt u r n e d . The change o f v i e w p o i n t induced b y such a t r a n s f o r m a t i o n f i s g i v e n by

M a t h e m a t i c a l l y , o u r n o t i o n o f d a t a g i v e s a p r e s e n t a t i o n o f a ZocaZe. Change o f s c a l e i s r e p r e s e n t e d by a continuous f u n c t i o n between l o c a l e s . A b s t r a c t l y we w r i t e such a change f: Y X. -f

We now c o n s i d e r a supplement t o o u r n o t i o n o f j u s t i f i c a t i o n . Suppose, we consider, t h a t c o n s i d e r a t i o n o f a p a r t i c u l a r t y p e o f d a t a would j u s t i f y 0, t h e n C$ i s j u s t i f i e d . T h i s i s t h e r e f l e c t i o n on w h i c h o u r whole p r o j e c t i s based: t h a t we can j u s t i f y t a l k o f i n c o m p l e t e o b j e c t s by r e f l e c t i n g on hypothetica2 i n d e f i n i t e l y c o n t i n u e d processes.

M.P. FOURMAN

166

We s h a l l f o r m u l a t e t h i s b y s a y i n g t h a t i f f : Y + X r e p r e s e n t s t h e i n t r o d u c t i o n o f new d i s t i n c t i o n s independent o f those r e p r e s e n t e d b y X t h e n

o r t h a t such an f i s a cover. Our f i n a l problem o f f o r m a l i s a t i o n i s t o c h a r a c t e r i s e t h e i n t r o d u c t i o n o f independent data. A s i m p l e example i s , g i v e n P a n d Q w i t h n o t i o n s o f c o v e r i n g , t o consider P x Q t h e product poset w i t h coverings

I I

1 1

E I } covers p i E I} covers q . The p r o j e c t i o n g i v e n by 5 n * ( p ' ) i f f p s p ' r e p r e s e n t s t h e i n t r o d u c t i o n o f d a t a o f t y p e Q i n d e p e n d e n t l y o f t h e d a t a IP under c o n s i d e r a t i o n . We s h a l l r e q u i r e t h a t a l l such p r o j e c t i o n s be covers.

I I

i

E

i

E

I 1 covers I } covers

p.q

when {pi

p,q

when

qi

i

I n general t h e r e a r e two c o n d i t i o n s we r e q u i r e t o view a change o f s c a l e as t h e i n t r o d u c t i o n o f independent data. The f i r s t i s obvious: no new covers s h o u l d be i n t r o d u c e d between e x i s t i n g o b s e r v a t i o n s . qi s f * ( r i )

{qi 1 i

{ri I i

E

E

I } covers each q I } covers r

5

f*(r)

The second i s s u b t l e : no new c o n d i t i o n a l r e l a t i o n s h i p s s h o u l d b e i n t r o d u c e d between e x i s t i n g o b s e r v a t i o n s . We e x p l a i n : i f w E Q i s such t h a t

r 5 w f*(p) r 5 f*(q) (we view w as e s t a b l i s h i n g a c o n d i t i o n a l r e l a t i o n s h i p between f * ( p ) and f * ( q ) ) , we demand t h a t w 5 f * ( s ) f o r some s E I P such t h a t r

5

r s p

r s s r s q

Technically, these require( t h a t t h e r e l a t i o n s h i p be a l r e a d y e s t a b l i s h e d i n IP). ments amount t o demanding t h a t t h e continuous map f : Y + X be a s u r j e c t i o n and t h a t i t be open. The s t r u c t u r e o f d a t a we have a r r i v e d a t may b e viewed as t h e c a t e g o r y o f l o c a l e s equipped w i t h t h e t o p o l o g y o f c o v e r i n g b y open maps. Before t u r n i n g t o a formal examination o f t h e i n t e r p r e t a t i o n o f l o g i c over t h i s s i t e , we sum up o u r i n t e n t i o n s . We i n t r o d u c e n o n - c o n s t r u c t i v e o b j e c t s b y e x p l a i n i n g t h e meanings o f t h e connecti v e s f o r statements c o n c e r n i n g them. T h i s i s not a m a t t e r o f c h a r a c t e r i s i n g a domain o f q u a n t i f i c a t i o n . We have t o e x p l a i n t h e c o n n e c t i v e s anew i n terms o f t h e way such an o b j e c t i s g i v e n t o us. Moreover, i t i s n o t s u f f i c i e n t t o m e r e l y paraphrase t h e new q u a n t i f i e r s Wa and 3u. Such a paraphrase e n t a i l s a r e v i s i o n o f t h e i n t e r p r e t a t i o n s o f -+ and v . Our aim i s t o show t h a t i t i s p o s s i b l e t o d e r i v e r i g o r o u s l y p r o p e r t i e s o f v a r i o u s domains o f i n c o m p l e t e o b j e c t s by g i v i n g a f o r m a l r e p r e s e n t a t i o n o f t h e d a t a which p r e s e n t s them as a s i t e . We c o n s i d e r t h a t t h e passage f r o m an i n f o r m a l n o t i o n o f data t o t h e c o r r e s p o n d i n g s i t e i s s i m p l e and n a t u r a l . (Indeed, f o r us, t o have a c l e a r c o n c e p t i o n o f a t y p e o f d a t a i s t o be a b l e t o d e s c r i b e t h e c o r r e s p o n d i n g s i t e . ) Once t h i s passage i s made, t h e d e r i v a t i o n o f p r o p e r t i e s ( c h o i c e and cont i n u i t y p r i n c i p l e s , f o r example) i s a mathematical m a t t e r . Our hope i n p r e s e n t i n g these modeis i s L e i b n i t z i a n : t o e l i m i n a t e f u r t h e r d i s c u s s i o n o f t h e j u s t i f i c a t i o n o f such p r i n c i p l e s b y r e d u c i n g t h e m a t t e r t o c a l c u l a t i o n . I n o u r paper " N o t i o n s o f Choice Sequence" C19821 we presented v a r i o u s n o t i o n s o f

Continuous Truth I c h o i c e sequence, i n c l u d i n g ones purpose. U n f o r t u n a t e l y , as t h e i s i n t h e eye o f t h e b e h o l d e r . a t i o n o f our informal notion o f

167

s a t i s f y i n g t h e axioms o f LS and CS, w i t h t h e same l i t e r a t u r e on c h o i c e sequences makes c l e a r , c l a r i t y Hence t h e p r e s e n t a t t e m p t a t a more c a r e f u l explann o n - c o n s t r u c t i v e o b j e c t and i t s f o r m a l i s a t i o n .

CONTINUOUS TRUTH We s t a r t w i t h a c o n c r e t e p r e s e n t a t i o n o f t h e i n t e r p r e t a t i o n o f h i g h e r - o r d e r l o g i c i n a Grothendieck topos. T h i s m a t e r i a l (561-3) i s well-known t o cognoscenti ( tautologously), b u t i s otherwise accessible only through a study o f scattered r e f e r e n c e s . We g i v e some o f these sources b u t make no s y s t e m a t i c a t t e m p t a t a complete l i s t . Many i m p o r t a n t and h i s t o r i c a l l y s i g n i f i c a n t c o n t r i b u t i o n s a r e n o t mentioned. Our account i s f u l l e r t h a n i s l o g i c a l l y necessary f o r t h e sequel i n o r d e r t o p o i n t o u t some connections between d i f f e r e n t approaches. I t i s not, however, e x h a u s t i v e . 51

Frames and Locales

A frame i s a complete l a t t i c e w i t h f i n i t e A d i s t r i b u t i v e o v e r 1.1 D e f i n i t i o n . a r b i t r a r y V. Frame morphisms, "and-or maps", a r e maps p r e s e r v i n g these o p e r a t i o n s ; T,A,V. 1.2 Example. The l a t t i c e O(X) o f open subsets o f a t o p o l o g i c a l space i s a frame. I f f: Y + X i s a continuous map t h e n t h e inverse image f*: O ( X ) -r O ( Y ) i s an A,V-map. 1.3 D e f i n i t i o n . The c a t e g o r y o f Zocalos o r g e n e r a l i s e d spaces i s t h e dual o f t h e c a t e g o r y o f frames. We c a l l t h e morphisms continuous maps f: Y + X and w r i t e f*: U(X) + O ( Y ) f o r t h e c o r r e s p o n d i n g i n v e r s e image maps between t h e frames of opens o f X and Y ( a s i n t h e t o p o l o g i c a l case). Example 1.2 g i v e s a f u n c t o r 6: Top maps t o l o c a l e s .

+

LOC f r o m t o p o l o g i c a l spaces and continuous

1.4 D i s c r e t e spaces. S p a t i a l l y P(A) corresponds t o t h e d i s c r e t e t o p o l o g y on A. p o i n t space w i t h O( ll) = P( ll) . 1.5

Definition.

1.6

Lemma.

An example i s t h e one-

A t o p o l o g i c a l space X i s sober i f f Top[Il,Xl

1 LocCll,pX1.

On t h e f u l l subcategory o f sober spaces 6 i s f u l l and f a i t h f u l .

We t a c i t l y r e s t r i c t o u r a t t e n t i o n t o sober spaces and h e n c e f o r t h o m i t mention o f 8. We view l o c a l e s as generalised spaces. (The r e l a t i o n s h i p between LOC and Top i s b e t t e r expressed i n terms o f t h e r i g h t a d j o i n t , p t : LOC + Top, t o 6.) Q u o t i e n t maps o f frames i n d u c e congruences: i f f*: O(X) + U(Y) i s has a c a n o n i c a l p a q i f f f * p = f * q . Each congruence c l a s s !PI r e p r e s e n t a t i v e j p = V t q I p a q } . The maps j : O(X) + O(X) a r i s i n g i n t h i s way a r e monotone P 2 jp j2 = j idempotent ,

muttip l i c a t i v e

j(p

A

9) = j p

A

jq

.

Such maps a r e c a l l e d n u c t e i . The q u o t i e n t may b e i d e n t i f i e d as t h e image ( o r f i x e d p o i n t s ) o f j . The q u o t i e n t s o f O(X) a r e i s o m o r p h i c ( a s posets) w i t h t h e n u c l e i on U(X). S p a t i a l l y we view these q u o t i e n t s as g i v i n g r i s e t o subspaces o f x.

M.P. FOURMAN

168

1.8 Surjections. Dually, we view i n j e c t i v e inverse image maps as giving r i s e to surjections o f spaces. Each frame map f* has a r i g h t adjoint f,,

1.9 Right a d j o i n t s . given by The map q

p

A

f*p = V{q q has a r i g h t a d j o i n t r p ~ q s r iff

1

f*q

+p

where p + r = v!q I p A q 2 r } . morphisms are d i f f e r e n t ) .

d i r e c t image,

.

pl r defined by q s p + r 5

-f

Thus frames are complete Heyting algebras ( b u t the

Y i s o en i f the inverse image map 1.10 Definition. A map of spaces f : X f*: O(y) + O ( X ) has a l e f t a d j o i n t 3,: U ( X ) + O ( V 7 commuting with A : 3f(f*(Y) A x ) = Y A j f ( X ) or, equivalently, i f f* preserves +. -f

1.11 Proposition. The category o f locales i s complete and cocomplete. surjections a r e s t a b l e (under pull-back).

Open

The theory of locales i s developed extensively by Joyal and Tierney C19821. Johnstone 119821 uses locales systematically and has a comprehensive bibliography. 52

S i t e s a n d Sheaves

2.1 Definitions. Let 0 be a small categor . A cribte K of A E ! E l i s a suboil : t h a t i s , f o r each B E 181 a s e t functor of the representable functor A E S' K ( B ) 5 IB,AI, s t a b l e under composition; f o r each f E K ( B ) and g : C B in C , the composite f g E K ( C ) . -f

0

2 . 2 Lemma. The c r i b l e s of A form a frame, P ( A ) . If f : B + A i n B we have an inverse image map f*: P(A) + P(B) given by f o r the correspondf*K = Ig f g E K} f o r K E P ( A ) . By abuse we write f : ing continuous map. This map i s open.

B A

0

-f

2.3 Definition. (Lawvere-Tierney) A Grothendieck topology j on is a family of nuclei j A : P(A) P(A), natural i n A: t h a t i s f * o j A = jBof*for f: B

+

-f

A.

Lemma. I f j i s a Grothendieck topology on 0 , the quotient frames n(A) have induced inverse image maps f*: n(A) + n ( B ) and the corresponding map of locales, which we write f : B j +A', i s open. 2.4

2.5 Definitions. which i s

A pretopotogy J on 0 i s a family J(A)

A

reflerrive

K

multiplicative

K

stable

(For example, l e t K A crible K

E

E

E

E

J(A) i f f j K

J(A)

E

J(A) K n L J(A) f*K =

E

L E J(A) J(A) f: B + A

E

J(B)

T).

P(A) i s inductively c t o s e d f o r J i f f

c

P ( A ) f o r each A

E

B

169

Continuous Truth I f: B + A

f*K

E

J(B)

.

f c K

As A i s c l o s e d and an i n t e r s e c t i o n of c l o s e d c r i b l e s i s closed, each c r i b l e K E P(A) has a cZosure j A K . T h i s g i v e s a t o p o l o g y j on C. We say K inductiveZy covers A iffj K = A, and w r i t e t h i s K E J(A). U(X) be a frame viewed as a p o s e t viewed as ( I d e n t i f y i n g B + A w i t h i t s domain.) Then be a s m a l l c a t e g o r y o f l o c a l e s c l o s e d under f i n i t e l i m i t s (2) Let i n c l u s i o n s . L e t K E J(A) i f f K c o n t a i n s some f a m i l y Ifi: Bi + A l i E

2.6

Exam l e s . (1)

___R_ Let K J A i f f VK E

maps such t h a t

V

Let

= A.

= A.

3fi(Bi)

a category. n(X) 1 O ( X ) . and open 11 o f open

The c r i b l e generated b y each open i n c l u s i o n U T h i s assignment g i v e s an A;V map r*: U(A)

i s closed f o r t h i s topology.

+

4

A

n(A),

s p a t i a l l y a s u r j e c t i o n r : AJ + A . Each c l o s e d c r i b l e c o n t a i n s a l a r g e s t open i n c l u s i o n . T h i s assignment g i v e s an A V map i*:n(A) + U(A), s p a t i a l l y we have an i n c l u s i o n i: A c+JJ. Furthermore, adjoint retract o f

AJ

r

o

i = i d A and r * i * s id,(A)

so A i s an

(Fourman C19821).

2.7 D e f i n i t i o n . A presheaf on C i s a f u n c t o r X: Cop + Sets. I f f: B + A E C and a E X(A) we use t h e n o t a t i o n a l f , "a r e s t r i c t e d a l o n g f " , f o r X ( f ) ( a ) E X(B). Note t h a t a l f l g = a l f o g and a l i d = a. The a p p r o p r i a t e morpkisms between presheaves F: Y + X a r e n a t u r a l t r a n s f o r m a t i o n s , maps FA: Y(A) + X(A) w h i c h commute w i t h r e s t r i c t i o n s FB(a,lf) = ( F A a ) l f . 2.8

Exam l e s .

( 1 ) The r e p r e s e n t a b l e f u n c t o r [-,A], ( o r by abuse, j u s t i f i e d by i s a presheaf. R e s t r i c t i o n s are by composition g l f = gof. Yoneda's lemma t e l l s us t h a t f o r any o t h e r presheaf, X, we have X(A) [A,X]. In

Yone 7 a+ s emma, +A)

p a r t i c u l a r t h e embedding C + Stop i s f u l l and f a i t h f u l . Each c r i b l e K E P(A) i s a subpresheaf K H A . ( 2 ) P and n a r e presheaves w i t h r e s t r i c t i o n s g i v e n b y i n v e r s e images K l f = f * ( K ) . 2.9 D e f i n i t i o n . A p r e s h e a f X on C i s a sheaf f o r t h e ( p r e ) t o p o l o g y J i f whene v e r K E J ( A ) , each n a t u r a l t r a n s f o r m a t i o n x: K + X has a unique e x t e n s i o n a l o n g K >+ A . E q u i v a l e n t 1y, i f K E J(A) and we have a f a m i l y x f E X(B) f o r f: B -f A E K such t h a t xgf = x f l g f o r each g: C

xf = x l f f o r each f Grothendieck topos.

E

K.

+

8, t h e r e i s a unique x

E

X(A) such t h a t

The c a t e a o r v of sheaves and n a t u r a l t r a n s f o r m a t i o n s i s a

There i s , as y e t , no s t a i s f a c t o r y i n t r o d u c t o r y t e x t on topos t h e o r y . r e f e r e n c e s a r e SGA4, W r a i t h C19751, Johnstone C19771, F r e y d '19721.

53

The b a s i c

Forcing over a s i t e

Here we d e s c r i b e J o y a l ' s p r e s e n t a t i o n o f i n t e r p r e t a t i o n s i n t o p o i i n terms o f a n o t i o n o f f o r c i n g . L e t C and a ( p r e ) t o p o l o g y J b e f i x e d . The b a s i c s t r u c t u r e s we c o n s i d e r a r e diagrams o f preskeaves on C. Each p r e s h e a f A i n t e r p r e t s a type o r s o r t o f v a r i a b l e . A morphism f : A1 x ... x An + B i n t e r p r e t s an n-ary operation.

A subobject R

-

A1

x

...

x

An i n t e r p r e t s an n - a r y r e l a t i o n .

3.1 D e f i n i t i o n s . L e t L be a f i r s t - o r d e r language ( p o s s i b l y many-sorted) w i t h e q u a l i t y . An i n t e r p r e t a t i o n o f L i s g i v e n b y a s s i g n i n g t o each s o r t A o f L a p r e s h e a f A, t o each o p e r a t i o n F f r o m A1,. . .,An t o B a n a t u r a l t r a n s f o r m a t i o n F: A1 R

*

x

A1

... x An B, and t o each r e l a t i o n R on A1 x ... x An .. . x An. Given such an i n t e r p r e t a t i o n , f o r U E J C C J +

x

a subfunctor we l e t

LU b e t h e

M.P. FOURMAN

170

expansion o f L o b t a i n e d by a d d i n g c o n s t a n t s o f t h e a p p r o p r i a t e s o r t s f o r t h e e l e ments o f A(U). I f f : V + U t h e n f o r any term T o r f o r m u l a Q o f LU we o b t a i n a term

[lTnu

o r f o r m u l a @ I of f LV by r e s t r i c t i n g any new c o n s t a n t s which occur.

T l f

for U

E E

IC1 we d e f i n e f o r each c l o s e d term A(U) b y i n d u c t i o n : ucnu

Note t h a t

U T l f l "

=

c

=

. . ,Tn) 1,

[IF( TI,.

for =

c

E

A(U)

..,iTnj) .

F(lT,n,.

Now we d e f i n e i n d u c t i v e l y t h e r e l a t i o n , U f o r c e s Q,

[TDulf.

UIk Q f o r 4 a sentence o f Lu. INDUCTIVE DEFINITION OF FORCING

Vflt

a l l f: V f

@If

+

K K

E

J(U)

UIt @

f o r a l l f: V

+

U, i f VIE $ I f t h e n V l t $If V l t @ $ +

f o r a l l f: V

+

U, f o r a l l c E A(U), VIE ~ l f C c / x l U l t wx.+

We now g i v e some " d e r i v e d r u l e s " f o r f o r c i n g : 3.2

(PI

Now

o f s o r t A o f LU an interpretation

T

Lemma.

Basic properties o f f o r c i n g UIt 0

f : V + U

V l t +If

171

Continuous Truth I

K

f o r each f i n some

f o r each f i n some

K

E

E

U I t $ y Ji J(U) e i t h e r V f l t $ 4 f o r V f l t $ l f

Ult- 3x.g j ( U ) we have V f l t $ C c / x l f o r some c

+

*

V l t 8 U i f V l t g l f t h e n VIE +If +

f o r a l l f: V

A(Vf)

E

VIE W X . $ f o r a l l f: V

(Atomic)-

U and a l l c

-f

f o r each f i n some K

E

E

j(U)'we

A(U), we have Vlk $ l f [ c / x l

hav;


...,

[Tnllf>

E

R(Vf)

Our p r e s e n t a t i o n h e r e i s non-standard i n t h a t t h e d e f i n i t i o n o f f o r c i n g i s u s u a l l y g i v e n b y s t i p u l a t i n g b o t h p o s i t i v e and n e g a t i v e r u l e s f o r each c o n n e c t i v e , ( I ) and (P) a r e t h e n d e r i v e d . The r e s u l t i n g r e l a t i o n i s t h e same. 3.3 D e f i n i t i o n . pretation iff

r

A sequent

-

r 1 $)

i s uaZid ( w r i t t e n

UI~$CS(X)/XI a l l q u I t $CS(X)/Xl

E

i n the given i n t e r -

r

where E i s an i n t e r p r e a t i o n o f t h e v a r i a b l e s o f L by elements o f t h e a p p r o p r i a t e A(U). I f each s o r t i s i n t e r p r e t e d by an inhabited p r e s h e a f (each 3.4 P r o p o s i t i o n . A(U) i s i n h a b i t e d ) t h e n t h e axioms and r u l e s o f H e y t i n g ' s p r o p o s i t i o n a l c a l c u l u s a r e v a l i d f o r k.

( A d a p t a t i o n s f o r domains w h i c h a r e n o t i n h a b i t e d a r e discussed i n Fourman r19771, S c o t t 119781, J o y a l & B o i l e a u C19811, Makkai & Reyes 119771.) 3.5

Definitions.

A p r e s h e a f A i s separated i f f

'Ita

=

a = b

A subobject R

t+

f o r a,b

E

A(U)

for a

A(U).

A i s cZosed i f f U I t R(a)

FaquJ

E

A h i g h e r - o r d e r t y p e - t h e o r y i s m e r e l y a many-sorted f i r s t - o r d e r t h e o r y w i t h some s t r u c t u r e on t h e c o l l e c t i o n o f s o r t s and c e r t a i n d i s t i n g u i s h e d o p e r a t i o n s and r e l a t i o n s . One o f t h e i n s i g h t s due t o Lawvere and T i e r n e y i s t h a t t o p o i have such h i g h e r - o r d e r s t r u c t u r e . We c o n s i d e r languages where f o r any two s o r t s A and B we can f o r m t h e product A x B w i t h a p p r o p r i a t e pairing and p ro j e c t i o n operations, t h e f unction space BA w i t h an evaluation operation -( -), and a l s o t h e power t y p e

M.P. FOURMAN

172

P(A) w i t h a membership reZation E . An i n t e r p r e t a t i o n i s standard i f a l l t h i s s t r u c t u r e i s i n t e r p r e t e d by t h e c o r r e s p o n d i n g s t r u c t u r e on Sh((C). 3.6 P r o p o s i t i o n . I n any s t a n d a r d i n t e r p r e t a t i o n t h e f o l l o w i n g schemata, which combine comprehension and extensionality, a r e v a l i d . x

-

A 3!y E B.@(x,y) 3 ! z E P(A) W X

E

E

3!f

A

B A WX

E

(X

z

E

++

E

A.$(x,f(x))

@(x)).

0

Thus power-types and f u n c t i o n spaces behave as t h e y should. The c a t e g o r i c a l c h a r a c t e r i s a t i o n of t h i s h i g h e r - o r d e r s t r u c t u r e i n terms o f a d j o i n t s i s v e r y simple, p r o d u c t s a r e c a t e g o r i c a l p r o d u c t s ,

We s h a l l n o t d e s c r i b e t h i s s t r u c t u r e i n general h e r e . We s h a l l b e d e a l i n g p r i m a r i l y w i t h s o r t s i n t e r p r e t e d by r e p r e s e n t a b l e s . These a r e p a r t i c u l a r l y s i m p l e t o deal w i t h because t h e y have generic elements. A well-known consequence o f t h i s i s t h e Yoneda Lemma: OP F(U) [U,FI f o r F E I S c I and U E / c C 1 . We use t h i s t o c a l c u l a t e some examples o f t h e h i g h e r - o r d e r s t r u c t u r e . F o r t h i s e x e r c i s e , we suppose t h a t ci: has f i n i t e p r o d u c t s and t h a t each r e p r e s e n t a b l e f u n c t o r i s a sheaf. 3.7

Lemma.

(1) F'(U)

I f F i s a s h e a f and U,V a r e r e p r e s e n t a b l e F(U

C)

x

U and E

E

F(U

V) w i t h U l k a

E

R i f f Rl = V .

w i t h e v a l u a t i o n f o r u: V (2)

(PU)(V) = n(U

Proof.

U

F (V)

x

[V,F

(PU)(V)

U

1

+

V) g i v e n by ~ ( u )= sI.

x

[UxV,FI 2 F(UxV)

CV,PUI

0

S u b ( U x V ) = 6?(UxV).

A l o g i c a l c o u n t e r p a r t t o Yoneda's lemma i s t h e f o l l o w i n g . 3.8

Lemma.

Generic elements for representables.

I f U i s representable then

VIkWx

E

U.$

iff

V

Ulk $IT~[T~/XI.

x

Proof. + -

V

x

I n one d i r e c t i o n t h i s i s immediate f r o m I n t h e o t h e r , suppose V w i t h a: W + U E U(W) t h e n cf,a>: W + V x U and, b y p e r s i s t e n c e , i f U(k$fn1C.rr2/xl t h e n Wk I $ l f [ a / x l . So b y ( W ) ' we have V[kWx.$ . 0

We g i v e an example of t h e use o f g e n e r i c elements i n t h e s i m p l e case of a c a t e g o r y o f presheaves. 3.9 Proof.

Proposition.

Choice h o l d s f o r r e p r e s e n t a b l e s i n c a t e g o r i e s o f presheaves.

L e t U be a r e p r e s e n t a b l e and suppose

Vlk

WX

E

u . 3 ~E F.@(x,y)

then

U X VIk 3~

F.$IT~(T~,Y)

U x V l k $ 1 ~ ~ ( . r r ~ , Cf o) r some 6

E

F(UxV)

173

Continuous Truth I U r e g a r d i n g 5 as an element o f F (V) t h i s g i v e s u x

Since

51 = 5 .

Vlt

~l+1,(51n2)(~l))

vit

wx

.

Thus U.4(X,E(X)),

t

and s o

vlt-

0

3f.Vx.+(x,f(x)).

From a c a t e g o r y - t h e o r e t i c v i e w p o i n t t h i s r e s u l t i s well-known i n t h e form, "Representables a r e i n t e r n a l l y p r o j e c t i v e " .

54

POINTS, LOCAL CHOICE, CONTINUITY

Now we l e t CC be a c a t e g o r y o f l o c a l e s c l o s e d under f i n i t e l i m i t s and open i n c l u s i o n s , equipped w i t h t h e open c o v e r t o p o l o g y , J. We w r i t e E f o r t h e topos Sh(C,J). F o r each l o c a l e X we d e f i n e an i n t e r n a l l o c a l e X b y

O(X)(u);

O(XX U J ) .

T h i s i s generated i n t e r n a l l y b y t h e b a s i s g i v e n b y B()#)(U)E

U)

O(Xx

or even b y t h e c o n s t a n t b a s i s Bo(X)(U) = O(X),

with t h e i n c l u s i o n s go(#) 4 X

x

UJ

-f

X

x

U

-f

X.

B(X)

4 I)()#)

induced b y t h e p r o j e c t i o n s

( I n t h e t e r m i n o l o g y o f J o y a l & T i e r n e y X = P*(X).)

The i n t e r n a l space o f p o i n t s o f X i s g i v e n b y ( p t X ) ( U ) ,z cruj,x1. T h i s i s t h e space o f E-valued models o f X. iff

F o r a: UJ

+

X and W

E

O(XxUJ)

= T.

-'(w)

s i n c e i*:o ( u ~ )+ U ( U ) r e f l e c t s 4.1

Proposition.

Proof.

F o r any X

We must show f o r W

6

B I t h e i n t e r n a l l o c a l e X has enough p o i n t s

E

O(X) t h a t

U l t K covers p t W

U l t K i s an i n d u c t i v e c r i b l e E K

UIFW

.

We assume t h e hypotheses, and l e t

M

=

{Wi

x

Ui

I

UiItWi

C l e a r l y IK i s a d o u b l y i n d u c t i v e c r i b l e o f O(W) We show t h a t I K covers W x U t h a t i s t h a t W x U s i n c e t h e n U l t W E K.

By p e r s i s t e n c e W

x

UlF KAn2

covers n l

E

x E

KIUil U(U), t h a t i s an open o f W x U. K, which i s e v i d e n t l y s u f f i c i e n t

M.P. FOURMAN

174

that i s W

U(k3V

x

Kln2.rl

E

V.

U(W) we have Wi

x

Ui

E

so IK"

= {Wi

x

U i l f o r some Vi

c

It Vi

E

K1r2

A

rl c Vi}

covers W x U. [Because, i f p: X + Y i s an open s u r j e c t i o n and Xlk V E K J P A e l p E V then YlkV E K A e E V: t h e b a s i c opens o f )# a r e constant and thus descend open s u r j e c t i o n s . 1 B u t now we c l a i m IK* IK because, by d e f i n i t i o n wi x ui I t n1 E vi i f f Mi 5 Vi and, as p r o j e c t i o n s a r e covers, Wi x Uilb Vi

E

KIn2 i f f

UiIkVi

E

KIUi.

0

Special cases o f t h i s a r e worthy o f mention. When X i s B a i r e space NN, Cantor N space 2 , Dedekind r e a l s R , t o say t h a t X has enough p o i n t s i s t h e i n t e r n a l statement of Bar i n d u c t i o n , Fan theorem, Heine Bore1 theorem ( r e s p e c t i v e l y ) . For these cases i t i s s u f f i c e i n t t o t a k e t h e topology on Q generated by covering f a m i l i e s of open inclusions: s i n c e each o f these spaces X has a p o i n t t h e proe c t i o n s X x U + X are covers f o r t h i s topology. We c a l l t h i s topology the open inclusion topology. We introduce some more general spaces. L e t f: X + U i n LOC. We consider t h e i n t e r n a l l o c a l e X / f defined a t U by t h e b a s i s U(V) w i t h a l l i t s standard covers. More p r o p e r l y f o r 9: W + U we d e f i n e

=

W/f)lg 0(9*X) given by p u l l i n g back f along g. Any commuting t r i a n g l e

x

F Y

induces an i n t e r n a l map o f l o c a l e s 5 : X / f + Y/h defined a t U. Given by 5-l on b a s i s elements, t h i s c l e a r l y takes b a s i c covers t o covers. Furthermore, i f 5: X + Y i s open (and s u r j e c t i v e ) then 5: X / f + Y/h i s open, s i n c e i t s u f f i c e s t o d e f i n e comnuting w i t h A on b a s i s elements, (and

4

s u r j e c t i v e since i f 5: X + Y i s an open s u r j e c t i o n then so a r e a l l i t s pullbacks, so i n t e r n a l l y 5-l r e f l e c t s b a s i c open covers). These spaces i n c l u d e t h e spaces )K we introduced e a r l i e r as U1l-X

(X

x U)/v

.

We now s p e c i a l i s e t o t h e case where t h e o b j e c t s o f b a r e T.I an isomorphism (ptX)(U) CCUj,Xl CCU,Xl

Then U

4

U j induces

so t h a t X represents t h e f u n c t o r p t X . This happens i n p a r t i c u l a r f o r t h e spaces N, NN, pN, R and t h e i r b a s i c opens (see Fourman 119831.) Furthermore, any element o f p t ( X / f ) d e f i n e d a t U induces a comnuting t r i a n g l e

\/

UJ

175

Continuous Truth I

X

which f o r TI

spaces

X corresponds t o a section o f g

U

U

So we have a p r e s e n t a t i o n correspond t o commuting t r i a n g l e s

pt(X/f)'g

/ / d l /

W

.U

9

w i t h r e s t r i c t i o n g i v e n b y composition. We e x t e n d o u r e a r l i e r lemma on g e n e r i c elements: 4.2

Lemma.

I f objects o f

(I:

a r e TI

then

ulk Vx

E

Pt(X/f).$

iff

Xlk

0

($lf)(id).

O f course these g e n e r a l i s e d r e p r e s e n t a b l e s can b e d e f i n e d i n t e r n a l l y i n any Grothendieck topos and t h i s r e s u l t h o l d s .

e4.3

I f t h e o b j e c t s o f C a r e T, t h e n f o r any X

P r o o s i t i o n-.

I C I and any

+

A

Wx

+

Proof.

E p t ( X ) . 3 a c A.$(x,a) 3 open c o v e r p: Z ->> X and a f u n c t i o n f: p t Z

E

WX E A.WZ E p t ZCpz = x As p t X i s r e p r e s e n t a b l e ,

+

such t h a t

$(x,f(z))l.

U I k V x 3 a o(x,a) iff

X i f f f o r some open c o v e r p: Z

zlk

ulk 3a

x

X

-->

x

$ln2(nl,a)

U

$ln20~(nlo~,S)

f o r some 5 c A ( Z )

iff

Ult-'Jz

E

z

$(P(Z),dZ)).

0

We do n o t know under what c o n d i t i o n s 6 descends t o g i v e a f u n c t i o n d e f i n e d on a c o v e r b y open s e t s . We can ensure t h i s b y c o n s i d e r i n g t h e open i n c l u s i o n t o p o l o g y on C i n w h i c h case we o b t a i n

1 Wx +

Wx

E

p t X.3a

E

3 open c o v e r Ui E

A.$(x,a) E

Ui.$(X'fi(X))

U(X) and f u n c t i o n s fi:

Ui

+

A such t h a t

.

We now c o n s i d e r c o n t i n u i t y . 4.4.

Proposition.

I f X,Y

a r e TI

then

1Vf:

pt

)#

+

p t W,

f is continuous.

M.P. FOURMAN

176

Proof. I f Uik f : p t )# + p t W t h e n f i s r e p r e s e n t e d b y 5 : X V E ( ) ( Y ) a b a s i c open o f W , w: W + U and x: W + X we have

Wkl iff

[S

0

(t;lw)(x)

E

<x,W>l-l(v)

w It- x

iff

1 regarding 5- V

O(Xx U) as an open o f

E

5- 1(V) i s open.

Thus

Ulk

55

Iteration

E

U

+

Y in

(c.

For

v w

=

-1 -1 <x,w> 5 (V) =

iff

x

w

c5-1(v)lwl d e f i n e d a t U.

)#

0

We r e t u r n f o r a w h i l e t o c o n s i d e r a t i o n o f a general Grothendieck topos B = Sh(O,J). We c o n s i d e r t h e i n t e r n a l c a t e g o r y (I i n E g i v e n b y (E(U)

(c/u

w i t h r e s t r i c t i o n s g i v e n b y p u l l i n g back. [For those who w o r r y about coherence (one s h o u l d w o r r y ) , we remark t h a t a conc r e t e c a t e g o r y i n E w i t h an e q u i v a l e n t c a t e g o r y o f s e c t i o n s o v e r U i s g i v e n by c o n s i d e r i n g V / f t o be r e p r e s e n t e d as t h e element S o f (PV)(U) determined b y W / ~ V E S i f~f ~ f o v = g . So & i s an i n t e r n a l s m a l l f u l l subcategory o f E whose o b j e c t s a r e s u b f u n c t o r s o f representables.] We g i v e C_ a t o p o l o g y b y l e t t i n g

xi -x

\/

Now f o r A

E

I E I we d e f i n e

w i t h r e s t r i c t i o n s f o r g: V

and f o r 5: Y/h

+

cover X / f i n

A, E +

X/g i n a/U,

+

i f Xi

+

X cover X i n

ShE(C,J) b y UkA_(X/f) A(X) U given b y r e s t r i c t i o n along f*g

by r e s t r i c t i o n along 5 Y

Any morphism A

&

B i n E induces

A

X

U +,B i n ShE($,J).

c.

Continuous Truth I

177

For those who p r e f e r g l o b a l d e s c r i p t i o n s , we associate t o A functors 6/U + E/U n a t u r a l i n U (i.e.

comnuting w i t h g* f o r g: V

+

E

If[ (pseudo)

U) as f o l l o w s :

where

For Y

'

,E/U

nf

U X

a, B

+

X

31

we have nh

P

.

npE whence

nhS* * nf

(as E.*

4ng)

U

and nhAy

* nPx (as

Ay)

E*Ax

.

This gives t h e r e q u i r e d arrow nhayA functor

+

What we o b t a i n i s an ( i n t e r n a l )

nfAXA.

C+EC

OP *

We s h a l l show t h a t t h i s preserves f i r s t order l o g i c . liere we work c o n c r e t e l y f o r t h e sake o f computations. A simple b u t more a b s t r a c t treatment w i l l appear i n Fourman and K e l l y C19831. We now consider a f i r s t - o r d e r language L w i t h s o r t s f o r t h e o b j e c t s o f E and operations symbols f o r i t s morphisms. I n f a c t t o avoid s i z e problems, we consider an a r b i t r a r y small f r a g ment o f such a language. We may consider L a l s o as a language i n K as a constant object ( v i a A ) . Working i n E we consider t h e i n t e r p r e t a t i o n o f L given by i n t e r p r e t i n g t h e s o r t A by A and each o p e r a t i o n f: A + B by t h e corresponding morphism 4 + &. 5.1

Lemma.

For f: X

+

U and g: X

~ l xk/ f k

9

+

V

iff

vlt-

X/gl!-

+

Ulk X/flk *g i s defined t o mean 0 f o r a l l g: X v As no r u l e decreases t h e complexity o f then IF i s closed under t h e r u l e s o f 9 we say assume t h a t t h e r e s u l t holds f o r subformulae o f 9.

Proof.

By i n d u c t i o n , i t s u f f i c e s t o show t h a t i f

v ~ XF/ g l k

+

.

Only (+)+ and ( W ) ' present any d i f f i c u l t i e s . r e s u l t f o r @ and $.

Me consider (-+)+, and suppose t h e

Suppose t h a t f o r a l l E: W U and a l l h: Z + g*X, i f W Z/(E*f h) Ip*01(f*E 0 h) Then i f n: W ' + V and h ' : Z + rr*X a r e such then W l k Z / ( E * f h ) I p d ( f * E - h ) . that W ' Z ' / ( n * g h ' ) IF @ l ( g * n h ' ) then by i n d u c t i o n hypothesis -+

0

0

M.P. FOURMAN

178

It

U k - Z ' ( f 0 g*no h ' ) * $1(g*no h ' ) whence ( l e t t i n g 5 = i d and h = g*n h ' ) we have 0

U

Z'/(

f

0

g*n

0

It * $1 (g*n

h'

h') So V l k X / g / k ~ - t $ . The p r o o f

i n particularW'IkZ'(n*goh')lkJil(g*qoh').

0

I

for V 5.2

i s similar.

Theorem.

Proof.

0

F o r Q a f o r m u l a o f L w i t h a p p r o p r i a t e parameters

U IF'' X/flk Q" i f f xlk Q . F i r s t l y , t h i s i s w e l l formed: Parameters f o r Q a t X / f a r e elements o f which a r e g i v e n as elements o f A(X) and a r e t h u s parameters f o r $ a t X.

m)

We proceed by i n d u c t i o n .

T h a t i s , we show t h a t i f we d e f i n e

It* i n t e r n a l l y

It

Ulk X / f * Q iff X l t Q c l o s e d under t h e d e f i n i n g c l a u s e s o f l k i n t e r n a l l y , (whence UIk X / f 1 1@ X\k $) and i f we d e f i n e \I by + X $ i f f Ulc X / f Ik @ t h e n i s c l o s e d under t h e d e f i n i n g c l a u s e s o f (whence Xlk Q *VIE X / f l t - Q). then

by

-

it* i s

It+

\kt

As t h e o p e r a t i o n s A + B a r e j u s t t h o s e i n h e r i t e d f r o m E, terms a r e i n t e r p r e t e d a l i k e i n b o t h contgxts: Thus i f [ T I = Uo] t h e n UlkU-rl = Dull, so i s closed under ( = ) + and i f Ulk U ~ l l= Uol t h e n UIk T = a,

11'

so

IF*

i s c l o s e d under ( = ) +

It and \I-* a r e c l o s e d under ( A ) ' , (v)', (3.)' i s t r i v i a l . F o r I, suppose 1 1 ' $Ifi f o r fi: Xi X i n some cover o f X t h e n X I 1 Xi/fi $Ifi and by I i n t e r n a l l y Xik X / i d l k $. I n t h e c o n t r a r y d i r e c t i o n , suppose Ulk Xi/g fi IF* $Ifi f o r some c o v e r of X as above. Then Xi $Ifi so Xlk Q t h a t i s Ulk X/g Q. F o r (+)+, f i r s t suppose t h a t f o r a l l f: V U i f V I - - + ~ lt hfe n V I k + ~ l f Then . we c l a i m U I U / i d l k @ + I$,because f o r a l l g: W + U and a l l h : V + W , i f W @1g h, t h e n V It+$ l g h so V IF $19 h, t h a t i s W V/h v/h Jilg h. Conversely, iff o r a l l g: W + U and a l l h: Z + g*X, where f: X + U, i f WIE Z / g * f h \I* $ l f * g h , t h e n X I k @ + $, because f o r h: Z X i f Z \ k $ l h then U l t Z / f h It-* $ l h so Ulk Z / f h It-*Jl?h which g i v e s Zlk $Ih, so Ulk X / f I/-* Q Ji. That

+

Xi

+

0

+

IF

0

o

0

0

+

-t

0

0

0

+

0

The p r o o f f o r W+ i s s i m i l a r .

0

We view t h i s thorem as a s s e r t i n g t h a t i n t h e topos E t h e n a i v e n o t i o n o f t r u t h g i v e n by t h e e q u i v a l e n c e t h e s i s i s consonant w i t h t h e t h e o r y o f meaning g i v e n b y t h e n o t i o n o f f o r c i n g o v e r t h e s i t e &. O f course t h i s may seem vacuous as i t appears t h a t B i s manufactured w i t h t h i s r e s u l t i n mind. However, i n t h e case o f p r i m a r y i n t e r k t f o r t h i s paper, t h e r e s u l t s o f 84 a l l o w us t o r e g a r d (I i n t e r n a l l y as a f u l l subcategory o f Loc(E) equipped w i t h t h e open cover t o p o l o g y . I n f a c t , i f Q i s t h e c a t e g o r y o f s e p a r a b l e l o c a l e s , we may i d e n t i f y (I as a c a t e g o r y o f s e p g r a b l e l o c a l e s i n E. We s h a l l deal w i t h t h i s , among o t h e r t h i n g s , i n a sequel t o t h i s paper. Given f: X

-f

U we may view an element a o f A(X) as a f u n c t i o n : U

It a:

X/f

+

A,.

T h i s a l l o w s us t o r e p h r a s e o u r theorem. 5.3

Corollary.

ulkX/flk $(a)

iff

Ulk~tE

X/f@(a[t)).

0

We view t h i s as a g e n e r a l f o r m o f t h e e l i m i n a t i o n theorem ( c f . T r o e l s t r a C19771

Continuous Truth I

179

The appropriate theory o f continuous t r u t h CT has an axiom f o r each pp.33,79). clause i n t h e d e f i n i t i o n o f X/f/k$(a). For example, t h e clause f o r 3 gives the axiom o f l o c a l choice Y t E 3 y $ ( a ( t ) , y ) i f f 3 open cover p: Z ->> X and continuThe t r a n s l a t i o n T $ o f a formula $ w i t h ous f: Z + Y such t h a t W z $ ( a ( p ( z ) ) , f ( z ) ) . o u t f r e e lawless v a r i a b l e s i s given by T$ :def/k $.

X

CODA A general n o t i o n o f non-constructive o b j e c t i s given by i n t e r p r e t a t i o n s i n Grothendieck t o p o i . The process o f i t e r a t i o n described i n 55 shows how we may view ( i n t e r n a l ) t r u t h i n t h i s i n t e r p r e t a t i o n as given by a non-standard theory o f meaning. The clauses d e f i n i n g t h i s g i v e axioms f o r the corresponding theory o f continuous t r u t h CT and an " e l i m i n a t i o n " t r a n s l a t i o n . By construction, CT tf T$ and f o r formulae i n t h e l a w l i k e p a r t o f t h e language T $ 5 $. The p r o o f t h e o r e t i c content o f t h e e l i m i n a t i o n ;

I$

CT

$

iff

ID

T$,

requires f o r m a l i s a t i o n o f our treatment i n an appropriate theory I D o f i n d u c t i v e d e f i n i t i o n s . We do n o t undertake t h i s here. A f i n a l example o f an u n f i n i s h e d o b j e c t i s t h i s paper. Some o f t h e r e s u l t s , i n p a r t i c u l a r c o n t i n u i t y p r i n c i p l e s i n sheaves over s i t e s , go back t o 1978 and were much i n f l u e n c e d by discussions w i t h S c o t t and Hyland. Some r e s u l t s are s t i l l being r e f i n e d . Other p e r s i s t e n t i n f l u e n c e s have been those o f Joyal and Lawvere on t h e one hand and o f K r e i s e l , T r o e l s t r a and Dummett on the other. This research has been supported a t various times by the N.S.F. (U.S.A.), the S.R.C. (Netherlands), and t h e A.R.G.S. ( A u s t r a l i a ) , and made e a s i e r (U.K.), t h e Z.W.O. by t h e h o s p i t a l i t y o f many people n o t a b l y C h r i s t i n e Fox, I r e n e Scott, Karen Green, and Imogen K e l l y . I am g r a t e f u l .

REFERENCES A r t i n , M., Grothendieck, A., Verdier, J.L., ThGorie des Topos e t Cohomologie, E t a l e des Sch6mas (SGA4), (Lecture Notes i n Math. 269, 270, Springer-Verlag, B e r l i n , 1972). Beth, E.W., Semantical Considerations on I n t u i t i o n i s t i c Logic, Indag. Math., 9(1947), p.572-7. Boileau, Andr6 & Joyal, Andr6, La logique des topos, J.S.L.

46(1981), p.6-16.

Brouwer, L.E.J., Cambridge Lectures on I n t u i t i o n i s m , D. van Dalen, ed. (Cambridge U n i v e r s i t y Press, 1981). Dummett, Michael, Elements o f I n t u i t i o n i s m , (Oxford U n i v e r s i t y Press, 1977). Dummett, Michael, T r u t h and

o t h e r enigmas, (Duckworth, London, 1978).

Fourman, Michael P., The l o g i c o f Topoi, i n Handbook o f Math. Logic (ed. Barwise, J.), (North-Holland, 1977), p.1053-90.Fourman, Michael P., Notions o f Choice Sequence, Proc. Brouwer Symposium, (ed. T r o e l s t r a , A. and van Dalen, D.), (North-Holland, 1982). Fourman, Michael P. & Grayson, Robin J., Formal Spaces, Proc. Brouwer Symposium, (ed. T r o e l s t r a , A. and van Dalen, D.), (North-Holland, 1982). Fourman, Michael P.,

T1 spaces over t o p o l o g i c a l s i t e s , JPAA,

( t o appear), 1983.

180

M.P. FOURMAN

Freyd, P e t e r , Aspects of Topoi, Bull. A u s t r a l . Math. SOC., 7(1972), p.1-76. I s b e l l , John, Atomless p a r t s of spaces, Math. Scand., 31(1972), p.5-32. Johnstone, P e t e r T . , Topos Theory, (Acad. Press, London, 1977). Johnstone, Peter T . , Stone spaces, (Acad. Press, London, 1982). J o y a l , Andre, & Tierney, Myles, An extension of the Galois theory of Grothendieck, p r e p r i n t , 1982. Kreise!, Georg, Lawless sequences o f natural numbers. p .222-48.

Comp. Math. 20(1968),

Makkai , Michael & Reyes, Gonzalo, First-Order Categorical Logic, (Lecture Notes in Math. 611, Springer-Verlag, 1977). Moschovakis, Joan R., A topological i n t e r p r e t a t i o n o f second-order i n t u i t i o n i s t i c a r i t h m e t i c , Comp. Math., ( 3 ) , 26( 1973), p.261-75. S c o t t , Dana S., Extending t h e topological i n t e r p r e t a t i o n t o i n t u i t i o n i s t i c a n a l y s i s , Comp. Math. 20(1968), 222-48. S c o t t , Dana S . , I d e n t i t y and Existence i n I n t u i t i o n i s t i c Logic, Proc. Durham Symposium, (ed. Fourman e t a l . ) (Lecture Notes i n Math. 753, Springer-Verlag, 1978) , p. 660-96. T r o e l s t r a , Anne S . , Choice Sequences, (Oxford University P r e s s , 1977). Wraith, Gavin C . , Lectures on elementary t o p o i , Model theory and t o p o i , (ed. Lawvere F.W. e t a l . ) , (Lecture Notes i n Math. 445, Springer-Verlag. B e r l i n , 1975), p. 114-206. Wright, Crispin, W i t t g e n s t e i n ' s Philosophy of Mathematics, (Duckworth, 1981).

LOGIC COLLOQUIUM '82 G. Lolli, G. Longo and A . Marcia (editors) 0 Elsevier Science Publishers B. V. (North-Holland), 1984

181

HEYTING-VALUED SEMANTICS R.J. Grayson

*

Institut fur mathematische Logik und Grundlagenforschung Einsteinstrafle 6 4 ,

4400 Munster, West Germany

Introduction. Chapter I.

The Logic o f H-Sets.

5 1 . Complete Heyting algebras. § 2. Interpretations of propositional logic

5 5 9

3. H-sets. 4.

Interpretations of predicate logic.

5 . Number systems.

§ 6. Complete H-sets.

5

7. Interpretations o f higher-order logic.

Chapter 11. Mathematics in H-Sets. § 8.

5

Some internal constructions.

9. Internal topologies.

§ l0.Choice principles.

9

11.Continuity principles.

References

Introduction. In this paper we develop a semantics for intuitionistic systems in which sentences are given "truth-values'' in complete Heyting algebras (cHa), just as sentences of classical set theory are given values in complete Boolean algebras ([MD], for example). T h e use o f the lattice of open subsets of a topological space t o interpret intuitionistic propositional logic goes back t o Tarski ([Ta,RS]). Extensions t o predicate logic were made b y Beth and Kripke (ID]) and applied t o metamathematical results for arithmetic by Smorynski ([Tr]). Further interest was drawn t o the area by the topological interpretations of analysis in [Sl,Mo,VD], where it was shown that "Brouwer's Theorem", on the continuity o f all functions between reals o r the Baire space, could be modelled in this way. In addition, Bishop's book ([Bi]) showed the feasibility of constructivism and gave new impetus t o the investigation of constructive and intuitionistic systems. A t the same time, interest has arisen from the theory of topoi,

*

Research Fellow of the Alexander-von-Humboldt-Foundation

R.J. GRAYSON

182

which can be seen as a category-theoretical formulation of intuitionistic higher-order logic ( [ F l l , for example). Other kinds of semantics are also suggested by this approach, for example, sheaves over sites ([MR]). However, the level of generality of Heytingvalued semantics seems to provide a natural stopping-point: the notion of cHa is simply an algebraicisation o f the notion of "truth-value" for intuitionistic predicate logic, staying within the conceptual framework of topological, Beth and Kripke models. The general theory of sheaves over a cHa (here called H-sets) is worked out in great detail in [ F S ] , where it is shown how they model intuitionistic higher-order logic (the extension to set theory is made in [Gl]). This paper is designed as a self-contained introductory exposition of the basic definitions and results, which it is hoped will enable the interested reader then to come to grips with more detailed treatments as well as with more specialised papers in this area. The paper falls into two chapters. In Chapter I we describe successively the interpretations of propositional, predicate and higher-order logic over a cHa. In Chapter I1 we develop some analysis and topology in these models, with particular emphasis on topological models and on the interpretation of various principles of choice and continuity. We close with Joyal's very elegant proof, using topological models, of a derived rule of local continuous choice for intuitionistic higher-order logic. I have not attempted on the whole to assign credit too exactly, beyond references to the literature, but I should like to acknowledge here the contributions of Dana Scott, whose influence on the whole treatment should be clear, and of Mike Fourman and Martin Hyland, who have stimulated my interest in the subject over the years. I thank the Alexander-von-Humboldt-Foundation, Bonn, for financial support, and the Institut fur mathematische Logik und Grundlagenforschung, Miinster, for their hospitality.

CHAPTER I.

5

THE LOGIC OF H-SETS

1 . COMPLETE HEYTING ALGEBRAS

We begin by defining the structures which are to act as our domains of "truth-values". Although we will be mostly concerned with topological examples, this more general, algebraic setting seems to make the essential features clearer, besides providing further examples (see 9 . 7 for example). Much information on the classical theory of complete Heyting algebras (cHa) may be found in [ R S ] and on the constructive theory in [ F S , Chapter I]; for we want to be handle our models "constructively" too (see 7.8 for further discussion of this point). 1 . 1 Definition. A complete Heyting algebra

lattice (H,Z), with finitary and infinitary by h , l \ , v , V ,

is a complete

meet

and join denoted

satisfying the distributive law, for pEH and ASH, phVA

E

V(phq1qEA).

Hereafter H will always denote a cHa, with elements p,q,... also the notation T for VH. the "top" element, and 1 for "bottom" one.

AH,

.

We use

the

Heyting-Valued Semantics

Logically, the order relation tion. -

5

183

is read as the relation of implica-

In addition one may define in any complete lattice an

implication operation by

5

(p-9) = V I r t p A r

q}.

AS a special case we have negation ~p defined as (p+I), which equals V{r I pAr=I) 1.2 Lemma. In any cHa H the implication operator is characterised by the adjunction rZ(p+q)

iff

(phr)lq.

Proof. If ( p n r ) ~ q ,then rL(p+q) always holds, by definition of implication. If H is a cHa and rl(p+q), then the distributive law gives pAr

5 =

PAVCSIPAS5 V{pAS [ P A S

q}

5 qj

5 4.

Proof. (i)

-

(iii) follow at once from Lemma 1 . 2 .

Since ~p(~p,

(iii) gives ph7p=l and then p5-,-.p. From (i) and qATq=l we obtain ph(p+q)h~q=I, hence applying 1 . 2

(p+q)

5

~ ( p h ~ qand ) (p-tq1A-q 5 TP, by

again gives (p+q)

5

(iii)i

(-q+-p). The remainder is left as

an exercise. 1.4

Examples. a) The open subsets O(T) of any topological space T

form a cHa under inclusion, 5 . A , V , V are the set-theoretic fl,U,u while hA=Int ( O A ) and T

(U+V) = IntftI tEU + tEV).

is T , I is the empty set

In this context we use u , V ,

d , and

...

-rU is Int(T'U1.

for elements of O ( T ) , and s , t , . . .

for elements of T. We call such cHa topological; ways of obtaining

R.J. GRAYSON

184

nontopological examples may be found in [FS,S2]. b) As special cases of topological cHa we have those arising from partial orders ( K , c ) , where K is given the topology of upwards closed subsets (that i s , O ( K ) consists of those P such that Vi,jEK.j)iEP + j E P ) . This provides the connection between Kripkemodels based on partial orders and semantics with "truth-values" in topological cHa (see 3 . 3 (c))

.

c) For a similar connection with Beth-models based on a partial order ( K , Z ) , one takes T to consist of all maximal chains a in K , with O(T) having as subbasis the sets {UlfEa) for iEK. 1.5 Heyting Algebras. A lattice equipped with an implication having the property of Lemma 1.2 we may call simply a Heyting algebra. These are treated in [RSI under the name of "relatively pseudocomplemented" lattices; it is shown there that all such lattices satisfy & t Jfinitary distributive laws, as well as the infinitary one for such joins as exist.

For the purposes of 2.5 it is useful to note the following simple completion process for any Heyting algebra H: Let O(H) be the topology of downwards closed subsets of H (compare 1 . 4 (b)), and let J be the J-operator ( I F S , 2 . 1 1 ] ) defined by J ( U ) = the set of all joins of subsets of U which exist in H. Then p W [PI = {qlq
(i) (ii) (iii) (iv) (v)

.

T=pV-rp T=-,r+r -(Tnlr) 5 (T+r) -t(pAq) 5 lpv-q r+(pvq) 5 (r-rp)v(r-rq)

These follow straightforwardly from the q=lp and ~ r = @ .

observation

that p=lq,

2. INTERPRETATIONS OF PROPOSITIONAL L O G I C

The interpretation of intuitionistic propositional logic in a general cHa (or even, Heyting algebra-1.5) can perhaps hardly be counted a s "interpreting" at all; it i s more a matter of algebraicising logic, as i s made clear in [RS]. 2.1 Definition. An interpretation of a propositional language in a cHa H assigns an element "PI1 of H , the "truth-value'' of P , to each propositional letter P. Symbols t and f , for "true" and "false", are included in the language, and we require "t11 =T and [[f]l=I.

Given an interpretation we extend the evaluation to give a value [[A]] in H to each formula A of the language as follows: AAB 11 = [[ A 11 A [[ B and similarly for v,+ and

7 .

11

Heyting-Valued Semantics

185

2.2 Definition. A propositional formula A is valid in an interpretation iff [[A T. Further, A is universally valid iff it is valid in all interpretations.

I=

2.3 Definition. The system IPL of intuitionistic ro ositional logic is given by the following axioms and rules (takenPfrEm [Tr, 1.1.311, rules being indicated by the double arrow

*.

PL PL PL PL PL PL PL PL PL

1) 2) 3) 4) 5) 6)

7)

8) 9)

A+A A,A+B * B A+B,B+C i, A+C AhB+A, AhB+B, A+AVB, B+AVB A+C,B+C AVB+C A+B,A+C * A+BhC AhB+C 9 A+(B+C) A+(B-+C) * AhB+C f+A,A+t

2.4 Soundness Theorem. Every propositional formula provable in IPL is universally valid. Proof. Straightforward using Lemma 1.2 and its Corollary. Firstly, validity of A+A means that “A 11 + [[ A 11 = T, that is, that [[A]] 5 [[A]]. Closure under PL 2 means that, if [[A]] = T and “A11 + “ B l I = T , then “ B l I = T ; but “A11 + “ B l I = T iff “A 11 5 [[B]]. P L 3 is just the transitivity of 5 in H , while PL 4-6 express that A and V are respectively meet and join in H. PL 7-8 correspond exactly to Lemma 1.2, and PL 9 results from the requirements [[ t I] =T , [[ f 11 = 1. Note: By the soundness theorem we can give counter-examples to the provability of various assertions, by the method of 1.6. 2.5 Completeness Theorem. Every universally valid propositional formula is provable in IPL. Proof. We construct in fact a “universal” interpretation for which validity is exactly provability. Denote provability in IPL by Iand consider the “Lindenbaum-Heyting“ algebra of equivalence classes of formulae under the equivalence relation A-B

iff

I- A

++

B,

with the order relation given by [A]

5 [B] iff k A + B .

NOW complete this Heyting algebra as in 1.5 and interpret “PI] as the (image of the) equivalence class [PI. Then [[A]] = [A] for every formula, whence A is valid iff [A]=T=[tl iff I-A.

s

3. H-SETS

Preparatory to interpreting predicate logic over a cHa H, we describe the objects which are to provide the domains of interpretation of variables. They are sets with an H-valued equality relation, [I.= #I], which is not required to be reflexive, the value [[a=a]] rather giving a measure of the “existence” of an element a. Further discussion of these “partial” objects and their logic will be found in [FS,§41 and [S2].

R.J. GRAYSON

186

3.1 Definition. An H-set (given a cHa H) is a s e t A with a function [[. =*I] : AxA-G satisfying, for all a,b,cEA, (i) (ii)

[[ a=b]l [[ a=bll

= A

[[ b=a]l [I b=cl1

5

[[ a=c]l

We do not require “a=a]l= T, by [[Ea]] = [[a=a]]. If [[Eall

=

.

but define the existence predicate E T , a is called global.

A s motivation for this definition we have the following basic example. 3 . 2 Definition. Given topological spaces X and T , the O(T)-set X consists of all continuous functions a:U+X for UEO(T), with equality defined by

“a=b]]

= Int{tla(t)=b(t)}.

(Here external equality is also taken t o be “strict”, so that a(t)=b(t) implies tEdom(a) I7 dom(b) . ) Then [[ Eall = [[ a=a]] = dom(a), as this is open by definition. T h u s the existence predicate has a very natural interpretation as the “domain of definition” o f an object. Furthermore, equality on X is “local”, since tE[[ a=b]] iff a and b agree on some neighbourhoox o f t.

3.3 Further examples. a) F o r any set A the trivial or constant H-set f h a s A as its underlying set with “a=b

11

i

=

T

i f a=b

I

if a+b.

(For a constructive version, when equality o n A i s not decidable, one puts “a=bIl =V{Tla=b}.) If X,T are topological spaces, the constant O(T)-set ? can be identified with the subset o f XT consisting of all constant functions G=At.x, for xEx, since in x T ’ [[ $=$]]= {t I X=y} = V { T l x=y}. b) The product (A x...xA

)

o f H-sets A

* *

*,An is defined t o be

their set-theoretic product with equality

a=gll

=

& [ [ -

ai=bill

(Note that this differs slightly from the definition in [FS,4.8]: we are not concerned here t o k e e p the product “separated“ in the sense o f [FS,4.6].)

Heyting-Valued Semantics

187

c) The domain of a Kripke-model based on a partial order ( K , O ([D,Tr]) is given by assigning a set A(i) to each iEK, in such a way that j ’ i implies A(i) 5 A(j). Then with A = ,u A(i! we have iEK a natural existence predicate [[ E(-)]] : A-bO(K), K having the topology of 1.4(b), given by [[Ea]]

= {ilaEA(i)}.

Furthermore, if the model is equipped with equivalence relations we have an evaluation -i o n each A(i), with -i 5 --j for

icj,

[[ a=bl]

=

(ila-i

b}

making A an O(K)-set. The point we want to make here i s that “partial” objects are already latent in the growing domains of Kripke-models. 3.4 Definition. A predicate on an H-set A is a function [[ P(-)]] : A-bH, which is strict and extensional, that is, for a,bEA, (i) (ii)

tt

~ ( a ) l l 5 [kall P(a) 11

A

“a=bl]

5 “P(b)

11

A relation on an H-set (or on several H-sets) is taken to be a predicate on the appropriate product. 3.5 Remarks. a) The requirement of strictness for predicates, which says that a predicate can hold only of existing objects, is found to be technically convenient in handling the models. It should be noted though that (strict) predicates are not (in general) closed under logical operations, for example, negation: Xa,b. -[[a=b]] may very well not be strict. This point is discussed further in [52,3.11. b) In [FS] and [52], use is made of a (non-strict) relation of equivalence f , which may be defined in any H-set by “a~b]]

=

([hall

V

[[Ebll

-b

[[ a=bll )

and expresses (the value of) “a and b are equal insofar as either of them exists”. This relation is useful for talking about partial functions, and the logic of partial elements given in S 4 may also be neatly axiomatised using E and Einstead of =: in particular, in any H-set, equality can be reeovered by “a=bll

= [hall

A

“a=bl]

However we will make no further mention of this relation.

4.

INTERPRETATIONS OF PREDICATE LOGIC

We are now ready to formulate the notion of an interpretation in H-sets and to prove soundness for a formal system of predicate logic with “partial existence”. For a completeness theorem for this logic we fall back on the well-known theorem for Kripke-models. 4.1 Definition. An interpretation of a first-order (relational) predicate language, with equality and existence predicate, over a cHa H consists of an H-set A together with a relation on A (in the sense of 3.4) for each non-logical relation symbol of the language. (In general, constant and function symbols have to be interpreted as special kinds of relations; see 4.7-8.)

R.J. GRAYSON

188

Given an interpretation in an H-set A, we define the value “C]] in H for each sentence C o f the first-order language extended by adding constants for the elements of A , as follows: (i)

The atomic cases are given by the interpretation, equality and existence being interpreted by the basic structure of A as an H-set (3.1).

(ii)

The propositional connectives are dealt with as in 2.1.

(iii)

The evaluation of quantifiers is defined by Ujx. cII = V { “Ea A c [a/xlIl laeA} “Vx. C11

=A{

“Ea+

C[a/x]]l

laen}.

(Note: this i s the first point where w e have needed completeness o f H.) The idea in evaluating quantified formulae is that we only quanitify over existing objects; in other words, the quantifiers are relativised to the predicate E. On the other hand, free variables need not refer t o existing objects, and we make the following definition. 4.2 Definition. A formula C with free variables x in an interpretation in an H-set A iff

.

V = E A ~ .[[ c [ a l / x l ,. . ,an/xnI 11

=

T

I

.

.

. .xn

i s valid

.

C i s universally valid iff it i s so in all interpretations. We return to give examples of interpretations and o f the evaluation of formulae in § 5. Before that w e want to deal briefly with a formal system (essentially that o f [S2]) which i s natural for the above notion of interpretation. 4.3 Definition. T h e system IQL o f predicate logic with equality and existence consis t s o f the propositional system I P L (2.3) together with (i)

- -

For equality and existence the axioms El) Ex ++ x=x E 2 ) x=y y=x E3) x=y A y = z x=z E4) R(5) + EX h...hEX 1 E5) R(x) - A x l = y l A Axn=yn + R(x)

...

(ii)

For Ql) Q2) Q3) Q4)

(for each symbol R)

the quantifiers the axioms and rules E X A C + 3x.C EX AVX.C + C ( E X A C - D ) * (3x.C-D) ( D A E x + C ) 9 (D+VX.C)

(In Q3 and Q 4 , x should not be free in D.) 4.4 Soundness Theorem. Every predicate formula provable in IQL i s universally valid. Proof. The validity o f axioms El-5 is simply built into the definition o f interpretation, since (by 1.3) an implication C-D is valid 5 “011 T o check Q 1 , for example, in an interpretation iff [[C]] in an H-set A , we need to show that

.

VaEA.

“Ed]

A

“C[a/xll]

<_ “3X.Cll

(assuming for convenience that x i s the only variable free in C).

Heyting-Valued Semantics

189

But this is immediate from the definition of all the terms on the left.

I]

“3x.C

as the join of

To show closure under Q4 note first that, for any p in H , p<-[[’dx.cn

VaEA. I[DlI A whence

“Dl1

5

iff VaEA.pA [[Ed]

<_ “Vx.C]]

[[

c [a/xlIl

again. Now the validity of the hypothesis

making use of Lemma 1 . 2 of Q4 reads

;

5

“Ed

[[ C [a/Xll]

that is, (D+Vx.C) is also valid.

4.5 Completeness Theorem. Every universally valid predicate formula is provable in IQL. We do not give a proof of this result, since it follows straightforwardly from the completeness theorem for Kripke-models, which can be seen as special cases of interpretations in our sense (according to 1.4(b) and 3.3(c)). 4.6 These interpretations extend immediately to many-sorted languages, with sorts a , B , each interpreted by an H-set Aa,A B , . . .

...

acting as the range f v riables xa,xB,. fiers of the form Vx8,3x

’,...

..

of each sort, with quanti-

Actually, as may already have been noticed, we are tailoring our logic to our interpretations, rather than vice-versa. So we will tend to start with H-sets A,B, with various relations (and functions - see 4 . 7 ) on them, and interpret the language of these structures, with a sort for each H-set, writing, for example, the quantifiers now as VxEA,3yEB,

...

...

4.7 Function Relations. In order t o interpret predicate languages with constant and function symbols we need in general to treat them as special kinds of relation, namely singletons and functional relations respectively. A singleton on an H-set A is a predicate P such that

Va,bEA. [ [ P(a) 11

A

“P(b)

11

<_ “a=bll

.

A functional relation between H-sets A and B is a relation R on AxB such that VaEA Vb,b’EB. “R(a,b)

11

A

“R(a,b’)

11 <_

“b=b’l]

These conditions clearly correspond respectively to the validity of Vx,yEA (P(x) AP(y) + x=y) and VxEA Vy,zEB (R(x,y) A R ( x , z ) + y=z). In general, functional relations will serve to interpret partial functions (and singletons partially existing constants). To interpret total ones the relation R should also be total, that is, VaEA.

[I

Eall

<_v{ “R(a,b) 11

I bEB);

in other words, VxEA i’yEB.R(x,y) i s valid. 4.8 Functions. In particular cases, however, function symbols may be interpreted by functions F:A+B which are strict and extensional, that is, for a,a‘EA, (i) (ii)

“E(Fa)ll [[ a=a’I1

5 A

“Eall [[ E (Fa)I1

.(

[[ Fa=Fa’I1

F then gives rise to a functional relation R by “R(a,b)]] = “Fa=b]l, which is total just in case for all a

=

R.J. GRAYSON

190 (iii) [[ Eall

5

E(Fa)

11 .

The property of completeness for B , which ensures that every functional relation arises from a function in this way, is discussed in

5

6.

4.9 If function symbols occur in a many-sorted first-order language and are interpreted as in 4.8, we now have a class of terms u , ~ , . . . each of a particular sort; these are evaluated as elements R u 11, [[TI] of the H-sets interpreting those sorts, by the clause

,...

“F(U l,...,Un)ll

bn]]

= F ( ~ ~ U l ~ ~ , . . ) ,~ ,

and atomic sentences containing terms are evaluated by

,...

~ ~ R ~ U l , ~ .= . “R(“UIII ~ U n ~ ~ ~

“On]]

)I1

The logic IQL should then be extended by the addition of a rule of substitution. Full discussion of this logic may be found in r.521.

5

5. NUMBER SYSTEMS

In this section we consider structures which interpret the firstorder theories of the natural numbers, integers, rationals and real numbers in H-sets. The higher-order theory (in particular, the induction axiom for the natural numbers) will be dealt with in 8, where we also see that these structures are the (standard) interpretations of the number systems.

s

G,?,$

5 . 1 We start by giving the constant H-sets (3.3(a)) the trivial or constant first-order structure of the natural numbers, integers and rationals. That is, on each we use the standard arithmetic functions to interpret Successor, sum, product (as in 4.81, while the order relation (on Q, say) is given by

Then it is an easy induceion to show that in each case the terms are interpreted by themselves, ~[u]]=u, and the sentences by

For a constructive treatment we show “CIl= VITlC), and for this H should be an open cHa in the sense of [G4]. In any case this indicates that these interpretations have in themselves no interest for intuitionistic mathematics; they are needed however to provide a basis for the more interesting interpretations of analysis. 5.2 T o interpret the first-order theory of real numbers we specialise (as defined in 3.21, where to the case of O(T)-sets and consider R R denotes the (external) real numbers. ‘?his O(T)-set has a very rich structure, as first exploited (essentially) by [Sl]; see 5.7 for a discussion of his model. We consider first relations of order and apartness on R [[a
T:

= {tla(t)< b(t)l =

{tEdom(a)

n dom(b) 1

a(t)

b(t))

Heyting-Valued Semantics

191

the right-hand sets being always open as a,b are continuous. As in 3.2 the occurrence of < on the right is taken to be “strict“; for a constructive treatment we put on the right too in the definition of apartness. Here is a picture in the case T=R:

[[ a
T

5.3 Proposition. The intuitionistic theory of order and apartness on the reals (eg [He,Sl]) is valid in every RT. That is, the following are valid: (i) vx,y. x=y ++ 7x*y (ii) Vx,y. xty -P yylx (iii) Vx,y,z. xPy + (x#z v zky) (iv) Vx,y. xyy ++ ( x < y v y<x) (V) vx,y,z. X < y A y < Z + X
7

“arb]]

“Ed]

n

[[ Ebll

Since [[a=b]] n [[a*]] is clearly empty, we obtain the inclusion from left to right (Cor. 1.3). On the other hand, if U I7 aabl] = @ and tEU n dom a Il dom b, we have -r(a(t) # b(t)), so a(t)=b(t), and the other inclusion follows. For (iv) we need [[ affb]] E [[ aCb]] u “Ma]] b(t) iff a(t) < b(t) v b(t) < a(t). a(t)

+

, which follows from

For (vi), let tEdom(a) n dom(b) f l domtc) and a(t) < b(t); a(t)< c(t) or c(t)< b(t), so tE[[a
then

For (viil , if act\ < b(t\ , L e t x be chosen so t h a t a \ t > < x( b \ t ) . Then for the constant function x s . x we have t € “ a < k b I l . Hence I cERT} = [[3z.a
=

(tl,to)

= (to,t2)

“atbll

= (tl,to) U (to,t2)

“a=bll

=

@.

192

R.J. GRAYSON

Hence the following sentences C are not valid in both cases): (i) (ii)

(in fact to @ "Cll

a
Furthermore, by "shifting" a and b about we can arrange to omit any point of T , so that the universally quantified forms of (i) and (ii) get value @, and we obtain the stronger result that (i)' -rVx,y. x
V

yCx V X=y x#y

+

are both valid in RR. 5.5 Arithmetic structure on RT is also given "pointwise", by functions into RT, namely a+b = At.(a(t)+b(t)) a.b = At. (a(t).b(t)) a

-1

= ~t.a(t)-'

In each case the arithmetic functions on the right are taken to be "strict" also, so that dom(a+b) = dom(a) fl dom(b) = dom(a.b) and dom(a-') = ftEdom(a)la(t)+O), taking the external inverse function to be defined just at non-zero reals. The inverse function i s thus a typical example of a partial function, and satisfies in R V X . E ( X - ~ ) ~ + +x

8.

Together with the other usual identities this shows that R models the theory of an apartness field [He], with unit elements The constant functions 8 and I. AS

in 5.4 we can give counter-examples in R

Let a be At.max{O,t) and b be At.max{O,-t). a(t).b(t) = 0, so that [[a.b=OI]

R: Then, for all t ,

= R

while [ [ a s v b a l l = (--,O) U ( 0 , ~ )so that (a.b=O is not valid.

+

a=O

V

b=O)

A

5.6 A s in 3.3(a) we can treat the O(T)-set Q as the subset of R T consisting of all constant functions $ for ~ E Q .We can now prove one half of a cheorem characterising RT as the Dedekind cuts in the (The other half is proved in 8.5.) rationals, interpreted as

3.

Theorem. The subset 6 i s dense in RT and each element of RT acts as a Dedekind cut in 6 . T2at is, the following are valid (using p,q,... to range over Q , and x,y,... over R ) :

T

(i) (ii) (iii) (iv) (V)

(vi)

vx,y. xX 3q. X
Proof. The proof of ti) proceeds exactly as that of 5.3(vii), except that we now choose a rational p between a(t) and b(t). The remaining assertions may be straightforwardly deduced from the properties in 5.3 and the denseness of Q , arguing within the system IQL.

Heyting-Valued Semantics

193

To explain our sense of "cut" here briefly: thinking of the rationals below (resp. above) an element x as the left-hand (resp. right-hand) elements of a cut in Q , these conditions say successively that the two halves of the cut are inhabited, disjoint, closed downwards (resp. upwards), open (in Q) , and close together. This notion of cut is due originally to Tierney. 5 . 1 It may be asked why we consider artial functions at all in our models RT, and not only total ones, t s was done in [Sll (and, for Baire space, in h o ] ) . The point is that in those papers only special cases of the space T are considered, for which every partial function is locall extendable to a total one; that is, for each a and tEdom(a), theri is a total b w i t h a = b ] ] .

In general there may be very few total continuous functions but many partial ones, so that the total ones do not at all provide a representative picture. For example, take T = R U { * I , where R has its usual topology and has asneighbourhoods just the complements of finite sets. Then all total elements of RT are constant, while there are many non-constant elements defined just on R itself.

*

s

6

COMPLETE H-SETS

It is immediate that the O(T)-sets XT are always complete in the following sense, which we formulate for an arbitrary H-set A.

6.1

(i) For each aEA and pEH there is a (unique) restriction, alp, of a to p, with the property VbEA. [[ alp=b]]

=

[[ a=b]I

A

p.

From this follows [[ E(a1p) 11 = [[ Ea]] A p = [[ a=alp]I , that is, the "existence" of a is restricted to p , but where alp exists it equals a. In X the restriction alU is simply the set-theoretic restriction of a toTa smaller domain. (ii) For each compatible subset B of A , i.e. such that Vb,b'EB. [[ Ebll A [[ Eb'N 5 [[ b=b']I , there is a (unique) join, V B , in A, with the property VaEA. [[ a=VB]I

=

v f "a=b

v{[[

11 I

bEB).

.

EbII 1 bEB), and for bEB, From this follows [[ E(VB) 11 = [[ Eb]] 5 [[ b=VB]], that is, V B is a "glueing together" of the elements of B. A subset B of X is compatible iff all its elements agree pairwise on the intersecTions of their domains. Then the join of B i s the set-theoretic union of B, which is again an element of X T' 6 . 2 Remarks. a) Alternative definitions of "completeness" and proofs that the notions all coincide may be found in [FS,§4]; in particular, complete H-'sets are equivalent to sheaves over H. In [FS] also the completion of an H-set is constructed. Our position here is that completeness is useful (as in 6 . 3 ) when it arises naturally, but not worthwhile introducing specially.

b) Constant H-sets are never complete (unless H is trivial), since all elements are global, so that the "non-existent" element (a7 1) is missing, at least. However, when X is treated as a subset of X T as in 3.3(a), we can identify its completion as the locally constant elements of X In particular, this is useful for understanding the completions 8 and Q^ as subsets of RT.

OF

.

R.J. GRAYSON

194

6.3 The consequence (indeed, equivalent) of completeness that interestsus most here is that, when B is complete, every (total) functional relation R on AxB, for any A , arises from a (total) function F:A+B (as in 4.8), related to it by VaEA,bEB. [[ R(a,b) 11

=

[[ F(a)=bll.

Namely, given R , define F(a) = v { b l "

R(a,b)ll I b€BI.

The compatibility of the set on the right-hand side, for each a , is precisely the functional character of R. This representation of arbitrary functional relations will be especially useful in 9.8. 6.4 Another interesting equivalent of completeness is that, if A is a complete H-set, we can evaluate description terms (1x.C) ("the x such that C " ) in them, according to [[ Ix.CI1

since "Vx.x=a such that C".

++

=

V{al [[ Vx.x=a

++

Cll 1 a€A}

C " expresses exactly that "a i s the unique element

The corresponding logic of descriptions [52,§61 has the one extra axiom I)

Vy[y=Ix.C

f*

Vx(x=y

f-t

Cll.

As an example, if R is a functional relation on AXB, one obtains the the corresponding funytion (6.3) as F(a) = [[I~€B.R(a,x)ll. In particular,'the inverse a in RT (5.5) is obtained as (the interpretation of) IX. (a.x='i). These are examples where existence of a solution may be partial, but uniqueness is guaranteed. T o give an example of "partial uniqueness" we refer back to the picture in 3.2 and consider the term Ix.(x=avx=b): here a solution exists on the interval (to,t,) but is unique only on (t1, t2)

.

5

7

INTERPRETATION OF HIGHER-ORDER LOGIC

The final step, before doing "mathematics in H-sets" properly, is to interpret intuitionistic higher-order logic. We will here be exclusively concerned with "standard" interpretations, in that powersets will contain "all possible" subsets. Thus the interpretations will no longer be complete for the logic, and certain problems will necessitate recourse to other kinds of interpretation (realisability, On the sheaves over sites etc), to which we refer briefly in 1 1 . 1 - 2 . other hand, the mere fact of soundness of the interpretations is quite powerful, as will come out most clearly in 11. This is also the point at which one begins to reap the benefits of the generalisation from Kripke-models; as observed in the introduction to [VD], standard interpretations of, say, Baire space in Kripke-models over a partial order only yield constant structures, and nothing i s gained.

s

7.1 Definition. The power-set P(A) on an H-set A consists of all predicates (3.4) P,Q,... on A , with equality defined by P=Qll

= I \ { " Pa]]

+* [[ Qall

I aEA)

In this context we write [[ a€P]] for [[Pa 11 , and interpret bounded quantifiers 3x€P, etc., in the obvious way. Note that every element of P(A) i s global ( [ [ EP]] = T ) , so that P(A) is (almost) never complete (compare 6.2(b)).

195

Heyt ing-ValuedSemantics

7.2 Proposition. P(A) acts as an extensional power-set of A, satisfying full comprehension and with E strict (as a relation on AxP(A)). That is, the following are valid: (i) (ii) (iii)

VX,YEP(A). X=Y .++ VxEA(xEX ++ xEY) 3XEP(A) VxEA. xEX ++ C , for each formula C. xEX -b E x h E X .

Proof. (i) and (iii) are simply the definition of P(A). For (ii), given a formula C (in which we have assigned constants from some H-sets to all the free variables except x), we set

E[

a E ~ l 1= [[ Ea

A

~ [ a / x l l l,

so that P is clearly a predicate satisfying

(ii).

7.3 Definition. A many-sorted higher-order language is one in which, for each sort a , we have a power sort P(a), and, for any sorts a l , ...,a n , we have a product sort ( a x...xan). 1

In addition there should be function symbols for tupling and projection for the product sorts, a relatioE symbol for membership, E , on each a x P ( a ) , and abstraction terms {x IC} of sort P(a)for each formula C. The system IHL of intuitionistic higher-order logic is a many-sorted (4.6) version of the system IQL (4.3) with the addition of standard equations for tupling and projection, and axioms of extensionality, comprehension and strictness of E for the power sorts, as formulated in Proposition 7.2. The system IHLN is obtained by adding a sort N for natural numbers with a symbol for the successor function, and satisfying Peano's axioms including full induction (which we formulate below in 8.2). 7.4 Standard Interpretations. A standard interpretation of a manysorted higher-order language is one in the sense of § 4 in which the H-set A assigned to a power sort is always the power-set P(A ) P (a)

of that assigned to a, and in which the membership relation is interprehed as in 7.1. In particular then the interpretation [[{x lC}]]of an abstraction term is given as in the proof of 7.2(ii) by the predicate la.[[ EahC[a/x]]1 on Aa. Soundness of standard interpretations for the system IHL is immediate from 7.2; the extension to IHLN is dealt with in 9 8 . A more detailed description of a system similar to IHL may be found in [S2,§7] and of standard interpretations in H-sets in [FS,§5 and s71. 7.5 Types. A type in a many-sorted higher-order language is a term whose sort is a power sort. In particular, each abstraction term i s a type. We may think naively of types as " s ts" % =xa '} and use the notation Each sort a we identify with the type {x'lx VxEa etc. for the quantifiers. Conversely, each type can be treated in as a new sort, using the restricted quantifiers VxEu, 3xEr, the obvious way, and.relativising power sorts, product sorts and abstraction according to P(u) = {XEP(a) IXc_u} for u a term of sort P(a), and so on (see 152,571).

...

7.6 Interpreting Types. In parallel to the treatment of types as new sorts, we want to interpret each type (in a given standard interpretation in n-sets) as an H-set: A term u of sort P(a) is already interpreted as a predicate [[o]] on the H-set Aa. The H-set A is then defined as having the same

R.J. GRAYSON

196

but with equality relativised to [[u]]

underlying set as A [[ a=b]I

[[ a=bll a

=

aEol1

A

:

,

denoting by the subscripts u and a evaluation in A

and A

a'

Now an induction over the terms and formulae of the language shows that interpreting the relativised quantifiers etc. as speaking about the H-sets A amounts to the same thing as interpreting them as defined in tge original language, within the H-sets A a' 7 . 7 Exponents. A basic example of a type is the exponent U,B,

B a of sorts

given by the term {xEp(axB) lVxEa 3!yED.<x,y>Ex).

The interpretation of this term as an H-set, according to 7.6, is then the exponent of A and A B , which consists of all relations R,S on AaXAB with [[ E(R) 11

=

[[ R a total functional relation]]

Ba

and [[ R=Sl]

=

[[ E(R)II

Ba

aa

A

{ [ [ R(a,b)ll

aEA,,

*+

[[ S(a,btll 1

b€AB)

7.8 Now we ask the reader to look back over this first chapter and see that the definitions of H-sets and validity in them, and the proof o'f soundness of standard interpretations, can all be carried out within the system IHLN of 7 . 3 . This means, for example, that we can iterate the construction of the models inside any universe of H-sets, just as forcing is iterated in classical set theory (see [FS,§9] for example). More interestingly, perhaps, we can use the provable soundness of the interpretations to obtain derived rules for the system. We give an example of this, due to Joyal, in 1 1 . 5 ; other examples may be found in [Be,H2,FJ]. So in Chapter I1 we will be concerned to note, as we did in 5, what principles are needed to prove the validity of various assertions in various models, arguing so far as possible "constructively", i.e. within the system we are interpreting. In order to distinguish what is assumed to hold "on the outside" (or "in the ground model") from that which is valid in the interpretations, we use the terms external and internal.

The above cor.siderations all extend mutatis mutandis to systems of with the power-set axiom and full comprehension, as formulated in [Gl] and exploited in [HZ]. The general problem of interpreting a set theory, with only the axiom of exponents, within such a theory is dealt with in [G3]; appl-ications of this are made in [Be].

set theory -~

CHAPTER 11.

5 8.

MATHEMATICS IN H-SETS

SOME INTERNAL CONSTRUCTIONS

We are now ready to interpret constructions within the system IHLN of higher-order logic with a sort for natural members (7.3) in H-sets. The integers, rationals, real numbers, functions etc. appear as types (7.5) in this language, which we want to interpret as H-sets according to 7 . 6 . Such characterisations will generally be

Heyting-Valued Semantics

197

only "up to isomorphism", in the following sense. 8.1 Definition. An isomorphism between H-sets A and B is a total functional relation ( 4 . 7 ) on AxB which is internally one-one and onto. As in 4.8 and 6 . 3 , in particular cases an isomorphism may be given by a function from A to B which is internally one-one and onto, or even (for example, when both A and B are complete) by a pair of functions F:A+B and G:B+A which are inverse to one another:

VaEA.

[[ Ea]]

VbEB. [[ Eb]]

and

5 5

[[ a=G(Fa) 11 [[ b=F(Gb) 11

.

The extension to isomorphisms of structures is made in the obvious way. As in classical mathematics there is only one structure (up to isomorphism) satisfying Peano's axioms for arithmetic in any standard interpretation fn H-sets. We do not prove this fact but only show that the H-set N with constant structure ( 5 . 1 ) does satisfy the axioms, and hence can serve as the interpretation of the sort N. 8 . 2 Proposition. f j with the standard successor function S satisfies Peano's axioms for arithmetic, including induction in the form

VXEP(2). O E X A v x E X . SXEX

+

VXEN. XEX

Proof. Since the interpretation of first-order sentences is always absolute (5.11, the first-order axioms are trivial. To prove inguction one shows, by an external induction, for any predicate P on N. that OEP

if

q =

then

VnEN. q

By definition q ( " of 4.4,

5

5

5

:

95" VxEP.SxEP11,

[[ nEP11

A

[[ nEP11, q

VXEP. SXEP]], nEP11

OEPII and VnEN. q

hence, if q

A

5

that is, as in the proof

[[ SnEPl];

[[ SnEPl].

8.3 We leave the reader to check that, for some standard definitions of the integers and rationals as types obtained from products of N, the corresponding H-sets, according to 7.6, are isomorphic to the constant H-sets 2 and Alternatively 5 and with constant structure (5.1), can be shown to be the unique H-sets (up to isomorphism) with certain properties (e.9. 6 is a countable dense linear order without endpoints).

a.

0,

We are now ready to formulate and prove the "converse" of Theorem 5.6, giving a characterisation of real numbers in topological models. 8.4 Definition. A Dedekind cut (in the rationals) is a pair (L,U) of subsets of Q which are inhabited, disjoint, closed downwards (resp. upwards), open (in Q), and close together: that is, (i) (ii) (iii) (iv) (V)

3pEL A 3pEU L n u = @ (pqEU -+ pEU) (PEL + 3qEL. q>p) A (PEU -+ 3qEU. q
The conjunction of (i)-(v) we abbreviate as Cut (L,U). The type R of Dedekind reals (defined in IHLN) is the set of such cuts, with order, for example, defined by

R.J. GRAYSON

198

(L,u) < (L',u') iff 3pEu

n

L'.

R is understood to carry its order topology, with basis the rational

open intervals, the rationals being embedded in R by p +b (Iqlqpl). The main point about this notion of "real number" is that its interpretation in O(T)-sets is RT, as we now show; we consider reals given by sequences of rationals in 10.3. In the non-topological case the representation of the reals is not so concrete [FH,521. 8.5 Theorem. The standard interpretation of the type of Dedekind reals in O(T)-sets is isomorphic to the O(T)-set RT, with structure as in 5.2 and 5.5. (For a constructive treatment we regard the external reals as defined in the same way, as Dedekind cuts.) 2 Proof. A s a term, R is interpreted as the predicate C u t on P(Q) ; so, as a type, it is interpreted as the O(T)-set of pairs (L,U) of predicates on with [[ E(L,U) 11 = Cut ( L , U )11 and "extensional" equality (relativised to "Cut"):

6,

(L,u)

=

(L',u')II

=

[[ cutc~,u)ll

A

[[ L=L'IIA

uU=u'i,

the latter being evaluated in P(Q). Now, by Theorem 5.6, every element a of RT determines predicates L a' Ua on Q , for which [[ Cut(La,U )

and

11

Eall, according to

=

[t PEL,]]

=

[[ p
= {tlp
PEU,II

=

[[ a
= {tJa(t)
[[

Conversely, for any predicates L . U .

and

Lt = Ut =

if we set

IpltE"pELl11 IPltE" PEUIII,

we find that, for t€[[ Cut(L,U)l], Cut (Lt,Ut) holds; for example, if t€[[ 3pEL]], then, for some p,tE[[pEL]I, so 3pELt, and so on. Then, if we set a(t) = (Lt,Ut) for t€[[ Cut(L,U)]], we obtain an element a of RT with [[ Ea]] = [[ Cut (L,U)11. that, for all p,q,

To check continuity of a, observe

{tlp
n [tpE~lI n

[[ qEul1

which is open as L,U are predicates. We leave it to the reader to check that the two functions between O(T)-sets defined above constitute an isomorphism in the sense (at the end) of 8.1, which preserves all first-order structure (in particular, order). We now give a similar representation for Baire space in O(T)-sets. 8.6

Theorem. Baire space NN is interpreted in O(T)-sets by

N (N ) T'

Proof. NN is defined as the exponent type ( 7 . 7 ) , which is interpreted by the O(T)-set of all predicates R on ,'?i with [[ E(R) 11 = = [[ R a total functional relation]] and "extensional" equality. Now, as for the reals in 8.5, since NN is topologised by the subbasic opens V = {xlx(m)=n? each element a of (NN)T'determines a n,m

199

Heyting-ValuedSemantics

predicate R a , for which [[ R to [[ Ra(m,n)II

total functional]] =

= [[ Ea]]

,

according

{t(a(t)(m)=nI.

Conversely each predicate R determines a in (NN)T with domain [[ R total functional]] according to a(t) (m)=n iff tE" R(m,n) 11

.

8 . 7 Remarks. a) Theorems 8.5 and 8 . 6 are special cases of a general result for the spaces of models of arithmetically defined infinitary geometric propositional theories; this general theory is described in [FGI. b) In proving 8 . 6 we had to deal-with functional relations rather is far from complete than actual functions from 6 to N. since (6.2-3). Indeed functions from fi to 6 are simply standard, external functions from N to N , which ive rise just to the constant elements of (NN),. But, in case T is Nw itself, there are many non-constant elements, for example, the identity function; it is this richness which is exploited in [Mo,vD] (who however consider only the total elements - see 5.7).

a

5

9.

INTERNAL TOPOLOGIES

In this section we give each O(T)-set X a natural internal topology, as an example of how higher-order srructures are interpreted in our semantics. In the cases of the reals and Baire space we obtain the usual internally defined topologies. This approach is . generalised in [FS,§81 and exploited in [G2] to give results in general topology. 9.1 Notation. Let X and T be topological spaces. We use the variables s,t,... to range over T ; x , ~ , . . . over X ; a r b , over X (3.2); over O(T) ; V , V ' , over O ( X ) : W,W',.. over O ( T x X ) T the U,U',... product topology on TxX.

...

.

...

9.2 Definition. For each open set W in the product topology on TxX we define a predicate W on X T by " ~ E ~ I =I {tj (t,a(t))EwI, this set being open as a is continuous. The O(T)-set O(X ) conT sists of all such predicates, with "extensional" equality defined as for P(X-1 (7.1). In particular, each element is global.

9.3 Proposition. O(X

T

)

is internally a topology on X

Proof. For any W,W' and a in X [[

defining

zflc

I1

=

[[a Ex]]

T'

T

n

[[ a EL'

I1

=

aEznw' 11,

as an abstraction term in the standard way. Hence [[Wnw' = = T , while W' is again an element of O(X T 1 ; thus O(X ) is internally closed under intersections. Also the whole spzce XT is clearly represented by the predicate (E).

a']]

200

R.J. GRAYSON

To verify closure under unions we consider an arbitrary predicate P on O(XT), and define the open set

Wo = U{WplWEO(TxX)

wP

where

=

wn

( r r WEPII XX) .

Then iff iff iff iff Thus

"W0=

UP]] = T , as required. N

special cases we obtain i n this way topologies on R T and ( N )T' which turn out to coincide with the usual topologies defined internally, when we construe them as the internal reals and Baire space as in 8 . 5 - 6 . 9.4

As

This follows fairly immediately from the observation that, if B is a basis for O(X), the sets {(UxV) IVEB, U E O ( T ) 1 form a basis for O(TxX). Then, writing VT for the predicate (E). the elements V , for V in B, form an internal basis for O(XT), since, if (UxV)sWT u g VTGw]3. Now for the reals the basic opens are the rational intervals (p,q), , as basic opens of O ( R T ) ; which get interpreted as ( ~ , q ) ~hence similarly for Baire space. 9.5 Metrics. To show how topological structure on X carries over to X let X be a metric space with metric d, and define

d'

:T'2 XT -+ R T

by d'(a,bl

= At. d(a(t1 ,b(t)).

Thus d' reads off the distance between a and b pointwise, producing a real number in O(T)-sets, by 8.5. It is an easy exercise to check that d' is internally a metric function on X , and that the corresponding. metric topology coincides with that xefined in 9.2. 9.6 Compactness. A topological space is compact iff every open cover has a finite subcover, the constructive sense of "finite" used here being that of "enumerable by the natural numbers less than some natural number".

Proposition. I f Proof

x

is compact, so is XT'

Let P be any predicate on O(X ) and suppose that UP]]; then, by the proof 0 f ~ 9 . 3 , {t}xX 5 U{W,lWEO(TXX)}.

5

Thus for xEX, by the definition of the product topology, we can find U,V,W with tEU, xEV and (UxV)5Wp, that is, UxVcW and Uc_[[WEP]]

.

Now, if X is compact, we can find finitely many such opens U.,V 1 iewi for i=l,. n , with X E iynVi. Then with U = iTnUi we find

..,

that (UxX) tEU

5

at t.

[[ XT

5

iynWi and U

5

n [[ WiEP]l .

i bn iynWi icnwiEPn, thus A

Hence

giving a finite subcover of P

9.7 Remarks. The above proof is constructive. Furthermore, since, by an extension of Theorem 8 . 5 , the closed unit interval [0,11 is ' interpreted in O(T)-sets as [0,1lT, compactness of the unit interval

201

Heyting-ValuedSemantics

is valid in all topological models (when assumed externally); similarly for Cantor space 2N, which is interpreted as (2N)T. These principles are acceptable to intuitionists of the Brouwer school, but regarded as uncertain by the Bishop school of constructivism. Our models show at least that they are constructively "consistent", for example, with the completeness of intuitionistic predicate logic. That they are also inde endent of IHLN is shown by the non-topological counter-models

9.8 Continuous Functions. We take as the constructive definition of a continuous function F between spaces X and Y that, for VEO(Y), F-~(v)Eo(x). We now want to give an external representation of internal continuous functions between spaces X and Y T'

Since YT fs complete (6.1) we can treat arbitrary internal functional relations to YT as external functions ( 6 . 3 ) . So let F : X + YT be a function of O(T)-sets which is total and continuous T

"over U " , that is, U = F total continuous]]. We represent F by a continuous function f : UxX-tY given by the equation

(*I

f(t,a(t))

=

~ ( a(t), ) for t E U n d o m a.

First we must show that f is well-defined by ( * ) , that is, if a(t)=b(t), then F(a) (t)=F(b)(t); for this we need to assume that Y space. is a T Let VEg(Y); then V (as in 9.4) belongs to O(Y ) , so U C F-l(VT) open]], hence we cxn find WEO(TxX) with U 5 [[$=F-l(V )IT. Then we have the following chain of equivalences; for t E U I7 do; a , iff

(t,a(t))EW tE [[ a Ewll

iff

tE[[F(a)EvTl1

iff

F(a) (t)EV.

Thus, if a(t)=b(t), F(a) (t)EV iff F(b) (t)EV, for any VEO(Y); so the T -property ensures ~ ( a(t)=F(b) ) (t). Furthermore, for the special case of a=R, the above equivalences give (t,x)EW iff f(t,x)EV, whence f-l(V) = WEO(TxX); so f is continuous. Conversely, given a continuous f : UxX-tY, the equation ( * ) clearly defines a function F : X + Y T with U = [[ F total]]. Furthermore, for -1

VEO(Y), since W = f (V] is open, we obtain equivalences as above showing that U C_ [[ W=F- (V )]I. But by 9 . 4 the elements VT form a basis for O(YT), so F is ayso continuous over U. This representation allow us to draw "pictures" of arbitrary internal continuous fbnctions as continuous Y-valued surfaces over the TxX-plane. In this way one can draw simple counter-examples to classical theorems such as the Intermediate Value Thebrem, or the attainment of bounds on a closed interval. 9.9 Brouwer's Theorem. The principal result of [Sl] is that, for T the Baire space, the O(T)-set R T satisfies the so-called "Brouwer's Theorem":

All functions from reals to reals are continuous. A

similar result for (NN), is in [Mo,vDI and a result for more

R.J. GRAYSON

202

general spaces T is proved in [G2, 5 his theorem [Br] from stronger forms we discuss in S 11; he obtained also closed intervals, which holds too in such intervals are compact.

8.21. Brouwer himself deduced of continuity r.L;nciple, which uniform continuity on all our models since by 9.6 all

The scheme of Scott's proof is as follows: Given F : R -rR with U = [[ F total]] , define f : U x R + R by the equation ( * ) as in 9.8, now using the special properties of T to show that f is well-defined and continuous. This then implies continuity of F over U , as in 9.8.

5

10

CHOICE PRINCIPLES

A good deal of the "mathematics of H-sets" is now available in the literature [BM, vD, F H , F S , G1-2, MO, Mu, R , Sl], so having outlined the framework of the theory we concentrate on two types of principle of importance in intuitionism, principles of choice and continuity. We will find that validity in all topological models (in particular, in the models over Euclidean spaces) corresponds to a certain kind of continuity property, which conflicts with even the weakest of countable choice principles, denoted by AC-NN: AC-NN: VmEN 3nEN. R(m,n)

+

3f

:

N - + N VmEN. R(m,f(m)).

Thus the theory generally interpreted in these models turns out to be rather different from the traditional intuitionistic one, or that presented in Bishop's book [Bi]. We start however with a " po si tive " re s u 1t

.

10.1 Proposition. In O(NN)-sets the principle of (relativised) Dependent Choices holds: that is, for any O(NN)-set A, DC(A):

VxEA 3yEA. R(x,y) + VxEA 3f:N+A[f(O)=x

A

VmEN. R(f(m),f(m+l))].

Proof. See tMo,S3] for example. The proof uses the property of Baire space that every open cover has a disjoint refinement. It is of interest that this proof can be made constructive assuming Dependent Choices and Bar Induction externally. By contrast with the preceding result for O(NN)-sets, where internal Baire space 1"( is large and rich, we find that the local connecNN N tedness of the reals make (N trivial. R N 10.2 Proposition. Every element of (N )R i s locally constant, so ("1,

is just the completion of the constant set ("1-

(see 6.2 (b))

.

N

Proof. If a is a continuous function from R to N , the image under a of any rational interval (p,q) contained in dom(a) must be connected in N1, hence a singleton; that is, a must be constant on (Pt9). 10.3 Cauchy Reals. Now the reason for real numbers as cuts in the rationals lence classes of) Cauchy sequences of intuitionism and constructivism (e.g.

our choice of definition of (8.4) rather than as (equivarationals, as is more usual in [Bi]), becomes apparent:

Let us define a (Dedekind) real x to be Cauchy iff it can be approximated by a sequence of rationals, that is

203

Heyting-Valued Semantics 3f

:

N + Q VnEN.lx-f(n) I < l/n.

Then in O(R)-sets, since all sequences of natural numbers, hence also of rationals, are (locally) constant, every Cauchy real has to be (locally) constant (as an element of R ) , whereas R has a multitude of non-constant elements, for examplg, the identiay function Xt.t. In particular, since every Dedekind real x clearly does satisfy VnEN 3qEQ.Ix-ql
u c_

R f converges to all,

and fE(QN),

satisfies

then, for each tEU, f(t) converges to a(t), which equals t. Thus we must have found an approximating sequence f(t) continuously in the parameter t , and this is what 10.2 shows to be impossible. Thus failure of AC-NN over R is seen to fact that (even classically) one cannot quence to each real number continuously The relevance of local continuity comes example.

correspond to the simple choose an approximating sein the real (even locally). out more clearly in the next

10.5 Roots of Cubics. A well-known fact from elementary analysis is that one cannot find a total continuous function of t giving a root of the cubic x3-x+t, because one has to make a "jump" somewhere. On the other hand, one can easily choose a root locally continuously, in the sense that, for each t , there is a neighbourhood of t on which one can choose a root continuously.

What is perhaps less well-known is that one cannot choose a root of the cubic x3+sx+t even locally continuously in both s and t. A s the picture below of the surface x3+sx+t=0 over the (s,t)-plane indicates, there is no continuous choice of root on any neighbourhood of the origin.

27

To interpret this in our models: the parameter space is now R 2 and we have two "generic" elements of R 2 , a=X(s,t) . s and b=X(s,t) .t. The failure of continuity in parameaers then shows that 3 ( 0 , O ) B [[ 3x. x +ax+b=Oll,

R.J. GRAYSON

204

2 so that, in O ( R )-sets, the reals are not real-closed (in the simplest sense); on the other hand, the principle AC-NN suffices to prove real-closure. 10.6 A Derived Rule. We can also apply the above considerations in a more positive direction to any assertion of the form 3y. R(x,y), where R is polynomial equation. If this is valid in all topological models, it must be valid in the model over the parameter space of the parameters s , and hence a solution y must exist locally continuously in the parameters. Furthermore, all this is provable in the system IHLN, which leads to a derived rule of local continuous choice for this system ; we prove a general result of this form for arbitrary R in 11.5, but it seems helpful to have these simple examples in the background for motivation. On the other hand our cubic example shows that AC-NN prevents such a derived rule even for the special case of polynomial equations.

Vz

9

11

CONTINUITY PRINCIPLES

Among the most positive and "anti-classical" tenets of intuitionism are continuity principles of various degrees of strength. In this final section we indicate to what extent these can be interpreted in our models, and prove a derived rule of local continuous choice for our basic system IHLN using only the soundness theorem for topological models ( a proof due to Joyal). 11.1 Weak Continuity. We have already seen that the continuity of all functions from reals to reals, or from Baire space to Baire space, holds in certain topological models ( 9 . 9 ) . In intuitionistic treatments these are sometimes derived from the principle of weak continuity for N ~ WC, , namely wc: -

VUENN 3nEN. A(a,n) -t VUENN 3 m , n E N vgENN[6(m)=p(m) -tA(a,n)1

In topological terms this says that every countable cover of Baire space has an open refinement. In case the formula A is a formula of analysis without parameters other than a , WC can be shown (classically) to be valid in O(NN)-sets [vD]. In its full generality however WC can never hold in topological models [ G Z , s 8.11. It is an open question whether WC might hold over some non-topological cHa, but Krol' [K] has given a permutation submodel of the "full" model over O(NN) in which WC is valid (see also [ G Z , Appendix]). 11.2 Continuous Choice. A much stronger group of principles are those of continuous choice, which we may formulate generally as CC(X,Y) for (definable) spaces x and Y. VxEX 3yEY. A(x,y) CC(X,Y) : 3f

:

X

*

N

+

Y VxEX. A(x,f ( X I .

The special case CC(N ,NN) is also known as Va 3B-continuity. This is inconsistent with Kripke's Schema and hence fails in the model . of Krol'. Fourman [F2] has shown how to model this principle in sheaves over a site; as worked out in [HM] one obtains in this way in fact a model for the f u l l theory CS of choice sequences. The

205

Heyting-Valued Semantics

consistency of such strong principles had of course already been shown by other means, for example, realisability [Tr]. On the other hand, realisability does not appear so useful in dealing with continuity principles for the real numbers, which we now consider. 11.3 Local Continuous Choice. The principle CC(R,R) is simply inconsistent as is shown by the first example f 10.5, since one may easily prove (without AC-NN) that Vt 3x.xq-x+t=0. The more traditional counter-example is given by the provability of VxER jnEN.x
+

%Y

Vx'EU.A(x',f(x')).

Fourman's models in sheaves over sites IF21 show the consistency (relative to IHLN) of LCC(X,Y) for any complete separable metric spaces X,Y (definable in IHLN). On the other hand, the cubic example of 10.5 shows LCC(Rp) to be inconsistent with AC-NN. Discussion of the relations between various continuity principles (in the presence of countable choice) may be found in [Be]. 11.4 Derived Rules. T o each continuous choice principle CC(X,Y) or LCC(x,Y) there corresponds a continuous choice rule, that provability of the hypothesis implies provability of the conclusion. We denote these rules by CCR(X,Y) and LCCR(X,Y). Derived rules of this kind are proved in [Be], using realisability, and [HI] by proof-theoretic means, for various systems. We give here an exceedingly elegant proof of LCCR(X,Y) for IHLN, for any definable complete separable metric spaces X,Y, which is due essentially to Joyal and uses just the provable soundness (7.8) of topological models for IHLN; other applications of this technique are t o appear in [FJ]. The addition of AC-NN prevents LCCR(R,R), of course, as noted in 10.6. 11.5 Theorem. (Joyal, Hayashi) The system IHLN is closed under the rule LCCR(X,Y) for any definable (provably) complete separable metric spaces X,Y. Proof. This will be an informal proof within IHLN starting from the ( c 1 0 s ed) as sumption

1- VxEX 3yEY.A(x,y). By the soundness theorem, provable in IHLN ( 7 . 8 ) , this is provably valid in all topological models. We now define the particular space T , over which we want to use this validity: T = Xx{O,l]

and

wo w1

with open sets those WST for which

= {xExl<x,o> =

E wl

{XEXl<X,l> E

E O(X)

w} 2 wo

This has the effect that X is homeomorphic to the closed subspace T =Xx{O} of T , while X disc (the set X with the discrete topology) is homeomorphic to the open subspace T =Xx{l). as a "glueing" of x disc to X along the'identiy

T can be regarded map.

The assumptions on X and Y ensure that, just as for R and NN in 8.5 and 8.6, when we interpret the definitions of X and Y in O(T)-sets,

R.J. GRAYSON

206

we obtain the O(T)-sets XT and Y with topologies as in 9.2. (The general theory appears in [FG, 3T81.1 Now we apply the internal validity of VxEX 3yEY.A(x,y), in O(T)-sets, to the projection a=(X<x,i>.x), which belongs to XT and so is treated internally as an "element of X". Then we get [[ 3bEYT.A(a,b)]1 = T, hence, for any point x of X, we may find b in YT with <x,O> E [[ A(a,b)]]

fI dom(b) = W ,

where we may suppose without I D S S that W =W,=U€O(X). W e next want to transfer into the universe over X disc, i.e. P(X)-sets,

a) Since

into

for which we need three observations. Xdisc is (homeomorphic to) the open subspace T 1 of T,

evaluations [[

lldisc

in P(X)-sets

are obtained simply by

"restriction" to T 1 of evaluations over O(T)

.

b) Since Y is a T1-space, for any x'EU, we find that b(<x',O>) = b(<x',l>), so that the restrictizn of b to T gives rise to a continuous function 2 from U to Y with b(x') = b(<xl , l > ) . Similarly the restriction of a gives rise simply to the identity function id on X. Hence (a) yields 'disc

C -

A

[[ A(id,b)lldisc

-

c) Finally, interpretations of formulae over discrete spaces are always obtained "pointwise" in terms of "external truth", in particular [[ A(id,b)]Idisc

= {x'EXIA(id(x') ,%(xu))).

Together with (b) this shows thzt we have found a neighbourhood U of x and a continuous function b:U+Y such that Vx'EU. A(x',%(x')), giving the conclusion of LCCR(X,Y).

REFERENCES. [Be] M. Beeson: Principles of continuous choice, Annals of Math. Logic 1 2 (19771, 249-322 [Bi] E. Bishop: Foundations of constructive analysis, McGraw-Hill, 1967. [ B r ] Brouwer's Cambridge Lectures on intuitionism, ed. D. van Dalen, Cambridge University Press, 1981.

[BM]

Burden and C . Mulvey: Banach spaces in categories of sheaves, in Applications of Sheaves, Springer Lecture Notes 753 (1979), 169- 196.

C.

[vD] D. van Dalen: An interpretation of intuitionistic analysis, Annals of Math. Logic 1 3 (19781, 1-43.

207

Heyting-Valued Semantics

[D]

M. Dummett: Elements of intuitionism, Oxford University Press, 1977.

[Fi] M.P. Fourman: The logic of topoi, & Handbook of Mathematical Logic (ed. J. Barwise), North-Holland, 1977, 1053-1090 [F2]

------------:

Continuous truth, to appear (1982).

e Proceedings

[FG] M.P. Fourman and R.J. Grayson: Formal spaces, of the Brouwer Symposium, North-Holland, 1982.

[FH] M.P. Fourman and J.M.E. Hyland: Sheaf models for analysis, & Applications of Sheaves, Springer Lecture Notes 753 (1979), 280-301. [FJ] M.P. Fourman and A. Joyal: Metamathematical applications of sheaf theory, to appear. [FS] M.P. Fourman and D.S. Scott: The logic of sheaves, Applications of Sheaves, Springer Lecture Notes 753 (1979). 302-401. [Gl] R.J. Grayson: Heyting-valued models for intuitionistic set-theory, 9Applications of Sheaves, Springer Lecture Notes 753 (1979), 402-414. [G2]

- - - - - - - - - - - - :C oncepts of general topology in constructive mathematics and in sheaves, Annals of Math. Logic 20 (19811, 1-41. Ditto, 1 1 , to appear in the Annals of Math. Logic.

[G3]

------------:

[G4]

. Constructive properties of complete Heyting algebras and related structures, pre-print (1982).

[Hl]

S . Hayashi: Derived rules related to a constructive theory of metric spaces, Annals of Math. Logic 19 (19801, 33-65.

[H2]

- - - - - - - - - - :A

F orcing in intuitionistic systems without powerset, to appear in Journal of Symbolic Logic (1981).

note on the bar induction rule, & Proceedings of the Brouwer Symposium, North-Holland, 1982.

[He] A. Heyting: Intuitionism, An Introduction, North-Holland, 1956. rHm3 G. van der Hoeven and I. Moerdijk: Sheaf models for choice sequences, pre-print (1982). [Hy] J.M.E. Hyland: Aspects of constructivity in mathematics, Oxford Logic Colloquium '76 (eds. Gandy and Hyland), NorthHolland, 1977. [K]

M.D. Krol': A topological model for intuitionistic analysis with Kripke's Schema, ZMLG 24 (19781, 427-436.

[MR] M. Makkai and G. Reyes: First-order categorical logic, Springer Lecture Notes 611, 1977. [MDI R. Mansfield and J. Dawson: Boolean-valued set theory and forcing, Synthese 3 3 (1976), 223-252.

208

R.J. GRAYSON J.R. Moschovakis: A topological interpretation of second-order intuitionistic arithmetic, Comp. Math. 26 ( 1 9 7 3 ) , 2 6 1 - 2 7 5 . C.J. Mulvey: Intuitionistic algebra and representations of rings, in Mem. Amer. Math. S O C . 1 4 8 ( 1 9 7 4 1 , 3 - 5 7 . H. Rasiowa and R. Sikorski: The mathematics of metamathematics, Warsaw, 1 9 6 3 . Rousseau: Topos theory and complex analysis, in Applications of Sheaves, Springer Lecture Notes 7 5 3 ( 1 9 7 9 ) , 6 2 3 - 6 5 9 .

C.

D.S. Scott: Extending the topological interpretation to intuitionistic analysis - I , Comp. Math. 2 0 ( 1 9 6 8 1 , 1 9 4 - 2 1 0 . - 11, & Intuitionism and Proof Theory (eds. Kino, Myhill, Vesley), North-Holland ( 1 9 7 0 ) .

_--______-.

. Identity and existence in intuitionistic logic, Applications of Sheaves, Springer Lecture Notes 7 5 3 ( 1 9 7 9 ) , 660-696.

A. Tarski: Der Aussayenkalkul und die Topologie, Fund. Math. 3 1 ( 1 9 3 8 ) , 1 0 3 - 1 3 4 . A.S. Troelstra: Metamathematical investigation of intuitionistic arithmetic and analysis, Springer Lecture Notes 3 4 4 ( 1 9 7 3 ) .

Address for correspondence: Church Cottage, Benenden, Cranbrook, Kent, England.

LOGIC COLLOQUIUM '82 G . Lolli, C.Long0 and A. Marcia [editors) 0 Elsevier Science Publishers 8.V. (North-Holland), 1984

209

LAMBDA C A L C U L U S ANU I T S MODELS

Henk Eatendhegt M a t h e m a t i c a l I n s t i t u t e , Budapestlaan 6 3508 TA U t r e c h t , The N e t h e r l a n d s .

INTRODUCTION The Lambda c a l c u l u s was i n t r o d u c e d by Church around 1930 as a f o r m a l t h e o r y about r u l e s ( i . e .

f u n c t i o n s as g i v e n by a l g o r i t h m s ) .

The r e l a t e d t h e o r y o f c m b i -

n a t o r s was i n i t i a t e d by S c h S n f i n k e l and C u r r y s m e y e a r s e a r l i e r . The t h e o r y was c o n c e i v e d as t y p e f r e e : a l l o b j e c t s can be used b o t h as argument and as r u l e t o be a p p l i e d t o o t h e r o b j e c t s . Perhaps t h e subconscious w i s h was t o have a u n i v e r s e

U such t h a t a l l ( o r a t l e a s t many) f u n c t i o n s f r o m U t o U b e l o n g t o U. S i n c e by Cantors theorem t h e c a r d i n a l i t y o f Uu i s l a r g e r than t h a t o f U,

i t was n o t c l e a r

how t o c o n s t r u c t such a U. I n s p i t e o f t h i s , t h e r e were i n t e r e s t i n g r e s u l t s i n t h e s u b j e c t . Kleene showed t h a t t h e r e c u r s i v e f u n c t i o n s can be r e p r e s e n t e d i n t h e A - c a l c u l u s .

Rosser c l a r i -

f i e d t h e r e l a t i o n between t h e A - c a l c u l u s and t h e t h e o r y o f c o m b i n a t o r s . The cons i s t e n c y o f t h e A - c a l c u l u s was p r o v e d v i a t h e Church-Rosser theorem. As a consequence o f t h i s c o n s i s t e n c y t h e r e a r e t h e open o r c l o s e d t e r m models c o n s i s t i n g o f t h e open or c l o s e d terms modulo p r o v a b l e e q u a l i t y . n a t o r s , e.g.

I n t e r e s t i n g work on t h e c m b i -

by BZjhm and h i s s c h o o l , can be viewed as r e s u l t s on t h e t e r m models.

I n 1969 S c o t t c o n s t r u c t e d n o n - s y n t a c t i c a l models o f t h e A - c a l c u l u s . A l t h o u g h t h e f u l l f u n c t i o n space Uu c a n n o t be i s o m o r p h i c t o U, some s u b s e t can be, e.g. t h e s e t o f c o n t i n u o u s f u n c t i o n s w i t h r e s p e c t t o sane c o n v e n i e n t t o p o l o g y . Because o f S c h l i n f i n k e l s i d e n t i f i c a t i o n o f Uuxu w i t h (UU)'

i t i s n a t u r a l t o use a class o f

t o p o l o g i c a l spaces t h a t f o r m a C a r t e s i a n c l o s e d c a t e g o r y ( c c c ) . F o r t h i s reason S c o t t worked w i t h i n t h e c a t e g o r y o f -with c o n s t r u c t e d an o b j e c t ,D

i s o m o r p h i c t o , ,D:

c o n t i n u o u s maps and thus y i e l d i n g an e x t e n s i o n a l model o f

the A-calculus. Some r e l a t e d C a r t e s i a n c l o s e d c a t e g o r i e s a r e a l s o o f importance. F i r s t t h e c o n t i n u o u s l a t t i c e s have a more n a t u r a l r e l a t i o n between t h e i r l a t t i c e s t r u c t u r e and t o p o l o g y ; e.g.

the topology o f a product i s the product o f the respective

t o p o l o g i e s , something t h a t i s f a l s e f o r c o m p l e t e l a t t i c e s . Then t h e r e a r e t h e ( c p o ' s ) o f w h i c h t h e r e a r e many more than t h e complete l a t t i c e s . P l o t k i n s model T W i s a cpo and n o t a c o m p l e t e l a t t i c e . Another u s e f u l c a t e g o r y i s t h a t o f fo-spaces as d e f i n e d by Ershov. These o b j e c t s have t h e advanP

H. BARENDREGT

210

t a g e o f n o t h a v i n g t o be complete,

e.g.

the set o f r.e.

s e t s p a r t i a l l y o r d e r e d by

i n c l u s i o n i s an f -space.

0

I t t o o k some t i m e a f t e r S c o t t gave h i s model c o n s t r u c t i o n u n t i l t h e r e was an agreement what i s t h e g e n e r a l n o t i o n o f a model o f t h e A - c a l c u l u s .

See Koymans

[1983] f o r t h e h i s t o r y . P r e s e n t l y one c o n s i d e r s two k i n d s o f models, v i z . t h e A a l g e b r a s and t h e A-models.

The A - a l g e b r a s s a t i s f y a l l p r o v a b l e e q u a t i o n s o f t h e

A - c a l c u l u s and form an e q u a t i o n a l c l a s s ( a x i o m a t i z e d by k x y = x , s x y z = x z ( y z ) and t h e f i v e c o m b i n a t o r y axioms o f C u r r y ) . T h e r e f o r e t h e A - a l g e b r a s a r e c l o s e d under s u b s t r u c t u r e s and homomorphic images. The A-models on t h e o t h e r hand s a t i s f y a l l p r o v a b l e e q u a t i o n s and moreover t h e a x i o m o f weak e x t e n s i o n a l i t y Vx(M=N)

-*

Ax.M=Ax.N.

I t t u r n s o u t t h a t A-models can be d e s c r i b e d by some f i r s t o r d e r axioms, b u t n o t

by e q u a t i o n s .

Indeed A-models a r e n o t c l o s e d u n d e r s u b s t r u c t u r e s n o r u n d e r homo-

morph i c images. Next t o t h e f i r s t o r d e r d e f i n i t i o n o f A - a l g e b r a s and A-models,

there i s a

s y n t a c t i c a l and a l s o a c a t a g o r i c a l d e s c r i p t i o n o f t h e s e c l a s s e s . The s y n t a c t i c a l d e s c r i p t i o n i s c o n v e n i e n t when c a l c u l a t i n g t h e i n t e r p r e t a t i o n o f terms i n a model The c a t e g o r i c a l d e s c r i p t i o n o f A - a l g e b r a s

i s r a t h e r n a t u r a l and u n i f i e s t h e two

I t c o n s i s t s o f a C a r t e s i a n c l o s e d c a t e g o r y t t o g e t h e r w i t h a so c a l l e d U r e f l e x i v e o b j e c t U E Q , i . e . U i s a r e t r a c t o f U: t h e r e a r e maps F:U+Uu and

concepts.

G:UU+U

such t h a t FOG = idUU. A s shown i n Koymans [19831,

i n t h i s context a

A-

model i s a A - a l g e b r a t h a t a r i s e s f r o m a c a t e g o r y t w i t h an o b j e c t U t h a t has "enough po i n t s"

.

Because of t h e p r e s e n t d e s c r i p t i o n o f lambda c a l c u l u s models, c h a p t e r

5 of

B a r e n d r e g t 119811 becomes somewhat o u t o f d a t e . T h i s p a p e r may be c o n s i d e r e d as a replacement o f t h a t c h a p t e r . U s i n g t h e c a t e g o r i c a l d e s c r i p t i o n o f t h e A - c a l c u l u s models, S c o t t [ 1 9 8 0 ] makes t h e f o l l o w i n g p h i l o s o p h i c a l remarks. 1 . The models f o r t h e t y p e f r e e A - c a l c u l u s come f r o m c c c ' s w i t h a r e f l e x i v e o b j e c t . The

CCC'S

-

themselves c o r r e s p o n d t o t h e t y p e d A - c a l c u l u s .

There-

f o r e t h e t y p e d A - c a l c u l u s has p r i o r i t y o v e r t h e t y p e f r e e t h e o r y .

2. Let

E

be a ccc w i t h r e f l e x i v e o b j e c t U. By t h e Yoneda lemna d: can be em-

bedded i n t o a t o p o s D = Setcop, U s i n g t h e K r i p k e - J o y a l semantics, i n s i d e PJ i t i s s a t i s f i e d t h a t Uu i s t h e f u l l f u n c t i o n space o f U and t h e r e f o r e

t h e axiom o f weak e x t e n s i o n a l i t y i s s a t i s f i e d by U i n

ID. The p r i c e one

has t o pay i s t o use i n t u i t i o n i s t i c l o g i c , s i n c e c l a s s i c a l l o g i c i s n o t sound f o r t h e K r i p k e - J o y a l

interpretation.

Some comnents. A s t o 1 , t h e r e a r e c e r t a i n l y n i c e r e s u l t s i n t h e t y p e d A - c a l c u l u s , f o r i n s t a n c e Statman [19801,

[19821. However we d i s a g r e e w i t h S c o t t s grounds f o r

21 1

Lambda Calculus and its Models

c o n c l u d i n g t h a t t h e typed t h e o r y has p r i o r i t y o v e r t h e t y p e f r e e one. Even i f t h e r e a r e f o r example more semigroups t h a n groups,

i t does n o t f o l l o w t h a t t h e

t h e o r y o f semigroups i s more fundamental t h a n t h e t h e o r y o f groups. As t o 2, S c o t t s s u g g e s t i o n t o make t r u e t h e o l d dream o f Church and C u r r y , namely U u E U , i n s i d e a topos,

i s indeed v e r y i n t e r e s t i n g . One has t o w a i t and see what a p p l i -

c a t i o n s t h i s can g i v e . T h a t A - a l g e b r a s a r e i n t e r n a l l y a l r e a d y A-models does n o t mean t h a t t h e c l a s s o f A - a l g e b r a s

i s d e v o i d o f i n t e r e s t . Compare t h i s w i t h t h e

n o t i o n o f a r e g u l a r r i n g . Viewed i n s i d e a t o p o s , r e g u l a r r i n g s a r e ( i n t u i t i o n i s t i c ) f i e l d s . B u t r e g u l a r r i n g s m e r i t a t t e n t i o n b y themselves and n o t j u s t as g l o b a l s e c t i o n s o f a f i e l d i n a topos. The same a p p l i e s t o A - a l g e b r a s . models a r e A - a l g e b r a s ,

b u t n o t A-models

Closed term

i n g e n e r a l , due t o w-incompleteness.

N e v e r t h e l e s s t h e s e s t r u c t u r e s have i n t e r e s t i n g p r o p e r t i e s , e.g.

t h e y a r e precom-

p l e t e numbered s e t s i n t h e sense o f Ershov, see V i s s e r [19801. N o t a t i o n s and r e f e r e n c e s t h a t a r e n o t g i v e n i n t h i s paper may be found i n B a r e n d r e g t [19811. I n p a r t i c u l a r we u s e t h e v a r i a b l e c o n v e n t i o n t h a t i d e n t i f i e s terms d i f f e r i n g o n l y i n t h e names f o r t h e i r bound v a r i a b l e s (e.g.

Xx.x=hy.y)

and r e q u i r e s t h a t a bound v a r i a b l e i n some m a t h e m a t i c a l c o n t e x t i s d i f f e r e n t f r o m the free variables i n t h a t context.

Sopfie

a b s t r a c t i o n i n t h e rneta language.

I thank K a t L b t KoljmuMn f o r many u s e f u l d i s c u s s i o n s on t h e sub-

Acknowledgements. j e c t and

1 denotes

WUM

Stakenbu%

51.

COMBINATORY ALGEBRAS

1.1

DEFINITION.

(i)

I=(X,.)

f o r her carefu1,nice typing o f the manuscript.

i s an a p p l i c a t i v e s t r u c t u r e i f

.

i s a b i n a r y oper-

a t i o n on X. ( i i ) Such a s t r u c t u r e i s e x t e n s i o n a l i f f o r a , b E X one has (VxEX

Notation.

-+

( i ) As i n a l g e b r a , a.b

( i i ) If

a=b.

i s u s u a l l y w r i t t e n as ab.

-+

If b=bl,

..., bn.

then

.. ( a b l ) b 2 . . . b n ) .

a b = a bl...bn=(

1.2

*

a.x=b.x)

I=

(X,.)

DEFINITION. L e t

t h e n we w r i t e a E W i n s t e a d o f a E X .

I be

an a p p l i c a t i v e s t r u c t u r e .

( i ) The s e t o f terms o v e r m , n o t a t i o n S b ) , i s i n d u c t i v e l y d e f i n e d as f o l l o w s . vo’vl’v2’‘’’ aE l A,BES@) N o t a t i o n . A,B,

. . . denote

E

SC)

*

ca

+

(AB)ESC)

€%(I)

(va r i ab 1 es ) (cons t a n t s )

a r b i t r a r y terms and x,y,

...

arbitrary variables i n

Sh).

H. BARENDREGT

212

ID?

( i i ) A valuation i n

i s a map p : v a r i a b l e s + 1.F o r a v a l u a t i o n p i n m t h e

i n t e r p r e t a t i o n o f A € % @ ) i n m under p ( n o t a t i o n (Ap'or

(A)

P

P

o r (A)m i f m o r p

i s c l e a r from t h e c o n t e x t ) i s i n d u c t i v e l y d e f i n e d as u s u a l :

( i i i ) A=B i s true i n U ! t under t h e v a l u a t i o n p ( n o t a t i o n m , p C A = B ) i f

m

n

( A l p = (B)p. i s t r u e i n J!R ( n o t a t i o n 9l

(iv) A = B

-

( v ) The r e l a t i o n

b A = B ) i f 8 ? , p I= A = B f o r a l l v a l u a t i o n s p.

i s a l s o used f o r f i r s t o r d e r f o r m u l a s over!U?. The d e f i -

n i t i o n i s as u s u a l . FV(A) i s t h e s e t o f ( f r e e ) v a r i a b l e s i n A. v a l u e s of p on FV(A). a t i o n (A)

1.3

P

C l e a r l y (A)

I n p a r t i c u l a r f o r c l o s e d A ( i . e . FV(A)

P

depends o n l y on t h e

=0)

the interpret-

i s independent o f p and may be denoted by ( A ) .

DEFINITION ( C u r r y ) . An a p p l i c a t i v e s t r u c t u r e 1 i s a c o m b i n a t o r y complete i f

f o r e v e r y A € % @ ) and xl...xn 3 f Vx,

...xn

w i t h FV(A)5{xl

f x l...x

,...,x

one has i n 1

= A

Note t h a t an e x t e n s i o n a l a p p l i c a t i v e s t r u c t u r e i s c o m b i n a t o r y complete i f f f o r a l l A € % @ ) one has

fz = A(;).

3 1 f Vz

1.4

NOTATION.

( i ) L e t p be a v a l u a t i o n i n m and l e t a E 1 . Then p ( x : = a )

i s the

valuation p' with

P'(x) = a

,

~ ' ( y )= P ( Y ) (ii)

if ygx.

+

-f

If x = x

a r e d i s t i n c t and a = a , , lI-.'Xn p ( x : = a ) = p(xl :=al) (X : = a 1. n n

( i i i ) A[x:=B]

1.5

...,an,

then

...

-f

i s t h e r e s u l t o f s u b s t i t u t i n g t h e t e r m B f o r x i n A.

LEMMA. L e t m b e an a p p l i c a t i v e s t r u c t u r e and A , A ' , B , B ' € S @ ? ) . ( i ) (A[x : = B I ) ,

=

:= (B)p)

(ii)wCA=A'hB=B' Proof.

(i)

Then

* 1 k

A[x:=BI

=A'[x:=B'].

I n d u c t i o n on t h e s t r u c t u r e o f A.

( i i ) By a s s u m p t i o n (A) (A[x : = B l )

P

=

P

=(A')

P

and (8)

,

:= (B)p)

= (A')p(x:=

(Bl)

P

= (8')

P

for a 1 p.

by ( i ) , = (A'[x:=B']) P

P

I t follows t h a t

213

Lambda Calculus and its Models

0

and we a r e done.

1.6

DEFINITION. L e t m = (X,.)

be an a p p l i c a t i v e s t r u c t u r e and l e t cp: X " + X

be

a map. ( i ) cp i s r e p r e s e n t a b l e o v e r !IN i f 3 f E X

V z € X"

+

+

f a = cp(a).

( i i ) cp i s a l g e b r a i c o v e r m i f t h e r e i s a t e r m A€X(!IR) w i t h

. . .x,l such 6 cp(2) = (A)p(;:=;).

FV(A) z { x l , . (1)

that

( C l e a r l y ( 1 ) does n o t depend on p ) .

Combinatory completeness says that all algebraic functions are representable. The converse is trivial. Schonfinkel showed that combinatory completeness follows from two of its instances. 1.7

DEFINITION. A cornbinatory a l g e b r a i s an a p p l i c a t i v e s t r u c t u r e 9 J = (X,.,k,s)

w i t h d i s t i n g u i s h e d elements s a t i s f y i n g k y = x sxyz = x z ( y 2 ) .

1.8

DEFINITION. L e t m be a c o m b i n a t o r y a l g e b r a . ( i ) Define the f o l l o w i n g constants: K = c k , S = c s , I ( i i ) For AE%(M) and a v a r i a b l e x , d e f i n e h * x . A E S m )

= c i with i

=

skk.

i n d u c t i v e l y as f o l l o w s :

,

X*X.X

= I

X*x.P

= KP,

i f P i s a v a r i a b l e g x o r a constant,

h*x.PQ = S(X*x.P) (h*x,Q).

+

( i i i ) L e t x = x,

1.9

, . . . ,x .

PROPOSITION. ( i ) FV(X*x.A) ( i ) (A*x.A)x=A, ( i i ) (X*;.A)z=A,

Proof. -

(i), (ii).

(ii )

ay

+

Then A*x.A

= ( h*xl

= FV(A)

. . . (X*xn.A).

. .) .

- {XI.

i n e v e r y cornbinatory a l g e b r a . i n e v e r y cornbinatory a l g e b r a .

I n d u c t i o n on t h e s t r u c t u r e o f A. Note t h a t l x = S K K x = K x ( K x ) = x .

(ii).

1.10 THEOREM. An applicative structure m i s combinatory completE iff it can be expanded to a combinatory algebra [by choosing k,s). Hence every combinatory algebra is combinatory complete.

Proof. By proposition 1.9(iii).

0

214

H. BARENDREGT REMARKS.

1.11

(i.e.

( i ) Note t h a t a c o m b i n a t o r y a l g e b r a % = (X,.,k,s)

iff k f s .

Card(W)>l)

i s non t r i v i a l

Indeed, k = s i m p l i e s a = s ( k i ) ( k a ) z = k ( k i ) ( k a ) z = i

f o r a l l a, S O W i s t r i v i a l ( i i ) When c o n s i d e r i n g c o m b i n a t o r y a l g e b r a s , we u s u a l l y t a c i t l y assume t h a t t h e y a r e non t r i v i a l .

1.12 D E F I N I T I O N .

Then c p : X

+X2

Let2Qi = ( X i , . i , k i , s i ) ,

i = 1 , 2 , be two combinatory a l g e b r a s .

i s a homomorphism ( n o t a t i o n cp:Wl+211)

2

1 and k and s , i .e. cp(x.,y)

i f cp p r e s e r v e s a p p l i c a t i o n

cp(y), cp(kl) = k 2 and cp(sl) = s 2 .

= cp(x).,

( i i) I +I2 l i f cp : w1 +I2 f o r some cp. i s ernbeddable i n m 2 (9111GW2) i f cp:YJ?,+?D2 f o r some i n j e c t i v e cp.

( i i i ) Wl (Wl (iv)

i s a s u b s t r u c t u r e o f 2112 (Ilc2R2)

W,

i f cp:Wl+W2

w i t h cp t h e i d e n t i t y . )

i s i s o m o r p h i c t o m 2 @J?l~9112)i f c p : 1 1 + W 2 f o r some b i j e c t i v e cp.

1.13 DEFINITION. ( i ) Q i s t h e s e t o f terms o f c o m b i n a t o r y l o g i c ,

i.e. applicative

terms b u i l t u p from v a r i a b l e s and K , S o n l y . Go = { P € Q l F V ( P ) = 0 } . ( i i ) L e t I be a c o m b i n a t o r y a l g e b r a . Then T h m ) = { P = Q l % C P = Q ,

1.14 PROPOSITION. L e t UJ:W,+SJI,. '

I

P,Q€&

0

1.

Then f o r P , Q € % @ ? * )

L

I 1 ( i ) cp(l[PJJ ) = [cp(P)? , w h e r e cp(P) r e s u l t s i r o m P by r e p l a c i n g t h e conP WP stants c a by 'cp(a). 0 ( i i )W, P = Q * W + s ( P ) = q ( Q ) , provided P , Q E & o r cp i s s u r j e c t i v e .

+

( i i i ) Th(9111)

2

Thb2).

( i v ) Th(9111) = Th@J$), Proof. -

p r o v i d e d t h a t cp i s i n j e c t i v e .

( i ) I n d u c t i o n on t h e s t r u c t u r e o f P€X(9ll).

k P=Q

( i i ) Wl

* *

-

[ P I p = UQn,

UPII,,

=

* W2 C I f P,Q€Q

0

f o r a l l p,

UPDcpop= ~ Q ~ c p of po r a l l P by ( i ) ,

[Qn,,

f o r a l l p ' i f cp i s s u r j e c t i v e ,

P=Q.

t h e n t h e i r v a l u e s do n o t depend on a p.

( i i i ) By ( i i ) .

0

( i v ) As f o r ( i i ) .

The f o l l o w i n g r e s u l t i s due t o Grzegorczyk.

1.15 THEOREM. Consider t h e f o l l o w i n g f i r s t o r d e r t h e o r y (CL) i n t h e language o f combinatory algebras.

1

Vxy

(CL)

Kxy=x

,

vxyz sxyz = xz(yz) K+S.

,

215

Lambda Calculus and its Models

( i ) (CL) i s e s s e n t i a l l y u n d e c i d a b l e ,

i .e. has n o

consistent decidable

extension. ( i i ) (CL) has no r e c u r s i v e models. Proof. -

(i)

I f T i s a consistent extension o f

A = IP

I

( C L ) , then

P=SET}

i s a non t r i v i a l s e t ( S E A , K ~ A o) f terms c l o s e d under p r o v a b l e e q u a l i t y . But then as i n B a r e n d r e g t [19811, theorem 6 . 6 . 2 ( i i )

i t f o l l o w s t h a t A and t h e r e f o r e

T i s not recursive. then T h b ) = { P = Q l W t = P = Q I i s a

( i i ) I f W i s a r e c u r s i v e model o f (CL), r e c u r s i v e c o n s i s t e n t e x t e n s i o n o f (CL),

contradicting ( i ) .

The axioms f o r c o m b i n a t o r y a l g e b r a s a r e i n s p i r e d by t h e a n a l y s i s o f r e c u r s i v e p r o c e s s e s , n o t by a l g e b r a . The f o l l o w i n g shows t h a t t h e s e s t r u c t u r e s a r e i n fact algebraically pathological.

1.16 PROPOSITION. Combinatory a l g e b r a s ( e x c e p t t h e t r i v i a l one) a r e ( i ) n e v e r commutative, ( i i ) never a s s o c i a t i v e , ( i i i ) never f i n i t e , ( i v ) never recursive. Proof. -

( i ) Suppose i k = k i . Then k = i k = k i , hence a = k a b = k i a b = i b = b f o r a l l a,b

and t h e a l g e b r a i s t r i v i a l . ( i i ) S i m i l a r l y t r i v i a l i t y follows from ( k i ) i = k ( i i ) . = kk,.

( i i i ) D e f i n e kl = k , kn+l ( i v ) By l . l s ( i i ) .

Then t h e kl,k2

,...

are a l l distinct.

0

The f o l l o w i n g r e s u l t , due t o B a r e n d r e g t , Dezani and K l o p , shows t h a t combinatory algebras a r e universal f o r recursive a p p l i c a t i v e structures.

1.17 THEOREM. Given a c o m b i n a t o r y a l g e b r a 8. Then e v e r y r e c u r s i v e a p p l i c a t i v e s t r u c t u r e !XI can be embedded i n t o 8. Proof. Let -

?I be g i v e n and !XI = ( I N , . )

a

= [A,'n']

with

.

recursive.

D e f i n e i n 91

= A*z.zArn'

w i t h A E B t o be d e t e r m i n e d and ' n ' E 8

t h e nth numeral. Then

a a = [A,'n'l[A,'m'] n m = [ A , rml]Arn' = AArml'n'

= [A,F'n"m''], = [A,'n.m'],

provided A=A*pqr.[p,Frq], provided F represents

. ,

216

H. BARENDREGT

.

= a n .m The e x i s t e n c e o f F f o l l o w s f r o m t h e A - d e f i n a b i l i t y : %+‘I1

Moreover hn.a a

= a

n

m

o f the recursive functions.

i s injective:

+ ‘n’

= ‘m‘

n=m.

0

The n e x t c o r o l l a r y i s a s t r e n g t h e n i n g o f a r e s u l t o f E n g e l e r due t o F. H o n s e l l .

1.18 COROLLARY. F o r e v e r y a p p l i c a t i v e s t r u c t u r e B t h e r e i s an e x t e n s i o n a l combinatory algebra

B

such t h a t

P r o o f . Given 8 , c o n s i d e r -

91CB.

the theory

T = (CL) + Diag(21) + E x t where Diag(B) = { P = Q I P,Q€Y€(a),F V ( P Q ) = 0 , 8 b P = Q } , E x t = Vab((Vx a x = b x ) Every f i n i t e p a r t T

D

+

a=b).

o f T i s c o n s i s t e n t : by 1.17 e v e r y non t r i v i a l e x t e n s i o n a l

c o m b i n a t o r y a l g e b r a , e.g.

D-,

can be made i n t o a model f o r T o . B u t t h e n by com-

pactness T i s c o n s i s t e n t and has a model a l g e b r a and

BCB.

B. T h i s i s an e x t e n s i o n a l c o m b i n a t o r y

0

The c o n s t r u c t i o n i n E n g e l e r [ 1 9 8 1 ] i s m o r e i n f o r m a t i v e i n a n o t h e r sense: f o r each s e t A t h e r e i s a c o m b i n a t o r y a l g e b r a DA such t h a t e v e r y a p p l i c a t i v e s t r u c t u r e w i t h u n i v e r s e A can be embedded i n t o DA.

92.

LAMBDA ALGEBRAS AND LAMBDA MODELS. S i n c e i n a c o m b i n a t o r y a l g e b r a B a b s t r a c t i o n can be s i m u l a t e d by k and s ,

i t i s p o s s i b l e t o i n t e r p r e t e A-terms

2.1

in 8.

NOTATION. L e t C be a s e t o f c o n s t a n t s . A(C)

i s t h e s e t o f A-terms u s i n g

p o s s i b l y c o n s t a n t s f r o m C . The A - c a l c u l u s axioms and r u l e s e x t e n d i n t h e o b v i o u s way t o e q u a t i o n s M = N w i t h M , N € h ( C ) .

For these M,N we s t i l l w r i t e X I - M = N .

!lX i s an a p p l i c a t i v e s t r u c t u r e , then A(YR)

2.2

If

i s A({cala€YR}).

DEFINITION. L e t m be a c o m b i n a t o r y a l g e b r a .

(i) Let

*

: A @ R ) + S @ l ? ) be t h e map t h a t r e p l a c e s e v e r y

M* f o r *(MI x* = x , c* = c , (MN)* =

M*N*

A

by

A*,

i.e.

writing

Lambda Calculus and its Models

(Ax.M)*

217

= A*x.M*.

( i i ) For M,N€A@J?)one d e f i n e s

I k

M=N

-

I , p b M = N f o r a l l p.

I f I i s a c o m b i n a t o r y a l g e b r a and UAx.xcaI B

.

aEI,

then we w r i t e e.g.

Ax.xa

for

Not a l l e q u a t i o n s p r o v a b l e i n A - c a l c u l u s a r e t r u e i n a c o m b i n a t o r y a l g e b r a . E.g.

i f II)) i s t h e t e r m model o f CL, then

I#Xz.(Ax.x)z since

2.3

S(K1)I and (Xz.z)*

(Xz.(Ax.x)z)*E

DEFINITION.

= 12.7. 5

I; b u t X!-Az.(Ax.x)z

=

Az.z.

( i ) A c o m b i n a t o r y a l g e b r a Yl? i s c a l l e d a A - a l g e b r a i f f o r a l l

M,N€A(!JJ~)

A

CM=N

=D

D?b M = N .

A - a l g e b r a homomorphism i s j u s t a c o m b i n a t o r y a l g e b r a homomorphism.

(ii) A

The n o t i o n o f A - a l g e b r a seems t o depend on t h e d e f i n i t i o n o f A*. of

2.4

B u t because

o t h e r d e s c r i p t i o n s , see 2 . 5 and § § 3 , 4 , t h i s i s n o t t h e case.

PROPOSITION. ( i ) I f q:iV?l+9J$,

p a r t i c u l a r q[MT

nn

= [M]m2

( i i ) Let!U?l+12. ( i i i ) 9R,C.m2 Proof. -

By 1.14.

9

thendM1:

=[q(M)IZfp

f o r M€A@).

then a l s o I 2.

Then Th@l)c_Th@J?2). So ifYl?, i s a A - a l g e b r a , Th@,)

In

f o r MEAO.

= Thm2).

0

By u s i n g C u r r y ' s c o m b i n a t o r y axioms AB one can a x i o m a t i z e t h e c l a s s o f

A-

algebras.

2.5

THEOREM. L e t I be a c o m b i n a t o r y a l g e b r a . Then

f i e s the f o l l o w i n g set of equations A

(Ag)

P r o o f . By -

I

(A.I)

Ii s

a A-algebra i f f % s a t i s -

R:

K = S ( S ( K S ) (s(KK)K)) ( K ( s K K ) ) ,

(A.2)

S = S(S(KS) (S(K(S(KS)))

(S(K(S(KK)))S)))

(A.3)

S ( S ( K S ) (S(KK) ( S ( K S ) K ) ) )

(KK) = S ( K K ) ,

(A.4)

S ( K S ) (S(KK))

(A.5)

S(K(S(KS)))

(K(K(SKK)))

= S ( K K ) (S(S(KS) (S(KK) ( S K K ) ) ) ( K ( S K K ) ) ) , (S(KS) ( S ( K S ) ) )

the fact t h a t the theories

= S ( S ( K S ) (S(KK) (S(KS) (S (K ( S (KS))

A

and CL+A

B

S) )

are equivalent,

(KS

.

i n t h e sense

H. BARENDREGT

218 that

A

I-

M=N

CL+A

see B a r e n d r e g t [19811 7.3.10

I- M* = N*,

B and 7.3.15.

0

The lambda a l g e b r a s u s u a l l y a r i s e as s u b s t r u c t u r e s o f a more n a t u r a l c l a s s o f A - c a l c u l u s models, t h e so c a l l e d lambda models. For t h e s e s t r u c t u r e s t h e r e i s a u n i f o r m method t o f i n d t h e elements r e p r e s e n t i n g a l g e b r a i c f u n c t i o n s , i n d e pendent o f t h e way t h e s e f u n c t i o n s a r e g i v e n (by t e r m s ) ; c f . theorem 5.8.

2.6

DEFINITION. L e t W be a c o m b i n a t o r y a l g e b r a . Ell i s c a l l e d weakly e x t e n s i o n a l

i f f o r A,B<X@i)

W C

Vx(A=B)

Ax.A = Xx.B

+

.

The c o n d i t i o n o f weak e x t e n s i o n a l i t y i s r a t h e r s y n t a c t i c a l . Meyer [19801 and S c o t t [19801 r e p l a c e i t as f o l l o w s .

2.7

DEFINITION.

( i ) I n a combinatory algebra d e f i n e 1 = S ( K I ) .

( i i ) A A-model

W such t h a t t h e f o l l o w i n g M e y e r - S c o t t axiom

i s a A-algebra

h o l d s i.n !JX Vx(ax=bx)

2.8

+

la = lb.

LEMMA. L e t Ell be a c o m b i n a t o r y a l g e b r a . Then i n Ell ( i ) lab = ab;

I f moreover!JX i s a A - a l g e b r a , (ii

1 = Axy.xy,

(iii

l(1x.A)

(iv

11 = 1 .

Proof. -

then

hence l a = X y . a y ;

= Xx.A,

f o r a l l A€%@)

i ) l a b = S ( K l ) a b = K l b ( a b ) = ab.

(ii

1 = S ( K I ) = (Xxyz.xz(yz)) (KI) = hyz.Klz(yz)

(iii

l ( A x . A ) = Ax.(Xx.A)x

= Ax.A,

PROPOSITION.

Proof. -

(-)

= 1yz.y~

by ( i i ) .

0

( i v ) By ( i i i ) and ( i i ) .

2.9

;

W i s a A-model

-

i s a weakly extensional A-algebra.

L e t m be w e a k l y e x t e n s i o n a l . Then

Vx ax = bx

* =b

Ax.ax = Ax.bx l a = l b , by 2 . 8 ( i i ) .

(*)

L e t W be a A-model.

Vx

A = B

=b

* =b

Then

Vx(Ax.A)x = ( h x . 6 ) ~ 1 (1x.A)

= t(Xx.B)

Ax.A = hx.8,

by 2 . 8 ( i i i ) .

0

Lambda Calculus and its Models 2.10 PROPOSITION. L e t % ?be a A - a l g e b r a .

ID) i s e x t e n s i o n a l Proof. -

(*)

* *

A=B

Then

ID) i s w e a k l y e x t e n s i o n a l and s a t i s f i e s I = 1

Q

(Ax.A)x = (Ax.B)x Ax.A = Ax.B,

by e x t e n s i o n a l i t y .

Moreover I x y = x y = l x y , so by e x t e n s i o n a l i t y ( t w i c e )

(*)

By 2.9

.

W i s a A-model

Vx a x = bx

219

*

I=1.

Hence

la = lb

0

a = b since 1 = I .

An e x t e n s i o n a l c o m b i n a t o r y a l g e b r a i s a u t o m a t i c a l l y a A - a l g e b r a .

A I-

because

*

M=N

CL+ext

t

M*=N*,

This i s

see B a r e n d r e g t [19811 7 . 3 . 1 4 .

TERM MODELS, INTERIORS. 2.11 DEFINITION. L e t T be an e x t e n s i o n o f t h e t h e o r y

A,

i.e.

o f the A-calculus.

( i ) Define M = N Q T [MIT = “€A

T

h / T = {[MI, [MIT.[NlT

k M=N

; t h i s i s a congruence r e l a t i o n on

A.

I M =T N}. 1 MEA}.

= [MNIT

; t h i s i s welldefined.

The open term model o f T i s ID)m(T) =

(

A/T,.,[KIT,[SIT

).

( i i ) By r e s t r i c t i n g e v e r y t h i n g t o c l o s e d terms one d e f i n e s t h e c l o s e d t e r m

-

model o f

T 0

ID) (T)

=(A

0

0 0 /T,.,[KIT,[SIT

Clearly i f T i s consistent, T

# K=S,

).

i.e.

does n o t p r o v e e v e r y e q u a t i o n , then 0 I n p a r t i c u l a r ID)(A) and ID) (A) a r e

so I ( T ) and d ( T ) a r e non t r i v i a l .

non t r i v i a l s i n c e i t f o l l o w s f r o m t h e Church-Rosser theorem t h a t t h e t h e o r y

A

consistent.

2.12 PROPOSITION. L e t T be an e x t e n s i o n o f t h e A - c a l c u l u s and l e t ( i ) F o r M w i t h FV(M) = { x ,

=

[MI;

(ii) T CM=N (iii) T Proof.

t-

M=N

[M [ x := +

* es

PI1

+

, . . . ,xn} (0)

* *

= ID)(O)(T).

has

.

ID)+ M = N

Wb

M = N , p r o v i d e d t h a t W = W ( T ) o r t h a t M,N a r e c l o s e d .

( i ) I n d u c t i o n on t h e s t r u c t u r e o f M * ,

(ii) TCM=N

W

and p w i t h p ( x i ) = [ P ]“’one i T

VF

TCM[Z:=bI

VP

[M[x : = P I I T = “ [ x

+

+

+

+

u s i n g TI-M

+

=N[x:=P]

+

+

:=PIIT

=

M*.

is

H. BARENDREGT

220

* Vp U M ~ , = [ N J P * I + M=N . = 9i?(T). L e t Po ( x ) = [XI,. Then

( i i i ) For W

!lJ?+M=N

* U MI * [MIT *

=

1N I

= [NIT,

by ( i ) ,

T+M=N.

F o r M,N c l o s e d .

!lJ?b M = N * UMJ,

=IN]

P [MIT = [NIT,

* *

by ( i ) ,

0

TkM=N.

2.13 COROLLARY. ( i ) W(')(T)

i s a A-algebra.

( i i ) I ( T ) i s a A-model. P r o o f . W r i t e I= I ( T ) . ( i ) By 2 . 1 2 ( i i ) . (ii )

B? b

* IC * WC

Vx a x = bx

Vx[M]x = [ N I X where a = [ M I and b = [M][z]

*

TCMz=Nz

* *

T

*

rlx+ l a = l b .

I- Az.Mz = k 1M = 1 N

T

"11~1,

=

"1,

f o r some f r e s h v a r i a b l e z ,

1z.N~

I0 (T)

Remarks. ( i ) ( J a c o p i n i [ 1 9 7 5 ] ) .

i s i n g e n e r a l n o t a A-model. Consider 0 0 T z A a x i o m a t i z e d b y {RKZ = RSZ Z E A 1 where R ~ ( A x . x x ) ( A x . x x ) . Then V Z E A 0 T RKZ = RSZ, hence I (T) l= Vx RKx = RSx. B u t I o ( T ) # 1 (RK) = 1 ( a s ) , s i n c e

1

+

otherwise T

t RKx

= RSx, w h i c h i s f a l s e .

0

( i i ) P l o t k i n [1974] shows t h a t e v e n I

(A)

and

I0 (An)

a r e n o t A-models.

( i i i ) By ( i ) i t follows t h a t p r o p o s i t i o n 2 . 1 2 ( i i i ) does n o t h o l d i n g e n e r a l 0 f o r I ( T ) : t a k e M=RKx, N=RSx.

conibinatory a l g e b r a . B ( n o t a t i o n B0 ) i s t h e s u b s t r u c t u r e

2.14 DEFINITION. L e t 91 be a ( i ) The i n t e r i o r o f

of

k,s. (ii)

II

is

hard i f

21'

= 21.

0 Note t h a t u p t o i s o m o r p h i s m W (T)

i s the i n t e r i o r o f %?(T).

2.15 PROPOSITION. L e t 8 be a A - a l g e b r a . (i) d(Th(8))

zWo

( i i ) L e t Th(%) = { M = N Then !$(Th(Yl))

zB.

M,N€Z(II),

c l o s e d and

B I= M = N l .

B

generated by

22 1

Lambda Calculus and its Models

Proof. -

i s a w e l l d e f i n e d isomorphism o n t o B

( i ) ( P ( [ M ] ~ ~ ( ~=~[ M ) j)'

( i ) Similarly.

0

.

0

f o l l o w s t h a t a l l A - a l g e b r a s a r i s e as a s u b s t r u c t u r e o f a A-model.

I

2.16 PROPOSITION. ( i ) ( B a r e n d r e g t , Koymans [ 1 9 8 0 ] ) .

Every

A - a l g e b r a can be em-

bedded i n t o a A-model.

( i i ) (Meyer [ 1 9 8 1 ] ) . P r o o f . ( i ) VI%J?

0

(Th

Every X - a l g e b r a

i s t h e homomorphic image o f a X-model.

(E)) c m(Th (X)). 0-

by t h e s u r j e c t i v e map t h a t r e p l a c e s

( ii ) Moreover %l?(Th(Z)) + %l? (Th(2)) e v e r y f r e e v a r i a b l e by say K.

0

The f o l l o w i n g i s proved i n B a r e n d r e g t and Koymans [19801. Here we s t a t e t h e result without a proof.

2.17 THEOREM. ( i ) T h e r e i s a A-model

t h a t c a n n o t be embedded i n t o an e x t e n s i o n a l

A-model. ( i i ) There i s a c o m b i n a t o r y c o m p l e t e a p p l i c a t i v e s t r u c t u r e t h a t cannot be made i n t o a A - a l g e b r a

(by c h o o s i n g k , s ) .

i i i ) There i s a A - a l g e b r a t h a t cannot be made i n t o a A-model

(by changing

k,s ( i v ) There i s a A-model CO 1

t h a t c a n n o t be made i n t o an e x t e n s i o n a l one (by

apsing i t ) . The t e r m models make i t p o s s i b l e t o g i v e t h e f o l l o w i n g p r o o f s o f some ccm-

pleteness r e s u l t s .

2.18 THEOREM. ( i )

A C

M=N

-

M = N i s t r u e i n a l l A-models

(or A-algebras).

( i i ) L e t T be an e x t e n s i o n o f t h e A - c a l c u l u s . Then T+-

M=N

(iii) Let (A)c

M = N i s t r u e i n a l l A-models s a t i s f y i n g T .

94

be t h e c l a s s i c a l f i r s t o r d e r t h e o r y a x i o m a t i z e d by t h e u n i v e r -

sal closure of Kxy = x syxz = xz(yz),

KZS V x ( a x = bx)

+

Then M=N

la = lb

-

xt

M=N.

H. BARENDREGT

222 Proof. -

(i)

(+) By d e f i n i t i o n .

t r u e i n W ( A ) ; hence

A t

(e)I f M = N i s t r u e i n a l l A-models,

then i t i s

M = N by 2 . 1 2 ( i i i ) .

( i i ) Similarly. ( i i i ) (a)Note t h a t W(X) C ( A ) c . T h e r e f o r e

(XIc (-)

53.

t

*

M=N

W(A) C M = N

A CM=N.

+

0

Trivial.

SYNTACTICAL MODELS I n t h i s s e c t i o n a s y n t a c t i c a l d e s c r i p t i o n o f t h e A - a l g e b r a s and A-models

w i l l be g i v e n , w h i c h i s e q u i v a l e n t t o t h e f i r s t o r d e r d e s c r i p t i o n i n 52. For some models,

i n p a r t i c u l a r t h e f i l t e r model o f B a r e n d r e g t e t a l .

[19831,

t h i s syntac-

t i c a l d e s c r i p t i o n i s more c o n v e n i e n t t h a n t h e f i r s t o r d e r . The method i s due t o H i n d l e y and Longo [19801.

3.1

DEFINITION. L e t W = ( X , . )

be an a p p l i c a t i v e s t r u c t u r e .

( i ) Val @I i) s the s e t o f valuations i n

1.

l ( i i ) A s y n t a c t i c a l i n t e r p r e t a t i o n i n 1 i s a map I : h @ ? ) x V a (sn) the f o l l o w i n g conditions; 1.

nxn P

I(M,p)

i s w r i t t e n as [MD

-f

X satisfying

P'

= p(x)

2. U c a l p = a

3. UPQD, = UPIIp.UQn P 4. UAx.Pl .a = UP] p ( x : = a ) P

5. pFFV(M)

EM1

= p'FFV(M)

P

= [MI

Note t h a t by t h e v a r i a b l e c o n v e n t i o n , 4

P"

i m p l i e s t h a t f o r y @ FV(M(x))

one has

41. u M ( x ) n p ( x : = a ) = [Ax.M(x)],a = UAy.M(y)l,a

:=a).

= uM(Y)np(y

( i i i ) A syntactical applicativestructure

[l 3.2

is a syntactical interpretation i n

i s of the formW=(X,.,[D)

where

W.

DEFINITION. L e t 1 be a s y n t a c t i c a l a p p l i c a t i v e s t r u c t u r e .

--

( i ) The n o t i o n o f s a t i s f a c t i o n i n 1131 i s d e f i n e d as u s u a l :

W,p C M=N WCM=N

[MI, Vp

= [NIP

W,pCM=N

-

and t h i s i s extended t o a r b i t r a r y f i r s t o r d e r f o r m u l a s o v e r t h e A - c a l c u l u s . ( i i ) IJ31 i s a s y n t a c t i c a l A - a l g e b r a (iii) i .e.

W i s a s y n t a c t i c a l A-model i f Va 'M'p(x

: = a ) = "'p(x

:=a)

if A k M = N

(5) W C +

[Ax.MI

WC

Vx(M=N) P

= UAx.ND

M=N. -f

P'

Ax.M = Ax.N,

223

Lambda Calculus and its Models

3.3

LEMMA. LetIIR be a s y n t a c t i c a l A-model.

= Vp

(p(M,N)

Consider t h e s t a t e m e n t

6M[x : = N l I p = uMIIp(x : = [ N I P ) .

Then f o r M,NEA6J?)

*

(i) z@FV(M) (ii) ~ M , N )

cp(M,z)

W(AY.M,N);

9

( ii i ) Q(M,N).

Proof.

( i ) W r i t e M=M(x).

uwnP

then

= uM(z)np(z : = p ( z ) )

= nM(X)ip(x :=p(z))

by 4 ’ . ( i i ) F i r s t assume x B F V ( N ) . By t h e v a r i a b l e c o n v e n t i o n y s x , y f F V ( N ) . f o r p* = p ( x : = [ N ]

P

) and a r b i t r a r y aEIIR : = a ) = UM[x : = N I I p ( Y : = a )

= :N(,I

(note that I N ]

P

:=a)).

=

UAy.M[x : = N I I p *

=

:=a)(x

=

:=a)

T h e r e f o r e by

= UAy.MB

:=[NIP)’

s i n c e (D(M,N),

;

(5)

P*

and hence [Ay.M[x := N l n

P

= [Ay.M[x

=‘Ay‘M’p(x I f xEFV(N),

P

= =

( i i i ) Now cp(M,N)

:=[NIP)’

[R[x

IMLx

:= Z ] [ Z := N]]

IIR

I-

P

= ”p(z

:=EN] P ) ( x

= u‘np(x

:=[NIP ) ’

by ( i ) , :=I”],)’

f o l l Q w s b y a s i m p l e i n d u c t i o n on t h e s t r u c t u r e o f M.

M=N

=P

IIRC M = N ,

i s a s y n t a c t i c a l A-algebra,

P r o o f . By

-

: = z ] I p ( z :=KN],)

THEOREM. L e t W be a s y n t a c t i c a l A-model.

A i.e.

P*

t h e n l e t z be a f r e s h v a r i a b l e . We have f o r M-Ay.M

[%[x := N l J

3.4

:= N l I

i n d u c t i o n on t h e l e n g t h o f p r o o f .

Then

Then

H. BARENDREGT

224 The axiom (Ax.M)N = M [ x : = N ]

f

i s sound:

= [M[x : = N ] ] Soundness o f t h e r u l e M = N

*

P'

by 3 . 3 ( i

i).

Ax.M = 1x.N f o lows from

(5). The

DEFINITION. A homomorphism between s y n t a c t i c a l A - a l g e b r a s

cp:XWl

i s a map

+XV2 such t h a t f o r a l l M € A @ ) one has

dM1,1

= I[cp(M)ncpop 2

where i n cp(M) t h e c

3.6

other rules are

0

trivial.

3.5

by 3,

(Ax.M)Nlp = !IAx.Ml,!IND,,

a r e r e p l a c e d by c

cp(a)

'

THEOREM. The c a t e g o r i e s o f s y n t a c t i c a l A - a l g e b r a s and homomorphisms and

t h a t o f A-algebras and homomorphisms a r e i s o m o r p h i c . Moreover s y n t a c t i c a l A models c o r r e s p o n d e x a c t l y t o A-models under t h i s isomorphism. Proof.'Easy. For a s y n t a c t i c a l X - a l g e b r a f o r c p : YX

1

+YX2

l e t Fcp = c p :

Pm,

+

XW=(X,.,II)

d e f i n e FBI = (X,.,UKl,USl);

Pmz. Then one has [MlPm

Conversely f o r a A-algebra B = (X,.,k,s)

P d e f i n e GB = (X,.,[]')

= [ M c f o r MEAm).

above. Then F, w i t h i n v e r s e G , i s t h e r e q u i r e d isomorphism.

54.

and Gcp = cp as

0

CATEGORICAL DESCRIPTION OF THE MODELS. I n t h i s s e c t i o n t h e c l a s s o f A-algebras w i l l be d e s c r i b e d i n a n a t u r a l c a t e -

g o r i c a l way. The A-models a r e t h e n t h o s e A - a l g e b r a s t h a t c m e f r o m c a t e g o r i e s " w i t h enough p o i n t s " .

The method i s due t o Koymans [1983] and i s based on work

o f Scott.

4.1

DEFINITION. L e t

0

be a c a t e g o r y . The i d e n t i t y map on an o b j e c t A E U i s de-

n o t e d by i d A . ( i ) 0 i s a Cartesian closed category (ccc) i f f

1.

iI

has a t e r m i n a l o b j e c t T such t h a t f o r e v e r y o b j e c t A E B t h e r e e x i s t s a

u n i q u e map ! A : A + T .

2. For A ,A

E C t h e r e i s an o b j e c t AIXAz

1 2 p i : A1xA2+Ai

u n i q u e map Notation.

(

( C a r t e s i a n p r o d u c t ) w i t h maps

( p r o j e c t i o n s ) such t h a t f o r a l l f i : C + A i

fl,fz)

: C + A 1 x A 2 w i t h p i n ( fl,f2)

If gi:Ai+B.

B1xBz, see f i g u r e .

( i = 1,2)

there i s a

= f i , see f i g u r e .

( i = l , 2 ) , then g l x g z = ( g l o p l ,

g20pz):A,XA2+

225

Lambda Calculus and its Models

I

exponcnt

pmduct

3 . For A,BE C t h e r e i s an o b j e c t

A

B E b (exponent) w i t h map e v = e v A , B :

A

3 x A + B such t h a t f o r a l l f : C x A + B t h e r e i s a u n i q u e A f : C + B A s a t i s f y i n g f = e v o ( A f X i d A ) , see f i g u r e . ( i i ) L e t C have a t e r m i n a l o b j e c t T. A

point

o f A€

i s a map x : T + A .

The

s e t o f p o i n t s o f A i s denoted by I A l . An o b j e c t A has enough p o i n t s i f f o r a l l f,g : A + B

one has f#g

*

3xE IAl

fox#gox.

Note t h a t i n a c c c one has A(hogxidB) = A(h) o g f , g ) o h = ( f o h , goh)

(

= ( f o h , gok)

fxgo (h,k)

DEFINITION. L e t C be a

4.2 o f U,

i.e.

CCC.

An o b j e c t U E B i s r e f l e x i v e i f Uu i s a r e t r a c t U and G : U + U such t h a t

t h e r e a r e maps F : U + U u

FOG = i d UU’

4.3

DEFINITION. L e t 0 be a c c c w i t h r e f l e x i v e o b j e c t U ( v i a t h e maps -F,G).

these data determine a s y n t a c t i c a l a p p l i c a t i v e s t r u c t u r e

m(e)

( = W(O,U,F,G))

Then as

follows: ( i ) The domain o f

m(C)

is

IUI.

( i i ) L e t Ap : U 2 + U be t h e map evU,Uo F x i d For f , g : A + U d e f i n e f a A g = Apo ( f , g ) . X.Y

= x .T y = Apo

(

IUI

x,Y).

As a p p l i c a t i v e s t r u c t u r e I ( Q ) ( i i i ) Uo = T, Un”

U’ In particular for x,yE

is (IUI,.)

= UnxU. L e t A = x l ,

...,x

. be a sequence o f d i s t i n c t v a r i -

a b l e s . W r i t e UA = U”. (iv)

nXA

: UA+U

i

i s t h e c a n o n i c a l p r o j e c t i o n on t h e i - t h c o o r d i n a t e .

226

H. BARENDREGT

,..., f n : A - t U ,

(v) I f fl

then ( f l

= '' A

0

(fl,...,fn+l)

= ((f,,

,..., f n ) : A + U n

i s d e f i n e d by

. . . ,f n ) , fn+l).

Clearly 7rA

0

(f,,

..., f n ) =

fi

xi ( v i ) Let

r

. . ,Y,

= yl,.

with

{q}~{z}.

Define

rlA = ( 7 r A

) :

,...,AA

y1

ua+ur.

ym

This i s the canonical "thinning",

IM(x,y)ll X?Y

[XIA

,. .. , x n ) .

b e i n g "X(x,y).[M(cx,c

(

y1

,.. . ,y,

)".

U A + U ( w i t h intended i n t e r p r e t Y

) l " ) as

follows.

;

= ITA

[canA = a o upon,

"h(xl

d e f i n e i n d u c t i v e l y [Ml,:

( v i i ) For AZFV(I1) a t i o n o f e.g.

i.e.

'

,

= upnA.ua

(for a €

tul) ;

nuA;

I A x . P I A = G O A ( [ P I I ~ , ~ )where , by t h e v a r i a b l e c o n v e n t i o n we assume x!€{Al. ( v i i i ) For a v a l u a t i o n p i n I U I l e t

[MI

P

=

[MILO

pA w i t h A=FV(M).

EM1 E I U I . P ( i x ) Finally%V(G) i s the s t r u c t u r e

Clearly

[ M I X :=N]lr

Proof. ( i ) , ( i i ) (i) [xy.Plro

(lUl,.,[l).

=

I n d u c t i o n on t h e s t r u c t u r e o f M. We o n l y t r e a t M=Xy.P. A = GoA(UPlr,y) o A

rfr

= GoA([P]r,yo = G~A([PI,,~~

= GoA(UPIA,y)

IH = niy.pnA.

n f x i d U ) , by 4 . 2 ( 1 ) ,

221

Lambda Calculus and its Models Here I H denotes “by

the induction hypothesis”.

+

+

3

= [~y.~[;,y

( i i ) u(A~.P)[x:=NII,

:=~,yIn, 3

= GoA(UP[;,y

:= N , ~ l l , , ~ )

+

(uN1,,y,

= GoA(IIPIA,yo IH

z

GoA(UPIA,ya(Exlr)x

= GoA(l[PIA,y)

0

*

i d U ) , see below,

([xl,)

= [[Xy.PD,o(U$), where

[yl,,y)),

;

i s shown as follows

(~ih,,~, u ~

+

I ~ , ~= ) ( ~NI,

0

n:.Y,

ni’y

(i) 3

p,,

= (UNIr,o

+

i d U o ( p1 ,p 2 )

= ([Nl,)x =

( i i i ) Apply ( i i ) t o A’=A,x

4.5

3

((“1,)

and

idUO pg

x idU.

+

r,

w r i t i n g A=y

PROPOSITION. L e t M,NEA(!V?(O)) and { A l z F V ( M N ) .

Proof. -

+

and M [ x : = N ] G M [ y , x

Then

I n d u c t i o n on t h e l e n g t h o f p r o o f of M = N . We t r e a t t h e e s s e n t i a l axiom

and r u l e . Axiom (Xx.P)Q = P [ x :=Ql. (hx.P)QIA = (GoA(uPIA,x))

0

UA

Rule P = Q

=

BPI,

--. *

*

Xx.P =

[ a , , by

[PlA0

uAx.PnA

=

[aA

AX.Q.

the induction hypothesis,

rIp

upnA,x =

= nQl,o

n p

an,,^. nxx.~,.

4.6

THEOREM. Every c c c

mc)

=

Proof. -

+

:=y,N].

t

0

w i t h r e f l e x i v e o b j e c t U determines a A-algebra

(iui,.,un). Immediate f r o m

4.5 and t h e d e f i n i t i o n of 111

P’

0

0

H.BARENDREGT

228

4.7

PROPOSITION. L e t

I=IIR($,U,F,G).

( i ) Let {A}?FV(M).

-Then

( i i ) U has enough p o i n t s U ( i i i ) U Z U v i a F,G I (iv) Proof.

u . U z U via

( i ) [lM],

inm.

[ l M ] I A = GoFo[M]I,

k 1

IIR

i s a A-model.

= I.

F,G and U has enough p o i n t s

-

Sn i s e x t e n s i o n a l .

= UAy.MyjA = GoA(evo ( Fo [ M I

UylA,y))

A,Y’

= GoA(evo ( F o [ M ] l A o IIA”, nA”))

A

Y

= GoA(evo(FoIIMDAopl,pz)) = GoA(evo (Fo [MI,)

x id)

= GoA(ev) o Fo [ M j A = GoFo [MI,.

(ii)

(-1

for a , b € I

Vx€Iax=bx

evo(Foa,x) = evo(Fob,x)

*

evo (Foa)x i d o ( id,x)

9

e v o (Foa)

* Hence

-

1 L e t U have enough p o i n t s . Then t h e same i s t r u e f o r U ( g U ) .

IIR

(-1 Then

= evo (Fob)x i d o ( id,x)

i d = e v o (Fob) x i d , s i n c e U’ has enough p o i n t s ,

X

l a = l b , s i n c e by ( i ) l c = GoFoc = GoA(evo ( F o c ) x i d ) . i s a A-model. Suppose

---

Ii s

a A-model

VxE I U l f O X = gox

and l e t f , g : U + U .

*

Vx i . x = g . x , where

1.f =

=9

7

= GoA(fopZ) and s a t i s f i e s f . x = f o x ,

1.9

G ~ F ~= TG

~

F

~

~

A ( f o p 2 ) = A ( g o p z ) , s i n c e FOG = i d , fop* = SOP2 f=g.

9

T h e r e f o r e U has enough p o i n t s . U ( i i i ) (*) I f U E U v i a F,G, t h e n GoF = i d U , hence by ( i )

UIMIA =

UMII,

i n p a r t i c u l a r 11x1,

[in, i.e.

II.

(-1

=

= [xi,.

[Ax.ixnA

Then as i n t h e p r o o f o f = U A ~ . ~ =~ 11111, ,

1 = I. Assume

II.

1 = I . Now

8 1 1 = [Axy.xyB

= GoA(llXy.xyJx)

= GoA(GoFo [ x I x ) ,

= G O A ( G O F ~ P; ~ )

by ( i ) ,

,

4.5

i t follows that

Now

229

Lambda Calculus and its Models

1 ID

and

= 60A(p2). Therefore

G O A ( G O F O=~ G ~ )O A ( ~ ~ )

*

A(GoFop2) = A(p2)

=s

GoFop2 =

(use F) (Ah u n i q u e l y d e t e r m i n e s h )

P2 GoF = idU

*

,

(use ( !

( i v ) By 2.10 and ( i i ) ,

idU)).

0

(iii).

L e t B be a A - a l g e b r a t h a t a r i s e s f r o m a c a t e g o r y t h a t i s " c o n c r e t e " , r o u g h l y one t h a t i s based on s e t s . Then

B

B

i.e.

i s a A-model and t h e i n t e r p r e t a t i o n i n

has a s i m p l e form.

4.8

DEFINITION. A c c c Q i s s t r i c t l y c o n c r e t e i f t h e r e i s a f u n c t o r 0 :@ + S e t

such t h a t 1. 0 i s f a i t h f u l 2. 0 i s f u l l

3.

(i.e.

(i.e.

i n j e c t i v e on a r r o w s ) .

s u r j e c t i v e ) o n Hom (T,A)

Q.

for AEC.

# p r e s e r v e s t h e t e r m i n a l o b j e c t , p r o d u c t s and p r o j e c t i o n s .

4. For a l l A,BE Q

Note t h a t t h i s i m p l i e s t h a t e v e r y o b j e c t i n

where ASet

,...,a n )

g(al

=Ad. g(al

,..., an,d)

C

has enough p o i n t s . Moreover

f o r g : Xn+l+Xn

i n Set. W r i t e

f o r t h e t e r m i n a l o b j e c t i n Set. Complete p a r t i a l o r d e r s o r c o m p l e t e l a t t i c e s w i t h c o n t i n u o u s maps a r e s t r i c t l y concrete c c c ' s .

4.9

DEFINITION. L e t Q be a s t r i c t l y c o n c r e t e c c c w i t h r e f l e x i v e o b j e c t U. ( i ) cp: ( i i)

IUI

-+

: #(Uu)

#(U) +

i s t h e b i j e c t i o n cp(x) = # ( x ) ( * ) .

#(U)

( i i i ) a.b = #(F)(a)(b)

i s t h e map

=

#(GI.

f o r a,bE#(U).

= V ( U ~ - ' ( M ) ~ ~ - ~f ,o~r ) MEA(O(U)).

( i v ) [MI:

. ,u

( v ) XQ# = ( ~ ( u ) ,

no).

4.10 THEOREM. (Koymand [ 1 9 8 3 ] ) .

( i ) The map

# 1. uxn, = P W ,

2. ucaj:

3.

= a,

f o r aE#(U);

UPQP = uPi#.ud'; P P P

4. U A X . P I ~= 0 6 d . P

:=d)).

'1

i n 4.9 s a t i s f i e s

{*I

H. BARENDREGT

230 ( i i ) a @ i s a A-model

isomorphic t o a ( c ) .

P r o o f . ( i ) As an example we show b . L e t p o = cp-lop and A = FV(Ax.P).

A

(*), where f o r s i m p l i c i t y we assume cp-'(P) = P ,

UAx.PIQ' = @ ( G O A ~ ( ~ P ~ , o, ~p,) ) P = ~ ( A s e p D ( ~ P I A , x )()P

A

1)

= o ( A d . O(UPJA,x)(p(x : = d ) A ' x ) ) = o()Sd. Q'(UPl

op0(x : = ~ - ' ( d ) ) ~ ' ~(*I) )

A.x

= o ( X d * (P(Up1 cp-lop(x : = d ) 1) = o(Xld. I P l O p ( x : = d ) ) .

+ : I i' s by 4 . 9 ( i v ) ( i i ) The map @::I($) enough p o i n t s , t h e s t r u c t u r e : I ( $ )

an isomorphism. S i n c e U i n i t f o l l o w s that!#

i s a A-model;

C

has

i s a A-model.

0 0

i s c a l l e d t h e c o n c r e t e v e r s i o n of:I(t)

v i a t h e f u n c t o r 0.

Now i t w i l l be p r o v e d t h a t e v e r y A - a l g e b r a can be o b t a i n e d f r o m a ccc w i t h a reflexive object.

4.11 DEFINITION.

L e t B be a A - a l g e b r a .

The Karoubi e n v e l o p o f 8, n o t a t i o n

i s t h e c a t e g o r y d e f i n e d as f o l l o w $ . L e t aob = A x . a ( b x ) ,

Objects: { a € $

I ao a

= a}.

Arrows: Hom(a,b) = I f € 8 Identity: i d Composition:

$(a),

for a,b€8.

1

bofoa = f } .

= a.

fog.

I t i s easy t o v e r i f y t h a t t ( 8 ) i s indeed a c a t e g o r y . Karoubi [1978] d e f i n e d t h e e n v e l o p f o r a d d i t i v e c a t e g o r i e s u n d e r t h e name " d e r i v e d pseudo a b e l i a n c a t e g o r y " .

T h i s can be g e n e r a l i z e d t o a r b i t r a r y categories.

The n o t i o n t h e n a p p l i e s t o a A - a l g e b r a

w(a)

= ({aes

Ia

F

by i n t r o d u c i n g t h e monoid

=la},o,~)

c o n s i d e r e d as c a t e g o r y w i t h one o b j e c t and as arrows t h e a E w ( 8 ) w i t h c o m p o s i t i o n . We need some n o t a t i o n f r o m t h e A - c a l c u l u s .

L e t [M,N]

= Az.zMN be p a i r i n g i n

a .a.), f o r i = 1 ,Z. Let 1 2 I w i t h n? t h e c a n o n i c a l A-terms such [MI] = M1, EM1 ,...,Mn+ll = "M1,.. *~MnI;Mn+lI n+l n+l n that n7[M1 Mn] = Mi f o r l < i < n . [ T I , = I , nn+l = TI^, ni = TI. o n l f o r t h e A - c a l c u l u s w i t h p r o j e c t i o n s ni = Ay.y(Aa

'

,...,

1S iSn.

1

4.12 PROPOSITION. ( S c o t t 119803). ( i i ) I i s a reflexive object Proof.

( i ) t ( 8 ) i s a ccc. n

t(n)

v i a the arrows F = G = l .

( i ) 1. T e r m i n a l o b j e c t . Th s i s t = A x y . y .

Note t h a t f : a + t

f=t.

23 1

Lambda Calculus and its Models

2 . P r o d u c t s . L e t a ,a

1

2

E C ( 8 ) . Then al X a 2 = X Z . [ ~ ~ ( T ~ aZ2)( ,n 2 z ) ] i s t h e

Cartesian product w i t h projections a a pi1 = a . o 71

i '

I

(f,g)

.

= Xz. [ f z , g z l .

3. Exponents. L e t a , b E C ( 8 ) .

Then

ba = Xz.b o z o a ev

a,b

= Xz.t(Tlz(a(T2z)))

A(f) = Xxy.f[x,y]. The c a l c u l a t i o n s t h a t show t h a t e v e r y t h i n g works a r e s t r a i g h t f o r w a r d and a r e l e f t t o t h e reader. ( i i ) Note t h a t 1 1 = 1 , 1 : l + l ,

1 : 1 + 1 and l o l = l = i d

4.13 THEOREM. (Koymans [ 1 9 8 3 1 ) . %V(U(%),l,l,l)

1'

0

=fl.

P r o o f . L e t I =YX(Q(8)). By i n d u c t i o n on t h e s t r u c t u r e o f M E A one can show

m

[MJ3 =

(+)

Xz. M

[x,

,... ,xn

:=

7172

,...,71:zl.

As an example we t r e a t M E 1 y . P . [Xy.PDz = GoA(lIPng

?Y

= loXpq.(Az.P[x,

,...,xn,y

n+ 1

:= ITl

z,..

IH = Xpq.P[x l,...,xn,y

hp. (Xy.P) [x,,

=

:= 71p ;

. . . ,x

... ,xn

hz.M[xl,

L e t 0 be a p p l i c a t i o n i n

,...,r;p,q

:= nyp,.

:= 7172,

I.Note

...,i

. . ,lr; T

p

that

a Q b = Ap o ( a , b )

= ev

o ( 1 o a,b) 1 9 1

= Xz.ev

[ l (az) ,bz] 1 3 1

= Xz.az(bz)

= Sab. Now d e f i n e c p : fl+m by cp(a) = Ka. Then cp i s c l e a r l y i n j e c t i v e .

x E I1

I =

t+

If

I , t h e n x i s c o n s t a n t s o x = K ( x l ) = c p ( x l ) ; t h e r e f o r e cp i s s u r j e c t i v e .

F i n a l l y cp i s a homomorphism: 1 . cp(xy) = K ( x y )

= S(Kx) ( K y ) , s i n c e 8 i s a X - a l g e b r a ,

H. BARENDREGT

232 = cp(x) o c o ( y ) .

2.

w(K)

=

KK = [K]"

by (+) and s i m i l a r l y f o r 5.

T h e r e f o r e cp i s an isomorphism and I

Zm.

C

I t f o l l o w s t h a t e v e r y A - a l g e b r a (A-model) can be o b t a i n e d f r o m a c c c w i t h r e f l e x i v e o b j e c t U ( h a v i n g enough p o i n t s ) . Remarks.

( i ) I t i s n o t h a r d t o show t h a t i f 91 i s a A-model,

then e v e r y o b j e c t o f

$ ( ? I ) has enough p o i n t s , see Koymans [ 1 9 8 3 ] . ( i i ) I t i s n o t t r u e t h a t C(%V(t,U))

t.

The c a t e g o r y

0:

may have many more

objects. The n o t i o n o f A - a l g e b r a homomorphism can be c h a r a c t e r i z e d i n a c a t e g o r i c a l way.

4.14 DEFINITION. A f u n c t o r @ between two

i s C a r t e s i a n i f Q, p r e s e r v e s t h e

CCC'S

t e r m i n a l o b j e c t , p r o d u c t s and e x p o n e n t s .

4.15 PROPOSITION. ( i ) F o r i = 1 , 2 t h e maps F. ,Gi.

L e t 0 :t 1 + C 2

l e t It. be a ccc w i t h r e f l e x i v e o b j e c t s U. v i a

be a C a r t e s i a n f u n c t o r w i t h Q(U ) = U 2 , Q ( F 1 ) = F2, 1

Q,(G1) =G2. Then @ i n d u c e s a homomorphism @ : %V(d,) ( i i ) I f cp:%+912

cp+ : C ( ? I 1 )

+

+* = c p

o v e r cp

i s a homomorphism,

then

(0

2 up t o isomorphism.

( i ) For x € I U

( i i ) For a an o b j e c t o f $ ( ? I , ) d e f i n e cp+(f) = cp(f). S i n c e

(0

T h i s i s a homomorphism s i n c e

d e f i n e cp+(a) = cp(a) and f o r f€Homd(al)(a,b)

p r e s e r v e s a l l c l o s e d A-terms,

f u n c t o r p r e s e r v i n g I and 1 . C l e a r l y

55.

m(C,).

Q(91 ) p r e s e r v i n g t h e r e f l e x i v e e l e m e n t s I and r e t r a c t i o n map 1 . More-

I d e f i n e @ * ( x ) = Q ( X ) € IU21. 1 p r e s e r v e s F,G and t h e C a r t e s i a n s t r u c t u r e .

Proof. -

+

induces a C a r t e s i a n f u n c t o r

+* = c p

(0

t h i s i s a Cartesian

o n m(C(21)) 2 8.

0

OTHER MODEL DESCRIPTIONS; CATEGORICAL MODELS. Lambda models were d e f i n e d as lambda a l g e b r a s s a t i s f y i n g t h e M e y e r - S c o t t

axiom. S i n c e t h e c o m b i n a t o r y axioms d e s c r i b i n g A - a l g e b r a s a r e n o t memorable, on'e may wonder w h e t h e r t h e s e c a n be s i m p l i f i e d i n p r e s e n c e o f t h e new axiom. T h i s i s indeed t h e case;

5.1

t h e r e s u l t i s due i n d e p e n d e n t l y t o Meyer and S c o t t .

DEFINITION. D e f i n e t h e f o l l o w i n g c o m b i n a t o r y terms.

l 1 = 1 = S(KI);

ln+l = S(Kl)(S(Kln)).

= 1 , In+, = S ( K I n ) 5.2, Remark. U s i n g t h e s i m p l e r d e f i n i t i o n l 1

5 . 3 and 5 . 6 ( i )

233

Lambda Calculus and its Models remain v a l i d .

5.2

LEMMA. ( i ) I f !D? i s a c o m b i n a t o r y a l g e b r a , t h e n

LY?C

( i i ) If!D?i s a A - a l g e b r a ,

then

. ab l...bn.

A k In= Aab l . . . b Proof. -

(i) (ii). 1,

5.3

.

= ab l . . . b

lnabl . . . b

I n d u c t i o n on n, e.g.

i n a A-algebra

= S(KI) = (Axyz.xz(yz))KI

= Ayz.Klz(yz)

THEOREM. (Meyer [19801; S c o t t [ 1 9 8 0 ] ) .

0

= Xyz.yz.

L e t A = (X,.,k,s).

Then!D? i s a

A-

model i f f !D? s a t i s f i e s 1 . Kxy = x ,

2 . sxyz = xz(yz),

3. Vx ax

*

= bx

l a = lb,

4. 1 2 K = K , 5. 1 Proof. -

(*)

s=

5.

I f A i s a A-model,

then by d e f i n i t i o n 1,2,3

hold. MoreoverA i s a

hence s a t i s f i e s 4,s s i n c e t h e s e e q u a t i o n s a r e p r o v a b l e i n

A-algebra,

(*)

3

A.

F i r s t show t h a t f o r a l l a , b E A l ( K a ) = Ka and l ( S a b ) = Sab.

Indeed, Ka = l Z K a = S(K1)Ka = l ( K a ) and s i m i l a r l y f o r S. S i n c e 1x.A i s always o f t h e f o r m KP o r SPQ i t f o l l o w s t h a t

(*)

l(Ax.A)

= Ax.A.

T h e r e f o r e A i s weakly e x t e n s i o n a l : Vx

* * *

A=B

Vx(Ax.A)x = (Ax.B)x l(Ax.A) = l(Ax.B), Ax.A = Ax.B,

by

by 3,

(*I.

I t remains t o show t h a t !D? i s a A - a l g e b r a ,

f o l l o w s by i n d u c t i o n on t h e p r o o f o f M = N , rule P=Q

9

Xx.P

i.e.

A I-

M=N

*

(JR

C M = N . This

weak e x t e n s i o n a l i t y t a k i n g c a r e o f t h e

= Ax.Q.

The f o l l o w i n g , d e f i n i t i o n o f Meyer [1980]

s i m p l i f i e s even more t h e d e s c r i p -

t i o n o f t h e essence o f a A-model.

5.4 fying

DEFINITION.

( i ) A c o m b i n a t o r y model i s a s t r u c t u r e A = (X,-,k,S,E)

(1)

Kxy = x ,

(2)

sxyz = xz(yz),

(3)

EXY

=

(4)

Vx

ax = b x

XY,

+

Ea = Eb.

satis-

H.BARENDRECT

234

( i i ) A c o m b i n a t o r y model i s s t a b l e i f moreover

(5) (6)

EE = E ,

E ~ K= K,

(7)

5.5

s,

E3S =

Here, o f c o u r s e ,

E

1

= E and E

-

n+l

=S(KE)(S(KE~)).

LEMMA. L e t 9 R = ( X , . , ~ , S , E >

(i) ~ (ii)

E

~ = +a a = a

be a c o m b i n a t o r y model.

~ A V X E(ax) =

vx l . . . x i

~9

(iii) l i s stable

Proof.

Ea a =

~

...x . )

€(axl

ax. = ax l . . . x . ,

O
~ , k , k a , s , s a , s a b a r e f o r a l l a,b

c9

f i x e d p o i n t s o f E.

( i ) (a)By a s s u m p t i o n

a = S(KE)(S(KEn))a

(1)

= E(S(KEn)a).

Hence by 5.4(3) ax = S ( K E ) a x = E (ax) ; t h e r e f o r e by 5 . 4 ( 4 )

and (1)

Ea = E(S(KEn)a) = a. (e)I n a c o m b i n a t o r y model one has cab = ab,

therefore

(2)

E(Ea) = Ea

Now ~ ~ + E~( Sa( K=E ~ ) ~ )t,h e r e f o r e by (2)

(3)

ax

Hence by 5.4(4)

= ~

E(En+la)

but also

=

E

~

,+

~

a

(ax) = ax.

and assumption E ( E ~ + ~ = ~Ea) = a .

Together w i t h

(3) t h i s i m p l i e s ~

~ = +a .

~

a

( i i ) By i n d u c t i o n on n .

( i i i ) (=+) As t o Sab: (4)

Sab =

E

3

Sab = E(Sab).

As t o Sa: Sa =

and i t f o l l o w s by (4)

E

3

Sa = S(KE) (sa)

,

and ( 2 ) t h a t Sa i s a f i x e d p o i n t of

E.

S i m i l a r l y i t f o l l o w s t h a t S, Ka and K a r e f i x e d p o i n t s . By assumption E i s a f i x e d p o i n t o f E. (e)By a s s u m p t i o n and ( i i ) .

5.6

~

0

PROPOSITION. ( i ) L e t B = ( X , . , ~ , S , E )

= and l( X , . , k , s )

i s a A-model.

(ii) I f l = (X,.,~,S,E)

be a s t a b l e c o m b i n a t o r y model. Then

Moreover k , s a r e u n i q u e l y d e t e r m i n e d by E .

i s a c o m b i n a t o r y model, t h e n W = ( X , * , k ' , s ' , ~ ' ) i s

a s t a b l e cornbinatory model, where k ' = e g k , s ' = E k and E '

3

=EE.

235

Lambda Calculus and its Models

Proof. -

( i ) Note t h a t

*

xy = l x y ; by 5.4(4),

EE = € 1

, , ,

E = 1

,

by s t a b i l i t y and 5 . 5 ( i i i ) .

Ex = E(1X)

-

= lx

* T h e r e f o r e (X,.,k,s)

by 5 . 5 ( i i i )

since l x = S ( K l ) x ,

by 5 . 4 ( 4 ) ,

i s by 5.3 a A-model.

As t o u n i c i t y ,

l e t ( X , . , k o , s 0 , ~ ) be a l s o a s t a b l e c o m b i n a t o r y model,

in

o r d e r t o show k = k o, s = s o . Then kxy = x = k 0x y ss

E(kx) = E(kOx)

+

s(kE)kx = s(kE)kox

+

E(s(kE)k) = E(s(kE)ko)

+

~

=+ Similarly s = s

k

2

=

~

2 0

k = k

k

0'

0'

( i i ) F i r s t note

(1)

EYZ

= yz, therefore

E(EY) = EY.

Now l e t x E { ~ ' , k ' , k ' a , s ' , s ' a , s ' a b } . By 5 . 5 ( i i i ) since then

E'X=EEX=EX=X.

i t s u f f i c e s t o show t h a t E X = X

S i m p l e c a l c u l a t i o n s show t h a t x = ~ yf o r some y ( e . 9 .

x = k ' = ~k = s ( k e ) ( s ( k E ) ) k = E ( s ( k s ) k ) ) . 2 0 E X = E ( E Y ) = EY = x.

Then by ( 1 ) i t f o l l o w s t h a t

A l t h o u g h i n A-models k,s a r e u n i q u e l y d e t e r m i n e d by 1 = s ( k i ) , p r e s e r v e s a p p l i c a t i o n and c o n s t a n t map Ax. 1 ' : I +

5.7

DEFINITION.

a map t h a t

1 i s n o t n e c e s s a r i l y a homomorphism: t a k e e.g.

the

I' .

( i ) L e t m = (X,.)

be a c o m b i n a t o r y c o m p l e t e a p p l i c a t i v e s t r u c -

t u r e . An e x p a n s i o n o f I i s o f t h e f o r m (?V,k,s)

= (X,.,k,s)

which i s a combinatory

a1 gebra. (ii)

I= (X,.)

i s a c a t e g o r i c a l A-model

t h e r e i s a u n i q u e e x p a n s i o n (9?,k,s)

making

( A - a l g e b r a , cornbinatory a l g e b r a ) i f

I into

a A-model

( A - a l g e b r a , combi-

natory algebra). ( i i i ) An element E o f I i s c a l l e d a s t a b l e E i f E E = E Vxax=bx

5.8

A

cab=ab

A

Ea=Eb.

+

THEOREM.@Let !Dl = (X;)

be c o m b i n a t o r y complete.

f r e p r e s e n t a b l e } and d e f i n e F : X + [ X + X l t o a A-model

i f f there exists a G : [ X + X ] + X

1.

FOG = i d

2.

GoFE [ X + X ] .

Let [ X + X ]

= {f : X+X

such t h a t

[x+xl;

( i i ) The G ' s s a t i s f y i n g 1,2

I

by F ( x ) ( y ) = x y . Then !Dl can be expanded

i n ( i ) correspond e x a c t l y t o s t a b l e

E'S.

H.BARENDREGT

236

( i i i ) VJt i s a c a t e g o r i c a l A-model

i f f the G i n ( i ) i s unique i f f there i s a unique s t a b l e Proof.

(i)

(-1

E.

L e t (m,k,s) be a A-model.

Oef i ne G(f) = l a f f o r some af r e p r e s e n t i n g f . G i s w e l l - d e f i n e d :

i f a x = f ( x ) = a ' x f o r a l l x, then

l a = l a ' by t h e M e y e r - S c o t t axiom. C l e a r l y F ( l a ) = F ( a f ) = f , so FOG= i d . Moreover f GoF(a) = l a , s i n c e a r e p r e s e n t s F ( a ) ; hence GoF i s r e p r e s e n t a b l e . (e)L e t k O , s O E X s a t i s f y t h e k,s

axioms. D e f i n e E,,=G(GoF).

Then

( X , . ,kO,sO,EO) i s a c o m b i n a t o r y model:

EOab = G(GoF)ab

= F(FoG(GoF) (a)) (b) = F(a)(b),

since FoG=id,

= ab.

Vxax=bx

=e

F ( a ) = F(b)

=e

~

GoF(a) = GoF(b) a

0

0

=

~

b

s i n c e E~ r e p r e s e n t s GoF. I t f o l l o w s by 5 . 6 ( i i )

t h a t R can be expanded t o a A-model.

( i i ) As i n ( i ) d e f i n e G E ( f ) = E a f actually stable:

~ =E

E

and E ~ = G ( G o F ) .F i r s t n o t e t h a t

F (~ E ~ ) ( E ~= ) F(G(GoF)) (G(GoF))

E~

is

= E ~ .

Moreover AG.E

and XE.G a r e i n v e r s e o f each o t h e r : G ( f ) = EGaf = F O G ( G O F ) ( ~ ~ )

G EG

= GoF(af) = G ( f ) ; = GE(GEoF) = Ea

E

GE

= EE,

since

E

GEoF

represents G O F :

GEoF(b) = EaF(b) = Eb, = E.

---

( i i i ) R i s a c a t e g o r i c a l A-model

c)

t h e r e a r e u n i q u e k , s m a k i n g W i n t o a A-model t h e r e a r e u n i q u e k , s , ~m a k i n g W i n t o a s t a b l e c o m b i n a t o r y model there i s a unique s t a b l e E t h e r e i s a u n i q u e G s a t i s f y i n g 1,2

in (i).

0

F i n a l l y some r e s u l t s a b o u t t h e c a t e g o r i c i t y o f two A-models. The argument is taken from Longo C19831, where the result i s shown in a general setting.

5.9

THEOREM. ( i ) (Bruce, Longo).

Pw i s a c a t e g o r i c a l A-model.

231

Lambda Calculus and its Models ( i i ) Pw i s n o t a c a t e g o r i c a l c o m b i n a t o r y a l g e b r a . Proof. -

( i ) D e f i n e x E P w t o be s a t u r a t e d i f ( n , m ) E x

A

enEen,

=)

(n',rn)Ex.

Step 1 . Assume x , x '

a r e s a t u r a t e d and Vy xy = x ' y , Then x = x ' .

Proof.

*

mExen=x'e

*

3en, c e (n'm) E x '

*

(n,m) E x ' , by s a t u r a t i o n .

(n,m)Ex

T h e r e f o r e x c x ' ; hence x = x ' by symmetry. Now l e t G s a t i s f y 1,2 o f 5.8 f o r Pw. Step 2.

( i ) a={(n,m)}.)aEiG(F(a)).

(ii) Vxf(x)zg(x)

Proof.

01

*

G(f)zG(g).

( i ) G(F(a))en = aen = { m l

*

3en, s e n ( " ' ,m) E G(F(a))

*

m E G ( F ( a ) ) e n ' = ae

*

n = n ' , o t h e r w i s e aen,

*

(n,m) E G(F(a))

n'

=0,

.

( i i ) Assume Vx f ( x ) _ c g ( x ) . Note t h a t

G(f)

= lG graph(f).

I mEf(en)}

where g r a p h ( f ) = {(n,m)

i s t h e s t a n d a r d g r a p h f o r Pw.

Therefore G ( f ) = lG graph(f)

5

lG graph(g)

uz

= G(g). Step 3. G ( f )

i s saturated.

P r o o f . Assume Then (n',m)

( n , m ) E G ( f ) and e n c e n I .

E GoF({(n',m)}),

-

c GoF({(n,m)}),

5

GoF(G(f)),

= G(f). F i n a l l y , l e t G,G'

by 2 ( i ) , by 2 ( i i ) ,

by 2 ( i i ) ,

0,

s a t i s f y 1,2 o f 5.8.

Then

G ( f ) ( x ) = f ( x ) = G ' ( f ) (x)

*

G(f) = G ' ( f ) ,

=)

G = G'.

T h e r e f o r e by 5 . 8 ( i i i )

by 3,1

,

Pw i s a c a t e g o r i c a l A-model.

3 , k = [ K J P w i s s a t u r a t e d . C l e a r l y k f 0 , s a y (n,m) E k . L e t e n 9 e n l . Then ( n ' , m ) E k and k ' = k - { ( n ' , m ) } a c t s e x t e n s i o n a l l y t h e same as k . ( i i ) By ( i ) s t e p

Hence k i s n o t u n i q u e .

0

H. BARENDREGT

238

5.10

[19831). ( i ) DA i s n o t a c a t e g o r i c a l A-model.

(Longo

THEOREM.

( i i ) As c m b i n a t o r y a l g e b r a s Pw and D Proof.

A

a r e n o t isomorphic.

I bEf(B)}.

( i ) I n DA t h e map G i s d e f i n e d by G ( f ) = { ( B , b )

But

G ' ( f ) = G ( f ) U A a l s o works. ( i i ) By ( i ) and

5.9(i).

119833

I n Longo

0

i t i s a l s o shown t h a t as a p p l i c a t i v e s t r u c t u r e s Pw and D A

a r e f o r c o u n t a b l e A m u t u a l l y embeddable i n t o each o t h e r .

REFERENCES Barendregt, H.P.,

119811

The Lambda c d c u l ~ t b ,k2 6yntaV and 6emaM.tic6, N o r t h H o l l a n d , Amstei.dam.

B a r e n d r e g t , H.P.

and Koymans, C . P . J . ,

[19801 Compaihing dome [ 19803, 287-302. Barendregt, H.P.,

dU4deA 06

Coppo, M.,

h b d a cdclLeus mod&,

Dezani-Ciancaglini,

i n H i n d l e y and S e l d i n

M.,

[19831 A d a e h Lambda model and Rhe compLeeteneA6

06

t y p e a6ignment, J. Symbolic

L o g i c , t o appear. Engeler, E.,

[1981]' Aegebhab and combinatom, A l g e b r a U n i v e r s a l i s , 13, 389-392. H i n d l e y , J.R.

[1980]

and Longo, G.,

Lambda cdcu&6 B d 2 6 , 289-310.

mod&

and e.xtenbionu&ty,

H i n d l e y , J.R. and S e l d i n , J.P. (Eds.). [19801 To H.B. C u r r y : EdsUyb on combinatoty Academic P r e s s , New Y o r k and London.

2. Math. L o g i k Grundlag. Math.,

Logic, Lambda-cdcLLeus and d o h m d h m ,

Jacopini, G.,

[19751

P h i n d p i o di M t e n b i o n W n e l cafkolo d e i combinatohi, C a l c o l o 1 1 , no. 4 , 465-471.

K a r o u b i , M.,

[1978]

K-theoty,

Koymans, C.P.J.,

[1983] Mode&

WI

in&odu&on,

06 .the

S p r i n g e r , B e r l i n and New York.

Lambda CaecLLeUd, I n f o r m a t i o n and C o n t r o l , t o appear.

Longo, G . ,

[19831 set-Rhtheoh&d mod& 06 h b d a c d c u R u s : theohicb, expavlbionb, homo%pkism, An. Math. L o g i c , t o appear. Meyer, A., i.4 a model 06 the h b d a C a e c l l h P, p r e p r i n t , L a b o r a t o r y f o r Comp u t e r Science, 545 Technology Square, Cambridge, Massachusetts 02139, USA.

[19801 What [1981]

Expanded v e r s i o n o f Meyer

[1980], Information and Control 52, 87-122.

Statman, R . ,

[I9801 On Xhe & f e n c e

d o s e d .tmi n -the t y p e d A-cdclLeud I , i n : [ l 9 8 0 ] , 511-534. 119821 CompLetenedd , i n v h n c e and A-de~inabLLiAy, J S y m b o l i c L o g i c , v o l . 47, 1 7-26. 04

H i n d l e y and S e l d i n

.

S c o t t , D.,

[1980]

R e l a t i n g -theohie~06 t h e h b d a Caecu&~, i n : H i n d l e y and S e l d i n t19801, 403-450.

Lambda Calculus and its Models Visser, A . , [19801 Numehatiom, X-cdcu&h 259-284.

and ahithmetic, in: Hindley and Seldin [19801,

239

LOGIC COLLOQUIUM'82 G. Lolli, G . Long0 and A. Marcia (editors) 0 Elsevier Science Publishers B. V. (North-Holland). 1984

24 1

EXTENDED TYPE STRUCTURES AND FILTER LAMBDA MODELS M. COPPO (1) F. HDNSELL ( 2 )

M. DEZANI-CIANCAGLINI (1) G. LONGD ( 3 )

D i p a r t i m e n t o d i Scienze d e l l ' I n f o r m a z i o n e , C. M. D ' A z e g l i o 42, 10125 T o r i n o . (2) Scuola Normale S u p e r i o r e , Pisa. (3) D i p a r t i m e n t o d i Scienze d e l l ' I n f o r m a z i o n e , U n i v e r s i t a d i Pisa. ITALY Research p a r t i a l l y s u p p o r t e d b y M.P. I . (Comitato p e r l a Matematica, f o n d i 40%). (1)

An Extended A b s t r a c t Type S t r u c t u r e i s a p r e - o r d e r e d s e t X which i n c l u d e s a l a r g e s t element and a + b, a A b, whenever a, b a r e i n X. Extended Type S t r u c t u r e s (ETS) may be g i v e n o v e r a p p l i c a t i v e s t r u c t u r e s , b y i n t e r p r e t i n g t h e p r e - o r d e r and " A " by s e t i n c l u s i o n and i n t e r s e c t i o n , r e s p e c t i v e l y . F o r any ,D model o f 1 - c a l c u l u s , t h e c l a s s o f b a s i c open s e t s , w i t h r e s p e c t t o t h e S c o t t t o p o l o g y , forms an ETS. The s e t o f f i l t e r s o f an ETS ( f i l t e r domain) i s an a l g e b r a i c complete l a t t i c e and may be t u r n e d i n t o a continuous a p p l i c a t i v e s t r u c t u r e . Domain w h i c h a r e models o f A-calculus ( f i l t e r A-models) a r e c h a r a c t e r i z e d . A c h a r a c t e r i z a t i o n i s a l s o g i m A-models which a r e r e f l e x i v e domains, t h a t i s which a r e domains where t h e s e t o f t h e c o n t i n u o u s f u n c t i o n s i s a r e t r a c t i o n . As a m a t t e r of f a c t , n o t any f i l t e r ?-model t u r n s o u t t o be a r e f l e x i v e domain. I n any f i l t e r h o d e l t h e i n t e r p r e t a t i o n o f a t e r m i s an element o f a t y p e ( s e t ) o f d a t a , as usual, as w e l l as a c o l l e c t i o n o f types; namely t h e f i l t e r o f t y p e s assigned t o i t by t h e t y p e assignment t h e o r y determined by t h e a s s o c i a t e d ETS. Moreover the f i l t e r &model i n [ 2 1 i s shown t o be i s o m o r p h i c , as a f i l t e r model. Also, domain, t o an e x p l i c i t l y g i v e n s u b s t r u c t u r e o f a D, any ( c o u n t a b l e ) a p p l i c a t i v e s t r u c t u r e may be i s o m o r p h i c a l l y embedded i n t o t h i s f i l t e r domain.

0.

INTRODUCTION

Type symbols a r e used i n v a r i o u s areas of Mathematical L o g i c and Computer Science as a f o r m a l r e p r e s e n t a t i o n o f c o l l e c t i o n s o f f u n c t i o n s , o f f u n c t i o n a l 5 o v e r f u n c t i o n s etc... I n Recursion Theory i n h i g h e r t y p e s one u s u a l l y b e g i n s w i t h j u s t one atomic t y p e symbol, 0 say, t o be i n t e r p r e t e d as t h e s e t o f n a t u r a l numbers N , and t h e n d e f i n e s t h e s e t o f t y p e symbols as t h e s m a l l e s t s e t c l o s e d under " + ' I . The meaning o f 0 0' i s t h e s e t o f f u n c t i o n s f r o m N t o N (see, f o r example, [131). Thus t h e i n t e r p r e t a t i o n o f t y p e s i s f i x e d and i t g i v e s t h e Type S t r u c t u r e as c o l l e c t i o n s o f f u n c t i o n s o v e r N e t c . i n any f i n i t e type. The use o f t y p e s f o r s t u d y i n g f u n c t i o n a l p r o p e r t i e s o f terms o f (untyped) 1 - c a l c u l u s i s due t o C u r r y (+) [51. C u r r y ' s t y p e s (which s h o u l d be b e t t e r c a l l e d " t y p e schemes") a r e j u s t s y n t a c t i c o b j e c t s , b u i l t f r o m a s e t o f v a r i a b l e s by an o p e r a t o r " + I ' o f t y p e f o r m a t i o n . They a r e assigned t o 1 - t e r m s

..,

(+)

While we were w r i t i n g t h e f i n a l v e r s i o n o f t h i s paper, t h e was saddened by t h e d e a t h o f P r o f e s s o r H a s k e l l B. Curry. P r o f e s s o r C u r r y s e v e r a l times. Besides h i s c o n t r i b u i t i o n s , a landmark f o r L o g i c , we c o u l d a l l admire h i s l i v e l y presence, h i s enthus iasm.

l o g i c community Most o f u s met which have been and encouraging

2 42

M.COPPO ET AL.

by formal assignment r u l e s . In t h i s way d i f f e r e n t types can be assigned t o t h e same 1-term. This i s the main d i f f e r e n c e between Curry's approach and t h e typed A-calculus 1111, where types a r e b u i l d a s in Recursion Theory and each term has a uni ue type. In [ 21 a n d ? 3 I a conservative extension of Curry's system i s g i v e n . This i s done by allowing e x p l i c i t l y t h a t t o a l - t e r m i s assigned more than one type. More formally a new o p e r a t o r " 4 ' ' of type formation and t h e " u n i v e r s a l " type w a r e introduced. The main f e a t u r e of t h i s extension i s t h a t types c h a r a c t e r i z e completely t h e functional behaviour of A-terms, see [ 3 I ( n o t i c e t h a t any 1-term has a t l e a s t one t y p e ) . S c o t t [181 gave a mathematical semantics f o r Curry's types which can be n a t u r a l l y extended t o t h e new types in [ 2 1 . Given an a p p l i c a t i v e s t r u c t u r e , types a r e i n t e r p r e t e d a s s u b s e t s of 0 , where, f o r A,BcD, A + B = { dcD PdecA d .eeB 1. Moreover " A " i s i n t e r p r e t e d a s set i n t e r s e c t i o n and I, w" a s t h e whole s e t D. By t h i s i n t e r p r e t a t i o n one has i n c l u s i o n r e l a t i o n s between t y p e s , which a r e represented in [ 2 1 by t h e (formal) r e l a t i o n " L " . A t y p i c a l case i s t h a t i f 0'50and T(T' then U+TLO' + T I ( c f r . 1.1.7): i f 0' i s smaller than u and r i s s m a l l e r than T', then t h e "functions" from u t o T a r e l e s s than those from u ' t o T I . On t h e o t h e r hand one may think i n a dual way. S t a r t i n g from t h e f a c t t h a t a type r e p r e s e n t s , in some sense, a domain-range information about a A-term, one may c o n s t r u c t a 1-model in which t h e i n t e r p r e t a t i o n of a term i s t h e s e t of i t s types. Since t h e s e s e t s a r e closed under " A " and " 5'' (upward), they turn o u t t o be f i l t e r s . This c o n s t r u c t i o n i s e x p l i c i t l y done in [ 2 1 , but i t i s , t o some extend, a common f e a t u r e of a l l 1-models defined by P l o t k i n ' s technique [141. Take, say, S c o t t ' s Pw model [191 o r Engeler's D A model [ 6 1 . There any d c DA i s a s e t of " i n s t r u c t i o n s " such a s ( B e b ) , where B E D A i s a f i n i t e s e t . I f ( g * b ) ~ d , then BLecD implies b f d - e , t h a t i s d c & + { " b } & D A , A where =Id cDAl c c _ d > . Thus ( B * b ) i s c l e a r l y r e l a t e d t o t h e type k + { i ~ l .This w i l l be used in several places i n t h i s paper and more formally s t u d i e d , p a r t i c u l a r l y when dealing with D, models. In t h e present paper t h e notion of types and t h e i r i n c l u s i o n p r o p e r t i e s a r e a b s t r a c t l y formalized i n t h e d e f i n i t i o n of Extended Abstract Type S t r u c t u r e (EATS). Actually EATS a r e information systems i n t h e sense of S c o t t [211 ( c f . 1 . 9 ) . I t i s i n t e r e s t i n s t o consider EATS given over a p p l i c a t i v e s t r u c t u r e s (they a r e c a l l e d ETS). ETS can be viewed a s i n t e r p r e t a t i o n s ( i n t h e sense of S c o t t ) of formal types. Continuous a p p l i c a t i v e s t r u c t u r e s over EATS ( f i l t e r domains) a r e defined. We then i n v e s t i g a t e embeddings and isomorphisms between f i l t e r domains. In p a r t i c u l a r , t h e f i l t e r domain defined in [21 i s shown t o be " u n i v e r s a l " in t h a t any f i l t e r domain i s isomorphic t o the range of a c l o s u r e operation which i s an element of IFI. Moreover we o b t a i n simple r e l a t i o n s between t h e p r o p e r t i e s of "2 of an EATS and the c l a s s of r e p r e s e n t a b l e f u n c t i o n s over the a s s o c i a t e d f i l t e r domain ( c f . 2.13). Some f i l t e r domains can a c t u a l l y be turned i n t o models of type f r e e 1-calculus ( f i l t e r A-models). An i n t e r e s t i n g c l a s s of them (which has a simple c h a r a c t e r i z a t i o n in terms of " < " ) i s t h e c l a s s of f i l t e r domains i n which a l l continuous functions a r e representable. However t h e r e e x i s t a l s o f i l t e r A-models i n which not a l l continuous f u n c t i o n s a r e r e p r e s e n t a b l e ( s e e 4.11). Embeddings and isomorphisms between r e f l e x i v e domains ( c f . [ l n ) and f i l t e r h o d e l s a r e s t u d i e d : i n p a r t i c u l a r any Dmspace i s isomorphic t o a ( s u i t a b l e ) f i l t e r h o d e l . L a s t l y j u s t using ( e a s i l y axiomatizable) a b s t r a c t 'I<'' r e l a t i o n s between formal type, we c o n s t r u c t some f i l t e r 1-models isomorphic to-D, spaces. In one of these D ,spaces t h e " u n i v e r s a l " f i l t e r isomorphically embedded.

1-model

can be

2 43

Extended Type Structures and Filter Lambda Models 1.

EXTENDED TYPE STRUCTURES

1.1. D e f i n i t i o n . An Extended A b s t r a c t Type S t r u c t u r e (EATS) S i s a s t r u c t u r e < X , ~ , A . + , ~ > , where X i s a set,weX,"A"and 14diare t o t a l f u n c t i o n s f r o m X x X t o X and k " i s a p r e o r d e r r e l a t i o n on X s a t i s f y i n g : 1. a L W 2. w 5 w+w 3. a L a A a 4. h b l a aAb5b 5. (a;b) A +c) L a +(bhf) 6. a s . , b d * a h b c a ' h b 7. a ' z a , & ' * a + b c a ' + b ' e X. where a,b,c,a',b'

(7

L e t a ?.b i f f a5bQ. Observe t h a t w w + w

and a

A

(bAC)

(ahb)hc.

...,in}, then

NOTATION: I and J w i l l always be f i n i t e s e t s o f i n d i c e s . I f I= {il, ,A a . means a . A a

I

'

11

i2/\

i;

1.2. EXAMPLES. ( i ) L e t T be t h e s e t o f f o r m a l t y p e s b u i l t f r o m w a n d a ( c o u n t a b l e ) s e t At= {$o, 4 l,. 1 o f t y p e v a r i a b l e s by t h e ( s y n t a c t i c ) o p e r a t o r s

..

" +'I

and

"A"

o f 1.1 t h e n

o f type formation. F=is

I f ''zO" i s t h e minimal p r e o r d e r s a t i s f y i n g 1-7 t h e f r e e EATS o v e r g e n e r a t o r s

...

4 0 . 4 ~ ~

defined i n [ 21. (ii)Consider $ = where P i s t h e s e t o f w . f . f . of ( p r o p o s i t i o n a l ) d e r i v a t i v e + A - l o g i c [16, p. 2851 , aeP and "(I' i s d e f i n e d by " p l q i f f p t q " . 9' i s an EATS s i n c e 1-7 t r i v i a l l y h o l d . and

Our " c o n c r e t e " EATS w i l l always be g i v e n o v e r an a p p l i c a t i v e s t r u c t u r e and " A " w i l l be ( i n t e r p r e t e d b y ) s e t t h e o r e t i c i n c l u s i o n and i n t e r s e c t i o n .

"2'

1.3. D e f i n i t i o n . ( i ) L e t be a ( p a r t i a l ) a p p l i c a t i v e s t r u c t u r e and A , B g . Define then A+B = { d r 01 VeeA d.eeB 1 €PO (if i s a p a r t i a l o p e r a t i o n , b y d.eeB we mean: I'd-e i s d e f i n e d and d e B " ) . ( i i ) L e t be an a p p l i c a t i v e s t r u c t u r e . An Extended T e S t r u c t u r e (ETS) ( o v e r D ) i s an EATS S = < p, 5 , n, +,D>, where PSPD, C_ a n d K p a r e s e t i n c l u s i o n and s e t i n t e r s e c t i o n and B t P .

"."

I n o t h e r words an ETS i s a s e t o f subsets o f an a p p l i c a t i v e s t r u c t u r e , n o t c o n t a i n i n g t h e empty s e t , and c l o s e d under " n " and ' I + ' ' . I t i s t h e n easy t o check t h a t t h e c o n d i t i o n s o f 1.1 a r e s a t i s f i e d (indeed, "L'" i s a p a r t i a l order). 1.4. EXAMPLE. ( i ) L e t
i n [ 1 9 1 ) and P = I A ' d d e A } . T h e n < P , E , n , + , D > i s an ETS. (ii)Each c o l l e c t i o n o f a l g o r i t h m s as d e f i n e d i n [ 1 2 1 i s an ETS c l o s e d under i n f i n i t e intersection. L e t T be t h e s e t o f f o r m a l t y p e s as d e f i n e d i n 1 . 2 ( i ) . Given any EATS S o v e r a c o u n t a b l e s e t X we can d e f i n e t h e f o r m a l t h e o r y o f S b y a ( s u i t a b l e ) o r d e r r e l a t i o n on T. 1.5. then

Definition.

( i ) L e t T be t h e s e t o f t y p e s o f 1 . 2 ( i ) . If u ,TET, i s a formula ( 1 ) . (ii)A t y p e t h e o r y T i s any s e t o f f o r m u l a s c l o s e d under 1-7 o f 1.1 p l u s t r a n s i t i v i t y and r e f l e x i v i t y . u c T 'I s t a n d s f o r ULTET. UZT

M.COPPO ET AL.

2 44

( i i i ) I f 1 i s any s e t o f f o r m u l a s t h e n T ( z ) i s t h e t h e o r y generated by z . < -z s h o r t f o r i T [ 2) (iv) S (T)

.

.

For any t y p e t h e o r y 1.2(i). 1.6.

is

T, S ( T ) i s t r i v i a l l y an EATS. L e t To be t h e f r e e t h e o r y o f

A , + ,w> Definition. ( i ) L e t Ss < X , ( , i s a f u n c t i o n V: T + X such t h a t :

be

an

EATS.

Then

a

2. V( O A T ) = V ( U) A V ( T) 3. v(U*T)= v ( U)* V(T). We say t h a t < S,V> i s a t e model (a c o n c r e t e t y p e model when S i s an ETS). (ii)If<S.V> i s a t y p h t s t h e o r y TV i s g i v e n by TV = { O T I V ( O ) G v(T)}*

E.VT

stands f o r

U ~ T

Tv.

N o t i c e t h a t g i v e n any EATS S, we can always f i n d many V:T+X such t h a t <S,V> i s a t y p e model. C l e a r l y , i f X i s c o u n t a b l e , V can be made s u r j e c t i v e . O f course, t h e c o n d i t i o n on c o u n t a b i l i t y may be dropped i f one t a k e s t h e s e t A t o f atoms o f the desired cardinality. F i.n a l l y , i f V i s o n t o , Obviously Tv= T (Tv) and ToETV ( i . e . ''2; extends " 3 " ) one c l e a r l y has


z v , , -+ A

,w>

< X,L,

I

A

+ ,

@>

.

Some more work can be done w i t h EATS, l o o k i n g a t c o l l e c t i o n s o f t h e i r subsets. be an EATS. D e f i n i t i o n . L e t S= <X, < A , *,w> An a b s t r a c t f i l t e r x o f 3 i s a non empty subset o f X such t h a t : 1. a.be x =. h b e x 2. a; x, a 3 * k x . ( i i ) If AGX, t A i s t h e a b s t r a c t f i l t e r generated by A. I f A= {a), f o r fIa1. ( i i i ) IS1 i s t h e s e t o f a b s t r a c t f i l t e r s o f S ( f i l t e r domain o f S ) . 1.7. (i)

I f S i s an ETS, IS1

i s clearly the set o f f i l t e r s o f

f a stands

S i n t h e u s u a l sense.

1.8. LEMMA. < IS1 & > i s a complete a l g e b r a i c l a t t i c e , where f w and X a r e t h e l e a s t and t h e l a r g e s t elements ( r e s p e c t i v e l y ) . Moreover i f x,ye I S I : (i) x w = t (xL5) ( i i ) xny = X ~ Y ( i i i ) IfA G l S l i s a d i r e c t e d s e t , t h e n U A = U A . ( i v ) The finite elements are exactly the principal filters, i.e. x = u { f a 1 facx 1 Proof. Easy. 0 1.9. REMARKS. (i) EATS a r e i n f o r m a t i o n systems i n t h e sense o f S c o t t 1211. I n f a c t , an EATS s<X,< , A , *, w> i s an i n f o r m a t i o n system (X,u,Con,+) where Con c o n s i s t s o f a l l - f i n i t e subsets o f X and, i f A = {al, a n } , At-b iff a,&

... ~a~

...,

5 b (and +I-b i f f

w~

b ) . Moreover

IS1 i s t h e s e t o f elements o f

t h e c o r r e s p o n d i n g i n f o r m a t i o n system. ( i i ) Any ETS < p , c,n , *,D > i s a neighbourhood system i n t h e sense o f [ Z O I . Moreover i f we d e f i n e : AfdB * deA+B (where deD and A , W P ) t h e n f d i s an approximable mapping, as d e f i n e d i n [201.

245

Extended Type Structures and Filter Lambda Models FILTER DOMAINS

2.

This s e c t i o n mainly deals w i t h p r o p e r t i e s o f f i l t e r domains ( o f EATS), viewed as a p p l i c a t i v e s t r u c t u r e s . I n t h e sequel complete l a t t i c e s w i l l always be considered w i t h t h e S c o t t topology ( c f . [17]). D e f i n i t i o n . ( i ) I f D i s a complete l a t t i c e ( w i t h respect t o "I")and " ' " : DxD+D i s continuous, then is a continuous a p p l i c a t i v e s t r u c t u r e . ( i i ) A continuous a p p l i c a t i v e s t r u c t u r e i s a l g e b r a i c i f f D i s algebraic.

2.1.

Given any EATS structure. 2.2.

S, one may t u r n Is1 i n t o an a l g e b r a i c continuous a p p l i c a t i v e

D e f i n i t i o n . L e t S be an EATS. Define: x IS1 + IS1 by x-y = { b 13aEy a + bexl.

"'":ISI

2.3. (ii)

LEMMA. ( i ) x , y r I S I * x ' y ~Is! i s an a l g e b r a i c continuous a p p l i c a t i v e s t r u c t u r e .

Proof. Routine (cf.Lemma 1.8.).

0

REMARKS. ( i ) L e t T be a type theory and S ( T ) as i n 1.5. Using 2.4. one can e a s i l y show t h a t T i s t h e theory o f a concrete t y p e model. J u s t d e f i n e vT (oi)= I X E I S ( T ) I 1 oiex 1 S * ( 1) = <;(T) , C , n ,+,IS ( T ) I > Then an easy i n d u c t i o n shows t h a t V T ( 0 ) = I x E [S(T)[I x 1

(2).

and t h a t <S* (T), VT > i s a type model whose theory i s e x a c t l y T ( c f . Theorem 1.10 o f [ 201). as defined i n [ 61). Given a s e t A, l e t XA be t h e ( i i ) (Connections w i t h , c l o s u r e o f A " h ) ( w h e r e o s A ) under ' ' + I ' and "n."-Then, i f SA =<XA, 5, A , +,w >, i t can be e a s i l y proved t h a t < I S A ~ : ,E>~
,C > ( 3 ) .

Define v : XA +DA by v (W)' P v (a)= {a} f o r a l l aeA v (b +c)= { v ( b ) + d I d c V ( C ) 1 v (br, c ) = V(b)uv(C) and v* : I S A ~ DA by +

v*(x)="

cc x

v(c).

A r o u t i n e c a l c u l a t i o n shows t h a t v* i s an embedding. As usual, i f i s an a p p l i c a t i v e s t r u c t u r e t h e s e t o f representable f u n c t i o n s over i s given by: .(DiD)= { f : D'+D 13x'D VyeD x-y = f ( y ) } . Clearly, i f < D 1 - , L > i s a continuous a p p l i c a t i v e s t r u c t u r e , then (D+D) C C(D,D), t h e s e t o f continuous functions from D t o D. I f we d e f i n e F ( x ) ( y ) = x - y then F i s a continuous map o f D i n t o C(D,D) (onto (D+D)). Notice t h a t ( D + D ) i s a complete l a t t i c e by t h e c o n t i n u i t y o f F.

2.5.

D e f i n i t i o n . (i) A r e - r e f l e x i v e domain i s a t r i p l e D i s a complete*lattice (2) FE C(D,C(D ,O)) and GcC((D+D), 0 ) (where (D+D)=F(D)) ( 3 ) F O G = i d (4).

(1)

such t h a t

M.COPPO ET AL.

246

!

i s algebraic i f f D i s algebraic.

domain -

REMARK. I f i s a p r e - r e f l e x i v e domain t h e n G O F i s a r e t r a c t whose 2.6. range i s i s o m o r p h i c t o ( D + D ) . I f i s a d d i t t i v e ( c o a d d i t i v e ) t h e n G O F i s a closure (projection). < I S I , * > can be t u r n e d ( i n more t h a n one way i n g e n e r a l ) i n t o a p r e - r e f l e x i v e domain. However, i t i s u s e f u l t o c o n s i d e r a p a r t i c u l a r c h o i c e o f G. B u t we f i r s t need a lemma.

2.7. (1) (2) (3)

LEMMA. L e t S be an EATS and X E I S I . Then t h e f o l l o w i n g a r e e q u i v a l e n t : a+bex bex - t a a+ b e t { c -4 I dex. t c }.

Proof. (1) * ( 2 ) . By d e f i n i t i o n o f ' I - ' ' . c + bex * a+ bex ( s i n c e c + b j a + b ) . (2) * ( 1 ) . box - ? a 3ccta ( 3 ) * ( 1 ) . By assumption f o r some I ( 5 ) A c i + d i i a + b =)

a n d V i € 1 di (2)

=.

EX

.fci

. Thus,

I by ( 2 ) * ( l ) ,V i

E

I

ci +diex

and t h e n a+ bex.

(3). T r i v i a l . 0

The lemma suggests how t o o b t a i n , g i v e n an EATS, a c a n o n i c a l G. 2.8.

THEOREM. L e t S be an EATS. D e f i n e

feC(ISI

, Isl),

G,(f)=tIa+b

L e t Go be t h e r e s t r i c t i o n o f ,G

I

".'I

bef(ta)}

( a n d F ) as above a n d s e t , f o r

. Then

FoG,:id.

t o (IS1 +IS1 ) , i s a c o a d d i t i v e

p r e - r e f l e x i v e domain. Proof. L e t s = < X , < , A , + , W > and f e C ( I S I , I S I ) . Since { t a l a e X } i s t h e s e t o f f i n i t e elements oPIS1 by 1 . 8 ( i i i ) one has f ( x ) = u { f ( t a ) ] . Thus, f o r a l l x e IS1 aex f ( x ) ={bl 3aex bef(fa)} c{bl 3aax a+be G (f)) G*(f).x. That i s fLF,G,

(f).

Note t h a t i f f e ( I S 1 + I S I ) , f = F ( z ) say, then, by Lemma 2.7, b e f ( t a ) = z * t a . Thus, i n t h i s case, one a c t u a l l y has f = FaG,,(f). Moreover, t a k e bi

E X

-?ai

CE

, ieI.

Eo

F ( x ) = ?{a+ b

By Lemma 2.7

I bex-fal.

Vie1 ai+bi

I t i s a r o u t i n e c a l c u l a t i o n t o show t h a t the Scott topology. n

Then, f o r some 1 , A a . +b I

E X

&

a + b e G o ( f ) =.

which implies

CEX.

i

Thus

< c with

i$0

FLid.

and F a r e c o n t i n u o u s w i t h r e s p e c t t o

I f S i s such t h a t , f o r some G ' , i s a non t r i v i a l r e f l e x i v e domain t h e n we would o b t a i n a A - c a l c u l u s model ( c f . [ 1, 18.11 ). We w i l l say t h a t i s a f i l t e r A-model i f f i t i s a &model. A f i l t e r X-model i s n o t n e c e s s a r i l y a r e f l e x i v e domain, c f . 4 . 1 1 ( i i ) . EXAMPLE. L e t F= be as i n 1 . 2 ( i ) , t h e n i s a c o a d d i t i v e r e f l e x i v e domain (see t h e remark a f t e r Lemma 2.13). I f we d e f i n e G'(f)= G o ( f ) u A t where f e ( l FI +I FI ) and A t i s d e f i n e d i n 1 . 2 ( i ) we can e a s i l y p r o v e

2.9.

t h a t

i s an a d d i t t i v e r e f l e x i v e domain ( j u s t mimic [ 9 ] f o r a p r o o f . ) .

247

Extended Type Structures and Filter Lambda Models

2.10.

REMARKS.

( i ) Theorem 2.8 a c t u a l l y proves t h a t

i s a continuous

r e p r e s e n t a t i o n between C ( ISl,ISI) and IS1 a c c o r d i n g t o t h e d e f i n i t i o n o f 1151, i . e . F o G& i d and 0 F 5 id.

(ii)F o & i s a c l o s u r e o f C(ISI ,IS1 ) whose range i s (IS1 +IS1 ) (use f L F o C, ( f ) f o r feC(IS1 ,I 9 ) and f = F 0 G, ( f ) f o r f e (IS1 + IS I)). ( i i i ) L e t Dk be as i n 2 . 1 1 ( i ) .

I t i s easy t o show t h a t i f is

i( Dk )s( D +Dlk. (G^(

(coaddi t t i v e ) a1 g e b r a i c p r e - r e f 1 e x i ve domain t h e n

i0((

an a d d i t i v e

(D

+

D l k )GDk 1.

Therefore I SI+lSI ) ) c Is f o r a l l EATS S. Moreover t a k e F as i n 2.9, t h e n k f( I Flk)cC( IF1 , IF1 ) k y s i n c e i s an a d d i t t i v e r e f l e x i v e domain. I t i s

k

easy t o see t h a t t h i s i s n o t t r u e f o r a l l EATS S . There a r e some s i m p l e c o n d i t i o n s on EATS which correspond t o t h e d e f i n a b i l i t y o f c l a s s e s o f c o n t i n u o u s f u n c t i o n s (among them, t h e c l a s s o f a l l continuous functions). 2.11. D e f i n i t i o n . (i) L e t D be an a l g e b r a i c complete l a t t i c e . D e f i n e Dk'ICeDI c i s f i n i t e I .

(ii)L e t D and D ' be a l g e b r a i c complete l a t t i c e s . A s t e p f u n c t i o n f a b : D + D ' d e f i n e d by

1

fab(c)'

is

b i f a& I' o t h e r w i s e

where aeDk, I X D l k and

1'

i s t h e l e a s t element o f D ' .

The f i n i t e elements o f C ( D , D ' ) a r e e x a c t l y g i v e n by t h e f u n c t i o n s Uf,.,, where a.eD b. E D ' i e I . Note t h a t faibrc)=Y {bil ai cl. I ii 1 k' k' Thus (*) iff J = { i / a i r c l # m a n d dL Ubi. J 2.12. D e f i n i t i o n . L e t S an EATS. We d e f i n e t h e f o l l o w i n g c o n d i t i o n s on S : C1) ai +bi z c + d h bi 'd

y

fcdbyfaibi

+

C2) C3)

C

I

a +b_u -td and dl;w=r c a and b y A ai+bi 5 c+d=dl;w*J={i Iczai

I

l#S@3

biId.

C o n d i t i o n (*) i s c l e a r l y e q u i v a l e n t t o C3, where we t a k e ISlk and ,G

as d e f i n e d

i n 2.8. A d i f f e r e n t f o r m u l a t i o n o f C3 w h i c h w i l l be used i n many p r o o f s i s : a 1. + b i z c +d afidl;w93J#@ C I c z f ai and A bi(d. J C l e a r l y C3*C2*Cl.

9

A t y p e t h e o r y T s a t i s f i e s Cl(C2 o r C3) i f f S ( T ) s a t i s f i e s Cl(C2 o r C3). 2.13.THEOREM.Let S be an EATS. Then ( i ) satisfies C I * ( 1.~1) contains a l l constant functions. ( i i ) S s a t i s f i e s c2 9 ( I S [ + ISI) c o n t a i n s a l l s t e p f u n c t i o n s . ( i i i ) S s a t i s f i e s C3 0 (1Sl-t I S l ) = C ( ISI, 1st) ( i . e . < IS1 ,F,Go> i s a r e f l e x i v e domain and, t h u s , a f i l t e r A-model). P r o o f . We p r o v e o n l y ( i i i ) . The p r o o f s o f (i) and (ii) a r e s i m i l a r and e a s i e r . Let ai+bi(c+d. Take f e C ( I S I , 6 I) d e f i n e d by f ( x ) = $ K t b i I t a i s x l = L e t now t { b i l a + x , i e I } . I t i s t h e n easy t o show t h a t G o ( f ) = t / t a i + b i .

+

1

J = { i Ic Lai

c +de Go(f).

(*).

1 ( t h u s C 5 9 a i ).

Then

L a s t l y , d7.w i m p l i e s J # a.

Go(f)* tc= t A b J i

and

3 9Id,

since

I t i s enough t o p r o v e t h a t a l l sups o f f i n i t e s e t s o f s t e p f u n c t i o n s a r e

M. COPPO ET AL.

248

r e p r e s e n t a b l e . Then t h e p r o p e r t y f o l l o w s f r o m t h e f a c t t h a t ( IS I + I S I ) i s a complete l a t t i c e . ) . We p r o v e t h a t L e t f be d e f i n e d as above (observe t h a t f = & I taifbi x f = k ( f ) = f f a i * bi r e p r e s e n t s f, i . e . t h a t 'dye IS1 xf * y = f $ bi, where

J =Iila.ey,

In

'ieI}.

1

fact

dexf'y

* 3cey

;aTbi:c+d.

Now,

if

J ' = {iI c i a . 1,' we have t h a t J'c_ J , s i n c e c L a i by C3. T h e r e f o r e

aie y. Thus J ' P O and biid d. Moreover i t i s easy t o p r o v e t h a t A b. < x * y and t h e J I- f

3 bi

result follows.0

C o n d i t i o n C3 i s t h e c o n d i t i o n o f Lemma 2 . 4 ( i i ) o f [2]. Thus t h e r e p r e s e n t a b l e f u n c t i o n s o v e r < I F I , . > a r e e x a c t l y t h e c o n t i n u o u s ones. We can now g i v e some examples o f s t r u c t u r e s which s a t i s f y o n l y C 1 ( o r C2). B u t we f i r s t need a lemma. 2.14. LEMMA. L e t S- be an ETS. I f t h e r e e x i s t A,BieP t h a t ASyB and Vie I A 5Z B i, t h e n S does n o t s a t i s f y C3.

( i e I ) such

(y

Bi does n o t

Proof. Observe t h a t , g i v e n any Ce p, n e e d t o belong t o P ) . n

Bi+

C=("I B i) + C G A + C

2.15. EXAMPLES. ( i ) L e t be an a p p l i c a t i v e s t r u c t u r e such t h a t Vd,eeD d - e = e. Then any (non t r i v i a l ) ETS S o v e r does n o t s a t i s f y C1. One a c t u a l l y has t h a t VA,BcD: A + A = B + B = D . (ii)L e t be such t h a t d-e=d. Then any (non t r i v i a l ) ETS S o v e r does n o t s a t i s f y C2, s i n c e vA,BED A+ B = D +B. C l e a r l y S s a t i s f i e s C1. ( i i i ) L a s t l y , we show an ETS which s a t i s f i e s C2 b u t n o t C3. L e t < x , - >be t h e Kleene a p p l i c a t i v e s t r u c t u r e d e f i n e d by n-m = { n 1 (m)

s a t i s f i e s C2. L e t A,B,C,E

. Actually

be non empty subsets o f 2 and EP 2. Then,

i f A + B G C + E , c l e a r l y BGE. Moreover l e t p e g \ E and q E " I k l ( x ) = i f x= r t h e n p e l s e q " i s such t h a t ke A+B b u t i s g i v e n by t h e C2 i s s a t i c f i e d . A X p l e T S o v e r < & , a > 1.4. Namely, by t h e I 1 Recursion Theorem t a k e no such t h a t

m e 2 . Then

{ A G yl

any ETS o v e r

B. I f r e C \ A, t h e n k t C+E. Thus G A and same argument used i n { n o X m ) = no, f o r a l l

no E A } i s an ETS and does n o t s a t i s f y C3 by Lemma 2.14.

A l s o t h e e x t e n s i o n a l i t y p r o p e r t y o f qISI,.> has an easy c h a r a c t e r i z a t i o n i n terms o f t h e p r o p e r t i e s o f S As u s u a l , an a p p l i c a t i v e s t r u c t u r e i s e x t e n s i o n a l i f f VG D a. c = b - c * a=b f o r a,beD.

.

2.16. THEOREM. L e t S*X, 2, A , +,u> be an EATS. ( i ) V'ze I S l x . z = y - z 0 (Va,beX a + b e x * a + b E y ) , f o r x , y l S I . ( i i ) . ISl,*>is e x t e n s i o n a l i f f VaeX 31 a * A b + ci.

I

Proof. ( i )

(-)

a+bex

i

b e x - f a (by Lemma 2.7) * be . f a * a-+iey. Immediate f r o m t k e d e f i n i t i o n o f ''.-''. ( * ) Let x a = f { b + c l a
(r).

(ii)

since and

< I S I , * > is

9 bi+

( c )Easy

3.

extensional.

Thus

ae xa,

that

is

By ( i ) x a = f a ,

3 1 Vi € 1 bi+ ci e xa

c. 5 a. 1

from ( i ) . 0

EMBEDDINGS AND ISOMORPHISMS

D e f i n i t i o n . L e t D be an a l g e b r a i c complete l a t t i c e . D e f i n e (i) 'c= {x I c L x } f o r c EDk, t h e cone o v e r a ( f i n i t e ) element.

3.1.

2 49

Extended Type Structures and Filter Lambda Models (ii)

K ( D ) = { z l ceD2.

( i i i ) C ( D ) as t h e c l o s u r e o f K ( D ) under f i n i t e union. As w e l l known, K(D) i s a b a s i s f o r t h e S c o t t t o p o l o g y on D. 3.2. REMARK. C(D) i s i f f C ( D ) i s c l o s e d under

c l o s e d under 'In''.,
, n ,+

,D >

is

an

ETS

'I+

THEOREM. L e t be an a l g e b r a i c c o n t i n u o u s a p p l i c a t i v e s t r u c t u r e . IfS = i s an ETS t h e n < D , * ,&>c+< I s c [ , -,c_> C < I S I,.,g>. ( i i ) If S K = < K ( D ) , E , n , + , D > i s an ETS t h e n < D , * , G K Proof. ( i ) L e t Emb : D + I S C I be Emb(d)= { X E C(D) I dsX}. The embedding o f D 3.3. (i)

i n t o I S c [ as

lattices

is

trivially

verified.

To

prove t h a t

Emb p r e s e r v e s

a p p l i c a t i o n observe t h a t : X E Emb(a'b) * a*beX

* 3ceO d b

a.ceX,

f o r { a * c I ceDk

ccb} i s d i r e c t e d

w D k d c ae*c+ x k =. X E Emb(a) Emb(b). =)

-

The r e v e r s e i n t r i v i a l . (ii) L e t Iso: D-+lSKI be I s o ( d ) = & K ( D ) I The isomorphism as l a t t i c e s i s a p p l i c a t i o n goes as i n case (i). 3.4.

crd}.

immediate.

The

REMARKS. ( i ) I t can be e a s i l y proved t h a t

defined

as

i n 161,

N o t i c e t h a t K(P,)

are

closed

under

"+

"

that

C(Pu) and

and

and K(DA) a r e n o t c l o s e d under

proof

so

Is0

preserves

C(DA), where D A i s

Theorem 3 . 3 ( i )

applies.

'I+''.

.

(ii) K(I S ( T ) I ) i s c l o s e d under ' I + " f o r a l l t y p e t h e o r i e s T I n t h i s case Theorem 3 . 3 ( i i ) j u s t amounts t o say t h a t i f we b u i l d f i l t e r s o f f i l t e r s we do n o t change t h e c o n t i n u o u s a p p l i c a t i v e s t r u c t u r e (modulo isomorphisms). ( i i i ) I f . K There i s a s i m p l e c o n n e c t i o n between t h e c o a d d i t t i v i t y o f and t h e c l o s u r e o f K(D) under ' I + ' ' . 3.5.

THEOREM. L e t be an a l g e b r a i c r e f l e x i v e dom;in. i s c o a d d i t t i v e , , * K ( D ) i s c l o s e d under I' K(D) i s c l o s e d under + ' I * 3 6 ' such t h a t r e f l e x i v e domain.

(1) (11)

Proof. de

+

( i ) We p r o v e t h a t Va,trDK

*a +b*=)F(d):

".

.

i s a coaddittive "

I

a + b = G(fab). O b v i o u s l y G ( f a d s a + b .

fab

*G(F(d)) !G(fab) (by c o a d d i t t i v i t y ) . *dlG(f

,d

(ii)D e f i n e a r b by : a L b = >+E, Notice t h a t b d e f i n i t i o n (1) f,bfF(d) * deZ+"b * arbgd

where a , b D

K'

and G ' ( f ) - C H a r b l f a , C f ) .

Moreover

M.COPPO ET AL.

250 and

f abC f * a c b LG'(f) (2) Thus F(G' ( f ) ) c K f a'b I f a b & F(G' ( f ) ) 1 = U [ f a b l a + b L G ' ( f ) l , by ( 1 ) CKfabl fabEf} f.

by ( 2 )

=

G'(F(d))=Uabb I f

CF(d) } abU a r b I a+b E d } , b y (1) d.

3.6.

REMARK. Since any

o f Theorem 3.5 i t

i s an e x t e n s i o n a l r e f l e x i v e domain, f r o m t h e p r o o f

f o l l o w s t h a t Va,b e (D,

)k a c b = f a E

From t h e p r e v i o u s r e s u l t s we o b t a i n t h e isomorphism o f any a l g e b r a i c c o a d d i t t i v e r e f l e x i v e domain w i t h t h e f i l t e r A-model, b u i l t on i t s compact cones. 3.7. D e f i n i t i o n . [ 151 An isomorphism between t h e r e f l e x i v e domains and i s a p a i r , where w,w> i s an isomorphism between D and D ' such t h a t (1) F'(d)= v 0 F (w(d)) o w (2) G ' ( f ) = v(G(w0 f a v ) ) where d E D ' and f e C ( D ' , D ' ) . (1) i m p l i e s t h a t v and w p r e s e r v e " * ' I . By ( 1 ) and ( 2 ) V d e D v ( G o F ( d ) ) = G ' o F ' ( v ( d ) ) ( c f . t h e n o t i o n o f isomorphism i n [ l , 5.3.21). 3.8. THEOREM. L e t be an a l g e b r a i c c o a d d i t t i v e r e f l e x i v e domain and S = . Then d),F',G'> = < I S I , F,G,>. K K Proof. S Ki s an ETS b y Theorem 3.5. Since < D , F ' > - < I S K I , F > (by Theorem 3 . 3 ( i i ) ) t h e range o f F i s C(IS 1,l Sd ) I i . e . < I S I,F,GO> i s a r e f l e x i v e domain. K K I n 1151 Sanchis n o t i c e s t h a t g i v e n two c o a d d i t t i v e r e f l e x i v e domains, o n l y one o f t h e c o n d i t i o n s o f 3.7 s u f f i c i e s t o have t h e isomorphism. So we a r e done, s i n c e < I SKls,F,Go> i s c o a d d i t t i v e by Theorem 2.8 and c o n d i t i o n ( 1 ) o f 3.7 h o l d s , b y Theorem 3 . 3 ( i i ) . 0 Another i n t e r e s t i n g c l a s s o f embeddings i s d e f i n e d c o n s i d e r i n g f i l t e r s o f EATS b u i l t from tvoe theories. F o l l o w i n g [ f i ' l an element U E I FI i s a c l o s u r e o e r a t i o n i f f i t s a t i s f i e s : G , ( i d ) L u = u 0 u ( where u 0 u =G,(Az. u.(u.z)T). 3.9. (i (ii)

THEOREM. L e t T be a t y p e t h e o r y . Then one has % < I S ( T ) I , - , c _ > i s i s o m o r p h i c t o t h e range o f a c l o s u r e o p e r a t i o n u e I F I .

Proof.

.

.

( i ) Observe t h a t a b s t r a c t f i l t e r s o f

S ( T ) are abstract f i l t e r s o f

then, a r e c l o s e d u n d e r a p p l i c a t i o n . Thus < I FI , * ,g >.

F and,

i s a substructure o f

25 1

Extended Type Structures and Filter Lambda Models

( i i ) As p o i n t e d o u t i n 1.6, z T extends 2 0 . D e f i n e u = t Io+TI O -< T T I E I FI a n d + A d S ( T ) I as t h e f i l t e r generated by t h e s e t o f t y p e s A ( n o t i c e t h a t u i s c l o s e d under lo w h i l e + A i n c l o s e d under

3.We p r o v e t h a t u*A=+A.

+Ac_ u . A i s t r i v i a l . F o r t h e r e v e r s e observe t h a t T E

u.A-3Ue

A

*

u

s

O+T

* 30s A * 30 E A * 30 E A =. 3 0 c A

31 V i e 1 u ~ ~ ~ T o ~ ~& + 31 V i e 1 0 ~ i ~ ~ a n d 3 J uz0 c _ I3 u i

33 ui

U L ~ /Jui T

T ~ A .

,

OT+ T

~

~

~

,I~T~ , f (o r~F T satisfies

C3

zT$~iiO~

s i n c e < extends lo -T

Obviously u _ > i = t { o + . r l u i o T I

and u = u

-

3

u. C l e a r l y S ( T ) i s t h e range o f u. 0

Theorem 3.9 proves t h a t i s " u n i v e r s a l " ( i n t h e sense o f [ 191) f o r a l l f i l t e r domains. R e c a l l i n f a c t t h a t each such domain i s t r i v i a l l y isomorphic t o an EATS g i v e n by a s u i t a b l e t h e o r y (see what p o i n t e d o u t a f t e r 1.6). We can a c t u a l l y p r o v e t h a t any a p p l i c a t i v e s t r u c t u r e can be embedded i n t o . 3.10. THEOREM. L e t % .

b e a ( c o u n t a b l e ) a p p l i c a t i v e s t r u c t u r e . Then

Proof. L e t A = { a . I i L l } and x A = { $ i i $ j + $ d a i * a j = Emb: A + I S ( T z

)I by

Emb(ai)=

t$i

a h } . Define

(iL1).

A

Emb(ai)* Emb(a.)= Emb(ai. a .). _> i s t r i v i a l and C_ i s g i v e n J J by t h e m i n i m a l i t y o f iZA, as d e r i v e d f r o m x A ( f o r t h i s some b o r i n g c a l c u l a t i o n s

We c l a i m t h a t

a r e needed. We l e a v e them as an e x e r c i s e ) . Moreover < I S(Tz ) I ; > k < l A

FI

>;

by Theorem 3 . 9 ( i ) .

0

The c o n d i t i o n on t h e c a r d i n a l i t y o f A may be dropped j u s t t a k i n g enough atoms, i . e . t a k i n g A t l a r g e enough and c o n s t r u c t i n g T f r o m i t as i n 1 . 2 ( i ) . 4.

FILTER A-MODELS

As a l r e a d y p o i n t e d o u t , any EATS s a t i s f y i n g C3 y i e l d s a f i l t e r A-model. A c t u a l l y any such A-model i s g i v e n by a r e f l e x i v e domain, i.e. i t has t h e s t r o n g p r o p e r t y t h a t any c o n t i n u o u s f u n c t i o n i s r e p r e s e n t a b l e . T h i s i s more t h a n what i s r e q u i r e d by an a p p l i c a t i v e s t r u c t u r e t o y i e l d a A-model. Theorem 4.8 c h a r a c t e r i z e s EATS S such t h a t < S,F,G > i s a f i l t e r A-model. Theorem 4.11 g i v e s a f i l t e r A-model, which i s n o t a r e f l e x i v e domain. F o r t h e n o t i o n o f (expanded) combinatory a l g e b r a and A-model we m o s t l y r e f e r t o [ I 1 s [ 9 1 9 [I01

.

4.1. D e f i n i t i o n . L e t S= <X,(,A,+,W> be an EATS ( r e c a l l t h a t a -+b+ c stands f o r a -+ ( b + c ) ) . D e f i n e -K = t { a + b +c I CE t a 1 S = t { a + b +c+d I dcta.tc-(tb.tc)} E * t { a - . b +c I CE t a - t b l .

-

Note t h a t

K,

S a n d 5 have been d e f i n e d j u s t u s i n g G*

o f 2.8.

M. COPPO ET AL.

252

4.2. LEMMA. L e t S be an EATS. Then x - z -(y.z) ~ _ S - x . y - z and x.y@-x*y. ( i ) Vx,y,ze I S I x g : x . y , i s a combinatory a l g e b r a , t h e n KSK and S C S . Moreover, f o r ( i i ) I f < lSI,-,S,K> I=SKK, a+ b € 1 0 a 2 b. Proof. ( i ) By FOG*,

i d (see 2.8).

( i i ) Observe t h a t K . t a - t b = t a i m p l i e s by 2.7 a+b+ccK f o r S. Moreover a + b e I 0 b E I - t a = t a . 0

-

f o r a l l ceta. S i m i l a r l y

-

4.3. THEOREM. L e t s be an EATS. I f t h e r e a r e S, K such t h a t < I S I , ,S,K > i s an expanded combinatory a l g e b r a t h e n a l s o d s l , , $,E > i s an expanded combinatory algebra.

Proof. Immediate f r o m 4.2. THEOREM. L e t S = < X , L , A , + , w > 4.4. ( i e 1 ) o n e has(*) /\(bi+ci)+bi+ciza+b*3J

be an EATS. Assume t h a t f o r any a,b,bi,ci a 2 A d . e .&I. J J 3I Then, i f i s a combinatory a l g e b r a , < l S I , . , E > i s a 1-model.

EX

P r o o f . F o l l o w i n g [ l o ] , we j u s t need t o show t h a t ( 1 )g.x.y = x - y ( 2 ) vz x.2 = y.z =$ g. x =g.y (3) E.4 = E . As f o r (17, n o t e t h a t g=S(K(S&)). Then use 4.3. As f o r ( 2 ) , observe f i r s t t h a t a + b e E- 31 ? a i + b i - + c i ( a + b f i V i E I a 1. <- b . 1+ c i

,3I

A(bi+ci) +bi+ci5a+b I =, 35 a L A d .+e by ( * ) J J JTake now begax, t h e n 3 a ~ xa + b E s Use t h e p r e v i o u s argument t o o b t a i n J such

.+,

t h a t a z A d .e.:h.

J J J

d j +e j e y . NOW,

Since a

E

x,

by 2 . 1 6 ( i )

and t h e assumption i n ( 2 ) , V j E J

A ( d j - t e j ) + d j + e j E E , by d e f i n i t i o n .

J

Thus VjEJ dj+ e . E E - Y and b EE_.Y. J ( 3 ) follows from t h e d e f i n i t i o n s . n Note t h a t , i f < IS I . . , E > i s a x-model, t h e n E =S(K(SKK)) ( c f . 4.1) and, by d o e s n ' t need t o 4.2,s C _ E . However, a l s o i f < l S I , - , ~ > i s a x-m<delS be a m o d e l , f o r < IS I , -,K,S_> may j u s t be a combinatory a l g e b r x ( o r a x - a l g e b r a ) . Each t y p e t h e o r y T induces a system o f t y p e assignment, i n t h e sense o f [ 2 1 , f o r t h e s e t A o f x-terms. By t h i s , Theorem 4.8 c h a r a c t e r i z e s t h e t y p e t h e o r i e s which y i e l d 1-models. N o t a t i o n and concepts a r e m o s t l y f r o m [ 21. I n p a r t i c u l a r i f CJ E T and M E A , then UM i s a statement, where u is t h e p r e d i c a t e and M t h e s u b j e c t . A basis i s a s e t o f statements w i t h o n l y v a r i a b l e s as s u b j e c t s . 4.5. D e f i n i t i o n . L e t 7 be a t y p e t h e o r y . The (extended) t y p e induced by T i s d e f i n e d by t h e f o l l o w i n g n a t u r a l d e d u c t i o n system

assignment

253

Extended Type Structures and Filter Lambda Models

( + ) i f x i s n o t f r e e i n assumptions on which T M depends o t h e r t h a n T

W r i t e B t-

oM i f

OX.

oM i s d e r i v a b l e f r o m t h e b a s i s B i n t h i s system.

4.6. D e f i n i t i o n . L e t T be a t y p e t h e o r y and S ( T ) be t h e EATS d e f i n e d i n 1 . 5 ( i v ) . F o r any map 5 f r o m v a r i a b l e s o f A t o IS( ~ ) and 1 M E A. define: (i) B =Ioxloe s(x)}

5

(ii) (II b y i n d u c t i o n on t h e s t r u c t u r e o f M) u x f =

UPQ 1 = F ( EPII [Ax. P

T

( UQII

T

I T = ( ,G he E 1 S( T ) I .[ P 1 ) 5 c [ x/el

(see 2.8).

( T h i s i s w e l l d e f i n e d , by t h e c o n t i n u i t y o f F and Note t h a t i f
I f B i s a basis, l e t B h = { o y 4.7.

> i s a x-model,

I oy

Thus V i

C_

. By i n d u c t i o n on M.

I

I

, by

induction

T

, by

rule ( < ) -T

Bd x / t a . l ~ BiP 1

* BPUIaix}t-~;P T

* B t-a.+Bi 5 1

Xx.P

, by (

The r e s u l t f o l l o w s b y u s i n g (

II J

i s the

[MI

T

5

= Io l B

T

5

t-oM}.

The o n l y non t r i v i a l case i s M - X x . P .

r

E

then

E B and y z x l .

THEOREM. L e t T be a t y p e t h e o r y . Then

Proof.

).

A

+I).

I ) and ( z T )

(standard)

M. COPPO ET AL.

254

.

3

T By i n d u c t i o n on t h e d e d u c t i o n B g t - oM. We j u s t check when ( + I ) i s used.

The r e s t i s t r i v i a l . Note t h a t i f

t ax1

t h e n we have, by a s h o r t e r d e d u c t i o n , gives the result. 0

BF[x/+alg 8p. The

i n d u c t i v e hypothesis

L e t G o be as i n 2.8. THEOREM. L e t T be a t y p e t h e o r y . Then i s a x-model * T T lB/X t- U+T xx.M * B / x ~ (UXI F TM). Proof. =*. R e c a l l t h a t G o i s t h e r e s t r i c t i o n o f G, t o ( I S ( T ) I + I S( T ) I ) , t h e

4.8.

IS( T ) I t o

IS( T ) I,

i s representable.

Thus t h e

representable functions.

By assumption,

which i s d e f i n e d by a

x-term

semantics o f x-terms i n

IS( T ) I i s d e f i n e d e x a c t l y as i n 4 . 6 ( i i ) ,

.

use G o i n s t e a d o f G, Let 4.7

g ( x ) = f { u J u z ~o r B

r

B/x+u+T

T

oxeB}. O b v i o u s l y B F p N * B I\x.M]

XX.M-U+TE

any f u n c t i o n f r o m

(using constants),

T

I-~N gB

Te

F(Go(f))(fu) f o r T

I

f=AeeIS

(T)

T

* B

/X U { U X } t- T M . gB =. The p r o o f o f Theorem 3.5 i n [ 2 ] remains v a l i d , c which r e q u i r e s t h e g i v e n c o n d i t i o n .

4.9. Oefine

except f o r

point

(iii),

we can now g i v e a c l a s s o f f i l t e r M o d e l s , which a r e n o t

D e f i n i t i o n . ( i ) Choose 0

c*=

b y Theorem

r-

q x / t u 1‘:’?ince

Using Theorem 4.8, r e f l e x i v e domains.

. Then,

where one may

{ ULU[ $/p]

IUE

A t and PcT such t h a t

T}.

( i i ) I * a n d ? i s short f o r (iii)u ~ i Tf f e i t h e r (1) u 3 T

zz* and &(E*),respectively.

0 does n o t o c c u r i n

p

.

Extended Type Structures and Filter Lambda Models 4.10. (ii)

ui*

LEMMA. ( i )

T

=)

o ~ A a + 6< * A y

. ,...,

d O / p I ~ * T[ $/PI + 6 = ~ * 3pl

pn

255

LT.

o b p l z . . . < p

I i i- J i in and each p (1 I h I n ) i s an i n t e r s e c t i o n o f arrows. Fi B&M~=) BL +/PI P T [ 4/01 M . ( i v ) vie1 B/XU{a.X} 6 f3.M and ai+Bi & A y . 4 . * VjeJ B / x U { y . x } p 6 . M . 1 1 J J J J J
.

.

.

(2

2.4(ii) i n [2]). Bi

I n f a c t v j ~ J f a ~ + By.+a.implies ~ i ~ J J

i t j . Therefore

a.X 1

:

from t h e derivations

3K?@SIy

j 3

pi and

f o r a l l i e K one may o b t a i n :

6 fM

ujx a.x

(lo1

1

K

i t

6iM

-

tBiM

(A1)

6.M

(50)

J

I f case (2.1) YiX

:

obtain

a.X 1

i n 4.9 a p p l i e s , t h e n f r o m t h e d e r i v a t i o n YiX BiM j u s t u s i n g (i*)i n and

a.x

6;M

one may

.

1

I f case (2.2) i n 4.9 a p p l i e s t h e n yi+6i-(ai+Bi) QyPdO/Ply f o r a l l 0.y. B h U{ai[ $/PI X } P Bi[ 4/01 4.11. (ii)

[$/p

1.

By d e f i n i t i o n

T h e r e f o r e f r o m ( i i i ) B/xu(aixl@BiM

implies

M.

THEOREM. (i)B~-*u+TAx.M -BUIUX}PTM. L e t =S , < T , i * , A , +, w > . Then i s

a f i l t e r A-model.

Proof.

(i)L e t D be t h e d e d u c t i o n showing B ~ U + T A X . M . Assume t h a t , f o r some I,ai+BiAx.M,icI, a r e a l l t h e statements i n D on which u+TXX.M depends and which

a r e c o n c l u s i o n s o f (+I). I t i s t h e n easy t o p r o v e t h a t +ai+6izu+~ i s d e r i v e d f r o m t h e ai+Bi 2.7(iii)

i n [ 21).

,Since u+TAx.M

Ax.M u s i n g o n l y r u l e s ( A I ) , ( A E ) and (2,). (Cf. Lemma

By 4 . 1 0 ( i i )

above,

each uh i s an i n t e r s e c t i o n o f arrows.

3p1

,..., 11, + a i + .1%. <~l~...~p n-


,

where

The r e s u l t f o l l o w s by i t e r a t e d a p p l i c a -

tions o f 4.10(iv).

(ii)By (i) and 4.8.

0

S+ does n o t s a t i s f y C2 (and hence C3), w h i l e i t i s easy t o check t h a t p i* $.

f o r 4+ 6

*I

P

+

P

and

0

I* P ,

M. COPPO ET AL.

256

4.12. REMARK. ( i ) Theorem 4.7 g e n e r a l i z e s 3.5 i n 121. By t h i s a l s o t h e completeness r e s u l t i n [2,3.1Dl can be g e n e r a l i z e d . Given a 1-model nl = , an environment 5 and a t y p e i n t e r p r e t a t i o n V: T +PD, l e t m,5, VCoM and m,E, V I - B be as i n 1.2 o f 1 2 1 . Now, g i v e n a t y p e theorVy,= , d e f i n e BI=ToM i f f f o r a l l m , s,V, such t h a t T c T one has m, 5, m,E,V k oM. V By t h e same t e c h n i q u e as i n 121, u s i n g 4.7 and 2 . 4 ( i ) one can e a s i l y p r o v e

;

r

r

B I= UM * B F uM. (ii)(Comparing t h e completeness r e s u l t s i n [ 7 1 , [a1 and 121). The p r o o f i n [ 71 uses a t e r m model o f k a l c u l u s , namely t h e s e t o f 1-terms up t o 16-convertib i l i t y , i.e. m ( A g ) = { [ M l I M i s a A-term 1 where [ M ] = { N l x p . F M = N }. Then, f o r a g i v e n b a s i s 8, ty es a r e i n t e r p r e t e d by (1)

vH={

mn 5 0 IB+@P

c_

O M }

rn(xe1

where c 0 i s t h e t r i v i a l environment d e f i n e d by c 0 ( x ) = [ o f B, see [ 6 1 . I n 121, t y p e s a r e i n t e r p r e t e d by Vo(o) = I d / o e d } E I F [ . However, g i v e n a b a s i s B, Theorem 4.7 see t h e p r o o f o f 4.8) Vo(o) = ( [ M I

%+

IB'F

implies t h a t ( f o r the d e f i n i t i o n o f

5B'

UM 1.

As a m a t t e r of f a c t , n o t e t h a t V o 5.

and B + i s a v a r i a n t

To

To (2)

XI

and V

EXTENDED TYPE STRUCTURES AND D-

H

a r e v e r y much a l i k e .

A-MODELS

I n t h i s s e c t i o n we p r e s e n t some i n t e r e s t i n g t y p e t h e o r i e s o b t a i n e d by i n t e r p r e t i n g t y p e s i n well-known a p p l i c a t i v e s t r u c t u r e s , i . e . D, 1 -models o f S c o t t [ 171. F i r s t , we show how t o o b t a i n e x a c t l y t h e t h e o r y To by i n t e r p r e t i n g t y p e s i n an i n v e r s e l i m i t space,

DZ

,

c o n s t r u c t e d by s t a r t i n g w i t h t h e l a t t i c e P,

.

As immediate consequence, we have t h a t 'the completeness theorem f o r C u r r y ' s t y p e assignment system (and f o r i t s c o n s e r v a t i v e e x t e n s i o n i n 1 2 1 ) i s proved by t h e use o f a "mathematical" A-model. Dl

T h a t i s , by a model (which i s a s u b s t r u c t u r e o f

) c o n s t r u c t e d by means which a r e n o t s y n t a c t i c i n n a t u r e ,

f i l t e r A-model

i n 1 2 1 o r t h e t e r m model i n 171.

assignment f o r t n e model P, by Theorems 3.9 and 3.10, 0

i n t o Dm

has been proved,

such as t h e

(Completeness o f C u r r y ' s t y p e

i n d e p e n d e n t l y , i n [ 41). Moreover,

any ( c o u n t a b l e ) a p p l i c a t i v e s t r u c t u r e can be embedded

.

I n t h e second p a r t t h e r e l a t i o n s between t y p e t h e o r i e s and some 0,

A-models

are studied. We use s t a n d a r d n o t a t i o n on i n v e r s e l i m i t spaces ( c f . [ 17 ] o r [ 11). An i n v e r s e l i m i t D:

(i,j) of

i s c o m p l e t e l y determined by a complete l a t t i c e D D 1

=C(Do,Do)

on Do. As u s u a l we i d e n t i f y dt.0

t h a t , i f D o i s a l g e b r a i c , a l s o D, 0

L e t D,

i s a l g e b r a i c and (D,)

be t h e i n v e r s e l i m i t space determined by

p r o j e c t i o n (il,j,),

where i,(d)=AeeD

0

0

and a p r o j e c t i o n

w i t h an,

n

k

=

u (Dn) ne

(d)eDm. R e c a l l

k.

Do= P, and t h e s t a n d a r d

.d and j l ( f ) = f ( l ) .

Extended Type Structures and Filter Lambda Models 5.1.

Definition. ( i )

(ii)

v!

&=
D:>. "

T+K(D:)

s a t i s f i e s C3.

LEMMA. (i)V u e T V n V o ( u ) c _ { n l .

(ii) vC~(D:)

Proof. deD,.

I

i s t h e t y p e i n t e r p r e t a t i o n g i v e n by Vo($n)={nl+{Ol.

Note t h a t , by 2 . 1 3 ( i i i ) and 3 . 4 ( i i i ) , S m 5.2.

257

3 p e V ~

(vo(~)~vo(T)G*

(i)

Observe

that,

{%}=f

Then

with

l e t u'(flah+Bh)A(a$

i, ( d ) = f

t h e standard projection,

= $ + { n'}=D:+{

i } ( c f . 3.6).

@{n} The p r o o f i s by i n d u c t i o n on u . I f Else,

O ( ~ )~ v o ( T ) + v o ( ~ ) fi v o ( p ) & ) .

U ~ Ut h e

for all

proof i s t r i v i a l .

~ ' ( ~ ) s { " n > . Then,

k ) and assume

i d

by ~ 3 , f o r some HIGH

and

D:G(H",Vo(ah))n(n,IkI) (n,Vo(Bh))n(;7, [ O l ) ~ { % } . K H which i s i m p o s s i b l e by t h e i n d u c t i o n hypothesis.

and K ' S K such t h a t K ' U H ' P 0 ,

I f K ' = 0, t h e n n, V '(6 ) C { n ' l H h - , If K'#@, then V F K ' DoE{kl, which i s impossible. m

(ii)

.

Again by i n d u c t i o n on u

Thus t a k e

If

UEU

t h e assumption may h o l d o n l y f o r c=@.

PEW.

I f u? ( $ a h + B h ? A ( A $ ) , by C3, f o r some H'GH and K'EK, H ' U K ' f Q , V'(T)C_ K k ((H",V "(a h) )"($ [k I) and ( V O(8 h) In( I f K'=@, take p- A B H' I f K ' # @ , t h e n VkeK V O( T)c_Ik 1, which i s i m p o s s i b l e by ( i ) .

pl{61)s;.

5.3.

THEOREM. (i) T o :T V o .

(ii)c,l.
m

,-,E>

.

Proof. (i)By d e f i n i t i o n T o G T V O . Conversely, we p r o v e t h a t TSU. T r i v i a l . ~-4,.

I f u'u

U ~ ~ T * U % T

, t h e n u~ VO $ n

L e t t h e n u:(fiah+Bh)

A

;

O(

.

does n o t h o l d .

and

,; 611S{b I,

bh) 1

T

(2 $ k ) -

05 V ~ $ n * 3 H ' c H 3K'CK H ' U K ' # $ ( ,V

b y i n d u c t i o n on

{ n k ( f l , V o ( a h ) ) n ( n , [ i } ) and K by C3. I

",

By Lemma 5.2 we have K ' = 0 ( e l s e

H and

"

V o ( ~ h ) C { O I ) . Thus V k e K ' { n } g { k } ,

i.e.

VkeK' n=k. T h e r e f o r e A , $ =$ OU'A$ f o r some U ' E T. T h i s g i v e s u%$ K k- n T:T~+T~. The p r o o f goes as i n p r e v i o u s case a p p l y i n g i n d u c t i o n . T C T ~ A T ~ Use .

(ii)

We

U ~ V ~ T ~ A TU 2L *V ~ T i

have t o

show

that

and

ULVoT2

there

Define

io:IFI+IK(D_) I ^vO

by i o ( d ) = ? { V o ( u ) l u e d }

an embedding

as

lattices

.

i s i n j e c t i v e . It i s c l e a r l y i n c l u s i o n preserving.

We p r o v e t h a t ? ' ( d ) * i o ( e ) L i o ( d - e ) .

.

.

exists

a p p l i c a t i v e s t r u c t u r e s . The isomorphism h o l d s by 3 . 4 ( i i i ) . BY ( i ) ,

n

The r e v e r s e i n c l u s i o n i s e a s i e r .

and as

M. COPPO ET AL.

258

u)g.

N o t i c e f i - r s t t h a t E E V q d ) =+ 3ue d V,"( * , . ( d ) * V o ( e ) =+ 31be\j0(e) b + C E V o ( d ) *CE V * " * 3Tee 3 u Ed V0(r)E b a&vVo(~)E b+c * 3 ~ e e3 0 E d VO(u)c VO(,)+ c ;+ 3 r e e 3 0 Ed 3peT Vo(u) c_ VO(,)+ Vo(p) V 0 ( p ) c c, by 5 . 2 ( i i ) * 3 p t T 3 ~ ee T+$d V o ( p ) C c, b y (i) =+ 'ceVo(d-e). n

Then

and

T h i s theorem has an immediate consequence. SOROLLARY. 5.4. i n t o D, .

Any ( c o u n t a b l e ) a p p l i c a t i v e s t r u c t u r e

Proof. By Theorems 3.10 and 5 . 3 ( i i ) .



can be embedded

0

As a l r e a d y mentioned, t h e h y p o t h e s i s an t h e c o u n t a b i l i t y o f A can be dropped by t a k i n g enough t y p e v a r i a b l e s i n t h e d e f i n i t i o n o f T i n 1 . 2 ( i ) and u s i n g Pa i n s t e a d o f Pw , f o r a l a r g e enough c a r d i n a l a , i n the construction o f Dana S c o t t ( p e r s o n a l communication)

.D:

has a d i r e c t argument f o r a s i m i l a r

embedding r e s u l t . By Theorem 3.5, 0m and type

IK(D_)[

any D,

a r e i s o m o r p h i c as r e f l e x i v e domains.

interpretations

V:

domains. < I S ( T V ) I ,F,G t h e c o n s i d e r e d 0,

space y i e l d s an ETS o v e r K ( D _ ) . Moreover, by 3.8, T+K(Dm ) ,

Thus f o r a l l s u r j e c t i v e

and D,are

IS ( T V ) I

isomorphic r e f l e x i v e

> can be used as a t o o l f o r i n v e s t i g a t i n g p r o p e r t i e s o f

. For t h i s

purpose, however, we need a t h e o r y TV easy t o

handle (we ask, f o r example, t h a t T V i s r e c u r s i v e l y a x i o m a t i z a b l e ) . I n t h e r e s t of t h i s s e c t i o n we d e f i n e some s i m p l e t y p e t h e o r i e s w h i c h y i e l d f i l t e r ?,-models i s o m o r p h i c t o i n t e r e s t i n g i n v e r s e l i m i t spaces. We f i r s t need a r e s u l t which i s w o r t h d i s p l a y i n g by i t s own i n t e r e s t f o r t h e axiomatization o f type theories. 5.5.

THEOREM. L e t T be a t y p e t h e o r y s a t i s f y i n g C3. Assume t h a t

and 3J $.+ A $ . + $ . where VjeJ V@ieAt "i.;L? 1 T J J J Define Z T [ $ $ j ~ @ i l ,"~j~~i}Ut"+13'"ij+"jI, QT Then T=T(Zl. Proof. We o n l y need t o p r o v e t h a t TC_T (

x

$.EAtU[w},$.

3 J$.+Q. J J 1-

), i.e.

that

0

€At

u and

T

.

TEW.

Trivial.

TZ@ E

A t . I f uzw

or u

E

. .

u ~ ~ T ; + u ( ~ T

The p r o o f i s by i n d u c t i o n on t h e number o f arrows i n cases on

J

We work by

A t t h e r e s u l t i s obvious.

Otherwise, l e t u = (Aah+Bh )A((?k $1 ' s a r e as i n t h e assumption.

) , w i t h @k+T

t$pl, where t h e

$

1

' s and

Extended Type Structures and Filter Lambda Models 0 5T T

-

uLT

34 .++j

I'A

f o r some J, by h y p o t h e s i s L ~ ++j* ~

h ) ~$1+( t+1)

-vje

J

=. V j

J 3H'L H 3 L ' c L

E

259

ah+3

( A , 6 )A(;,

H'UL'#

0

and $ .<

- J-r'

fl, ah)A(t, $1) and

$1 IT+ by c 3

- V j e J 3 H ' zHH 3 hL 1 & L '

,9

H ' U L ~ #o

and

$ ;~filah)A(~,+l)and

($,~~)h((!,+~)5~ +jj. b y t h e assumption i f H ' i s empty and by t h e i n d u c t i o n hypothesis otherwise

* The case

, by

5< T

-z

T E T ~4 T~

i s t r i v i a l w h i l e t h e case

the induction hypothesis.

5.6.

LEMMA.

Let

1.1.

D m be any

t h a t V C ~ ( D 31 ~ )V(+$l)=c ~

T S T ~ + T ~e a s i l y

f o l l o w s f r o m C3 and

n inverse

limit

space.

Let

V:T+

K ( D m ) be such

(where $ i e A t f o r a l l i d ) . Then V i s s u r j e c t i v e .

P r o o f . R e c a l l t h a t i f ce(Dm ) k , t h e n c ~ ( D , ) ~ f o rsome n. The p r o o f i s by i n d u c t i o n on n. n=O. By t h e assumption.

-

.

L e t c ~ ( D ~ + ~Then ) ~ . f o r some I and ai,b;Dn, c = y fa,b. Now, =; ii+6 i, by Remark 3.6 1 1 c=nf I a.b. *;= ?V(ui)+V(ri) f o r some I, by t h e i n d u c t i o n h y p o t h e s i s JE=v(Au.+T

1 1

). i

(i)( P r o j e c t i o n s ) Given a complete l a t t i c e D o , d e f i n e t h e f o l l o w i n g p r o j e c t i o n s o f C(Do,Do) on Do : ie(d)= f je(f)=f(e) where eeDo ed i f Do= Pw,set i*(d)uf{nl{nll nedl j*(f)U(dl i*(d)Lfl. , and N L i s t h e f l a t (ii) ( I n v e r s e l i m i t spaces) L e t 6 = {1 , T ) , where I(T l a t t i c e o f i n t e g e r s ( i . e . x y i f f x = l o r Y=T o r x=y) Do = and p r o j e c t i o n s Then se:, w i t h 5.7.

Definition.

P

w

pw

NL NL 4 Dm

5 Dm

( iii) ( Formul a s )

6 0

M. COPPO ET AL.

260

(iv)

(Type I n t e r p r e t a t i o n s ) D e f i n e , f o r q=O,

... ,5,

Vq:l

vo(Qn)=v'(@n)=Iil V 2 ( Qn)=V3(Qn)=il

.

v4(Qn)=v5(@n)= i

The p r o j e c t i o n s ( i j ) have been i n t r o d u c e d by Park e' e It i s easy t o check t h a t ( i * , j * ) i s a c t u a l l y a p r o j e c t i o n o f C(Pw,Pw) on Pw

.

It

has been d e f i n e d u s i n g P r o p o s i t i o n 3.10 o f [ 17 1. 5.8.

THEOREM. L e t q= 0

(i)

V'

(ii)

T(Zq) = T v q

,... ,5.

Then

i s onto K ( D ~ ) .

.

P r o o f . By 5.5 and 5.6. 5.9.

REMARK. The s e t o f t y p e s i n T(Cq) and T ( z , ) a c t u a l l y c o n t a i n s j u s t one t y p e

v a r i a b l e (up t o " ^ . " ) , w h i l e t h e o t h e r t h e o r i e s above d e f i n e d , c o n t a i n i n f i n i t e t y p e v a r i a b l e s . T h i s corresponds t o t h e f a c t t h a t 6 has o n l y one compact element, T,

d i f f e r e n t from

elements.

I,

while NL

and Po have c o u n t a b l y many incomparable compact

I n general a complete a l g e b r a i c l a t t i c e , by t h e same t e c h n i q u e ,

would

g i v e a t y p e t h e o r y w i t h as many non e q u i v a l e n t t y p e v a r i a b l e s as t h e c a r d i n a l i t y o f t h e compact elements i n i t ( e x c e p t f o r The isomorphism between f i l t e r

I

).

A-models

i n i n v e s t i g a t i n g t h e t h e o r i e s o f D,

spaces.

and D _ s p a c e s may be o f some h e l p

I n p a r t i c u l a r we c o n j e c t u r e t h a t

n o t any D, A-model has a maximal t h e o r y . A n a t u r a l g e n e r a l i z a t i o n o f EATS i s o b t a i n e d by a l l o w i n g " A " t o be a p a r t i a l f u n c t i o n which s a t i s f i e s some ( n a t u r a l ) c o n d i t i o n s ( f o r example, i f U / \ T i s d e f i n e d and U < ( J ' , TLT' t h e n a l s o d AT' must be d e f i n e d , c f . t h e s e t Con o f r 2 1 1 ) . I n t h T s case t h e f i l t e r domains t u r n o u t t o be c.p.0.s. This i s also t h e approach o f S c o t t ' s i n f o r m a t i o n systems [ 2 1 ] . We c l a i m t h a t most o f t h e p r o p e r t i e s o f t h i s paper s t i l l h o l d . FOOTNOTES

"I" i s

(1)

Note t h a t i n D e f i n i t i o n 1.1 r e d as a s y n t a c t i c o b j e c t .

a r e l a t i o n w h i l e , here,

(2)

As u s u a l , i f f: A+B and CCA, f ( C ) = I f ( a ) I aeC1.

(3)

As u s u a l , L, ( t h e r e i s an i n j e c t i v e ( b i j e c t i v e ) homomorphism v:
"I" i s

conside-

means t h a t D';,E'

>.

Extended Type Structures and Filter Lambda Models

261

(4)

I d denotes always t h e i d e n t i t y f u n c t i o n on some s e t which i s c l e a r from the context.

(5)

We keep u s i n g t h i s abuse o f language: 3 I.. stands f o r " t h e r e e x i s t f i n i t e l y many elements o f t h e i n t e n d e d s e t , indexed i n I , such t h a t . . . " .

.

"2' between t y p e s i s i n t e n d e d modulo p e r m u t a t i o n s and and EUAW ).

(6)

" A W "

( f o r example

BATETAU

ACKNOWLEDGEMENTS A few d i s c u s s i o n s we had w i t h Henk Barendregt and K a r s t Koymans, i n t h e b e a u t i f u l s u r r o u n d i n g s o f Mount Gran Paradiso, were v e r y h e l p f u l i n c l e a r i f y i n g o u r view p o i n t and some o f o u r r e s u l t s . REFERENCES

[l]

Barendregt, H., The Lambda C a l c u l u s , i t s Syntax and Semantics ( N o r t h Hol 1and, Amsterdam, 1981).

[2 ]

Barendregt, H., Coppo, M. and D e z a n i - C i a n c a g l i n i , M., A F i l t e r Lambda Model and t h e Completeness o f Type Assignment, J. Symbolic L o g i c ( t o appear).

[3 ]

Coppo,M., D e z a n i - C i a n c a g l i n i , M. and Venneri, B., F u n c t i o n a l Characters o f S o l v a b l e Terms, Z. Math. L o g i k Grundlag. Math. 27 (1981) 45-58.

1

Coppo, M., Completeness o f Type Assignment i n Continuous Lambda Models, Theor. Comput. S c i . ( t o appear).

[4

[5 1

Curry, H.B.

and Feys,R.

%

Combinatory L o g i c

I (North-Holland, Amsterdam,

1958). [6 1

Engeler,

E.,

Algebras and Combinators,

Algebra U n i v e r s a l i s 13 (1981)

389-392. [7]

H i n d l e y , R., The Completeness Theorem f o r Typing X-terms, Theor.Comput. S c i . 22 (1) (1983).

[81

Hindley, R., The Simple Semantics f o r Coppo-Dezani-Sall@ Type Assignment, in: Dezani-Ciancaglini, M. and Montanari, U. (eds.), I n t e r n a t i o n a l Symposium on Programming (LNCS, 137, Springer-Verlag, B e r l i n , 1981).

[9 I

Longo, G., S e t - t h e o r e t i c a l Models o f A - c a l c u l u s : T h e o r i e s , Expansions, Isomorphisms ( r e v i s e d v e r s i o n ) , LCS-MIT Tech. Mom. 207 (December 1982), Ann. Math. L o g i c ( t o appear).

[lo 1

Meyer, A., What i s a Model o f X - c a l c u l u s ? (expanded v e r s i o n ) , I n f o r m a t i o n and C o n t r o l 52 (1982) 87-1 22.

[111

M i l n e r , R., F u l l y A b s t r a c t Models o f Typed X - c a l c u l u s , S c i . 4 (1977) 1-22.

[12 1

Nolin, 8, R-1,

L., A l g o r i t h m e s U n i v e r s e l s , R.A. I.R.O. (1974) 6-17.

Theor.

Comput.

I n f o r m a t i q u e Th@orique,

M. COPPO ET AL.

262 [131

Norman, D., Recursion on t h e Countable F u n c t i o n a l s (LNM, 811, S p r i n g e r Verlag, B e r l i n , 1980).

[141

P l o t k i n , G.D., A s e t - t h e o r e t i c a l D e f i n i t i o n o f A p p l i c a t i o n , School o f A r t i f i c i a l I n t e l l i g e n c e , Memo MIP-R-95, Edinburgh Univ. (1972).

[151

Sanchis, L.E., R e f l e x i v e Domains, i n : H i n d l e y , J.R. and S e l d i n , J.P. (eds.), t o H.B. C u r r y : Essays on Combinatory L o g i c , Lambda C a l c u l u s and Formalism (Academic Press, London, 1980).

[161

Schmidt, H.A., 1960).

Mathematische Gesetze d e r L o g i k ( S p r i n g e r - V e r l a g , B e r l i n ,

17 1

S c o t t , D.S., Continuous L a t t i c e s , i n : Lawvere, F.W. (ed.), Toposes, A l g e b r a i c Geometry and L o g i c (LNM, 274, S p r i n g e r - V e r l a g , B e r l i n , 1972).

181

S c o t t , D.S., Open Problem n. 11, 4 i n : Bijhm, C. ( e d . ) , A-calculus and Computer Science Theory (LNCS, 37, S p r i n g e r - V e r l a g , B e r l i n , 1975).

191

S c o t t , D.S.,

201

S c o t t , D.S., L e c t u r e s on a Mathematical Theory o f Computation, Lab. Tech. Monograph PRG-19, O x f o r d Univ. (1981).

211

S c o t t , D.S., Domains f o r D e n o t a t i o n a l Semantics, i n : N i e l s e n , M. and Schmidt, E.M. (eds.), Automata, Languages and Programming (LNCS, 140, S p r i n g e r - V e r l a g , B e r l i n , 1982).

Data Types as L a t t i c e s , S I A M J. Comput. 5 (1976) 522-587. Comp.

LOGIC COLLOQUIUM '82 G . Lolli, G. Long0 and A. Marcja (editors) 0 Elsevier Seience Publishers B. V. (North-Holland], 1984

263

D E C I S I O N PROBLEMS I N PREDICATE LOGIC

Egon B o r g e r ( + ) Lehrstuhl Informatik I 1 U n i v e r s i t a t Dortmund P o s t f a c h 500 500 D-4600 Dortmund 50

ABSTRACT I n t h i s paper we s u r v e y fundamental methods and r e s u l t s a b o u t t h e d e c i s i o n problem f o r c l a s s e s o f f i r s t o r d e r l o g i c a l f o r m u l a e . We beg i n w i t h a kintotocal. account which t e l l s t h e main s t e p s i n t h e deve: lopment o f t h e f i e l d f r o m H i l b e r t ' s f o r m u l a t i o n o f t h e Entscheidungsproblem t o today. We t h e n d i s c u s s i n more d e t a i l n meMod due t o Aanderaa and m y s e l f which b u i l d s upon and extends i d e a s o f T u r i n g and Buchi and i s p a r t i c u l a r l y w e l l s u i t e d 60. logical. &chipLLvMo 0 6 compuRnLLvml. pmblem,; we e x p l a i n how b y t h i s method (and var i a n t s t h e r e o f ) s t r u c t u r a l p r o p e r t i e s o f c o m p u t a t i o n f o r m a l i s m s and o f t h e i r d e s c r i b i n g f o r m u l a e a r e i n t i m a t e l y c o r r e l a t e d i n such a way t h a t many r e c u r s i o n and c o m p l e x i t y t h e o r e t i c a l p r o p e r t i e s by t h i s r e d u c t i o n a r e e a s i l y c a r r i e d o v e r f r o m t h e c o m b i n a t o r i a l d e c i s i o n problems t o t h e c o r r e s p o n d i n g l o g i c a l d e c i s i o n problems. As example we produce by s l i g h t and n a t u r a l v a r i a t i o n s o f t h a t method u ~ d o m(and e a s y ) pmo@ f o r : NP-resp. x 1 r e s p . n 1- r e s p . n 2 - c o m p e t e n u n 0 6

t h e d e c d i o n pmblem f o r p r o p o s i t i o n a l (Cook) resp. f r i s t o r d e r l o g i c (Church, T u r i n g ) r e s p . o f t h e emptiness ( T r a c h t e n b r o t , B u c h i ) r e s p . t h e i n f i n i t y problem f o r f i r s t o r d e r npectm, t h e chamctehizuLLon o f t h e l a t t e r ( S c h o l z ' s problem) as t h e NEXPTIME-acceptable s e t s ( B e n n e t t , Rodding, Schwichtenberg, Jones, Selman) r e s p . o f t h e gener a l i z e d s p e c t r a as t h e NP-sets ( F a g i n ) , s i m p l e axioms f o r e s s e n t i a l l y u n d e c i d a b l e and i w o m p l & t e t h e o h i u resp. n a - t h @ a b l e d o m u h e w i t h o u t fiecumive mock& d e s c r i b i n g enumeration programs f o r xo-unseparab l e r . e . s e t s , Loweh compeexity bounds and i n d e e d completeness r e s u l t s f o r many n a t u r a l s o l v a b l e cases o f f i r s t o r d e r l o g i c a l d e c i s i o n problems as s u b r e c u r s i v e analogues t o t h e u n d e c i d a b l e r e d u c t i o n c l a s s e s , and o t h e r c o m p l e x i t y r e s u l t s f o r f i r s t o r d e r o r p r o p o s i t i o n a l l o g i c problems l i k e a n a t u r a l l o g i c a l c h a r a c t e r i z a t i o n o f n e t work o r T u r i n g machine campLexity o BooLmn duncLLo~owhich i s s t r o n g l y r e l a t e d t o t h e P = NP-prob em. Our main concern i s t o r e v e a l t h e deep s t r u c t u r a l and c o m b i n a t o r i a l s i m u l a r i t i e s between computat i o n s and l o g i c a l d e d u c t i o n s , which b r i n g o u t e x p l i c i t e l y t h e fundamental and u n i f o r m reason f o r many u n d e c i d a b i l i t y and c o m p l e x i t y r e s u l t s f o r c o m b i n a t o r i a l and f o r l o g i c a l d e c i s i o n problems (see t h e above c i t e d examples). HISTORY OF IDEAS From t h e v e r y b e g i n n i n g of mathematics a g r e a t amount o f mathematical r e s e a r c h has been d e v o t e d t o f i n d i n g a l g o r i t h m i c s o l u t i o n s t o g i v e n problems. An i m p o r t a n t subc l a s s o f such problems a r e t h e so c a l l e d d e c i s i o n problems c o n s t i t u t e d b y a c l a s s X of o b j e c t s t o g e t h e r w i t h a p r o p e r t y ( o r r e l a t i o n ) P on X; such d e c i s i o n problems a r e c a l l e d s o l v a b l e if t h e r e e x i s t s an a l g o r i t h m which e n a b l e s t o d e c i d e f o r e v e r y o b j e c t i n X whether i t shares t h e p r o p e r t y P o r n o t . The e f f o r t t o s o l v e problems +) P r e s e n t l y a t I s t i t u t o d i M a t e m a t i c a , I n f o r m a t i c a e S i s t e m i s t i c a o f U n i v e r s i t y o f U D I N E I I t a l y , on l e a v e f r o m U n i v e r s i t y o f Dortmund.

4

E. BORGER

264

i n an a l g o r i t h m i c manner on t h e one hand r e p l i e s t o t h e p r a c t i c a l need o f mechanizing a p p l i c a t i o n s o f mathematical reasoning, o f breaking down complicated mathematical processes i n t o a succession o f elementary steps which can be performed i n a p u r e l y mechanical way w i t h o u t deeper mathematical understanding and w i t h o u t any need f o r i n t e l l i g e n t e x t e r n a l c o n t r o l o r i n t e r v e n t i o n . On the o t h e r hand t h i s e f f o r t o f a l g o r i t h m i s i n g mathematical t h i n k i n g i s c o r r e l a t e d t o and has been s t i m u l a t e d by an o l d dream o f humanity t o f i n d a general r e l i a b l e method by which a l l p h i l o s o p h i c a l and s c i e n t i f i c problems and a l l d i s p u t a t i o n s on them could be s e t t l e d i n an e f f e c t i v e and d e f i n i t e way. This Ldidea 06 a u n i v e m d pmbLem n o t u h g a.tgo&LZhn, present f o r ex. i n Raimundus L u l l u s ' reasoning about an ars magna, has been made more p r e c i s e by L e i b n i z f i r s t l y i n h i s d i s t i n c t i o n between an ars inveniendi - an a l g o r i t h m f o r searching and l i s t i n g s y s t e m a t i c a l l y s o l u t i o n s t o a l l problems - and an ars i u d i c a n d i - an a l g o r i t h m f o r deciding f o r every p a r t i c u l a r problem posed whether the answer t o i t i s yes o r no ; secondly by r e a l i s i n g t h a t such a u n i v e r s a l search o r d e c i s i o n method presupposes a mathem a t i c a l l y precise u n i v e r s a l language i n which a l l problems can be unambigously expressed. This Leibnizean p r o j e c t o f a chamc.tehid.tiCa u n i v e a ~ a L ihas ~ been f u l l y r e a l i z e d o n l y by Frege's f o r m u l a t i o n o f what we c a l l today c l a s s i c a l f i r s t order l o g i c ( p r e d i c a t e c a l c u l u s ) and Godel's (1930) completeness theorem t h a t t h e l a t t e r i n deed describes e x a c t l y t h e n o t i o n o f u n i v e r s a l l o g i c a l v a l i d i t y . This r a t i o n a l calculus, t o say i t i n Leibnizean terms, enabled H i l b e r t t o t u r n the o l d human dream of an ars magna i n t o a s p e c i f i c mathematical problem, namely H i l b e r t ' s program. (1) H i l b e r t asked i n p a r t i c u l a r t o c o d i f y the various branches o f mathematics by f i r s t order axiom systems so t h a t t h e p r o o f o f a n y p a r t i c u l a r mathem a t i c a l statement comes up t o d e r i v i n g i t from t h e axiomsby the i n d i c a t e d p u r e l y l o g i c a l means; t h a t would t u r n d e r i v a t i o n s o f mathemacical r e s u l t s a t l e a s t i n p r i n c i p l e i n t o a mechanical game w i t h concrete o b j e c t s , namely s t r i n g s o f symbols representing f i r s t order l o g i c a l d e r i v a t i o n s . This i s the reason why the d e c i s i o n problem f o r c l a s s i c a l p r e d i c a t e l o g i c , i . e . t h e problem t o know i f t h e r e e x i s t s (and e v e n t u a l l y t o e x h i b i t ) an a l g o r i t h m by which f o r any w e l l formed statement o f p r e d i c a t e l o g i c i t can be decided i n a f i n i t e number o f steps whether i t i s l o g i c a l l y t r u e o r not, has been c a l l e d by H i l b e r t "WS ENTSCHEIDUNGSPROBLEM" t o u t court, considered one i f n o t the main mathematical problem o f t h a t period.

A l l attempts i n the Twenties and e a r l y T h i r t i e s t o solve the Entscheidungsproblem f a i l e d and came up w i t h n o l d o v l b o f the d e c i s i o n problem o n l y 6012 phticduh nubcasu. To formulate the most s i g n i f i c a n t o f these e a r l y p a r t i a l s o l u t i o n s ...) t h e c l a s s o f a l l closed prenex formulae o f r e s t r i c t e d denote by n(ml,m2,m3, predicate l o g i c ( i . e . w i t h o u t f u n c t i o n symbols o r i d e n t i t y s i g n ) having a p r e f i x o f form and c o n t a i n i n g a t most m. p r e d i c a t e symbols o f rank i. The f o l l o w i n g subcases o f p r e d i c a t e l o g i c have a'solvable d e c i s i o n problem w i t h respect t o s a t i s f i a b i l i t y ( 2 ) and a r e optimal t h e r e f n i n a sense t o be made p r e c i s e l a t e r : Monadic p r e d i c a t e l o g i c V. VAV V V... VMV V VA A V...

..

... ... ...

(Lowenheim (1915)) (Ackermann (1928)) (Godel (1932),Kalmar(1933),SchUtte (1934)) (Bernays fi Schonfinkel (1928))

(l) I t i s e p i s t e m o l o g i c a l l y i n t e r e s t i n g t h a t H i l b e r t ' s program was formulated w i t h t h e i n t e n t i o n t o defend mathematics against t h e a t t a c k o f t h e foundational c r i s i s due t o the discovery o f various paradoxes w i t h i n systems o f s e t theory, by p u t t i n g mathematical reasoning on a safe epistemological basis. ( * ) Mostly f o r t e c h n i c a l convenience we s h a l l speak o f l o g i c a l d e c i s i o n problems always i n terms o f s a t i s f i a b i l i t y instead of l o g i c a l v a l i d i t y ; t h i s i s wothout l o s s o f g e n e r a l i t y since a formula i s s a t i s f i a b l e i f f i t s negation i s n o t l o g i c a l l y v a l i d . For a c l a s s o f formulae we consider t h e r e f o r e i f {FIFEC,F i s s a t i s f i a b l e ) i s r e c u r s i v e o r not.

265

Decision Problems in Predicate Logic

I n 1936/7 i t has been p r o v e d t h a t H d b e m 2 EnXAcheidu~1gop4obbe.mas a whole noR nabuabbe. abgohithmicaUy. Indeed Church (1936) showed t h a t any p a r t i a l r e c u r s i v e f u n c t i o n can be r e p r e s e n t e d i n a f i n i t e e x t e n s i o n o f f i r s t o r d e r p r e d i c a t e l o g i c ; as a c o n c l u s i o n t h i s e x t e n s i o n and t h e r e b y a l s o D r e d i c a t e l o g i c cannot be a d e c i d a b l e t h e o r y . T u r i n g (1937) o b t a i n e d t h e r e s u l t i n d e p e n d e n t l y by an e x p l i c i t d e s c r i p t i o n of T u r i n g machine computations by l o g i c a l f i r s t o r d e r formulae t h e r e b y r e d u c i n g e f f e c t i v e l y an u n s o l v a b l e c l a s s o f p a r t i c u l a r word problems f o r T u r i n g machines t o t h e d e c i s i o n problem f o r corresponding f i r s t o r d e r formulae, c o n c l u d i n g t h e u n d e c i d a b i l i t y o f t h e l a t t e r f r o m t h e undecidab i l i t y o f t h e former. Since a t t h a t t i m e a l a r g e number o f s p e c i a l c l a s s e s o f f i r s t o r d e r formulae was known a l r e a d y t o have a s o l v a b l e Entscheidungsproblem, T u r i n g found i t i n t e r e s t i n g t o observe i n o p . c i t . t h a t f o r h i s r e d u c t i o n he r e a l l y needed o n l y a small p o r t i o n o f p r e d i c a t e l o g i c , namely t h e subclass V A V A6(0,-). By T u r i n g ' s c o n s t r u c t i o n , Godel I s completeness theorem t e l l i n g t h a t f i r s t o r d e r l o g i c a l v a l i d i t y i s r e c u r s i v e l y enumerable and t h e u n i v e r s a l i t y o f T u r i n g machines t h e above c l a s s i s even a "ke&LLon claoo", i . e . a c l a s s X o f f o r m u l a e w i t h a d e c i s i o n problem t o which t h e whole Entscheidungsproblem i s many-one r e d u c i b l e , i n o t h e r words f o r which a procedure e x i s t s a s s o c i a t i n g t o e v e r y f i r s t o r d e r formulaF a f o r m u l a F i n X which i s e q u i v a l e n t t o i t i n t h e sense t h a t F i s s a t i s f i a b l e i f f F i s s a t i s f i a b l e . A f t e r t h e appearence o f T u r i n g ' s paper, many e f f o r t s were spent t o i m prove t h i s r e s u l t and t o 4e@%2H d b s 4 - f ~EnXAcheidcngsp&obbem, on t h e one hand by p r o d u c i n g " s m a l l e r " such " r e d u c t i o n c l a s s e s " - t h e r d o r e w i t h undecidable, even many-one complete Entscheidungsproblem, see Suranyi (1959) - on t h e o t h e r hand by e x h i b i t i n g d e c i s i o n procedures f o r l a r g e r and l a r g e r subclasses, see Ackermann (1954) f o r t h e s t a t e o f t h e a r t i n t h e F i f t i e s . I n measuring t h e l o g i c a l c o m p l e x i t y o f formulae m a i n l y t h e above mentioned two c l a s s i f i c a t i o n p r i n c i p l e s were pursued c o n s i d e r i n g t h e s t r u c t u r e o f t h e p r e f i x - l e n g t h and number (and k i n d ) o f q u a n t i f i e r changes and/or number and a r i t y o f o c c u r i n g p r e d i c a t e symb o l s . Most r e d u c t i o n s proceeded by s k i l f u l and o f t e n v e r y c o m p l i c a t e d f i r s t o r d e r f o r m a l d e s c r i p t i o n s o f d i r e c t t r a n s f o r m a t i o n s o f models f o r g i v e n formulae F ( o f w e l l known l o g i c a l s t r u c t u r e ) i n t o models o f a " c o d i n g " F o f F such t h a t f r o m any model f o r F a model f o r F can be e x t r a c t e d . These a x i o m a t i z a t i o n s o f d i r e c t model t r a n s f o r m a t i o n s o f t e n y i e l d e d much more t h a n what was r e q u i r e d by t h e v e r y n o t i o n o f r e d u c t i o n c l a s s , namely second-order l o g i c a l equivalences; and indeed i t i s a w i d e l y open, i n t e r e s t i n g problem t o know what a r e t h e b e s t r e d u c t i o n c l a s s e s w i t h r e s p e c t t o second-order d e d u c i b l e e q u i v a l e n c e between ( t h e e x i s t e n t i a l c l o s u r e s o f ) F and i t s r e d u c t i o n f o r m u l a F ( i n s t e a d o f a s k i n g m e r e l y t h a t F i s s a t i s f i a b l e ( f i r s t o r d e r d e d u c i b l e ) i f f F i s ) . B u t i n s p i t e o f many new r e d u c t i o n s which were found up t o t h e end o f t h e F i f t i e s , n o t even t h e d e c i s i o n p r o blem o f a l l f o r m a l l y s p e c i f i e d c l a s s e s o f formulae l i k e t h e p r e f i x c l a s s e s (I o r t h e p r e f i x - s i m i l a r i t y c l a s s e s n(ml,m *,...) c o u l d be s e t t l e d .

-

S t r a n g e l y e n o u g h T u r i n g ' s i d e a t o l o o k f o r smooth and d i r e c t l o g i c a l d e s c r i p t i o n o f machine computations o r s i m i l a r processes was n o t r e a l l y pursued up t o 1962. Only t h e n Buchi (1962) t o o k up a g a i n T u r i n g ' s approach and combined i t w i t h s k i l f u l use o f *Lea theohems &e Ro Shabern. These theorems t e l l t h a t a prenex f o r m u l a o f r e s t r i c t e d p r e d i c a t e l o g i c ( i . e . w i t h o u t f u n c t i o n symbols and w i t h o u t i d e n t i t y s i g n ) i s s a t i s f i a b l e i f f i t s Skolem normal f o r m i s , and t h a t i n models f o r such Skolem normal forms one can r e s t r i c t a t t e n t i o n t o t h e domain o f terms b u i l t up f r o m t h e i n d i v i d u a l c o n s t a n t s and f u n c t i o n symbols o c c u r i n g i n t h e f o r m u l a and t o i n t e r p r e t a t i o n o f t h e terms by themselves. T h e r e f o r e i n t h e l o g i c a l d e s c r i p t i o n o f computation processes l i k e t h e one g i v e n i n T u r i n g ' s (1937) paper one has n o t t o c a r e any more about t h e f o r m a l r e p r e s e n t a t i o n o f t h e o b j e c t s o f computation - l i k e numbers, words, sequences, domino p o s i t i o n s and l i k e ; these d a t a a r e r e p r e s e n t e d j u s t as i n d i v i d u a l terms appearing i n f o r mulae i n Skolem normal form. A f o r m u l a F i n VAVA o f form VAVAG f o r example i s uxvy s a t i s f i a b l e i f f i t s Skolem normal f o r m M G U , V ( O , ~ ' ) - w i t h Hx 1'. . ,xn ( tl 3. . . I t n )

w

.

E. BORGER

266

d e n o t i n g t h e r e s u l t o f simultaneous s u b s t i t u t i o n o f xi by ti - i s s a t i s f i a b l e o v e r the naturals IO,O',O",O'", ...1 w i t h t h e i n d i c a t e d i n t e r p r e t a t i o n o f t h e zerop l a c e resp. one-place f u n c t i o n as t h e number 0 r e s p t h e n a t u r a l successor VA o f f o r m f u n c t i o n x + 1. S i m i l a r l y a f o r m u l a F i n AV

...

AV

xv1

. ..

VA G VnY

i s s a t i s f i a b l e i f f i t s Skolem normal f o r m G ; v.,l

. .,vn

(al,.

. . ,an)

i s s a t i s f i a b l e o v e r t h e domain of a l l words o v e r t h e a l p h a b e t

...,an)

{al.

w i t h t h e one-place f u n c t i o n symbols i n t e r p r e t e d as word successor f u n c t i o n s . S i m i l a r l y formulae o f p r e f i x f o r m M V t a l k about t h e F i t c h domain (Dyck language D1) o f a l l c o r r e c t p a r e n t h e s i s expressions b u i l t up f r o m t h e 2-ary p a r e n t h e s i s (,); V n M V corresponds t o t h e s t r u c t u r e o f b i n a r y t r e e s w i t h leaves l a b e l e d by symbols al,. . ,a ecc.

.

Buchi's s i m p l e b u t f u n d a m e n t a l o b s e r v a t i o n c o n s t i t u t e d a breakthrough. I t became c l e a r t h a t i n o r d e r t o show a c l a s s X t o be a r e d u c t i o n c l a s s one had t o l o o k above a l l f o r an a p p r o p r i a t e t y p e o f c o m b i n a t o r i a l system - l i k e T u r i n g machines, Thue systems, P o s t correspondence problems, domino games - where a p p r o p r i a t e n e s s means t h a t t h e d a t a s t r u c t u r e o f t h e c o m b i n a t o r i a l system can be c o n v e n i e n t l y encoded i n t o t h e t e r m s t r u c t u r e o f t h e (Skolem normal f o r m o f t h e ) f o r m u l a e i n X, and then d e s c r i b e an u n s o l v a b l e d e c i s i o n problem f o r t h e c o m b i n a t o r i a l system by a s a t i s f i a b i l i t y ( o r d e d u c i b i l i t y ) problem o f a f o r m u l a i n X.(1) By t h i s m t h o d Biichi (1962) showed t h e u n s e t t l e d p r e f i x c l a s s t o be a (even c o n s e r v a t i v e T 2 ) ) r e d u c t i o n c l a s s . Indeed f o l l o w i n g t h i s l i n e o f a t t a c k w i t h i n 4 y e a r s t h e d e c i s i o n problem o f a l l p r e f i x - s i m i l a r i t y c l a s s e s n(ml,m 2,...) could be s e t t l e d by proving the f o l l o w i n g t o be c o n s e r v a t i v e r e d u c t i o n c l a s s e s : AVA(-,l)

Kahr (1962), i m p r o v i n g t h e r e d u c t i o n c l a s s AVA (0,m) Kahr, Moore, Wang (1962)

in

A V ~ A ( O , ~ K) o s t y r k o (1964), Genenz (1965) ; s t r e n g t h e n e d i n Deutsch (1981) AVAVm(O,l) Gurevich (1966)

(1) I t i s i n t e r e s t i n g t o n o t e i n t h i s c o n t e x t t h a t v a r i o u s small u n i v e r s a l combinat o r i a l systems o r s t r o n g c o m b i n a t o r i a l t o o l s have been developed i n t h e a t t e m p t t o d e c i d e a g i v e n l o g i c a l d e c i s i o n problem. Good example a r e : Ratmey'n (1928) theanem developed i n t h e course o f t h e s t u d y o f t h e d e c i s i o n problem f o r t h e SchGnf i n k e l - B e r n a y s c l a s s V" f i m , a n d o t h e r s ; R a d d i t t g ' o (1969) p t h pmbLem i n t h e f i r s t (Gaussian) q u a d r a n t w i t h n a t u r a l c o o r d i n a t e s developed f o r an e l e g a n t p r o o f o f t h e u n d e c i d a b i l i t y o f t h e AVA (-,1) case - t h i s u n s o l v a b l e problem, a k i n d o f g e o m e t r i c a l model encoding a r b i t r a r y machine computations, t u r n e d o u t t o be extreml y u s e f u l f o r c o n s t r u c t i o n o f s m a l l u n i v e r s a l T u r i n g machines (see K l e i n e Buning & Ottmann (1977) and f o r d e c i s i o n problems i n g e n e r a l i z e d v e c t o r a d d i t i o n systems (see K l e i n e Buning (1980)); t h e fineah nampfing pnobLem d e v i c e d by Aanderaa d u r i n g t h e s t u d y o f subcases o f t h e AVA - d e c i s i o n problem (Aanderaa (1966) see below t h e AVA -Subclass Theorem) which prompted Lewis (1979) t o g i v e a new p r o o f f o r B e r g e r ' s (1966) theorem of t h e u n s o l v a b i l i t y o f t h e u n c o n s t r a i n e d domino problem. (2) A reduction class X i s c a l l e d conservative i f also f i n i t e s a t i s f i a b i l i t y i s preseri f t h e r e d u c t i o n procedure a l s o f u l f i l l s t h a t F i s s a t i s f i a b l e i f f F i s . ved,i.e.

267

Decision Problems in Predicate Logic

These t h r e e fundamental r e s u l t s t o g e t h e r w i t h t h e d e c i d a b l e cases mentioned a t t h e b e g i n n i n g (and w i t h some a d d i t i o n a l easy r e d u c t i o n s , see f o r ex. Gurevich (1966) o r K o s t y r k o (1966) y i e l d t h e f o l l o w i n g p r e f i x - s i m i l a r i t y theorem, f o r which we g e n e r a l i z e s l i g h t l y o u r

No-t~~tition.F o r any c l a s s n o f p r e f i x e s and any c l a s s u o f p r e d i c a t e symbols l e t n ( u ) be t h e c l a s s o f c l o s e d prenex f o r m u l a e o f r e s t r i c t e d p r e d i c a t e l o g i c w i t h p r e f i x i n n and p r e d i c a t e symbols i n U. L e t V"A" : = I V ~ Am,n~ ~= 0,1, ... I mn) be a c l a s s o f ml monadic,m2 and s i m i l a r l y f o r V m ~ V m e t c . L e t (ml,m2, b i n a r y ,...,mn n-ary and no o t h e r p r e d i c a t e symbol.

...,

Phe6ix-SimiJhhity Theohem doh k e n h i c t e d ptedicute

F o r any c l a s s p r e f i x e s and any c l a s s u o f p r e d i c a t e symbols t h e f o l l o w i n g h o l d s :

n

of

1. e i t h e r n(~) i s a c o n s e r v a t i v e r e d u c t i o n c l a s s o r has a s o l v a b l e d e c i s i o n problem 2. n(o) has a s o l v a b l e d e c i s i o n problem i f f e i t h e r u c o n t a i n s o n l y monadic p r e d i c a t e symbols o r II 5 V"A" or

u V"AV"

u V"MV"

and ~ - ( v " A " u V"AV" u V"MV")

are f i n i t e

( i . e . e s s e n t i a l l y o n l y t h e subcases mentioned a t t h e b e g i n n i n g have s o l v a b l e d e c i s i o n problem). 3. Any p r e f i x - s i m i l a r i t y r e d u c t i o n c l a s s i s more c o m p l i c a t e d o r equal t o one o f the f o l l o w i n g nine minimal conservative reduction classes: f i n i t e p r e f i x : AvA(m,l)

A3V (", 1

i n f i n i t e prefix: A"V(0,l)

A 3 ~ " ( ~1),

v"A3v( 0,1)

V"AVA(O,~)

q u a n t i f i e r changes :1

AVA"(O, 1), AV"A (0, 1

AVAV"

(o,~)

:2 :3

where n ( u ) i s n o t more c o m p l i c a t e d t h a n ( i . e . < ) n ' ( u ' ) i f f t h e elements o f n can be o b t a i n e d f r o m t h e elements o f n' by d e T e t i o n o f q u a n t i f i e r s and t h e r e i s a 1-1-mapping f r o m u t o U ' which does n o t decrease t h e a r i t y o f p r e d i c a t e symbols i n u . An analogous theorem h o l d s when a l s o f u n c t i o n symbols and t h e e q u a l i t y s i g n a r e allowed, denote by ( ~ ; u ; T ) t h e c l a s s o f c l o s e d prenex f o r m u l a e w i t h p r e f i x i n n

(2)

A good source f o r a p r o o f a r e f o r p a r t 2 t h e papers by Lowenheim, Godel, Bernays E S c h o n f i n k e l c i t e d a t t h e beginning, f o r p a r t 3 t h e papers Gurevich (1966), Kostyrko(1964) and f o r t h e AVA(m,l) case e i t h e r K o s t y r k o ( l 9 6 6 ) o r Rodding (1969).

E. BORGER

268

and p r e d i c a t e s resp. f u n c t i o n s as i n d i c a t e d by 0 r e s p . T. We t h e n can f o r m u l a t e t h e f o l l o w i n g v a r i a n t o f t h e above theorem which a g a i n has been e s t a b l i s h e d piecemeal by v a r i o u s a u t h o r s :

P m & i x - S U n i t y Theoaiem doh. @&?. ptredicate L o g i c . 1. F o r i d e n t i t y f r e e p r e d i c a t e l o g i c a c l a s s ( 7 ; O ; T )

w i t h a t l e a s t one f u n c t i o n symbol e i t h e r has a s o l v a b l e d e c i s i o n problem and i s i n c l u d e d i n a t l e a s t one o f t h e c l a s s e s o f a l l c l o s e d prenex f o r m u l a e w i t h - o n l y monadic p r e d i c a t e and f u n c t i o n symbols or - a t most one u n i v e r s a l q u a n t i f i e r o r t h e c l a s s i s u n s o l v a b l e and i n d e e d a c o n s e r v a t i v e r e d u c t i o n c l a s s and i n c l u d e s a t l e a s t one o f t h e c l a s s e s o f a l l c l o s e d prenex f o r m u l a e w i t h p r e f i x M a n d one b i n a r y p r e d i c a t e and one monadic f u n c t i o n symbol o r v i c e versa one b i n a r y f u n c t i o n and one monadic p r e d i c a t e symbol.

2. F o r f u l l p r e d i c a t e l o g i c w i t h = a c l a s s ( T ; O , T ) w i t h a t l e a s t one f u n c t i o n symbol e i t h e r has a s o l v a b l e d e c i s i o n problem and i s i n c l u d e d i n a t l e a s t one o f t h e c l a s s e s o f a l l c l o s e d prenex f o r m u l a e w i t h

-

-

o n l y monadic p r e d i c a t e and a t most one, a m o n a d i c > f u n c t i o n symbols or o n l y one u n i v e r s a l q u a n t i f i e r and a t most one, a monadic, f u n c t i o n symbol or only existential quantifiers

o r i t i s a c o n s e r v a t i v e r e d u c t i o n c l a s s and i n c l u d e s a t l e a s t one o f t h e classes w i t h

-

prefix A

,2

-

prefix A

,

p r e f i x A\

monadic f u n c t i o n s and no p r e d i c a t e besides = or 1 b i n a r y f u n c t i o n and no p r e d i c a t e besides = or 1 monadic f u n c t i o n , 1 b i n a r y p r e d i c a t e besides =

The same theorem h I d s i f f u n c t i o n s may be i n t e r p r e t e d n o t as t o t a l b u t as p a r t i a1 f u n c t i o n s .?I) A t t h e b e g i n n i n g o f t h e S e v e n t i e s Krom (1970), Aanderaa (1971) and m y s e l f i n 8 o r g e r (1971) i n d e p e n d e n t l y came up w i t h a t h i r d i d e a which pursued B u c h i ' s approach t o i t s l a s t consequences. Whereas Buchi and h i s f o l l o w e r s i n t h e 6 0 - i e s d e s c r i b e d c o m p u t a t i o n processes w i t h e x p l i c i t r e f e r e n c e t o t h e t i m e component, we r e a l i z e d t h a t t h i s i s n o t necessary i f one aims a t a d e s c r i p t i o n o f p r o p e r t i e s o f computations where t h e t i m e needed a t a d e s c r i p t i o n o f p r o p e r t i e s o f computations where t h e t i m e needed t o reach t h i s p r o p e r t y i s i r r e l e v a n t . Such a p r o p e r t y i s f o r ex. t h a t a computation j u s t h a l t s , w i t h o u t w o r r i n g a b o u t how many s t e p s t h i s may take. T h e r e f o r e we t r i e d and succeeded i n b c ~ i b i n c jcompu*a.tion p o c e s n e s &thou2 ae6eaencing .time. T h i s method a l l o w e d enormous s i m p l i f i c a t i o n s o f r e d u c t i o n f o r m u l a e d e s c r i b i n q machine and l i k e problems and r e s u l -

(1)

F o r p r o o f s see Gurevich (1969),(1973),(i976) - f o r a s i m p l i f i c a t i o n o f t h e main r e d u c t i o n i n t h e l a s t paper a l s o Borger (1978) o r W i r s i n g (1977 b ) ; t h e decidab i l i t y o f t h e o n e - q u a n t i f i e r c l a s s w i t h = and o n l y one, a monadic, f u n c t i o n symb o l i s due t o Shelah (1977); t h e case f o r p a r t i a l f u n c t i o n s i s t r e a t e d i n Abramsky (1980).

Decision Problems in Predicate Logic

269

ted i n almost t r i v i a l i z i n g many proofs(')and i n much s h a r p e r reduction c l a s s e s as before, defined by imposing r e s t r i c t i o n s not only on p r e f i x and s i m i l a r i t y , b u t a l s o on t h e t r u t h - f u n c t i o n a l s t r u c t u r e of reduction formulae, on the s t r u c ture of atomic subformulae i n them and on t h e number of occurences of atomic subformulae. F u r t h e r a n a l y s i s using s t r o n g l y the idea of describing computations without bothering about t h e time d e s c r i p t i o n and based on these new s t r o n g e r c l a s s i f i c a t i o n p r i n c i p l e s i n some cases again brought t o t h e border l i n e between decidable and undecidable cases, although not in such a complete and natural way a s i n the case of t h e p r e f i x - s i m i l a r i t y problem. For a systematic ( b u t not complete) account we r e f e r t o t h e two r e c e n t books Dreben & Goldfarb (1979) and Lewis (1979) and l i m i t ourselves here t o mention only f o u r t y p i c a l and outstanding examples f o r such (almost) minimal undecidable cases (indeed reduction c l a s s e s ) . The f i r s t example i s about Khom ,pmunLLeae, i.e. formulae in prenex conjunctive normal form with a matrix containing only binary d i s j u n c t i o n s . The i n t e r e s t f o r t h i s c l a s s of formulae comes from two f a c t s : a ) the proof by Herbrand (1930: pg. 118), (1931: pg. 33 sq.) t h a t t h e decision problem f o r formulae in prenex conjunctive normal form where the matrix i s a conjunction of atomic o r negated atomic formulae i s s o l v a b l e ; b) Chang's & K e i s l e r ' s (1962) normal form theorem showing t h a t any i d e n t i t y - f r e e f i r s t order formula can be p u t i n t o prenex conj u n c t i v e normal form with a l t e r n a t i o n s of length a t most 3. Krom (1964), (1966), (1967), (1967 a ) , (1968) studied then formulae with binary d i s j u n c t i o n s from various points of view and obtained in Krom (1970) the r e s u l t t h a t t h e i r decision problem i s unsolvable. Thourough research based on the method developed in Aanderaa (1971) and Borger (1971) r e s u l t e d i n the following &om and H o m ~Pnehix Theohem. R e s t r i c t e d t o K m m (prenex conjunctive normal form w i t h matrices containing only binary d i s j u n c t i o n s ) and t o Hohn formlilae ( i . e . no d i s j u n c t i o n contains more than one non-negated atomic subformula) a l l p r e f i x c l a s s e s except f o r t h e c l a s s e s AVAVn f o r n = 1,2, ....m - whose decision problems a r e s t i l l open, but we c o n j e c t u r e them t o be solvable - a r e e i t h e r conservative reduction c l a s s e s o r have a s o l v a b l e decision problem. In p a r t i c u l a r the following a r e minimal undecidable c l a s s e s : VAVA AVVA AVM M V A : r e s t r i c t e d t o Krom & Horn AVAV A3V : ( r e s t r i c t e d t o Krom allowing = ) o r ( r e s t r i c t e d t o Horn) whereas t h e following c l a s s e s have a s o l v a b l e decision problem: AVA (even with = allowed) VmAmVm : r e s t r i c t e d t o Krom AVA : r e s t r i c t e d t o Horn The two Krom and Horn c l a s s e s MV nand A V V A a r e h i s t o r i c a l l y t h e f i r s t minimal ones which have been proved t o posses an unsolvable decision problem by the method developed in Aanderaa (1971) and Borger (1971); f o r t h e sake of exempli-

(2) Recently Jones & Matijasevich (1982) applied t h e same idea t o a d i r e c t desc r i p t i o n of r e g i s t e r machine h a l t i n g problems by exponential diophantine equat i o n s ; t h i s r e s u l t e d in a tremendous s i m p l i f i c a t i o n of t h e proof f o r the DavisPutnam-Robinson theorem t h a t every r e c u r s i v e l y enumerable set i s exponential diophantine avoiding completely use o f the Chinese remainder theorem and t h e t r i c k y number t h e o r e t i c a l c o n s t r u c t i o n s involved.

270

E. BORCER

f i c a t i o n t h i s p r o o f w i l l be reproduced i n t h e n e x t s e c t i o n . o t h e r undecidable cases have been o b t a i n e d by l a t e r r e f i n e m e n t s o f t h i s method. Up t o today t h e p r e f i x - s i m i l a r i t y problem r e s t r i c t e d t o Krom and f o r Horn c l a s s e s i s neverthel e s s s t i l l open; o n l y s c a t t e r e d p a r t i a l r e s u l t s ( a l t h o u g h o b t a i n e d by i n t e r e s t i n g refinements o f t h e above mentioned method) a r e known l i k e t h e u n d e c i d a b i l i t y o f the classes VAVmA(O,l),AVmA( 1,1),AVmA(0,2) r e s t r i c t e d t o Krom and Horn (see Lewis (1976),(1979) based on a d e s c r i p t i o n o f P o s t correspondence problems f o l l o w e d by f u r t h e r r e d u c t i o n s by a technique i n s p i r e d by Sh n o n ’ s (1956) cons t r u c t i o n o f a u n i v e r s a l T u r i n g machine w i t h o n l y 2 s t a t e s . ) ? ? )

“7,

I n t e r e s t i n g l y enough i n t h e Krom case, d i f f e r e n t l y from t h e n o t t r u t h - f u n c t i o n a l l y r e s t r i c t e d c l a s s i c a l case, p r e d i c a t e s o f rank b i g g e r than 2 may p l a y an e s s e n t i a l r o l e f o r t h e ( u n - ) s o l v a b i l i t y o f t h e d e c i s i o n problem o f a c l a s s ; i n f a c t t h e c l a s s AVA-(-,-) r e s t r i c t e d t o Krom has a s o l v a b l e d e c i s i o n problem as proved i n Borger (1973) by r e d u c t i o n t o t h e AVA-Krom case, whereas f o r some k t h e c l a s s e s MVA(0,-,k) and AVM(0,-,k) r e s t r i c t e d t o Krom a r e c o n s e r v a t i v e r e d u c t i o n c l a s s e s . F o r r e a l l y small k l i k e k < 7 n o t h i n g i s known and t h e a c t u a l l y a v a i l a b l e methods do n o t seem t o be s u f f i c i e n t t o s e t t l e these q u e s t i o n s . W i t h o u t c o n s i d e r a t i o n o f t h e p r o p o s i t i o n a l f o r m one has f o r one o f t h e minimal f i n i t e - p r e f i x r e d u c t i o n c l a s s e s t h e f o l l o w i n g i n t e r e s t i n g sharp c l a s s i f i c a t i o n o f subclasses: AVA Subc&eans Theohem. Subclasses o f t h e minimal undecidable f i n i t e - p r e f i x c l a s s AVA(m,l) s p e c i f i e d by any o f t h e 212-1 combinations o f a l l o w e d atomic subformulae b u i l t up f o r m t h e v a r i a b l e s i n t h e p r e f i x AVA have an u n s o l v a b l e deXVY c i s i o n problem ( i n d e e d c o n s t i t u t e r e d u c t i o n c l a s s e s ) i f f a t l e a s t t h r e e forms o f atomic sirhfnrmiilae(jrc1uding e i t h e r Rxy t o g e t h e r w i t h Ryv o r Ryx t o g e t h e r w i t h Rvy a r e allowed.

Fw Atomic SubgomLLeae Theomm. ( G o l d f a r b ( 1 9 7 4 ) ) ( 4 ) The subclass o f AVA“ o f a l l formulae w i t h m a t r i c e s o f t h e form (Ao A 1A1) v (A2 A l A 3 ) where A . a r e atomic formulae i s a r e d u c t i o n c l a s s ; t h e c l a s s o f a l l formulae c o n t a i n i n g ( e v e n t u a l l y an a r b i t r a r y number o f d i f f e r e n t occurences o f ) o n l y two d i s t i n c t atomic subformulae has a s o l v a b l e d e c i s i o n problem. The case where t h r e e atomic subformulae a r e a l l o w e d i s open. ( l ) AVAA,AAVA a r e due t o Lewis (see Aanderaa & Lewis (1973)), AVAV and A3V f o r H w n a r e a l s o ( s e e Ph.D. T h e s i s ) and f o r Krom w i t h allowance o f = t o Aanderaa & F r g e r & Gurevich (1982). The d e c i s i o n procedures a r e due r e s p e c t i v e l y t o Aanderaa & Lewis (1973) ( f o r t h e i n c l u s i o n o f = see a g a i n Aanderaa & Borger & Gurevich ( 1 9 8 2 ) ) , Maslov (1964) and G o l d f a r b (1974). F o r c o n s e r v a t i v i t y o f t h e r e d u c t i o n s see Aanderaa & Borger & Lewis (1982). F o r AVA i n Krom w i t h = see G o l d f a r b ’ s Ph.D. Thesis.

(’) An e a r l y r e s u l t o f an u n s o l g a b l e p r e f i x - K r o m c l a s s w i t h a small number o f b i n a r y p r e d i c a t e symbols was AV A (0,4) r e s t r i c t e d t o Krom and Horn and was proved i n Rodding & Borger (1974) by a much s i m p l e r method t h a n t h e one used by H a r r y Lewis f o r h i s s h a r p e r r e s u l t s . Krom (1970) o b t a i n e d t h e u n s o l v a b i l i t y o f t h e Krom c l a s s hVmA(O,k) f o r some ( b i g ? ) k by d e s c r i b i n g d e d u c t i o n s i n P o s t ’ s t a g systems. ( 3 ) F o r t h e d e c i d a b l e cases see Dreben & Kahr & Wang (1962), f o r t h e undecidable cases Aanderaa & Lewis (1974) which i s based on Aanderaa (1966) and t h e i n t e r e s t i n g and v e r y d i f f i c u l t l i n e a r sampling problem e s p e c i a l l y devised f o r t h i s case. T h i s c l a s s i f i c a t i o n was suggested a l r e a d y by Buchi (1962) and a l s o appears i n Wang (1962). K o s t y r k o (1966) proves t h e u n s o l v a b i l i t y w i t h any t h r e e o f t h e f o u r atomic subformulae Rxy,Ryx,Ryv,Rvy b e s i d e s o n l y monadic subformulae. ( 4 ) F o r t h e l o n g t r a d i t i o n t o c l a s s i f y formulae w i t h r e s p e c t t o forms o f t h e i r atomic subformulae see t h e l i s t o f r e f e r e n c e s i n Lewis (1979), pg. 155 i n c l u d i n g among o t h e r s Skolem, Church, Friedman, and Maslov.

271

Decision Problems in Predicate Logic W h i n g ' n T h e o k m . W i r s i n g (1977). The c l a s s o f formulae o f form A

x1

... 9

(S1 = S 2 A S 3 # s 4 )

6

w i t h terms si b u i l t up from a monadic f u n c t i o n symbol f and t h e v a r i a b l e s x ,..., x i s a c o n s e r v a t i v e r e d u c t i o n c l a s s (The cases w i t h 5,4 o r 3 u n i v e r s a l q i a n t i f i h - s a r e s t i l l open problems.) The i n t i m a t e s t r u c t u r a l and c o m b i n a t o r i c a l connections between programs M and the l o g i c a l formulae LX d e s c r i b i n g t h e e f f e c t o f t h e e x e c u t i o n o f M on g i v e n data, once r e v e a l e d by o u r method f o r l o g i c a l d e s c r i p t i o n o f computation processes t o be e x p l a i n e d i n t h e n e x t s e c t i o n , immediately y i e l d as b y - p r o d u c t t h a t v i a t h i s implementation m c k i n e haPLing pmbLemn and Logical? & c h i o n pMbl?emn a m mcwrniueLy inomo?Lpkic. From t h i s isomorphism many c o m p l e x i t y t h e o r e t i c a l consequences can be drawn; some examples w i l l be g i v e n i n t h e t h i r d s e c t i o n below. I n part i c u l a r l e t us mention here two fundamental theorems which w i l l be proved parap h r a s i n g t h e many-one completeness p r o o f g i v e n i n t h e n e x t s e c t i o n f o r H i l b e r t ' s Entscheidungsproblem:

Aandenaa'n Theohem (1971). I f a program M enumerates r e c u r s i v e l y unseparable s e t s then t h e f i r s t o r d e r t h e o r y w i t h t h e program d e s c r i p t i o n a M as n o n l o g i c a l axiom has r e c u r s i v e l y unseparable theoremhood and l o g i c a l f a l s e h o o d and i s t h e r e f o r e e s s e n t i a l l y undecidable and incomplete. (A v a r i a n t o f ) a,,, i s a s a t i s f i a b l e formula

w i t h o u t r e c u r s i v e models.

(We w i l l g i v e t h e p r o o f w i t h t h e s i m p l e r formulae

aM

found i n Borger (1975),

(1982) which y i e l d an analogous statement f o r En-unseparable En+l-sets Grzegorczyk's h i e r a r c h y . )

in

Ttiacktenbhot'n Theomm. The c l a s s e s o f c o n t r a d i c t o r y r e s p . f i n i t e l y s a t i s f i a b l e ( r e s p . n o n - c o n t r a d i c t o r y b u t n o n - f i n i t e l y s a t i s f i a b l e ) f i r s t o r d e r formulae a r e r e c u r s i v e l y i n s e p a r a b l e . As a c o n c l u s i o n t h e same h o l d s r e s t r i c t e d t o a r b i t r a r y c o n s e r v a t i v e r e d u c t i o n classes. Note t h a t by t h e u n s o l v a b i l i t y o f t h e Entscheidungsproblem, G o d e l ' s completeness theorem and t h e obvious f a c t t h a t t h e f i n i t e l y s a t i s f i a b l e formulae form a r e c u r s i v e l y enumerable c l a s s ( 1 ) t h e c l a s s o f i n f i n i t y axioms ( i . e . o f noncontrad i c t o r y b u t n o t f i n i t e l y s a t i s f i a b l e formulae) i s t r i v i a l l y n o t r e c u r s i v e l y enumerable. Most known formulae c l a s s e s w i t h s o l v a b l e d e c i s i o n problem have t h e p r o p e r t y t h a t any s a t i s f i a b l e f o r m u b i n them admits a l s o f i n i t e models; t h i s so c a l l e d f i n i t e c o n t r o l l a b i l i t y p r o p e r t y i s s t u d i d e x t e n s i v e l y i n Dreben & Goldf a r b (1979); see a l s o Ash (1975). An i n t e r e s t i n g counterexample i s t h e c l a s s A V A r e s t r i c t e d t o Krom which c o n t a i n s t h e f o l l o w i n g i v f i n i t y axiom (Gxy means: x i s b i g g e r t h a n y; r e d v as successor x o f x ) :

I ::

A V A (Gvx & (GXY

+

Gvy)

A

~GXX)

X V Y

T h i s f i r s t c o n j u n c t asks f o r a " g r e a t e r " element x

~ t +o any ~ g i v e n xn, t h e

second c o n j u n c t l i n k s these elements t o g e t h e r by ( a k i n d o f ) t r a n s i t i v i t y ( 2 ) i n t o a c h a i n (xo,xl,x 2,...) where every xi i s " g r e a t e r " t h a n x . i f i i s b i g g e r than j, J t h e t h i r d c o n j u n c t exludes t h a t an x . may be equal t o some xi f o r i < j . We w i l l J use t h i s s i m p l e i n f i n i t y axiom i n t h e n e x t s e c t i o n t o assure c o n s e r v a t i v i t y o f reductions. ( 1 ) F i n i t e s a t i s f i a b i l i t y i s c o m p l e t e l y a x i o m a t i z e d i n B u l l o c k & Schneider (1973); c f . a l s o H a i l p e r i n (1961) f o r a complete a x i o m a t i z a t i o n o f formulae which a r e i n v a l i d i n some f i n i t e domain. ( 2 ) NB. F u l l t r a n s i t i v i t y cannot be expressed by a Krom formula, see Krom (1966).

212

E. BORGER

The o r i g i n a l proofs in Trachtenbrot (1950), (1953) a r e much more involved, they y i e l d i n t e r e s t i n g a p p l i c a t i o n s t o normal form theorems f o r r e c u r s i v e l y enumerable p r e d i c a t e s . Indeed t h e 1950-proof i s based o n a construction s h o w i n g t h a t p4ecDeLq t h e gnaphn ad pLhtiaL necuhbive @nc.tiou have a npectmL hepmenah.tivn, i . e . admit a f i r s t o r d e r formula F with =, without function symbols and with cert a i n monadic p r e d i c a t e symbols Pi such t h a t f o r a l l x ~ , . . . , x ~ + ~ : f ( xl , . . . . x ) = x ~ +i f f ~ t h e r e i s a f i n i t e model of F where t h e monadic prea r e i n t e r p r e t e d a s s e t s of d i c a t e symbols P 1 , . . . , P n + l c a r d i n a l i t y x l , ... , x ~ +r e ~s p e c t i v e l y ( ' ) . Related t o these techniques developed by Trachtenbrot and o t h e r s i s the SpectnaLpmblem formulated i n Scholz (1952), i . e . the pmbLem t o chanactehize npectm f o r f i r s t o r d e r formulae including t h e i d e n t i t y symbol, where t h e spectrum ( F ) of a formula F i s defined a s t h e c l a s s of a l l those natural numbers n f o r which F has a f i n i t e model of c a r d i n a l i t y n . (Note t h a t by well known p r o p e r t i e s of f i r s t order l o g i c i t i s reasonable t o formulate t h i s problem a s done above f o r f i n i t e c a r d i n a l i t i e s of models f o r f i n i t e l y axiomatizable l o g i c a l t h e o r i e s with = , where without l o s s of g e n e r a l i t y functions a r e represented by t h e i r g r a p h s . ) Already e a r l y i n v e s t i g a t i o n s i n t o t h i s problem - s e e Asser (1955), Mostowski (1956) - showed t h a t t h e SpekxhzLpmbLem h a ~-13 do w L t h compcLta.tivmL p m b L m a n d t h e i 4 cvmpLexity a t lower l e v e l s of t h e Grzegorczyk-hierarchy ( E n : 0 5 n ) of p r i m i t i v e r e c u r s i v e functions: f o r ex. every Grzegorczyk-E2 ( r e a d : by a determin i s t i c polynomial-time-bounded r e g i s t e r machine a c c e p t a b l e ) set i s a spectrum and the c l a s s of s p e c t r a i s s t r i c t l y included in t h e c l a s s of a l l E3-sets ( r e a d : of a l l s e t s acceptable by a d e t e r m i n i s t i c r e g i s t e r machine in exponential t i m e ) , whereas u p t o today i t i s not known whether every spectrum of a f i r s t o r d e r f o r mula i s a l s o an E2-set nor what i s t h e answer t o Anne4'n pmbLem whether the complement of every spectrum i s a l s o a spectrum. Considering in the same way f o r any f i n i t e order n the c l a s s SPECTRAn : = {spectrum(F)I F l o g i c a l formula of order n} of s p e c t r a of n - t h o r d e r formulae i t turned o u t t h a t n - t h order s p e c t r a form a s t r i c t hierarchy SPECTRAn+l SPECTRAn+2exhausting e x a c t l y t h e c l a s s of a l l Kalmar-elementary s e t s ( E 3 - s e t s ) of p o s i t i v e n a t u r a l numbers (Bennett 1962), i n particular: T h e v m (Rodding & Schwichtenberg 1972): y n c SPECTRAn+l 5 y n + l Here y n denotes the c l a s s of a l l s e t s which a r e accepted by a r e g i s t e r machine an(x) within time bound a n ( p ) f o r some polynomial p and a o ( x ) : = x , a n + l ( x ) := 2 B e n n e t t ' s t h e s i s already contained among o t h e r s an au-totmah Lhevnetic chamctehiza.tivn ad d i h b t o4dehspecxhz, b u t s i n c e i t was never published t h i s r e s u l t became not known t o t h e s c i e n t i f i c community. Rodding's and Schwichtenberg's above c i t e d paper (which was submitted on february 26, 1971) rediscovered among o t h e r s - without s t a t i n g i t e x p l i c i t e l y a s such - t h i s c h a r a c t e r i z a t i o n as s p e c i a l case obtained by a smooth n = 1 of the proof f o r t h e f i r s t i n c l u s i o n y n 5

( l ) Oeutsch (1975) s t r e n g t h e n s this r e s u l t t o closed prenex formulae with p r e f i x AV ...VA, only one occurence of t h e i d e n t i t y symbol and besides the P . only one, a binary, p r e d i c a t e symbol. His proof i s based on t h e Davis-Putnam-Rabinson (1961) exponential diophantine normal form f o r r e c u r s i v e l y enumerable predic a t e s , s e e f o o t n o t e 2 on page 7 of t h i s paper and c f . Deutsch (1975). See a l s o t h e reduction i n Fagin (1975) of a r b i t r a r y s p e c t r a with only one e x t r a pred i c a t e , a binary one.

Decision Problems in Predicate Logic

273

n - t h order l o g i c a l d e s c r i p t i o n of time-bounded r e g i s t e r machine computations over f i n i t e domains. Since t h a t paper was w r i t t e n i n german i t needed a t h i r d independent rediscovery of this r e s u l t by N.D.Jones and A.L.Selman - announced i n AMSNotices 19,2(1972) under number *72T-E28 and published in Jones & Selman (1974) -, and a n a t u r a l extension of i t t o f i r s t o r d e r f i n i t e l y axiomatizable p r o j e c t i v e c l a s s e s of f i n i t e type formulated e x p l i c i t e l y i n Fagin (1974), t o become widely known and c e l e b r a t e d . The fundamental idea underlying a l l these d i f f e r e n t proofs i s simple: t o give an a p p r o p r i a t e f i r s t - o r n - t h order L o g i c a t denchiption 0 4 &m i t e d hnLtLng p m b l m o d nrrckined OWeh 4inite ( n o t any more i n f i n i t e ) danrrim. In f a c t we can e x t r a c t t h i s idea e x p l i c i t e l y by showing t h a t a natural adaption of our l o g i c a l d e s c r i p t i o n of machine computations - namely t o f i n i t e computations t o be described over corresponding f i n i t e domains - y i e l d s almost t r i v i a l l y (and following t h e same proof p a t t e r n a s explained f o r the Church/Turing, the Aanderaa and t h e Trachtenbrot theorem) t h e above mentioned: .btomta RheomaZc chamctehizaaZon 06 & i & t - O h & h specfm : With r e s p e c t t o unary (resp. binary) r e p r e s e n t a t i o n SPECTRAl coincides with the NP- ( r e s p . NEXPTIME-) s e t s of p o s i t i v e n a t u r a l numbers. As usual P,NP,DEXPTIME,NEXPTIME denote the c l a s s of a l l s e t s which a r e accepted by a d e t e r m i n i s t i c resp. nondeterministic Turing machine within polynomial resp. exponential time in t h e ( b i n a r y ) length of the input. C.A. Christen i n his doctoral d i s s e r t a t i o n "Spektren und Klassen elementarer Funktionen" (ETH Zurich, 1974) has l i f t e d this Bennett-Rodding-SchwichtenbergJones-Selman-characterization t o higher-order s p e c t r a completing Rodding's and Schwichtenberg's inclusions yn 5 SPECTRAn+l 5 Y ~ t +o ~ SPECTRAn+l = NTIME( an+l)-sets of p o s i t i v e numbers. Fagin (1974) observed t h a t t h i s c h a r a c t e r i z a t i o n of s p e c t r a a p p l i e s equally well t o f i n i t e l y axiomatizable c l a s s e s of f i n i t e s t r u c t u r e s : one only needs t o add t o the c o n s t r u c t i o n a n a p p r o p r i a t e encoding of s u b s e t s of f i n i t e s e t s . Remember t h a t a f i n i t e l y ( f i r s t o r d e r ) axiomatizable pmjectiwe c . k s b of f i n i t e type i n the sense of Tarski i s a c l a s s of p r e c i s e l y those f i n i t e s t r u c t u r e s ( i . e . w i t h f i n i t e domain and f i n i t e l y many f i n i t e r e l a t i o n s over t h a t domain) which a r e models of a formula V ... V a without f r e e individual v a r i a b l e s , with t h e bounded p r e d i c a t e '1 'r and some f r e e occuring p r e d i c a t e symbols R1,...,Rd, sometimes v a r i a b l e s P1,...,Pc such a c l a s s i s a l s o c a l l e d R1,...,Rd -spectrum of V ... V o r simply genem'1 'r &zed ( f i r s t o r d e r ) specinurn. Assuming t a c i t l y t h a t a l l model c l a s s e s we a r e t a l l i n g about a r e closed under isomorphisms and r e f e r r i n g t o a standard encoding of f i n i t e s t r u c t u r e s i n t o binary words the proof of the above c h a r a c t e r i z a t i o n of f i r s t order s p e c t r a i s e a s i l y extended t o a proof f o r the following:

Chamctehizafion 0 6 genekxfized dih&t-ohakh s p e c t m : The (encodi ngs o f ) generalized s p e c t r a a r e p r e c i s e l y t h e NP-sets (of nonempty words). The method of l o g i c a l d e s c r i p t i o n of f i n i t e computations over f i n i t e domains does not depend on the p a r t i c u l a r machine model. I f we apply i t mutatis mutandis d i r e c t l y t o f o r ex. t h e rudimentary p r e d i c a t e s i n t h e sense of Smullyan we o b t a i n d o h ewehy mckmentahy pmciicate a 6 i ~ O th & h mphedenaiztion i n ,5inite domim and thereby from the e x i s t e n c e of a rudimentary Kleene-T-predicate a s c o r o l l a r y the n;-completenedn 06 t h e emptineds pmbLem d o h 6imi-ohdeh &pectm (Buchi 1962) and the ~2-compLetenedb 06 .thein: indinity pmblem. The Bennett-Rodding-Schwichtenberg-Jones-Selma,i-Fagin-Christen-characterization o f ( g e n e r a l i z e d ) s p e c t r a shows a very c l o s e connection between t h e Spektralpro-

E. BbRGER

274

blem resp. A s s e r s ' s ( s t i l l unsolved(')) complement problem f o r f i r s t order s p e c t r a and the fundamental and outstanding complexity theone.tical wahiant ad HLLobent'n E n h c h e i d u n g n p ~ ~ b l ewhich m can be formulated a s follows and i s known a s the P = NP-problem o r Cook'n ptroblsm I s t h e r e a d e t e r m i n i s t i c Turing machine wich recognizessatisfiability f o r a r b i t r a r y formulae of propositional l o g i c w i t h i n polynomial time bound ( i n the length of t h e i n p u t formulae)? Indeed t h i s open problem plays a c e n t r a l r o l e f o r the important question about t h e f e a s i b i l i t y o r not of a huge v a r i e t y of algorithms designed f o r combinatorial dec i s i o n problems occuring in almost every area of computer a p p l i c a t i o n s . This i s not the r i g h t place t o d i s c u s s i n more length the importance and the wide range of ram i f i c a t i o n s of Cook's problem i n t o many a r e a s of computer s c i e n c e , operations research, mathematics and l o g i c ; s e e f o r ex. the book by Garey & Johnson (1979) e want t o mention however one important a s p e c t of o r the survey Hartmanis (1982). W t h i s problem which i l l u m i n a t e s t h e analogy between Cook's problem and H i l b e r t ' s Entscheidungsproblem and echoes the analogy between the P = NP-question and P o s t ' s problem in degree t h e o r y . ( 2 ) The P = NP-problem i s e q u i v a l e n t t o the above given formulation of Cook's problem because the decL5inion pmoblem doh plwpobLtiomf? l o g i c 0 complete d o h NP with r e s p e c t t o polynomial-time computable reductions in t h e same sense a s t h e p r e d i c a t e l o g i c decision problem i s many-one complete f o r t h e r e c u r s i v e l y enumerable s e t s : Cook'n theonem (1971). The s a t i s f i a b i l i t y problem f o r propositional l o g i c i s NP-complete. The i n t e r e s t i n g p o i n t i n our context i s t h a t n o t only the r e s u l t i s analogous t o the cl-completeness of H i l b e r t ' s Entscheidungsproblem (with r e s p e c t t o deducibil i t y ) , b u t t h a t v i r t u a l l y t h e same proof can be given f o r both theorems: we j u s t eventually i n f i n i t e com u t a t i o n s in r e i n t e r p r e t e our firs_t~crdeyd e s c r i p t i o n ptopon-i.tioml Logic terms doh d i n i t e compu*a.tionn. E s s e n ~ i i l T y - W ~ - w ~ n T ~ - ~ ~ ~ ~ r t amounts t o "look a t " f i r s t o r d e r atomic formulae - representing machine configur a t i o n s - a s propositional v a r i a b l e s and t o use f i n i t e conjunctions i n s t e a d of universal q u a n t i f i c a t i o n s f o r t h e d e s c r i p t i o n of p o s s i b l e machine t r a n s i t i o n s . We w i l l give the d e t a i l s i n the l a s t s e c t i o n . Apart from r e v e a l i n g in a n a t u r a l and s t r i k i n g l y simple way the j u s t one fundamental reason f o r computational completeness of f i r s t order and propositional

for

~-

( l ) F o r generalized s p e c t r a where only monadic p r e d i c a t e symbols a r e allowed Fagin (1975 a ) shows t h a t not f o r every spectrum t h e complement i s a l s o a spectrum. See a l s o Fagin (1975 b ) and Yasuhara (1971) where o the c l a s s of number theor e t i c a l f u n c t i o n s d e f i n a b l e from successor and maxfny by composition and t h e folloYing max-Qounded p r i m i t i v e recursion: f(x,l) = g(x) f(;,n+l) = max(')(h(;,n,f(;,n)),n+l) i t i s shown t h a t both t h e range and ( i f not empty) a l s o i t s complement a r e firstorder s p e c t r a f o r every element i n t h i s class.Examples of t h u s obtained s p e c t r a a r e the s e t of Fermat resp. o f Mersenne primes and t h e i r complements.Fagin (1975) gives an i n t e r e s t i n g c h a r a c t e r i z a t i o n of those s e t s X of n a t u r a l s where X and i t s complement a r e f i r s t - o r d e r s p e c t r a . (*)For i l l u m i n a t i n g remark on t h i s l a s t analogy s e e f o r ex. Fagin (1974:pp.86 sq.), Hartmanis (1982), Specker & S t r a s s e n (1976). I n t e r e s t i n g degree t h e o r e t i c a l ams i d e r a t i o n s about r e l a t i o n s between t h e complexity of s e t s of n a t u r a l s A,the s e t of formulae v a l i d i n a l l s t r u c t u r e s of c a r d i n a l i t y i n A and r e l a t i v i z e d h a l t i n g problems can be found i n Hay (1973), (1973 a ) , (1975), Selman (1973), (1974).

Decision Problems in hedicate Logic

275

l o g i c , f o r Aanderaa's and f o r T r a c h t e n b r o t ' s theorem, f o r complexity t h e o r e t i c a l c h a r a c t e r i z a t i o n s of s p e c t r a , t h i s way t o prove those r e s u l t s permits a l s o t o loc a t e e x p l i c i t e l y and in a p r e c i s e way how the LagLcal h t m a e of program desdeXemnLnhLLc to nondetemnLdLLc comp&c r i p t i o n s .LA addected by pIbAing . t i o ~ n . Cook (1971) proved t h a t the s a t i s f i a b i l i t y problem f o r propositional Krom formulae i s i n P , t h e i r u n s a t i s f i a b i l i t y problem i s even complete f o r nondetermin i s t i c logarithmic space a s shown by Jones & Laaser & Lien (1976) and an analogous completeness r e s u l t holds f o r t h e i r s a t i s f i a b i l i t y problem i n q u a n t i f i e d Boolean l o g i c , s e e Aspvall & Plass & Tarjan (1979). While i n t h e realm of f i r s t order d e s c r i p t i o n s of computations Kmm h ~ c t u mcan in some cases be obtained by an a p p r o p r i a t e choice of computation model and using s u f f i c i e n t l y rich o t h e r l o g i c a l expressive means - p r e f i x s t r u c t u r e , s i m i l a r i t y type e t c . -, in some o t h e r cases i t c u t s down t h e complexity of decision problems from undecidable t o decidable a s can be seen from t h e Krom and Horn p r e f i x theorem; i n the propositional case f o r Cook's theorem Krom s t r u c t u r e cannot be obtained unless P = NP: in f a c t a choice between a t l e a s t two possible next s i t u a t i o n s following a given one does not seem t o be d e s c r i b a b l e by a binary d i s j u n c t i o n ( b u t with a t e r n a r y one i t i s ) ; note t h a t t r a n s i t i v i t y cannot be expressed by Krom formulae, c f . Krom 1966). S i m i l a r l y H a m ~ L u c t u ~seems e t o belong t o d e t e r m i n i s t i c computations: the s a t i s f i a b i l i t y problem f o r propositional Horn formulae i s in P and l i k e Krom formulae shares c e r t a i n completeness p r o p e r t i e s , s e e Jones & Laaser (1977) and Aanderaa & Borger (1979). In f a c t t h e Horn s t r u c t u r e can be preserved in going from f i r s t order t o propositional l o g i c d e s c r i p t i o n of the program formulae of d e t e r m i n i s t i c programs, but ( u n l e s s P = NP) t h i s i s impossible f o r t h e i n p u t d e s c r i p t i o n which can however be given by a Krom formula: Cook's theorem can be shown with a conjunction of a program formula which i s not Krom b u t Horn except f o r those conjuncb describing nondeterministic moves and a Krom formula which i s n o t Horn f o r desc r i p t i o n of input and s t o p condition. T h u s i t seems natural t o measure the comp l e x i t y o d Boolean ~unc.titiolzn in terms of minimal length of propositional formulae defining them and having (almost) Horn s t r u c t u r e . This y i e l d s a complexity measure which i s s t r o n g l y r e l a t e d t o Cook's problem and which by a natural adaption of the Aanderaa-Borger reduction method t o propositional d e s c r i p t i o n s of logical networks has been shown t o be equivalent t o network and Turing machine complexity f o r Boolean f u n c t i o n s , a s w i l l be discussed i n t h e l a s t s e c t i o n . The Spektralproblem and i t s r e l a t i o n t o computational complexity problems i s only one example i n t h e s t i l l growing f i e l d of complexity theory where smooth logical d e s c r i p t i o n s of combinatorial (computational) problems play a d e c i s i v e r o l e . From the examples i n t h e following s e c t i o n s i t should become c l e a r t h a t i f the logical d e s c r i p t i o n i s such a s t o "show" an i n t i m a t e s t r u c t u r a l c o r r e l a t i o n between the combinatorial system described and the l o g i c a l expressive means used, then t h i s link w i l l a l s o c a r r y over complexity phenomena from the computational system t o the corresponding l o g i c a l system; indeed the optimal s i t u a t i o n i s t h a t Rhe l o g i cal nynteni can be u i w e d v i a Rhe -tmMnk.tion jut an a naiutuml i.mplemevLta.tion 0 6 the denchibed compu;ting nyhtem. In a way t h i s d e s i r e underlies a l s o t h e many approaches t o d e f i n e semantics of programming languages by l o g i c a l o r a b s t r a c t a l g e b r a i c a l means; i t underlies numerous simulation techniques between various computation models. Success r e s u l t s here not only i n b e t t e r and deeper understanding of t h e s i t u a t i o n , b u t a l s o in b e t t e r technical s o l u t i o n s of given problems. (Take a s a b s t r a c t example t h e simulation techniques developed in Borger 1979.) In any way i t w i l l become c l e a r from t h e next s e c t i o n s t h a t and why the techniques developed f o r e s t a b l i s h i n g lower complexity bounds f o r decision procedures f o r dec i d a b l e l o g i c a l decision problems resemble s t r o n g l y those developed i n t r a d i t i o n a l reduction theory. The reader i s i n v i t e d t o compare t h i s w i t h t h e i l l u m i n a t i n g d i s cussion i n the book of Machtey & Young (1978) where i t i s shown how lower comp l e x i t y r e s u l t s can be derived by methods invented by Godel t o give h i s incompleteness theorems, j u s t by proving them through a p p r o p r i a t e r e p r e s e n t a b i l i t y ( r e a d : expressabi l i t y ) statements.

om

By careful l o g i c a l d e s c r i p t i o n s of a p p r o p r i a t e l y chosen computation models many lower complexity bound r e s u l t s have been proved f o r l o g i c a l t h e o r i e s which in some cases meet e x a c t l y known upper bounds ( i . e . complexity of e x i s t i n g algorithms f o r

E. BORCER

216

t h e s o l u t i o n o f t h e problem under c o n s i d e r a t i o n ) . I n t h e r e a l m o f pure p r e d i c a t e l o g i c t h e f o l l o w i n g two theorems resume e s s e n t i a l l y t h e s t a t e o f t h e a r t :

d x n p l e x i t y Theomri d o h P t l e d i x - S U h i t g C h n n e n . The d e c i s i o n problem w i t h r e s p e c t t o s a t i s f i a b i l i t y o f t h e f o l l o w i n g s o l v a b l e subclasses o f p r e d i c a t e l o g i c has t h e i n d i c a t e d p r e c i s e ( l o w e r and upper bound) c o m p l e x i t y where c denotes some c o n s t a n t and n t h e i n p u t l e n g t h : Monadic p r e d i c a t e l o g i c : N T I M E ( c ~ " ~ ~1 VmAVm n Monadic : DTIME( " " ) vm*2vm

: NTIME(

V"A"

: NTIME(C")

"

"

)

Examples o f c l a s s e s w i t h NP-complete s a t i s f i a b i l i t y problem a r e VpAq w i t h i y t q , V"Aq w i t h OZq,A"

and VAm.

Complexity Theotlem d o h K m m nubc&zannen. The s a t i s f i a b i l i t y problem f o r Krom f o r mulae r e s t r i c t e d t o t h e f o l l o w i n g c l a s s e s has t h e i n d i c a t e d c o m p l e x i t y : Monadic i s complete f o r P 2 V ~ (even A ~ v A") i s complete f o r p o l y n o m i a l space V ~ A (even ~ V ~ A ~ V )i s complete f o r DEXPTIME

b u t V"AkVm i s i n P f o r e v e r y f i x e d k AVA i s i n P The u n s a t i s f i a b l e Krom f o r m u l a e i n AVA a r e complete f o r n o n d e t e r m i n i s t i c l o g a r i t h m i c space. The u n s a t i s f i a b l e Herbrand formulae ( i . e . prenex normal forms whose m a t r i x i s a c o n j u n c t i o n o f a t o m i c o r negated atomic f o r m u l a e ) a r e a l s o comp l e t e f o r n o n d e t e r m i n i s t i c l o g a r i t h m i c space. T h e r e f o r e n o t o n l y f o r t h e u n d e c i d a b l e cases, b u t a l s o f o r t h e c o m p l e x i t y o f dec i d a b l e cases o f t h e Entscheidungsproblem Krom s t r u c t u r e p l a y s a d e c i s i v e r o l e . Complete ( r e f e r e n c e s and) p r o o f s f o r t h e above two theorems can be found i n Lewis (1980), F u r e r (1981), Denenberg & Lewis (1982), Lewis & Statman (1983); we l i m i t o u r s e l v e s h e r e t o a s h o r t comment on how t h e l o w e r c o m p l e x i t y bounds a r e o b t a i n e d t o g i v e t h e r e a d e r a f e e l i n g t h a t i n a s t r o n g sense u n d e c i d a b i l i t y and l o w e r comp l e x i t y bound r e s u l t s a r e s i m i l a r i n n a t u r e . The l o w e r NEXPTIME bounds f o r t h e monadic, t h e Godel-Kalmar-Schutte and t h e Schonfinkel-Bernays case i n Lewis (1980) a r e o b t a i n e d by a d i r e c t d e s c r i p t i o n o f t h e acceptance problem f o r n o n d e t e r m i n i s t i c exponential-time-bounded T u r i n g machine computations as s a t i s f i a b i l i t y q u e s t i o n f o r f o r m u l a e o f t h e f o r m VAAVAM w i t h o n l y monadic p r e d i c a t e s i n t h e f i r s t two cases,of t h e f o r m V...VA...A i n t h e t h i r d case where p a r t i c u l a r c o m p l i c a t i o n s a r i s e f o r an a p p r o p r i a t e d e s c r i p t i o n o f t h e successor r e l a t i o n between t h e encod i n g s o f n a t u r a l numbers ( t h e s e r e p r e s e n t a t i o n resemble by t h e way those which had t o be i n t r o d u c e d by Jones & Selman (1974) f o r t h e i r automata t h e o r e t i c c h a r a c t e r i z a t i o n o f s p e c t r a . ) F u r e r ( p r i v a t e communication) o b a t i n e d t h e l o w e r c o m p l e x i t y bound N T I M E ( c ~ " ~') ~ even f o r t h e subclass A V n M f r o m a r e d u c t i o n t o i t o f t h e n o t o r i g i n c o n s t r a i n e d bounded domino problem which he has shown t o be o f e x a c t (upper and l o w e r ) c o m p l e x i t y NTIME(cn). The l o w e r DEXPTIME bound f o r t h e monadic Ackermann case has been o b t a i n e d i n d e p e n d e n t l y by F u r e r (1981) and Lewis (1980); Lewis achieves t h e r e s u l t by a d e s c r i p t i o n o f t h e non-acceptance problem f o r t h e a l t e r n a t i n g push-down automata ( i n v e s t i g a t e d i n Chandra & Stockmeyer (1976) and Ladner & L i p t o n & Stockmeyer (1978) and a c c e p t i n g p r e c i s e l y t h e s e t s i n DEXPTIME), v e r y much i n t h e s p i r i t o f t h e r e d u c t i o n t e c h n i q u e e x p l a i n e d i n t h e n e x t s e c t i o n based on t h e f a c t t h a t t h e p r e f i x s t r u c t u r e i n t h i s case a l l o w s t o speak d i r e c t l y about t h e words t o be memorized i n t h e s t a c k ; a s i m i l a r d e s c r i p t i o n o f a l t e r -

Decision Problems in Predicate Logic

271

nating stack automata accepting p r e c i s e l y t h e s e t s in DZEXPTIME y i e l d s a lower n/log n f o r another i n t e r e s t i n g subclass of t h e p r e f i x c l a s s complexity bound cc \p'~\p'/\determined by imposing r e s t r i c t i o n s on the form of occuring atomic s u b formulae b u i l t u p from only binary p r e d i c a t e symbols, see again Lewis (1980); Fiirer describes linear-time-bounded a l t e r n a t i n g Turing machines using t h e f a c t t h a t DTIME ( c f ( " ) ) ~ASPACE(f(n)) f o r every f and o b t a i n s thereby even formulae k in t h e subclass AV where k comes from t h e number of successors in universal branch s t a t e s . The complexity theorem f o r Krom c l a s s e s i s e n t i r e l y due t o Denenberg & Lewis (1982). They o b t a i n t h e i r lower complexity b o u n d s ( r e a d : completeness r e s u l t s ) in the monadic case from t h e observation t h a t Lewis'(1980) construction f o r the Ackermann c l a s s y i e l d s monadic formulae and t h a t t h e only conjuncts in the f o r mulae describing t h e non-acceptance problem of an a l t e r n a t i n g push-down automaton which a r e not Krom a r e those used t o describe t h e a c t i o n of t h e machine a t universal branch s t a t e s ; b u t without universal branch s t a t e s one i s describing the non-acceptance problem of nondeterministic ( r a t h e r than a l t e r n a t i n g ) push-down automata which i s hard f o r P . For the Bernays-Schonfinkel Krom case a d e s c r i p t i o n of polynomial-space-bounded Turing machine computations i s given again in the same s p i r i t a s the method explained in t h e next s e c t i o n using t h e f a c t t h a t the p r e f i x s t r u c t u r e in t h i s case allows t o r e p r e s e n t s t a t e s and symbols of the machine d i r e c t l y by constant terms; t h e space needed f o r t h e computation i s taken i n t o c a r e by t h e number of arguments of t h e (unique) p r e d i c a t e symbol which determines a l s o t h e number of universal q u a n t i f i e r s needed. The subcase with only 2 e x i s t e n t i a l q u a n t i f i e r s comes from t h e f a c t t h a t i t i s s u f f i c i e n t t o have machines working over an alphabet with only 2 symbols. S i q i l a r l y t h e lower DEXPTIME bound f o r t h e Maslov case i s achieved bv a d e s c r i o t i o n ofcomoutations of 1 inear-space-bounded a ] t e r n a t i n q Turinq machjnesiwhich accept p r e c i s e i y t h e s e t s in DEXPTIME,the r e s u l t i n a formulae a r e even i n A V . A l p t comment has t o be-made on t h e G 6 d d c&eanean w L t h identity, i . e . t h e c l a s s V"A V" w i t h t h e e q u a l i t y symbol allowed. I t i s not known whether t h i s c l a s s has a recursive decision problem o r not whereas without i d e n t i t y i i s f i n i t e l y cont r o l l a b l e a s shown by Godel (1933) and Schutte (1933), (1934)tl;. Goldfarb (1981) has shown t h a t a t l e a s t t h e r e can be no primitive r e c u r s i v e decision procedure. His proof c o n s t r u c t s formulae F n ~ M Vw i t h e q u a l i t y describing i n i t i a l p a r t s of t h e graph of t h e Ackermann function; t h e Fn a r e (even f i n i t e l y ) s a t i s f i a b l e b u t not over domains with l e s s than a(n,O) elements f o r t h e Ackermann function (Y. Recently f o r a subcase of t h e Godel c l a s s with = , Goldfarb & Gurevich & Shelah (1983) gave a proof of f i n i t e c o n t r o l l a b i l i t y ; t h i s i s t h e s u b l a s s of formulae Q YF in N\V which r e q u i r e only f o r every unordered p a i r Ix,y) ( i n s t e a d of every ordered p a i r ( x . y ) ) a v such t h a t F(x,y,v) holds,formally speaking t h e Class of a l l formulae of form A A V((KXy -+ ~ K y x )& (KXY -+ G ) )

0

X Y

v

with a binary p r e d i c a t e symbol K and a q u a n t i f i e r f r e e formula G ; t h e a l l e g e d decision procedure i s not primite r e c u r s i v e , but i t i s not known whether t h e r e can be no p r i m i t i v e r e c u r s i v e one: u p t o now no formulae a r e known i n t h a t c l a s s which allow only "big" models l i k e Goldfarb's Fn mentioned above. We conclude t h i s panorama of main i d e a s , methods and r e s u l t s in c l a s s i c a l reduction and complexity theory f o r l o g i c a l decision problems by a h i n t t o an area of research which has not y e t found broader a t t e n t i o n d e s p i t e i t s n a t u r a l n e s s s many f a s c i n a t i n g open problems and t h e a v a i l a b i l i t y o f strong methods which could

(l)Gurevich & Shelah (1983) have p u t foreward a very e l e g a n t and s t r a i g h t forward p r o b a b i l i s t i c argument showing f i n i t e c o n t r o l l a b i l i t y of A2V" with = .

278

E. BORCER

e v e n t u a l l y be p u t t o use t o s e t t l e those problems. I am speaking a b o u t t h e comp l e x i t y o f t h e d u h i o n p4obLem 6011. decidxbee &COJL&A h.enZLicted t o 6om&e with 4impLe p X d d i e h O h r m W x oh atomic 4ub6om&e na3tuctu4e : a huge amount o f r e s u l t s i s known a b o u t d e c i s i o n problems f o r d e c i d a b l e f i r s t o r (weak) (monadic) second o r d e r t h e o r i e s and t h e c o m p l e x i t y o f d e c i s i o n procedures f o r d e c i d a b l e cases(1); we have seen how much work has been done i n p u r e p r e d i c a t e l o g i c t o determine t h e non- o r s u b r e c u r s i v e c o m p l e x i t y o f t h e d e c i s i o n problem f o r f o r m u l a e c l a s s e s determined b y r e s t r a i n t s on t h e a v i l a b l e e x p r e s s i v e means l i k e q u a n t i f i e r , m a t r i x o r a t o m i c subformulae s t r u c t u r e . I n p a r t i c u l a r we have seen t h a t such s t r u c t u r a l c o n s t r a i n t s p l a y an i m p o r t a n t r o l e i n c u t t i n g down huge ( u p p e r and l o w e r ) c o m p l e x i t y bounds. The s i t u a t i o n t h a t a l m o s t a l l a c t u a l l y known l o w e r combounds f o r d e c i d a b l e t h e o r i e s a r e s o h u g e ( 2 ) may w e l l depend on t h e f a c t t h a t no n a t u r a l s t r u c t u r a l c o n s t r a i n t s a r e imposed on t h e formulae; f o r ex. what a b o u t a huge l o w e r c o m p l e x i t y bound f o r a t h e o r y i f a r b i t r a r y q u a n t i f i e r l e n g t h s and q u a n t i f i e r a l t e r n a t i o n s a r e a l l o w e d which h a r d l y appear i n mathematical p r a c t i c e ? To b r i n g t o a e t h e r t h e s e two l i n e s o f r e s e a r c h w i l l be a f r u i t - and s u c c e s s f u l enterprise.( 3) A LOGICAL DESCRIPTION OF MACHINE COMPUTATIONS

F o l l o w i n g Aanderaa (1971) and B o r g e r (1971) we b e g i n w i t h a d e s c r i p t i o n o f how machines M can be encoded smoothly i n f o r m u l a e aM. Through an a p p r o p r i a t e c h o i c e of M and aM we w i s h t o a c h i e v e two t h i n g s : t o g e t s y n t a c t i c a l l y s i m p l e f o r m u l a e aM whose l o g i c a l s t r u c t u r e r e f l e c t s t h e s y n t a c t i c a l s t r u c t u r e o f M, and - based on such 9 r e l a t i o n - t o make t h e p r o o f o f e q u i v a l e n c e o f t h e M - d e c i s i o n problem t o ( l ) See t h e e x c e l l e n t surveys Ershov e t a l . (1965), Rabin (1977), t h e book F e r r a n t e & Rackow (1979) and Kozen (1979) f o r r e s u l t s and r e f e r e n c e s . Joseph & Young (1981) c o n t a i n a d i s c u s s i o n o f r e l a t i o n s o f q u e s t i o n s o f p r o v a b i l i t y i n weak t h e o r i e s o f a r i t h m e t i c t o such computational q u e s t i o n s as whether P = NP o r NP = coNP. I f may be n o t e d a l s o t h a t ( u n - ) d e c i d a b i l i t y r e s u l t s f o r ( f i r s t o r d e r ) l o g i c a l t h e o r i e s can i n t u r n y i e l d r e s u l t s i n p u r e p r e d i c a t e l o g i c ; f o r ex. H e i d l e r (1973) o b t a i n s t h e s u r p r i s i n g r e s u l t t h a t p u r e e q u a t i o n a l l o g i c w i t h o u t any o t h e r p r e d i c a t e symbol a p a r t f r o m = and w i t h o u t any f u n c t i o n symbol b u t a l l o w i n g H i l b e r t ' s c h o i c e o p e r a t o r E t o b u i l d terms f r o m f o r m u l a e has an u n s o l v a b l e d e c i s i o n problem, by r e d u c i n g t o i t t h e ( u n d e c i d a b l e ) t h e o r y o f one symmetric r e l a t i o n .

( * ) Ex: n o n - K a l m a r - e l e m e n t a r i t y o f weak monadic second o r d e r t h e o r y o f one successor o r o f f i r s t o r d e r t h e o r y o f l i n e a r o r d e r (Meyer ( 1 9 7 5 ) ) , t h e t r i p l e exp o n e n t i a l l o w e r bound f o r t h e d e c i s i o n problem o f m u l t i p l i c a t i v e a r i t h m e t i c ( F i s c h e r & Rabin ( 1 9 7 5 ) ) , d o u b l e e x p o n e n t i a l f o r P r e s b u r g e r a r i t h m e t i c ( F i s c h e r & Rabin ( 1 9 7 4 ) J and f o r r e a l a d d i t i o n t h e completeness i n t h e c l a s s o f problems s o l v e d by a l t e r n a t i n g T u r i n g machines i n t i m e bound 2Cn u s i n g n a l t e r n a t i o n s f o r some c o n s t a n t c, see F i s c h e r & Rabin (1974), F e r r a n t e & Rackow (1975) and Berman (1977). ( 3 ) Some i n t e r e s t i n g r e s u l t s i n t h i s d i r e c t i o n : see G u r e v i c h (1965) and S c a r p e l l i n i (1982) where t h e r o l e o f q u a n t i f i e r r e s t r i c t i o n s on (un-) d e c i d i a b i l i t y f o r some t h e o r i e s i s analysed, and B o r g e r & K l e i n e Buning (1980) where f o r e x t e n s i o n s o f m u l t i p l i c a t i v e a r i t h m e t i c i t i s shown t h a t r e s t r i c t i o n s on p r e f i x - s i m i l a r i t y type, q u a n t i f i e r , Krom, Horn and t e r m s t r u c t u r e c u t down a r b i t r a r i l y complex u n d e c i d a b i l i t y t o d e c i d a b i l i t y . One h a l f o f t h i s c l a i m i s p r o v e d b y a p p l i c a t i o n s o f t h e r e d u c t i o n method o u t l i n e d i n t h i s s e c t i o n t o v a r i o u s o t h e r c o m p u t a t i o n f o r m a l i s m s l i k e r e s t r i c t e d P o s t c a n o n i c a l forms, P e t r i n e t s ( d e s c r i b e d as f a c t o r replacement systems)etc. ( N o t e t h a t i n t h e meantime i t has been p r o v e d i n K o s a r a j u (1982) t h a t t h e r e a c h a b i l i t y problem f o r P e t r i n e t s i s i n d e e d r e c u r s i v e , a r e s u l t which i s needed f o r t h e above c l a i m . )

Decision Problems in Predicate Logic

279

t h e a M - d e d u c i b i l i t y q u e s t i o n i n f i r s t o r d e r p r e d i c a t e l o g i c a t r i v i a l one. As exp l a i n e d i n t h e course o f t h e p r e c e d i n g s e c t i o n v i a Skolem prenex normal form t h e data of M a r e r e p r e s e n t e d by t h e l o g i c a l terms c o n s t i t u t i n g t h e u n i v e r s e o f t h e i n t e n d e d model o f aM; t h e s t a t e s o f t h e f i n i t e c o n t r o l o f M a r e encoded by p r e d i c a t e symbols o c c u r i n g i n ab,. The q u a n t i f i e r f r e e m a t r i x o f aM w i l l be a conj u n c t i o n o f i m p l i c a t i o n s where t o each p o s s i b l e t r a n s i t i o n s t e p ( " i n s t r u c t i o n " , " r u l e " ) Ii d e f i n e d by t h e program M corresponds a c o n j u n c t pi o f aM a s s u r i n g t h a t i f an i n s t a n c e o f t h e p r e m i s e s i n

pi

r e p r e s e n t s a c o n f i g u r a t i o n C o f M,

then t h e c o r r e s p o n d i n g i n s t a n c e o f t h e c o n c l u s i o n s i n

pi

r e p r e s e n t s t h e immediately

succeding c o n f i g u r a t i o n a c c o r d i n g t o Ii. Consider a r e g i s t e r machine program M w o r k i n g o v e r 2 r e g i s t e r s . S i n c e t h e i r i n v e n t i o n by Minsky (1961) and Shepherdson & S t u r g i s (1963) t h e s e machines have become w i d e l y known, i n p a r t i c u l a r t h e f a c t t h a t t h e y a r e u n i v e r s a l f o r t h e computation o f a l l p a r t i a l r e c u r s i v e f u n c t i o n s . F o r convenience o f e x p o s i t i o n we assume w i t h o u t l o s s o f g e n e r a l i t y t h a t M c o n s i s t s o f i n s t r u c t i o n s Ii = ( i : do oi,go

t o (p,q))

oiEIal,a2,s1,s2,stop}

w i t h i n s t r u c t i o n numbers l ~ i ~o pre,r a t i o n symbols and numbers l s i , q i z r

o f t h e n e x t i n s t r u c t i o n t o be executed.

Execute Ii means: t e s t , i f t h e r e g i s t e r c o n s i d e r e d i n oi equals z e r o o r n o t ,

- i . e . +1 i n t h e j - t h r e g i s t e r i n case oi = aJ. ' -1 i n t h e j - t h r e g i s t e r i n case oi = s . - and t h e n go o v e r t o t h e n e x t i n s t r u c t i o n w i t h number J pi r e s p . qi i f t h e r e g i s t e r t e s t e d was equal z e r o r e s p . n o t . Assume a l s o t h a t

execute oi

Ir = ( r , s t o e go t o r ) , t h a t t h e s t a r t i n s t r u c t i o n i s I1 and t h a t t e s t s a r e executed o n l y i n s u b t r a c t i o n i n s t r u c t i o n s ( i . e .

oiEIal,a2)

i m p l i e s pi = qi

in

which case we w r i t e pi i n s t e a d o f (pi,pi)).

To

defines\,

we r e p r e s e n t n a t u r a l numbers n by means o f l o g i c a l terms b u i l t up

from a symbol 0 f o r an i n d i v i d u a l c o n s t a n t and a symbol ' o f a monadic f u n c t i o n ; f o r a l a t e r a p p l i c a t i o n we w r i t e these terms b o t h i n p r e f i x - and i n p o s t f i x n o t a t i o n , i . e . i n t h e forms -

n : = O& n times

:E

or

-

I...I v 0

n - t i mes

Any s t a t e i o f M i s r e p r e s e n t e d by a b i n a r y p r e d i c a t e symbol, c a l l i t a g a i n Ii. T h e r e f o r e an M - c o n f i g u r a t i o n C = (m,i,n) o f s t a t e i and c o n t e n t s m o f t h e f i r s t and n o f t h e second r e g i s t e r i s coded by t h e l o g i c a l a t o m i c f o r m u l a (m,i,n)

::

iii I. n 1 -

t o be i n t e r p r e t e d as meaning t h a t t h e c o n f i g u r a t i o n C can be reached through an M-computation s t a r t i n g f r o m a p r e v i o u s l y g i v e n i n i t i a l c o n f i g u r a t i o n . D e f i n e uM : :A A ( P ~ A . . . A ~ ~w i- t ~h ) t h e c o n j u n c t s XY i n s t r u c t i o n Ii on c o n f i g u r a t i o n s d e f i n e d by: x'1.y J (OIiy -+ O I j y ) xIiy

f o r I.= ( i , d o al,

-f

A

(x'Iiy

-f

pi

describing the e f f e c t o f

go t o j )

x I k y ) f o r Ii=(iSl,

go t o ( j , k ) )

S y m m e t r i c a l l y f o r i n s t r u c t i o n s w i t h o p e r a t i o n s a2 o r s2 by i n t e r c h a n g i n g t h e p o s i t i o n s o f x and y and u s i n g ' x i n s t e a d o f X I .

E. BORGER

280

I t should already be c l e a r from the above t h a t r e a l l y u i s nothing e l s e t h a n ( a definition of the e f f e c t o f ) the program of instructrons M formulated in f i r s t order logical terms; in f a c t one has t r i v i a l l y the following

RechcLLoian Cmm. For any M-configurations C , D holds: C +M D iff u M h i5 + D

bL

where C +M D means t h a t M , s t a r t e d in C , a f t e r a f i n i t e number of steps reaches D, and $L denotes deducibility in predicate logic. Phood. Based on Skolem's theorem(') on canonical interpretations of s a t i s f i a b l e formulae over the corresponding term domains discussed i n the preceding section one obtains the implication from r i g h t t o l e f t j u s t by observing t h a t the above indicated intended canonical interpretation over the natural numbers of m1.n as meaning C 3 (m,i,n) 1-

obviously yields a model f o r the premisse uM A c, from which in t h a t model the follows meaning t h a t C +M 0. The other implication follows from the conclusion f a c t t h a t by the very definition of uM as e f f e c t of program M , any model for uM simulates every possible computation step of M ; t o say i t more precisely in any canonical model of uM over the naturals a n d f o r every single M-computation step producing D from C by l e t us say instruction I . , the corresponding conf r h C in the given model. junct pi assures the t r u t h of the inference of

n

Since 2-register machine programs a r e known t o be universal f o r the computation of p a r t i a l recursive functions the above reduction lemma already contains a complete proof f o r the

C a t o U u h q . The Entscheidungsproblem f o r ( r e s t r i c t e d ) f i r s t order predicate logic i s complete f o r the recursively enumerable s e t s (Church(1936), Turing (1937)) even when r e s t r i c t e d t o the class VAVA(0,m) (Buchi (1962)) and Krom and Horn formulae (Aanderaa (1971), Borger ( 1 9 7 1 ) ) ( 2 ) ; thereby t h i s class i s a reduction class f o r sa t i s f i abi 1 ity . Indeed by contraposition a ( f o r ex. halting) configuration D cannot be reached by M from C i f f the formula uM A A ID i s s a t i s f i a b l e , and t h i s formula i s via prenexing equivalent t o the Skolem normal form of a Krom and Horn formula in VAVA(0,m).

As referred t o in the preceding section many other r e s u l t s on strong reduction classes have been obtained by variations and f u r t h e r refinements of the above method to describe over term domains suited t o the data s t r u c t u r e of the given machine model three things: - the e f f e c t uM of the program M - a s t a r t condition C - a (non-) stoo condition ij (1) See f o r example Kreisel & Krivine (1967: pp. 18-20) (L)

With respect t o prefix and Krom structure t h i s r e s u l t i s already optimal as i t i s with prefix AVVA; the l a t t e r i s obtained in l i t e r a l l y the same way by substituting 0 by a term *x f o r a new Skolem function symbol * t o be used as constant zero function.

28 1

Decision Problems in Predicate Logic

We want t o g i v e h e r e d e t a i l s f o r j u s t one o t h e r case t o i l l u s t r a t e t h a t such r e d u c t i o n s can y i e l d n o t o n l y u n d e c i d a b i l i t y phenomena f o r l o g i c a l d e c i s i o n problems based on t h e u n d e c i d a b i l i t y o f t h e c o m b i n a t o r i a l system described, b u t by s i m p l e c o n t r a p o s i t i o n a1 so Logical dechiniotz phoccddwren doh cornbimtotohiaL ciecOiun pmbLem reduced t o d e c i s i o n problems o f a s o l v a b l e c l a s s o f formulae. T h i s approach has been i n v e s t i g a t e d by Lewis (1975) f o r ( t h e emptyness problem o f ) c o n t e x t f r e e grammars, f i n i t e ( t r e e ) automata ( o f f i n i t e o r d e r ) and push-down automata and f o r ( t o t a l i t y and t o t a l - e q u i v a l e n c e problem o f ) f u l l schemas; i n t h e l a s t case an i m provement o f Manna's (1968) r e d u c t i o n o f these problems f o r a b s t r a c t programs t o t h e d e c i s i o n problem f o r p r e d i c a t e l o g i c w i t h f u n c t i o n symbols ( b u t n o t = ) i s g i v e n which a v o i d s use o f f u n c t i o n s and r e s u l t s f o r f u l l schemata i n formulae o f a d e c i d a b l e c l a s s . F o r t h e o t h e r cases r e d u c t i o n s t o f o r m u l a e o f Monadic p r e d i c a t e resp. p r o p o s i t i o n a l l o g i c a r e g i v e n whose d e c i s i o n problem i s known t o be s o l v a b l e . We show h e r e t h e d e s c r i p t i o n o f c o n t e x t f r e e grammars. L e t a c o n t e x t - f r e e grammar M be g i v e n w i t h o u t l o s s o f g e n e r a l i t y w i t h r u l e s Sk + S . S and S . + ak i n Chomsky normal f o r m and axiom S1. We g i v e a d e s c r i p t i o n 1 J J o f M by a Skolemized f o r m u l a uM w i t h prenex normal f o r m i n t h e monadic subclass 2 o f V"A V; t h e t e r m i n a l symbols ak a r e r e p r e s e n t e d by i n d i v i d u a l c o n s t a n t s (denoted a g a i n b y ) ak t h e o p e r a t i o n o f c o n c a t e n a t i o n o f symbols t o words b y a b i n a r y f u n c t i o n symbol denoted by

'I()":

any t e r m

T

b u i l t up f r o m t h e a k by use o f " ( ) "

corresponds t o t h e u n i q u e l y determined word t o b t a i n e d f r o m T by j u s t c a n c e l l i n g a l l parentheses. Every v a r i a b l e 5 . o f M i s r e p r e s e n t e d by a monadic p r e d i c a t e 1

symbol (denoted a g a i n b y ) Si w i t h t h e f o l l o w i n g i n t e n d e d i n t e r p r e t a t i o n :

Sit

means

Si +Mt

t h e q u e s t i o n i f S1 -fMt f o r some t e r m i n a l word t i s f o r m u l a t e d as q u e s t i o n

i.e.

whether some t can be parsed s u c c e s s f u l l y by M (read: whether i n v e r s e a p p l i c a t i o n s o f grammatical r u l e s o f M t o some t f i n a l l y y i e l d S1). D e f i n e uM :=

A A ( P ~ A . . . ~w~i t)h t h e c o n j u n c t s B~

describing the parsing e f f e c t

(i.e.

t h e e f f e $ t Y o f an i n v e r s e a p p l i c a t i o n ) o f t h e L - t h grammatical r u l e S.S o f M t o words d e f i n e d by: i j Six A S.y + S k ( x y ) f o r e v e r y r u l e Sk + S . S . i n M and l e t tMbe t h e J 1 J c o n j u n c t i o n o f a l l formulae:

Sk

+

f o r every r u l e S .a J k Then t h e f o l l o w i n g Ireduc.tion p ~ o p e h t yh o l d s :

S1 +Mtf o r some t e r m i n a l word t i f f +

PL

S. J

+

uM A tM+ VSlx

ak

in

1.1

as one can show paraphrasing

X

t h e a r g u m e n t a t i o n g i v e n f o r t h e p r e c e d i n g r e d u c t i o n o f r e g i s t e r machines. Theref o r e uM A tMA 1 V S1x i s a monadic f o r m u l a which i s s a t i s f i a b l e i f f M generates X

t h e empty language; and a prenex normal f o r m o f i t i s indeed i n t h e c l a s s V.. . V M V w i t h number o f b e g i n n i n g e x i s t e n t i a l q u a n t i f i e r s depending on t h e number o f t e r m i n a l symbols. P r e c i s e l y t h e same c o n s t r u c t i o n a p p l i e s t o f i n i t e t r e e automata w i t h f i x e d o r d e r k where S i f expresses t h a t M w o r k i n g on t reaches t h e r o o t i n s t a t e i; t h e i m p l i c a t i o n o f u , , c o r r e s p o n d i n g t o an i n s t r u c t i o n ( i ( l ) , Si!l)xl~...~Si(k)xk j-th letter.

-+

Sefjx l...xk

...,i ( k ) , j )

+

L

o f M i s then

w i t h a k - a r y Skolem f u n c t i o n f . r e p r e s e n t i n g t h e J

E. BORGER

282

COMPLEXITY RELATIONS BETWEEN PROGRAMS AND PROGRAM FORMULAE The i n t i m a t e c o n n e c t i o n between programs M and t h e i r l o g i c a l d e s c r i p t i o n ',,, ( b e t t e r : t h e l o g i c a l f o r m u l a t i o n o f t h e i r e f f e c t on g i v e n data, o f t h e i r semant i c s as i s s a i d i n computer s c i e n c e ) which has been e s t a b l i s h e d by t h e method exp l a i n e d i n t h e p r e c e d i n g s e c t i o n f o r t h e examples r e g i s t e r machines, c o n t e x t f r e e grammars and f i n i t e automata i s such as t o c a r r y o v e r n o t o n l y ( u n - ) d e c i d a b i l i t y p r o p e r t i e s b u t a l s o t o p r e s e r v e t h e c o m p l e x i t y o f c o r r e s p o n d i n g d e c i s i o n problems, a g a i n i n such a way t h a t t h e p r o o f o f t h i s c o m p l e x i t y p r e s e r v a t i o n i s o b v i o u s f r o m t h e c o n s t r u c t i o n . We i l l u s t r a t e t h i s f e a t u r e by two examples c o n s i d e r i n g a ) deghee compeexity o f d e c i s i o n problems, b ) i M e p m b i U t y phopehtien f o r machine problems and t h e i r t r a n s l a t i o n i n t o complexity bouna2 ( o h f i t h e mode,& o f t h e desc r i b i n g formulae. S i n c e t h e program

M

and t h e program f o r m u l a aM a r e " t h e same"

-

even more: i n t h e

r e d u c t i o n lemma e v e r y c o m p u t a t i o n s t e p o f M i s i n 1-1-correspondence w i t h a l o g i c a l d e d u c t i o n s t e p u s i n g an i m p l i c a t i o n o f aM - i t i s n o t s u r p r i s i n g t h a t b y t h e r e d u c t i o n lemma any 0 - h a l t i n g problem HD(M) : = IC

I

C

+M 01

i s 1-1 e q u i v a l e n t and t h e r e f o r e hec~vrcrivedyb o m o h p k i c t o t h e d e c i s i o n problem w i t h r e s p e c t t o deduci b i 1 it y o f t h e f o r m u l a e c l a s s FD(M) : = {aM

A

+

D

1

C arbitrary}. (1)

We say: any 0 - h a l t i n g problem " i s " v i a o u r t r a n s l a t i o n a l o g i c a l d e c i s i o n problem ( w i t h respect t o d e d u c i b i l i t y ) . Since registermachines ( w i t h 2 r e g i s t e r s ) are universal f o r the computation o f a l l p a r t i a l recursive f u n c t i o n s a l s o the deducib i l i t y d e c i s i o n problem o f any c y l i n d r i c a l r e c u r s i v e c l a s s o f f o r m u l a e F can be shown " t o be" ( r e c u r s i v e l y i s o m o r p h i c t o ) a 0 - h a l t i n g problem o f a 2 - r e g i s t e r mac h i n e M w i t h a s i m p l e r e l a t i o n between F and M. We c o n c l u d e t h a t f i r s t o r d e r l o g i c a l d e c i s i o n problems a r e as " n a t u r a l " as h a l t i n g problems o f 2 - r e g i s t e r machines. B u t th.ere i s more t o say a b o u t t h e degree t h e o r e t i c a l r e l a t i o n between l o g i c a l and c o m b i n a t o r i a l d e c i s i o n problems. F i r s t observe t h a t b y i g n o r i n g t h e v a r i a b l e s i n t h e d e f i n i t i o n o f oM,uMcan he viewed as a semi-Thue system - w i t h s u b s t i t u t i o n rules

Pi

and an a l p h a b e t c o n s i s t i n g o f l e t t e r s 1,O and I.( f o r e v e r y s t a t e o f M) J

- w o r k i n g on " c o n f i g u r a t i o n words" C j u s t l i k e M on c o n f i g u r a t i o n s C i n t h e p r e c i s e sense o f t h e r e d u c t i o n lemma above, which i s a g a i n o b v i o u s f r o m t h e semiThue " i m p l e m e n t a t i o n " uM o f M and now reads: C +M 0

i f f C +u

-

0. M I n t h i s sense t h e l o g i c a l d e c i s i o n problem f o r c l a s s e s o f f o r m u l a e a M ~ + CD

i s ( u p t o r e c u r s i v e isomorphy) a l s o t h e same as t h e word problem

{(SJD) IC

+o

Dl

M f o r t h e semi-Thue i n t e r p r e t a t i o n uM o f M.

(1) Note t h a t t h e s e c l a s s e s FD(M) as w e l l as t h e f o r m u l a e uM can be c h a r a c t e r i z e d by p u r e l y s y n t a c t i c a l l o g i c a l means w i t h o u t any r e f e r e n c e t o programs M o r c o n f i g u r a t i o n s D.

Decision Problems in Predicate Logic

283

In Borger (1979) (1983) f u r t h e r e q u a l l y natural and simple " i n t e r p r e t a t i o n " of M have been given !n terms of Thue-systems - j u s t add t o uM t h e inverses of a l l i t s PrOdUCtionS--, Markov algorithm , Post normal c a l c u l i , Post correspondence problems, Turing machines, p a r t i a l implicational propositional c a l c u l i , Wang's non e r a s i n g Turing machines and o t h e r s and i t i s shown t h a t through a l l these i n t e r p r e t a t i o n s of M t h e many-one degrees i f not the r e c u r s i v e isomorphy types of corresponding decision problems l i k e h a l t i n g , word, confluence and so on a r e preserved; t h e proofs of these equivalences follow a general p a t t e r n developed t h e r e ; they a r e easy i f not t r i v i a l once one has grasped t h e "good implementation" of M i n t h e system considered. Therefore we have p u t foreward strong evidence f o r the following epistemological STATEMENT. LogicaL d e c h i a n p m b k m axe ab "mtuhae" M a n y aMeh k i n d a6 cambimto hiad d e c i b i o n p h o b L m . Let us mention two consequences of these c o n s i d e r a t i o n s . The f i r s t concerns Wang's (1962: pg. 54) problem "Whether t h e r e i s some natural undecidable s e t of formulas of the p r e d i c a t e c a l c u l u s w i t h a decision problem t h a t i s not of t h e maximum r e c u r s i v e l y enumerable degree". Our reply on t h e b a s i s of t h e above shown " i d e n t i t y " of (machine and o t h e r ) h a l t i n g problems - t r a d i t i o n a l l y taken a s representing ( r e c u r s i v e l y enumerable) degrees - with l o g i c a l decision problems i s t h a t Wang'h p m b L e m h n o t a problem about l o g i c a l decision problems but t h e pobLem whetheh thehe ate m t m k i & m e d i a t e degheen; t h e f a c t t h a t u p t o now a l l p r e f i x - s i m i l a r i t y and o t h e r " n a t u r a l l y " c l a s s i f i e d c l a s s e s turned o u t t o have e i t h e r a r e c u r s i v e o r a decision problem of maximum degree i s only just another among many examples in mathematics suggesting t h a t intermediate degrees do not ( y e t ? ) c o n t r i b u t e t o s a t i s f a c t o r y c l a s s i f i c a t i o n s of t h e complexity of construct i o n s occuring in mathematical p r a c t i c e . A second s i m i l a r conclusion can be drawn about many attempts in the l i t e r a t u r e

t o study decision problems of formal grammars with r e s p e c t t o t h e i r degree comp l e x i t y . In Borger (1983) we develop a method showing t h a t Post correspondence problems - t h e s e a r e most f r e q u e n t l y used f o r reductions t o show formal language decision problems t o be unsolvable - and any formal language decision problem t o which t h e former have been reduced e f f e c t i v e l y in a very strong s y n t a c t i c a l sense ' b e the same"; indeed t h e i r r e c u r s i v e isomorphy types coincide. Therefore here again i t turns o u t t h a t degree complexity does not c o n t r i b u t e t o t h e i n s i g h t i n t o formal language decision problems: j u s t d e c i d a b i l i t y o r u n d e c i d a b i l i t y of maximal degree i s t h e only r e l e v a n t question, a p a r t from i n v e s t i g a t i o n s on subrecursive complexity i n decidable cases. Let us conclude t h i s argument however with a p o s i t i v e example: the f a c t t h a t the h a l t i n g problems H D ( M ) and t h e decision problems of FD(M) a r e " t h e same" implies t h a t well known complexity r e s u l t s f o r metadecision problems of h a l t i n g problems i n t h e Kleene-Mostowski a r i t h m e t i c a l hierarchy c a r r y over automatically t o logical metadecision problems a s exemplified i n the following

C o h a U a h y . (Borger & Heidler (1976)) With r e s p e c t t o d e d u c i b i l i t y t h e following metadecision problems f o r l o g i c a l decision problems a r e of t h e i n d i c a t e d ( p r e c i s e ) a r i t h m e t i c a l complexity: - t h e emptyness problem i s nl-complete - t o t a l i t y and i n f i n i t y problem a r e n2-complete - c o f i n i t e n e s s , r e c u r s i v i t y and reduction c l a s s problem a r e n3-complete A b h o h t and n&p&

pmoa doh T m c k t e n b m t ' n Meohem, promised i n the preceding s e c t i o n a s consequence of our reduction method, w i l l be given now. By t h e r e duction lemma we have already shown t h a t t h e c l a s s e s No := I F / C , F } F i n : = IFlF has a f i n i t e model}

284

E. BORCER

a r e r e c u r s i v e l y i n s e p a r a b l e : t h e r e c u r s i v e u n s e p a r a b i l i t y o f two h a l t i n g problems

HE(M) i s c a r r i e d o v e r t o t h e no-model and t h e f i n i t e - s a t i s f i a b i l i t y prob-

HD(M) and

lems through o u r r e d u c t i o n :

(1) C +MD

implies

( 2 ) C +ME

"

+

1(uM

A

PL

uM

C

A

A

c

l n

A

la)

has a f i n i t e model

I n f a c t f o r ( 2 ) n o t e t h a t C +ME i m p l i e s t h a t n o t C i n t h e r e d u c t i o n lemma i n t h a t case f o r uM

A

C

+MD;

A 1D

t a k e t h e model c o n s t r u c t e d

and c u t i t down t o t h e i n i t i a l

domain o f a l l numbers zt t 1, where t denotes t h e maximal r e g i s t e r c o n t e n t o c c u r i n g i n t h e t e r m i n a t i n g computation from C t o E, by d e f i n i n g ( t + l ) ' : = t t l . T h i s i s s t i l l a model f o r u,, A C A 1 D . The c l a s s e s No and I n f o f a l l i n f i n i t y a x i o m a r e r e c u r s i v e l y i n s e p a r a b l e because No and F i n a r e r e c u r s i v e l y enumerable b u t p r e d i c a t e l o g i c i s undecidable. To show a l s o t h e r e c u r s i v e u n s e p a r a b i l i t y o f t h e f i n i t e - s a t i s f i a b i l i t y and t h e i n f i n i t y - a x i o m p r o p e r t y we have t o m o d i f y s l i g h t l y o u r d e f i n i t i o n o f uM f o r M t o assure t h a t a l l n o n - p e r i o d i c computations o f M g e t i n f i n i t y axioms as t h e i r desc r i p t i o n ; because t h e n t h e d e s i r e d i n s e p a r a b i l i t y p r o p e r t y i s c a r r i e d o v e r f r o m t h e corresponding i n s e p a r a b i 1 it y p r o p e r t y o f a p p r o p r i a t e l y chosen machines M. The problem i s e a s i l y s o l v e d b y t h e o b s e r v a t i o n t h a t n o n - p e r i o d i c r e g i s t e r machine computations must have l a r g e r and l a r g e r numbers o c c u r i n g i n a t l e a s t one r e g i s t e r ; such " b i g " elements i n t h e models can be assured by t h e " g r e a t e r as" axioms discussed i n t h e p r e c e d i n g s e c t i o n , r e l a t i v i z e d however t o p o s s i b l e r e g i s t e r contents. F o r m a l l y d e f i n e 6 as b e f o r e uM w i t h t h e a d d i t i o n a l c o n j u n c t s ( f o r e v e r y M - i n s t r u c t i o n Ii): (xIiy

+

x'Gx) (yIix

+

x ' G x ) (XGY

+

x'GY)

ixGx

As b e f o r e t h e r e d u c t i o n lemma and f o r a p p r o p r i a t e D, E a l s o t h e above r e l a t i o n ( 2 ) h o l d ; i n a d d i t i o n we have t h e d e s i r e d r e l a t i o n t h a t i f C does n o t d e r i v e D i n M, then 6 A A l n has a model as b e f o r e b u t no f i n i t e one. F o r more s u b t l e quest i o n s !!bout c o n s e r v a t i v i t y o f s p e c i f i c r e d u c t i o n procedures see Aanderaa & Borger & Lewis (1982). F o r a n o t h e r i11 u s t r a t i o n ( l ) o f how o u r r e d u c t i o n technique c a r r i e s o v e r unseparab i l i t y p r o p e r t i e s f r o m machine problems t o formulae problems d e f i n e ( t h e "ThueV C ~ A ~ O M " T~ ) as uM w i t h ++ i n s t e a d o f +. I t i s easy t o adapt t h e p r o o f o f t h e r e d u c t i o n lemma f o r oM t o a p r o o f f o r t h e f o l l o w i n g

ReducGtion Lemm

6 0 ' ~Rhe

Thue uemian

T~

05

uM. F o r any c o n f i g u r a t i o n s D,E

of M

which a r e n o t r e a c h a b l e one f r o m t h e o t h e r i n M h o l d s :

(1) C

+MD

iff

I-

T~ A

PL

( 2 ) C +ME

iff

+

PL

T ~ ,A

ln ,n

IT

A

+

A

+c

( l ) Taken f r o m Aanderaa (1971) and Borger (1975). Aanderaa's method i s d i f f e r e n t f r o m mine i n t h e r e s p e c t t h a t Aanderaa does n o t s e p a r a t e c o m p l e t e l y t h e i n p u t d e s c r i p t i o n f r o m t h e program f o r m u l a t i o n . Due t o t h i s f a c t h i s formulae a r e more complex than o u r s and h i s p r o o f o f what i n o u r f o r m u l a t i o n reduces a g a i n t o t h e r e d u c t i o n lemma becomes more i n v o l v e d and does n o t show t h a t t h e same argument works a l s o i n s u b r e c u r s i v e c o m p l e x i t y l i k e f o r En-unseparable En+l-sets.

285

Decision Problems in Predicate Logic

Pmod. In ( 1 ) from l e f t t o r i g h t the claim follows from t h e case f o r r i g h t t o l e f t assume t h a t C 7LMD: then a canonical model f o r i s given by defining F a s meaning F fMD

T~

A

JD

uM; A

E

A

from

C

I n ( 2 ) from l e f t t o r i g h t conclude a s i n t h a t case f o r uM but using the implithe to infer c a t i o n s from r i g h t t o l e f t i n T~ and s t a r t i n g w i t h given computation C +ME. In the o t h e r d i r e c t i o n defining F a s meaning F +ME

r

y i e l d s a canonical model f o r which means: C +ME. The reduction lemma f o r

T~

T~

A

ID

A

c

F where t h e r e f o r e a l s o

c must

be t r u e

immediately implies my 1975-version o f Aanderaa's

Theonem. For a r b i t r a r y r e c u r s i v e l y unseparable h a l t i n g problems H o ( M )

and H E ( M ) , in theory T w i t h T~ A TEA F a s non-logical axiom theoremhood and l o g i c a l falsehood a r e r e c u r s i v e l y unseparable. Therefore t h i s theory i s e s s e n t i a l l y undec i d a b l e and consequently incomplete. Phood. Any r e c u r s i v e s e t R s e p a r a t i n g I F I h F ) and IF1 pT1F) would y i e l d the rec u r s i v e s e p a r a t i o n s e t {TITER) f o r H D ( M ) and H E ( M ) . For the same reason T can have no r e c u r s i v e supertheory. B u t then i t must be incomplete because otherwise i t would be r e c u r s i v e .

S u f f i c i e n t l y d i f f i c u l t h a l t i n g problems f o r a machine M(1ike t h e above r e c u r s i v e s t a t i n g t h e e f f e c t of t h e u n s e p a r a b i l i t y ) generate "program formulae" T~ A ID A program on i t s p o s s i b l e data t o g e t h e r with a commitment on one s t o p s t a t e which may and one which may not be reached and y i e l d i n g thereby e s s e n t i a l l y undecidable and incomplete t h e o r i e s . I t would be i n t e r e s t i n g t o analyze how simple such program formulae could become generating s t i l l incomplete t h e o r i e s . ( l ) .

r

Note t h a t by r e l a t i v i z i n g our program formula -tM A A 'E t o a new "successor" r e l a t i o n S defined by A V Sxv a formula i s obtained which by t h e same arguments

x v as above can be seen t o have r e c u r s i v e l y enumerable b u t no r e c u r s i v e models i f H D ( M ) and H E ( M ) a r e r e c u r s i v e l y unseparable, and En+l - but no E n - models in the Grzegorczyk hierarchy i f H O ( M ) and H E ( M ) a r e En-unseparable En+l-sets. (The comp l i c a t i o n by r e l a t i v i z i n g t o a successor r e l a t i o n i s needed because T~ A A i s a KrOm formula and a l l s a t i s f i a b l e Krom formulae a r e known t o posses r e c u r s i v e models ( s e e Aanderaa & Jensen (1973), Ershov ( 1 9 7 3 ) ) ) . T h u s we have a s h o r t proof e x h i b i t i n g a very simple s a t i s f i a b l e formula excluding recur-

ln

(1) A very i n t e r e s t i n g r e s u l t . r e l a t e d t o t h i s question i s i n German0 (1976) where i t i s shown t h a t any theory i s incomplete i f i t i s r e c u r s i v e l y enumerable, c o n s i s t e n t and admits term r e p r e s e n t a t i o n s f o r a d d i t i o n and m u l t i p l i c a t i o n ( u s i n g =,O, Successor).

E. BdRGER

286

s i v e models, improving considerably much more involved e a r l i e r s o l u t i o n s by Kreisel (1953), Mostowski (1953), (1955), Rabin (1958) t o t h a t problem r a i s e d by H i l b e r t & Bernays (1939).

DESCRIPTION OF BOUNDED MACHINE COMPUTATIONS: COOK'S PROBLEM, SPEKTRAL-PROBLEM Take again t h e technique of f i r s t o r d e r d e s c r i p t i o n of e v e n t u a l l y unbounded machine computations a s explained i n s e c t i o n 2 and consider i t f o r f i n i t e computat i o n s : look a t the atomic formulae PI .q representing t h e t - t h configuration 1t C reached by M s t a r t i n g with Co = C a s propositional v a r i a b l e s := I t . with P31 ,q the same intended i n t e r p r e t a t i o n of

ct

I t . = 1 a s meaning CO c~. PY1.9 I f you d e f i n e now uM f o r given computation length a. and given i n p u t (memory) bound k a s before but using f i n i t e conjunctions over time 0 5 t 5 a. and r e g i s t e r contents 0 5 p. q 5 k i n s t e a d of universal q u a n t i f i c a t i o n , then t h e r e s u l t i n g formula u M,ge,k again f u l f i l l s the &duc.tion pmpehty t h a t -f

c

~

-f

iD i f f

u,,,,a,k

A

' C

-+

na.i s

a tautology

However the length of u grows exponentially because t h e formula d e s c r i b e s M,k,k g l o b a l l y the e f f e c t of every machine i n s t r u c t i o n , i . e . r e f e r r i n g always t o the whole configuration say ( p , i , q ) and not only the content of the p a r t i c u l a r memory p o s i t i o n a f f e c t e d by I i . Indeed i n t h a t way we succeeded in giving compact formul a t i o n s of M - namely by Krom formulae - and i n t r i v i a l i z i n g completely t h e equivalence proofs showing t h a t t h e reductions work. I t i s easy however t o modidy 0u4 g l o b a l duchip-tion techniyue t o a Local one s i n c e f o r every reasonable notion of algorithm t h e execution of an elementary computation step ( i n s t r u c t i o n ) has a local c h a r a c t e r . I t i s s u f f i c i e n t t o consider the r e g i s t e r s not any more a s s t a c k s but a s containing s t r i n g s of symbols displayed i n l i n e a r order with a p o i n t e r showing the p o s i t i o n of the symbol on which t h e program a c t u a l l y i s working (Tur i n g t a p e ) . What we w i l l gain i s t o o b t a i n from our c o n s t r u c t i o n i n the same way a l s o a proof f o r Cook's theorem t h a t any computation in time bound a. , s t a r t i n g with input of length n , of any (even nondeterministic) Turing machine M can be described by a propositional formula u M,n,a. of polynomial length in M,n,a. Since the s a t i s f i a b i l i t y problem of Krom formulae i s in P(see Cook 1971), unless P=NP we have t o g i v e uv ,the K4om b t ~ u c t u h ef o r u ~ , ~ , ~ . W e can preserve however the Horn s t r u c t u r e f o r our program formulae, a p a r t i c u l a r l y i n t e r e s t i n g f e a t u r e s i n c e P contains a l s o t h e s a t i s f i a b i l i t y problem f o r Horn formulae. This comes o u t from the following ( c f . Aanderaa & Borger 1979): T h e o m o n ( a h o n t - ) How &chipZion 0 6 ~ i n ; R ecompuhtioiovld. There i s a P-comput a b l e function a s s o c i a t i n g t o every d e t e r m i n i s t i c T u r i n g machine program M and any natural numbers n ( f o r i n p u t l e n g t h ) and a. ( f o r computation l e n g t h ) a Horn formula u ("pmgmm ~ o h m u h " ) , a Krom Formula aM ("n&~taht & ~ m u h "with so c a l l e d M, a. ,n "input" v a r i a b l e s x l , .. . , x n ) and a Herbrand formula wM,% ("&top ~ o t ~ n & ~ ' s) a t i s fying t h e following &duca%on pmpehty: l e t Co(q) be t h e s t a r t configuration of M with i n p u t q . Then

-+i accepting config.

i s s a t i s f i a b l e f o r every M,a. 0-1-sequence q of length n where a M,n ( 9 ) denotes t h e r e s u l t of simultaneous substit u t i o n of every x i by qi f o r 1 5 i 5 n . Co(q)

i f f uM,k

A

aM,n(q) A w

281

Decision Problems in Predicate Logic ( A Hehbmnd formulae. )

60m&

i s a conjunction of atomic formulae o r negations of atomic

Proof. F i x a r b i t r a r i l y Q , n and a d e t e r m i n i s t i c Turing machine M with s e t s of i n s m i o n s e t s I . = ( i , j , o . .,$. .) f o r 1 5 i 5 r , 0 5 j 5 m over the alphabet 1 - J 1,J {ao,...,am}. To execute I i means a s usual: i f in s t a t e i the l e t t e r under the reading head (pointed t o by t h e p o i n t e r ) i s a . , t h e n do o i , j - which i s one o f : a k . J ( " w r i t e " a ) o r r resp. a.(move p o i n t e r 1 p o s i t i o n t o the r i g h t resp. l e f t ) - and k go t o execute the i n s t r u c t i o n w i t h index $i .. Let a. = 0 , al = 1, a2 = b ("blank" ,J symbol) and without l o s s of g e n e r a l i t y l e t 0 be t h e accepting s t a t e of M . Let the i n i t i a l c o n f i g u r a t i o n s CO(q) f o r 0-1-input sequences q of length n be defined by the i n i t i a l s t a t e 1 and t h e following f i n i t e tape w i t h p o s i t i o n s numbered by -Q ,...,0 , l ) . .., I.: b. ..b b q b . . b

+

c-rc'+

Q times

1 - n times

i . e . t h e p o i n t e r (reading head) i s in p o s i t i o n (with number) 0. To encode a r b i t r a r y M-configurations we introduce f o r a l l numbers t , k , j , i with 0 5 t 5 1 , -Q < k 5 R , 0 5 j 5 m, 0 5 i 5 r pairwise d i f f e r e n t propositional t-Tt where Ct denotes with t h e following intended i~~Xeqme,?izfitn variable I i , Pk, k,j t h e configuration reached by M in t s t e p s f o r given Co = Co(q):

I t. = 1 t Pk = 1 Ti,j

i f f t h e i n s t r u c t i o n t o be executed i n Ct i s i i f f the p o i n t e r C t i s in p o s i t i o n k 1 i f f the tape c e l l ( w i t h number) k i n C t contains a . .

=

J

a s conjunction of t h e following formulae desDefine t h e pmgmm @un& c r i b i n g l o c a l l y - i . e . f o r any s i n g l e tape c e l l - f o r every M-instruction I i how execution of I i on C a f f e c t s the i n s t r u c t i o n address, the p o i n t e r p o s i t i o n and t h e content of the tape c e l l (we w r i t e f o r s h o r t n e s s 6 -+ y1 A y 2 i n s t e a d of (6

-+

(i)

Y,)

A

(6

+

~2)):

f o r any w r i t e - i n s t r u c t i o n ( i , j , a h , $ i , j ) i n M:

IF

A

t

I.

P:

A

t

A

Pk

T:,~

+

T~+' +i,j

t

A

Tk1.j'

A

P:+'

A

klh

tt 1 -+

T~+'

(successive s t a t e , pointer position, c o n t e n t of working c e l l ) ( " c o n t e n t of nonworking c e l l s remains unchanged")

T k i , j i

f o r any -Q 5 k , k' 5 R and 0 5 j ' 5 m w i t h k' # k ( i i ) f o r any right-move j n s t r u c t i o n ( i , j , r , $ . . ) : 1 ,J t t t+l t+l A T t + l A 'k A T k , j A ':',j' . A 'k+l k ,j ,J

A

T t+ k, 1 , j'

( i i i ) f o r any left-move i n s t r u c t i o n analogously w i t h -I 5 k - 1.

E. BORGER

288

Define the n f ~ ~ t a h~tomunUaaa

M,n I; { i n i t i a l s t a t e i s 11 P:

encoding Co a s conjunction o f : { i n i t i a l p o i n t e r p o s i t i o n i s 01

Ti,o

-

G,2

f o r - 2 5 k 5 0 o r n < k 5 a. Inoninput c e l l s a r e blank}

f--t

';,l

lXk

xk f o r 1 5 k 5 n

{ i n p u t in c e l l s 1, ..., n}

of l I .9i f o r 1 5 i 5 r expressing t h a t Define the n t o p 6omunUaa w ~ a, s conjunction ~ a t time a. M could be in accepting s t a t e 0 b u t i n no o t h e r s t a t e . The reduction property f o r u A aM,,(q) A w ~ i s, e a~ s i l y proved following the M,9. approach explained in s e c t i o n 2: i f M reaches i n 9. s t e p s an accepting configurat i o n ( i . e . with s t a t e 0 ) from given Co = C o ( q ) , then t h e above i n d i c a t e d intended t r u e . Conversely any model i n t e r p r e t a t i o n obviously makes u M , % A ~ ~ , ~ A( q ) f o r t h i s formula simulates any M-computation of length (a. s t a r t i n g with Co(q) i n the sense t h a t f o r any t r u t h assignement A making our formula t r u e and f o r any t t C and t 5 .f with Co(q) +; C t where C has i n s t r u c t i o n i, p o i n t e r p o s i t i o n k and tape a . ... a . , A a s s i g n s truth-value 1 t o the encoding v a r i a b l e s 1.; and L

J-9.

J9.

2 9.. I t follows t h a t C9. must have s t a t e 0 because

T k , j h f o r -9. 5 h

A(1:)

Pi

L

=

0 for

a l l 1 <. i < r. This c o n s t h c t i o n e s t a b l i s h e s a l s o C o o k ' 6 Rheohm about NP-completeness of the prop o s i t i o n a l l o g i c decision problem s i n c e i f M i s nondeterministic, proceed a s above b u t f o r every p a i r ( i , j ) take in u a s conclusion of the corresponding impliM,a. c a t i o n ( s ) t h e d i s j u n c t i o n over a l l p o s s i b l e 1 - s t e p t r a n s i t i o n s ( i , j , o . .,$. .) 1.J 1 , J of M. (Note t h a t then uMP9. i s not any more a Horn formula.) We now show t h a t applying t h e same c o n s t r u c t i o n t o machine d e s c r i p t i o n s over f i n i t e domains y i e l d s t h e famous automata t h e o r e t i c c h a r a c t e r i z a t i o n of s p e c t r a . In f a c t the only thing t o show i s t h e following h g i c a l denchipLLan 0 6 ,jivzite cornpuhLLouzb Oweh divzite domcLim : For every n we a s s o c i a t e t o every r e g i s t e r (or Turing) machine program M and any s a formula ci of order n + 1 f u l f i l l i n g t h e following /reduction p m p e h t y : M accepts k in 'an(ks) steps i f f s f k a f o r every 2 5 k . Proof. We paraphrase our proof given f o r Cook's theorem. Let n,M,s be a r b i t r a r i l y fixed. To d e s c r i b e over a domain w i t h k elements - say k := { O , l , ..., k - 1 ) - t h e M-computation of length a n ( k s ) s t a r t e d with i n p u t k , we need an encoding of a,(ks) many successive "time moments" t o g e t h e r with t h e corresponding s i t u a t i o n of t h e computation. The i d e a i s t o c m t e by nuccenniwe poloeh n e t c o m t m t i o n - s t a r t i n g from t h e s-ary Cartesian product over k - Rhe needed a n ( k 6 ) o b j e c h of a type u n of order n , t o ohdeh thene o b j e c h in a & n U h m y and then t o d e s c r i b e t h e M-computation in the same way a s done before but using now a zero p r e d i c a t e Z and a successor r e l a t i o n S r e l a t i v e t o the previously defined l i n e a r ordering K, and an embedding F of k i n t o a segment of these un-type o b j e c t s f o r d e s c r i p t i o n of t h e input. Formally the power s e t types over k S a r e defined by u1

:=w u

s-times

~ :=+ ( ~ ui)

IU n I

=

n

289

Decision Problems in Predicate Logic

...,

Over k = 10, k - 1 ) t h e r e a r e e x a c t l y a n ( k s ) o b j e c t s of type u n where by o b j e c t s of type uo we understand s - t u p l e s of elements from k . We use x,y,z a s v a r i a b l e s of 1 resp. a s s-tuples and t . u , v , w , t ' , u ' , a s v a r i a b l e s of type o n f o r n type of v a r i a b l e s of type I f o r n = 0 ( u s i n g u = v a s abbreviation f o r u1 = V ~ ... A A u = vs, s i m i l a r l y A e t c . ) .

...

S

U

Dedine t h e formula Ord(Z,S) of o r d e r n + 1 - expressing t h a t Z represents "zero" ( t h e f i r s t element) and S the "successor" r e l a t i o n w i t h r e s p e c t t o a l i n e a r ordering K of a l l type -u - o b j e c t s over any ( f i n i t e ) domain - a s conjunction o f the n following formulae: ( ( K u v v Kvu v u

A A A

=

v)

A

A TKuu

(Kuv

A KVW +

K uw ) )

u v w VZU

A

u A

A ( Z U -+

u

1VKvu)

A(Suv + + ( K u v

u v

{"zero" has no "predecessor")

V A

lV(Kuw

A

Kwv)))

{no element between successors}

W

Using Ord(Z,S) we can d e f i n e t h e program formula uM a s before using Z resp. S (predicate) f o r 0 resp. t + 1 and (almost) t h e same i n t e n d e d r n u n i n g doh v a r i a b l e s I i , P and T . encoding over any domain k = {O, k - 1 ) resp. s t a t e i , J the p o i n t e r ( r e a d i n g head) p o s i t i o n and tape c e l l i n s c r i p t i o n a . f o r any time J moment t and any tape p o s i t i o n u of any M-computation s t a r t e d with C o ( k ) ( l e t - 2 be t h e number of t h e leftmost c e l l v i s i t e d during t h e given computation and It( t h e o r d e r number of t i n t h e given ordering K ) :

...,

I i t i s t r u e i f f a t time It1 i n s t r u c t i o n i i s executed P t u i s t r u e i f f I u I = ( p o i n t e r p o s i t i o n a t time It1 )+ I T . t u i s t r u e i f f a t time It1 the tape c e l l with number 1.1 J letter a .

+ n.

contains t h e

J

Formally l e t M have i n s t r u c t i o n s I i ( O 2 i 5 r ) over the alphabet a.(O 5 j 5 m ) J with a. = b ( " b l a n k " ) , a l = 1, accepting s t a t e 0 of M and i n i t i a l configurations C o ( k ) defined by the i n i t i a l s t a t e 1 and unary encoding of k i n t o the tape k-times ... b'l l'b

...

...

f

with the p o i n t e r (reading head) in p o s i t i o n (numbered) 0. For technical reasons which w i l l become c l e a r l a t e r choose s such t h a t f o r any k , i f M accepts k , then during t h e computation s t a r t e d w i t h input k i t w i l l never v i s i t the tape c e l l numbered -a (ks) + k. Without l o s s o f g e n e r a l i t y we assume a l s o t h a t in the accept i n g s t a t e !ny computation becomes c o n s t a n t (formally I. = (O,j,aj,O) f o r a l l 0 < j < m; remember t h a t we t h i n k of Turing i n s t r u c t i o n s I i a s of s e t s of quxdruples. )

Dedine the program formula of t h e following formulae:

uM

a s universal q u a n t i f i c a t i o n of t h e conjuncticn

9 s

(i)f o r any w r i t e - i n s t r u c t i o n s e t ( i , j , a h , + i , j ) i n M: t ' A Pt'u A Tht'u I i t A P t u A T . t u A S t t ' -+ I J +i,j {successive s t a t e , p o i n t e r p o s i t i o n unchanged,new working c e l l content}

E. BORGER

290

I i t A P t u A T . , t v A v f. u A S t t ' + T t ' v f o r 0 5 j' J j' {content of non working c e l l s remains unchanged) ( i i ) f o r any right-move i n s t r u c t i o n s e t ( i , j , r , @ i .) in M:

zm

,J

1.t 1

A

Ptu

A

T.,tv

+

I

T.tu

A

J

A

J

t'

$i,j

Stt' A

A

SUU'

Pt'u'

T.t'u

A

J

A

T.,t' J

for 0 5 j '

zm

( i i i ) f o r left-move i n s t r u c t i o n s analogously To d e f i n e t h e i n p u t r e p r e s e n t a t i o n we make use of the following embedding dohmula s t a t i n g t h a t t h e given domain (namely k = {O, ...,k-1) f o r some k ) i s embedded i n t o a segment of the ordering of t h e on-objects by some function ( w i t h graph)F:

V Fxu { e x i s t e n c e } x u A A A (Fxu A Fyu -t x A

A

A

A ( F X U A Fxv

A A

+

u

=

v ) {uniqueness]

x u v =

y) {injectivityl

X Y IJ A

A(KUV u v w x y

A A A A

A

Kvw

A

Fxu

A

Fyw

-+

VFzv)

{range i s a segment}

Z

Ued-ine t h e n-taht domula

aM,s

a s conjunction of the above embedding formula and

the following formulae: A ( Z t + I l t ) {read: a t time p o i n t 0 i n s t r u c t i o n 1 i s t o be executed} t A ( Z t -t V V ( P t u A Fxu A A ( K v u -t 1VFyv))) t u x V Y {read: a t time 0 t h e p o i n t e r p o s i t i o n i s encoded by the f i r s t F-value} A ( ( J V F x u ) -+ T o t u ) ) u x X {read: a t time 0 every tape c e l l (with number) i n the range of F has tape i n s c r i p t i o n a l = 1 and any o t h e r the blank symbol a. = b . 1

A ( Z t + A((VFxu + Tltu)

t

by saying t h a t a t t h e l a s t moment no i n s t r u c t i o n De6-ine t h e n t o p 60hmLLPa w M,s d i f f e r e n t from the a c c e p t i n g - s t a t e - i n s t r u c t i o n I. can be executed:

...

+ 111th A lIrt) t u We now show t h a t M e ConjuncLLon a := Ord(Z,S) A oM,s A aM,s A wM,s

A((1VKtU)

pmgmm, n.taht and n t o p 6omlLeae w i t h hebpect t o M e ckr&Lned by Ord(2,S) ~ U R d mM e heduction pmpehty.

06

ze.ho-AWCebnOh-bth4kJW

Indeed i f M accepts k in a t most a n ( k s ) . then {remember t h a t t h e configuration sequence ( C t : 0 < t 5 a n ( k s ) ) defined by Co(k) and M becomes c o n s t a n t once the i s c a l l e d } t h e above i n d i c a t e d accepting i n s t r u c t i o n set I. = (O,j,aj,O)o < intended meaning of K,Z,S,Ii,P,T. over k = TO,.T.,k-ll y i e l d s a model f o r a over k J together w i t h Fxu meaning I u I = p. + x (where p. and I u / a r e defined a s i n d i c a t e d above f o r the given k ) .

29 1

Decision Problems in Predicate Logic I n v e r s e l y any model M o f c a r d i n a l i t y k s a t i s f y i n g a, say o v e r t h e domain t k-11, encodes any M-computation ( C :O 5 t 5 a ( k ' ) ) s t a r t e d f r o m n t Co = Co(k) i n t h e sense t h a t i f C has i n s t r u c t i o n number i, p o i n t e r p o s i t i o n k k = {O,l,

...,

and t a p e a .

...

~ - e

, then

a

Iiut,

Putue+k and Tjhutuh+e

for

-e 5

h <

j a n ( ks)-a.

an(ks) a r e t r u e i n t h e model where u t denotes t h e t - t h o b j e c t o f t y p e on o v e r k w i t h r e s p e c t t o t h e g i v e n i n t e r p r e t a t i o n o f K and i s t h e o r d e r number o f t h e consequently M a t t i m e p o i n t K - s m a l l e s t F-value. Due t o t h e s t o p f o r m u l a w M,s an(ks) c a n n o t be i n any o f i t s s t a t e s 1 5 i 5 r, t h e r e f o r e a t t h a t moment i t must be e x e c u t i n g t h e a c c e p t i n g i n s t r u c t i o n Io. From t h e above g i v e n c o n s t r u c t i o n and w e l l known f a c t s a b o u t t h e Grzegorczyk- and t h e y n - h i e r a r c h y ( c f . Rodding 1967) f o l l o w s : yn 5 SPECTRAn+l, whereas SPECTRAn+l 5 yn+l f o l l o w s f r o m an easy g o d e l i z a t i o n o f formulae w i t h types bounded by n + 1 i n t o a r i t h m e t i c a l statements o v e r +,.,2' w i t h q u a n t i f i e r s bounded by an; see o p . c i t . S i n c e as e x p l a i n e d f o r Cook's theorem t h e same c o n s t r u c t i o n a p p l i e s t o n o n d e t e r m i n i s t i c machines f r o m t h e s p e c i a l case n = o f o l l o w s :

Fimt-ohdm Spectm a m ULith m p e c t t o 06

u n a q mpmentaLion pmc.hdy

the

NP-be.12

pobLtLve nwnbehs.

S i n c e a f u n c t i o n i s p o l y n o m i a l i n t h e l e n g t h k o f t h e unary r e p r e s e n t a t i o n k-times o f k i f f i t i s e x p o n e n t i a l i n t h e l e n g t h o f t h e b i n a r y r e p r e s e n t a t i o n o f k, t h a t c h a r a c t e r i z a t i o n can a l s o be s t a t e d i n t h e more f r e q u e n t l y used form:

WLth hebpect t o b i m q m p w e n t a f i o n t h e 6ih6t-ohCleh 6peCtM am phec.hdy those netn 0 6 pOb&iVe numbea which am accepted by a nondetemini6Lic Tuhing mckine i n e x p o n e m e (namely 2c"(x) f o r some c o n s t a n t c ) time. Using t h e s t a n d a r d encoding e(S) o f a f i n i t e s t r u c t u r e S = (k;R l,...,Rd) t e n a t i o n o f t h e encodings e(R1) d e f i n e d as {al,a2}-word

,...,

e(Rd) o f R1,...,Rd

as conca-

where f o r r - a r y R e(R) i s

o f l e n g t h kr which has i - t h d i g i t a2 resp. al

if R is

t r u e resp. f a l s e f o r t h e i - t h element o f kr w i t h r e s p e c t t o l e x i c o g r a p h i c a l o r d e r i n g , t h e domain c a r d i n a l i t y k need n o t t o be encoded s i n c e i t can be computed n o n d e t e m i n i s t i c a l l y i n p o l y n o m i a l t i m e f r o m e ( S ) . T h i s i s t h e reason why t h e above g i v e n p r o o f f o r t h e c h a r a c t e r i z a t i o n o f f i r s t - o r d e r s p e c t r a proves a l s o F a g i n ' s e x t e n s i o n t o g e n e r a l i z e d s p e c t r a : F o r any LENP n o t c o n t a i n i n g t h e empty word h o l d s L = e ( [ I } -spectrum(a)) where a i s t h e f o r m u l a d e f i n e d i n t h e main cons t r u c t i o n f o r a n o n d e t e r m i n i s t i c T u r i n g machine M a c c e p t i n g L i n t i m e ks, b u t w i t h a d d i t i o n a l l y Ord(O,xx,y.y = x + l ) - i n o r d e r t o r e s t r i c t a t t e n t i o n t o models o v e r k = [O,l, k-11 - and t h e f o l l o w i n g replacements i n t h e s t a r t f o r m u l a aM . ,s' r e p l a c e t h e embedding f o r m u l a by t h e new embedding 60hmLLen e x p r e s s i n g t h a t t h e domain k = [O,l, k-11 'is embedded v i a F i n an o r d e r p r e s e r v i n g way i n t o a segment o f t h e s - t u p l e s : A V Fxu A A A A A(Fxv A Fyw -P ( S xy++ Svw)) x u x y v w

...,

...,

{Note t h a t t h e o r d e r - p r e s e r v a t i o n i m p l i e s uniqueness and i n j e c t i v i t y o f F and t h e f a c t t h a t t h e range o f F i s an K-segment}; r e p l a c e t h e i n i t i a l - t a p e - d e s c r i p t i o n by t h e f o l l o w i n g new i n i t i a l - h p e 60mLLen e x p r e s s i n g t h a t M s t a r t s a t t i m e 0 w i t h t h e encoding o f t h e monadic " i n p u t " p r e d i c a t e I - which i s o f l e n g t h k and i s i n s c r i -

E. BORGER

292

bed i n t h e tape c e l l s numbered by F-values u: A ( Z t + A A(FXU + ( ( I x

A

+

Tptu)

(1Ix

A

-+

Tltu))

x u

t

A(Zt t

-+

A((~VFXU) Totu)) u x -f

{ b l a n k a.

o u t s i d e range ( F ) }

F i n a l l y bound a l l p r e d i c a t e symbols e x c e p t I by an e x i s t e n t i a l q u a n t i f i e r . Again t h e same c o n s t r u c t i o n a p p l i e s t o r u d i m e n t a r y p r e d i c a t e s , i . e . those number t h e o r e t i c a l r e l a t i o n s which can be d e f i n e d e x p l i c i t e l y f r o m t h e graphs o f "+" and a number t h e o r e t i c a l ' I - ' ' u s i n g Boolean o p e r a t i o n s and bounded q u a n t i f i c a t i o n s : p r e d i c a t e R i s s a i d t o have a ,$&5.t ohdeh mpkueevLtaLLtion i n 4.inite d o d m i f f some f i r s t o r d e r f o r m u l a % ( c o n t a i n i n g i n p a r t i c u l a r a p r e d i c a t e symbol T? o f t h e same a r i t y as R and e v e n t u a l l y a b i n a r y p r e d i c a t e symbol K) i s s a t i s f i a b l e o v e r e v e r y domain k := { O , l , ..., k ~ l }w i t h K i n t e r p r e t e d as < and i n e v e r y such model o f "p; t h e i n t e r p r e t a t i o n o f R i s t h e r e s t r i c t i o n o f R t o k. O b v i o u s l y t h e zero p r e d i c a t e xx.x = 0 and t h e successor r e l a t i o n xx,y.y = x + l have r e p r e s e n t a t i o n Ord(Z,S)Z resp. Ord(Z,S)S w i t h Ord(Z,S) as d e f i n e d above, whereas one can d e f i n e aR : z Ord(Z,S)

A

add f o r R = G+ : = hx,y,z.x+y

%

A

add

:: Ord(Z,S)

A

= z and

m u l t f o r R = G.

where add and m u l t a r e t h e r e c u r s i v e d e f i n i t i o n s o f

"+"

resp.

'I-"

add : z A A ~ ( z + y (G+xyz +-+x = z ) ) { r e a d : x + o = XI X Y z A A A A A ( s y y ' + ( G + X y ' Z ' t t v(-d+XyZ A S Z Z ' ) ) ) { X + y ' = (X x y ' 2' y Z m u l t :E A A A(Zy XY z A

+

(c.xyz++

A A

A

A(Syy'

x y'

2'

y

+

y)}

+

x}

{ r e a d : x0 = XI

y = 2))

+ (G.Xy'Z'tt

f r o m Z,S:

v(G.XyZ

A

G+ZXZ')))

{xy' = Xy

Z

Since K l e e n e ' s T - p r e d i c a t e can be c o n s t r u c t e d as r u d i m e n t a r y p r e d i c a t e (see Smullyan 1961) t h e r e i s i n p a r t i c u l a r a f i r s t o r d e r r e p r e s e n t a t i o n ar o f T i n f i n i t e domains where f u r t h e r m o r e T(i,x,y)

i m p l i e s i , x < y. Therefore t h e

cl-

complete non-emptiness problem I i I V V T ( i , x , y ) l f o r t h e r . e . s e t s W . = XY I x 1 3 y T ( i , x , y ) ) resp. t h e i r n2-complete i n f i n i t y problem { i I g x 3 y T ( i , x , y ) }

is

1-1-reduced t o t h e nonemptiness resp. i n f i n i t y problem f o r spectrum (nonemptyi) resp. spectrum ( i n f i ) nonemptyi

where

:: V V V(a z x y Xz.z=i

A

Z=i) A

(ar

TZXY))

{read: machine i f o r some i n p u t x has an a c c e p t i n g computation y l infi

:= machine i f o r some x has an a c c e p t i n g computation y A

A 1KYYl Y1

A

A (fzxyl

YI

{ y i s t h e l a s t element i n t h e model]

-+

lKyly)

{no a c c e p t i n g computation f o r i n p u t x i s s h o r t e r than y l

293

Decision Problems in Predicate Logic

Note t h a t i n t h e a l m o s t Horn d e s c r i p t i o n o f f i n i t e computations non-Horn i m p l i c a t i o n s appear o n l y f o r t h e i n p u t d e s c r i p t i o n o r f o r n o n d e t e r m i n i s t i c i n s t r u c t i o n s . T h i s suggests t h e f o l l o w i n g LogicaL complexity meanurn do& Boolean dune.tiom: say t h a t a f o r m u l a F d e f i n e s a Boolean f u n c t i o n s f ( w i t h r e s p e c t t o i t s " i n p u t " v a r i a b l e s x = xl, x ) i f f f o r e v e r y 0-1-sequence q: n f ( q ) = 1 i f f F(x/q) i s s a t i s f i a b l e .

...,

F i s c a l l e d pseudo-Horn o r Horn i n i t s w o r k i n g v a r i a b l e s i f F ( x / q ) " i s " a Horn f o r m u l a f o r e v e r y 0-1-sequence q. D e f i n e Horn c o m p l e x i t y o f f as l e n g t h o f a s m a l l e s t pseudo-Horn f o r m u l a F d e f i n i n g f. T h i s c o m p l e x i t y measure i s n t m n g l y connected t o Cook'n pmbLem: The Horn c o m p l e x i t y o f any Boolean f u n c t i o n can be p o l y n o m i a l l y bounded by i t s a r i t y n and program s i z e and maximal r u n t i m e on any i n p u t sequence o f l e n g t h n o f any d e t e r m i n i s t i c T u r i n g machine computing t h e funct i o n ; s i m i l a r l y i t can be proved t h a t P # NP i f f o r e v e r y p o l y n o m i a l p t h e r e i s a f u n c t i o n f such t h a t i t s s m a l l e s t pseudo-Horn d e f i n i t i o n i s a t l e a s t p - b i g g e r than i t s s m a l l e s t p r o p o s i t i o n a l d e f i n i t i o n . S i m i l a r r e s u l t s a r e known f o r network o r T u r i n g machine c o m p l e x i t y o f Boolean f u n c t i o n s ; i n f a c t we have t h e f o l l o w i n g

T h e o m (Aanderaa & Borger 1979) F o r any Boolean f u n c t i o n f, i t s Hohn complexity and i t s ( l o g i c a l ) nehuonk compLexity - and t h e r e f o r e T u r i n g machine c o m p l e x i t y ahe poLynomiaLtq e q l L i v a L e k W i t h o u t g i v i n g t h e whole p r o o f we want t o i l l u s t r a t e t h a t one d i r e c t i o n o f t h i s e q u i v a l e n c e , namely C H ( f ) 5 O(C,4(f)), amounts t o a l o g i c a l d e s c r i p t i o n o f a r b i t r a r y l o g i c a l network computations which can and has been done a g a i n by an approp r i a t e m o d i f i c a t i o n o f t h e r e d u c t i o n technique e x p l a i n e d i n t h e second s e c t i o n o f t h i s paper: S i n c e C r r ( f ) can be bounded by some l i n e a r e x p r e s s i o n i n t h e network c o m p l e x i t y o f f w i t h r e s p e c t t o l o g i c a l networks b u i l t up w i t h any complete. s e t o f b i n a r y Boolean o p e r a t i o n s , we need t o c o n s i d e r o n l y networks computing f w i t h b i n a r y o p e r a t i o n s say v , A and I ( S h e f f e r ' s s t r o k e ) . We show how one can a s s o c i a t e t o an a r b i t r a r y such l o g i c a l network N (computing Boolean f u n c t i o n f ) a Horn netw h k domanLLea on, a Kmm i n p u i dom& an which i s Horn i n t h e i n p u t v a r i a b l e s and a Hehmnd ouipLLt darn&

w such t h a t uN

A

an

A

w defines f w i t h r e s p e c t t o i t s i n -

p u t v a r i a b l e s and i s o f l e n g t h l i n e a r l y bounded by t h e c o m p l e x i t y o f N. By such a construction CH(f) CN(O(f)) i s proved. L e t N be an a r b i t r a r y l o g i c a l network w i t h nodes No,...,Nm

where N1,

...,Nn

are

e n t r i e s , No i s t h e node w i t h r e s p e c t t o which N computes f and t h e n o n - e n t r i e s a r e l a b e l e d w i t h v,

A

1.

or

Every node Nk i s encoded by v a r i a b l e s yk.uk w i t h t h e in-

tended iuztehpm&ztivn Yk = f,i,Nk(q)

and

k

-

1Y k

f o r p r e v i o u s l y g i v e n values q t o t h e i n p u t v a r i a b l e s xl,

..., xn.

Define therefore

aN as c o n j u n c t i o n o f t h e f o l l o w i n g formulae f o r e v e r y node Nk l a b e l l e d w i t h op(Nk) a p p l i e d t o t h e d i r e c t l y preceeding nodes Ni,N.

-

0 5 i,j,k

p u t e d a t Ni,N.: J Case 1. op(Nk) = A : (yi Case 2. op(Nk) = v : yi Case 3. op(Nk) = Define

an

w := yo

A

i n t h i s order

J

5 m -, d e s c r i b i n g t h e computation a t node Nk f o r t h e arguments com-

I:

u.

1

A -+

A

yj) yk

+

yk

yj u . + yk

-+

J

as c o n j u n c t i o n o f a l l yi

ui yk

Yi

+

uk u j + uk (ui A u . ) + uk J Uk Y j Uk

+

xi and ui

+

++

l x . f o r 1 5 i 5 n and

The n e c h t i o n pmpehty f o r a l l qe{O,ll

A

reads:

294

E. BORGER

fN,No(q) = 1 iff

uN

A

an(q)

A

yo

A

l u o is satisfiable

where a s before a n ( q ) denotes an a f t e r s u b s t i t u t i o n of q i f o r x i . The proof of the reduction property follows the now well e s t a b l i s h e d p a t t e r n : from l e f t t o r i g h t the above i n d i c a t e d intended i n t e r p r e t a t i o n s a t i s f i e s a N A a n ( q ) A w . Conversely any truth assignement f o r which t h a t formula i s t r u e simulates t h e network computation i n t h e sense t h a t f o r every node N k of N: (i)

f N , N k ( q )= 1 implies A(yk) = 1

from which f

N,No

( 4 ) = 1 follows because A ( u o ) = 0 by

W.

The simulation property

i s shown by induction along t h e computation process of N: the base of t h e induct i o n a t e n t r i e s N i ( l 5 i 5 n ) i s assured by a n ( q ) , whereas f o r every node N k with d i r e c t l y preceding nodes Ni,N. i n t h i s order t h e claim follows from t h e inducJ t i v e hypothesis, the formulae corresponding t o t h i s node and f N , N k ( q ) = o p ( N k ) ( f N , N , ( q ) ,f N , N ( 9 ) ) . j 2 3 For a proof of the o t h e r claim C N ( f ) 5 O(CH(f) ( l g C H ( f ) ) ) s e e Aanderaa & Borger 1981.

Added 1 9 8 3 , j u l y . W.D. G o l d l a r b has j u s t shown t h a t t h e case w i t h = i s u n s o l v a b l e , e v e n when r e s t r i c t e d t o Krom

GGdel-Kalmar-SchOtte

o r t o formulae w i t h

o n l y d y a d i c p r e d i c a t e l e t t e r s . See t h e p a p e r “The U n s o l v a b i l i t y o f t h e GGdel c l a s s w i t h i d e n t i t y ” s u b m i t t e d t o t h e J . o f Symbolic Logic.

Decision Problems in Predicate Logic

295

References A complete l i s t o f papers published i n t h e area o f (complexity o f ) l o g i c a l d e c i s i o n problem i s o u t o f t h e scope o f t h i s survey. However t a k i n g t h e union o f a l l papers c i t e d i n any o f t h e papers c i t e d i n t h i s l i s t w i l l g i v e you a r a t h e r complete picture o f the situation. Aanderaa, S t i l 0. (1966): A New Undecidable Problem w i t h A p p l i c a t i o n s i n Logic. Ph.0. t h e s i s , Harvard U n i v e r s i t y . Aanderaa, S.O. (1971): On t h e d e c i s i o n problem f o r formulas i n which a l l d i s j u n c t i o n s are b i n a r y . Proc. o f t h e Second Scand. Log. Symp., pp. 1-18. Aanderaa, S.O., Borger, E. (1979): The Horn complexity o f Boolean f u n c t i o n s and Cook's problem. i n : F.V. Jensen, B.H. Mayoh, K.K. M o l l e r (Ed.): Proceedings from 5 t h Scandinavian Logic Symposium, Aalborg U n i v e r s i t y Press, pp. 231-256, Aanderaa, S.O., Borger, E. (1981): The equivalence o f Horn and network complexity f o r Boolean f u n c t i o n s . Acta I n f o r m a t i c a 15, 303-307. Aandera, S.O. & Borger, E. & Gurevich, Yu. (1982: P r e f i x classes o f Krom formulae w i t h i d e n t i t y . i n : Archiv f. math. Logik 22, pp, 43-49. Aanderaa, S.O., Borger, E., Lewis, H.R. (1982): Conservative r e d u c t i o n classes o f Krom formulas. The Journ. o f Symb. Logic 47, 110-129. Aanderaa, S.O. & Jensen, F.V. (1973):On t h e existence o f r e c u r s i v e models f o r Krom formulas. Manuscript, U n i v e r s i t y o f Oslo. Aanderaa, S d l 0. and Harry 4. Lewis (1973): P r e f i x classes o f Krom formulas, Journal o f Symbolic Logic pp. 628-642.

38,

Aanderaa, S . O . G1 Lewis, H.R. (1974): Linear sampling and t h e v j v case o f t h e dec i s i o n problem. Journal o f Symbolic Logic 39, pp. 519-548. Abramsky, M o t t i (1980): The c l a s s i c a l d e c i s i o n problem and p a r t i a l f u n c t i o n s . i n : Archiv math. Logik 20, pp. 3-12. Ackermann, W. (1928): Ober d i e E r f u l l b a r k e i t gewisser Zahlausdrdcke. Mathematische Annalen 100, pp. 638-649. Ackermann, W. (1954): Solvable Cases o f t h e Decision Problem, North-Holland, Amsterdam. Ash, C.J. (1975): Sentences w i t h f i n i t e models. i n : Z e i t s c h r . f . math. Logik und Grundlagen d. Math. 21, pp. 401-404. Asser, G. (1955) : Das Reprasentantenproblem i m Pradikatenkal k u l der e r s t e n Stufe m i t I d e n t i t a t . Z e i t s c h r i f t fir math'ematische Logik und Grundlagen der Mathematik, 1, pp. 252-263. Bennett, J. (1962): On spectra, Doctoral D i s s e r t a t i o n , Princeton U n i v e r s i t y , P1.J. Berman, L. (1977). Precise bounds on Presburger a r i t h m e t i c and t h e r e a l s w i t h add i t i o n : p r e l i m i n a r y r e p o r t . Proceedings o f 18th Annual Symposium on Foundations o f Computer Science, IEEE Computer Society, pp. 95-99. Bernays, Paul & Schonfinkel, Moses (1928): Zum Entscheidungsproblem der mathemat i s c h e n Logik, Mathematische Annalen 99, pp. 342.372. Borger, E. (1971): Reduktionstypen i n Krom- und Hornformeln. D i s s e r t a t i o n . Minster see B e i t r a g z u r Reduktion des Entscheidungsproblems auf Klassen von Hornformeln m i t kurzen Alternationen. i n : Archiv f u r math. Logik und Grundlagenforschung 16 (1974), 67-84. Borger, E. (1973): Eine entscheidbare Klasse von Kromformeln. Z e i t s c h r . f u r math, Logik und Grundlagen d e r llathematik 19, 117-120.

E. Borger (1974): La z3-completude de l'ensemble des types de reduction, in: Logique e t Analyse 65-66, 89-94.

E. BORGER

296

Borger, E. (1975): On t h e c o n s t r u c t i o n o f simple f i r s t - o r d e r formulae w i t h o u t r e c u r s i v e models. Proc. Coloquio sobra l o g i c a simbolica, Madrid, 9-24. Borger, E. (1979): A new general approach t o the theory o f t h e many-one equivalence o f d e c i s i o n problems f o r a l g o r i t h m i c systems, i n : R. Kaerkes, J. flerkwitz and 1.4 Oberschelp, Eds., S c h r i f t e n z u r I n f o r m a t i k u. Angew. Math. 30, RWTH Aachen Also i n : Z. Math. Logik Grundlagen Hath. 25, 135-162. E. Borger (1983): U n d e c i d a b i l i t y versus degree complexity o f d e c i s i o n problems f o r formal grammars. Submitted f o r p u b l i c a t i o n . P r e l i m i n a r y v e r s i o n i n : Proc. o f GTI, Paderborn 1932. Borger, E. and H e i d l e r , K. (1976): Die m-Grade l o g i s c h e r Entscheidungsprobleme. i n : Archiv f. math. Logik und Grundlagenforschung 17, 105-112. Borger, E. and K l e i n e Buning, H. (1980): The r e a c h a b i l i t y problem f o r P e t r i nets and d e c i s i o n problems f o r Skolem a r i t h m e t i c . i n : Theoretical Computer Science 11, 123-143. Buchi, J.R. (1962): Turing machines and t h e Entscheidungsproblem, Mathematische Annalen 148, pp. 201-213. Bullock, A.M. & Schneider, H.H. (1973): On generating t h e f i n i t e l y s a t i s f i a b l e formulas. i n : Notre Dame J . o f Formal Logic X I V , pp. 373-376. Chandra, A.K. & Stockmeyer, L.J. (1976): A l t e r n a t i o n , Proc. 17th Symposium on Foundations o f Computer Science, pp. 98-108. & Stockmeyer, L.J. (1978): A l t e r n a t i o n , IBM Thomas Kozen, D.C. Chandra, A.K., J . Natson Research Center Technical Report, RC 7489. See J . o f t h e ACM 28 (1981), pp. 114-133. Chang, C.C. and H.J. K e i s l e r (1962): An improved prenex normal form. Journal o f Symbolic Logic pp. 317-326.

7,

Church, Alonzo (1936): A n o t e on t h e Entscheidungsproblem. Journal o f Symbol. Logic 1, pp. 40-41; C o r r e c t i o n i b i d , pp. 101-102. Cohen, Daniel C. (1980): Degree problems f o r modular machines, i n : The Journal o f Symbolic Logic 45, pp. 510-528. Cook, S,A. (1971): The complexity o f theorem p r o v i n g procedures. Conf. Rec. o f 3 r d ACM Symp. on Theory o f Computing, 151-158. Davis, M. & Putnam, H. & Robinson, J. (1961): The d e c i s i o n problem f o r exponent i a l diophantine equations. i n : Annals o f Math. 74, 425-436. Oenenberg, L. & Lewis, H.R. (1982): The complexity o f t h e s a t i s f i a b i l i t y problem f o r Krom formulas. manuscript, Harvard U n i v e r s i t y . Deutsch, M. (1975): Zur D a r s t e l l u n g koaufzahlbarer Pradikate b e i Verwendung eines einzigen unbeschrankten Quantors. in: Z e i t s c h r . f. math. Logik und Grundlagen der Mathemati k 21, pp. 443-454. Deutsch, Michael (1975 a . ) : Zur Theorie der spektralen Darstellung von Pradikaten durch Ausdrucke der P r a d i k a t e n l o g i k I . Stufe. i n : Archiv math. Logik 17, pp. 9-16. Deutsch, rlichael (1981): Zur Reduktionstheorie des Entscheidungsproblems. i n : Zeitschr. f . math. Logik und Grundlagen d. Mathe., 27, pp.113-117. Oreben, Burton (1961): Solvable Suranyi subclasses: An i n t r o d u c t i o n t o t h e Herbrand theory. Proceedings o f a Harvard Symposium on D i g i t a l Computers, 3-6 A p r i l 1961; Annals o f t h e Computation Laboratory o f Harvard U n i v e r s i t y 2, Harvard U n i v e r s i t y Press, Cambridge, Plassachusetts, pp. 32-47. Dreben, 6. 4r Goldfarb, W.D. (1979): The Decision Problem: Solvable Classes o f Q u a n t i f i c a t i o n a l Formulas, Rddison4esley, Advanced Book Program, Reading, Massachusetts, 1979.

297

Decision Problems in Predicate Logic

Oreben, Burton, A.S. Kahr, and Hao Wang (1962): C l a s s i f i c a t i o n o f AEA formulas by l e t t e r atoms. B u l l e t i n o f t h e American Mathematical S o c i e t y 68, pp. 528-532. Ershov, Yu. L . (1973): Skolem f u n c t i o n s and c o n s t r u c t i v e models. Algebra y Logika 12, 644-654. Ershov, Y.L., Lavrov, I . A . , Taimanov, A.D., & T a i t s l i n , M.A. t h e o r i e s . Russian Math. Surveys, 20, pp. 35-105,

(1965). Elementary

Fagin, R. (1973): C o n t r i b u t i o n s t o t h e model t h e o r y o f f i n i t e s t r u c t u r e s . Doctoral d i s s e r t a t i o n , U n i v e r s i t y o f C a l i f o r n i a , Berkeley. Fagin, Ronald (1974): Generalized F i r s t - O r d e r Spectra and Polynomial-Time Recogn i z a b l e Sets. i n : R. Karp (Ed.): Complexity o f Computation, SIB1 ANS Proceedings Vol 7, pp. 43-73.

-

.

Fagin, R. (1975): Ronadic generalized spectra. i n : Z e i t s c h r . f. math. Logik und Grundlagen d. Math., 21, pp. 89-96. Fagin, R. (1975 a ) : A two-cardinal c h a r a c t e r i z a t i o n o f double spectra. i n : Z e i t s c h r i f t fiir math. Logik und Grundlagen d. Math., 21, pp. 121-122. Fagin, R. (1975 b ) : A spectrum hierarchy. i n : Z e i t s c h r . f . math. Logik und Grundlagen d. Math., 21, pp. 123-134. Ferrante, J. & Rackoff, C. (1975): A d e c i s i o n procedure f o r t h e f i r s t order theory o f r e a l a d d i t i o n w i t h order. Ferrante, J. & Rackoff, C.W. (1979): The Computation Complexity o f Logical TheoSpringer-Verlag Lecture Notes i n Mathematics. New York.

ries.

Fischer, F1.J. d Rabin, M.O. (1974). Super-exponential complexity o f Presburger a r i t h m e t i c . Complexity o f Computation, ed. R.M. Karp, pp. 27-41. Proceedings o f SIAM-AMS Symposium i n Applied Mathematics. F i r e r , M. (1981): A l t e r n a t i o n and t h e Ackermann case o f t h e d e c i s i o n problem. i n : L'Enseignement math'eniatique X X V I I , 1-2, pp. 137-162. Garey, 1,I.R. & Johnson, D.S. (1979). Computers and I n t r a c t a b i l i t y : A Guide t o the Theory o f NP-Completeness, H. Freeman, San Francisco. Genenz, J. (1965) : Untersuchungen zum Entscheidungsproblem i m P r l d i katenkal kU1 der e r s t e n StUfe. D i s s e r t a t i o n , I n s t i t u t f. math. Logik und Grundlagenforschung, Univ e r s i t a t Miinster i.W. Germano, G. (1976): An a r i t h m e t i c a l r e c o n s t r u c t i o n o f t h e l i a r ' s antinomy using a d d i t i o n and m u l t i p l i c a t i o n . i n : Notre Dame J. o f Formal Logic X V I I , 457-461. Godel, Kurt (1930): Die V o l l s t a n d i g k e i t der Axiome des logischen Funktionenkalkuls. Flonatshefte f u r Mathematik und Physik 2, pp. 349-360; E n g l i s h t r a n s l a t i o n i n van H e i j e n o o r t (1971), pp. 582-591. Godel, K u r t (1932): E i n S p e z i a l f a l l des Entscheidungsproblems der theoretischen Logik, Ergebnisse eines mathematischen Kolloquiums 2, pp. 27-28. Godel, K u r t (1933): Zum Entscheidungsproblen des logischen Funktionenkalkuls. Monatsh. Math. Phys. 40, pp. 433-443. Goldfarb, Warren 0. (1974): On Decision Problems f o r Q u a n t i f i c a t i o n Theory. Ph.D. t h e s i s , Harvard U n i v e r s i t y

.

Goldfarb. W.D. (1981): On t h e Godel c l a s s w i t h i d e n t i t y . i n : The Journal of Synbol i c Logic 46, pp. 354-364. Goldfarb, Warren 0. and Harry R. Lewis (1975): Skolem r e d u c t i o n classes. Journal o f Symbolic Logic 40, pp. 62-68. Goldfarb, W.D. Gurevich, Yu. & Shelah, S. (1983): On t h e Godel c l a s s w i t h ident i t y . ( t o appear).

E. BORGER

298

Grzegorczyk, A. (1953): Some classes o f r e c u r s i v e f u n c t i o n s . Rozprawy Matematyczne I V , Warzsaw. Gurevich, Yuri (1965): E k z i s t e n t i a l ' n a y a i n t e r p r e t a t s i y a . Algebra i Logika 71-84.

4,pp.

Gurevich, Yuri (1966): Ob effektivnom raspozriavanii v i p o l n i m o s t i formul U I P . Algebra i Logika 5, 25-55. Gurevich, Yuri (1969): Problema razresheniya d l y a l o g i k i predikatov i operatsiy. Algebra i Logika 8 , pp. 284-308; E n g l i s h t r a n s l a t i o n i n Algebra and Logic pp. 160-174.

8,

Gurevich, Yuri (1973): Formulas w i t h one v. i n : I n Memory o f A . I . Russian).

Maltsev. ( i n

Gurevich, Yuri (1976): The d e c i s i o n problem f o r standard classes. Journal o f Symb o l i c Logic pp. 460-464.

5,

Gurevich, Yu. & Shelah, S. (1983): Random models and t h e Godel case o f t h e d e c i s i o n problem. The Journal o f Symbolic Logic ( t o appear). H a i l p e r i n , Th. (1961): A complete s e t o f axioms f o r l o g i c a l formulas i n v a l i d i n some f i n i t e domain. i n : Z e i t s c h r i f t f . math. Logik und Grundlagen der Math. 7, pp. 84-96. Hartmanis, J. (1982): On t h e s t r u c t u r e o f f e a s i b l e computations o r t h e search f o r what i s f e a s i b l y computable. Series o f l e c t u r e s d e l i v e r e d t o t h e AMS Summer School i n "Recursion Theory," Cornell U n i v e r s i t y , I t h a c a N.Y. Hay, L. (1973): The h a l t i n g problem r e l a t i v i z e d t o complements. i n : Proc. AMS 41, pp. 583-.587. Hay, L. (1975): Spectra and h a l t i n g problems. i n : Z e i t s c h r i f t f. math. Logik und Grundlagen d. Math. 21, pp. 167-176. Heidler, K (1973). Untersuchungen zur Reduktionstheorie des Entscheidungsproblems i n der Pradikaten- und Termlogik. D i s s e r t a t i o n , U n i v e r s i t a t Freiburg. J . Herbrand (1930): Recherches sur l a th'eorie de l a d'emonstration. Travaux de l a Socigt'e des Sciences e t des L e t t r e s de Varsovie. Classe 111. Warszawa, pp. 128 Herbrand, Jacques (1931): Sur l e probl&ne fondamental de l a l o g i q u e mathgmatique. Sprawozdania z posiedzen Towarzystwa Naukowego Warszawskiego, Wydzial 111, 24, pp. 12-56; E n g l i s h t r a n s l a t i o n i n Herbrand (1971), pp. 215-271. H i l b e r t , 0. & Bernays, P. (1939): Grundlagen der Mathematik, Vol. 11. B e r l i n Jones, N.D. & Laaser, W.T. (1977): Complete problems f o r d e t e r m i n i s t i c polynomial time. i n : Theor. Computer Science 3, 105-117. Jones, N.G. & Selman, A.L. (1974): Turing machines and t h e spectra o f f i r s t - o r d e r formulas, Journal o f Symbolic Logic 39, pp. 139-150. Joseph, Deborah & Young, Paul (1981): A survey o f some r e c e n t r e s u l t s on computat i o n a l complexity i n weak t h e o r i e s o f a r i t h m e t i c . Proc. MFCS, pp. 46-60. Kahr, A.S. (1962): Improved reductions o f the Entscheidungsproblem t o subclasses o f AEA formulas. Proceedings o f a Symposium on t h e Mathematical Theory o f Automata, Brooklyn P o l y t e c h K c I n s t i t u t e , New York. Kahr, A.S., Moore, E.F. & WangAao (1962): Entscheidungsproblem Reduced t o t h e AEA case, Proc. Nat. Acad. Sci. USA 48, pp. 365-377. KalmBr, Laszlb (1933): Uber d i e E r f u l l b a r k e i t derjenigen Zahlausdrucke, welche i n der Nomalform zwei benachbarte A l l z e i c h e n enthalten, Mathematische Annalen 108, pp. 466-484. K l e i n e Buning, Hans (1980): Decision problems i n generalized vector a d d i t i o n systems. i n : Annales S o c i e t a t i s Mathematicae Polonae, Series I V : Fundamenta I n f o r matica 111. 4., pp. 497-512.

Decision Problems in Predicate Logic

299

K l e i n e Buninq, H. & Ottmann. Th. (1977): K l e i n e u n i v e r s e l l e mehrdimensionale Turingmaschinen: i n : Elektronische informationsverarbeitung und Kybernetik E I K 13, pp. 179-201. Kosaraju, S. Rao (1982): D e c i d a b i l i t y o f r e a c h a b i l i t y i n v e c t o r a d d i t i o n systems. ACM Symposium on Computing, pp. 267-281. Kostryko, V.F.

(1964): Klass cvedeniya

v$v.

Algebra i Logika

Kostyrko. V.F. (1966): Klass cvedeniya v3v. K i b e r n e t i k a t r a n s l a t i o n , Cybernetics 2, pp. 15-19. Kozen, D.C. Science.

2,

3,

pp. 45-65.

pp. 17-22; E n g l i s h

(1979): Complexity o f Boolean Algebras. i n : T h e o r e t i c a l Computer

K r e i s e l , G. (1950): Note on a r i t h m e t i c models f o r c o n s i s t e n t formulae of p r e d i c a t e c a l c u l u s . i n : Fundamenta rlathematicae 37, 265-285. K r e i s e l , G. (1953): Note on a r i t h m e t i c models f o r c o n s i s t e n t formulae o f t h e pred i c a t e c a l c u l u s 11. i n : Proc. X I - t h I n t . Congr. Philosophy, v o l . 14, pp. 39-49. K r e i s e l , G. fi K r i v i n e , J.L. (Dunod)

.

(1967): Elements de l o g i q u e mathhnatique, P a r i s

Krom, M.R. (1964): A d e c i s i o n procedure f o r a c l a s s o f formulas o f f i r s t order p r e d i c a t e c a l c u l u s . i n : P a c i f i c J . o f Math. 14, 11, pp. 1305-1319. Krom, M.R. (1966): A p r o p e r t y o f sentences t h a t d e f i n e quasi-order. J. of Formal Logic 7, pp. 349-352.

i n : Notre Dame

Krom, M.R. (1967 a.): The d e c i s i o n problem f o r a c l a s s o f f i r s t - o r d e r formulas i n which a l l d i s j u n c t i o n s a r e binary. i n : Z e i t s c h r . f u r math. Logik und Grundlagen der llath. 13, 15-20. Krom, P1.R. (1967 b): The d e c i s i o n problem f o r segregated formulas i n f i r s t - o r d e r l o g i c . i n : Mathematica Scandinavica 21, 233-240. Krom, H.R. (1968): Some i n t e r p o l a t i o n theorems f o r f i r s t - o r d e r formulas i n which a l l d i s j u n c t i o n s a r e binary. i n : Logique e t Analyse 43, 403-412. Krom, M.R. (1970): The d e c i s i o n problem f o r formulas i n prenex c o n j u n c t i v e normal form w i t h b i n a r y d i s j u n c t i o n s . i n : The J. o f Symbolic Logic 35, 210-216. Ladner, R.E., Lipton, R.J. & Stockmeyer, L.J. (1978): A l t e r n a t i n g Pushdown Automat a , Proc. 1 9 t h Symp. on Foundations o f Computer Science, pp. 92-106. Lewis, H.R. (1975): D e s c r i p t i o n o f r e s t r i c t e d automata by f i r s t - o r d e r formulae. i n : Math. Systems Theory 9, 97-104. Lewis, H.R. (1976): Krom formulas w i t h one dyadic p r e d i c a t e l e t t e r . i n : Journal o f Symbolic Logic 41, pp. 341-362. Lewis, H.R.

(1979): Unsolvable classes of q u a n t i f i c a t i o n a l formulas. Reading(Mass.)

Lewis, H.R. JCSS.

(1980): Complexity r e s u l t s f o r classes o f q u a n t i f i c a t i o n a l formulas.

Lewis, H.R. & Statman, R. (1983): U n i f i a b i l i t y i s complete f o r CO-NLOG space. i n : I n f o r m a t i o n Processing L e t t e r s ( t o appear). LovBsz, L. & GBcs, P. (1977): Some remarks on generalized spectra. i n : Z e i t s c h r . f . math. Logik und Grundlagen d. Math., 23, pp. 547-554. Lowenheim, Leopold (1915): Uber Moglichkeiten i m R e l a t i v k a l k u l , Mathematische Annalen 76, pp. 447-470. Machtey, M. & Young, P. (1978). An I n t r o d u c t i o n t o t h e General Theory o f Algorithms. North-Holland, New York. Manna, Z. (1968): Termination o f Algorithms. Doctoral D i s s e r t a t i o n , CarnegieMe1l o n U n i v e r s i t y .

300

E. BORGER

Maslov, S. Ju. (1964): An i n v e r s e method f o r e s t a b l i s h i n g d e d u c i b i l i t i e s i n t h e c l a s s i c a l p r e d i c a t e c a l c u l u s . i n : Dokl. Akad. Nauk SSSR 159, 17-20 = S o v i e t Math. Doklady 5, 1420-1424. Meyer, A.R. (1975). Weak monadic second order theory o f successor i s n o t element a r y - r e c u r s i v e . Proceedings o f Boston U n i v e r s i t y Logic Colloquium, Springer-Verlag, pp. 132-154. Minsky, M.L. (1961): Recursive u n s o l v a b i l i t y o f P o s t ' s problem o f 'Tag' and o t h e r t o p i c s i n theory o f Turing machines, Ann. o f Math. 74, 437-455. Mostowski, A. (1953): On a system o f axioms which has no r e c u r s i v e l y enumerable model. i n : Fundamenta Mathematicae 40, 56-61, Mostowski, A. (1955): A formula w i t h nor r e c u r s i v e l y enumerable model. i n : Fundamenta Mathematicae 43, 125-140. Mostowski, A. (1956): Concerning a problem o f H. Scholz, Z e i t s c h r i f t f u r mathemat i s c h e Logik und Grundlagen der blathematik, 2, pp. 210-214. Rabin, MO. (1958): On r e c u r s i v e l y enumerable and a r i t h m e t i c models o f s e t theory. i n : The Journal o f Symbolic Logic 23, pp. 408-416. Rabin, 1.1.0 (1977): Decidable Theories. Handbook o f Mathematical Logic, ed. 3 . Barwise, pp. 595-629. North-Holland, Amsterdam. Rackoff, C.W. (1975). Complexity o f some l o g i c a l t h e o r i e s . Doctoral D i s s e r t a t i o n , Dept. o f E l e c t r i c a l Engineering, MIT, Cambridge, Mass., P r o j e c t MAC Technical Report TR-144. Ramsey, F.P. (1930): On a problem o f formal l o g i c . i n : Proc. London Math. SOC. ( 2 ) , 30, pp. 264-286. Rodding, D i e t e r (1970): Reduktionstypen d e r Pradikatenlogik. Lecture notes, Uni1969/70. See review i n : Z e n t r a l b l a t t f u r Mathematik v o l . v e r s i t a t Munster i . W . , 207, No. 02034. Rodding, 0. & Borger, E. (1974): The u n d e c i d a b i l i t y o f b i n a r y d i s j u n c t i o n s . Journ. o f Symb. Logic 39, 412-413.

AV

~(0,4)

-

formulae w i t h

Rodding, D., Schwichtenberg, H. (1972): Bemerkungen zum Spektralproblem. Z e i t s c h r i f t f u r math. Logik und Grundlagen der Mathematik, 18, pp. 1-12 S c a r p e l l i n i , B. (1982): Constructive q u a n t i f i e r e l i m i n a t i o n i n Presburger A r i t h metic. Manuscript, U n i v e r s i t y o f Basel. Scholz, H e i n r i c h (1952): E i n ungelostes Problem i n der symbolischen Logik. i n : The Journal o f Symbolic Logic 17, pp. 160. Schutte, K u r t (1933) : Untersuchungen zum Entscheidungsproblem der mathematischen Logik, Mathematische Annalen 109, pp. 572-603. Schutte, K u r t (1934): Ober d i e E r f u l l b a r k e i t e i n e r Klasse von logischen Formeln. i n : Plathematische Annalen 110, pp. 161-194. Selman, A.L. (1970): A r i t h m e t i c a l r e d u c i b i l i t i e s and s e t s o f formulas v a l i d i n f i n i t e c a r d i n a l i t y . Ph.0. Thesis, Pennsylvania S t a t e Univer., U n i v e r s i t y Park, Pa. Selman, A.L. (1973): Sets o f formulas v a l i d i n f i n i t e s t r u c t u r e s . i n : Transactions AMS 177, pp. 491-504. Selman, A.L. (1974): R e l a t i v i z e d h a l t i n g problems. i n : Z e i t s c h r . f . math. Logik und Grundlagen d. Math., 20, pp. 193-198. Sevjakov, V.S. (1973): Formulas o f t h e r e s t r i c t e d p r e d i c a t e c a l c u l u s which d i s t i n g u i s h c e r t a i n classes o f models w i t h simply computable predicates. i n : Soviet Nath. Dokl. 14, 743.745. Shelah, S. (1977): D e c i d a b i l i t y o f a p o r t i o n o f t h e p r e d i c a t e c a l c u l u s . i n : I s r a e l J. Math. 28, pp. 32-44.

Decision Problems in Predicate Logic

30 1

Shepherdson, J.C. (1965): Machine configuration and word problems of given degree of unsolvability. Z. Math. Logik Grundlagen Math. 11, 149-175. Specker, E. & Strassen, V. (1976): Komplexitat von Entscheidungsproblemen. Springer Lecture Notes in Computer Science Vol. 43. Suranyi, J. (1959): Reduktionstheorie des Entscheidungsproblems im PradikatenkalkU1 der ersten Stufe. Verlag der Ungarischen Akademie der Wissenschaften, Budapest Trachtenbrot, B.A. /1950): Im ossibility of an a1 orithm for the decision problem in finite classes. in: Dokl. /$kad. Nauk SSSR70, f950, pp. 569-572. English translation in: AMS Transl. Ser. 2 Vol. 23 (1963) pp. 1-5. Trachtenbrot, B.A. (1953): 0 recursivno otdelimosti. in: Dokl. Akad.. Nauk SSS R 88, p p . 953-955.

Turing, Alan M. (1937): On computable numbers, with an application to the Entscheidungsproblem. Proceedinqs of the London Mathematical Society, 2nd series 42, pp. 230-265; correction ibid., 43, pp. 544-546. Wang, Hao (1962): Dominoes and the AEA case of the decision problem. Proceedings of a Symposium on the Mathematical Theory of Automata, Polytechnic Institute of Brooklyn, New York, pp. 23-55. Uirsing, M. (1977): Das Entscheidungsproblem der Klasse von Formeln, die hochstens zwei Primformeln enthal ten. manuscripta math. 22, 13-25. Wirsing, Martin (1978): A proof by Turing machines o f the undecidability o f the class of first order formulas with only one quantifier. Manuscript, Techn. Universitat Munchen, pp. 7. Yasuhara, M. (1971): On a problem o f Mostowski on finite spectra. in: Zeitschr. f. math. Logik und Grundlagen d. Math. 17, pp. 17-20.

LOGIC COLLOQUIUM '82 G . LON< G. Long0 and A. Marcia (editors) 0 Elsevier Science Publishers B. V. (North-Holland), 1984

MODEL THEORETIC ISSUES IN THEORETICAL COWUTER SCIENCE, PART I: RELATIONAL DATA BASES AND ABSTRACT DATA TYPES.

J.A.Makowsky Department of Computer Science, Technion - Israel Institute of Technology, Haifa, Israel

Table of Contents: Introduction. 1. Abstract model theory and computer science. 1.1 From syntax to semantics and back. 1.2Finding axioms. 1.3Comparing logics.

2.Data base theory. 2.1. What it is all about. 2.2.Safety (definiteness,domain independence). 2.3.Typed dependencies. 2.4.Implicational dependencies. 2.5. Decision problems. 2.6. Query languages. 2.7.Conclusions and some open problems. 3. Specification of abstract data types. 3.1.Introduction. 3.2. The axiomatic framework. 3.3. A complete specification language for rich semantical systems. 3.4. Typical models and initial algebras. 3.5. A complete language for semantic systems which admit initial semantics. 3.6. Relevance for specification of abstract data types. 3.7.A word on other applications.

Supported by t h e Swiss National Science Foundation Grant No. 82.820.0.80

303

304

J.A. MAKOWSKY

Introduction The following paper is an account of experiences I had in several attempts to capture problems posed in computer science, or more precisely by computer scientists. In most of the cases the computer scientist already thought that logic might help in stating problems more precisely, and ultimately, also in solving them, though they were usually suspicious about the impact such solutions would have on their direct practical involvement with programming, program analysis, program design or program veriflcation. Maybe a word on impact of foundational studies on applied science and engineering is needed here: Most electricians are not aware how much 19th century physics has contributed in making the portability of electrical appliances possible. The fact that two or three numeric parameters (voltage, power and the number of cycles in alternating current) contain all the information needed to decide, whether a given appliance can be used by plugging it into a given outlet, has become too common in every day life to be reflected upon. To remember that many years of research were needed to clarify this situation, is by now safely forgotten. Ultimately the problems posed by computer scientist to logicians, or for that matter to any one willing to spend time on foundational questions, is similar: What are the parameters needed t o e m r e portability and reliability of software? Needless to say, we are still very far from satisfactory answers. The current progress in technology even prevents computer manufacturers from reaching agreements, a s they were reached rather quickly, say, by manufacturerers of phonographs and, to some extent, videotape systems, on speed and size of the records (tapes) to be produced. But a deeper reason behind the problem consists in the absence of a definite model of the real world, here the programming environment. Though models of "computability" have been sufficiently clarified for deterministic sequential algorithms, provided their task can be unambiguously specified in some form of "natural scientific language", it is much less clear what "specipcation", "implementation" and "correctness" should mean. The problems involved are not exclusively problems of computer science. Any large scale design and implementation of a big organisational complex, from industrial to social engineering, touches upon the same fundamental questions. The only difference with computers stems from the fact, that they execute programs very quickly, and programs, which are used only once or a few times, become very soon obsolete. But we are generally inclined to expect that the time needed t o develop a program stays within proportions to the time i t runs and remains useful. This leads some to think, that also the foundational questions can be solved quickly. However, a short glance a t the history of mathematics shows us, that something like fifty years were spent till the basic notions of, say, point set topology were safely

Model Theoretic Issues in Theoretical Computer Science

305

established and had their impact not only on mathematicians but also physicists and practitioners of applied statistics. In the following paper I will try to explain how I have learned to view certain problems in the foundations of computer science and I will do this with three apparently different aspects of current computer science research: Data base theory, algebraic specification of abstract data types and algorithmic logic. As I will try to show, they have much more in common than widely believed, a t least when looked a t from the point of view of " a b s t r a c t m o d e l theory". But I a m fully aware that the challenge of applied science is not met by declaring that some ready made theory captures all its problems. This is never so. The difference between pure and applied, say, in differential equations, consists rather in the motivation of the results than in the results itself. The pure mathematician is content with knowledge which contributes to his understanding of the internal problems of differential equations as such, and solving a particular one is seen by him as a challenge of his general understanding. To the physicist, most of the work actually consists in justifying his particular differential equation, its parameters and its solutions in terms of his physical problem, and a large part of the work of good applied mathematics consists in reaching an understanding between the two perspectives. In computer science, we meet the same situation and both sides are often tempted to underestimate the work involved in Listening t o e a c h o t h e r . This paper is also an attempt to illustrate this work. For reasons of space (in this proceedings) and time (the Damocles sword of the deadlines), the paper had t o be cut into two parts. The first deals with data base theory and specification of abstract data types, and the second one with various approaches to semantics of programming languages. The first part also includes a general introduction on abstract model theory and its potential use in theoretical computer science. The second part [Makowsky 19831 will also include a n expository chapter of some more technical parts of abstract model theory. Here is an outline of both parts: In chapter 1 we t r y to give a description of what abstract model theory is all about, and how it is connected to the fundamental questions of computer science cited above. In chapter 2 we t r y to exemplify this in the case of data base theory. As it turns out there are various intimate connections between f r n i t e m o d e l t h e o r y and data base theory, which have led people to think that either data base theory is just undergraduate logic or that the logicians try to sell i t as such. But the real problems in any applied science are neither defined by their mathematical difficulty nor by the methodologies used to solve them, but rather by the questions they try to answer. Data base theory tries to answer the questions about design, design criteria, optimization and specification of data bases and their queries. This chapter could not have been written without the patience of C.Beeri, M.Vardi and A.Zvieli.

306

J.A. MAKOWSKY

In chapter 3 we turn our attention to the problem of specifying abstract data types. There are various approaches to this problem, but the most successful, a t least in terms of fashion, is the one called algebraic. Here w e flnd an i n t e r e s t i n g i n t e r p l a y b e t w e e n category theory and universal algebra, which has led people to think that this is just disguised "abstract nonsense", but again, as above, the problem we address here is the clarification of concepts such as data type, implementation, specification, modular programming and their ramifications and only a close, sympathetic analysis of these problems can lead to satisfactory answers. The results reported here are joint work with B.Mahr. As it turns out, there is a common theme in these last two chapters: In both u n i v e r s a l Horn formulas play a n eminent role. In the last section of chapter 3 we try to give some explanation of this phenomenon. It seems to support some of the arguments put foreward by the proponents of logic p r o g r a m m i n g as the programming style appropriate for the fifth generation computers. However, some recent complexity results, such a s [Itai-Makowsky 19821 still nourish some scepticism with respect to the unrestricted use programming languages like PROLOG. The remaining two chapters form part two ([Makowsky 19831): In chapter 4 we turn to the technical parts of abstract model theory a s we see them fit the needs of various branches of program semantics and program verification. We attempt to give a general definitjon of predicate transformers, as they appear in the context of program correctness. The definition is parallel t o the definition of generalized quantifiers, which will turn out to be a special case. On the basis of a set of predicate transformers one can build various algorithmic logics, of which again the classical examples of dynamic logic, process logic and others are special cases. In chapter 5 we turn our attention to program correctness and programming logics. We use the various logics from the previous chapter to introduce a new type of semantics, which in contrast to operational or denotational semantics, maps programming languages into subsets of logics. Here the meaning of a program is the set of all statements in a predicate transformer logic which are true about it. This is clearly not new as such, but has never been defined in a general context. One of the advantages of such a general approach is, that this allows us also to compare various approaches to program semantics which hitherto were considered incomparable. This last chapter is to report about work which is still in progress, mainly in collaboration with N.Francez and S.Katz. I would like to thank the Swiss National Science Foundation, who supported me generously during the two years in which the material presented came into being. I would like to thank also E.Engeler and E.Shamir who encouraged me to look into foundational problems in Computer Science and to C.Beeri, A.Meyer and V.Pratt. whose interest and criticism in early stages of the work was extremely stimulating. I

Model Theoretic Issues in Theoretical Computer Science

307

would also like 'to thank R.Fagin and J.Thatcher, who read and commented the almost final version. A last note on the references: There are over one hundred titles listed as I saw them fit my presentation. I did not try to give historic remarks, nor did I attempt completeness. I tried t o give the reader pointers to the vast literature, a s I came accross it during my random walk in the world of theoretical computer science. It serves as a basis for further backtracking: The transitive closure of this reference list surely covers much.

1. Abstract model theory and computer science. 1.1 From syntax to semantics and back. In the early days of modern logic, logic was perceived mainly syntactically. Propositional logic, first order logic and second order logic were given as formal languages and a main topic of research was the study of deduction rules and proof systems. There are various philosophical, sociological and even political reasons for this, usually subsumed under the name "Hilbert's program". G.Kreise1 has written extensively about Hilbert's program and the way it failed. From his analysis in [Kreisel 1968,19701 he drew several conclusions relevant for computer science which inspired the theses of R.Statman [Statman 19743 and C.Goad [Goad 19801. The former added a new dimensions to our understanding of the complexity of proofs and the latter used his experience gained in proof theory to speed up the synthesis of special purpose programs for hidden surface elimination, [Goad 19821, Hilbert's program wanted to reduce mathematics, and therefore all exact sciences, to the formal (or, as we would say today: algorithmic) manipulations of symbols. The ultimate hope behind this was, to find general purpose algorithms, which would solve all formally stated problems. A s we know today, Godel showed that this is impossible. But a t the same time modern semantics was born. The fashion had changed, and instead of the "God given" Natural Numbers, Tarski and his contemporaries moved to accept naive set theory as the basis of mathematics and proposed to explain logic in terms of set theory. The meaning of logical formulas was explained in terms of structures, relations, functions and in the case of first order logic this was justified by the celebrated conipleteness theorem. One of the corollaries of the completeness theorem is the compactness theorem, which was extended to uncountable sets by Mal'cev in 1936 and independently by Henkin in 1949. Among the many consequences of the compactness theorem is the existence of various "non-standard" models of arithmetic and analysis, which later led to a very fruitful branch of logic called non-standard analysis, which was first pursued by

308

J.A. MAKOWSKY

A.Robinson and led to various impressive results in analysis, Banach space theory, the theory of Brownian motion and even mathematical economics. But one of the first non-trivial applications of the compactness theorem was the characterization of the universal first order formulas in terms of an algebraic preservation property: A first order formula cp is logically equivalent to a universal formula if and only if cp is preserved under substructures. This result was followed by a n intensive program exploring the relationship between semantic properties of formulas and syntactic characterizations of such formulas. Motivated by algebraic practice the notion of substructure was replaced successively by unions of chains, products, reduced products, factors and many others and sufficient experience was gained to delegate this direction of research to the level of master theses and difficult exercises. The mathematical tools used to solve such problem, if there is a clean solution, usually are interpolation theorems, ultraproducts and Back and Forth arguments. In [Chang-Keisler 19731 the reader may find what ever is known in this direction. But there is another way in looking a t this program: We could reverse the problem and start with any syntactically defined class of formulas together with their meaning functions and ask for a characterization of this class in terms of the preservation properties it has. Looked a t it this way, what we really are asking for is giving meaning to syntactic categories. It is this aspect of preservation theorems which I think is relevant to foundational questions in computer science. Very often the computer scientists start with syntactic restrictions and later try very hard to remove them, without understanding their significance. But, a s will be shown in chapters 2 and 3, those restrictions, originally imposed for technical reasons, can be characterized by preservation properties, which show that they are intimately connected with the implicit assumptions the computer scientists have made. The use of powerful set theoretic methods led also to another development. Already in the fifties Mostowski in Warsaw, Engeler in Zurich and Tarski and Henkin in Berkeley started to look a t various generalizations of first order logic involving infinitary constructs and generalized quantifiers and a n abundancy of logics appeared. It was Engeler, however, who first noticed the possible relevance of infinitary logics to computer science([Engeler 1967,1970]).This has since led to the development of dynamic logic, and we shall return to this topic in chapter 4 and 5 of this paper. In the sixties much of model theory was generalized to those newly discovered logics, and, based on earlier work by Mostowski, Lindstrom defined an axiomatic framework, sometimes called abstract model theory or higher model theory, in which we can study logics in general, compare their expressive power and prove characterization theorems for logics in terms of their model theoretic properties. The latter has very striking parallels with the above mentioned preservation theorems both in content a s well as in methodology. In the rest of this chapter we shall briefly describe this framework and give some key

Model Theoretic Issues in Theoretical Computer Science

309

results, which we will apply in chapter 4 and 5 to the study of various dynamic logics. An account of the state of art in abstract model theory can be found in the forthcoming book [Banvise-Feferman 19831, and an introductory survey in [Flum 19751. The purpose of abstract model theory can be summarized as follows: We want to be able to quantify over all possible logics satisfying certain properties and prove theorems about them. The theorems we want to prove can be (A) characterization theorems. (B) presentation theorems. (C) theorems relating various properties of logics. Examples for (A) are the Lindstrorn theorems, ([Lindstrom 1969]), characterizing first order logic in terms of the Lowenheim-Skolem theorem together with various properties such a s compactness or axiomatizabilty. The latter has been closely analyzed for its usefulness in computer science in [Manders-Daley 19831 and in [Makowsky 19801. Examples for (B)are Birkhoff's theorem characterizing the varieties a s the equationally definable classes of algebras (cf.[Graetzer 1979]), Mal'cev's characterization of the quasi-varieties (cf. [Mal'cev 19711) and Cudnovskii's theorem that every class of structures closed under substructures can be axiomatized by a class of infinitary clauses (cf. theorem 1 in chapter 3 and [Cudnovskii 19681).In some sense the result in [Meyer-Parikh 19811, showing that most dynamic logics for finitely branching programs can be embedded in the recursive part of countably infinite logic, also fits this category. An example for (C), finally, is that axiomatizability implies recursive compactness or that for countable logics the amalgamation property is equivalent to compactness. The former is a corollary of the Lindstrom theorems and the latter is in [Makowsky-Shelah 19831. The use of abstract model theory lies in its limitative character. It tells us that certain requirements are incompatible o r entail other limitations. It can tell us to what extent seemingly different approaches are nevertheless the same. Or it can give is a framework in which we can precisely compare concepts which hitherto appeared incomparable. 1.2 Finding axioms. Our first problem is to find axioms for logics. Logics will consist of quadruples L = ( T , % r , F l n l , i= ), where T is a class of v o c a b u l a r i e s or s i g n a t u r e s and Str is a function mapping every T E T into a subclass * ( T ) of all structures of vocabulary T such that if T,T' E T,T c T' then S t r ( ~c) S t 7 ( ~ ' ) . Here we assume that in T we have a partial order denoted by c . In all the cases we consider the elements of T are sets of symbols and c just is the subset relation. The usage of the term v o c a b u l a r y for what is called similarity type ( o r signature or even language) seems to capture what we really have in mind. The vocabulary is the most elementary part of logic, and it determines about what we will talk. In the case of first order logic it consists just of sets of relations symbols, function symbols or constant symbols,

310

J.A. MAKOWSKY

together with their arities, and in the case of many sorted logic, with their sort specifications. In other cases it may also specify variables to be second order, or make certain distinctions between logic with or without equality, or other special symbols. Sometimes it is convenient to think of T as a c a t e g o r y of v o c a b u l a r i e s rather than just a set or a class. This is especially the case when want to consider vocabularies which are more complicated than usually and we have to define the partial order on T in a more complex way. But these cases are still not very well developed. Now given T,the class % ( T ) tells us which structures of vocabulary 7 we are interested in. This may, in logic, often comprise all the 7-structures, but in applications we very often impose restrictions. In chapter 2 we shall see that for data base theory we only consider finite structures, or finite reducts of expansions of the standard model of arithmetic, and in chapter three only certain countable structures, the reachable structures are of interest. Finally in chapters 4 and 5 various more complicated structures will enter the picture, including certain models of tense logic, probability logic etc. Again it may be convenient to think of % ( T ) as subcategories of a big category S r u c t = u S ~ T ( Tand ) of a functor T E

T

mapping T into S t m c t . We refer the reader to [Barwise 19741 for a detailed presentation. Also i+ni is a function which maps every T E T into a set of objects called f o n n t d a s . Again we require that for T , T ' E T , T C T ' we have that M ( Tc)M ( 7 ' ) .When choosing the set of formulas, we have to bear in mind to contradicting aspects: We want to say much about our structures, certainly as much as we need in our particular context. But we do not want to say too much, because we want to keep our model theory out of the difficulties of full second order logic. Finally I= is a relation on % ( T ) x ~ ( T ) ,which satisfies certain axioms: IsomOrphismAxiom If A B E ST(T),(P E M(T and ) AEB then At= iff BI= rp. ReductAxiomlf ( p ~ m n l ( ~ ) , ~ c ~ ' a n d thenAl=rpiff A~St~(~A ' )r r l = ( p . Renaming Axiom Let U,T E T and p : +~u be a r e n a m i n g , i.e. an isomor) is phism in the category of vocabularies. Then for each p E M ( Tthere ++ E Rd (u) such that for all A E S ~ T ( Twe ) have that Al= rp iff MI= p". Those axioms do not require too much. All examples which we shall encounter in this paper satisfy them. From the theorems in abstract model theory cited in the previous section, however, only example (B) can be proved with these axioms alone. In chapter 3 we shall use similar axioms to axiomatize the behaviour of sets of formulas rather than formulas. What makes abstract model theory into a theory are various additional closure properties, which we impose on the formulas, or rather on their models Mod,(rp)=IA E S t r ( ~ ) : A l =pi. Many model theoretic properties of various logics can be stated by only referring to the model classes Mod,((p) definable by their formulas. The compactness theorem and the Lowenheim-Skolem theorem are among them, and also various

Model Theoretic Issues in Theoretical Computer Science

31 1

definability theorems. But here are some of the closure axioms: Atomic Axiom For every T E T the usual r-atomic formulas are contained ill M

( T ) .

Basic Axiom For every

T E T the usual 7-basic formulas (i.e. atomic and negated atomic formulas) are contained in M ( T ) . Boolean Axiom Ftnl(7) is closed under the boolean operations A , V , .. with their usual meaning, i.e. heir model classes are defined by intersection, union and complement respectively. QuantificationAxiom h z i ( T ) is closed under existential quantification 4 z with its usual meaning. The next two axioms assume some knowledge of the structure of the formulas if we want to state them naturally. They are the relativizatian axii~rnand the substitution axiom. We will discuss them in more detail in chapter 4. For the first part of the paper their exact definition is irrelevant. Examples of logics are first order logic, infinitary logics, logics with generalized quantifiers etc. All the classes of dependencies in chapter 2 can be viewed a s logics (though without all the closure properties) and in some sense also the s e m a n t i c a l systems of chapter 3. Behind the choice of closure properties lies the problem of iteration under various formation rules for formulas, in other words the choice of primitives for our logics. Already in the early days of infinitary logics did fieisel point out in [Kreisel 19681 that we need definability criteria to evaluate such choices, rather than just adding various constructs ad libitum. He advocated a line of research which not only led to unifying results for infinitary logics and generalized recursion theory but also to a deeper insight in general. It led to the very rich theory of admissible sets, as presented in [Barwise 19751. The corresponding problem we face in computer science has not even been formulated generally. In data base theory only [Chandra 1901] questions the choice of programming primitives and [Chandra-Hare1 19801 define general criteria for query languages. We shall study the latter in section 2.7. and show that in this case "Kreisel's program" can be followed to a large extent. For specifications of abstract data types [Burstall-Goguen 19831 and [Makowsky-Mahr 19831 attack this problem In the second part of the paper we shall outline what can be done for semantics of programming languages, but we are still f a r from a general understanding. When we want to apply the framework of abstract model theory t o foundational problems in computer science, we observe quickly that what we hope to be logics are usually not closed under all the closure operations we have mentioned above. The striking example here is Hoare logic, which consists of statements about programs of a particular form, the correctness statements, but is not closed under any iteration or boolean operation. This leads t o an abundancy of logics which are hard t o compare, cf. [Meyer-Tiuryn 19811 and [Meyer 19801. The reasons for

312

J.A. MAKOWSKY

the absence of closure properties are sometimes not clear, but in other cases motivated by practical experience. Logics are usually closed under substitutions of predicate symbols by formulas, but in programming some formulas occur as test in programs, and those should remain simple, and it is clear that we do not allow termination statements of other programs to occur as test. The choice of the correct closure properties for logics affects the applicability of results from abstract model theory very much. Sometimes, however, the absence of closure properties can be compensated by a weakening of the theorems. OIten a theorem states that every formula p in a given logic satisfying certain conditions is equivalent to another formula in a different logic. In the absence of the Boolean axiom, this may be conveniently rephrased by stating that (p is equivalent to a boolean combination of such formulas. In more complicated cases, however, we have to allow the use of additional predicates. To be more precise we need some definitions: Definitions: (i) Let cp E M ( T u IRj) R E T be a formula and u c T . We say that (p deflnes R implicitly over u i f : Every u-structure A can be expanded to a T u [Rj-structure A’ such that A’ I=p and given two structures A,B E ~ T ( uT {I?{) with Al= (p,BI=(p such that At u = E l u, then RA=RB,i.e. R is uniquely determined by (p and u. (ii) Let p E M ( T [Rj) Ube a formula which defines R implicitly over u, R n-ary. We say that $(vl,vz,...,v,) E ~ ( U )with n free variables defines R explicitly, if for every A E ~ T ( T with ) Al= (p we have that Al= v 111.v~ ....,vn(R<=>$(v i,vz,...,

v,)).

With these definitions we can state a n even stronger closure property: A-closure Axiom: Every implicitly defined relation has an explicit definition. Examples: (i) (Many-sorted) first order logic satisfies all the closure axioms. The A-closure is a variant of Beth’s definability theorem first stated in [Feferman 19741. (ii) First order logic without function symbols and with all structures finite, as we shall use it for data base theory in chapter 2 , does not satisfy the A-closure axiom, a s pointed out in [Hajek 19761. (iii) Note that A-closure is a stronger property than closure under substitution. (iv) I t is this A-closure property which made various ideas of [Kreisel 19681 more precise. Kreisel’s work led to the definition of admissible sets, and H.Friedman showed a deep connection between A-closed logics and logics built on admissible sets, cf. [Makowsky-Shelah-Stavi 19761. (v) An interesting application of Beth’s theorem to data base decomposition problems may be found in [Vqrdi 19821. More examples will be studied in chapter 4.

Model Theoretic Issues in Theoretical Computer Science

313

1.3 Comparing logics. When we want to compare logics, we want to compare their expressive power, i.e. what subsets or relations of its structures are definable. We say that a logic L1 is reducible to a logic Lz if every formula of L , can be translated into a formula of Lz. Again our notion of comparability will depend on the various closure properties the logics in question have. For positive results we are usually interested in the highest possible degree of precision on the nature of this translation, whereas for negative results, on the contrary, we prefer ample freedom. Let us propose some definitions: Definitions: Let &=(7i,&t,ml,I= *) be logics for i=1.2. (i) L , is explicitly reducible to L,, if T , c Tz,for each T E T, Str2(r)c S T , ( T ) , and for every p E M , ( T there ) is 11 E M z ( r )such that MoG((o)n S t ~ ( ~ ) = M o d ~ ( l ( ' )

We write for this L , < L ~ . (ii) L1 is implicitly reducible to Lz, if TI c Tz, for each T E TI . % T ~ ( T )c str,(~),and for every implicit definition over T via c E i%i,(r') with T c T' there is an implicit definition over T via @ E F&(T') such that ~%d,~(p)n * z ( T ' ) 1 mod,(@) 1 T We write for this L , h , L 2 . (iii) We say that L , and Lz are explicitly (implicitly) equivalent, if both L I c L z and Lz
314

J.A. MAKOWSKY

2. Data Base Theory for the Relational Model. 2.1 Introduction One of the most frequent application of computers nowadays is in data bases. There are various ways of modeling data bases, such as the network model or the hierarchical model, but the most widely studied for theoretical purposes is the relational model. An excellent reference is the textbook [Ullman 19821. The entity-relationship model, cf. [Cheri 1976,1981] has not yet been really studied from a theoretical point of view. As much as I understand it, most of the theoretical results for the relational model carry over to the entity-relationship model, such as computable queries and dependency theory, but, to the best of my knowledge, no serious attempt has been undertaken to carry out such a task. In what follows we concentrate on the relational model. It consists of families of data base states, which are divided in acceptable or consistent states and inconsistent states. Those are distinguished by constraints or dependencies. The consistent states are the models of the dependencies. With data base states we can do two things: We can ask queries or we can perform transactions. To complicate matters this is usually done by many users a t the same time; we speak therefore of concurrent users. Transactions map consistent data base states into consistent data base states. They are usually decomposed into smaller operations which map consistent data base states sometimes into inconsistent data base states. There are two kinds of simple transactions: read only and write only. More complex transactions can be built from simple transactions by composition. Needless to say, all these operations should be computable. To sort out this mess a theory of transactions and concurrency control is in the making. The state of the art is described in [Date 19821, [Casanova 19811 and in the forthcoming book [Maier 19831. An excellent survey is [Bernstein-Goodman 19821. In this chapter we are only concerned with a special case of read only transactions, queries and dependencies. Neither general transactions nor concurrency control play a direct role. Indirectly, however, they serve as a motivation in our presentation of dependency theory. Queries map data base states into relations. In [Chandra-Hare1 19801 an abstract definition is given, the computable queries, which is the basis of our presentation in this chapter.

Data base states are structures like for first order logic, but for practical purposes some restriction are necessary. First of all, the structures are finite. Second, we distinguish between the relations representing the tuples in the data bases and the aggregate functions such as arithmetic operations or linear order on the entries. And third, we are not really interested in the underlying universes but only in the relations a s such. In this chapter we shall not talk about the aggregate functions a t all. To ensure that it makes no difference whether we talk about

Model Theoretic Issues in Theoretical Computer Science

315

relations or structures we introduce an invariance condition, called safety, which is discussed in section 2.2.

Dependencies are classes of data base states, usually the models of some first order sentences. They are grouped and classified according to syntactic criteria: Functional dependencies (FD), full implicational dependencies (FID), embedded implicational dependencies (EID), template dependencies (TD), multi valued dependencies (MVD) and embedded multi-valued dependencies (EMVD) can be conveniently described as classes of first order sentences with specific syntactic restrictions. In the sense of chapter 1, dependencies usually form a logic, with the vocabularies ranging over sets of relation symbols only and all the structures being finite. Queries are definable relations in this logic, and we shall see in section 2.6. that the knplicit definition play an important role here. Preservation theorems in logic are theorems which characterize classes of first order formulas having some semantic properties by showing that those sentences are exactly the ones which allow a special syntactic normal form. They are special cases of presentation theorems in the sense of chapter 1. But in contrast to universal normal form theorems (such as every first order formula is equivalent to a prenex formula) which usually are constructive, the normal form theorems coming from preservation theorems are often non-constructive. What we get is the following: The set of sentences Shaving some semantic property P is not recursive, but there is a recursive set So such that every sentence u E S is equivalent ( over some first order theory ) to a sentence u o ~ S O . Though the theorem is non-constructive this has two advantages: (i) We can, with no loss of generality, restrict ourselves - or for that matter the programmer of a data base system - to dependencies of the form So,and (ii) by doing so, we know that property P is a priori ensured. If the property p is one which is of intrinsic importance to our database system, then the restriction to sentences from So will free the programmer from the correctness proof - or rather - force him t d choose his dependencies carefully and prove then correctness before he is allowed to write them down.

Now in logic, the choice of the semantic properties P is usually given in a natural way, say from algebraic considerations , and the problem is to find So. In data base theory the situation is reversed: We are given various candidates s,, a s the FD, FID, EID, MVD, TD, EMVD etc, and the problem we pose, is to define the corresponding properties P which both characterize so’and a r e g e n u i n e l y m o t i v a t e d by data base considerat i o n s . Tt is our firm belief, that the syntactic restrictions given to various classes of dependencies are only meaningful iff they correspond to a semantic property which reflects d a t a base p r a c t i c e . And it is such a property which should be called the meaning of a syntactic restriction. What we show here is giving meaning to being safe, typed and being a

316

J.A. MAKOWSKY

typed implicational dependency. What we propose furthermore is a program which consists of searching for the meaning of various other syntactic definitions of dependencies such as template, embedded etc. Fagin went a good way to do this for typed embedded implicational dependencies [Fagin 19821 by showing that they are faithful (i.e true in a product of finite non-empty relations iff true in each factor) and my previous [Makowsky 19811 proposed several such characterization, but their relevance for data base practice was not yet satisfactorally shown. In a forthcoming paper [Makowsky-Vardi 19831 more such results are collected. Our main results here are: (i) The complete characterization of equality generating dependencies based on separable dependencies wich have the subrelation property and are preserved under products. (ii) The complete characterization of full typed tuple generating dependencies based on separable dependencies with the intersection property and the duplicate extension property. The intersection property had been previously characterized in [MaierMendelzon-Sagiv 19791 a s the property which guaranties the uniqueness of the completion operation in connection with the chase. Separability, however, is introduced here to give meaning to the restriction to typed formulas. It is discussed in detail in section 3 and captures the idea of separation of sorts, or attributes. This chapter is organized a s follows: In section 2.2 we discuss a well known example from the above point of view, the definite formulas from [Kuhns 19691 and their syntactic characterization a s permissible f o n n d a s , as described by many authors, e.g. [Cooper. 19801, or as safe formulas ,as described in Ullman’s book [Ullman 19821. We also note that the definite formulas are not recursively enumerable, as was shown by [Di Paola 19691. In section 2.3 we follow the same pattern to propose a semantic characterization of typed, formulas. We also show that this class is not recursively enumerable. The results in this section are drawn from [Makowsky-Vardi 19831. In section 2.4 we discuss FID’s and FD’s and connect the typed FID‘s (TFID) to the intersection property of relations. The results here are continuations of our previous work [Makowsky 19811. We end our presentation with some final remarks and open problems. In section 2.5 we discuss decidability and complexity results for the consequence problem for various classes of dependencies. In section 2.6 we give a brief presentation of the theory of computable queries and in section 2.7 we draw some conclusions and present some more open problems.

Model Theoretic Issues in Theoretical Computzr Science

317

2.2. Safety (def3niteness.domai.n independence). Already in 1967 [Kuhns 19671 it was realized that first order formulas, which are relevant for data base dependencies or queries, should satisfy an invariance condition. Intuitively this condition says that it does not matter if we speak of a relation or of a first order structure containing this relation. Definitions: Let R be a finite relation on and let A = c A , , A 2 , ' ' ,&,R>

n,A(

be the corresponding relational structure. Let A * be the relational structure obtained from A by addition of exactly one element 4 to every sort A( and not extending R . Kuhns calls the formulas u which are true in A iff they are true in A* definite. A class r of data base states is called definite if it is closed under the formation of A*. Fagin [Fagin 19821 independently looked a t this property and called it &main independence. Let S denote the class of definite first order formulas, and 9 be the class of definite formulas with a t most k a o free variables. Di Paola showed Theorem 1: 9 is not recursively enumerable for any kro. For a proof one may also consult [Vardi 19811. Note that if we allow infinite relations, we only get that s* is not recursive. Here we have a non-recursive set of formulas S and we would like to find a recursive set So such that every formula u E S is equivalent to a formula uo E s,. Let So be the set of safe formulas a s in Ullman's book [Ullman 19821 ( or equivalently the set of permissible formulas from [Cooper 19801 ). If we allow infinite relations, it follows easily from results in model theory that every formula of S is equivalent to a formula in So. In fact we have even more: Theorem 2: Let C be a first order theory. Call a formula C -definite if i t is definite on the class of finite models of 8. Then the following are equivalent: (i) u is C-definite and (ii) In all finite models of C is u is equivalent to a formula in So. For the proof we define a n algorithm based on r e l a t i u i z a t i o n , which maps arbitrary first order formlas into safe formulas and which preserves equivalence (for models of C) if and only if the original formula was safe. This does not contradict theorem 1, since it merely says that the set of first order formulas, on which this algorithm does preserve equivalence, is not recursive. Formulas with 'free variables define relations. For first order formulas this gives us a special case of first order (explicitly) definable queries. The definition of definite is naturally extended to this case. We will return to definite formulas in the section on query languages.

318

J.A. MAKOWSKY

2.3.Typed dependencies. In this section we look at the class of typed and s a f e formulas, which we denote by To. Clearly this a recursive set of first order formulas. We propose to define a n operation on finite relations, which intuitively corresponds to the introduction of different attributes (sorts) for the arguments of the relation. Let R c A n be a finite n-ary relation over some domain A . Let rri(R) be the i-th projection of R onto A . and q=ni(R)x[i] We define a new relation R on in the following way: ((a1,1).(az,2) ,...,(%,TI)) E R if7 ( a l , a z,...,u,,) E R .

n& i

We say that a first order formula u admits separation of attributes (sorts) or, shortly, is separable, if u is true about R iff it is true about R. A class of data base states r is called separable if it is closed under the formation of R. Remarks: (i) If all the rr,(R) are disjoint then R is isomorphic to R . (ii) Using (i) we see that separable formulas are definite. This is due to our definition of A, which is a projection. Had we defined it just to be a new copy of A , the results below had to be slightly modified. (iii) Functional dependencies are separable. Let T denote the class of separable first order formulas, and P be the class of separable formulas with a t most k r o free variables. Using a similer argument as in [Vardi 19811 one gets: Theorem 3: P is not recursively enumerable for any kto.

Problem: Is the class of EID’s which are in T recursive ? That separable formulas really capture the separation of attributes (sorts) is shown in the following theorem: Theorem 4: Let Z be a first order theory. Call a formula E-separable if it is separable on the class of finite models of X . Then the following are equivalent: (i) u is X-separable and (ii) X proves that u is equivalent to a formula in To. The proof is similar to the proof of theorem 2. 2.4. Implicational Dependencies. We are now in a position to define more classes of dependencies: Definitions: (i) A first order formula over a set T of relation symbols is a full implicational dependency (FID),if it is of the form Yf Aib,(z)

4

b (z)

where each bi is an atomic formula not containing the equality symbol, b is atomic possibly containing equality and each variable which occurs in b also occurs in some b,. Note that we do not allow the empty conjunction.

Model Theoretic Issues in Theoretical Computer Science

319

If

b is an equality we also speak of equality generating dependencies (EGD), and if b is an instance of a relation symbol we speak of tuple generating dependencies (TGD). The functional dependencies (FD)are the EGD’swith only two hi's. (ii) The classes TFID of typed full implicational dependencies, typed tuple generating dependencies TTGD and typed equality generating dependencies TEGD are defined analoguously. (iii) The class of embedded implicational dependencies EID, consists of first order formulas of the form VZ A+@)

+

gV

A;j(T.V)

where the hi's are a s for the FID and the ci’s are atomic with all the variables from d occurring already in the b i g s . (iv) The class of embedded template dependencies ETD8 consists of the EID’s with only one formula cr, which is not a n equality. In contrast to. some papers in the literature we allow EID’s to be untyped. A special case of template dependencies are the inclusion dependencies IND, where there is also only one formula b i . (v) The classes TFID,TEID,TETD,TINDof typed embedded dependencies are defined similerily. An important subclass of TID are the Functional Dependencies FD. Let X be a set of first order formulas and E(X) denote the set of first order formulas which are equivalent to some formula in X. The followihg is a useful observation: Proposition 5: (Beeri and Vardi) Every typed full implicational dependency is equivalent to a conjunction of a TGD and a EGD. As was observed by Vardi and the author we have Theorem 6:Both E(FID)and E(TGD) are not recursive. Neither is E(FD).

A proof, due to Vardi, may be found in [Makowsky 19811. I t could also be proved using methods similar to [McNulty 19791. U n d e r what conditions can we axiomatize classes r o f data base states by dependencies of prescribed syntactic form ? Let us look first at

TFID’S.Clearly they are again definite and separable. They also are preserved under Cartesian products (the p r o d m t property). Given u E FD and a relation R and a subrelation Ro c R then u is true about R iff it is true about Ro. (This is not generally true for FID.) Let us call this last property the subrelation property, both a s a preservation property for formulas u as well as a closure property for classes of data base states r. The subrelation property is very strong and dependencies which satisfy it are invariant under losing any portion of your data bases. Its integrity can not be destroyed by deleting data. Note that the subrelation property is stronger than the substmcture property in model theory, because here we really take subsets of the relation, whereas in model theory we take subsets of the domains and consider the relation naturally induced on them. The substructure property is true for FfD..h

320

J.A. MAKOWSKY

fact we have ([Makowsky-Vardi19833): Theorem 7: A class r of data base states, closed under isomorphisms. is axiomatizable by a set of typed equality generating dependencies TEGD i f fr i s (i) separable, (ii) has the subrelation property and (iii) contains the trivial structure and (iv) is closed under products. The triwial s b t u r e , is the structure which has exactly one element of each sort and all the tuples satisfy all the relations. Since the compactness theorem is not true, if we only consider finite structures, theorem 7 can not be stated, like theorem 2 and 4, for single formulas. Similarily we can define the intersectionproperty, which requires that if u is true in two relations Rl,R2 then it is also true in RI n Ra. Again we have two versions of it, one as a preservation property and the other as a closure property for classes of structures. Clearly the subrelation property implies the intersection property, but the intersection property s e e m more natural: Not every subset of a library catalogue is necessarily a catalogue, but we definitely expect the intersection of two catalogues t o be a catalogue. The intersection property is true for TGD and for FD but not for FID. Proposition 8: If a separable formula has the intersection property, then it has the substructure property (but not necessarily the subrelation property). The proof is purely semantical and uses the fact that we can represent every substructure as the intersection of two relations by renaming. A last such property we want to consider is preservation (closure) under duplicate eztensions. This is like logic without equality, i.e. we allow multiple occurrence of elenients. More precisely, let a E A , b L A and h be a mapping such that it is the identity on A+] and h ( a ) = b . We have a natural extension of h to R. Now C A u f b 1,R u h(R)> is a duplicate extension of cA.R>. With this we have ([Makowsky-Vardi 19831): Theorem 9: A class r of data base states, closed under isomorphisms, is axiomatizable by a set of tuple generating dependencies TGD iff r is (i) definite (ii) closed under duplicate extensions, (iii) the intersection property and (iv) contains the trivial structure. For typed dependencies we have the following analogue to theorem 9, also from [Makowsky-Vardi 19831. Similar theorems can also be stated for the other cases. Theorem 1 0 Let E be a set of first order formulas such that (i) E is true in the trivial structure, (ii) is separable,

Model Theoretic Issues in Theoretical Computer Science

32 1

(iii) has the intersection property and (iv) preserves duplicating extensions. Then C is equivalent to a set of typed tuple generating dependencies TTGD'S.

All the properties above but the closure under products have natural justifications in terms of data base practice. We had previously characterized TFfD in terms of the Armstrong property (cf. [Fagin 1982, Makowsky 19811 and a strong form of the finite model property [Makowsky 19811,but theorem 10 seems more natural. The Armstrong property is version of the weak generic structures, a s dealt with in chapter 3, adapted t o database theory, The finite model property in question is related to the class of securable formulas as defined in section 2.5. 2.5. The Consequence Problem

For various classes of dependencies the consequence problem has been studied. In general it is stated as follows: Given a finite set C of dependencies in D , and a single dependency o F D , can we decide whether o is true in all (finite) relations satisfying 2? This is closely related to the consequence problems in logic, with the difference that here we are mainly interested in finite models over relation symbols only and that the class of formulas D , is of very low quantifier rank. Additionally, if we look a t typed dependencies, we can not use variables repeatedly in different positions. Though a logician would expect undecidability results, if the formulas involve both existential and universal quantifiers, there is still place for many decidable subcases. Clearly if the dependencies are boolean combinations of purely universal and purely existential formulas, the consequence problem is decidable ([Bernays-Schonfinkel 19281,see also [Lewis 1979,1980]).This class of formulas has also been extensively studied in model theory (cf.[Tharp 19741 and [Makowsky 1975]), and they were called securable or c o n t i n u o u s formulas. They have many nice properties: Proposition 11: Let s be the class of securable formulas. Then (i) S is closed under boolean operations and (ii) The class of valid and of finitely valid formulas in S coincide. Clearly we get from this that the consequence problem for securable formulas is decidable. In fact, the exact complexity of this consequence problem is known. The reader not familiar with complexity classes should consult [Garey-Johnson 19791. Theorem 1 2 ([Lewis 19801)) There are constants c > d > l such that the consequence problem for securable formulas without function symbols or equality can be solved in N71AfE(cn)but not in NTIME(dn). Securable formulas have the finite model property and are closed under boolean operations. This leads us t o the following problem: Problem: Do the properties (i) and (ii) proposition 1 1 characterize S up t o logical equivalence, i.e. given s satisfying (i) and (ii), is i t true t h a t

322

J.A. MAKOWSKY

every formula in s' is equivalent to a formula in S ? If this is not the case, i s there such a maximal class for (i) and (ii), or what properties have to be added to get maximality? Note that by a folklore result in model theory [Shoenfield 1967, problem ~ O C p.971 , S is characterized by the fact that every formula (p E S both (p and -(p are preserved under unions of chains. Some applications of securable formulas in characterizing dependencies may be found in [Makowsky 19811, cf. also section 2.4. The first undecidability result for a class of dependencies, which was introduced previously in data base theory, appeared in [ChandraLewis-Makowsky 19811. There it is shown that the consequence problem over finite relations for typed embedded implicational dependencies TEID- is not even recursively enumerable. This has been later improved to the class of typed template dependencies TTD by [Vardi 19821 and [Gurevich-Lewis 19821. Theorem 13: ([Gurevich-Lewis 19831) Let C range over finite sets of TETD's and u over elements of TETD. Then the following two sets are effectively inseparable: (i) The pairs (E,u) such that E I= u ( C I= (ii) The pairs ( C p ) such that IT fails in some finite data base which satisfies C. I t follows immediately hat neither set is recursive and that there is no for TETD. recursive axiomatization for I= As we know from theorem 12, the consequence problem for FID is decidable in exponential time. The most successful algorithm for this is the CHASE introduced in [Maier-Mendelzon-Sagiv 19791. Some of its limitations are discussed in [Goodman-Shmueli 19811. Its popularity derives from the fact that it runs rather fast on interesting subclasses of FID. An abundancy of complexity results for such cases may be found in [Maier-Sagiv-Yannakakis 19811. However, for the general case of typed and untyped FID's we have:

Theorem 14: ([ Chandra-Lewis-Makowsky 19811) ~W) (i) The consequence problem for TFID can be solved in D T I M E ( C ~ / ~ but for some constants c . d > l . not in DTIME(d-) (ii) The consequence problem for FID can be solved in DTIME(cn) but not in DTIME(dn) for some constants c > d > l . These results apply in particular to the CHASE algorithm. It is sometimes argued, e.g. by V.Pratt, that simply exponential algorithms are suitable for computers. If we accept this we view we might argue that the FID's are the largest class of dependencies which are reasonable for data bases. Further evidence for this view stems from the fact that they were independently proposed in various disguised forms, e.g. in [Papadimitriou-Yannakakis 19821, [Fagin 19821, [Beeri-Vardi 19811 or [Paredaens 19821. The first three papers also introduce the EID's. A good source for the history is [Fagin 19821. The question which remains open till today, is whether every reasonable subclass of embedded

323

Model Theoretic Issues in Theoretical Computer Science

dependencies has an undecidable consequence problem. The hard case s e e m to be the following: Definition: A dependency is a embedded multivalued dependency if it is of the form 't/abibzci~zd~dz((P(a,bi.~~.di) A P ( a , b z . c z , d z ) )-a

4 dsP(e.b,,cz.d3)).

Here all the a , b , c . d ' s are vectors of variables. We do not enter here the discussion of the importance of multivalued dependencies. The reader is referred to [Fagin 19771 or [Ullman 19821. The reason we introduce them here is the following Problem: Is the consequence problem for embedded multivalued dependencies decidable ? To show how delicate such problems can be, let us look at the case of inclusion dependencies IND, which are a very special case of EID's where both the hypothesis and the conclusion have length one and no equality is allowed. The precise definition was stated in section 2.4. The following resumes what is known on the consequence problem for IND alone and for IND u FD.

Theorem 15: (i) [Casanova et al. 19821:The consequence problem for hVD is

PSPACE-

complete. (ii) [Mitchell 19831 and [Chandra-Vardi 19831:The consequence problem for IND u FD is undecidable. This is an example where a subset of EID which is not in F1D has a decidable consequence problem, but where a very small extension leads to an undecidable consequence problem. 2.6. QueryLanguages. A query q of type 7 , where

7=s0 u [I?,,] is the similerity type of a class of data base states augmented by a new n-ary relation symbol, is a function which maps states(r,,) into n-ary relations on the domain of these states. Since both states and relations are finite objects, we can code them in arithmetic and it makes sense to require that (i) this function q is a parEial recursive function on these codes, i.e. there exists a Turing machine T&, which computes the query on the codes. On the other hand we do not want this function to be dependent on the particular codes, so we require also that (ii) if two data base states D1,Da are isomorphic, i.e. D,EDz, so

P (DiIrq

(4).

In [Chandra-Hare1 19801 queries satisfying condition (i) and (ii) are called computable queries. They argue convincingly that every reasonable query should be computable and say that a query language Q is complete i f every query in Q is computable and for every function satisfying (i) and (ii) there is a term q E Q representing it. They also

324

J.A. MAKOWSKY

construct a complete query language QL which is based on relations only, i.e. without aggregate functions, but which leaves the arity of the relation symbol dynamic. QL also has wtijkstaternent built in. Query languages actually used in practice or studied in the literature are not complete. However, two such languages SQL and QBE can be completed, without violating their main design principles. In [MakowskyZvieli 19831 we show how to complete those two languages with static arity of R and by only adding a recursive insed-procedure, rather than a while-statement. The basic idea behind this comes from realizing that there is a n alternative, more model theoretic, definition of the computable queries. Let us look a t the graph of the above function q . It is given by a class of structures of type T such that (iii) for every finite so-structure A there is a t most one, up to isomorphism, relation *RA> in the graph of q . We realize that this is exactly a statement of the form "The graph of q defines R implicitly", as we know it from model theory. Now there are various forms of implicit definitions, depending on the use of additional predicates or even extensions of the domains. However, if we allow unrestricted extensions of the domains, even the first order implicitly defined relations on finite structures are not in general computable. On the other hand, without additional predicates and extensions of the domains, only a restricted class of computable queries can be obtained. In fact, it follows from [F'agin 19741 that the queries which arise from first order implicit definitions without extensions of the domains are exactly the .VPrecognizable classes of finite structures which are closed under isomorphisms. To get things under control we introduce a notion of implicit definition with recursively bounded extensions of the universe, which we parametrize by families of recursive functions. With the help of these concepts we ([Makowsky-Zvieli 19831) can show that Theorem 16: The computable queries are exactly the first order recursively bounded implicitly definable queries. This theorem can be viewed as another presentation theorem in the sense of chapter 1. I t is also another illustration of Kreisel's program, since it says, that on finite structures the computable queries are hclosed for recursively bounded implicitly definable queries. In other words, it exhibits the connection between definability and computability, which deepens the justification for the approach in [Chandra-Hare1 19801. The language QL turns out to be an analogue of the recursive infinite extension of first order logic, which also plays the role as the unifying logic for various versions of dynamic and algorithmic logics, a s discussed in chapter 5 . The theorem also says that all the Coddcomplete, i.e. containing all explicit definable queries, and computable query languages are implicitly equivalent.

Model Theoretic Issues in Theoretical Computer Science

325

To measure how they differ explicitly, one can now classify query languages according to the complexity of the evaluation of its queries. This leads to various hierarchies, with the complete languages on the top and the first order explicitly definable queries on the bottom. Query languages which are in this hierarchy are called Codd-complete. In [Chandra-Hare1 19821 and [Vardi 19821 this hierarchy is investigated along traditional complexity measures, which are not obviously connected to the way those queries are expressed. In [Immerman 19821 a different approach to complexity is suggested, which is based on the complexity of the definitions of the queries. Theorem 15 is useful especially for this latter approach. However, from a practical point of view, implicitly definable queries have one serious drawback: Proposition 17: I t is undecidable, given a first order formula (p over a vocabulary T containing a predicate letter R, whether (p(R)represents an implicit definition of R . In first order logjc over arbitrary models Beth’s Definability theorem tells us that every implicit definition is equivalent to an explicit definition. Over finite models this is false. To illustrate the power of implicit definitions we ([Makowsky-Zvieli 19831) have an interesting application of an old result of [Friedman 19761. Theorem 18: For every recursive function there exist an explicitly definable query q such that (i) there is an implicit definition p of q but every explicit definition 9 of q has length 1 ($)>I(1 ((p)) In this context we can return to our theme of preservation theorems. Like for dependencies, we can look at queries, whose implicit definitions are given by formulas with certain syntactic restrictions. For instance, if the implicit definition is given by a formula from FID, treating the relation variable like an ordinary relation symbol, we call them the f u l l implicational implicit queries (FIIQ).

Proposition 19: Every query q E FIIQ is an explicitly definable query The queries in FIIQ are queries written in the language of programming logic, as exemplified by the language PROLOG, with the additional property that they have a unique solution. In programming with PROLOG one is not interested in this case, but rather in the least fixed point. Proposition 18 justifies this point of view. In [Chandra-Hare1 19821 the complexity of PROLOG queries is investigated: Though they do not form a complete query language, they go far beyond the explicitly definable queries. Problem: Show that, for the explicit FIrg from theorem 19, there is an exponential lower bound. Show also, that in theorem 18 we could restrict the implicit definition to be an embedded implicational implicit query EIIQ. In other words, like for the consequence problem, the difference between embedded and full consists in not recursively bounded versus exponential lower bound.

326

J.A. MAKOWSKY

2.7.Conclusionsand some open problems. What is the meaning of a syntactic restriction on first order dependencies , we asked in the beginning of this section. In cases like prenex normal forms the meaning is that one can always do it. In the case of safe formulas the answer is definiteness, as it was observed by many authors before. In the case of typed formulas the meaning is separation of attributes (separability), a s we showed in section 2.3, in the case of equality generating dependencies the meaning is separability, the subrelation property and closure under productsr, and in the case of full typed tuple generating dependencies ( T G D ) the meaning is separability, the duplicate extension and the intersection property. In the latter case we could also answer separability , the substructure property and faithfulness, but this can only satisfy an algebraist, and is not relevant for data base theory. The intersection property is not only conceptually more appealing, but is also justified by its usefulness in connection with the "chase", as pointed out in [Maier-Mendelzon-Sagiv 1979, lemma 71.

In each of the above cases the meaning of the syntactic restriction was exhibited by proving a n appropriate preservation theorem. The tools to prove such theorems are taken from model theory or a t least derive very.strongly from proof techniques well known to model theorists. But the properties which are preserved are directly derived from data base practice. In each of the above cases the syntactic restriction also gives us a recursive set of formulas (dependencies) whereas the set OF formulas equivalent to a restricted formula is not even recursively enumerable. I t is now natural to ask whether such characterization can also be given to other classes of dependencies. In [Makowsky 19811 we proposed such characterizations, but they are not satisfactory enough. Problem: Give characterizations for embedded multi-valued dependencies, typed embedded implicational dependencies and template dependencies. We have not yet studied'the way normalization theory gives rise t o syntactically defined dependency classes. But it s e e m natural to guess that there are preservation theorems stemming from normal f o r m of data base schemes. A step in this direction may be found in [GinsburgHull 19821 and [Ginsburg-Spanier 19821. Another direction along these lines are characterizations which describe more closely the structure of the vocabulary (similarity type). Being typed is such a property, but more relevant and fruitful is the distinction between cyclic and acyclic data bases, a topic which we unfortunately could not cover here, cf. [Goodman-Shmueli 19821. A good survey is [Fagin 19831. The distinction between those two types of data bases is also reflected in various complexity results. For a n excellent survey of normal forms (in contrast to syntactic normal forms), cf. [Beeri-Bernstein-Goodman 19781, [Bernstein-Goodman 19801 and [Ullman 19821.

Model Theoretic Issues in Theoretical Computer Science

327

Queries were defined as computable functions of data base states which preserve isomorphisms In the spirit of our approach i t is natural to ask if there other invariance properties which should be considered. In [Cooper 19801 it is suggested that definite formulas play also a role in query languages. His point is that, whereas queries are in general only partial recursive function, queries explicitly defined by definite formulas lead to total functions. This line of thought deserves more attention. The way one could make use of separable queries is obvious: but not very promising. A modest step for queries definable by FID’s was done in proposition 18. A related approach, though different in spirit, may be found in [Goodman-Shmueli 19821, where aspects of query processing used for syntactic characterizations. But what we have is rather a Problem: Which invariance properties for queries give meaning to interesting syntactically defined classes of computable queries. And which of these classes reflect also some complexity issue ? The importance of the work in [Chandra-Hare1 19801 lies not only in the clarity of their concepts. The complete query language QL they propose is the decisive step to provide data base theory with a notion of operational semantics for all reasonable query languages. The completeness of QL enables us to provide all other query languages which precise definitions of their semantics, by interpreting them in QL. The usefulness of QL is only theoretical. But it also lead in [Makowsky-Zvieli 19831 to extensions of existing query languages such as SQL and QBE, cf. [Ullman 19821 for their definition. Those extensions turned out to be complete in the sense of [Chandra-Hare1 19801. The success of the approach of [Chandra-Hare1 19801 suggests some further research: One can try to mimick these definitions in the context of the entity-relationship model for data bases, as proposed in [Chen 19761.This would lead to a precise comparison of the power of the two models and probably to a precise notion of their equivalence. On the other hand one can t r y to generalize the concept of a computable query to other transactions and develop a theory of computable transactions. A transaction will also be a computable function mapping data base states into data base states, but it is less clear what kind of isomorphisms o r other properties i t should preserve. We suggest that the methods we have illustrated in this section may lead to interesting developments in a general theory of computable transactions.

328

J.A. MAKOWSKY

3. Specification of Abstract Data Types.

3.1.Introduction. Data structures are structures, usually finite, sometimes potentially infinite. Their main purpose in programming sterns from the need to organize algorithms transparently, saving space and time wherever appropriate. This is particularily important when we want t o build polynomial, especially linear time algorithms. As a modest example for logicians let us pose the following problem: Given a set of propositional Horn formulas, we want to test satisfiability. It is easy to find an algorithm which uses 0(n3)time, and working a bit harder, even one that uses O ( n i o g n ) time. But only a careful choice of the data structure will give a 0 ( n ) algorithm, cf. [Itai-Makowsky 19821. Logicians are usually not trained to express their ideas that way. Much more dramatic results of this type, involving deep mathematics, were obtained e.g. by R.Tarjan for graph theoretic algorithms, for which he was awarded the newly created Nevanlinna Prize 1982. Good introductions to combinatorial algorithms using various data structures are [Even 19791 and [Reingold-NivergeltDeo 19771. The latter contains also an annotated bibliography. Abstract data types are abstract structures like in algebra, category theory or model theory. They arise in attempts to axiomatize the properties of the data structures, which are needed to prove correctness of the so designed algorithms. The abstraction stems from the need to distinguish between the intended data type and its implementation. Especially in modular programming or in correctness proofs of programs one has to distinguish further between what is t r u e in a data type and what jbllows from the assumptions which were made explicit. All other assumptions should be considered implementation dependent. From our remarks in the introductory chapter it should be clear, that we try to separate between correctness of a program in a particular implementation and correctness which is provable from explicit assumptions, and therefore enhances portability. aeciflcation theory is the model theory of these explicit assumptions, or rather of what we allow to be such assumptions, since what follows from them is true in all its models. Usually such assumptions are expressed as equations, or, more generally, as universal Horn fomulas, and strict universal Horn formulas (i.e. inondegenerate implications) cf. [Goguen-Burstall 19831. But the model theory is modified to the extent that not all algebras, but only the initial algebras are considered, or at least, play a special role. We t r y to attack the problem here from the point of view of abstract model theory, as described in [Barwise-Feferman 19831. The computer scientist's point of view of specification of abstract data types is widely discussed in the literature of the last decade, and many formalisms and semantical approaches have been proposed. A very useful dydactic essay on the "software engineering viewpoint" is [Bjorner 19801.

Model Theoretic Issues in Theoretical Computer Science

329

Data types are generally considered t o be many sorted structures or algebras, and methods from universal algebra, category theory and model theory have more or less successfully been applied to study the various questions concerning modularization techniques and expressive power (see for example [Goguen et al. 19781, [Burstall-Goguen 19801. [Wand 19781, [Kamin 1980,1983],[Bergstra et al. 19811, [Ehrig et al. 19801, [Ehrig et al.lQ82], [Ernst-Ogden 19801). While most of the work in this area is intended to contribute to the design of specification techniques of languages, or studies of recursiveness in connection with specifications, little is known about the consequences of the implicit assumptions which underly the proposed semantical concepts. Studying the implicit assumptions of algebraic specification theory is a problem which is conveniently expressible as a problem of a b s t m c t m o d e l t h e o r y : We have to axiomatize the universe of discourse when we want t o prove something about "all passible a p p m a c h e s t o algebraic speci$cations". The main difference here is, that we have to be much more careful about the choice of closure condiCions imposed o n t h e specifications. It is not a priori clear that we have closure under negation. d i s j u n c t b n or any sort of quantification. And in fact our results will show that such closure conditions are n o t appropriate. Here we want to exhibit some of these assumptions common to all of the above mentioned approaches, and shows that, surprisingly, they essentially determine the character of the possible specification languages. Our results confirm the particular choice of specification languages in the literature in the following sense: Each of the languages we discuss is complete for a set of implicit assumptions; in other words, each such language satisfies the assumptions, and any other language which satisfies these assumptions has no more expressive power. In other words, we have here another instance of a presentation theorem in the sense of chapter 1. But it is evident that, to complete Kreisel's program, we still have to go much further in isolating more definability criteria and in extending the framework to allow more general concepts of data structures, such as envisaged in [Goguen-Burstall 19831. Our results rely havily on the assumptions we made on the vocabularies (similarity types) of the data structures. Though they hold also for many-sorted relational structures with heterogeneous function and relation symbols, covering for instance all of the examples in [BroyWirsing 19831, we do not know, a t present, if they can be extended to other similarity types, a s allowed in [Bloom-Wright 19821. The results in this chapter are inspired by Mal'cev's characterization of free classes [Mal'cev 19541 and extend [Mahr-Makowsky 19821. They are essentially taken from [Mahr-Makowsky 19831, which is an adaption of this characterization to framework of specification of abstract data types.

330

J.A. MAKOWSKY

The notions used in this chapter are standard in universal algebra and logic and can be found for example in [Monk 19761. Explicitly we assume signatures, (vocabularies) to be of the form T = (S,C,F,R), so including sorts, sorted names for constants, functions and relation symbols. Rnitary signatures, (finitary vocabularies), are those where function and relation symbols have finite arity Structures (including relations) and algebras are defined in the usual way. Renaming T u + T for signatures a and T denotes the bijective assignment of T to u which is compatible with the sorting of the components of T and u. Renaming carries over to structures, and we denote by A[r), with respect t o a renaming 7 , the structure which is identical to A except that its universe, constants, functions and relations are renamed according to T . Basic formulas consist of atomic and negated atomic formulas (including equations and inequalities) with free variables. Free variables could also be treated as uninterpreted constants. Basic sentences are variable free. For a structure A , the set of basic sentences holding true in A is called the (basic)diagram of A If E is a set of formulas (possibly infinite) we denote by A E , V C the conjunction (resp. disjunction) of all the formulas in Z.

3.2.The Axiomatic Framework. To prove statements about all possible specification languages we have to make precise what we mean by "all possible specification languages". In this section we give such a definition. The only objection it could provoke is being too general. But since our theorems hold for it, they will a fortiori hold for any narrower concept of specification languages, so we do not have t o be bothered by this discussion here. Definition:A semantical system is given by a pair ( T , A ) consisting of: a class T of finitary signatures and a family A=(C,),EI of classes of type(i)- structures where type:/-+ T associates with each index a EI a signature type (i) such that the following axioms four hold: l.bomorphism Axiom Given T - structures A.B, and a class C, from A. Then A E B implies that A E C, iff B E C, . This axiom merely says that we deal with abstract data types, i.e. we are only interested in its isomorphism type and not in its particular representation. This is just the isomorphism axiom from abstract model theory, as defined in chapter 1. 2.Renaming Axiom Given a renaming T : T - . U and a class C, with type(i)=r, then there exists j E I with type U ) = u such that : A E C, iff A(') E Cj for all rstructures A . This axiom just says that we can change names of relations or functions without affecting the structures. For example we can change from additive to multiplicative notation when dealing with a group without affecting the group itself. This is just the renaming axiom from chapter 1.

Model Theoretic Issues in Theoretical Computer Science 3. Intersection Axiom: For all indices i.j

E

I there is index k t~ I

33 1 such that

C;=C, n Cj.

This axiom ensures that the union of two specifications is again a specification. Note that here, in contrast to abstract model theory, we actually axiomatize the notion of "sets of sentences", rather than sentences. We get therefore conjunctions for free, but can avoid the other closure operations in our basic definitions. 4. Empty Class Axiom For each T E T there is i E I such that type(i)=r and q = g . This axiom merely says that we can specify the empty class of r structures. The next step in our definition ensures that we are always allowed to add new constant symbols. We could be more liberal and also allow free use of new relation and function symbols, but our main results shows that this does not change anything. Definition: A semantical system (T,A)is r i c h e n o u g h if additionally the following axiom (5) holds: 5. Richness Axiom: If T=(s,c,F,R) E T , then for all families of constant symbols C ' over the same sorts S such that C c C' also s'=(S,C',F.R)E T ; and for all T E T and all sets B of basic rsentences Mod(B)E A. In other words, we can extend a signature by arbitrary sets of constant symbols and every set of basic (variable free) sentences defines a specification. In computer science, unlike in model theory, the intended models have to be countable, and preferably also r e a c h a b l e . Recall that a structure A is reachable if every element in a r s t r u c t u r e A is the interpretation of a term over r. (Clearly every structure can be made reachable by adding enough constant symbols.) Since we want to use model theoretic methods, we do not restrict ourselfs to such models, but demand, that every specification which has a model, has also a reachable model. More precisely: Definition: A semantical system (T,A)admits r e a c h a b l e structuTes if it is rich enough and additionally the following axiom: 0. ReachabilityAxiom For all indices i E I there is a reachable structure A EC,.

Remark: If (T,A) is rich enough, then for any reachable rstructure A with r E T there is a n index i c I with A E c,. g can be chosen to be the class of models of the b a s i c d i u g r u m of A , i.e. the set of atomic and negated atomic (variable free) sentences true in A . Definitions: (i) Given a class C of r-structures.'Then C is basic c o m p a c t if for all sets B of basic r-sentences C n Mod(R)# $ iff for all finite Bo c B C n Mod(BO)# 9. (ii) We call a semantical system (T,A)basic c o m p a c t (or o f f i n i t e support), if for all i E I the class C, is basic compact.

332

J.A. MAKOWSKY

Note, if C is first order definable, then C is basic compact. Basic compactness just says that i€ a set of basic sentences makes a specification inconsistent, then there is already a finite subset of basic sentences which makes i t inconsistent. Clearly, any system axiomatizable by finitary rules has this property. EXampleS: (1) Let Te be the class of signatures containing only function symbols and constant symbols and let AE be all the equationally definable classes. This gives us a basic compact semantic system which admits reachable structures. (2) Let TE be as above and A,$ be the all the classes definable by finite sets of equations. This gives us a basic compact semantic system, but it is not rich enough. (3) Let ' ' 7 be as above and AH be the quasi-varieties (i.e. classes definable by sets of finite first order Horn formulas, cf. [Monk 19761). This again gives us a basic compact semantic system which admits reachable structures. (4) Let T be the class of all signatures and A" be the classes definable by first order formulas. Then we get a basic compact semantic system which is rich enough but does not admit reachable structures. However, if we restrict ourselfs to classes definable by universal formulas, then it does admit reachable structures. If we allow infinitary clauses (cf.section 3.2) then we destroy compactness, but still get a semantic system which admits reachable structures. (5) If TR contains only relation symbols and A ~ Bconsists of classes definable by full implicational data base dependencies (F'D), as in chapter 2, then we get a basic compact semantical system, provided we allow infinite data base states. If we allow only finite data base states, a s in chapter 2, compactness fails. However, in both cases the system admits reachable structures. (6) We still get a semantical system which is rich enough if take T a s above and let Am be the classes definable by sets of statements expressing partial correctness of program, i.e statements of Hoare 1,ogic. For terminology cf. [Harel 19791. But here we loose both compactness and the reachable structures. (7) If L = ( T , S r , m ,I= ) is a logic, then the classes of the form Mod,(@) with @ c % ( T ) form a semantic system. If L satifies additionally the Basic Axiom then the resulting semantic system is rich.

Model Theoretic Issues in Theoretical Computer Science

333

3.3. A complete specification Language for rich semantical systems admitting reachable structures. In this section we show that the existence of reachable structures together with the axioms of rich semantical systems already determines fairly well, what kind of syntax is appropriate for specification languages. The reader, however, should be warned: the infinitary language we first present is not the ultimate specification language. Our first. theorem just illustrates how little we need to get our first presentation theorem. If we add basic compactness (a model theoretic substitute for axiomatizability), a s in theorem 2, we get the more familiar finitary logics. Definition: The language of i n f l n i t a r y c l a u s e s is given by LO=(To,ALg,mnlo,I= ), with: To the class of all finitary signatures and Alg(r) the class of all 7-structures. mnlO(r) consists of all infinitary clauses over T , i.e. formulas which are infinite disjunction of basic r-formulas, possibly with infinitely many free variables. Formally, if B is any set of basic rformulas (possibly with free variables) , then V B is a infinite clause. Finally,for A E AZg(7) and c M o ( ~I= ) ,is defined by Al= oQ if for every V B E the universal closure of V E holds in A , We denote by L, the system which we get from Lo by restricting it to finite sets B of basic formulas, and call it the language o f f i n i t e clauses. L, is logically equivalent to the system given by sets of universal first order sentences (cf. Example (4)). Both Lo and L, are logics satifying the Basic Axiom. Our next two theorems show that the language of infinite (finite) clauses is universal for semantic systems which admit reachable structures (and are basic compact). More precisely: Theorem 1: Let (To,&)be the semantical system resulting from the logic Lo of infinitary clauses, i.e. C, E & iff there is c b ( t y p e ( i ) )with C,=hlolod(O) . Then admits reachable structures; (i) ( To,Ao) (ii) If ( T , A ) is a semantical system which admits reachable structures, then A is a subfamily of &#i.e. for all C E A we have C E &, and thus C is Lo- definable. Theorem 2: Let ( To.AJ) be the semantical system resulting from the logic Lo of finite clauses, i.e. C, €AJ iff there is @ c L , ( t y p e ( i ) ) with G=&lod(Q) . Then (i) ( To,A,) admits reachable structures and is basic compact; (ii) If (T,A) is a semantical system which admits reachable structures and is basic compact. then A is a subfamily of A, i.e. for all C E A we have C E A,, and thus C is L,- definable. Proof The first. theorem is proved using the method of diagrams and the second follows from the first using compactness. A reader with no background in model theory should consult [Monk 19761 , or any other beginning text in model theory. A complete proof may be found in [Mahr-Makowsky 19821.

+

+

+

334

J.A. MAKOWSKY

3.4. Typical models and initial algebras. The notion of a semantical system is meant to capture the semantics of a specification language and interpretes a class C in the system as the semantics of a single specification. However, specification of abstract data types often attaches a single structure as semantics to a specification, like the initial algebra approach [Goguen e t a1.19781 or the final algebra approach [Wand 19781. In both cases the single structures have a distinguished position in the "specific" class C, which characterizes them uniquely up to isomorphism. Note, that one could also propose to vary the notion of "isomorphisms", i.e. pass to different categories, than just the category of rstructures. In [Wirsing-Broy 19801 it was proposed to require that the category of finitely generated structures form a complete lattice, to allow other than universal first order formulas in the specification language. Our theorems below illustrate, why such a n approach must run into certain difficulties. On the other side there more possible choices of structures which are unique in their class, so additional arguments should be put forward when one chooses initial or final algebras. One such argument may be found in the notion of g e n e r i c algebras or, what this really amounts to, the concept of proof by e x a m p l e . If we write do% a specification Z of a data structure in some formally defined specification language L , the intended data structure should satisfy C, but nothing else. However, this is not possible, since some other statements in L might be logical consequences of C. So the best we can hope for is a structure (algebra) A wich satisfies C together with all the consequences of C, but whenever some statement u E L is not a consequence of C, then it is false in A . In algebra such a structure is called g e n e r i c f o r C . In data base theory such structures are called A m t r o n g r e l a t i o n s , (cf.[Fagin 19821). The usefulness of this concept is that truth in the generic structure (an example) is equivalent to being a logical consequence of C, i.e. it formalizes the notion of proof by example.This idea has recently also been M exploited for testing programs ,cf [RowlandDavis 19811. What we try to argue for here, is that behind the notion of the initial a l g e b r a lies a similar concept, and that the uniqueness of the initial algebra is just one of the many nice properties it has. The following notion of D f r e e s t r u c t u r e captures the intention behind these two approaches. Definition: Given a class T of signatures, and let PT and NT denote the atomic, respectively negated atomic, rsentences for

T E

T. Then a class

D c PT u NT is called a preference s y s t e m f o r T if (i) D is consistent, i.e. M O ~ ( D ) I, # (ii) D is maximal, i.e. any D with D c D c PT u NT is inconsistent, in other words Mod(D)=#, Note that if D contains free variables, then D is consistent if the existential closure of D has a model.

Model Theoretic Issues in Theoretical Computer Science

335

Example: Let A be a r-structure and D be the set of all basic r-sentences true in A (the diagram of A ) . Then D is a preference system. In fact, every preference system can be obtained in this way. To obtain a preference system D with free variables in this way we just look at the free variables as distinct new constant symbols (or generators) and take as A the reachable model described by D and the new constant, symbols. Preference systems may also be useful to handle parametrizations without mechanisms for "parameter passing". as was pointed out t o me by J.Thatcher, cf. [Goguen et al. 19781 and [Thatcher et al. 19821. Definitions: (i) Given a class T of signatures and a preference system D for T. Let C be a class of rstructures with T E T and A E C, then A is D-MiCd in C if A I= u implies C I= u for all u E D . If u contains free variables, we mean by A I= IJ that the universal closure of E holds in A and sirnilwily €or Cl= u. D-typical structures are a weak form of generic structures, as far as basic, variable free sentences are concerned. If D contains free variables and D c PT,T has no relation symbols, then they are exactly the generic structures. (ii) A is D-free in C if A is reachable and D-typical in C. D-free structures combine the requirements of reachability and genericity, as far as they are compatible. For the usual definition of generic algebras, it may well be that there are no reachable generic structures, even if both separately exist. More on generic structures may be found in [Gratzer 1979, Appendix 41. FXElmpleS: (1) (initial) Let 7' be arbitrary and D=PT. Then A initial in C iff A D-free in C. (2) (Anal) Let T be arbitrary and D c NT.Then A final in C iff A D-free in c.

(3) In general , if D is the diagram of some structure A then a D-free structure 3 in C is as different from A as C permits, i.e. for u E D B I= u only if for all B E c B I= U. This is why we call D a preference system. Facts: (1) If A.A' are D-free in C then A r k . (2) Let D be an arbitrary preference system for T and C the class of all rstructures for given T E T , then A is D-free in C iff the restriction of D to T is exactly the diagram of A . (Recall that the diagram of A is the set of all basic sentences holding in A ) . Definition: A semantical system (T,A) admits D-pee S ~ r U c t U r e sfoor a given prejerence s y s t e m D ~ O T if (T,A) is rich enough and additionally satisfies the following axiom: 6'.Preference Axiom: For all indices i E I there is 'a D-free structure in C,.

336

J.A. MAKOWSKY

3.5. A complete language for semantic systems which admit initial f3emantics.

In this section we show that the existence of D-free structures determines even more, what kind of syntax is appropriate for specification languages. Again, it does not suggest any particularily useful syntax, but it shows how few assumptions allow us to get much information. If we assume, furthermore, basic compactness, then we shall get the expected finitary Horn clauses. Definition: Given T=T,, the class of all finitary signatures, and D a preference system for T . Then the logic of infinitary D - H o r n clauses Lf=(To,AZg , M DI= , is defined like Lo,except that for a set B of basic Tformulas V B E L # ( T ) iff there is a t most one formula in B n D. We denote by Lj) the set of f i n i t e Horn clauses. Note that if B is a set of basic sentences with B n D=$ and d E D then - d = b o is not in D and the clause V ( B u [ d j ) is equivalent to the infinitary formulas A B + b o , which is indeed a n infbitary Horn formula. Theorem 3: Let ( T o , @ be the semantical system resulting from the logic L# for a given preference system D , i.e. C, E @iff there is c FhzlD(type(i)) with C, =Mod (a) . Then (i) If D=(PT then Lo.@ admits D-free structures and (ii) If (T,AD is a semantical system which admits D-free structures, then AD is a subfamily of 4D, i.e. for all C E AD we have C E @, and thus are Lfdefinable. Remark: (i) of the theorem has a n additional assumption, which we conjecture not to be necessary. However, in the case of basic compact semantic systems this additional assumption is not needed. Theorem 4: Let (T, , A f i be the semantical system resulting from the logic L? for a given preference system D , i.e. C, E A: iff there is c FhzlD(twe(i)) with C, =Mod (a) . Then (i) I,,,Ap) admits D-free structures and (ii) If ( T , A is ~ a semantical system which admits D-free structures and is basic compact, Lhen AD is a subfamily of A?, i.e. for all C e A D we have C E A?, and thus are L?- definable. Proof: Part (ii) in both theorems follows from a result due to G.V.Cudnovskii [Cu68] which was independently rediscovered via methods of category theory in [Andreka-Nemeti 19751 and in [Banaschewski-Herrlich 19761. Part (i) in the infinitary case with D=PT may also be found there. To prove part (i) for general D one has to prove a lemma: Lemma 5: Let C be a set of finite D-Horn formulas and u1n2E D . Then C u [u,,bzjis consistent iff for each i=1,2 C u tu,j is consistent. Proof of lemma: This follows from a close analysis of the resolution method to check satisfiability of sets of clauses, together with compactness. For more details on resolution we recornmend [Robinson 19791.

Model Theoretic Issues in Theoretical Computer Science

337

3.6. Relevance for Specification of Abstract Data Types. The completeness results in the previous sections talk about the limitations in defining classes of structures by specifications. These limitations are not determined by the properties of particular specification languages, but are caused mainly by the assumption of admitting reachable or D-free structures. That such assumptions are reasonable will be discussed below. What should be pointed out here is, that we have turned the tTad%hnalquestion of finding reasonable semantics for given syntactic approaches, upside down: We have first defined axiomatically how our semantics should look like by extracting some of the key ideas and intuitions behind the [Goguen e t al. 1978l-approach and then we proved that this determines, up to logical equivalence, pretty well what kind of a syntax is well suited for specification of abstract data types. I t remains open, how this approach can successfully be extended to the more general framework as envisaged in [Goguen-Burstall 19831. But I a m convinced that a careful implementation of “Kreisel’sprogram” will lead t o various clarifications in the relative chaos of too many practical proposals. And the experience gained from abstract model theory may help us in asking a t least the right type of questions. We did not deal for instance with the problem of ”hidden functions”, as stated e.g. in [Thatcher e t al. 19821. But it may well be that our notion of A-closure and implicit definability from section 1.3 points in the right direction to clarify this concept. In section 2.6 w e have already given a n example of how t o use implicit definitions. Let us now look, retrospectively, a t two special cases, initial and final structures, and discuss the semantic assumptions more closely. Specification of Abstract Data m s w i t h Initial Semantics. The so-called algebraic approach to data type specification originates in the work of [Liskov-Zilles 19741, [Guttag 19751 and [Goguen et al. 19781 and considers specifications to be sets of equations or implicational equations (=strict universal Horn formulas). The definable classes are varieties or quasi-varieties of many-sorled algebras which contain, uniquely up to isomorphism, an initial algebra. Several attempts to extend this approach have been made, namely to use arbitrarty first order formulas (including relation symbols), see [Carvalho et al. 19801 and [Wirsing-Broy 19801. The last theorem shows that any extension beyond universal Horn clauses is unsafe in the sense that it does not guarantee the existence of initial structures. In the spirit of chapter 2 we could say that the meaning of universal Horn formulas is exactly given by the initial semantics. Since equivalence to a set of universal Horn sentences is generally undecidable (see section 2.4), a specification language which admits initial structures, and which allows a reasonable syntax analysis, therefore should be the language Lp with D = P ~ . This observation also applies to RequiTement specifications as introduced in [Ehrig 19811. There a set of requirements (in a typical case a set of first order sentences) is meant to precondition the data

338

J.A. MAKOWSKY

type to be specified, or to restrict the class of structures. That such a set of requirements allows initial structures is thus of great importance. A language for such requirements again is bounded in its expressive power by Lywith D = P p Specificationof Abstract Data m e s with Final Semantics. As a reaction to [Goguen e t al. 19781 final semantics is proposed in [Wand 19781 to determine by a specification not only a single data type, but also its possible implementations. Specification techniques for the so-called final semantics approach are not equally well developed. See, however, [Kamin 1980, 19831 and [Hornung-Raulefs 19811. The possibilities of specifying a class of implementations for the, up to isomorphism, uniquely existing final structure a r e bounded by L g or Lj' with D=NT, in a sense just like above.

3.7.A Word on Other Applications. Some of our results may have other applications a s well. A s it turns out the programming language PROLOG, (cf. [Clocksin-Mellish 19821) gains popularity and is even considered by some as the language of the fifth generation of computers. In PROLOG, one can specify data types directly in Horn formulas, and our results show that this choice is appropriate. In [Chandra-Hare1 19811 the connection between PROLOG and query languages, as we described them in the previous chapter, is studied. They show that PROLOG, taken as a query language, is not complete, and determine exactly its position in their complexity hierarchy for query languages. In PROLOG, the data types specified by Horn clauses, are always realized as the initial structures, provided space considerations allow it. Not provable is equated with not true. Our notion of preference systems suggests interesting variations for that. I can think of applications in domains, which usually are captured by "non-monotonic logics", a s suggested in [Artificial Intelligence 19801. Here D can be thought of a s a description of a n ideal world, or some default assumptions, and the D - t y p i ~ d models a s best approximations to or least deviations from D. Specifications expressed in D-Horn formulas give the description of the real world. Our characterization of D-Horn formulas implies that such best approximations always exist, and that, if they always exist, the specifications have t o be written in D-Horn formulas. We plan to explore this aspect of our work in the future.

Model Theoretic Issues in Theoretical Computer Science

339

References. (Note that the references have no claim for either completeness nor historic accuracy concerning priorities. They merely reflect the authors accidental readings.) Andreka,H. and Nemeti.1.; Generalisation of variety and quasivariety concepts to partial algebras through category theory, Dissertationes Mathematicae (Rozprawy Math.) 204 (1982) Artificial Intelligence vol13 (1980), Special Issue on Non-Monotonic Logic. J3anaschewski.B. and Herrlich,H.; Subcategories deflned by implications, Houston Journal of Mathematics 2.2 (1976)pp.149-171 Barwise,J. and FefermanS.; Higher Model Theory: Logic o! Mathematical Concepts, Springer 1983 BarsriseJ.;Axioms for abstract model theory, Ann.Math.Logic vo1.7 (1974) pp. 221-265. BarwiseJ.; Admissible sets and structures, Springer Verlag 1975. Beeri,C. and Vardi,M.Y.;The implication problem for data dependencies, Proceedings of the 8th ICALP. LNCS vol. 115 (1981) pp.73-85. Beeri,C., Bernstein,P.A. and Goodman,N.;A sophisticate’s introduction to database normalization theory, 1978 W E Conf. pp 113-124 Bergstra. J.A., Broy, M., Tucker. J.W., Wirsing, M.; On the Power of Algebraic Specifications, Proceedings of the MFCS’81, Springer Lecture Notes 118 (1981) Berman,P.,Halpern.J.Y. and Tiuryn,J.;On the power ot nondeterminism in dynamic logic,ICALP1962, LNCS vol.140 pp 48-60 I3ernays.P. and Schoenflnke1.M.; Zum Entscheidungsproblem der mathematischen Logik, Math.Annalen 99 (1926) pp.342-372. Bernstein, P.A. and Goodman, N.; What does Boyce-Codd normal form do ?, VLDB 1980,pp 245-259 Bernstein,P.A. and Goodman,N.;A sophisticate’s introduction to distributed database concurrency control, Harvard University TR +l8-82 ,46p. B1oomS.L. and Wright.J.B.; Finite quasi-varieties, Journ.of pure and applied algebra 25 (1982) pp. 121-154. Bjorner, D.; Formal description of progr~ammingconcepts, Mathematical foundations of computer science, Springer LNCS vol. 66 (1980) pp.1-21. W.Oy,M. and Wirsing,M.: Generalized heterogeneous algebras and partialinterpretations. CAAP ‘83,to appear in Springer LNCS. Burstal1,R.M. and Goguen,J.A.;The semantics of CLeAR, a specification language, Proc. of 1979 Copenhagen Winter School on Abstract Software Specifications. Pequeno, T.H.C., Pereda, A.A. and Veloso, P.A.S.; A Carvalho, R.L.de, Maibaum, T.S.E., Model Theoretic Approach to the Theory of Abstract Data Types and Data Structures, Research Report CS-80-22,Waterloo, Ontario, 1980, Casanova. M.A.; The concurrency control problem tor database systems. LNCS 116. S p r m e r 1981 Chandra, Ashok and’Harel, David; Computable queries for relational data bases, JCSS 21.2 (1980) pp 156-178 Chandra, Ashok and Harel, David; Horn clauses and the fixpoint query preprint 1982 .I1 p

hierdrchy,

Chandra. Ashok and Harel. David; Structure and complexity of relational queries, JCSS to appear, preprint 30p

340

J.A. MAKOWSKY

Chandra, Ashok; Programming primitives for database languages, Proceedings of the 8th symposium on Principles of Programming languages (1981) pp.50-62. Chandra,A.K.,Lewis,H.R.and Makowsky.J.A.;Embedded Implicational dependencies and their inference problem, abstract in SIGACT 1981. to appear in JCSS 1983. Chang,C.C.and Keis1er.H.J.;Model theory, North Holland 1973. Chen.P.: The entity-relationship model - Towards a unified view of data, ACM TODS 1.1 (1976) pp.9-38 Chen,P.ed.; Entity-Relationship Approach t o Information Modelling and Analysis, ERInstitute Publications, 1981. Clocksin,W.F. and Mellish,C.S.;Programming in PROLOG, Springer 1981. Cooper,E.C.;On the expressive power of query languages for relational databases, TR1480, Harvard University 1980 Cudnovskii,G.V.; Some results in the theroy of inbitely long expressions, Soviet Math.Dokl.9 (1968) pp.556-559 Date,C.J.;An Introduction to Data Base SystemsLvol.l,3rded., Addison-Wesley 1981. Date,C.J.;An Introduction to Data Base Systems,vol.2,Addison-Wesley 1983. Di Pao1a.R.A.; The recursive unsolvability of the decision problem for the class of definite formulas, JACM 16.2 (1969) pp 324-27 Ehrenfeucht,A.; An application of games to the completeness problem of formalized theories, Fund.Math, 49 (1961) pp.129-141. Ehrig, H.. Kreowski, H.J., Mahr, B. and Padawitz. P.: Algebraic Implementation of Abstract Data Types,t o appear in TCS. Fall '82. Ehrig. H., Kreowski, H.J., Thatcher, J.W., Wagner, E.G. and Wright, J.B.; Parametrized Data Types in Algebraic Specification Languages, Proceedings of the 7th ICALP'80, LNCS 85 pp.157-168 Springer 1980 Ehrig. H.: Algebraic Theory of Parametrized Specifications with Requirements, Proceedings of the CAAF"81,Springer - Lecture Notes, No.112 (1981) Engeler,E.; Algorithmic properties of structures, Math.Syst.Theory 1 (1967) pp.183-195. Engeler,E.: Proof theory and the accuracy of computations, In: Symposium on automatic demonstration, Springer LNM vol. 125 (1970) pp.62-71. Emst,G.W. and Ogden,W.F.;Specification of abstract data types in MODULA,ACM TQPLAS 2.4 (1980) pp 522-543. Eken,S.;Graph Algorithms, Computer Science Press 1979. Fagin.R.; Armstrong Databases For Functional and Inclusion Dependencies, RJ3500 8.7.1982 IBM Report RJ 3500 (1982) Fagin,R.; Horn clauses and database dependencies, t o appear J.ACM, preprint March 1982. 58 p ( extended abstract ln SIGACT 1980) Fagin,R.; Multivalued dependencies and a new normal form for relational data bases, ACM TODS 2.3 (1977) pp.282-278. FaginJ?.; Acyclic data base schemes (of various degrees): A painless introduction, Proceedings of the 8th CAAP, L'Aquila 1983. to appear. FefermanS.; Applications of many-sorted interpolation theorems, In: Proceedings of the Tqrski Symposium, L.Henkin ed. AMS Proceedings of symposia in pure mathematics V01.25 (1974), pp. 205-224.

FlumJ : First order logic and its extensions, In: ISILC Logic Conference proceedings, G.H.Muller et al. ed.. Springer LNM vo1.499 (1975) pp.248-310.

Model Theoretic Issues in Theoretical Computer Science

341

Francez,N.,Katz,Sand Makowsky,J.A.;The expressive power of nondeterministic control and random assignments, preprint April 1982 Friedman,H.; The complexity of explicit definitions, Advances in Math. vol. 20 (1976) pp.18-29. Garey M.R and Johnson D.S.;Computers and intractability, San Francisco 1979. Ginsburg. Seymour and Hull, Richard Characterizations for functional dependency and Boyce Codd normal f o r m f a d i e s . preprint Feb 1982,65 p Ginsburg, Seymour and Spanier, Edwin H.; On completing tables to satisfy functional dependencies, preprint July 1982, 13 p Goad,C.;Automatic construction of special purpose programs, Report No. STAN-CS-82897, Stanford University 1982. Goad,C.; Computational uses of the manipulation of formal proofs, Report No. STANCS-80-819, Stanford University 1980. Goguen, J., Thatcher, J., Wagner, E.; Abstract Data Types as Initial Algebras and Correctness of Data Representations, Current Trends in Programming Methodology, Vol. 4 (R.Yeh,ed.). Prentice Hall, N.Y., 1978, pp. 80-149. Goguen.J.A. and Burstal1,R.M.; Institutions: Abstract model theory for program specification, unpublished draft, Stanford SRI, 1983. Goodman,N.and Shmueli.0.;hmitations of the Chase, IFP 13.4-5 (1981) pp.154-159. GoodmanJ and Shmueli,O.;Syntactic characterization of tree database schemas, preprint, March 1982, Goodman,N. and Shmueli, 0.; Tree queries: a simple class of queries, ACM TODS 7.4 (1982) pp.653-677. Gratzer,G.: Universal Algebra, 2nd ed., Springer 1979 Gurevich,Y. and Lewis,H.R.; The inference problem for template dependencies, to appear in Information and Control 1983, extended abstract in Proceedings of the 14th annual ACM Symposium of Computing. 1982, pp.137-142. Guttag, J.V.; The Specification and Application to Programming of Abstract Data Types, TR. CSRG-59, Toronto, 1975. Hajek.P.; Some remarks on observational model theoretic languages, In: Set theorey and hierarchy theory,W.Mareket al. ed. , Springer LNM vol. 537 (1976)pp. 335-345. Harel,D.;First-Order Dynamic Logic, LNCS 68, Springer 1979 Hornung, G. and Raulefs, P.; Initial and Termlnal Algebra Semantics of Parametrized Abstract Data Type Specifications with Inequalities. Proceedings of the CAAF"B1.LNCS No. 112, Springer 1981 Immerman,N.; Number of quantifiers is better than number of tape cells, JCSS 22.3 (1981) pp.384-406. Immerman,N.; Relational queries computable in polynomial timaproceedings of the 14th annual ACM Symposium of Computing, 1982, pp.147-152. Itai,A. and Makowsky,J.A.;On the complexity of Herbrand's theorem, TR-243, May 1982, Technion-Israel Institute of Technology, Haifa, Israel. Kamin. S.;Final Data Type Speciflcations: A New Data Type Speciflcation Method, Proc. of the 7th POPL-Conference, 1980. Kamin, S.;Final data types and their specification, ACM TOPLAS 5.1 (1963) pp.97-124. Kreise1,G.: Hilbert's programme and the search for automatic proof procedures, In: Symposium on automatic demonstration, Springer LNM vol. 125 (1970) pp.128-146.

3 42

J.A. MAKOWSKY

Kreise1,G.;Survey of proof theory, JSL vo1.33 (1968) pp.321-38E. Kuhns,J.L.; Answering questions by computer: A logical study. RM-5428-PR,Rand Corp. Santa Monica, Ca. Dec. 1967. Lehmann.D.,Pnueli,A. and StaviJ.; Impartiality,justice and fairness: The ethics of concurrent termination,Proceedings of the Eth ICALP, LNCS vol.115 (1981) pp.264-277. Lewis.H.R.;Complexity results for classes of quantificational formulas, JCSS 21 (1980) pp, 317-353. Lewis,H.R.;Unsolvable classes of quantificational formulas, Addison-Wesley 1.979. Llndstrom,P.; On extensions of elementary logic, Theoria vo1.35 (1969) pp.1-11. Liskov, B.H.and Zilles, S.M.; Programming with Abstract Data Types, Proc. ACM Symp. on Very High Level Languages, SIGPLAN Notices, 9, 1974, pp. 50-9, McNulty,G.F.;The decision problem for sentences preserved under relations between structures, abstract, JSL 44.4 (1979) p.452. Mahr, B. and Makowsky, J.A.; Characterizing Specification Languages with admit Initial Semantics, Tech. Report #232, Technion, Haifa, Israel, February 1982, to appear in TCS, extended abstract in CAAP ‘83, to appear also in LNCS. Mahr,B and Makowsky,J.A.; An axiomatic approach to semantics of specification languages, Theoretical Computer Science, 6th GI-Conference, Dortmund, January 1983, LNCS vol. 145 (1983) pp. 211-220. Maier,D., Mendelzon,A.O.and Sagiv,Y.;Testing implications of data base dependencies, ACM TODS, 4.4 (1979) pp 455-469 Maier,D.,Sakiv,Y.and Yannakakis,M.; On the complexity of testing implications of functional and join dependencies, JACM 28 (1981) pp.680-895. Maier,D.;Theory of Relational Data Bases, Computer Science Press, to appear (1983?). Makowsky,J.A.;Characterizing Data Base Dependencies, Proceedings of the 8th ICALP ‘81. LNCS 115, (1981) pp.86-97 Mak0wsky.J.A.;Measuring the expressive power of dynamic logics,in: Proceedings of the 7th ICALP. LNCS vol.85 (1980) pp.409-421. Makowsky.J.A.;Securable quantifiers, k-unions and admossible sets, In “Logic Colloquium ‘73” R0se.H.E and Shepherdson,J.C.ed.. North l-iolland 1975, pp. 409-428. Mak0wskyJ.A.; Model theoretic issues in theoretical computer science, part 2: Assigning logics to programming languages, in preparation. Positive results in abstract model theory: A theory of Mnk0wsky.J.A. and Shelah,S.; compact logics, to appear in Ann.Math.Logic (1983). Makowsky J.A.,Shelah,S. and StaviJ; Ann.Math.Logicvol.10 (1976) pp.155-192.

A-logics

and

generalized

quantifiers,

Makowsky,J.A.and Vardi.M.Y.;in preparation (1983) Makowsky,J.A.and Zvieli, A,; Defhable queries, in preparation (1983). Mal’cev. A.I.; Quasiprimitive Classes of Abstract Algebras,in the Metamathematics of Algebraic Systems, Studies in Logic, Vol. 86. North-Holland, 1971, pp, 27-31. Manders,K.L. and Daley,R.F.;TheComplexity of the validity problem for dynamic logic, to appear in 1nf.and Contr. Meyer,A.R.;Ten thousand and one logics of programming. FATCS Bull. (January 1980) pp. 11-20, Meyer,A.R. and Tiuryn.J.; A note on equivalence among logics of progranwpreprint MlT Dec 1961

Model Theoretic Issues in Theoretical Computer Science

343

Meyer.A.R., and Parikh,R.; Definability in dynamic logic, JCSS vo1.23.2 (1981) pp.279298. Monk,J.D.;Mathematical Logic, Springer 1976 Papadimitriou,C.H. and Yannakakis,M.; Algebraic dependencies, JCSS 25 (1982) pp.2-41. Paredaens,J.; A universal formalism to express decompositions, functional dependencies and other constraints in a relational data base, Th.C.Sc.vol.19.2 (1982) pp. 143160.

Reingold,E.M.,Nievergelt,J. and Deo,N.;Combinatorial Algorithms, Prentice Hall 1977. Robinson,J.A.;Logic: Form and Function, North Holland 1979 Rowland,J.H. and Davis,P.J.; On the use of transcendentals for program testing, JACM 28.1. (1981) pp.181-190 Sadri,F. and Ullman.J.D.; Template dependencies: A large class of dependencies in relational databases and its complete axiomatization, Journal ot ACM 29.2 (1982) pp 363372 Shmueli.0.; The fundamental role of tree schemas in relational query processing, Harvard University TR-16-81, 160 p Shoenfield,J.R.; Mathematical logic, Addison-Wesley 1967. Statman,R.; Structural complexity of proofs, Ph.D. Department of philosophy, Stanford University, 1974. Tarski,A,; Some notions and methods on the borderline of algebra and metamathematics, Proc.Int.Congr.of Mathematicians, Cambridge Ma., vol 1 (1952) pp 705-720. Tharp,L.H.; Continuity and elementary logic, JSL 39 (1974) pp.700-716 Thatcher,J.W., Wagner,E.G. and Wright,J.B.; Data type specification: Parametrization and the power of specification techniques, ACM TOPLAS 4.4 (1982) pp.711-732. Ullman,J.D.; Principles of Data Base Systems,second edition, Computer Science Press 1982 Ullman,J.D.;Principles of data base systems, Computer Science Press 1980 VardiMY.; On decompositions of relational databases, FOCS 1982 Vardi,M.Y.; The complexity of relational query languages, Proceedings of the 14th annual ACM Symposium of Computing, 1982, pp.137-146. Vardi,M.Y.;The decision problem for data base dependencies, IPL 12.5 (1981) pp. 251254. Vardi,M.Y.;The implication and finite implication problem for typed template dependencies, Proceedings of the ACM Symposium on Principles of Data Base Systems, Los Angeles 1982, pp. 230-236. Wand,M.; Final Algebra Semantics and Data Type Extensions, Indiana TR65 (1978) Wirsing,M. and Broy,M.; Abstract data types as lattices of flnitely generated models. Mathematical foundations of computer science, Springer LNCS vol.88( 1980) pp.673685.

LOGIC COLLOQUIUM '82 G. Lolli. G. Long0 and A . Marqa (editors) 0 Ekevier Science Publishers B. V. (North-Holland), I984

345

NP AND C R A I G ' S INTERPOLATION THEOREM

Daniele Mundici

.

L oc Romol a N .76 50060 Donnini

F1 orencc-I t a l y

The t r u t h - v a l u e s of two renowned c o n j e c t u r e s about NP (namely, P f NP and, NP i s n o t c l o s e d under complementation) depend on t h e d i f f i c u l t y i n w r i t i n g down C r a i g ' s i n t e r p o l a n t s i n s e n t e n t i a l l o g i c . The g e n e r a l connection between NP and i n t e r p o l a t i o n i s s t u d i e d by b l e n d i n g i d e a s and t e c h n i q u e s from both model t h e o r y and computation t h e o r y .

0.

Introduction.

W e f i x throughout an a1 habet

c

and r e g a r d Boolean e x p r e s s i o n s a s p a r t i c u l a r words over I f B-+C i s a tautology, then Craig's i n t e r p o l a t i o n theorem y i e l d s an i n t e r p o l a n t I , t h a t i s , a Boolean e x p r e s s i o n I such t h a t B + I and IC are t a u t o l o g i e s , and t h e v a r i a b l e s o c c u r r i n g i n I are e x a c t l y t h o s e which j o i n t l y occur i n B and C. I n Theorem 2 and C o r o l l a r y 6 w e prove t h e f o l l o w i n g r e s u l t (where TAUT&* i s t h e set of t a u t o l o g i e s and is the set of words over

2.

c*

c):

A t l e a s t one of t h e f o l l o w i n g s e n t e n c e s i s t r u e :

(I)

TAUT

i s accepted i n d e t e r m i n i s t i c polynomial t i m e (viz..P=NP);

(11)

TAUT

i s n o t a c c e p t e d i n n o n d e t e r m i n i s t i c polynomial t i m e NP i s n o t c l o s e d under complementation);

(viz., (111)

-

-

I n t e r p o l a t i o n i s polynomially i n t r a c t a b l e , v i z . , f o r every f u n c t i o n Cp if $J i s computable i n d e t e r m i n i s t i c polynomial t i m e , t h e n f o r some t a u t o l o g y B C , $(B,C) f a i l s t o be an i n t e r p o l a n t f o r B+ C.

:c*xc*

c*,

For t h e proof w e use a m i x t u r e of t e c h n i q u e s from computation and model t h e o r y . Assume now t h a t t h e upper bounds f o r computations are r e l a x e d from t h e set 3 of polynomials t o any set 3 2 9 c l o s e d und e r sum and composition. Then t h e above r e s u l t s t i l l h o l d s r e l a t i v e perhaps w i t h a d i f f e r e n t d i s t r i b u t i o n of t o t h e new upper bounds t r u t h - v a l u e s among ( 1 ) - ( 1 1 1 ) . T h i s i s proved i n Theorem 7. Thus i t might be i n t e r e s t i n g t o f i n d two sets 3 and 3' ( i f any) ass i g n i n g d i f f e r e n t t r i p l e t s of t r u t h v a l u e s t o ( I ) - ( I I I ) : as a matter of f a c t , when 3 i s t h e set of a l l f u n c t i o n s , o r even, when '5 c o n t a i n s t h e e x p o n e n t i a l , t h e n t r i v i a l l y ( I ) becomes t r u e and (11)

--

3 46

D.MUNDICI

and (111) f a l s e . A t t h e o p p o s i t e extreme, when '5 is restricted t o t h e set of p o l y n o m i a l s , i t i s w i d e l y c o n j e c t u r e d t h a t ( I ) i s f a l s e , hence, e i t h e r of (11) or (111) i s t h e n t r u e . The above r e s u l t s y i e l d j u s t one more c o n n e c t i o n between t h e modelt h e o r e t i c a l n o t i o n of i n t e r p o l a t i o n , and computation t h e o r y . For a n o t h e r such c o n n e c t i o n , i n c61 i t i s proved t h a t i f t h e r a t e of growth of t h e l e n g t h of i n t e r p o l a n t s f o r any t a u t o l o g y B + C can be kept below some polynomial i n t h e l e n g t h of B and C , t h e n every f u n c t i o n which i s computable i n d e t e r m i n i s t i c polynomial time h a s c i r c u i t d e p t h growing p r o p o r t i o n a l l y t o t h e l o g a r i t h m of t h e input l e n g t h . T h i s as w e l l as o t h e r r e s u l t s c o n c e r n i n g t h e complexity of C r a i g ' s i n t e r p o l a t i o n theorem i n s e n t e n t i a l and i n f i r s t - o r d e r l o g i c a r e surveyed i n t h e f i n a l s e c t i o n of t h i s p a p e r . For t h e g e n e r a l r o l e of C r a i g ' s i n t e r p o l a t i o n theorem i n ( a b s t r a c t ) model t h e o r y , s e e , e . g . , [I1 , [8J and [91.

1.

Preliminaries.

For

A

an a r b i t r a r y s e t ,

d e n o t e s t h e set of words o v e r

A*

t h e set of a l l f i n i t e s t r i n g s of symbols from length

For paper

of

1w1

aCA,

an

A.

For

i.e.,

A,

the

w€A*,

i s t h e number of o c c u r r e n c e s of symbols i n

w

stands f o r

aa...a

(n

times).

Throughout

w.

this

d e n o t e s t h e f o l l o w i n g set of symbols:

c

{ A , v , T p ) .( , X , 0, 1 ) .

=

c,

Boolean e x p r e s s i o n s are u n d e r s t o o d as p a r t i c u l a r words over

accord-

i n g t o t h e f a m i l i a r f o r m a t i o n rules s t u d i e d i n s e n t e n t i a l l o g i c . p o s i t i o n a l v a r i a b l e s a r e words o v e r where t h e s u b s c r i p t

bl...b

i n binary notation.

For

,

var B

E

c*

I var

and

number of e l e m e n t s i n t h i s s e t .

B

I

The v a r i a b l e s i n

t h e o r d e r g i v e n by t h e i r s u b s c r i p t s . Letting

now

XI=

we mean t h a t

B,

B

b = lvar Bl (read:

. . .bn

'

an a r b i t r a r y Boolean e x p r e s s i o n ,

r e s p e c t i v e l y d e n o t e t h e set of v a r i a b l e s o c c u r r i n g i n

bits.

Xbl

i s t h e sequence of d i g i t s of a number

n B

of t h e form

{X,O,I)

Pro-

x

Elements of and

var B {0,1)

x €{O,l}b

satisfies

and

B,

,

inherit

are c a l l e d by

B)

c o n s i d e r e d as a Boolean f u n c t i o n

the

B:{O,l}b-+{O,I),

NP and Craig's Interpolation Theorem

takes value

on i n p u t

1

.. ,xb)).

( =(xl,.

x

341

One can r e c o v e r

f a m i l i a r s e m a n t i c s of s e n t e n t i a l l o g i c upon i d e n t i f y i n g "true"

and

Mod B =

For a r b i t r a r y

tautology

terminology,

( t o the f i r s t

n

Mod B r ( f i r s t n b i t s ) of t h e models of

bits)

Mod B = { O , l } b

f o r any two Boolean e x p r e s s i o n s t a u t o l o g y , an i n t e r p o l a n t var I

Here, as u s u a l ,

logies. In case

var B

n var

C

and

C

a

$

and

either i

B

g:

-

c- p,ll3

c

f o r every

{o,i)",

=(c

1

(resp.,

polynomial t i m e . strings

with regard whether tion

T u r i n g machine i n time

P=NP

one-one

. .. , g ( c n ) ,

A s usual,

P

we s h a l l b r i e f l y

s t a n d s f o r t h e c l a s s of sets

(that is,

f o r any

whether S E A * ,

NP if

say

of

whose c h a r a c t e r i s t i c f u n c t i o n

i n d e t e r m i n i s t i c polynomial t i m e ;

o r even,

bounded

(resp., nondeterministic)

t o n o n d e t e r m i n i s t i c polynomial t i m e .

(that is,

c

i s computable by a determi-

f

nondeterministic)

which are a c c e p t e d

i s computable)

(but otherwise arbi-

,...,CJEC*.

i s computable i n d e t e r m i n i s t i c

f

i s a tautology.

n a t u r a l l y i n d u c e s a one-one map:

g

by a polynomial i n t h e l e n g t h of t h e i n p u t , that

C

r ( c ) = c o n c a t e n a t i o n of g ( c l ) ,

with

I n s t e a d of s a y i n g t h a t nistic

((iB)V(C)).

mapping t h e symbols of

o n t o t h e s e t of t r i p l e t s of b i t s ;

r :C*

a

I such

a r e tauto-

I-C

or

Throughout t h i s p a p e r w e a l s o u s e a f i x e d function

is

B+C

(never occurring i n t h i s paper) Craig's

i n t e r p o l a t i o n theorem s t a t e s t h a t

trary)

) yields,

Boolean e x p r e s s i o n

B-I

a

Mod B f $.

(see [l]

such t h a t

is

B

i s an a b b r e v i a t i o n of

B-C =

and

B

that is

I,

var B f l var C,

=

by

i s t h e set

B.

is satisfiable iff

B

;

C r a i g ' s i n t e r p o l a t i o n theorem i n s e n t e n t i a l l o g i c

that

i s defined

w e a l s o set

n,
iff

Mod B

{ x € [ O p l ] b l x k B}.

I n model-theoretical of r e d u c t s

The set

"false".

with

0

the

with

1

NP

is the

same,

I t i s n o t known

i s c l o s e d under complementaSbNP

then

A*\

S€NP).

I n t h e f o l l o w i n g theorem t h e s e problems are r e l a t e d t o t h e d e g r e e of

D.MUNDICI

3 48

d i f f i c u l t y i n w r i t i n g down

Craig's interpolants i n sentential logic.

A t l e a s t one of t h e f o l l o w i n g s t a t e m e n t s h o l d s t r u e :

2 . Theorem.

(i)

P=NP;

(ii)

NP

i s n o t c l o s e d under complementation;

( i i i ) ( I n t r a c t a b i l i t y of s e n t e n t i a l i n t e r p o l a t i o n ) :

function

:

c*x c*-

c*,

$ is

if

computable i n d e t e r m i n i s t i c

polynomial t i m e , t h e n t h e r e i s a t a u t o l o g y and

var B A var C

for

B-C.

f

!if, such t h a t

f o r every

with

B-C

B,CEX*

i s n o t an i n t e r p o l a n t

$(B,C)

For t h e proof w e p r e p a r e t h e f o l l o w i n g lemma: 3.Lemma.

For

an a r b i t r a r y nonempty set t h e f o l l o w i n g

S C{O,I}*

are e q u i v a l e n t : (a)

SENP;

(b)

there e x i s t s a function

F:{l}*

-c*

d e t e r m i n i s t i c polynomial t i m e , such t h a t , have, f o r e a c h

Proof of Lemma 3 .

1x1 and

s

Fn;

and (b)

+

(a).

i s , on i n p u t

F~

and

Cook's

as f o l l o w s :

XE{O,I)*,

XI

=yl ,

(b)

.

Fnl ;

n

bits).

f i r s t compute

f i n a l l y guess a

We s h a l l o b t a i n t h e r e q u i r e d NP-complete

(see [2J ) .

,...,h)

lal,a2,...,a k )

Let

A =

and

ak=the b l a n k

v a r i a b l e s of 1

j Gk,

p

Fn

O,
symbol.

S

i n t h e l e n g t h of t h e i n p u t .

b e t h e set of s t a t e s of

T,

with

be t h e a l p h a b e t of Fix

are, f o r each

nB 1 t,i,j,q

t h e following:

We

Theorem 7 . 3 . 9 J .

be a n o n d e t e r m i n i s t i c T u r i n g machine a c c e p t i n g

bounded by a polynomial {O

by

F

well-known argument t o p r o v e t h a t t h e s a t i s f i a b i l i -

s h a l l c l o s e l y f o l l o w t h e n o t a t i o n and t e r m i n o l o g y of [5, T

n =

y E [O,l}r

...,xn=yn.

t y problem f o r Boolean e x p r e s s i o n s i s

So l e t

~ ( i ~w )e ,

=

A f a s t nondeterministic algorithm f o r

r = lvar

(a) modifying

= Mod F n r ( f i r s t

Sn{O,l)n

then let y != Fn

such t h a t

letting

n 2 1:

l v a r Fnlgn,

accepting

which i s computable i n

in t i m e

Let

h

the halting state.

T,

with

and l e t

a =1, a = O 1

m = p(n).

(OGtgm,

2 The

O,
349

NP and Craig's Interpolation Theorem

HEAD(t,i),

SYMB(t,i,j),

(1)

and

STATE(t,q),

respectively saying t h a t a t t i m e square

i,

Consider t h e

t,

a . i s p r i n t e d on t a p e J i , and T i s i n s t a t e q.

has t a p e p o s i t i o n

T

c o n j u n c t i o n of t h e f o l l o w i n g s e n t e n c e s , which uniformly

d e s c r i b e t h e b e h a v i o r of a t each t i m e

T

over any i n p u t of l e n g t h

t , each tape square

p r i n t e d on i t ,

n:

h a s p r e c i s e l y one symbol

i

i s scanning p r e c i s e l y one s q u a r e , and

T

T

i s i n p r e c i s e l y one s t a t e ; a t e a c h time

i

symbol on

if

t,

i s n o t over s q u a r e

T

t',

T

the

halts;

t h e computation s t a r t s i n s t a t e

o v e r t h e l e f t hand

0

end of

w i t h t h e t a p e o n l y c o n t a i n i n g b l a n k symbols ( w i t h

m

t h e p o s s i b l e e x c e p t i o n of t h e s q u a r e s initially

or

then

does n o t change;

a t some t i m e

the input,

i,

each t a p e square

m

through

through m+n-I

m+n-1);

has e i t h e r

1

p r i n t e d on i t ;

0

t h e changes i n t a p e symbols, head p o s i t i o n and s t a t e obey

T's

i n s t r u c tions.

w e assume t h a t t h e i n p u t i s p r i n t e d

N o t i c e t h a t , f o l l o w i n g c5],

rn

t h e tape squares and on

through

m, h, k

and t h e i n s t r u c t i o n s of

among t h o s e d i s p l a y e d i n l y w r i t t e n down i n

our

(A)

(F)

is

m+n-I.

I t i s well-known

can be w r i t t e n down as Boolean e x p r e s s i o n s

(F)

[5

t a k e c a r e of

(D)

and

t h a t (A),(B),(C) o n l y depending

are

and whose v a r i a b l e s

indeed, such expressions a ~ e x p l i c i t -

(1):

, p.236,237, e x p r e s s i o n s ( 1 ) - ( 5 ) , ( 7 ) , ( 8 ) , ( 9 ) ] :

corresponds t o ( 7 ) A (8)A (9)

T,

on

(1)A ( 2 ) A ( 3 ) , therein

.

( B ) i s (41, ( C )

i s (5).

and

The f o l l o w i n g Boolean e x p r e s s i o n s

(E):

m- 1

2m

STATE( 0 , O ) A HEAD(0.m) A ( i& SYMB( 0 , i ,k ) A (i=c+n SYMB(0, i ,k ) ) ;

(D)

mcn-I

i.A = m (sYMB(o,i,l)v S Y M B ( O , ~ , ~ ) ) .

Let

F,

=

(A)A

...A (F);

let

s = I v a r Fn I

-

I.

I t i s no

loss

of

D. MUNDICI

350

g e n e r a l i t y t o assume t h a t t h e first SYm(O,m,l), riable

Xr

x r t {O,l), F,;

...,SYMB(O,m+n-1,l)

of

n

r = 0,.

.. , s ,

are

Fn

i n t h e g i v e n o r d e r . Assume e a c h

i s assigned a b i t

F,

v a r i a b l e s of

va-

( i n t u i t i v e l y , a truth-value)

i n such a way t h a t

xo,

.. . , x

S

satisfies

i n symbols,

(2)

X

o'.

,x

*

t

Then i n t h e l i g h t of

we obtain a collection

(1)

" a t time

of t h e form

-

Fn

t ,

such and such s t a t e , and t a p e s q u a r e p r i n t e d on i t " ,

f o r each

of s t a t e m e n t s

h a s such and such p o s i t i o n , T i s i n

T

OGtLm

i

and

h a s such and such symbol OGiC2m

A

.

Now t h e d e f i n i -

Fn. t o g e t h e r w i t h ( 2 ) e n s u r e t h a t i s not a brute l e c t i o n OP i n c o h e r e n t s t a t e m e n t s , b u t r a t h e r d e s c r i b e s a l e g a l t i o n of

c e p t i n g computation of

m

tape squares

on some i n p u t

T

through

m+n-1

.

y E {O,lIn

colac-

p r i n t e d on t h e

To f i n d t h e symbol

y . originally J (j = 0 , . , n - l ) , observe t h a t i f x . = l p r i n t e d on t a p e s q u a r e m + j J t h e n SYMB(O,m+j,l) i s " t r u e " and, by ( 1 ) w e have t h a t y . = 1 .

..

J

x . = 0 , t h e n SYMB(O,m+j,l) J i s " t r u e " , s o t h a t by ( l ) ,

On t h e o t h e r hand, i f by ( E ) , that 0

SYMB(O,m+l,2)

a =1

1 ,... ,n-1,

and

a = 0).

Mod F~

Conversely ,i f

V

of

T

xo,.. /=

, xs

XO'...'xn-l'xn,... (3)

In definitive,

2

so t h a t

..

on i n p u t

E S

1

T.

xo

i n t i m e a t most

m,

s a f e l y assume t h a t t h e i n p u t i s p r i n t e d on t a p e s q u a r e s m+n-1, Using x

t h e whole t a p e (1)

,...,x n

S

having

Therefore, I n symbols,

t h e n t h e r e i s an a c c e p t i n g computation

... , xn-1

X0'

J

,...,xn- 1 E S. b i t s ) E s n { 0 . 1 ) ~.

n

*Xn-l

y. = x.

y . = 0 (recall J f o r each j =

i s a c c e p t e d by

implies

Fn

r (first

xo..

. ,xn-1

i s " f a l s e " and,

2m+l

many s q u a r e s ,

0.1.

and w e

m

can

through

...,2 m .

w e can unambiguously a s s i g n t r u t h v a l u e s t o all the

v a r i a b l e s of

of i n f o r m a t i o n c o n t a i n e d i n p u t a t i o n w e t h e n have, (4)

X0'.

Therefore,

xo,

.. ,xS

...,xn-1

v.

F

n ' Since t h e l a t t e r i s a

by d e f i n i t i o n of !=

E S

X O p . . . l X n-1 ' by j u s t coding t h e p i e c e s

legal

com-

Fn :

Fn

implies that there is

yo,

...,y s I=

Fn

35 1

NP and Craig's Interpolation Theorem

with

yo=xo,.

n

s

(5)

.. , Y , - ~ = X ~ - ~ . C _ Mod Fn

{O,ljn

Now ( 3 ) and ( 5 )

I n symbols,

r (first n

j o i n t l y y i e l d a f i r s t d e s i r e d c o n c l u s i o n about

(a) 3 (b)

To complete t h e proof t h a t

[5,

But t h i s i s well-known

( A ) A ( B ) A (C)A(F) ;

t o hold f o r

portional t o

p4(n),

t h e r e i n i s a t worst pro-

(1)-(9)

i s so simple.

and t h a t ( 1 ) - ( 9 )

The same

c l u s i o n h o l d s i n t h e p r e s e n t c a s e , w i t h t h e same argument. c l u d e s t h e proof t h a t

+ (b)

(a)

Assume s t a t e m e n t s

Mod I n

with

r

the

( i i i ) i n Theorem 2 a r e both

c*

I: {If*+

which i s computable

NP

(6)

s

=

I?

is satisfiable},

{ x ~ j o , ~ ) ~-'(XI l

map d e f i n e d i n s e c t i o n 1 . S i n c e t h e set of s a t i s f i a b l e Boolean e x p r e s s i o n s t h e n s o i s t h e set

( s e e r5, 7.3.51) =

In = I ( 1 " ) . w e

n21:

Proof of L e m m a 4. is in

T h i s con-

I

i n d e t e r m i n i s t i c polynomial t i m e , such t h a t , l e t t i n g have f o r each

con-

and completes t h e proof of lemma 3

( i i ) and

Then t h e r e i s a f u n c t i o n

false.

see

i s claimed t o h o l d ,

m8

where an upper bound of t h e form

p.2381

a f t e r o b s e r v i n g t h a t t h e l e n g t h of

4.Lemma.

F.

w e must show t h a t t h e map

d e s c r i b e d above i s computable i n d e t e r m i n i s t i c polynomial

I n I----, Fn

time.

bits).

r

{ x € {O,l]*]

S

g i v e n by

is satisfiable].

-'(x)

T h i s c l e a r l y f o l l o w s from t h e d e f i n i t i o n of

-S =

assumed t o be f a l s e , t h e n t h e set

r

.

Since

is in

S

{O,l{*\

(ii) i s

NP, t o o .

Withcut l o s s of g e n e r a l i t y t h e r e are n o n d e t e r m i n i s t i c T u r i n g machines T

T'

and

-

accepts

S

and a polynomial

p

i n t i m e bounded by

exist functions

H':

H,

such t h a t p.

~I}*-b~*

(7)

s

n 1o.1)~

s n {o.i)."

accepts

n

Hn = H ( 1 ) ,

n21:

~~r

=

Mod

=

Mod H,:

=+

and

S

TI

(b)) t h e r e

which are computable i n d e t e r -

m i n i s t i c polynomial t i m e such t h a t , l e t t i n g

w e have f o r each

T

B y Lemma 3 ( ( a )

(first

n

bits),

I (first

n

bits).

and

HI

n

= H'(ln),

D. MUNDICI

352

Since

and

T

b o t h a c t i n time bounded by

T'

t i o n of t h e proof of

p , by an e a s y i n s p e c -

w e can s a f e l y s t i p u l a t e t h a t , i n

Lemma 3

ad-

dition, =

v a r H r\ v a r HA

(8)

n

[SYMB(O,m,l)

,...,SYMB(O,m+n-1,l))

w e j u s t u s e t h e same symbols f o r t h e f i r s t

HA,

t h e n rename t h e o t h e r v a r i a b l e s of

condition

is satisfied.

(8)

Hn

n

v a r i a b l e s of

w e have t h a t

xO , . . . , ~ n - l E S

and

so that

Consider now t h e Boolean e x p r e s s i o n

..., xn - l , x n ,...,x

xo,

Hn

( i f necessary)

i f t h e l a t t e r c o n j u n c t i o n were s a t i s f i a b l e

H n h HA:

t h e s i s ) , say

;

n5

(absurdum hypo-

HnA HA , t h e n by ( 7 )

satisfies

, which i s i m p o s s i b l e .

Therefore

we get: (9)

Hn-iHv

i s a tautology, f o r each

n

S i n c e w e a r e assuming t h a t

(iii) i s f a l s e , l e t

that is,

( i i i ) i n Theorem 2 :

example t o

n& 1.

$

$

be a c o u n t e r -

i s computable i n

t e r m i n i s t i c polynomial t i m e , and misses no i n t e r p o l a n t s .

,

= $(Hn

1HA);

then

So l e t

deIn

has t h e following properties:

In

(10)

H n I In

and

(11)

v a r In

{SYMB(O,~,I)

(12)

t h e mapping

In-

a r e t a u t o 1o g i e s ;

i H A

,...,S Y M B ( O , ~ + ~ - ~ , ~ ) ;) .

=

i s computable i n d e t e r m i n i s t i c

In

In

polynomial t i m e . Clause

(12)

i s a consequence of our assumptions about

w i t h t h e f a c t t h a t t h e maps

n

+ Hn

and

computable i n determin s t i c polynomial t i m e . in (13)

( 1 0 ) . and from Mod H~

Hence, by (14)

and

(8

r (first

n

(11)

bits)

C_

w e have Mod I

n

.

Mod In.

S i m i l a r l y , from t h e second t a u t o l o g y i n

(15)

a r e both

From t h e f i r s t t a u t o l o g y

(7) we get

S n{O,lJnC

and from

n w HA

together

(8),(11),(7)

s n {O,l)n

C_

we obtain

ModlIn

.

( l o ) , w r i t t e n as

HI

n

-

71

n'

NP and Craig's Interpolation Theorem From

(14)

and

(15), recalling

(16)

Mod I n = S n j 0 , l ) "

353

we get

(6)

is satisfiable

{ X € { O , I } ~ ~r - ' ( x )

=

w h i c h c o m p l e t e s t h e p r o o f o f our L e m m a .

5 . End of t h e p r o o f of Theorem 2. W e s h a l l prove t h a t if

(iii)

I:{l)**C*

To t h i s p u r p o s e , l e t

M

and

a r e b o t h f a l s e , t h e n P=NP.

(ii)

b e as g i v e n by L e m m a 4 , and l e t

b e a d e t e r m i n i s t i c T u r i n g machine computing e a c h

bounded by a p o l y n o m i a l

q

i n the length

B E E *a,s compute

(D2)

w r i t e down

n =

(D3)

using

w r i t e down

(D4)

check whether

cess

M

Ir(B)l r(B)

3lBI;

=

In; In.

c a n b e c a r r i e d o u t i n d e t e r m i n i s t i c p o l y n o m i a l time

(D4)

q ( n ) , and

r(B)

i n t h e l e n g t h OP

L e t now

B.

\In(

i s bound-

b i t s . The a b o v e p r o -

B

i n t i m e bounded by a p o l y n o m i a l

Therefore we conclude t h a t , under our assumptions,

T h i s c o m p l e t e l y provets

TAUTSZ*

t a u t o l og i e s

n

provides t h e required d e c i s i o n procedure f o r s a t i s -

(Dl)-(D4)

holds.

ensure t h a t

M

i s a sequence of

f i a b i l i t y of any Boolean e x p r e s s i o n

P=NP

on

r(B);

as c l a i m e d , s i n c e t h e p r o p e r t i e s o f e d by

of t h e i n p u t .

follows:

(Dl)

Notice t h a t

i n time

for satisfiability is,

A fast(deterministic)decision p r o c e d u r e

input

n

In

t h e Theorem.

d e n o t e t h e set o f B o o l e a n e x p r e s s i o n s w h i c h

are

. A t l e a s t o n e of t h e f o l l o w i n g s t a t e m e n t s h o l d s t r u e :

6.Corollary. (I)

TAUT

i s accepted i n d e t e r m i n i s t i c polynomial time;

(11)

TAUT

i s n o t accepted i n n o n d e t e r m i n i s t i c polynomial time;

(111)

same a s s t a t e m e n t

P r o o f . I t i s well-known

(iii) that

i n Theorem

TAUT

is i n

2.

P

iff

P=NP

(see [2]).

S i m i l a r l y , TAUT i s i n NP i f f NP i s c l o s e d u n d e r c o m p l e m e n t a t i o n (see, e.g.,

[3.

1.11

).

Now a p p l y Theorem

2.

D.MUNDICI

354

The above C o r o l l a r y i s s t a b l e under r e l a x a t i o n of t h e upper bounds for

computations, a s w e s h a l l

( d e t e r m i n i s t i c and n o n d e t e r m i n i s t i c )

7

s e e i n Theorem functions

below.

f : PI-

.A

El

%

A s usual,

3

set

C

N

5

all

i s c l o s e d under composition

N

5

i f f t h e composition of any two f u n c t i o n s i n in

t h e set of

denotes

i s still a function

c l o s u r e under sum i s s i m i l a r l y d e f i n e d .

;

7. Theorem.

9C%

Let

be an a r b i t r a r y set c o n t a i n i n g t h e polynoThen a t l e a s t one

m i a l s and c l o s e d under composition and sum.

3

( p e r h a p s depending on

)

of t h e f o l l o w i n g s t a t e m e n t s h o l d s t r u e :

TAUT i s a c c e p t e d by some d e t e r m i n i s t i c T u r i n g machine i n t i m e

(Ig)

3

bounded by a f u n c t i o n of (113)

( i n t h e l e n g t h of t h e i n p u t ) ;

i s n o t a c c e p t e d by any n o n d e t e r m i n i s t i c T u r i n g machine

TAUT

- c*

i n t i m e bounded by a f u n c t i o n of For every

(1113)

p

:c * X

c*

3.

;

,

4,

if

a

i s computable by

,

d e t e r m i n i s t i c T u r i n g machine i n time bounded by a f u n c t i o n of then t h e r e i s a tautology such t h a t

C , with

B--,

-

n var

var B

B,CEC*,

i s n o t an i n t e r p o l a n t f o r

H(B.C)

B

f @,

C

C.

For t h e proof w e modify L e m m a s 3 and 4 as f o l l o w s :

8. Lemma.

@

Assume

f

Then t h e r e e x i s t s

F:

i s accepted by a

non-

i n t i m e bounded by a f u n c t i o n

f e z

S E{0,1}*,

d e t e r m i n i s t i c T u r i n g machine (1)"-

T

and

z*

S

by a d e t e r -

which i s computable

3 ,

m i n i s t i c T u r i n g machine i n t i m e bounded by a f u n c t i o n of letting lvar Proof.

w e have f o r each

Fn = F ( I n )

F

~

n, I

and ~

s

n {o,i)"

=

Write down

Fn

T

i s now g i v e n by

and n o t e t h a t

of t h e p r e s e n t Lemma.

Fn

f

such t h a t

n>l: Mod

~~r

Argue e x a c t l y as i n t h e proof of Lemma 3

upper bound f o r

.

(first

n

bits).

( ( a ) =3 ( b ) ) ;

( i n s t e a d of

p

the

therein).

s a t i s f i e s t h e second r e q u i r e m e n t

T o see t h a t t h e mapping

n W F n

i s computable

by a d e t e r m i n i s t i c T u r i n g machine i n t i m e bounded by a f u n c t i o n of by analogy w i t h t h e f i n a l o b s e r v a t i o n i n t h e proof of Lemma 3,

'5 ,

first

note that lFnl i s a t most p r o p o r t i o n a l t o m4 , t h a t i s , p r o p o r t i o n a l 4 can be w r i t t e n down i n time n o t much g r e a t e r t o f ( n ) . Again, than

IFn)

,

F,

say f o r definiteness lFn12

.

But t h e f u n c t i o n

f8(n)

355

NP and Craig's Interpolation Theorem

2 !FA2 still is in 9.Lemma.

(119) and

).

r

{0,1)

€

*\ z

Z

be defined by

c*\TAUT

. 3Clearly,

1°C

=

p ( x ) is

{O,l>*I

r

,

is accept-

2

in time bounded by

Y

is in

.

w

we can safely assume

c*

HI : {I}*-

w'.

=

By Lemma 8

3

)

zn{o,I)n

=

Mod

zn(O,l}n

=

Mod

in

Hl;r

Since

(1113)

sum,

deterministic Turing

such a way

5

(using the

that

(first n

bits),

(first n

bits).

and

-

Arguing now as in the proof of Lemma 4 one shows that is a tautology.

W' in

there are functions

which are computable by

3

-

Z=

is closed under

machines in time bounded by the same function U E closure properties of

a

NP, hence the set

is accepted by a nondeterministic Turing machine

time bounded by a function ~ ' € 3 Since

H,

such that letting

is a tauto~ogy).

By assumption, and by definition of

ed by a nondeterministic Turing machine €unction ~

3 ,

n21

(x E {O,l}n

z c_ {o,I)*

Let

a taut 01ogy

a

which is computable by a deterministic

we have for each Mod In =

Proof.

(1119) are both false. Then there is

machine in time bounded by a function of

In = I(ln)

.

3

by the assumed closure properties of

- x*

Assume

function I: {I}* Turing

3,

Hn is assumed to be false, let

- I Hn '

$

be

computable by a deterministic Turing machine in time bounded by some function

bE

4

tautology, $(Hn

3

, 1 HA);

, with the property that whenever B -C is a (B,c) is an interpolant for B + C. Let In

the mapping

n !-+

In

Turing machine in time bounded by some function d € g tained as a suitable composition of the functions u,b some polynomial).

The mappingnI,n

(d

can be ob-

together with

is now proved to satisfy all

our requirements by the same argument as in the end of the proof Lemma

4.

This completes the proof o€ Lemma

3

Turing machine

accepting

Thus Theorem 7

is proved.

of

9.

Arguing now as in section 5 , using Lemmas 8 and 9 closure properties of

=

is computable by a deterministic

and the

, one easily produces a deterministic TAUT

in time bounded by a function of

5.

D. MUNDICI

356

10.

F u r t h e r Topics.

I n t h i s f i n a l s e c t i o n w e survey

what i s known on t h e complexity

of

e s h a l l s t a t e a number of r e s u l t s C r a i g ' s i n t e r p o l a t i o n theorem. W c o n c e r n i n g t h e r a t e of growth of i n t e r p o l a n t s , b o t h i n s e n t e n t i a l Boolean e x p r e s s i o n s for s e n t e n t i a l l o g i c

and i n f i r s t - o r d e r l o g i c .

are p a r t i c u l a r words over a l p h a b e t

c

as d e f i n e d i n s e c t i o n 1 .

Sentences of f i r s t - o r d e r l o g i c are u n d e r s t o o d as p a r t i c u l a r over some s u i t a b l e a l p h a b e t ).

( s ee [I]

rules

c' ,

words

according t o t h e f a m i l i a r formation

I n s e n t e n t i a l l o g i c t h e precise determination

of t h e r a t e of growth

of i n t e r p o l a n t s i s an open

(and i m p o r t a n t )

The Pollowing Theorem s t a t e s t h a t if s e n t e n t i a l i n t e r p o l a n t s

problem.

t u r n out t o grow polynomially, i n d e t e r m i n i s t i c polynomial

t h e n e v e r y f u n c t i o n which i s computable

(Turing) t i m e ,

has c i r c u i t depth

p o r t i o n a l t o t h e l o g a r i t h m of t h e i n p u t l e n g t h .

pro-

T h i s would p r o v i d e

a p o s i t i v e s o l u t i o n t o a c e n t r a l open problem of computation t h e o r y (see [I I] )

.

Assume t h e r e e x i s t s a polynomial

10.1 Theorem. whenever

an i n t e r p o l a n t

with

I

111

t h e r e s t r i c t i o n of

f

d e p t h Fn See [6

Recall function

,

< p ( 1BI +

F1 , F 2 ,

to

,

2.3.23)

f : {O,l}n-

I

tO.1)

,

Then f o r e v e r y f u n c t i o n

c i r c u i t s , with

d e l a y complexity of

computing

n

c>O,

.

n = 1,2,...

. that the delay

complexity

of a Boolean

i s t h e d e p t h oP t h e smallest d e p t h

(over our f i x e d b a s i s

Boolean e x p r e s s i o n

F

such t h a t , f o r some

f o r each

)

{A, V

. As u s u a l ,

any

i s r e g a r d e d as a Boolean f u n c t i o n o v e r i t s own

B

variables, via the identification

propagate t o

ICl).

... of

[O,lJn,

< c - l o g2 n

Theorem 2 . g

(from b 0

circuit for

such t h a t

which i s computable i n d e t e r m i n i s t i c polynomial

(0'1)

t i m e t h e r e i s a sequence

ProoP.

p

i s a t a u t o l o g y i n s e n t e n t i a l l o g i c , one can f i n d

B-C

f : {O,l]*-

for t h e n e c e s s a r y background:

See f l 0 , 2.21

B

1 = 'ltrue'l

i s , roughly,

and

0 = "false"; the

t h e t i m e needed f o r i n p u t s t o

t h e o u t p u t , i n t h e Pastest c i r c u i t computing

B.

The

f o l l o w i n g Theorem t h e n s t a t e s t h a t t h e t i m e needed by t h e f a s t e s t

NP and Craig's Interpolation Theorem

c i r c u i t t o compute terpolant

( t h e Boolean f u n c t i o n c o r r e s p o n d i n g t o ) any

needed t o compute e i t h e r of 10.2 Theorem. d<620) and I

C

or

B

C:

F o r i n f i n i t e l y many

t h e r e i s a tautology

d€N

(and s t a r t i n g w i t h

some

i n s e n t e n t i a l l o g i c , with

B+C

B

having t h e i r d e l a y complexity S d , such t h a t e v e r y i n t e r p o l a n t

h a s a d e l a y complexity

Proof.

in-

B - + C may happen t o be g r e a t e r t h a n t h e time

for

I

351

17,

See

d

I

>d +

(1/3)log2(d/2).

Theorem 2.51

As remarked above, i n s e n t e n i a l l o g i c t h e r e i s a t p r e s e n t no deEini-

t i v e e s t i m a t e oP t h e r a t e oP growth of ICl

, where

Theorem

a s a f u n c t i o n of IBI and

111

i s a smallest l e n g t h i n t e r p o l a n t € o r

I

(See [6,

B-C.

1.93 f o r an upper bound, and t r y t o improve i t ) .

By c o n t r a s t

i n f i r s t - o r d e r l o g i c w e have: 10.3 Theorem. ( i ) In t h e arithmetical hierarchy t h e r e i s a

TI1 - f u n c t i o n

g i v i n g an upper bound f o r t h e l e n g t h oP P i r s t - o r d e r (*)

whenever

B+C

interpolant ( i i ) No

c

1

I

-function

with

B+C

as i n

(*)

[ 6 , Theorem 3.13

Theorem

b

ICl). can g i v e an

.

Proof.

,

b(lBI+

111

( i . e . , no r e c u r s i v e f u n c t i o n )

111

[4

i n t e r p o l a n t s , i.e..

i s v a l i d i n f i r s t - o r d e r l o g i c , t h e r e i s an for

upper bound f o r ( i ) See

b: N+pl

.

(11)

T h i s c a n be e x t r a c t e d from

11.

Due t o i t s a s y m p t o t i c c h a r a c t e r , t h e above Theorem

10.3 ( i i ) g i v e s

no i n f o r m a t i o n on t h e p o s s i b l e l e n g t h s of i n t e r p o l a n t s € o r s h o r t plications.

The 'following i s a non-asymptotic r e s u l t :

We c a n w r i t e down a v a l i d i m p l i c a t i o n i n f i r s t - o r d e r

10.4 Theorem.

logic,

with

B---+C

l B ( .lC(<1145

pol a n t w e have : .*2

III> 2'

Proof.

See

im-

&,

Theorem 3 . 4 .

1

such t h a t whenever

seven t w o ' s

.

I

i s an i n t e r -

D.MUNDICI

358

References Chang C . C . and Keisler H . J . , Model Theory (North-Holland, Amsterdam, second e d i t i o n 1 9 7 7 ) . Cook S . A . , The complexity of theorem p r o v i n g p r o c e d u r e s , I n : P r o c e e d i n g s of t h e T h i r d Annual ACM Symp. on t h e Theory of Computing, Yay 1 9 7 1 , pp.151-158. Cook S . A . and Reckhow R.A., The r e l a t i v e e f f i c i e n c y of p r o p o s i t i o n a l proof s y s t e m s , J o u r n a l of Symb.Logic 44 ( 1 9 7 9 ) 36-50. Friedman H., The complexity of e x p l i c i t d e f i n i t i o n s , Advances i n Mathematics 2 0 ( 1 9 7 6 ) 18-29. Machtey M. and Young P . . An I n t r o d u c t i o n t o t h e General Theory of Algorithms (North-Holland, Amsterdam, t h i r d p r i n t i n g 1 9 7 9 ) . Mundici D . , Complexity of C r a i g ' s i n t e r p o l a t i o n , Annales SoC. Math.Pol., Series I V : Fundamenta I n f o r m a t i c a e V.3-4 (1982) 261- 278. Mundici D . , A lower bound f o r t h e complexity of C r a i g ' s i n t e r p o l a n t s i n s e n t e n t i a l l o g i c , Archiv math.Logik ( 1 9 8 3 ) t o appear. Yundici D . , D u a l i t y between l o g i c s and e q u i v a l e n c e r e l a t i o n s , T r a n s a c t i o n s A m e r Math SOC 27 0 ( 1 98 2) 1 1 1 -1 29.

.

. .

Mundici D . , Compactness, i n t e r p o l a t i o n and F r i e d m a n ' s t h i r d problem, Annals of Mathematical Logic 22 ( 1 982) 197-21 1

.

Savage J.E.,

The Complexity of Computing (Wiley, New Y o r k 1 9 7 6 ) .

Schnorr C.P., The network complexity and t h e T u r i n g machine complexity of f i n i t e f u n c t i o n s , Acta I n f o r m a t i c a 7 (1976) 95-107.