Words, Languages & Combinatorics III

WORDS, LAN

COMBINATO

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Proceedings of the International Conference

WORDS, lR#GURGES ft COMBINATO Kyoto, Japan

14 - 18 Mavch 2000

Editors

Masarni I t o Kyoto Sangyo University, Japan

Teruo Imaoka Shimane University, Japan

b

World Scientific NewJersey London Singapore Hong Kong

Published by World Scientific Publishing Co. Re. Ltd. 5 Toh Tuck Link, Singapore 596224 USA once: Suite 202, 1060 Main Street, River Edge, NJ 07661 UK once: 57 Shelton Street, Covent Garden, London WC2H 9HE

British Library Cataloguing-in-PublicationData A catalogue record for this book is available from the British Library

WORDS, LANGUAGES & COMBINATOFUCS III Proceedings of the Third InternationalColloquium Copyright 0 2003 by World Scientific Publishing Co. Re. Ltd. All rights reserved. This book, orparts thereof; may not be reproduced in any form or by any means, electronic or mechanical, includingphotocopying, recording or any information storage and retrieval system now known or to be invented, without wrirten permissionfrom the Publisher.

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ISBN 981-02-4948-9

Printed in Singapore.

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Preface The Third International Colloquium on Words, Languages and Combinatorics was held at Kyoto Sangyo University from March 14 to 18, 2000. The colloquium was a continuation of the previous two International Colloquiums on Words, Languages and Combinatorics held in Kyoto in 1990 and 1992. The colloquium was organized under the sponsorship of the Institute of Computer Science at Kyoto Sangyo University and with the financial support of the Asahi Glass Foundation and the Japan Society for Promotion of Science. The program committee consisted of the following members:

J. Almeida ( U . Porto, Portugal), J . Brzozowski (U. Waterloo, Canada), C. Calude ( U . Auckland, New Zealand), J . Dassow (U. Magdeburg, Germany), K. Denecke (U. Potsdam, Germany), V . Diekert ( U . Stuttgart, Germany), F. G6cseg (U. Szeged, Hungary), T . Hall (Monash U., Australia), T. Head (Binghamton U., USA), J . Howie (U. St Andrews, UK), T . Imaoka (Shimane U., Japan), M. Ito (Kyoto Sangyo U., Japan, chair), H. Jurgensen (U. Western Ontario, Canada & U. Potsdam, Germany), J . Karhumaki (U. Turku, Finland), M. Katsura (Kyoto Sangyo U., Japan), S. Marcus (U. Bucharest, Romania), J. Meakin (U. Nebraska, USA), M. Nivat (U. Paris VI, France), Gh. P5un (Ins. Mathematics, Romania), J . Reif (Duke U., USA), N . Reilly (Simon Fraiser U., Canada), G. Rozenberg (U. Leiden, Netherlands), J . Sakarovitch (ENS des Telecommunication, France), B. Schein (U. Arkansas, USA), G. Thierrin ( U . Western Ontario, Canada), P. Trotter (U. Tasmania, Australia), Do Long Van (Ins. Mathehmatics, Vietnam), M. Volkov (Ural State U., Russia) The topics of the colloquium were: ( a ) semigroups, especially free monoids and finite transformation semigroups, ( b ) codes and cryptography, ( c ) automata, ( d ) formal languages, ( e ) varieties of semigroups and languages, ( f ) word problems, ( 9 ) word- and term-rewriting systems, ( h ) ordered structures and categories, (i) combinatorics on words, ( j ) complexity and computability, ( k ) molecular computing, especially DNA computing, (/) quantum computing The number of participants was 92, from 19 different countries. There were 69 lectures (5 plenary lectures among them) during the sessions. The colloquium was arranged by the conference committee consisting of the following members:

P. Domosi (U. Debrecen, Hungary), Z. Esik ( U . Szeged, Hungary), U . Knauer (U. Oldenburg, Germany), Y. Kobayashi (Toho U., Japan), T. Imaoka (Shimane U., Japan, co-chair), B. Imreh (U. Szeged, Hungary), M. Ito (Kyoto Sangyo U., Japan, cechair), M. Katsura (Kyoto Sangyo U., Japan), M. Kudlek (U. Hamburg, Germany), C. Nehaniv (U. Hertfordshire, UK), F. Otto (U. Kassel, Germany), K. Shoji (Shimane U., Japan), K.P. Shum (Chinese U. of Hong Kong, Hong Kong)

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This volume contains papers based on lectures given at the colloquium. All papers have been refereed. The editors express their gratitude to all contributors of this volume including the referees. The organizers would like to express their thanks to the Institute of Computer Science, the Asahi Glass Foundation, the Japan Society for Promotion of Science and the World Scientific Publishing Company for providing the conditions to host the colloquium. We are also grateful to Ms. Yuki Yasuda, Ms. Miyuki Endo, Ms. Tomomi Hirai, Ms. Chikage Totsuka, Mr. Kenji Fujii, Mr. Taro Nakamura, Mr. Jun-ichi Nakanishi, Mr. Tetsuya Hirose and Mr. Ryo Sugiura for their help to realize the colloquium. Finally, we would like to express our appreciation for the assistance of Mr. Christopher Everett during the editing procedure.

March 2003 Masami Ito Department of Mathematics Kyoto Sangyo University Teruo Imaoka Department of Mathematics Shimane University

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Scientific Program March 14, 2000 Plenary lecture 10.00-10.50 Gh. PYun (Instilute of Mathematics of the Romanian Academy),

P systems: An early survey Section A Invited lectures 11.00-11.40 J . Gruska (Masaryk University), Quantum challenges in automata theory 11.40-12.20 D.L. Van (Hanoi Institute of Mathematics), A unified approach to the embedding problem for codes defined by binary relations

12.20-13.00 N.H. Lam (Hanoi Institute of Mathematics), Finite maximal solid codes Contributed lectures 14.30-15.00 D.Y. Long & W.J. Jia (City University of Hong Kong), A new symmetric cripto-algorithm based on prefix codes 15.00-15.30 V. Brattka (FernUniversitat Hagen), The emperor’s new recursiveness 16.00-16.30 A. Yamamura (Communications Research Laboratory) & K. Kurosawa (Tokyo Institute of Technology), Key agreement protocol over a commutative group 16.30-17.00 G. Horvbth, K. Inoue, A. Ito & Y. Wang (Yamaguchi University), Closure property of probabilistic Turing machines and alternating Turing machines with subalgorithmic spaces

...

Vlll

Section B Invited lectures 11.00-11.40 M. Steinby ( University of Turku), Tree automata in term rewriting theory 11.40-12.20 G. Niemann & F. Otto (UniversitZt Kassel), Some results on deterministic restarting automata 12.20-13.00 E. Csuhaj-Varj~(Computer and Automation Research Institute, Hungarian Academy of Sciences) & A. Salomaa (Turku Centre for Computer Science), Networks of Watson-Crick DOL systems

Contributed lectures 14.30-15.00 G. Horvith (Yamaguchi University), Cs. Nagylaki (University of Debrecen) & 2. Nagylaki (Hiroshima University), Visualization of cellular automata 15.00-15.30 H. Nishio, Cellular automata with polynomials over finite fields 16.00-16.30 R. Schott (Universitd Henri Poincard) & J.-C. Spehner (Universitd de Haute Alsace) Two optimal parallel algorithms on the commutation class of a word 16.30-17.00 I. Inata (Toho University), Presentations of right unitary subiiionoids of monoids

March 15, 2000 Plenary lecture 10.00-10.50 J . Meakin (University of Nebraska), One-relator inverse monoids and rational subsets of one-relator groups

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Section A Invited lectures 11.00-11.40 L. Kari (University of Western Ontario), Computation in cells 11.40-12.20 S.W. Margolis (Bar-Ilan University), J.-E. Pin (Universitk Paris 7) & M.V. Volkov (Ural State University), Words guaranteeing minimal image 12.20-13.00 C. Campbell (University of St Andrews), The semigroup efficiency of groups and finite simple semigroups

Contributed lectures 14.30-15.00 T. Buchholz, A. Klein & M. Kutrib (University of Giessen), Iterative arrays with limited nondeterministic communication cell 15.00-15.30 T. Saito, Acts over right, left regular bands and semilattices types 16.00-16.30 G. Mashevitzky (Ben Gurion University of the Neger), On definability of weighted circulants by identity 16.30-17.00 M. CiriC & T. PetkoviC (University of NiS), Syntactic and semantic properties of semigroup identities

Section B Invited lectures 11.00-11.40 J. Karhumaki (University of Turku), Remarks on language equations 11.40-12.20 A. Mateescu (University of Bucharest), Routes and trajectories 12.20-13.00 J . Dassow (Otto-von-Guericke-Universitat Magdeburg), On the differentiation function of some language generating devices

Contributed lectures 14.30-15.00 M. Ogawa (NTT Communication Science Laboratories), Well-quasiorders and regular w-languages

X

15.00-15.30 J.A. Anderson (University of South Carolina) & W. Forys (Jagiellonian University), Regular languages and seniretracts 16.00-16.00 K . Hashiguchi, Y. Wada & S. Jimbo (Okayama University), Regular binoid expressions and regular binoid languages 16.30-17.00 K. Shoji (Shimane University), On a proof of Okninski and Putcha’s theorem

March 16, 2000 Plenary lecture 10.00-10.50 J. Shallit (University of Waterloo), Number theory and formal languages

Section A Invited lectures 11.00-11.40 K . Denecke (University of Potsdam), Tree-hyper recognizers and treehyper grammars 11.40-12.20 Z. Esik (University of Szeged) & W. Kuich (Technische Universitat Wien), Inductive *-semirings 12.20-13.00 V. Diekert & & C. Hagenah (Universitat Stuttgart), A remark on equations with rational constraints in free groups

Contributed lectures 14.30-15.00 F. Bassino (Universitd de Marne La Vallde), A characterization of cubic simple beta-numbers 15.00-15.30 T. Poomsa-ard (Khon Kaen University), Hyperidentities in medial graph algebras 15.30-16.00 T. PetkoviC, M. Cirid & S. Bogdanovid (University of Nis), Nonregular varieties of automata

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Section B Invited lectures 11.00-11.40 S. Marcus (Institute of Mathematics of the Romanian Academy), From infinite words to languages and back: an expected itinerary 11.40-12.20 C. Choffrut & S. Grigorieff (UniversitC Paris 7), Rational relations on transfinite strings 12.20-13.00 T. Yokomori (Waseda University), On approximate learning of DFAs

Contributed lectures 14.30-15.00 S. Konstantinidis (Saint Mary’s University), Error-detecting properties of languages 15.00-15.30 M. It0 (Kyoto Sangyo University) & Y . Kunimochi (Shizuoka Institute of Science and Technology), On C P N languages 15.30-16.00 S.V. Avgustinovich, D.G. Fon-Der-Flaass & A.E. &id (Sobolev Institute of Mathematics), Arithmetical complexity of infinite words

March 17, 2000 Plenary lecture 10.00-10.50 J. Almeida (University of Porto) & A. Escada (University of Coimbra), Semidirect products with the pseudovariety of all finite groups

Section A Invited lectures 11.00-11.40 K.P. Shum (Chinese University of Hong Kong), On super Hamiltonian semigroups 11.40-12.20 G. SCnizerguesu (UniversitC Bordeaux I), The equivalence problem for a subclass of Q-algebraic series

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12.20-13.00 M. Ozawa (Nagoya University), Computational equivalence between quantum circuits and quantum Turing machines

Contributed lectures 14.30-15.00 0. Carton (Universitk de Marne-la-vallk), R-trivial languages of words on countable ordinals 15.00-15.30 N. RuSkuc (University of St Andrews), Some (easy?) questions concerning semigroup presentations 16.00-16.30 B. Steinberg (University of Porto), Polynomial closure and topology 16.30-17.00 H. Machida (Hitotsubashi University), Some properties of hyperoperations and hyperclones 17.00-17.30 R. Matsuda (Ibaraki University), Characterization of valuation rings and valuation semigroups by semistar-operations

Section B Invited lectures 11.00-11.40 A.V. Kelarev & P.G. Trotter (University of Tasmania), A combinatorial property of automata, languages and syntactic monoids 11.40-12.20 J. Sakarovitch (ENST), Star height of rational languages: a new presentation for two old results 12.20-13.00 P. Domosi (University of Debrecen) & M. Kudlek (Universitat Hamburg), An improvement of iteration lemmata for context-free languages

Contributed lectures 14.30-15.00 P. Domosi (University of Debrecen), M. Kudlek (Universitat Hamburg) & s. Okawa (University of Aizu), A homomorphic characterization of recursively enumerable languages 15.00-15.30 B. Imreh (University of Szeged), M. Ito (Kyoto Sangyo University) & A. Pukler (Istvin Szkchenyi College), On commutative asynchronous automata

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16.00-16.30 T. Imaoka (Shimane University), Some remarks on representations of orthodox *-semigroups 16.30-17.00 Z. PopoviE, S. BogdanoviC, M. CiriC & T. PetkoviC (University of NiS), On finite generalized directable automata 17.00-17.30 C. Choffrut (Universitd Paris 7), S. Horvith (Eotvos Lorind University) & M. Ito (Kyoto Sangyo University), Monoids and languages of transfinite word

March 18, 2000 Plenary lecture 10.20-11.10 J.-E. Pin (Universitb Paris VII) & P. Weil (Universitd Bordeaux I and CNRS) , Semidirect products of ordered semigroups

Section A Invited lectures 11.20-12.00 A. Atanasiu, C. Martin-Vide & V. Mitrana, On the sentence valuations in a semiring - An approach to the study of synonymy 12.00-12.40 K . Auinger (Universitat Wien), Join decompositions involving pseudovarieties of semigroups with commuting idempotents

Contributed lectures 14.10-14.40 T. Koshiba (Telecommunications Advancement Organization of Japan) & K. Hiraishi (JAIST), A note on finding one-variable patterns consistent with examples and counterexamples 14.40-15.10 M. Yasugi & M. Washihara (Kyoto Sangyo University), Rademacher functions and computability

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Section B Invited lectures 11.20-12.00 J.-E. Pin (Universitd Paris VII) & P. Weil (Universiti Bordeaux 1 and CNRS), Semidirect products of ordered semigroups - Applications t o languages 12.00-12.40 C. Mauduit (Institut de Mathematiques de Luminy), Pseudorandom words

Contributed lectures 14.10-14.40 E. Moriya & T. Tada (Waseda University), Relation between the space complexity and the number of stack-head turns of pushdown automata 14.40-15.10 M. Mitrovid, S. BogdanoviC & M. CiriC (University of NiS), Iteration of matrix decompositions

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List of Speakers Almeida, J. (University of Porto) e-mail: j [email protected] Anderson, J.A. (University of South Carolina) e-mail: [email protected] Auinger, K. (Universitat Wien) e-mail: [email protected] Bassino, F. (Universitk de Marne La Vallke) e-mail: [email protected] Brattka, V. (FernUniversitat Hagen) e-mail: [email protected] Campbell, C. (University of St Andrews) e-mail: [email protected]. uk Carton, 0. (Universitk de Marne-la-Vallke) e-mail: [email protected] Choffrut, C. (Universitk Paris 7) e-mail: [email protected] CiriC, M. (University of Nis) e-mail: ciricmebankerinter .net Csuhaj-Varj6, E. (Computer and Automation Res. Inst. Hung. Academy) e-mail: [email protected] Dassow, J. (0tto-von-Guericke-Universitat Magdeburg) e-mail: [email protected] Denecke, K. (University of Potsdam) e-mail: [email protected] Diekert, V. (Universitat Stuttgart) e-mail: [email protected] Domosi, P. (University of Debrecen) e-mail: [email protected] Esik, Z. (University of Szeged) e-mail: [email protected]

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Frid, A. E. (Sobolev Institute of Mathemathics) e-mail: [email protected] Gruska, J. (Masaryk University) e - mail: gruskaeinformati cs.muni. cz Hashiguchi, K. (Okayama University) e-mail: [email protected] Horvith, S . (Eotvos L o r h d University) e-mad: [email protected] Imaoka, T . (Shimane University) e-mail: [email protected] himane-u .ac.jp Imreh, B. (University of Szeged) e-mail: imreh0inf.u-szeged.hu Inata, I. (Toho University) e-mail: [email protected] Inoue, K. (Yamaguchi University) e-mail: [email protected] Ito, M. (Kyoto Sangyo University) e-mail: [email protected] Karhumaki, J. (University of Turku) e-mail: [email protected] Kari, L. (University of Western Ontario) e-mail: [email protected] Kelarev, A.V. (University of Tasmania) e-mail: [email protected] Konstantinidis, S . (Saint Mary’s University) e-mait S.Kons t [email protected] Koshiba, T. (Secure Computing Laboratory, Fujitsu Laboratory Ltd) e-mail: koshi baeyokohama. t ao .go .jp Kudlek, M. (Universitat Hamburg) e-mail: [email protected] Kutrib, M. (University of Giessen) e-mail: [email protected] Lam, N.H. (Hanoi Institute of Mathematics) e-mail: [email protected]

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Long, D,Y. (City University of Hong Kong) e-mail: [email protected] Machida, H. (Hitotsubashi University) e-mail: [email protected] Marcus, S . (Institute of Mathematics of the Romanian Academy) e-mail: [email protected] Mashevitzky, G. (Ben Gurion University of the Neger) e-mail: [email protected] Mateescu, A. (University of Bucharest) e-mail: [email protected] Matsuda, R. (Ibaraki University) e-mail: [email protected] Mauduit, C . (Institut de Mathematiques de Luminy) e-mail: [email protected] Meakin, J. (University of Nebraska) e-mail: [email protected] Mitrana, V. (University of Bucharest) e-mail: [email protected] Mitrovib, M. (University of Nis) e-mail: meli@junis .ni.ac.yu Moriya, E. (Waseda University) e-mail: [email protected] Nagylaki, Z. (Hiroshima University) e-mail: [email protected] Nishio, H. (Kyoto, Japan) e-mail: [email protected] Ogawa, M. (NTT Communication Science Laboratories) e-mail: [email protected] Otto, F. (Universitat Kassel) e-mail: [email protected] Ozawa, M. (Tohoku University) e-mail: [email protected] P b n , Gh. (Institute of Mathematics of the Romanian Academy) e-mail: [email protected]

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PetkoviC, T. (University of NiB and TUCS) e-mail: [email protected] Pin, J.-E. (Universitk Paris VII) . e-mail: [email protected] Poomsa-ard, T. (Khon Kaen University) e-mail: [email protected] Popovid, Z. (University of NiB) e-mail: [email protected] Ruskuc, N. (University of St Andrews) e-mail: [email protected] Saito, T. (Innoshima, Japan) e-mail: [email protected] Sakarovitch, J. (ENST) e-mail: [email protected] Schott, R. (Universitk Henri Poincark) e-maik [email protected]. Senizergues, G. (Universiti Bordeaux I) e-mail: [email protected] Shallit, J. (University of Waterloo) e-mail: shallit C3graceland.math .uwaterloo. ca Shoji, K. (Shimane University) e-mail: [email protected] Shum, K.P. (Chinese University of Hong Kong) e-mail: [email protected] Steinberg, B. (University of Porto) e-mail: [email protected] Steinby, M. ( University of Turku) e-mail: [email protected] Van, D.L. (Hanoi Institute of Mathematics) e- mail: dlvan @thevin h.ncs t .ac.vn Volkov, M.V. (Ural State University) e-mail: [email protected] Weil, P. (Universitk Bordeaux I and CNRS) e-,mail: WeilC3labri.u-bordeaux.fr

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Yamamura, A. (Communications Research Laboratory) e-mail: [email protected] Yasugi, M. (Kyoto Sangyo University) e-mail: [email protected] p Yokoinori, T. (Waseda University) e-mail: [email protected]

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Table of Contents Contributed Papers Semidirect Products with the Pseudovariety of All Finite Groups . J. Almeida (Porto, Portugal) and A. Escada (Coimbra, Portugal)

. . .

1

On the Sentence Valuations in a Semiring . . . . . . . . . . . . . 22 A. Atanasiu (Bucharest, Romania), C. Martin- Vide (Tarragona, Spain) and V. Mitrana (Bucharest, Romania) Join Decompositions of Pseudovarieties of the Form DH K. Auinger (Wien, Austria)

n ECom . . .

Arithmetical Complexity of Infinite Words . . . . . . . . . . . S. V. Avgustinovich (Novosibirsk, Russia), D. G. Fon-Der-Flaass (Novosibirsk, Russia) and A. E. Frid (Novosibirsk, Russia)

. .

The Emperor’s New Recursiveness: The Epigraph of the Exponential Function in Two Models of Computability . . . . . . . . . . . . V. Brattka (Hagen, Germany)

51

.

63

.

73

. . . . . . . .

88

Iterative Arrays with Limited Nondeterministic Communication Cell T. Buchholz (Giessen, Germany), A. Klein (Giessen, Germany) and M. Kutrib (Giessen, Germany) %Trivial Languagesof Words on Countable Ordinals 0. Carton (Marne-la- Vallke, fiance)

40

The Theory of Rational Relations on Transfinite Strings . . . C. Choflrut (Paris, France) and S. Grigorieff (Paris, Frunce)

. . . .

103

Networks of Watson-Crick DOL Systems . . . . . . . . . . . . . . 134 E. Csuhaj- Varjd (Budapest, Hungary) and A. Salomaa (lhrku, Finland) On the Differentiation Function of Some Language Generating Devices J. Dassow (Magdeburg, Germany)

151

xxii

Visualization of Cellular Automata . . . . . . . . . . . . . . . . 162 M. Deminy (Debrecen, Hungary), G. Horva'th (Debrecen, Hungary), Cs. Nagylaki (Debrecen, Hungary) and 2. Nagylaki (Debrecen, Hungary) On a Class of Hypercodes . . . . Do Long Van (Hanoi, Vietnam)

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

A Parsing Problem for Context-Sensitive Languages . . . . . P. Domosi (Debrecen, Hungary) and M. It0 (Kyoto, Japan)

. . . .

171

183

An Improvement of Iteration Lemmata for Context-Free Languages . . 185 P. Domosi (Debrecen, Hungary) and M. Kudlek (Hamburg, Germany) Quantum Finite Automata . . . . . . . . . . . . . . . . . . . . 192 J. Gruska (Brno, Czech Republic) and R. Vollmar (Karlsruhe, Germany) On Commutative Asynchronous Automata . . . . . B. Imreh (Szeged, Hungary), M. It0 (Kyoto, Japan) and A . Pukler (Gyor, Hungary)

. . . . . . . .

212

Presentations of Right Unitary Submonoids of Monoids . . . . . . . 222 I. Inata (Funabashi, Japan) A Combinatorial Property of Languages and Monoids . . . . . . . . 228 A. V. Kelarev (Hobart, Australia) and P. G. Trotter (Hobart, Australia) Error-Detecting Properties of Languages S. Konstantinidis (Halifax, Canada)

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

240

A Note on Finding One-Variable Patterns Consistent with Examples and Counterexamples . . . . . . . . . . . . . . . . . . . . . . T. Koshiba (Kawasaki, Japan) and K. Hiraishi (Ishikawa, Japan)

253

On the Star Height of Rational Languages: A New Presentation for Two Old Results . . . . . . . . . . . . . . . . . . . . . . . . S. Lombardy (Paris, fiance) and J. Sakarovitch (Paris, France)

266

Some Properties of Hyperoperations and Hyperclones

H. Machida (Kunitachi, Japan)

. . . . . . . .

286

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Words Guaranteeing Minimal Image . . . . . . . . . . . . . S. W. Margolis (Ramat Gan, Israel), J.-E. Pin (Paris, fiance) and M. V. Volkov (Ekaterinburg, Russia)

. . .

297

Power Semigroups and Polynomial Closure . . . . . . . . . . . . . 311 S. W. Margolis (Ramat Gan, Israel) and B. Steinberg (Porto, Portugal)

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

.

323

Characterization of Valuation Rings and Valuation Semigroups by Semistar-Operations . . . . . . . . . . . . . . . . . . . . . . R. Matsuda (Mito, Japan)

.

339

..

352

Routes and Trajectories . . . . . . A . Mateescu (Bucharest, Romania)

Further Results on Restarting Automata . . . . . . . . . . . . G. Niemann (Kassel, Germany) and F. Otto (Kassel, Germany) Cellular Automata with Polynomials over Finite Fields H. Nishio (Kyoto, Japan)

. . . . . . . 370

Generalized Directable Automata . . . . . . . . . . . 2. Popovic' (NiS, Serbia), S. Bogdanovic' (NiS, Serbian), T. Petkovic' (Turku, Finland) and M. CiriC (NiS, Serbia) Acts over Right, Left Regular Bands and Semilattices Types T. Saito (Innoshima, Japan)

. . . . ..

. . . . . 396

Two Optimal Parallel Algorithms on the Commutation Class of a Word . . . . . . . . . . . . . . . . . . . . . . . . . . R. Schott (Nancy, France) and J.-C. Spehner (Mulhouse, France) A Proof of Okninski and Putcha's Theorem K. Shoji (Matsue, Japan)

378

.

403

. . . . . . . . . . . . 420

Subdirect Product Structure of Left Clifford Semigroups . . . . K. P. Shum (Hong Kong, China), M. K. Sen (Calcutta, India) and Y. Q. Guo (Kunming, China) Tree Automata in the Theory of Term Rewriting M. Steinby (Turku, Finland)

. . . 428

. . . . . . . . . .

434

xxiv

Key Agreement Protocol Securer Than DLOG . . . . . . . . . . . 450 A . Yamamum (Tokyo, Japan) and K. Kurosawa (Hitachi, Japan)

A Note on Rademacher Functions and Computability . . . . . M. Yasugi (Kyoto, Japan) and M. Washihara (Kyoto, Japan) Authors Index

.

. . .

466

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

477

1

Semidirect products with the pseudovariety of all finite groups* Jorge Almeida

Ana Escada

Abstract This is a survey of recent results related to semidirect products of an arbitrary pseudovariety with the pseudovariety of all finite groups. The main flavour is the establishment of links between various operators on pseudovarieties, some obviously computable, others known not to be so. This not only leads to decidability results but does so in a sort of uniform way which has a structural tint even though the arguments are mostly syntactical.

1

Introduction

Many problems in computer science lead to decidability questions on pseudovarieties of finite semigroups. Often the problem involves some sort of decomposition process which in terms of pseudovarieties translates to the calculation of a semidirect product of pseudovarieties. When just two factors are concerned, the cases in which the second factor is the pseudovariety G of all finite groups or the pseudovariety D of all finite definite semigroups have attracted the most attention [18,3, 22, 43, 461. This paper is a survey of some recent work around the theme of the semidirect product with the pseudovariety G. It uses some powerful tools to deal with such semidirect products, particularly when the second factor is the pseudovariety G , to obtain syntactic proofs of equalities of the form V * G = &V,where &Vdenotes the pseudovariety consisting of all finite semigroups 'The authors gratefully acknowledge support by FCT through the Centro de Matemc'tica da Universidade do Porto and the Centro de Matemc'tica da Universidade de Coimbm, respectively, and by the FCT and POCTI approved project POCTI/32817/MAT/2000 which is comparticipated by the European Community Fund FEDER.

2

whose idempotents generate subsemigroups from V. Subpseudovarieties V of DS are considered, including all subpseudovarieties of LI,DA, DS itself, and J. The latter of these provides a new proof of a crucial step in a result of Henckell and Rhodes [23] which is their deduction from Ash's inevitability theorem [13] of the famous equality IPG = 'BG between the pseudovariety generated by all power semigroups of finite groups and the pseudovariety of all finite semigroups in which regular elements have a unique inverse [33]. The arguments are of a syntactical/combinatorial nature. They consist in suitable formal manipulations of words in the enlarged signature with a pseudo-inversion operation which is never nested. Most proofs are only sketched here. See the full paper [6] for further details.

2

Generalities

We gather in this section the necessary notation and background for the remainder of the paper. The reader is referred to [32] for a basic introduction to finite semigroup theory and to [3] for a more comprehensive treatment based on methods which are closer to those adopted here. See also these references for any undefined terms. By a pseudovariety we mean a class of finite semigroups which contains all homomorphic images, subsemigroups, and finite direct products of members of the class. The most active and successful area of finite semigroup theory is precisely the study of pseudovarieties, particularly some natural operations on them such as the semidirect product. Such operations are often obtained by applying some natural algebraic operator to semigroups from the argument pseudovarieties and closing up to the generated pseudovariety.

2.1

Various operators on pseudovarieties

Let V and W be pseudovarieties. The semidirect product pseudovariety V * W is defined to be the pseudovariety generated by all semidirect products S* T with S E V and T E W. It turns out that rather than using general semidirect products one may use specifically the wreath product which, in a suitable context, is associative, and so the semidirect product of pseudovarieties is also associative. A few other operators will play a role in this paper. The join VVW is simply the pseudovariety generated by the class VUW or, to use an algebraic operator, by all direct products S x T with S E V andT~W. Denote by &V the class of all finite semigroups S whose idempotents

3

generate a subsemigroup which lies in V. Note that the operator E is idempotent. The Mal'cev product V @ W is the pseudovariety generated by all finite semigroups S for which there is a homomorphism cp : S -+ T with T E W and p-'e E V for every idempotent e E T . As indicated below, the Mal'cev product has important links with the semidirect product. The power operator Y associates with V the pseudovariety IPV generated by all power semigroups Y ( S )with S E V. See [3] for an extensive study of this operator and [19,201 for recent improvements and extensions. Let S be a finite semigroup and D one of its regular 'D-classes. Let be the equivalence relation on the set of group elements of D generated by the identification of elements which are either 3 or C-equivalent. A block of D is the Rees quotient of the subsemigroup of S generated by a --class modulo the ideal consisting of the elements which do not lie in D. The blocks of S are the blocks of its regular 'D-classes. The block operator associates with V the class of all finite semigroups whose blocks lie in V, which can be shown to be a pseudovariety. For a semigroup S, denote by E ( S ) the set of its idempotents. The local operator C is defined by letting LV consist of all finite semigroups S all of whose submonoids of the form eSe, with e E E ( S ) ,lie in V. Note that C is also an idempotent operator. The class DV is defined to consist of all finite semigroups whose regular 'D-classes are subsemigroups which lie in V. It is again easy to see that DV is a pseudovariety. Among the above operators, which do not exhaust those of interest for the applications, some have explicit structural definitions while others involve taking the pseudovariety generated by a subclass which itself is defined explicitly in that sense. Note that membership in classes with such explicit structural definitions is relatively easy to test and, in particular, can be done algorithmically. Call a class of finite semigroups decidable if there is an algorithm to test membership in it. It is by no means obvious how to construct an algorithm for the pseudovariety generated by a decidable class. In fact, this task is not always possible. More precisely, the join [l] and the semidirect and Mal'cev products [37] of decidable pseudovarieties may not be decidable. Recently, Auinger and Steinberg [15] have announced that the power operator also fails to preserve decidability. Thus, any connections which may be found between operators defined by generators and structurally defined operators are particularly useful and often translate in an elegant manner into algorithms for computing values of the former. We review below some such connections which are of interest for the specific topic of this paper. For this purpose, we need to introduce some

-

4

further ideas and results.

2.2

Semidirect products with D

Tilson [46] introduced the notions of pseudovariety of (finite) categories and of pseudovariety of (finite) semigroupoids (categories without the requirement of local identities) and he showed that one is led to considering them by studying semidirect products of pseudovarieties of semigroups. In this context, semigroups are seen as semigroupoids by viewing elements as edges (or morphisms) at a virtual single vertex (or object). On the other hand, the edges of a semigroupoid with both ends at a particular vertex v, assuming there is at least one, form a semigroup which is called the local semigroup at v. See Tilson’s work (and the recent continuation [41]) for precise definitions and results. It is well known that pseudovarieties of semigroups may be defined by formal equalities between members of free profinite semigroups (this is basically Reiterman’s theorem [2, 351; see [lo] for a presentation in this language), which are called pseudoidentities. For pseudovarieties of semigroupoids there is an analogous result where free profinite semigroupoids freely generated by finite graphs play the role of generating sets of free profinite semigroups [ll, 261. Thus, pseudoidentities for pseudovarieties of semigroupoids are written over finite graphs. See Theorems 2.4 and 2.6 for specific examples. The global gV of a pseudovariety V of semigroups is the pseudovariety of semigroupoids generated by V. We say that V is local if gV is defined by pseudoidentities over 1-vertex graphs. Consider some frequently encountered pseudovarieties of semigroups, where [C]denotes the class of all finite semigroups which satisfy all members of a set C of pseudoidentities. In such pseudoidentities we adopt the convention that e, f,. . . stand for idempotents and 0 for a zero. So, for instance, a semigroup satisfies the pseudoidentity ex = xe if and only if its idempotents commute with all elements. S1 = {finite semilattices} = [x2 = z,zy = y z ]

B = {finite bands} = [x2 = x ] 0 = {finite orthodox semigroups} = [ ( e f ) 2 = ef] Corn = {finite commutative semigroups} R = {finite %trivial semigroups} L = {finite C-trivial semigroups} J = {finite &trivial semigroups} = R n L

5

D = {finite semigroups in which idempotents are right zeros} D, = [ Y Z ~" - z n = 2 1 ..-z,I K = {finite semigroups in which idempotents are left zeros} K, = [ZI.*-z,Y = ~1 .-.z,] N = {finite nilpotent semigroups} = K n D = [e = 01 G = {finite groups} Ab = {finite Abelian groups} LG = {finite left groups) = D1 V G = [ e z = z] RG = {finite right groups} = K 1 V G = [ z e = z] A = {finite aperiodic semigroups} CS = {finite simple semigroups} CR = {finite completely regular semigroups} = {finite unions of groups} S = {finite semigroups} I = {singleton semigroups} We warn the reader that the notation 0 is often found in the literature with a different meaning [24]. A pseudovariety of semigroups is said to be monoidal if it is generated by its monoids. The interest of the notion of a local pseudovariety of semigroups comes from the following result which is a simplified version of the so-called delay theorem.

Theorem 2.1 ([46]). A monoidal pseudovariety V of semigroups is local if and only if V * D = CV. Theorem 2.2 (1171). The pseudovariety S1 is local. Theorem 2.3 ([18, 421). The pseudovariety R is local (therefore so is L). Theorem 2.4 ([30]). The pseudovariety J is not local, 2.

gJ = [ ( z y ) " z t ( ~ t ) "= (zy)"(Zt)"; -.

z

1.'

Y, t

Theorem 2.5 ([43]). A s pseudovarieties of monoids, nontrivial pseudovarieties of groups are local.

--

Theorem 2.6 ([45]). The pseudovariety Com is not local,

gCom = [ z y z = z y z ;

-.

2,z Y

1.

An orthogroup is an orthodox completely regular semigroup. 'See Subsection 2.4 for a definition of the w-power.

6

Theorem 2.7 ([27]). Every monoidal pseudovariety of orthogroups which does not consist entirely of groups is local. Theorem 2.8 ([28]). The pseudovariety 'DS is local. Theorem 2.9 ([4]). The pseudovariety DA is local.

2.3

Semidirect products -

* H vs other operators

It is a well-known result that, for a pseudovariety V,

and the first inclusion is an equality if V is local and monoidal. Theorem 2.10 ([42, 181). R * G = ER. Theorem 2.11 ([12]). S1* G = ESl. Theorem 2.12 ([IS]). The equality V @ G = EV holds for every nontrivial monoidal pseudovariety V of bands.

Combining with Theorem 2.7, we deduce that the equality V * G = EV holds for every nontrivial monoidal pseudovariety of bands. In the same paper the authors of Theorem 2.12 claim to establish the following result but their arguments are flawed. Based on results of Szendrei [44], P. Trotter has shown in unpublished work that indeed the result is true. Theorem 2.13. CR * G = ECR.

The following is a combination of results of Margolis and Pin [31] and Henckell and Rhodes [23], the latter depending on a deep theorem of Ash [13] which is presented further below. Theorem2.14. I P G = J * G = J @ G = E J = ' B G .

A pseudovariety of groups V is said to be arborescent if (V n Ab) * V = V. Gildenhuys and Ribes [21] had shown that, for such pseudovarieties, their free profinite groups have Cayley graphs which are profinite trees (in a natural homological sense). Almeida and Weil [9] in turn showed that the converse also holds. The following result is a combination of parts of results of Steinberg [38, 391 who has done very extensive work on pseudovarieties of the form V * H and related pseudovarieties. Theorem 2.15. For an arborescent pseudovariety H of groups, IPH = J

H=J@H.

*

7

By further refining the study of the geometry of Cayley graphs of relatively free profinite groups, Auinger and Steinberg [14] have recently obtained a characterization of all pseudovarieties of groups for which the equality J *H = J @ H holds. The situation is however different from that of G.

Theorem 2.16 ([25]). The inequality J @ H # 'BH holds for every pseudovariety H G closed under extension.

2

5

Note that IBH # EV for every pseudovariety H G and every pseudovariety V since G E l but G IBH. From a result of Karnofsky and Rhodes [29] it follows that

c

A*GS&A. For a pseudovariety H of groups, denotes the class of all finite semigroups all of whose subgroups lie in H, which is easily shown to be a pseudovariety. Theorem 2.17 ([47]).For V = CS n Ab, we have the following strict inclusions: V * G V @ G EV.

5

2

Fkom work of Rhodes [36] it follows that V = A * G is also an example in which both inclusions in ( 1 ) are strict and in fact EV = &A. Among the various questions suggested by the above results, we consider in this paper the following problem.

Problem. For which pseudovarieties V do we have V * G = EV? At this point, we add a couple of elementary observations related with the above problem.

1. The pseudovariety &Vis the largest pseudovariety W such that W * G &V.

c

2. We have &'DOE &A[6].

3. If V g &Aand (V n A) * G = E(V n A), then V * G = EV

2.4

(w- 1)-words

By an (w-1)-word we mean a term in a free unary semigroup where the unary operation is denoted (-)"-l. The height of an (w - 1)-word is h(w) = 0 if w does not involve the operation (-)"-l, and is recursively defined by letting h((~)~= - ' h(w) ) 1 and ~ ( w ~ w=zmax{h(wl), ) h(wZ)}.

+

8

The natural interpretation of the unary operation (-)"-l in a finite semigroup S associates with an element s the inverse sW-l of se in the maximal subgroup K of the subsemigroup generated by s, where sw = e is the idempotent of K . The following constitute a Noetherian system of reduction rules which preserve equality in the free group:

a(wa)w-l -+ (w)"-l (aw)w-la -+ (w)"-l (1)-1 -+ 1 The system is not confluent since, for instance, the (w-1)-word (ab)w-la(ca)w-l may be reduced to bw-l(ca)w-l and also to (ab)w-lc"-l but both of these are irreducible (w - 1)-words. For words that reduce to 1, there is a more convenient set of reduction rules. L e m m a 2.18. Let w be an (w - 1)-word of height a t most 1 which i s equal to 1 in the free group. Then it is possible to reduce w to the empty word by applying a finite number of times rules of the form U(VU)W--1V

-+ 1

(2)

where u and v are possibly empty words. L e m m a 2.19 ( P r o d u c t Inverse Formula). In a finite semigroup, the following formula holds:

where a: = (ai+l. . .anal . . .ai)w-lai+l .. .anal . . .ai-1.

(3)

Let S be a finite semigroup and let s E S. By a weak inverse of s we mean an element t E S such that tst = t. Note that sW-l is the only power of s which is a weak inverse of s. Also, the element a: of Lemma 2.19 given by (3) is a weak inverse of ai. A weak conjugate of s is an element of the form asb where one of a and b is a weak inverse of the other. The self-conjugate core D ( S ) is defined to be the smallest subsemigroup of S containing the idempotents which is closed under weak conjugation.

Corollary 2.20. Let S be a finite semigroup and let ai E S and ui E D ( S )U (1) (i = 1,.. . , n ) . Then the product

a l u l . . .a,u,(a,+l. lies in D ( S ) .

. .anal . . .

%+iG+i. . .'%an

9

By a relational morphism p : S + T between two semigroups we mean a relation with domain S which is a subsemigroup of S x T. We shall call a graph what is usually called a directed multigraph, i.e., edges are directed and there may be several edges with the same end vertices. An edge-labeling of a graph by a semigroup S is a function which associates with each edge an element of the semigroup. An edge-labeling X of a graph by a group is said to commute if, for every circuit ( e l , . . . ,en) (which is turned into an oriented cycle if some of the edges in it are reversed), we have the equality (XS,)'~ . . . (Xsn)'- = 1where ~i = -1 or E, = 1according to whether the edge ei is reversed in the circuit or not. For a pseudovariety H of groups, an edge-labeling X of a finite graph I? by a finite semigroup S is said to be H-inevitable if, for every relational morphism p : S + G into G E H, there is a commuting edge-labeling of I? by G which is p-related with A. We may now formulate Ash's inevitability theorem [13] as follows taking into account some observations in [8]. Denote by AS the free profinite semigroup freely generated by a set A.

Theorem 2.21. An edge-labeling X of a finite graph I? by a finite semigroup S i s G-inevitable if and only if, for every (or for some) onto homomorphism q : AS + S , there is an edge-labeling p of I? by (w - 1)-words of height at most 1 such that q o p = X and p commutes over G .

In the terminology of J. Rhodes, an element s of a finite semigroup S is called a type 11element if, for every relational morphism p : S + G into a finite group, (s,1) E p. In other words, the l-vertex l-edge graph labeled s is G-inevitable. The set of all type I1 elements of S is denoted K ( S ) and is called the group kernel of S. By Theorem 2.21, s E K ( S ) if and only if there is some (w - 1)-word of height at most 1 that is equal to 1 in the free group and evaluates to s in S. From this observation it is now easy to establish the following result which was known in the 1980's as the type 1 1conjecture and which was proposed by J. Rhodes. Theorem 2.22 ([13]). For every finite semigroup S , K ( S ) = D ( S ) .

Proof. Note that aba = a =+ e E E(S) a

asb = a(ba)"-'sb, e=eW.

Hence D ( S ) E K ( S ) . Conversely, if s E S is a type I1 element, then it admits an expression as an (w - 1)-word w of height at most 1 which evaluates to 1 in the free group.

10

By Lemma 2.18, the (w - 1)-word w admits a factorization into factors of the form a l u l . . . a,u,(a,+l. . .anal . . . a,)w-lu,+lar+l . . .u,a, with the ui evaluating to 1 in the free group (i = 1 , . . . ,TI). Thus, assuming inductively that all ui E D ( S ) U {l},by Corollary 2.20 we deduce that s E D(S). 0

Bases for semidirect products V * G

2.5

The following result combines Theorem 2.21 with the special case of semidirect products with G of what has come to be known as the basis theorem. It was proved by Almeida and Weil [ll]as a combination of profinite techniques with Tilson's derived category theorem [46].

Theorem 2.23. Let V be a pseudovariety of semigroups and suppose g V admits a basis of pseudoidentities E involving only a bounded number of vertices. Then the semidirect product V * G is defined by the pseudoidentities of the f o m $p = $q where p = q is a pseudoidentity from C over a finite graph I' and cp is an edge-labeling of 'I by (w - 1)-words of height at most 1 which commutes over G . We present next a simple application of this result for which a further few preliminaries are needed. Denote by B2 the 5-element multiplicative matrix semigroup consisting of the 2 x 2 matrices over 2 / 2 2 with at most one nonzero entry. Observe that B2 is locally a semilattice, i.e., B2 E LSl. Let X be an alphabet and let X - l be a disjoint set of formal inverses of the letters. For a word w over X (or, more generally, a member of OxS) using all the letters, let -W denote the equivalence relation on Y = X U X - l generated by the pairs (2-',y) such that xy is a factor of w. Then the most general X-labeled graph rwwith initial and final vertices (which may also be seen as an automaton) supporting w, in the sense that the word w may be read along the graph from the initial to the final vertex, is obtained as follows: 0

Vertices(I',) = Y/mW;

0

Edge@,):

0

initial vertex: x/-, where x is the first letter of w;

0

final vertex: x-'/-,

x/wW-%z/-,

if y

-W

x and y-'

-, z;

where x is the last letter of w.

11

Based on results of Reilly [34] in the context of inverse semigroups, Almeida, Azevedo and Teixeira [5] have observed that a semigroup pseudoidentity u = v is valid in B:!if and only if u and v use exactly the same variables and J?, = r V .This in turn allowed them to prove the following result which explains why the calculation of globals concentrates on pseudovarieties excluding B2.

Theorem 2.24. If B2 E V and V = [C], then g V is defined by the pseudoidentities in C viewed over the most general graphs supporting them. Using this result and Theorem 2.23, we may now proceed to compute some specific semidirect products with G. Proposition 2.25. Let V be a pseudovariety containing S1 and suppose {ui = v, : i E I } is a basis of pseudoidentities f o r V . Then C V is local and (CV) * G is defined by the pseudoidentities of the f o r m

u i ( z w y l x w , . .. ,xWy,zW) = v2(zWy1xW,. . . ,xWy,xW)

(4)

with i E I , x a variable, and the y j (w - 1)-words of height at most 1 which are 1 in groups. Proof. Since V 2 S1, the local semillatice B2 belongs to C V . Note that C V is defined by the pseudoidentities of the form (4)where x , y1 , , .. ,Y n are distinct variables and u i , q depend on the same n variables. By Theorem 2.24, g C V is defined by the pseudoidentities of the form (4) viewed over the corresponding most general graph supporting both sides. But, for the variables x,yk the equivalence relation defining r identifies x with 2-l and y;', and also identifies Y k with x-'. Since this holds for every k E (1, .. . ,n } , the whole graph I' has only one vertex. Hence the pseudovariety CV is local. The remainder of the result follows from Theorem 2.23 by noting that no restriction needs to be imposed on z since it only appears in (4) as an w-power . 0 Note that C I is not local. It can be easily shown that its global is defined by a single pseudoidentity on a 2-vertex graph:

Using this observation, it is also easy to construct a 2-vertex semigroupoid which fails the above pseudoidentity but satisfies all 1-vertex pseudoidentities valid in gCI, i.e., whose local semigroups lie in C I .

12

Proposition 2.26 ([4012). Let V be a pseudovariety containing Sl. T h e n EV is local. Proof. Let {ui = v, : i E I} be a basis of pseudoidentities for V. Then EV is defined by the pseudoidentities of the form

with i E I. Since B2 E ESl E EV, it suffices to verify that the most general graphs over which such pseudoidentities may be written have only one vertex. Now, the (symmetrized) relation generating the equivalence relation -W which defines the most general graph I'w over which w = xy . . .x: may be read is determined by the following connected graph 21

Xn

Hence the graph I'whas only one vertex. Since V contains S1, ui and vi must involve precisely the same variables. It follows that the most general graph in which the two sides of (5) may be read has only one vertex. By Theorem 2.24, this shows that g E V is defined by pseudoidentities over 1-vertex graphs, that is EV is local. Again, in contrast, &I is not local and in fact

Stiffler [42] has shown that, if V is monoidal, then D * V E V * D . This allows us to obtain some curious inclusions relating the operators L and -* G under suitable hypothesis.

Proposition 2.27. Let V be a monoidal pseudovariety of semigroups such that V * G is local. Then, we have the following inclusions: a) L(V * G) * G = L(V * G). b) (LV) * G g L(V * G). Proof. (a) By [3, Exercise 10.2.4(a)], V * G is a monoidal pseudovariety. By Theorem 2.1, it follows that L(V * G) = V * G * D. Taking into account Stiffler's result, this leads to the following inclusions:

L(V * G) * G = V * G * D * G C V * G * G * D = V * G * D = L(V* G) 21n [40] one also finds the statement If V * G = EV and V i s a monoidal pseudovariety of bands, then V * G is local (Proposition 10.1). However, in view of Theorems 2.7 and 2.12, the hypothesis V * G = EV is only stating that V is nontrivial and therefore the result adds no new cases of locality to Proposition 2.26.

13

and the reverse inclusion is obvious. (b) Indeed, we have CV C(V * G),and so

(LV)* G C L(V * G)* G = L(V * G ) . Note that the 6-element semigroup with zero given by the following presentation s = (e,f ;e2 = e, f 2 = f, fef = 0) lies in C&S1= L(Sl* G)but not in &LSl 2 (LSl) * G.Hence, at least for V = S1, the inclusion of Proposition 2.27(b) is strict.

Some solutions of the equation V * G = €V For the remainder of the paper, we concentrate on the equation V * G = 3

&V.We have already mentioned some important solutions and also that the equation does not hold in full generality. A syntactical approach based on Theorem 2.23 allows us t o find similarly flavoured proofs that the equation holds for many pseudovarieties contained in IDS. We only sketch here some of those proofs. The details may be found in [6]. 3.1

Locally trivial solutions

Consider the following pseudovarieties:

Kk = 1x1 - - . z n e= z1 -..x,f] LI; = i.1. . . znef = ~ 1. .znf] . LI: = [ e f q - - - z= nezl...X J I u = 82. = 0 , z y = yz]. Note that &I= [e = f] = (N n Corn) V G where the last equality is easily established and may be found in [3, Section 9.11. It is also an easy exercise in the methods of [3] to show that KL = K, V &I= K, V N V G. The nilpotency index of a nilpotent semigroup S is the least positive integer n such that S satisfies the identity 2 1 . ..xn = 0. From results of Almeida and Reilly [7] it is easy to deduce that U is the smallest pseudovariety of nilpotent semigroups with no bound on the nilpotency index of its members. Using Theorem 2.23, one may show that U * G = EU = &I.Also using the same theorem, one observes that [ e~z l . . . z , = z l . . . z , e = z l . . . [zl---zn=O]*G

zn3

(6)

(where the inclusion is actually an equality) and so the semidirect product with G of a nilpotent pseudovariety V with a bound on the nilpotency index

14

of its members is not EV (which is equal to &I) since N is not contained in the right side of (6). This proves the first part of the following result. The other parts can be established similarly.

Theorem 3.1. Let V C_ ELI. Then V * G = EV if and only i f one of the following conditions holds:

a) U C V C_ €1 = Kb in which case EV

b) V

= €1;

E EK, and V p KL for every n in which case EV = EK;

c ) left-right dual of (ii) in which case EV = ED; d) V is not contained in any of the pseudovarieties LIL and LI:, in which case EV = ELI.

3.2

Some solutions in 'DS

It is well known that DS is the largest pseudovariety that excludes the semigroup Bz.It has been the object of much attention not only since it and some of its subpseudovarieties such M J, R and DA are found in various applications in theoretical computer science, but also because they are particularly amenable to syntactical methods, that is the investigation of their properties through the study of relatively free profinite semigroups and pseudoidentities. This is the case also for our equation. The following theorem summarizes the known results in this respect. Theorem 3.2. Every pseudovariety in the following four intervals (indicated by a bold line) together with 'DS is a solution of the equation V * G = EV: IDS

For the remainder of the paper, we sketch the proof of this result for the two extreme pseudovarieties in the above diagram, namely D S and J.

15

3.3

The case of DDS

Let S be a finite semigroup. We say that v E S is good if, for every x E S, ((VZ)"V)"+l

= (V2)"V

that is (vx)"v is a group element. Observe that an idempotent is always good.

Lemma 3.3. Let S E EDS. Then a)

if a , b E S are good then so is ab;

ii) if v E S is good and aba = a or bab = b, then avb is good.

Proof. (i) Let x E S and let e = (bza)". Then, by associativity, we have the following equalities: ( a b . x)"ab = aeb, ( a . bx)"a = ae, (b . xa)"b = eb. Since (bxa)"-lbz . aeb . ax(bza)"-' = e , all these elements lie in the same 9-class of S. Here is a sketch of their distribution in the corresponding egg box picture:

From the hypothesis that both a and b are good it follows that ae and eb are group elements and therefore their %-classes contain idempotents. Since S E EBS, the %-class of aeb must also contain an idempotent and, therefore, ab is good. (ii) Let x E S. We consider here the case aba = a , the other case being handled similarly. We show that (avbx)"avb is a group element by a sequence of equalities in which in each step we either simply use associativity or we underline the factors about which some property is being used to lead to the next step. At one of the steps we use the fact that, by (i), since bav is a product of an idempotent and a good element, it is good.

16

(gvbx)"gvb = (abavbz)"abawb

= = = = =

b

a . (&bza)"&.

a( (bavbxa) "bav)"+I

b a (bavbxa)"bav ((bavbza)w bav)" b (-ababz)" abav( b(awbzab)"av)"b (avbz)" av (b(gvbz&)"gw) "b = (avbz)"avb((avbz)wuvb)" = ((avbz)"avb)"f'. 17

We may now easily complete the proof of the equality D S * G = EDS. Note that D S is defined by the pseudoidentity ((zy>"z)"+l = (XY)" 2. Since D S is local, by Theorem 2.23 we obtain the following basis of pseudoidentities for D S * G:

9s * G = ~ [ ( ( ~ ~ ) " u =y +(uv)wu l

:

G

I=

= = 11.

Given S E EDS, a homomorphism cp : AS -+ S, and a pseudoidentity u = 1 valid in G , we have cp(u) E K ( S ) = D ( S ) (cf. Theorem 2.22) and so cp(u) is good by induction on the construction of elements of D ( S ) using Lemma 3.3 for the induction steps. This shows that S verifies all the pseudoidentities in the above basis (even those without any requirement on v). Hence EDS C D S * G and the reverse inclusion is valid for any pseudovariety in the place of D S .

3.4

The case of J

In the case of the pseudovariety J , the fact that it is not local makes things somewhat harder which lets us show how the (w- 1)-words of height at most 1 come handy. Let S be a finite semigroup and let s E S. Say that s is w-central if (st)" = (ts)" for all t E S. The following is the analog of Lemma 3.3. The proof is similar and is omitted.

Lemma 3.4. For S E EJ, the set of all w-central elements contains E ( S ) and is closed under multiplication and weak conjugation. Part (i) of the following proposition improves a result of Margolis and Pin [31] which characterizes EJ as the pseudovariety defined by the pseu= (xe)" in the sense that the proposition shows that EJ doidentity satisfies apparently much stronger pseudoidentities.

17

Proposition 3.5. Let w E AS be such that G the following pseudoidentities: 2)

w = 1. Then EJ satisfies

(wz)"= (zw)";

ii) (wz)"w = (wz)"; iii)

& J + 1=

w".

Proof. For (i), note that, in a given S E E J , w evaluates to an element of K ( S ) and therefore to an element of D ( S ) . Hence w is w-central by Lemma 3.4. Part (ii) is proved similarly, that is by establishing first a suitable 0 analog of Lemma 3.4. Part (iii) follows from (ii) by taking z = w. It is now a routine matter to prove the following corollary using Proposition 3.5 and associativity.

Corollary 3.6. Let u , v,w E GAS be such that G satisfies the pseudoidentities

+ uw = v = 1. Then EJ

(uvw)wuw = UW(UzIW)" = (uvw)w. Corollary 3.6 gives the first step in an induction procedure that gives, more generally, the following result for (w 1)-words of height at most 1. The proof is omitted.

-

Proposition 3.7. Let u,v be (w - l)-urords of height at most 1 such that u = 1 holds in G and u + v by rules of the form u(vu)"-~v + 1. Then EJ satisfies the pseudoidentities vu" = uw = uwv. With these tools at hand, one may now proceed to prove that J is a solution of our equation by establishing the following result.

Proposition 3.8. Let u1, u2, v1, v2 be (w - 1)-words of height at most 1 such that u1u2 = 211212 = u1v2 = 1 in groups. Then EJ satisfies the pseudoidentity ( ~ 1 ~ 2 ) w ~ l ~ 2 ( v=1 (~~ 2 1~ )2 w) w ( v 1 v 2 ) w .

We only sketch here the proof of Proposition 3.8. By Lemma 2.18, it is possible to reduce the (w - 1)-word ulv2 to 1 by a finite sequence of applications of rules of the form u(vu)W-1v

+ 1.

(7)

18

where u and v are possibly empty words. In such a reduction, part will be done entirely within factors descending from u1 or from v2 while some may require a factor descending from u1 and a factor descending from v2. The former are handled directly using Proposition 3.7. For the latter, some care has to be taken. By associativity, we may for instance assume that the word u is shortest possible and that the factor ( V U ) ~ - ~ Vcomes entirely from a descendant of v2 and that this factor cannot be erased from that descendant of 212 by application of rules of the form (7). The factor u in turn is a product u = yz where y comes from a descendant of u1 and z from a descendant of 0 2 . So there are descendants ui of u1 and vh of v2 under the reduction rules of the form (7) and factorizations ui = uyy and vh = , Z ( V U ) ~ - ~ V Vwith ~ , IyI smallest possible. Now there is a shortest suffix t of v2 from which 06 is obtained by application of rules of the form (7) and we let 02 = st. Because u was chosen to be shortest possible, and since v1svh = v l v 2 = 1 in G , it must be possible to reduce 01s to an (w - 1)-word of which y is a suffix. Moreover, such a reduction must come from the left factor v1,that is there is a factorization vl = xjjs' such that 5 reduces to y and s's reduces to 1. This shows that the reduction ~ Z ( V U ) ~ - ' V+ 1 that has to be performed in a descendant of u 1 v 2 to reduce it to 1 may also be performed in a descendant of v102 and so again Proposition 3.7 allows us to do it using the right factor ( ~ 1 7 ~ 2 ) For ~ . the reader's benefit, we depict the various factorizations in the following picture where the appearance in the second line of a factor below another in the first line means it is a descendant of the latter. '111 '112 '111 .. . .-

...

--.

'111

212

1 '1;

73'111

v2

s

't

v1

xlgls'

4 -x Y -

v2

s

I

...

t

4 ... -

Y

The equality J * G = EJ follows now from Theorem 2.23 taking into account the basis of pseudoidentities for gJ given by Theorem 2.4.

References [l] D.Albert,

R. Baldinger, and J. Fthodes, The identity problem for finite semigroups (the undecidability of), J . Symbolic Logic 57 (1992) 179-192.

[2] J. Almeida, The algebra of implicit operations, Algebra Universalis 26 (1989)

16-32. [31

, Finite Semigroups and Universal Algebra, World Scientific, Singapore, 1995. English translation.

19

[41

, A syntactical proof of locality of D A , Int. J . Algebra and Computation 6 (1996) 165-177.

[5] J. Almeida, A, Azevedo, and L. Teixeira, O n finitely based pseudovarieties of the forms V*D and V*Dn, J . Pure and Appl. Algebra 146 (2000) 1-15. [6] J. Almeida and A. Escada, On the equation V*G=EV, J . Pure and Appl. Algebra. To appear. [7] J. Almeida and N. R. Reilly, Generalized varieties of commutative semigroups, Semigroup Forum 30 (1984) 77-98. [8] J. Almeida and B. Steinberg, O n the decidability of iterated semidirect products and applications to complexity, Proc. London Math. SOC. 80 (2000) 50-74. [9] J. Almeida and P. Weil, Reduced factorizations in free profinite groups and j o i n decompositions of pseudovarieties, Int. J. Algebra and Computation 4 (1994) 375-403.

POI

, Relatively free profinite monoids: a n introduction and examples, in Semigroups, Formal Languages and Groups, J. B. Fountain, ed., vol. 466, Dordrecht, 1995, Kluwer Academic Publ., 73-117.

[111

, Profinite categories and semidirect products, J . Pure and Appl. Algebra 123 (1998) 1-50.

[12] C. J. Ash, Finite semigroups with commuting idempotents, J . Austral. Math. SOC.,Ser. A 43 (1987) 81-90.

~ 3 1 , Inevitable graphs: a proof of the type 11 conjecture and some related decision procedures, Int. J. Algebra and Computation 1 (1991) 127-146. [14] K. Auinger e B. Steinberg, T h e geometry of profinite graphs with applications to free groups and finite monoids, Tech. Rep. CMUP 2001-06, 2001. [15] K. Auinger and B. Steinberg, O n the extension problem f o r partial permutations, Tech. Rep. CMUP 2001-08, 2001. [16] J.-C. Birget, S. Margolis, and J. Rhodes, Semigroups whose idempotents f o r m a subsemigroup, Bull. Austral. Math. SOC.41 (1990) 161-184. [17] J. A. Brzozowski and I. Simon, Characterizations of locally testable events, Discrete Math. 4 (1973) 243-271. [18] S. Eilenberg, Automata, Languages and Machines, vol. B , Academic Press, New York, 1976. [19] A. Escada, T h e power exponent of a pseudovariety, Semigroup Forum. To appear.

POI

, Contributions f o r the study of power operators over pseudovarieties of semigroups, Ph.D. thesis, Univ. Porto, 1999. In Portuguese.

[21] D. Gildenhuys and L. Ribes, Profinite groups and Boolean graphs, J . Pure and Appl. Algebra 12 (1978) 21-47.

20

[22] K. Henckell, S. Margolis, J.-E. Pin, and J. Rhodes, Ash’s type 11 theorem, profinite topology and Malcev products. Part I, Int. J. Algebra and Computation 1 (1991) 411-436. [23] K. Henckell and J. Rhodes, T h e theorem of Knast, the PG=BG and Type I1 Conjectures, in Monoids and Semigroups with Applications, J . Rhodes, ed., Singapore, 1991, World Scientific, 453-463. [24] P. M. Higgins, Pseudovarieties generated by transformation semigroups, in Semigroups with Applications, including Semigroup Rings, S. Kublanovsky, A. Mikhalev, J. Ponizovskii, and P. Higgins, eds., St Petersburg, 1999, TPO “Severny Ochag”, 85-94. [25] P. M. Higgins and S. W. Margolis, Finite aperiodic semigroups with commuting idempotents and generalizations, Israel J. Math. 116 (2000) 367-380. [26] P. R. Jones, Profinite categories, implicit operations and pseudovarieties of categories, J. Pure and Appl. Algebra 109 (1996) 61-95. [27] P. R. Jones and M. B. Szendrei, Local varieties of completely regular monoids, J. Algebra 150 (1992) 1-27. [28] P. R. Jones and P. G. Trotter, Locality of D S and associated varieties, J. Pure and Appl. Algebra 104 (1995) 275-301. [29] J. Karnofsky and 3. Rhodes, Decidability of complexity one-half for finite semigroups, Semigroup Forum 24 (1982) 55-66. [30] R. Knast, S o m e theorems o n graph congruences, RAIRO Inf. ThBor. et Appl. 17 (1983) 331-342. [31] S. W. Margolis and J.-E. Pin, Varieties of finite monoids and topology for the free monoid, in Proc. 1984 Marquette Semigroup Conference, Milwaukee, 1984, Marquette University, 113-129. [32] J.-E. Pin, Varieties of Formal Languages, Plenum, London, 1986. English translation. [331

, BG=PG: A success story, in Semigroups, Formal Languages and Groups, J. Fountain, ed., vol. 466, Dordrecht, 1995, Kluwer, 33-47.

[34] N. R. Reilly, F’ree combinatorial strict inverse semigroups, J. London Math. SOC.39 (1989) 102-120. [35] J. Reiterman, T h e Birkhofl theorem for finite algebras, Algebra Universalis 14 (1982) 1-10. [36] J. Rhodes, Kernel systems - a global study of homomorphisms o n finite semigroups, J. Algebra 49 (1977) 1-45. [371

, Undecidability, automata and pseudovarieties of finite semigroups, Int. J. Algebra and Computation 9 (1999) 455-473.

[38] B. Steinberg, Inevitable graphs and profinite topologies: some solutions to algorithmic problems in monoid and automata theory, stemming f r o m group theory, Int. J. Algebra and Computation 11 (2001) 25-71.

21

WI ~401

, A note on the equation PH

= J*H,Semigroup Forum. To appear.

, Semidirect products of categories and applications, J. Pure and Appl. Algebra 142 (1999) 153-182.

[41] B. Steinberg and B. Tilson, Categories as algebras 11,Tech. Rep. CMUP 20004, 2000. [42] P. Stiffler, Extension of the fundamental theorem of finite semagroups, Advances in Math. 11 (1973) 159-209. [43] H. Straubing, Finite semigroup varieties of the form V * D,J. Pure and Appl. Algebra 36 (1985) 53-94. [44] M. B. Szendrei, T h e bifree regular E-solid semigroups, Semigroup Forum 52 (1996) 61-82. [45] D. Therien and A. Weiss, Graph congruences and wreath products, J. Pure and Appl. Algebra 36 (1985) 205-215. [46] B. Tilson, Categories as algebra: a n essential ingredient in the theory of monoids, J. Pure and Appl. Algebra 48 (1987) 83-198. [47] S. Zhang, An infinite order operator on the lattice of varieties of completely regular semigroups, Algebra Universalis 35 (1996) 485-505.

22

On the Sentence Valuations in a Semiring Adrian ATANASIU*

Carlos MARTiN- VIDE**

Victor MITRANA*

*University of Bucharest, Faculty of Mathematics Str. Academiei 14, 70109, Bucharest, Romania e-mail: [email protected] e-mail: [email protected] **Research Group in Mathematical Linguistics and Language Engineering Rovira i Virgili University Pqa. Imperial TBrraco 1, 43005 Tarragona, Spain e-mail: [email protected]

Abstract. This paper proposes an algebraic way of sentence valuations in a semiring. Actually, throughout the paper only valuations in the ring of integers with usual addition and multiplication are considered. These valuations take into consideration both words and their positions within the sentences. Two synonymy relations, with respect to a given valuation, are introduced. All sentences that are synonymous form a synonymy class which is actually a formal language. Some basic problems regarding the synonymy classes are formulated in the general setting but the results presented concern only very special valuations.

1

Introduction

A series of paper, see, e.g., [l],[a], [8], [9], and the references thereof, have dealt with homomorphisms h from a free generated monoid M into the monoid ((0, m),., l), so that the sum of all homomorphical images of generators of M equals 1, called 'Supported by the Direccih General de Enseiianza Superior e Investigacibn Cientifica, SB 97-00110508

23 Bernoulli homomorphisms (distributions, measures). Besides being homomorphisms, Bernoulli homomorphisms may be viewed as probability measures on the family of all languages over a given alphabet. Furthermore, they played an important role in developing the theory of codes [l].Some authors discarded the homomorphism property keeping the probability measure property as done in [8], [9] whilst others proceeded vice versa [6], calling them valuations. These valuations were used in the study of unambiguity representations of languages. Along these lines, in [7] the equation system zI = pi which identifies a context-free grammar, is transformed, via a valuation h, into a numerical equation system h(zi)= h(pi). Solving the former system one gets the context-free language generated by the given grammar while solving the later system one gets exactly the valuation of L , h ( L ) , defined as the sum of all h ( w ) with w E L. Close relations between the unambiguity of the given context-free grammar and the value h(L) are discussed. Moreover, new characterizations of unambiguity in regular expressions based on the same concept of language valuation are proposed. This way of assigning values to a sentence remembers also some devices introduced in the area of regulated rewriting: weighted grammars and automata, see, e.g., [4,5 , 12, 10, 131, where a given number in a group is associated with each computation step (derivation or configuration). A computation is valid iff the total value assigned t o that computation, computed in accordance with the operation of the group considered, is the neutral element of that group. A consistent extension to the basic paradigm of constraint satisfaction in parsing might make use of the penalty factors assigned to syntactic, semantic, and mapping constraints. Penalty factors, which may range from zero to one, are combined multiplicatively leading to confidence scores which indicate a sort of level of constraints violation. This extension can be used to model distance effects if one takes into consideration the local distance between two consecutive constraint violations. This suggests t o compute the confidence score depending also on the position of the constraint. In this paper, we introduce a generalization of the aforementioned valuations in the following sense. The value of a sentence depends not only on its words but also on their positions within the sentence. Furthermore, the valuation is computed in a richer structure that of a semiring instead of a monoid. Moreover, we consider valuations that allow a finite set of values for each sentence. More precisely, each word in a given vocabulary has a finite set of values (attributes) and each position (a natural number) has just one attribute. For a given sentence, the value associated to a position occupied by a certain word a is obtained by considering two attributes: one is that of position itself the other being one among the attributes associated to the word a. Thus, we need an operation for computing the value of every position in the sentence and one operation for computing the value of the whole sentence.

24

The latter should be, in our opinion, an additive type one. What structure might be the most relevant one for our purposes? We have chosen a very common and widely investigated structure in semantics, that of a semiring. More precisely, all the results we present concern a particular semiring, namely the (semi)ring of integers. Based on this valuations there are defined two types of synonymy relations. Two sentences are weakly synonymous, with respect t o a given valuation, if they have a common value computed in accordance with the given valuation; they are strongly synonymous if they lead t o a common value in between any contexts. Informally speaking, two sentences are weakly synonymous if they have a common meaning. However, if one adds the same contexts t o two weakly synonymous sentences, one may get two new sentences that have no common meaning (the new sentences are not weakly synonymous anymore). This undesired feature is avoided by the definition of the strong synonymy relation. We investigate the decidability of the finiteness problem of synonymy classes as well as the possibility of algorithmically deciding whether two given sentences are strongly synonymous (as we shall see, it is always decidable whether or not they are weakly synonymous). In our approach we consider two very special types of valuations depending on their position attributing function, that is the polynomial (constant and linear) and (restricted) exponential functions, respectively.

2

Definitions and examples

A vocabulary is a finite nonempty set whose elements are called words; if V = {al, a 2 , . . . , a,} is a vocabulary, then any sequence w = ailai, . . . a i k ,1 5 ij 5 TI, 1 5 j 5 k , is called sentence over V . The length of the aforementioned sentence w is denoted by l g ( w ) and equals k . The empty sentence is denoted by E , lg(&)= 0. As a rule, the words are denoted by small letters from the beginning of the Latin alphabet and the sentences are denoted by small letters from the end of the same alphabet, ~ the sentence obtained from excepting the empty sentence. Moreover, ( x ) delivers x by removing all words not in U. The set of all sentences over V is denoted by v* and V+ = V*- { E } . Any subset of V * is called language. A structure ( A ,+, -,0 , l ) is called a semiring iff the following conditions are satisfied for all a, b, c E A: (i) ( A ,

+,0) is a commutative monoid,

(ii) ( A ,-,1) is a monoid, (iii) a . ( b

+ c ) = a . b + a . c, ( a + b) . c = a . c + b . C,

(iv) O . a = a . O = O .

25

The semiring ( A ,+, -,0 , l ) is said to be commutative iff ( A ,., 1) is a commutative monoid. For further notions we refer to [ll].If B and C are two subsets of A and q E A, one defines

B .Q = {PqlP E B } ,Q . B = {qpb E B ) , B + C = { p + r l p E B,r E C}, B - C = {prlpE B , r E C } . Let V be a vocabulary and ( A ,+, .,0 , l ) be a commutative semiring. A valuation of V * in A is a pair of mappings

4 = (.,P), where : V --+ 2f, (the word valuating function; a ( a ) is the set of all values (attributes) of a ) ,

0

a

0

p : IN +A , (the position valuating function; p(n) is the value (attribute) of the position n ) .

Here 2; denotes the set of all finite subsets of A . Given a valuation a sentence x = a 1 a 2 . . a, E V * ,ai E V, 1 5 i 5 n, we define

4 as above and

n

val+(x)= Ccr(ai). p ( i ) . i= 1

Moreover, v a l + ( ~delivers ) always 0, for any valuation q5. A valuation as above is deterministic iff card(cr(a)) = 1 for all a E V . By our intuition, the beginning and the end of a sentence offer more information then its middle part. Even so, one may argue that beginning is still predominant, but for our further results this makes no difference since for any string x as above we can consider n vaZ4(x) =

C a(ui) .p(n - i + 1).

i=l

Two sentences x,y are weakly synonymous with respect to the valuation 4, written as x "+ y , iff val+(x)n v a l @ ( y )# 8. One may easily notice that this relation is reflexive and symmetrical but not transitive. The weak s y n o n y m y class of x is defined as [XI4 = {Y E V*lZ -4 9). Two sentences x,y are strongly synonymous with respect to the valuation 4, written as x "+ y , iff val+(uxv)f l val+(uyv) # 8, for any pair of sentences u, E V * . Again,

26

this relation is reflexive and symmetrical but not transitive. The strong synonymy class of x is defined as b$4 = {Y E V * b “4 Y}. Note that the strong synonymy always implies the weak synonymy but the converse does not hold.

Example 1. Let us consider the semiring +[XI of all polynomials with only one indeterminate and coeficients in Z together with addition and multiplication. W e consider the valuation of {a, b, c, d}* in Z [ X ] ,4 = (a,p), defined as follows:

a(.) = 2X2 a ( b ) = X 2 - 1 a(.) = 1 a ( d ) = 2X2 - 1 P(i) = X

+

2,

i E IN.

It is easy t o note that

valb(dacb) = val+(aba) = 5 X 3 + 10X2- X which implies dacb

wB

Example 2. Take V with

-

2

aba.

=

{a, b, c} and the valuation of V* in (Q, +, ., 0 , l )

4 = ( a ,p)

a ( a ) = {1/2}, a(b)= {1/3}, a ( c ) =z {-1/6} P(n) = 5 , for all n 2 1. The reader may easily verify that [&I4= {xi3ig((x)a)

+ 21g((x)b)= l g ( ( x ) c ) } .

Note that [&I4 is a context-free non-regular language. Moreover, both valuations are deterministic. We proceed t o investigate mainly the synonymy classes. A natural problem concerns the finiteness of these sets as well as the possibility t o decide on this problem. As we shall see in the sequel, a closely related problem concerns the decidability status of the next problem: For a given value q, are there sentences whose valuation set contains q? Furthermore, we are concerning with the problem of finding appropriate devices (automata, grammars, etc.) which characterize the synonymy classes. Since the above definitions were given in a very general setting, we should restrict our investigation to particular valuations. To this end, in this paper we shall only consider the valuations in the ring of integers Z with the usual addition and multiplication. Of course, other semirings may be considered as well,

27

but we have chosen this semiring because it is the most natural and simple one. Even so, the problems we considered appeared t o be very difficult. A similar investigation for other semirings remain to be done. The absolute value of an integer x is denoted by 1x1. In the sequel, we shall foccus our attention on valuations whose function p is either the constant polynomial, the linear polynomial, or the exponential function

an. We start with a lemma which will be useful in the sequel. Let x = ( 3 1 ~ 2 . .. a, be a sentence in V* and 4 = ( a ,p) be a valuation of V' in Z with

p(n)= Conk + C 1 n k P 1 + . . .

+

Ck.

Denote by

4i = (a,nZ),o 5 i 5 k . Clearly,

by a direct calculation one gets the desired equality.

3

0

The constant polynomial

As [x]4= V ' , for all x E V*, providing that p is the null function, we shall consider only non-zero position valuating functions in the rest of this section. Note that the relations stated by Lemma 1 and relation (l),respectively, may be combined in

+

valg(xy) = val@(x) val4(y).

(2)

The next result is an immediate consequence of relation 2.

Proposition 1. Let 4 = ( a ,p) be a valuation of V * in Z.Then, x

~4

y iff x "4 y .

Theorem 1. Let q5 = ( a ,p) be a valuation of V' in 72. Then, the following problems are decidable: I. Given q E Z,are there sentences x E V+ such that q E va14(x)? 2. Given q E Z,are there arbitrarily many sentences x such that q E va14(x)?

28

Proof. Assume that P(n) = k , n E W, for some integer k . Moreover we take a positive k , the case k < 0 may be treated similarly. 1. Obviously, there is no sentence whose valuation contains q if q is not a multiple of k . We distinguish two cases depending on the values of the words in V . Firstly, let us suppose that all values of the words in V are nonnegative; the reasoning is the same when all of them are negative. It follows that val4(z) contains only nonnegative integers, for each x E V+, hence q has to be nonnegative. Clearly, if q = 0, the answer is affirmative if and only if there is a word in V that has a null value. For q > 0 it suffices to restrict our search to sentences of length a t most q / k . Consequently, one can algorithmically find the answer in this case. Now, let us consider that the set C = { p # Olp E a ( u ) , a E V } contains both negative and positive integers. We claim that exists z E V+ such that q E val+(z)if and only if q is a multiply of k d , where d is the greatest common divisor of all integers in C. Obviously, if q E valg(z) for some x E V+,then q must be a multiple of k d . We prove now th a t for each multiple of kd there exists a sentence that contains it in its valuation set. Let q l , q l , . . . qn be all positive integers in C and p l , p 2 , . . . p,, be all negative integers in C. I t is known that

i=l

i=l

for some integers ki,rj, 1 5 i 5 n, 1 5 j 5 m. Moreover, one can choose either ki,rj 2 0 or kj, r j 5 0, for all 1 5 i 5 n, 1 5 j 5 m. The last claim requires a short discussion: we thought that we would find a reference for it but we were not able to find such a reference. In order to keep the proof easy to follow, we prefered to prove it in an appendix at the end of the paper. Let q be an arbitrary multiple of kd; consider the sentence x given by the next algorithm:

Algorithm 1. Procedure Findsynonymy-class-representative(q); begin 2

:= E ;

if q > 0 then choose k i , r j 2 0 in (3) else choose k i , r j 5 0 in (3); endif; for i:= 1 t o n do choose a E V with qj E a(a); 2

:= x , e ~ i / ( ~ 4 ;

29 endfor; for i:= 1 to m do

choose a E V with pi E a(a); x := xaVil(k4; endfor; if p=O then choose a, b E V such that 91 E a(a),pl E a @ ) ; 2

:= a l P l l p ;

endif; end.

It is easy to notice that q E valb(z) which concludes the reasoning of the first assertion. 2. The latter item follows from the first one as follows. Find, if any, a sentence z such that q E valb(z). If the sentence z exists, detect a sentence y whose valuation contains the value 0. When no sentence y exists, only a finite number of sentences might have q in their valuation sets. Indeed, if there is no sentence y E V + with 0 E val+(y), then all values of the words in V are either positive or negative. By the first part of this proof it follows that only a finite number of sentences might have q in their valuation set. If such a sentence exists, then by equation 2 all sentences 0 zym,rn 2 0, are in [x]b,which ends the proof. From the previous theorems one can infer the next result.

Theorem 2. Let qj = (a,p) be a valuation of V * in Z.The following problems are decidable: 1. Are two given sentences weakly/strongly synonymous? 2. Given a sentence z E V * , are [XI+ and (x)4finite? We recall now an operation on sentences that will turn out to be very useful for our investigation regarding the type of languages [z]b. This operation, called shufle is a well-known operation in formal language theory and in parallel programming theory. We define this operation on sentences, recursively, as follows: for two strings z, y E V * and two symbols a, b E V we write (2)

(ii)

zLU&=&IUz=2, uz u bg = a(z LU by) u b(az Lu y).

A shuffle of two strings is an arbitrary interleaving of the substrings of the original strings We naturally extend this operations to languages: L1LULz=

u

zuy.

ZEh,YELZ

The next theorem settles the position of synonymy classes with respect to valuations whose position valuating function is a constant in the Chomsky hierarchy.

30

Theorem 3. Let q5 = ( a ,p ) be a valuation of V* in Z and x be a sentence in V * . 1. The language [x]4is context-free. 2. If q5 is deterministic, one can decide whether or not [x]g is regular.

Proof. 1. The reader may construct a nondeterministic pushdown automaton that recognizes all sentences in [ x ] g . We prefer another proof, namely we use a slightly modified version of additive valence grammar. A right-linear additive valence grammar, see [3], is a construct G = ( N ,T, P, v), where N , T , P are the parameters of a right-linear grammar and v : P -+Z is the valence function. The valence associated t o a derivation

s,

D : wo

~1

s,

ar2 ~2 3 . . . arm wm

such t h a t at each step wi-1 ari wi, 1 5 z 5 m, the applied rule is ri is

i= 1

The language generated by G with the valence q E Z is the set

L(G,q) = {x E T*l there exists a derivation S =+*

x with v(S J*

x) = q ) .

It is known [3] t h a t all languages generated by right-linear additive valence grammars are context-free. Now, given a valuation q5 = (a,P) of V * in Z,with a constant function p, we construct a right-linear additive valence grammar G+= ( N , V,S, P, v) as follows:

v,Q E 44).

0

N = { S ) u { ( a ,q)la E

0

For each nonterminal (a,q) E N we have the following rules in P:

S (a,q)

(a,q)

-+ -+ +

(a,q), with the valence q, aS, with the valence 0, E , with the valence 0.

Obviously, the equality [XI4 =

u

L(GbtIP(1))

tEWQlg( 5 )

holds. Since the family of context-free languages is closed under union the first assertion is completely proved.

31

2. Let q5 = ( a ,p ) be a deterministic valuation of V * in

Z.Denote by

v = { a E Vla(a)= 0} and U = V \ v. It is easy to note that [TI4= [(z)v]g v*.

If card(U) 5 1, then [z]4 is regular for any z E V * . Indeed, if card(U) = 0, then [z]4= V * .If card(U) = 1, then all classes [ ( z ) ~ ]are + finite, hence all languages [z]4 are regular. Let us suppose that card(U) 2 2. If a(a). a ( b ) > 0 for all pairs (a, b) E V x V , then [(z)v]4 is a finite set for all z E V* (see the proof of Theorem l),therefore [z]4 is regular. If there exist a, b E V such that a ( a ) - a ( b ) < 0, then the language [ ( z ) U ] $ is context-free but not regular. This language is [(Z)UI4

= {z E U*l

c

l g ( ( z ) a ) . a ( a ) = Val&)).

QEU

As shuffling a context-free non-regular language with a regular language, the languages being over disjoint vocabularies, one gets a context-free non-regular language, it follows that [z]4is regular iff either card(U) 5 1 or a ( a ) . a ( b ) > 0 for all pairs 0 ( a , b) E V x V , both conditions being decidable. Remark. In the view of the last theorem, the decidability of the finiteness problem for synonymy classes follows directly from the finiteness problem for context-free languages. However, the proof of Theorem 1 offers a more easily testable condition and a less complex (time and memory) implementation.

4

The linear polynomial

In this section we shall consider only valuations whose position valuating functions are linear polynomials.

Theorem 4. Let q5 = ( a ,P ) be a valuation of V * over one can decide the finiteness of

[XI$.

+. Given a sentence

3:

E V*,

Proof. Let 4 = ( a ,p) be a valuation with @(n)= k n + p . By Lemma 1 and relation 1, one may write also

wUal&y)

= wal,$(z)

+ wal+(y) + k .l g ( z ) .val+,(y).

(4)

Assume that a ( a ) contains only positive integers, for all a E V ; the case when a ( a ) contains just negative integers may be treated analogously. We analyse what happens when k < 0; the reasoning may be carried over the case k > 0 with minor

32 changes. Clearly, there exists no E IN such that valb(z) has only negative values (vaZ+(z)< 0, for short), for all sentences z in V * longer than no. Let z be such a sentence. We claim that valb(y) < valb(z), for all y E V * such that Zg(y) 2 lg(z). maz{ Irl I r E valb(z)}. Due t o the length of y, one infers that walb(y)

5 maz{lrl I r

E valb(z)} . maz{val~(w)lwis a subsentence of y of length Zg(z)}

which is smaller than valg(z) because valb(w) is negative too, providing that w is a subsentence of y of length l g ( z ) . Consequently, [z]@is finite for all z E V * . Let us consider that exist a, b E V , possibly the same, such that &(a) . a(b) contains at least one negative integer. Take q1 E a ( a ) ,q2 E a ( b ) such that q1 -q2 < 0; the sentence y = alQzlblqllsatisfies the relation 0 E valb,(y). Moreover, we claim that 0 E valb, (,aR), for all z E V* with 0 E vaZgO(z).Indeed, if z = 2122.. . z,, zi E V , and ti E a(zi),1 5 i 5 m, so that CEl ti = 0, then (2m I) Ci=,(ti)E va$(zzR) holds. Note also that 0 E val+,(zzR),too. Now, as valg(zzR) = k . Val+, (2.") p . val@,(ZZR)

+

+

one gets 0 E valg(zzR). Due to the relation 4 one concludes that all sentences 0 z(zzR)q,q 2 0, with z as above, belong to [z]d. As far as the position of languages [ z ] in ~ the Chomsky hierarchy is concerned, we have:

Theorem 5 Let 4 = (a,p ) be a valuation of V* in Z.The language sensitive, for any z E V*.

[XI+

is context-

+

Proot Let us suppose that P(n) = k n p . We give the proof for k > 0 only; the proof may be carried over the case k < 0 with the appropriate changes. Take no the minimal natural number such that kn p > 0 for all n > no. For each z E V* one constructs the phrase-structure grammar G, which works accordingly with the next nondeterministic procedure:

+

1. The grammar generates a sentential form X a l a 2 . . . a,YZ, X, Y,Z being nonterminals, ui E V,1 5 i 5 n, and n > no. 2. If no > 0, choose q E valb(ala:!.. .ano) and transform the sentential form into either X b l b z . . .bnoano+l... a,Y(-l)lqlZ, iff q < 0, or X b l b 2 . . . bnoa,o+l.. .anYlqZ, iff q 2 0,

bl, b2,.

. . , bno being nonterminals.

33

3. While the current sentential form contains words in V and no trap nonterminal do 0

0

Assume that the suffix of the current sentential form is Y d Z , for some c E {-1, l}, and q 2 0. if cq E [rnin(valb(z)), rnas(val+(z))], then

- choose a word a, in between X and Y , - transform a, into a nonterminal b,, - choose T

T

E a(a,), and write either

lr(kz+p),

if

T

2 0, or ( - l ) l r l ( k z + P ) , if

< 0, before 2,

- remove iteratively all pairs of consecutive symbols

-1,1, or 1,-1, in

between Y and 2. 0

0

if cq < rnin(vaZb(z)),then look for a word a, in between X and Y such that a(@,)contains a positive integer; if no such position exists, then block the derivation by a trap nonterminal; otherwise

a, into a nonterminal b,, choose T E a(a,),r > 0, and write lr(kzz+P) before 2, remove iteratively all pairs of consecutive symbols -1,1, in between

- transform -

Y and 2. 0

0

if cq > rnaz(valb(s)),then look for a word a, in between X and Y such that a(a,) contains a negative integer; if no such position exists, then block the derivation by a trap nonterminal; otherwise - transform a, into a nonterminal b,,

- choose T E a(a,),r < 0, and write ( - l ) l r l ( k z + p ) before 2, - remove iteratively all pairs of consecutive symbols 1,-1, in between Y and Z . 4. If the current sentential form does not contain any trap nonterminal, check whether or not its suffix YcqZ satisfies the relation cq E vul+(z),q 2 0. In the affirmative, remove all symbols c and X , Y , 2,and rewrite all nonterminals b, into a,, 1 5 a 5 n, otherwise block the derivation by a trap nonterminal. Clearly, [Zlb = q G , )

u {Y E V*ls

Yl

k A Y ) 5 720).

34

Note that the working space [14] of each z E L(G,) is bounded as follows

W S ( z ,GZ)I m a x ( l g ( z ) + 3 + m ~l g,( z ) + 3

+ + 2 . m3(k . l g ( z ) + p ) ) , m2

where

ml = max{lsl : s E v a l g ( y ) , y E V * , l g ( y ) I n o } , m2 = max{lsl : s E valg(x)}, m3 = max{ls\ : s E a ( a ) , a E V}. It follows that L(G,) is context-sensitive, hence [x]g is context-sensitive, too. 0 Note that, by relation 1 and Lemma 1, if Zg(z) = l g ( y ) , then x -g y iff z z g y . We do not have any algorithm for deciding whether two sentences are strongly synonymous with respect t o valuations whose position functions are arbitrary polynomials. However, we present below an algorithm for a large class of valuations.

Theorem 6. If 4 = ( a , @is ) a valuation with p being a non-constant polynomial and there i s a sentence a with exactly one non-zero value in a ( a ) , theiz one can decide whether x =o g , for a n y sentences x,y.

+ +

Proof. Let 4 = ( a ,p ) be a valuation of V *over Z with @(n) = conm +clnm-l . .. h.Assume that x z g y ; it follows that 2 y as well as val+(xak)n d 4 ( y a k )# 0, for all k 2 0, a being the word in V with just one attribute in a(.) which is not zero. By Lemma 1 and relation 1 one gets N~

i=O

i=O

+ i=o c cz j=1C ( l g ( 5 )+ j)"-iiY(u). m

= valg(x)

k

(5)

Analogously, m

k

Consequently, l g ( x ) = lg(y) must hold, otherwise {s1-s2Is1 E v a l g ( x ) , s2 E valg(g)} would be infinite, a contradiction. Indeed, for a ( a ) is a non-null integer, if l g ( y ) > l g ( x ) , then the relations (5) and (6) may be written as:

35 for some integers q k , t k . One infers that t k E {s1 - s2lsl E val+(y), s2 E val+(x)} for all Ic 2 0, which is contradictory. Analogously, when lg(y) < lg(z). In conclusion, x ~6 y iff (x "6 y)&(lg(x) = lg(y)), conditions that may be algorithmically checked. 0

5

The exponential case an

The subject of investigation in this section is the class of valuations q5 = (a,,@ whose position valuating function P is a particular exponential function, namely p(n)= an, n 1 1,a E Z \ {0,1}. Clearly, val+(zy) = val+(z) a'g(")valb(y). (7)

+

Theorem 7.Let q5 = ( a ,P ) be a valuation of V * in P,P(i) = ail i E IN,a E +\{O}. One can decide whether there exist sentences x E V+ such that q E val+(x),for a given q E Z?

Proof. One can distinguish three cases: a = 1, la( 2 2 and a = -1. If a = 1, we are dealing with a valuations whose position valuating function is constant; this situation has been treated in the proof of Theorem 1. Let us analyse the case a = -1. Define d as being the greatest common divisor of all integers in the set { r - s l r E a ( b ) ,s E a ( c ) } ,where b and c are words (might be the same) in V . Given an integer q, there exists x such that q E va16(x) if and only if q = t( mod d ) , t E {o}Uub,,{lSl I S E a(b)lS < 0) or d - t E u b c v { S l S E a ( b ) ,.S > o}. The argument is similar to that used in the proof of Theorem 1; the reader may easily find out the slight modifications. Assume now that la1 12. It is easy to notice that exists a sentence z E V+ such that q E vald(z) if and only if there exists a polynomial P whose coefficients are in the set C = { p E a ( b ) ( b E V } such that P ( a ) = q/a. Suppose that z = blb2.. . b,, bi E V,1 5 i 5 m.For

+ ~ a ( b 2+) . . .a"-'a(b,))

VU~+(X)= a(a(b1)

and a # 0, it follows that the required polynomial P is an element in the set of polynomials a(b1) X a ( b 2 ) . . . Xrn-'a(bm).

+

+

Let p = max({ll( I 1 E C} U { q / a } ) . The following algorithm decides, for any given la) 2 2, whether there is a polynomial P with coefficients in C such that P - q/a has the zero a.

36 Algorithm 2. Procedure ExistxPolynornial(q5,q); begin := { p E a(b)lb E V } ; D1 := { - q l a } ; D := 0; repeat D := D U D1; R := 0; for each i E D do for each j E C do if i+j mod a=U then R := R U {i j dzv a } ; if 0 E R then “THE POLYNOMIAL EXISTS”; stop; else D1 := D1 U R; until D = D1;

c

+

“THE POLYNOMIAL DOES NOT EXIST”; end.

In order to finish the proof, we need a reasoning for the correctness of the above algorithm. Termination. We claim that at each step when a number i + j div a, i E D and j E C, is added to R, this number is between -p and p. Indeed, initially the assertion is valid. Assume that at an arbitrary moment, when entering the repeat ...until loop, all elements of D are bounded by -p and p, respectively. For la1 2 2, every multiple of a of the form i j , i E D , j E C, is in the interval [-2p, 2p], hence i j div a is in [-p,p]. Consequently, either 0 E R, during the loop or D = D1 after this loop has been performed at most 2 p times. Correctness. Assume that the algorithm provides 0 in R at some step. This implies the existence of some k 2 1 such that

+

or equivalently q / a E a(bil)ak-’

+

+ a(biz)ak-2+ . . . + a(bi,).

It follows that q E ual#(bikbik--l . . . bi,). Obviously, if the algorithm ends with D = 0 D1, then there is no sentence y such that q E vald(y).

It is worth mentioning here that the problem of deciding upon the strong synonymy between two given sentences can be algorithmically solved for the same class of valuation as that stated in Theorem 6. Theorem 8. If I$ = ( a l p ) is a valuation with p being a n exponential function p(n) = an, whose base a is any integer distinct of 0 and -1, and there is a word

37 b with exactly one value in a(b), then one can decide whether x z $ y, for any sentences x, y.

The proof is an immediate consequence of relation 7 being left to the reader as an exercise.

6

Final remarks

We briefly discuss here some considerations that seem to be in order. In the present paper we have considered the semiring of integers with the addition and multiplication. It appears to be interest to replace it by other semirings (or other structures) having linguistical relevance. Our approach tries to valuate all sentences over a vocabulary. A more natural approach might be the valuation of just those sentences which belong to a given language. An attractive class of languages seems to be the context-free one. As one can easily notice, there are plenty of natural questions without answer; all of them remain to be further investigated. We provide below a list of a very few of them which appear to be more attractive from our point of view. 1. Is it decidable whether or not two given sentences are strongly synonymous with respect to valuations whose position function is an arbitrary polynomial or exponential function? 2. Can we decide the finiteness of strong synonymy classes in the arbitrary polynomial case? What about the same problem for both classes in the exponential case? In our opinion, a natural direction of further work may consider this formalism as an algebrac backbone upon which other formalisms of semantical structure can be grafted.

References [l] J. Berstel and D. Perrin, Theory of Codes, Pure and Applied Mathematics, Academic Press, 1985. [2] J. Berstel and C. Reutenauer, Rational Series and Their Languages, EATCS Monographs on Theoretical Computer Science, vol. 12, Springer, Berlin, 1988. 131 J. Dassow, Gh. P lu n , Regulated Rewriting in Formal Language Theory, SpringerVerlag, 1989. [4] J. Dassow, V. Mitrana, Finite automata over free generated groups, Intern. Journal of Algebra and Computation, 10, 6(2000), 725-737.

38

[5] S. A. Greibach, Remarks on blind and partially blind one-way multicounter machines, Theoret. Comp. Sci. 7(1978), 311-324. [6] H. Fernau, Valuation of languages, with applications t o fractal geometry, Theoret. Comput. Sci. 137 (1995) 177-217. [7] H. Fernau, L. Staiger, Valuations and unambiguity of languages, with applications t o fractal geometry, ICALP'94, LNCS 820, Springer, 11-22. [8] G. Hansel and D. Perrin, Codes and Bernoulli partitions, Math. Systems Theory 16 (1983) 133-157. [9] G. Hansel and D. Perrin, Rational probability measures, Theoret. Comput. Sci. 65 (1989) 171-188. [lo] 0. H. Ibarra, S. K. Sahni, C. E. Kim, Finite automata with multiplication, Theoret. Comp. Sci.,2(1976), 271-294. [ll]W. Kuich and A. Salomaa, Semirings, Automata, Languages, EATCS Monographs on Theoretical Computer Science, vol. 5, Springer-Verlag, 1986. [12] V. Mitrana, R. Stiebe, Extended finite automata over groups, Discrete AppJ. Math., 108, 3(2001), 247-260. [13] Gh. P b n , A new generative device:valence grammars, Rev. Roum. Math. Pures et Appl., 25, 6(1980), 911-924. [14] A. Salomaa, Formal Languages, Academic Press, 1973.

39

Appendix Let S be a finite set of positive integers; we denote by gcd(S) the greatest common divisor of all elements from S . We now proceed to prove the following claim which is equivalent with the fact used in Theorem 1:

Claim 1 Let S be a finite set of positive integers. For any partition S1, S2 of S , with both sets S1,S2 nonempty, the following two conditions are satisfied: 1. There exist coeficients c(t), t E S , such that

gcd(S) = C c ( t ) t tES

and c(t ) 2 0 for all t

E 5’1,

and c( t) 5 0 for all t E S2

2. There exist coeficients c( t) , t E S , such that gcd(S) =

C c(t)t tES

and c(t ) 5 0 for all t E S1, and c( t) 2 0 for all t E

5’2

Proof. We prove the claim by induction on the cardinality of S . Let us assume that S = { t l , t z } . It is known that gcd(S) = ctl f t z for some integers c, f. Clearly, c 2 0 and f 5 0 or c 5 0 and f 2 0. Without loss of generality we may assume that f 5 0. I f f = 0, then c = 1, gcd(S) = t l , and t2 = g t l for some g 2 1. Now we can write gcd(S) = (1 - 2g)tl 2t2 and we are done. Suppose now that f < 0; then ctl = gcd(S)- ft2. Since t 2 = g d , for some g 2 1, one gets ctl = gcd(S) - fg(ct1 f t 2 ) which implies gcd(S) = c(f g l ) t l f 2 g t 2 . For f g 1 5 0 and f 2 g 2 0 the basic step of induction is completely proved. We assume that the assertion holds for any set S of cardinality s, consider a set S’ of cardinality s 1, and a partition of S’ in two nonempty sets S i and SL. Obviously, at least one of these sets, say SL, contains at least two elements. Consider an arbitrary element of S;, say 1 and set S = SIUS~,where Sl = S i and S2 = S;\{l}. By the induction hypothesis,

+

+

+

+

+

+

+

gcd(S’) = gcd({gcd(S),l})= d C c ( t ) t

+ f‘l

tES

holds, where the coefficients c( t) can be chosen as stated above. Again, either c’ 2 0 and f’ 5 0 or c‘ 5 0 and f’ 2 0. We now choose c ( t ) 2 0, 0 for all t E S1, and c(t ) 5 0, for all t E Sz, which completes the proof.

40

JOIN DECOMPOSITIONS OF PSEUDOVARIETIES OF THE FORM DH n ECom KARL AUINGER Institut f i r Mathemat&, Universitat Wien, Strvdlhofgasse 4, A-I090 Wien, Austria E-mail:Karl.Auingerhnavie.ac.at A constructive proof of the equation DH n ECom = (J n ECom) V H is presented where H denotes any arborescent pseudovarietyof groups. In addition, a larger class of pseudovarieties of groups is found for which that equation holds.

1

Introduction

The purpose of this paper is to present a constructive proof of the equation

DH n ECom = (J n ECom) V H

(1)

where H is a “sufficiently nice” pseudovariety of groups. As usual, for a group pseudovariety H, DH denotes the pseudovariety of all (finite) semigroups all of whose regular ID-classes are groups in H, while ECom is the class of all (finite) semigroups with commuting idempotents and J stands for the pseudovariety of all J-trivial semigroups. (In this paper, all semigroups except free semigroups A+ and free monoids A* are assumed to be finite.) A syntactic proof of the equation (1) has been found by Almeida and Weil in the case H being arborescent which means that (H n Ab) * H = H (where Ab is the pseudovariety of all abelian groups and * denotes the Mal’cev product, or, equivalently, the semidirect product of the involved pseudovarieties). Moreover, also in one can find (in terms of a “unique factorisation property”) a condition characterizing the set of all pseudovarieties H satisfying equation (1). From that condition it follows that this set is closed under taking joins (within the lattice of all pseudovarieties). The arguments in are based on a careful study of the free pro-H groups and some knowledge of the free proDS semigroups. (Here DS denotes the pseudovariety of all semigroups all of whose regular ID-classes are subsemigroups). The proof thereby obtained is not constructive in the sense that for a given S E DH n ECom it would effectively construct a semigroup C E J n ECom and a group H E H such that S divides the direct product C x H . From the proof we only know that suitable C and H do exist.

41

In contrast, our approach will prove equation (1) for a larger class of pseudovarietes H which can be characterized by a certain condition (P) (see Definition 2.3) and the proof will be constructive. It is based on a discovery by Ash, Hall and Pin which provides a convenient set of generators of the pseudovariety DH n ECom. In it is shown that each S E DH n ECom divides a precisely described finite direct product of transition semigroups of automata of a very special kind (these automata will be introduced in section 3); conversely, all such transition semigroups are in DH n ECom. The main idea of our proof then will be, given an automaton d of that kind, to consider a certain quotient automaton A/- ( N essentially eliminates the non-trivial group sub-automata of d) and whose transition semigroup is aperiodic (that is, it is a member of J n ECom). Then we construct a suitable finite group H such that the transition semigroup M(d) of d divides the direct product of the transition semigroup M(d/N) of d/- and H . The group H is especially designed to outweigh the “loss in accuracy” which comes from going from d to d/N. The prerequisites to construct such a group are developed in section 2 (without giving full proofs). An expanded version of the paper, containing full proofs and several refinements will appear elsewhere. For undefined notions in the theories of semigroups, pseudovarieties, automata, etc., the reader is referred to the books of Almeida and Pin ’; for background information about varieties of groups the book of Neumann is a good reference. Throughout, for a word w E A* (on a finite alphabet A), c ( w ) stands for the content of w, that is, the set of all letters occurring in w while I wI denotes the length of the word w. For any finite set S, IS1 stands for the number of elements of S. For any A-generated (semi)group S = (A) and any word w E A+ we will write, if emphasis is necessary, w ( S ) to denote the evaluation of the word w in S. 2

Groups

Here we present an auxiliary result which will be essentially used in the next section in the proof of the main theorem. The result is about semigroup identities (not) being satisfied by certain group varieties. (However, the result holds - mutatis mutandis - for group identities as well). The notation throughout this section will be as follows. For a finite alphabet A let ( z i ) i l l be a sequence of letters of A and ( u i ) i ? ~(vi)i>o , two sequences of words in A* (some of them may be empty) such that zz

$ C (I Ui-1) uC(Vi-1)

u c(u2) u (Vi).

42

We will mainly be interested in identities of the form uo21u1... znunN voz1w1.. .2,vn.

Let U ,V be varieties of groups and let of U and V , that is,

U * V = {G I GIN

EV

U * V be the usual (Mal'cev) product

for some normal subgroup N E U}.

It is well known that * is an associative operation on the lattice of group varieties, and that U * V is generated, as a variety, by all possible semidirect products U * V with U E U and V E V (see Neumann '). Moreover, there is a well-known representation of the A-generated free object in U * V as a subgroup of a semidirect product of an appropriate member of U by an appropriate member of V . We use here the version presented in Theorem 10.2.1 in '. More precisely, let F = FAV be the A-generated free object in V and let A' = F x A. Then F acts on A' by g ( h a) , = (gh,a ) Vg, h E F, Vu E A. Consequently, if G = FAIU is the A'-generated free object in U then F acts on G by automorphisms on the left via g[(hl,U1)*1.

. . (hn,an)*l] = (gh1,Ul)*l ... (ghn,an)*I.

So we may form the semidirect product G * F subject to this action. Then the free object on A in U * V is isomorphic to the subgroup of G * F freely generated by the elements of the form ((l,a), a) where a E A. For semigroup identities u N v with u, v E A* this means that if u = a1 . . .an and v = bl . . .b, then U * V u N v if and only if (i) V k u 'v v and (ii) U k (l,al)(al,a2). .. (a1.. .an-l,an)N (l,bl)(bl,b2). .. (bl . . .b,_l,b,). In the latter identity the variables are of the form (u,a ) with u E A* and a E A, and two such variables, say ( u ,a ) and (v,b) are the same if and only if a and b are the same letters from A and V u 'v v. We are going to formulate the main result of this section. For each positive integer n let ?in be a group variety which does not satisfy any non-trivial semigroup identity u E v with ) u ((,v J5 n and u,v E A*. For each n 2 0 let

+

Vn := '?&+I

*'+I!, * . . . * 3t2 * 3c1

and for convenience put V-1 := 7, the trivial variety. In the next results we assume that the words ui,vi and the letters xi satisfy the conditions imposed at the beginning of this section. The first result can be proved by induction on n.

43

Lemma 2.1 The variety V, does not satisfy any identity uo21u1.

..2,+1u,+1

?! wo21w1

. . .xtvt

for 0 5 t 5 n. The main result of this section is an easy consequence of this lemma, again to be proved by induction on n. Theorem 2.2 If V, satisfies uox1u1.. .x,an N V o z l V l . . .Z,W, then Vn-1 satiesfies u; E wi for all i. The following property (P) of a pseudovariey H of groups will turn out to be crucial for our purpose. Definition 2.3 A pseudovariety H of groups has the property (P) if f o r each G E H and for each positive integer n there is a group F E H such that 1. F does not satisfy any non-trivial semigroup identity u 'v v f o r IuI, IvI 5 71.7

2. ( F ) * (G) C H. Property (P) has already been pointed out to be of some interest in '. Here (. . .) denotes the pseudovariety generated by the group ". . ." and * is the Mal'cev (that is, semidirect) product of the involved pseudovarieties. Observe that a pseudovariety H enjoys property (P) if the seemingly weaker condition holds: for each group G E H there exists a prime p for which the wreath product q p z 1 G is in H (see '). The following corollary is a consequence of Theorem 2.2 in a form we shall use it in the next section. Corollary 2.4 Let H be a pseudovariety of groups satisfying the property (P) and let G = ( A ) be an A-generated group in H; then f o r each positive integer n there is an A-generated group G, = ( A ) in H such that 1 . for all u,v E A+, i f u(G,) = w(G,) then also u(G) = w(G),

2. whenever (uox1u1.. . znun)(G,) = ( w o z 1 w 1 . ..xnwn)(Gn) then ui(G) = wi(G) for all i (with the assumptions on the words ui, w; and the letters x; imposed at the beginning of this section). Proof. Let Go = G E H and n E N;choose a group F1 E H not satisfying any identity u E w with 1u1, lwl 5 2 such that (F1) * (Go) C H. Notice that (F1)* (Go)is locally finite, that is, it is the finite trace of a locally finite variety. In particular, in (F1) * (Go) all finitely generated free objects exist. Let G1 be the A-generated free object in (F1) * (Go). Suppose that F,-1 and G,-1 have already been constructed. Let F, be in H such that F, does not satisfy any non-trivial identity u N w with 1211, IwI I n 1 and (F,) * (Gn-l) C H.

+

44

Now let G , be the A-generated free object in (F,) * (Gn-l). By induction one can see that for all i 5 n, Gi and (Fi)* . . . * (F1) * (Go) satisfy the same identities in [A1variables. Consequently, G , is the A-generated free object in (F,) * .. . * (F1)* (Go) and so by Theorem 2.2 has the requested property.

Remark 2.5 Note that the property (P) is not a closure property in the sense that for each pseudovariety H there would exist a least pseudovariety V such that H G V and V has (P).

3 The main result As mentioned in the introduction, a pseudovariety of groups is arborescent if (H n Ab) * H = H . It has been shown in that for H arborescent free pro-H groups enjoy certain “unique factorisation conditions” (similar to the free group). These factorisation conditions were, in turn, one essential ingre dient of the proof that such pseudovarieties satisfy the equation (1); the other ingredient was a sufficiently precise description of the implicit operations on DS. This section gives a constructive proof for the join decomposition (1) which applies to a wider class of pseudovarieties, namely those which satisfy condition (P). As a preparation for this new proof we first shall recall a result of Ash, Hall and Pin presenting a set of “nice” generators of the pseudovariety DH n ECom. We require some definitions. Throughout, let A be a finite set of letters. A permutation automaton (or group automaton) A = (&,A,-) on A consists of a finite non-empty set of states Q together with a labelling of some permutations of Q by the letters of A, denoted by a : q C) q .a. Here two different letters may label the same mapping and the m e IQI = 1 is also included; in the latter case, each letter labels the identity mapping on Q. Let r E N;for each i 5 r let Ai C A and = (Qi, Ai, -) be a permutation automaton such that Qi n Qj = 0 if j # i. Moreover, let uo,u, E A* and u1,. . . ,u,-1 E A+ be such that the first letter of ui is not in Ai (1 5 i 5 r ) and the last letter of ui is not.in Ai+l (0 5 i 5 T - 1). For each i choose pi,qi E Qi; as in 2 , p. 393, let

be the automaton depicted in Figure 1 (with appropriately chosen states inside each path ui).

45

Figure 1: The automaton A

'

A formal definition of this automaton has been given in on p. 38. For convenience, let us call such an automaton good. Note that in good automata have been characterized as those admitting a linear quasiorder on the set of states which is compatible with the action of the letters. Let n = l u o u 1 . ..url be the length of that automaton. If all transition groups of the automata belong to the group pseudovariety H then the automaton d will be called H-good. The transition semigroup M(d) of an H-good automaton A is in DH n ECom (see 5 , Corollary 2.3 or 2 , Lemma 3.9). Moreover, the class of all transition semigroups of H-good automata generates the pseudovariety DHnECom. On the one hand, this follows from the proof of Theorem 3.8 in 2 : there it is shown that the class of all such transition semigroups does not satisfy more pseudoidentities than DH n ECom itself; therefore, by Reiterman's Theorem the result follows. On the other hand, a constructive proof of this assertion has been presented in '. That is, given any S E DHnECom, a precisely described finite set of H-good automata has been constructed such that S divides the direct product of the transformation semigroups of these automata (Proposition 3.5 in '). We shall discuss that construction in more detail. Crucial for it is an important lemma by Ash which holds, if appropriately reformulated, in the more general context of semigroups with commuting idempotents and which lemma has been an important step in Ash's famous proof that each semigroup with commuting idempotents divides an inverse semigroup (see The version we shall need is the following (see Propositions 3.3 and 3.4 in '): Proposition 3.1 Let S = ( A ) E DGnECom; then there is a positive integer K = K ( S ) (depending o n S only) such that each word w E A+ admits, for some n 2 0, a factorization w = goulgl .. .ungn such that

'

314).

I . all g,(S> are group elements (go,gn may be empty),

2. for each i, the first letter of ui is not in c(gi-1) and the last letter of ui is not in c ( g i ) , 3.

lUlU2..

.un-lu,I 5 K

46

This statement and its more general version is usually proved by the use of Ramsey's Theorem. For the above mentioned case we are interested in, semigroups in DG n ECom, we shall give an elementary proof, thereby getting a better bound for the number K . Recall that each S E DG n ECom is a semilattice of unipotent semigroups, that is, if q is the least semilattice congruence on S then S/q C E ( S ) and each q-class S, is an ideal extension of the group He (the group %-class containing e ) by a certain nilpotent semigroup N , U (0). Then S, = He U Ne and the product of any INeI 1 elements of S, lies in He. Notice also that for any u , v E A+ with c(u) = c ( v ) we have u(S)r] v(S), that is, u ( S ) and v(S) are in the same subsemigroup S,. Corollary 3.2 Let S = ( A ) E DG n ECom and let

+

N = N ( S ) = max l N e l + l . eEE(S)

Then the number K ( S ) of Proposition 3.1 can be chosen to be less than NIAl. More precisely, each w E A+ admits a factorisation as in Proposition 3.1 such that Ju1. . . u,1 5 Nlc(w)I- 1. Proof. We only have to prove the existence of a factorisation satisfying conditions ( 1 ) and ( 3 ) . Namely, if g(S) is a group element and if x € c(g) then J,(s) 2 J,(s) and therefore (zg)(S)and ( g e ) ( S )are group elements, as well. Likewise, if g(S) and h ( S ) are group elements then so is ( g h ) ( S ) . Therefore, each factorisation satisfying (1) and (3) can be reduced to a factorisation satisfying (1),(2) and (3). The proof now is by induction on the size Ic(w)l of the content of w. If 1c(w)1 = 1 then the claim follows immediately from the definition of N . So let w E A+ and suppose that the claim is true for all words v with Ic(v)I < lc(w)l. We factorize w as w = 'lllx1u2x2... 'Ilkxkuk+l

such that for all i 5 k , Ic(ui)l = lc(w)l-l and xi is a letter not being contained in c(ui), that is, c(uixi) = c ( w ) ; moreover, I C ( U k + l ) l < Ic(w)I and Uk+1 may be empty. If k 2 N then we are done because in this case,

is a product of N or more elements within the same semilattice congruence class Se so that this product lies in He. Since c(uk+l) C c(u1x1. . . ukxk) the element w(S) itself is a group element. So we may assume that k 5 N - 1. By induction hypothesis, each ui admits a factorization in group and nongroup parts such that the accumulated length of the non-group parts is by induction hypothesis - at most Nlc(w)l-l- 1. Thereby we have already found a factorisation of w in group and non-group elements: each uixi admits

47

such a factorisation with the accumulated length of the non-group elements being at most Nlc(w)l-l - 1 + 1 = NIC(w)l-l.Hence the accumulated length of the non-group elements in w is at most N l " ( w ) l - y N - 1) + N I C ( 4 - 1 - 1 = N I C ( 4 l - 1,

as required. Remark 3.3 For the semigroup S = ( A ) E DG n ECom and for any a C A, a # 8 put N , = {w(S) I w(S) is non-regular and c ( w ) = a } and let N ' ( S ) = m a x ~- # IN,I+l. ~ c ~ Then Corollary 3.2 still holds if N ( S )is replaced with N'(S). The proof of Proposition 3.5 in (combined with Corollary 3.2) now shows the following. Corollary 3.4 Let S = ( A ) E DG n ECom; then S divides a direct product of transition semigroups of good automata on A such that I . each of these automata has length at most IAIN(S)lAl; = ( H e ,A,, -) where 2. the incorporated group automata are of the form each He is a maximal subgroup of S (which can be regarded to be generated by a subset A, of A); the action . is just the multiplication on the right. The transition group of each such automaton is isomorphic to the group He.

Remark 3.5 1. In item 1 we could restrict ourselves to good automata on A of length (precisely) IAIN(S)IAI; but then, as another factor in the direct product, we have to mention the least group 'fl-class of S which is A-generated and which may be needed for the division but which need not be representable as (a divisor of) the transition semigroup of any automaton described in item 2 of Corollary 3.4and having positive length. 2. Observe that the number of distinct automata satisfying i t e m 1 and 2 above and also the size of each associated transition semigroup are bounded by a primitive recursive function in the cardinality of S. 3. In the proof of Proposition 3.5 in one can argue by induction on Ic(w)U c(w')Jinstead of Ic(w)l Jc(20')).This is the reason why the factor 2 which occurs in the sentence before Proposition 3.6 in does not occur in the expression IAIN(S)IAI. Now let d = d(uO,p1,dl,q1. .. , p , , A,q,., u,) be a good automaton with group automata = ( Q i ,Ai, Let M(d) be the transition semigroup of A. We intend to show that M(d) divides the direct product C x H for some

+

0 ) .

C E J n ECom and some group H . The aperiodic semigroup C is obtained by “factoring the groups out of d”: more precisely, consider the equivalence relation on the set of states of d which identifies all states within each group automaton A. The resulting automaton, denoted by A/- is the following (as on p.394 in ’):

-

A1

A’

Ar

Figure 2: The automaton A/The associated transition semigroup C = M ( d / - ) is in J n ECom (see Corollary 2.3 in or Lemma 3.9 in ’). Now we construct the group H . Thereby we shall use the ideas developed in section 2. Let 3tl be the (non-trivial, locally finite) variety of groups generated by the transition groups of the automata d k (1 5 Ic 5 r ) . Let n = luo.. .url be the length of d and for each i E (2,. . . , n 1) let Xi be a locally finite variety of groups not satisfying any non-trivial identity u 11 w * 3t1 and let with IuI, Iwl 5 i where u, w E A*. Let V , = %,+I * . . . * 3 t ~ H = (A) be the free group on A in V,. Then we have, with the notation introduced above, the main result of the paper: Theorem 3.6 The semigroup M ( d ) divides the direct product C x H . Proof. We show that the map (a,a) e a, a E A, extends to a morphism ((a,a) 1 a E A ) C C x H -+ M ( d ) . Let u,w E A+ be such that

+

u(C) = w(C) and u ( H ) = w ( H ) . We have to show that u ( M ( d ) )= w(M(d)); therefore, it suffices to show that for each state q of d, q . u = q . w. From u(C) = v(C) and from the definition of A / - we have the following: for each state p of d / - , p . u = p . w. Consequently, for each state p of d ,p . u is defined if and only if p - w is defined. Let i j = q-; assume that i j is somewhere on the path ui and ?j. u = i j . w is somewhere on uj. Then i 5 j and we may assume that i < j . Then there are factorisations of u and w, respectively, such that

and

49

where the words U k are the words occurring in d resp. d/N, u: is a (possibly empty) su& of ui, u>is a (possibly empty) prefix of u j , and for all possible 1, the first letter of uz is not in c(gz)uc(hz)[c(gj)Uc(hj)]while the last letter of uz [u:]is not in c(gz+l)Uc(hz+l) [c(gi+l)Uc(hi+l)J(some of the words gz, hz may be empty, as well). By our assumption, u ( H )= v ( H ) ,that is V, b u E v. By Theorem 2.2, 311 gz N hi for all 1 (notice that in Theorem 2.2 some of the segments U k , vk may be empty). In particular, for each I E {i 1, . .. ,j } and each group G = (Az) in 311 we have that gI(G) = hl(G). Let GI = ( A z ) be the transition group of the automaton dz. By the above argument we have gz ( G I )= hz (G I ) ,that is, gz (G I )and hz(G I ) are the same element of the group GI and consequently, gz and hz act in the same way on &I. This applies to each I E { i 1, . . .,j } . Therefore, q .u:gi+l = q .u:hi+l, and by induction we get:

[ui]

+

+

q * u:gi+l = q . u:hi+l* q . u:gi+lui+l = q . u:hi+lUi+l Q . u:gi+lui+lgi+2 = 4 . u:hi+lui+lhi+z

* * ... * q . u = q . v

and the theorem is proved. Using Corollary 2.4 we get the next result precisely in the same way. Corollary 3.7 Let H be a pseudovariety of groups satisfying condition (P) of Definition 2.3. Let M be the transition semigroup of a n H-good automaton. Then M divides the direct product C x H for some C E J nECom and some H E H. We have already remarked that, given an arbitrary semigroup S = ( A ) in DG n ECom then we can find transition semigroups M I , . .. ,Mk of good k automata such that S divides Hi=, Mi and such that k as well as all /Mil are bounded by a primitive recursive function in JSI. Moreover, the length of the involved automata is bounded by IAIN(S)IAI.Similarly as in 6 , Corollary 3.1, it can be shown that the cardinality of the group H in Theorem 3.6 can be bounded by a primitive recursive function in the length of the involved automata and the cardinalities of the involved subgroups. Summing up, we get the next result (answering problem 25 in for that particular join decomposition): Corollary 3.8 The decomposition DG n ECom = (J n ECom) V G is effective in the following sense: each S E DG n ECom divides a direct product C x H f o r some C E J n ECom and H E G such that the cardinalities of C and H are bounded by primitive recursive functions in the cardinality of S . We remark that the latter result more generally holds for each arborescent pseudovariety H. What we actually need is that in property (P) (Definition 2.3) the cardinality of F is bounded by a primitive recursive function in n and

50

the cardinality of G. References

1. J. Almeida, Finite Semigroups and Universal Algebra, World Scientific, Singapore, 1994. 2. J. Almeida and P. Weil, Reduced factorizations in free profinite groups and join decompositions of pseudovarieties, Int. J. Algebra Comput. 4 (1994) 375-403. 3. C. J. Ash, Finite idempotent-commuting semigroups, pp 13-23 in: S. M. Goberstein and P.M.Higgins (eds.), Semigroups and Their A p plications, Reidel Publishing, Dordrecht, 1987. 4. C. J. Ash, Finite semigroups with commuting idempotents, J. Austral. Math. SOC.43 (1987) 81-90. 5. C. J. Ash, T. E. Hall and J. E. Pin, O n the varieties of languages associated with some varieties of finite monoids with commuting idempotents, Inform. Computation 86 (1990) 32-42. 6. K. Auinger, Semigroups with Central Idempotents, pp 25-33 in: J. C. Birget et a1 (eds.), Algorithmic Problems in Groups and Semigroups. Birkhauser, Boston Base1 Berlin, 2000. 7. K. Auinger and B. Steinberg, The geometry of profinite graphs with applications to free groups and finite monoids, preprint. 8. H. Neumann, Varieties of Groups, Springer-Verlag, Berlin Heidelberg New York, 1967. 9. J. E. Pin, Varieties of Formal Languages, North Oxford, London and Plenum, New York, 1986.

51

Arithmetical Complexity of Infinite Words S. V. Avgustinovich,* D. G. Fon-Der-Flaass,t A. E. Fridl Sobolev Institute of Mathematacs, pr. Koptyuga, 4, Novosibirsk, Russia Email: {avgust,flaass,frid}Qmath.nsc.ru

Abstract

We introduce a new notion of the arithmetical complexity of a word, that is, the number of words of a given length which occur in it in arithmetical progressions. The arithmetical complexity is related t o the well-known function of subword complexity and cannot be less than it. However, our main results show that the behaviour of the arithmetical complexity is not determined only by the subword complexity growth: if the latter grows linearly, the arithmetical complexity can increase both linearly and exponentially. To prove it, we consider a family of DOL words with high arithmetical complexity and a family of Toeplitz words with low complexity. In particular, we find the arithmetical complexity of the Thue-Morse word and the paperfolding word.

1

Introduction

The famous 1927 theorem by Van der Waerden [9, 71 states that for each infinite word w = wow1 . . . w, . . . on a finite alphabet C there exist arbitrarily long arithmetical progressions k, k + p , . . . , k n p such that wk = wk+p =

... - wk+np.

+

In this paper we are interested in the following generalization of the problem: what can the words WkWk+p . . . Wk+np be in general for a given w and arbitrary k, p , and n ? What are the properties of the arithmetical closure *Supported in part by INTAS (grant 97-1001) and RFBR (grant 00-01-00916). +Supported in part by Netherlandish-Russian grant NWO-047-008-006 and RFBR (grant 99-01-00581). fsupported in part by RFBR (grant 99-01-00531) and Federal Aim Program “Integration” (joint grant AO-110).

52

FA(w), i. e., of the language consisting of all such words? In particular, we make an attempt to compute or estimate the arithmetical complexity f i ( n ) of w which is defined as the number of words of FA(w)of length n. The procedure of taking arithmetical closure reminds what is called decimations in the paper [6] by J . Justin and G. Pirillo. In terms of that paper, taking an arithmetical progression is a blind decimation. Note that the arithmetical closure can be defined not only for a word but also for a language, and the term “closure” is used just because for all w the equality FA(w)= FA(FA(w)) holds. As for the arithmetical complexity f i ( n ) ,it is somehow similar to the usual subword complexity f w ( n ) ,i. e., t o the number of factors of w of length n. For example, both subword and arithmetical complexities of a periodic infinite word are ultimately constant, and both functions are bounded by (#C)n. Clearly, for all w and n

We try to show in this paper that arithmetical subwords and the function of arithmetical complexity are worth studying. In Section 2, after introducing the needed notions, we show that an arithmetical subsequence of a uniformly recurrent word is uniformly recurrent. Then we pass to studying arithmetical complexity and first show in Section 3 that it is not obliged to grow as slow as the subword complexity. We describe a class of words containing the Thue-Morse word and having linear subword complexity and arithmetical complexity equal to ( # C ) n . On the other hand, arithmetical complexity of a non-periodic word can grow linearly, and in Section 4 we validate it by an example of some family of Toeplitz words. Finally, we use the latter result to compute the arithmetical complexity of the paperfolding word, which turns out to be equal to 8n 4 for all n 2 14.

+

2

Basic Notions

Let C be a finite alphabet. As usuai, the set of all finite words on C is denoted by C*, the set of all non-empty finite words is denoted by C+, the set of all words of length n is denoted by En, and the set of all (right) infinite words is denoted by Cw. For any t E C+, the word tt . . . t . . . is denoted by tW. A (finite) word u is called a factor, or subword of a (finite or infinite) word v if v = slusz for some words $1 and s2 which may be empty. Let us consider an infinite word w E Cw:

53

where wi E C. The set of all factors of w is denoted by F(w).The the wellknown function of subword complexity of the word w (or of the language F(w)) is the number of words in F(w)of length n; we denote it by f,(n) = f F ( w ) ( n ) . Let us call the infinite word wi = WkWk+pwk+Zp. . . ~ k . . .+ the~ arith~ metical subsequence of w starting with position k and having diference p . A factor of some wi is called an arithmetical subword of w,and the set FA(w) of all arithmetical subwords of w is called its arithmetical closure:

FA(w)=

u

I P > Lk,n20},

F(w~)={X}U{~k~k+p...wk+np

P21,kB

where X denotes the empty word. In these terms, the Van der Waerden theorem can be stated as follows:

Theorem 1 (Van der Waerden 1927) For each infinite word w and positive integer n there exists a symbol a E C such that an E FA(w). In this paper, we are interested in the properties of the language FA(w) and in particular in its subword complexity f F A),( ( n ) which is denoted also by ( n ) and called the arithmetical complexity of w. Let v = W k W k + l . . .Wk+n. Formally, an occurrence of v in w is the word v together with the number k of its first letter in w.Clearly, a word may have a finite or infinite number of occurrences in a n infinite word w.If v is a prefix of w ,we call its occurrence corresponding t o k = 0 the prefix occurrence. Recall that a n infinite word w is called uniformly recurrent if each of its factors occurs in w infinitely many times with bounded gaps, i. e., if there exists a finite recurrence function R,(n) such that each factor of w of length R,(n) contains all factors of w of length n. The following lemma does not seem t o be new, but since we did not find a reference t o it, here it is given with a proof.

fi

Lemma 1 An arithmetical subsequence of a uniformly recurrent word is uniformly recurrent. PROOF. Let us consider an arithmetical subsequence wf of a uniformly recurrent word w. Since a word obtained from a uniformly recurrent word by erasing a finite prefix is uniformly recurrent, it is sufficient to consider 1 = 0 and t o show that the prefix u' of wj of length n' 1 occurs in it once more with a gap bounded by a function of n' and p. Let n = n'p and u = wo. . . w,. The word u' is an arithmetical subword of u. To find another occurrence of u' in w;,we shall find an occurrence of u in w lying at a distance dividing p from the prefix occurrence.

+

54

For an occurrence B = W k . . .W I + ~of a word v in w, we define the function c(B) = Ic mod p . Our goal is t o find an occurrence fi of u not equal t o the prefix one and having c(ii) = 0. Denote u = uo. For all i = 1,.. . , p , we define ui inductively as the minimal prefix of w containing two occurrences of ui-1 (including the prefix occurrence): ui = ui-19-i = liui-1 for some Zi,ri E C+. Since w is uniformly recurrent, all ui are well-defined and Iuil 5 R,(Iui-ll) + 1: indeed, by definition, an occurrence of ui-1 is contained in each subword of w of length Rw(lui-ll)including w1 .. . w ~ ~ ( l ~and ~ -the ~ loccurrence ), of ui-1 is not the prefix one. Thus, ui is a prefix of wo. . .WR,,,(I~~-~ 1). For all j 2 i 2 0, let us denote the last occurrence of ui in the prefix occurrence of uj by ua. By the pigeon-hole principle, at least two of the numbers c(uz) = 0, c(uk), . . . ,c(uE) are equal: c(ui) = c(uA) for some 0 5 i < j 5 p . This implies the equalities c ( u i ) = c ( u i ) for all Ic 5 i; in particular, 0 = c(ui) = c(u{). Since u = uo is a prefix of ui,we have also c(fi)= 0, where fi is the first occurrence of u in u:. Since i < j, ii is not the prefix occurrence of u in w but the needed one. Clearly, since luol = n 1 and luil 5 R,(lui-ll)

+

lupl I R,(R,(. . . (R,(n

+ 1 for all i, we have

+ 1)+ 1).. . + 1)+ 1.

P

Consequently, two occurrences of u’are contained in the prefix of w: of length at most

1 1 - ( / u p] 1) 1 5 - R,(R,(. . . (R,(n

P

+

- P

+ 1)+ 1).. . + 1)+ 1)+ 1.

P

Since this upper bound does not depend on the choice of u‘ (to consider another word, we just erase another prefix of w in the beginning of the proof), there is an occurrence of each factor of w: of length n’ + 1 in each its factor of length ;R,(R,(. . . (R,(n 1) 1).. . 1) 1). So, this is an estimate

-

+ +

+ +

P

for the recurrence function:

R,; (n’

+ 1) 5 P--1R,(R,(.

. . (R,(p’

+ 1)+ 1).. . + 1).

P

We have estimated the recurrence function of an arithmetical subsequence using its difference and the recurrence function of the initial word. The lemma is proved. 0

55

High Arithmetical Complexity In this section we consider DOL words. Let cp : C* + C* be a morphism, i.e., 3

a mapping satisfying cp(xy)= cp(x)cp(y)for all x , y E C*. Clearly, a morphism is completely defined by the images of letters. If for some a E C the image cp(a) starts with a, and Icpi(a)l-b 00, then there exists a right infinite word called the fixed point w of cp starting with a and defined by the equalities

w = cp(w) = lim @ ( a ) . n+cc

Fixed points of morphisms are called also DOL words and are widely used as examples of infinite words with given properties. Here we consider only uniform morphisms, that are morphisms with all the images of letters having the same length denoted by m. Let C = C, = {0,1,. . . ,q - l},and let a @ b denote the symbol-to-symbol addition modulo q of the words a and b of equal length. In this section, for every i E {0,1,. . . ,q- l},the expression ij denotes the word i . . . i (and not the j t h power of the number i ) . To distinguish it from

v j

the arithmetical subsequence wi, the word

Wk

.. ' w k is denoted by

(wk)j.

j

We say that a morphism cp is symmetric if for all i E C we have cp(i)= cp(0) @ i". Clearly, a symmetric morphism is determined by the image of 0. A DOL word is symmetric if it is a fixed point of a symmetric morphism starting with 0. Theorem 2 Let cp be a symmetric morphism on C , , where q is prime, and let its jixed point w be non ultimately periodic. Then for all n 2 0 we have f 3 4 = qn.

PROOF. The words of the form Oilbi, where Ibil = n - i , constitute a basis of C y under the symbol-to-symbol addition modulo q. So, it is sufficient t o prove, first, that the set of arithmetical subwords of w having length n is closed under this addition, and second, that the words O i l are contained in FA( w)for all i 2 0. First, let us consider two arithmetical subwords a and b of equal length of w and prove that a @ b also belongs t o FA(w). Let a = W k W k + p . . . ~ k and b = w k ' w k ' + p ' * . . Wk'+np'. Let r be an integer such that k n p < mr, where m is the image length of a symbol. Note that for every i , j E C we have cp(i @ j) = cp(0) @ i" @ j m = p ( i ) @ j " . Thus, cpz(i)= cp(cp(0) @ i") = ( ~ ~ (@0im2, ) etc.; by induction, we have cp'(i) = cpr(0) el3 impfor all i. The symbol Wk'mP+k is the (k + 1)st

+

+

56

symbol of cp'(wk0 = cp'(0) @ (wkOmr; thus, it can be obtained from the (k 1)st symbol of cp'(0) (equal t o W k ) by adding W k ' modulo q. This means that W k l m r + k W k @ W k ' ; analogouSly, W(kt+pl)mP+k+p = wk+p @ Wkl+pl, etC. Thus, a @ b = wk'm'+kW(k'+pl)mP+k+p . . . 'W(k~+np')mP+k+np; We see that a @ b E FA(W). Now let us prove that Oil E FA(w)for all i 2 0. Since q is prime, it is equivalent t o Oik E F A ( w )for some k E C,, k # 0: indeed, kc 1 (mod q ) for some number c E {0,1,. . . ,q - l},and adding Oik t o itself c times as it is described above, we obtain O i l . Suppose the opposite: an arithmetical subsequence of the form Oi can be prolonged in FA( w ) only by 0 if i is sufficiently large. Since the morphism is symmetric and thus 0 and any other symbol are interchangeble in FA(w), and due t o the Van der Waerden theorem, infinitely long arithmetical progressions of 0's do occur in F A ( w ) . Let us consider such progression wa = 0". Since we always can pass from cp to some its power cpk without changing the fixed point w , without loss of generality we assume that the difference p of the progression is not greater than m. So, since we know w;, starting with some point, we know at least one symbol of each image of letter. But since the morphism is symmetric, this is sufficient to determine the images of letters themselves. Since the positions modulo m of known symbols (participating in wa) change periodically, w is itself ultimately periodic. A contradiction. 0

+

Remark 1 It is possible t o characterize all ultimately periodic symmetric DOL words using the fact that they must contain arbitrarily long factors whose position modulo m depends on an occurrence. In other terms, each ultimately periodic symmetric DOL word is uncircular (an explicit definition of this notion can be found e. g. in [5]). Using the criterion of circularity obtained in [5] and adding the condition that the morphism cp is symmetric, we can conclude that a fixed point of a symmetric morphism cp can be (ultimately) periodic only if the morphism has a very special structure. Namely, up t o cyclic renaming of symbols (of the form k -+ ck mod q for a fixed integer c), we must have p(0) = (01. . . ( q - 1 ) ) l O for some I > 0. The complete proof of this statement is not at all difficult but rather cumbersome and thus is omitted.

Remark 2 If we replace the condition of non-periodicity by that of occurrence of O i l in FA(w) for all i, Theorem 2 becomes true for all cardinalities of alphabets.

57

So, at least for the case when the cardinality q of the alphabet is prime (and in fact for arbitrary alphabets too), Theorem 2 concerns most symmetric DOL words. In particular, it is applied t o the most famous one, the ThueMorse word W T M = 0110100110010110~~~ (see [3]) which is the fixed point starting with 0 of the symmetric morphism ( P T M :

{

(PTM (0) (PTM(1)

= 01, = 10.

Since q = 2, the arithmetical complexity of the Thue-Morse word is 2".

4

Low Arithmetical Complexity

In this section, we consider a subfamily of Toeplitz words and prove that each word from it has a linearly growing arithmetical complexity. Let us consider an alphabet E and a symbol ? $! C called gap. A pattern is a finite word t E C(C U {?})*. For an infinite word w E (C U {?})" and a pattern t we define an infinite word Tt(w)as a result of replacing all gaps in w by corresponding in order symbols of the periodic infinite word t W . Consider a sequence of patterns tl ,t 2 , . . . ,t , . . . and the corresponding sequence of infinite words

.. Clearly, this sequence has a limit U(t1,. . . ,t,, . . .) E C". It is called the Toeplztz word generated by the sequence (tl , .. . ,t,, . . .). In the particular case when all the ti are equal t o the same pattern t , we say that the Toeplitz word is generated by t and denote it by U ( t ) . The subword complexity of Toeplitz words generated by one pattern was found in [4] (see also [S]); it always grows as a polynomial and is linear in the case when the number of gaps in the pattern divides its length. All such words are uniformly recurrent. In this paper we consider only patterns with gaps constituting an arithmetical progression of prime difference dividing the length of the pattern; i. e., patterns of the form

58

where 1 is prime and j E (1,.. . ,I - 1). The length of such pattern t is q l , and the gaps constitute an arithmetical subsequence in tW starting with j t h symbol and having difference 1. The set of all such patterns is denoted by T(1,q , j ) . A very close notion of regular patterns together with corresponding Toeplitz words (generated in general by different patterns) and their subword complexity were considered in [8]. Example 1 Let us consider a pattern t,f = 0?2?. Clearly, t,f E T(2,2,1). We have

uo =?”

=............................

... ,

Ul = Tt, (Uo) = 0?2?0?2?0?2?0?2?0?2?0?2?0?2?.. * , ., U2 = Tt, ( U i ) = Q02?022?QQ2?022?QQ2?022?QO2?~~ U, = Ttpf(U2) = 0Q20022?0022022?0020022?0022~. . ,

...

The limit of this sequence is called the (canonical) paperfolding word and denoted by U ( t , f ) :

.. . U ( t , f ) = 002002200Q220220002002220022~ The subword complexity of U(t,,) was found in [l]and is equal t o 4n for n 2 7. By a result of [4] (and also, indirectly, of [S]), the properties of the paperfolding word are not unique: the subword complexity of every Toeplitz word generated by a pattern of T(Z,q,j)is O ( n ) . We prove that the same property holds for its arithmetical complexity: Theorem 3 For every t E T(1,q, j ) the arithmetical complexity of U ( t ) grows linearly: f&t,(n)= Wn).

PROOF. First of all, without loss of generality we can consider only canonical patterns, i. e., patterns all whose symbols except gaps are distinct. Indeed, we can obtain any other pattern t’ E T(1,q, j ) (and consequently the Toeplitz word U(t’)) from a canonical pattern t E T(Z,q,j)(respectively, from U ( t ) ) by identifying symbols. Thus, factors of arithmetical subsequences of U (t’) are also obtained from factors of corresponding arithmetical subsequences of U ( t ) by identifying symbols. Since each symbol of the canonical pattern t always maps to the same symbol of t’, a word from F l ( U ( t ) )always maps t o the same word of F l ( U ( t ’ ) ) ,but different words of FA(U(t)) can give the

59

same word of F A ( U ( ~ ' )Thus, ). the Toeplitz word generated by the canonical pattern has the maximal arithmetical complexity: fvA(t) (n) 2 f $ t , ) (n). Without loss of generality we consider the word U ( t ) generated by the pattern t = TOTI . . . ~ ~ l -such 1 that the symbol ri is equal to i if i # kl + j , 0 5 k < q, and t o ? otherwise. Clearly, t is a canonical pattern from T(1,q , j ) . Let us consider a finite or infinite word u = uoul . . 'u,.. . with ui E C. We say that a position i (and the symbol ui)have nth order if i = k l n + j e . This definition is introduced so that if u is a Toeplitz word U(t1,t 2 , . . .) for ti E T(1,q, j ) , then i is of nth order if and only if ui appears from a gap not earlier than in Un+l. This is easy to prove by induction: its base is given by the fact that all positions are of order 0, and its step uses the fact that the symbols of (n 1)st order in u are exactly the symbols of 1st order in the arithmetical subsequence of u constituted by its symbols of n th order. Let us choose an arithmetical subsequence v = vovl. . . v, . . . , where vi E C, of the word U ( t ) and study the set of factors of v. Let the difference of v be equal t o p = mql p' for some m 2 0 and p' E (0,. . . ,ql - 1); in other terms, let it be equal to p' modulo ql. Suppose first that 1 divides p'. If the position of the first symbol of v in U ( t ) is equal t o j modulo I , then v consists of symbols having 1st order in U ( t ) ,and thus is equal to another arithmetical subsequence of U ( t ) having smaller difference and consisting of appropriate symbols of 0th order. So, it does not contain factors which do not occur in subsequences of smaller differences. And otherwise, if the position of the first symbol of v in U ( t ) is not equal t o j modulo 1, then there are no symbols having 1st order in U ( t ) in it, and thus v is periodic with the period not exceeding q. Such subsequences can add only a finite number of arithmetical subwords of each length. Now consider the main case of the difference p = mql p' not divided by 1. Since I is prime, it means that exactly one of each 1 consequtive symbols of v is of first order in U ( t ) ,exactly one of each 1' symbols is of second order etc.. For each n 2 0, let us consider the set S(n) of all factors of v of length 1" whose ( w j ) t h symbol is of nth order in U ( t ) ,i. e., is situated in a position s for some k. Clearly, for all n the set S(n) is not empty. number kln j Let us show that the prefix of length 1" of a word v ( n 1) E S(n + 1) belongs t o S(n). Indeed, for all j > i, the ith and j t h symbols of a word from S(n 1)are symbols situated at distance p ( j - i) in w. In particular, since its (-j)th symbol occupies the position number a ( k ,n) = k l n + l + j 11"+'-1 -1,

+

+

+

+

+

its ( ' 2 j ) t h symbol is situated at the position number

+

60

which is of nth order. Thus, there exists an infinite word s such that for each n its prefix of length 1" belongs t o S(n). By the construction of s , each its factor is a factor of v. Vice versa, since v is uniformly recurrent according t o Lemma 1, and s contains its arbitrarily long factors, it contains all its factors. So, F(v)= F(s). Let sk be the kth symbol of s having nth order in it. In the initial Toeplitz word U ( t ) ,sk+q' is situated at the distance pqZn+l from sk. This distance is divided by qln+', and thus, if only the position of sk in V ( t )is not of ( n + l ) s t order, sk = sk+ql. We see that the set of n t h order (but not ( n + 1)st order) symbols of s is defined by a pattern t, E T ( l , q , j ) . Consequently, s is a Toeplitz word: s = U ( t o , t l , . . . ,t,, . . .). Moreover, since the (k 1)th symbol of each ti is equal t o k p + ci modulo ql for some ci, each pattern ti is uniquely determined by its first symbol (equal t o ci) and by p or, more exactly, by the remainder p' from division p t o qz. That is why we can denote

+

s = V(t1, , .. ,t,, . ..) = V(p';c1,. , . ,c,, . . ,) = V ( p ' ;c ) , where c is the sequence (c1,. . . ,en,. ..), By the definition of s = U(p';c), each word equal t o some its prefix occurs in v so that symbols which had n t h order in V(p'; c) correspond t o n t h order positions in V ( t )for all n . In particular, the position a, = je (i.e., the first position of nth order) in s = V ( p ' ; c ) is occupied by symbol c, and corresponds t o the position number d, = kqZn+l cnZn j e in U ( t ) , where k 2 0. What symbol occurs at the position number a,-l = jin V ( p ' , c ) ? By the definition, it is equal t o c,-1. On the other hand, in V ( t )it corresponds t o the symbol in the position dn-1 = dn -p(an - an-1) = dn -pln-l. Sincep=mqE+p', we haved,-l = ( k l - j m ) q l n + ( j + l c n - j p ' ) l n - l + j ~ , and thus (mod qZ). j lc, - jp' f c,-1

+

+

+

We see that c,-1 is uniquely determined by c, and p': in the example 2 below we denote this fact by c,-1 = a(c,,p'). Hence, since the sequence c is infinite, it is periodic and uniquely determined by c1 andp'. It means that the sequence s = U ( p ' , c ) and, consequently, the factorial language F ( s ) = F ( v ) depend only on p' and c1: s = V ( p ' ,c) = V ( p ' ,c l ) . Thus,

61

where P is the set of factors of finite number of periodic words given by values of p divisible by i. We see that the arithmetical closure of U(t) is the union of a finite number of languages of factors of Toeplitz or periodic words. So, computing the arithmetical complexity is reduced to computing the subword complexity of several Toeplitz words. It follows from results of [4] that the subword complexity of each of them is linear, so the arithmetical complexity of U(t) also grows linearly. D Example 2 Let us find the arithmetical complexity of the paperfolding word U(tpf) defined in Example 1 (note that the pattern tpf is canonical). To do it, we find all the U(p';c\) and use Formula (1). First of all, as proved above, c n _i is uniquely determined by cn and p' and does not depend on n or p: cn-\ = a(cn,p') = j + lcn — jp' (mod ql). Here j = 1 and q = / = 2, so we easily find a(0,1) = a(2,1) = 0 and a(0,3) - a(2,3) = 2. That is why if p' = 1, then the only possible c is ( 0 , . . . , 0 , . . . ) , and £/(p',c) = 17(1,0) = I7(0?2?) = U ( t p f ) . Analogously, if pf = 3, t h e n c = (2,... , 2 , . . . ) , and U(p',c) =U(ipf), where ipf = 2?0?. The even (i.e., divisible by / = 2) values of the difference add the set P of factors of (T, 2 W , and (02) w , so that

FA(U(tpf}) = F(U(tpf))UF(U(tpf)}

U P.

It is not difficult to show (see [2]) that a word of length n > 14 can belong to at most one of these three sets. It is clear also that f u ( t p ! ) ( n ) = /c/(t p/ )(^) — 4n, so for n > 14 we have

/#(*„)(*) = fu(trf)(n) + fu(tpf)(n) + 4 - 8n + 4. The values of f£,t -. (n) for n < 14 can be found by manual comparing the three sets F ( U ( t p f ) ) , F ( U ( t p f ) ) , and P, and are given in the following table:

1 2 3 4 5 6 7 8 9 10 11 12 13 14 n fA(n) 2 4 8 16 24 32 44 52 64 76 86 96 106 116 We have found the arithmetical complexity of the paperfolding word.

5

Concluding remark

Like most new notions, arithmetical complexity poses a series of natural problems. As always, it would be interesting to investigate the case of low

62

complexity and, for instance, t o characterize the set of infinite words whose arithmetical complexity grows linearly. Another direction is examination of known families of infinite words, like DOL words and Toeplitz words from wider classes, Sturmian words etc., and finding their arithmetical complexity.

Acknoledgements We thank professor Masami Ito who made it possible to present this work at the 3rd ICWLC, and the referee for careful reading and checking calculations in Example 2.

References [l] J.-P. Allouche, The number of factors in a paperfolding sequence, Bull. Austral. Math. SOC.46 (1992) 23-32.

[2] J.-P. Allouche, M. Bousquet-Mklou, Canonical positions for the factors in paperfolding sequences, Theoret. Comput. Sci 129 (1994), 263-278. [3] J.-P. Allouche, J. Shallit, The ubiquitous Prouhet-Thue-Morse sequence, in: C. Ding, T. Helleseth, and H. Niederreiter (eds.), Sequences and their Applications, Proc. of SETA’98, DMTCS, Springer (1999), 1-16. [4] J. Cassaigne, J . Karhumaki, Toeplitz words, generalized periodicity and periodically iterated morphisms, European J. of Combinatorics 18 (1997), 497-5 10. [5] A. E. Frid, On uniform DOL words, in: M. Morvan, C. Meinel, and D. Krob (eds.), STACS’98, LNCS 1373, Springer (1998), 544-554.

[6] J . Justin, G. Pirillo, Decimations and Sturmian words, RAIRO Informatique 31 (1997), 271-290.

[7] A. Khintchine, Three Pearls in Number Theory, Graylock Press, New York, 1948. [8] M. Koskas, Complexite‘ de suites de ToepZitz, Disc. Math. 183 (1998), 161-183. [9] B. L. Van der Waerden, Beweis einer Baudet’schen Vermutung, Nieuw. Arch. Wisk. 15 (1927), 212-216.

63

THE EMPEROR’S NEW RECURSIVENESS: THE EPIGRAPH OF THE EXPONENTIAL FUNCTION IN TWO MODELS OF COMPUTABILITY VASCO BRATTKA Theodische Informatik I, FernUniversitat Hagen, D-5808d Hagen, Germany E-mail: vasco.brattkaofernuni-hagen.de In his book “The Emperor’s New Mind” Roger Penrose implicitly defines some criteria which should be met by a reasonable notion of recursiveness for subsets of Euclidean space. We discuss two such notions with regard t o Penrose’s criteria: one originated from computable analysis, and the one introduced by Blum, Shub and Smale.

1

Introduction

In his book “The Emperor’s New Mind” Roger Penrose raises the question whether the famous Mandelbrot set A4 g lR2 can be considered as recursive in some well-defined sense. Throughout his discussion of this problem Penrose uses an intuitive notion of recursiveness and he complains about the lack of a mathematically precise meaning of this notion. On the one hand, he argues that it is insufficient to define recursiveness of a set as decidability with respect to computable points, since in this case even a simple set like the unit ball B := {(s, y) E R2: x 2 + y2 5 1) does not become recursive. Since Penrose is convinced that the unit ball should become recursive we are led to introduce the following criterion.

Penrose’s first criterion. A reasonable notion of recursiveness for subsets of Euclidean space should make the closed unit ball recursive.

Figure 1. The closed unit ball B

64

On the other hand, Penrose argues that certain other ways to define recursiveness are also inappropriate, especially, because they do not handle the border of the sets under consideration in the right way. This aspect is important since the complexity of sets is often inherent in their border, as in case of Mandelbrot’s set. For instance, a definition of recursiveness as decidability with respect to rational or algebraic numbers is insufficient, since in this case sets like the closed epigraph of the exponential function E := {(x,y) E R2 : y 2 e Z } would not be handled appropriately. The border of this set does not contain any algebraic point besides ( 0 , l ) and thus the border is irrelevant to a decision procedure which is restricted to algebraic points. Of course, Penrose is convinced that a set, easily structured like the closed epigraph of the exponential function, should be recursive. This motivates the second criterion.

Penrose’s second criterion. A reasonable notion of recursiveness for subsets of Euclidean space should make the closed epigraph of the exponential function recursive.

...

L

.r Figure 2. The closed epigraph E of the exponential function

Apparently, there are several similar conditions and Penrose’s criteria are by no means sufficient conditions for a reasonable notion of recursiveness. They are just necessary conditions; a notion of recursiveness which does not meet Penrose’s criteria would be highly suspicious since it could be doubted whether it reflects algorithmic complexity in the right way. Since Penrose did not present any notion which fulfills all his requirements, it seems as if there exists no suitable notion of recursiveness. The aim of this paper is to compare two existing notions of recursiveness for subsets of Euclidean space and to find out which comes closest to Penrose’s requirements. The first notion is based on computable analysis and has been developed and investigated by several authors. The basic idea of recursive analysis is to call a function f : R n -+ R computable, if there exists a Turing machine which transforms Cauchy sequences of rationals, rapidly converging to an input 2, into Cauchy sequences of rationals, rapidly converging to the

65

output f(z).Moreover, a set A 2 R” is called recursive, if its distance funct i o n d A : R n +. R is computable.a Here d denotes the Euclidean metric. This notion of recursiveness straightforwardly generalizes the notion of recursiveness from classical Computability theory (see Odifreddi 3 ) : if we endow the natural numbers N with the discrete metric, then the distance function of a subset A N is equal to its characteristic function and computability of the characteristic function is equivalent to recursiveness of the set A. In Euclidean space the distance function is a LLcontinuous substitute” for the characteristic function. Although recursiveness of subsets of Euclidean space in this sense does not correspond to the intuition of “decidability”, it is a formal generalization of the classical notion of recursiveness. Especially, a subset A N considered as a subset of R is recursive, if and only if it is classically decidable. Finally, it is easy to prove that this notion of recursiveness meets Penrose’s criteria. As a second notion of recursiveness we will investigate the notion which has been developed by Blum, Shub and Smale.4i5In their theory a function f : R ” +. Iw is computable (we will call it algebraically computable for the following), if there exists a real random access machine which computes f . Such a machine uses real number registers, arbitrary constants, arithmetic operations, comparisons and equality tests. Moreover, a set A C R”is called recursive by Blum, Shub and Smale (we will call it algebraically recursive for the following), if its characteristic function is algebraically computable. If we restrict the class of constants appropriately (for instance to rational numbers), then a set A C N considered as a subset of R is algebraically recursive, if and only if it is classically decidable. In this sense the notion of algebraic recursiveness is a second generalization of the classical notion of recursiveness. Obviously, the unit ball is algebraically recursive and hence Penrose’s first criterion is met. Blum and Smale have proved that Mandelbrot’s set is not algebraically recursive and hence it seems as if they have given an answer to Penrose’s original question. But with a similar technique we will prove that the closed epigraph of the exponential function is not algebraically recursive and hence it is highly questionable whether Blum and Sinale’s answer to Penrose’s question is significant. If even a simple set like the epigraph of the exponential function is not algebraically recursive, we can conclude that algebraic non-recursiveness obviously does not reflect the intrinsic algorithmic complexity of a set.

aThe idea of using distance functions to characterize “located” sets has first been used in constructive analysis, see Bishop and Bridges.2

66

2

Recursive and Recursively Enumerable Sets

In this section we give the precise definitions of several classes of recursively enumerable and recursive sets and we will give a short survey on some elementary properties. Let d : R" x R" .+ R be the Euclidean metric of R", defined by d ( x , y) := dET'l Ixi - yi12 for all x , y E R". By B ( x , r ) := {y E R" : d ( x , y ) < T - }we denote the open balls and by B ( X , T:= ) {y E R" : d(x,y) 5 r } the closed balls with respect to d. For each set A 5 IR" we denote by d A : R" -+ R the distance function of A , defined by d A ( x ) := infaEAd ( x , u). Let a : N + R" be some standard enumeration with range(0) = Q", defined for instance by c x ( ( i ~ , Icl), j ~ , ..., (i,,j,, k,)) := ..., -1. Here, (.) : N2 -+ N denotes Cantor's Pairing Function, defined by ( i , j ) := + j ) ( i + j + 1)+ j , which can inductively be extended to a function (.) : N" + N. All these pairing functions are bijective and computable, as well as their inverses. We assume that the reader is familiar with the definition of computable real functions (see, for instance, Weihrauch17Pour-El and Richards,s KO 9 ) . We briefly recall the ideas: a function f :C IR" -, Iw is called computable, if there exists a Turing machine which transforms each Cauchy sequence (qi)iEn of rational numbers qi E Q" (encoded with respect to a ) , which rapidly converges to some x E dom(f) into a Cauchy sequence ( r i ) i Eof~ rational numbers ri E Q , which rapidly converges to f(x). Here, rapid convergence means d ( q i , q k ) 5 2-k for all i > Ic (and correspondingly for (ri)iEn). Of course, a Turing machine which transforms an infinite sequence into an infinite sequence has to compute infinitely long, but in the long run the correct output sequence has to be produced. It is reasonable to assume one-way output tapes for such machines since otherwise the output after some finite time would be useless (because it could be replaced later). Functions, such as exp, sin, cos, In and max are examples of computable functions. One of the basic observations of computable analysis is that computable functions are continuous. This is because approximations of the output are computed from approximations of the input and therefore each a p proximation of the output has to depend on some approximation of the input. Computable functions of type f : N R" can be defined similarly and are called computable sequences. Now we are prepared to define the notion of recursively enumerable and recursive subsets in the sense of computable analysis (see Brattka and Weihrauch lo for a survey). These notions are explicitly defined for open or closed sets, respectively.

(w,

i(i

67

Definition 2.1 (Recursively enumerable open and closed sets) 1. An open subset A IR" is called recursively enumerable, (r.e. for short), if there is a computable function f : N 4 N2 such that A = U ( i , j ) E r a n g e ( f ) B(a(i),2 - 9

2. A closed subset A En is called recursively enumerable, (r.e. for short), if A = 0 or there is a computable sequence f : N -+ IR" such that range(f) is dense in A. 3. An open (closed) set is called co-recursively enumerable (cer.e. for short), if its complement AC is r.e. 4. An open (closed) set is called recursive, if it is r.e. and cer.e. Recursively enumerable open sets have first been introduced and investigated by Lac0mbe.l' Equivalent definitions to the given ones have been investigated by several authors (see Weihrauch and Kreitz,l27l3KO et a1.,1419715 Ge and Nerode,16 Zhou,17 Zhong,l8 Brattka 19). The following characterization gives an impression of the stability of the definition of r.e. sets. For completeness we also mention the characterizations via semi-computable d i s tance functions. These notions are not used any further in this paper and the interested reader is refered to Brattka and Weihrauch lo for the definitions and proofs.

Lemma 2.2 (Characterization of r.e. closed sets) Let A

Rn be a

closed set. T h e n the following equivalences hold:

1. % %

2.

A i s recursively enumerable { (2, j ) E N2 : A n B ( a ( i )2, - j ) # 8) is recursively enumerable d A : R n IR i s upper semi-computable, -+

A i s co-recursively enumerable

* {(i, j ) E N2 : A n B(a(i),2 - j ) = 0 ) i s recursively enumerable % d A : R n --+

%

3.

A = f-l{O}

i s lower semi-computable f o r some computable f u n c t i o n f : R"

A i s recursive

-+

R,

d~ : Rn -+ R i s computable.

Using these characterizations and the fact that the exponential function is a computable function one can easily show that the notion of recursiveness of computable analysis fulfills Penrose's criteria.

68

Proposition 2.3 (Recursive sets and Penrose’s criteria)

1. T h e closed unit ball B := ((2,y) E R 2 : x2 + y2 5 1) i s a recursive set. 2. T h e closed epzgruph E := { (x,y) E R 2 : y 2 ex} is u recursive set. Proof.

d

m

1. We obtain d ~ ( ~ , =y max(0, ) - 1) for the distance function dg : R 2 --+ R . Thus, d~ is computable and B is recursive.

2. There exists a computable function f : N R 2 such that range(f) = {(z, e” y) E R 2 : z, y E Q, y 2 0 } , since the exponential function is computable. Since range(f) is dense in E it follows that E is an r.e. closed set. The function g : R 2 -+ R with g ( z , y) := max(0, e” - y} is computable and E = g-l{O}. Thus, E is a ccer.e. closed set. Altogether, 0 E is a recursive closed set.

+

+ .

More generally, the proof of 2. shows that the closed epigraph epi(f) = { (2,y) E R 2 : y 2 f(x)} of a computable function f : R +. R is a recursive set.b It is worth noticing that the notion of computability and the notion of recursiveness of computable analysis fit together very well: a continuous function f : Iw +. R is computable, if and only if its graph is recursive a s a closed subset of R 2 (see Weihrauch 20). 3

Algebraic Recursiveness

In this section we want to prove that the notion of algebraic recursiveness does not meet Penrose’s second criterion. We start with the definition of algebraically r.e. sets as halting sets of real random access machines, as they have been used by Blum, Shub and Smale.425These real random access machines use real number registers, arbitrary constants, arithmetic operations, comparisons and equality tests. We assume that the reader is familiar with the precise definitions. F’rom the point of view of computable analysis especially the comparisons and equality tests are problematic. From the point of view of classical Computability theory also the constants are suspicious since one can code an arbitrary function f : N +. N in such a constant.

bFora general discussion of computability properties of the epigraph, see Zheng et. a1.20,21

69

Definition 3.1 (Algebraically r.e. sets) Let A

R".

1. A is called algebraically r.e., if A is the halting set of some real random access machine. 2. A is called algebraically recursive, if A as well as its complement A" is algebraically r.e. If A is the halting set of a real random access machine which does only use rational constants, then we will say that A is algebraically r.e. with rational constants. Obviously, the unit ball B := ((5, y ) E R2 : x2 + y2 5 l} is an algebraically recursive set, even with rational constants. We just have to compute z2 y 2 and test z2 y2 5 1.

+

+

Proposition 3.2 T h e closed unit ball B := {(z, y ) E R 2 : x2 algebraically recursive with rational constants.

+ y2 5 1) i s

One can easily prove that the open epigraph of the exponential function > e 2 ) is an r.e. open set and hence it is also algebraically r.e. (as any other r.e. open set). On the other hand, we will show,that the closed epigraph E of the exponential function is not algebraically recursive. Indeed, we will prove that it is not even algebraically r.e. The proof uses some standard techniques of Blum, Shub and Smale's theory, especially their Path Decomposition Theorem, which states that each algebraically r.e. set is a countable (disjoint) union of basic semi-algebraic sets (see Blum et al.5). We recall some basic definitions and facts from real algebraic geometry (which can be found in Bochnak et a1.22 and Marker et al.23). The class of semi-algebraic subsets of R" is the smallest class of subsets of R" which contains all sets {x E R" : p ( z ) > 0 ) with real polynomials p : R" -+ R,and which is closed under finite intersection, finite union and complement. Each semi-algebraic set can be written a s finite union of basic semi-algebraic sets, which have the form {z E RIB": Pl(Z)= 0,.'.,p z ( z ) = 0,q1(x) > 0, ' " 1 qJz) > O } , where p l , . . . , p i , 41,..., qj : R" + R are real polynomials. A (partial) function f :C R" + R is called semi-algebraic, if its graph graph(f) := ((2, y ) E EXnf1 : f(z)= y} is a semi-algebraic set. Using the normal form given above, it is easy to show that each semi-algebraic function is algebraic, i.e. there exists some real polynomial p : Rn+l + R , p # 0 such that p ( z , f(z))= 0 for all z E dom(f). By the Theorem of Tarski-Seidenberg semi-algebraic sets are closed under projection and one can conclude that the interior A", the closure 71 and {(z,y) E R 2 : y

70

hence the border aA = 71\ A" of a semi-algebraic set A is semi-algebraic too (additionally, one uses the fact that the Euclidean metric is a semi-algebraic function). Correspondingly, one can see that the lower border A1 := {(x,y) E R 2 : (z, y) E A and (Vz E R ) ( ( x 2) , E A +z 2 y ) } is semi-algebraic if A C R 2 is. By f l u we will denote the restriction of a function f with dom(f 1 ~ = ) dom(f) n U . Now we are prepared to prove the following result.

Proposition 3.3 T h e closed epigraph E := { ( s l y ) E R 2 : y 2 e 2 } of the exponentaal function i s not algebraically T. e. Proof. Let E := {(z,y) E R 2 : y 2 f ( x ) } be the closed epigraph of the exponential function f : R -+ R . Let us assume that E is algebraically r.e. Then, by the Path Decomposition Theorem, E is a countable union of semi-algebraic sets Ai C R2, i.e. E = U Z o A ~ Since . the closure of a semi-algebraic set is semi-algebraic too, we can m u m e w.l.0.g. that all sets Ai are closed. Especially, we obtain aE = U&(dE n Ai)and since the border dE is a complete subspace of R 2 it follows by Baire's Category Theorem that there is some i E N and a non-empty open set U C R 2 such that 8 # dE n U 5 Ai. Since 8E = graph(f) and f is continuous, there are some non-empty open intervals I , J C R such that f(1)C_ J and V := I x J C U . Hence graph( f 11) = d E n V = A! n V is semi-algebraic, since A! and V are semi-algebraic. But using the Identity Theorem for real-analytic functions and the power series expansion of the exponential function, one can prove 0 that f l is ~ not algebraic. Contradiction! This proposition proves that algebraic recursiveness does not meet Penrose's second criterion. w e will call a function f : R + everywhere transcendental, if f Iu is not algebraic for each non-empty open set U R.The proof that the exponential function is everywhere transcendental can be found in basic texts on analysis (see, for instance, Erwe 24). Besides the fact that the exponential function is everywhere transcendental and continuous, we have not used any specific properties of the exponential function in the previous proof. By symmetry we obtain the following general result.

Theorem 3.4 Iff : R -+R i s a n everywhere transcendental and continuous function, t h e n neither the closed epigraph, nor the closed hypograph, nor the graph o f f i s a l g e b r a i d l y r.e. It is worth noticing that the notions of algebraic recursiveness and algebraic computability do not fit together in the same sense as the notions of recursiveness and computability of computable analysis. The square root

71

:c

function f R --t R , 2 c-) fi is an example of a function which is not algebraically computable but whose gra.pli is algebraically recursive. Hence, the algebraic non-recursiveness of the graph of the exponential fuiiction cannot simply be deduced from the fact that the exponential function is not algebraically computable.‘

4

Conclusion

We have seen that the notion of algebraic recursiveness does not meet Penrose’s criteria, while the notion of recursiveness from computable analysis does. The latter notion describes recursiveness in terms of computability of the distance function d A of a set A. In view of the fact that equality on the real numbers is undecidable, recursiveness in this sense is the best what one could expect. Recursiveness implies “decidability up to the equality test on the real numbers”: if we only could decide whether d ~ ( z=) 0, then we could decide whether 2 E A or not. An essential question remains open. We do not know whether the Mandelbrot set is a recursive closed set or not. It is easy to see that it is a c+r.e. closed set but it is still a challenging open question to find out whether it is also an r.e. closed set or not!

Acknowledgements The main result of this paper, Proposition 3.3, has been motivated by an inspiring discussion with Peter Hertliiig in Dagstuhl 1997. This work has been supported by DFG Grant BR 1807/4-1.

References 1. R. Penrose, The Emperor’s New Mind. Concerning Computers, Minds and The Laws of Physics (Oxford University Press, New York, 1989). 2. E. Bishop and D.S. Bridges, Constructive Analysis (Springer, Berlin, 1985). 3. P. Odifreddi, Classical Recursion Theory (North-Holland, Amsterdam, 1989). 4. L. Blum, M. Shub, and S. Smale, On a theory of computation and complexity

over the real numbers: NP-completeness, recursive functions and universal machines, Bul. Amer. Math. SOC.21:l (1989) 1-46. 5. L. Blum, F. Cucker, M. Shub, and S. Smale, Complexity and Real Computation (Springer, New York, 1998). “Over algebraically closed fields a function is algebraically computable, if and only if its graph is algebraically recursive, see Ceola and L e c ~ m t e . ’ ~

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6. L. Blum and S. Smale, The Godel incompleteness theorem and decidability over a ring, in M.W. Hirsch et al. (eds.), From Topology to Computation: Proceedings of the Smalefest (Springer, New York, 1993) 321-339. 7. K. Weihrauch, Computable Analysis (Springer, Berlin, 2000). 8. M.B. Pour-El and J.I. Richards, Computability in Analysis and Physics (Springer, Berlin, 1989). 9. K.-I KO, Complexity Theory of Real Functions (Birkhauser, Boston, 1991). 10. V. Brattka and K . Weihrauch, Computability on subsets of Euclidean space I: Closed and compact subsets. Theoret. Comput. Sci. 219 (1999) 65-93. 11. D. Lacombe, Les ensembles rkcursivement ouverts ou fermks, et leurs applications a 1'Analyse rbcursive, C.R. Acad. Sc. Paris 246 (1958) 28-31. 12. K. Weihrauch and C. Kreitz, Representations of the real numbers and of the open subsets of the set of real numbers, Ann. Pure Appl. Logic 35 (1987) 247260. 13. C. Kreitz and K. Weihrauch, Compactness in constructive analysis revisited, Ann. Pure Appl. Logic 36 (1987) 29-38. 14. K.-I KO and H. Friedman, Computational complexity of real functions, Theoret. Comput. Sci. 20 (1982) 323-352. 15. A. Chou and K . 4 KO, Computational complexity of two-dimensional regions, SIAM J. Comput 24 (1995) 923-947. 16. X. Ge and A. Nerode, On extreme points of convex compact Turing located sets, in A. Nerode and Y. V. Matiyasevich (eds.), Logical Foundations of Computer Science, vol. 813 of LNCS (Springer, Berlin, 1994) 114-128. 17. Q. Zhou, Computable real-valued functions on recursive open and closed subsets of Euclidean space, Math. Logic Quart. 42 (1996) 379-409. 18. N. Zhong, Recursively enumerable subsets of R' in two computing models: Blum-Shub-Smale machine and Turing machine, Theoret. Comput. Sci. 197 (1998) 79-94. 19. V. Brattka, Computable invariance, Theoret. Comput. Sci. 210 (1999) 3-20. 20. K. Weihrauch and X. Zheng, Computability on continuous, lower semicontinuous and upper semi-continuous real functions, Theoret. Comput. Sci. 234 (2000) 109-133. 21. X. Zheng, V. Brattka, and K. Weihrauch, Approaches to effective semicontinuity of real functions, Math. Logic Quart. 45:4 (1999) 481-496. 22. J. Bochnak, M. Coste, and M.-F. Roy, Ge'ome'trie alge'brique re'elle (Springer, Berlin, 1987). 23. D. Marker, M. Messmer, and A. Pillay, Model Theory of Fields (Springer, Berlin, 1996). 24. F. Erwe, Differential- und Integralrechnung (Bibliographisches Institut, Mannheim, 1973). 25. C. Ceola and P.B.A. Lecomte, Computability of a map and decidability of its graph in the model of Blum, Shub and Smale, Theoret. Comput. Sci. 194 (1998) 219-223,

73

ITERATIVE ARRAYS WITH LIMITED NONDETERMINISTIC COMMUNICATION CELL T. BUCHHOLZ, A. KLEIN AND M.KUTRIB Institute of Informatics, University of Giessen Arndtstr. 2, 0-35392 Giessen, Germany E-mail: [email protected] An iterative array is a line of interconnected interacting finite automata. One distinguished automaton, the communication cell, is connected to the outside world and fetches the input serially symbol by symbol. Sometimes in the literature this model is referred to as cellular automaton with sequential input mode. We are investigating iterative arrays with a nondeterministic communication cell. All the other cells are deterministic ones. The number of nondeterministic state transitions is regarded as a limited resource which depends on the length of the input. It is shown that the limit can be reduced by a constant factor without affecting the language accepting capabilities, but for sublogarithmic limits there exists an infinite hierarchy of properly included real-time language families. Finally we prove several closure properties of these families.

1

Introduction

Devices of interconnected parallel acting automata have extensively been investigated from a language theoretic point of view. The specification of such a system includes the type and specification of the single automata, the interconnection scheme (which sometimes implies a dimension to the system), a local and/or global transition function and the input and output modes. One-dimensional devices with nearest neighbor connections whose cells are deterministic finite automata are commonly called iterative arrays (IA) if the input mode is sequential to a distinguished communication cell. Especially for practical reasons and for the design of systolic algorithms a sequential input mode is more natural than the parallel input mode of so-called cellular automata. Various other types of acceptors have been investigated under this aspect (e.g. the iterative tree acceptors in [8]). In connection with formal language recognition IAs have been introduced in [7] where it was shown that the language families accepted by real-time IAs form a Boolean algebra not closed under concatenation and reversal. Moreover, there exists a context-free language that cannot be accepted by any &dimensional IA in real-time. On the other hand, in [6] it is shown that for every context-free grammar a 2-dimensional linear-time IA parser exists. In [lo] a real-time acceptor for prime numbers has been constructed. Pattern

74

manipulation is the main aspect in [I]. A characterization of various types of IAs by restricted Turing machines and several results, especially speed-up theorems, are given in [13,14,15]. Various generalizations of IAs have been considered. In [20] IAs are studied in which all the finite automata are additionally connected to the communication cell. Several more results concerning formal languages can be found e.g. in [21,22,23]. In some cases fully nondeterministic arrays have been studied, but up to now it is not known how the amount of nondeterminism influences the capabilities of the model. In terms of Turing machines bounded nondeterminism has been introduced in [ll]. Further results concerning cellular automata, Turing machines, pushdown automata and finite automata can be found e.g. in [3,5,16,17,18,19]. Here we introduce IAs with limited nondeterminism. We restrict the ability t o perform nondeterministic transformations to the communication cell, all the other automata are deterministic ones. Moreover, we limit the number of allowed nondeterministic transitions dependent on the length of the input. The paper is organized as follows. In section 2 we define the basic notions and the model in question. Section 3 is devoted to the possibility t o reduce the number of nondeterministic transitions by a constant factor. In section 4 by varying the amount of allowed nondeterminism we prove an infinite hierarchy of properly included language families. Due t o the results in section 3 we need sublogarithmic limits for the number of nondeterministic transitions in order t o obtain the hierarchy. Finally, in section 5 several closure properties of the real-time acceptors with such limits are shown.

2

Model and Notions

We denote the rational numbers by Q,the integers by 7, the positive integers { 1 , 2 , . . .} by N, the set N U (0) by No and the powerset of a set S by 2s. The empty word is denoted by E and the reversal of a word w by w R . We use C for inclusions and C if the inclusion is strict. For a function f we denote its i-fold composition by f [ i ] , i E N, and define the set of mappings that grow strictly less than f by o(f) = { g : NO-+ N limn+, = 0). The set R ( f )

I

#

> 0). The identity is defined according t o {g : No -+ N 1 liminf,,, function n F-+ n is denoted by id. An iterative array with nondeterministic communication cell is an infinite linear array of finite automata, sometimes called cells, where each of them is

75

connected t o its both nearest neighbors (one to the left and one t o the right). For convenience we identify the cells by integers. Initially they are in the socalled quiescent state. The input is supplied sequentially to the distinguished communication cell at the origin. For this reason we have two local transition functions. The state transition of all cells but the communication cell depends on the current state of the cell itself and the current states of its both neighbors. The state transition of the communication cell additionally depends on the current input symbol (or if the whole input has been consumed on a special end-of-input symbol). The finite automata work synchronously at discrete time steps. More formally:

Definition 1 An iterative array with nondeterministic communication cell (G-IA) is a system ( S ,S , S n d , SO, #, A , F ) , where 1. S is the finite, nonempty set of states, 2. A is the finite, nonempty set of input symbols, 3. F S is the set of accepting states, 4 . so E S is the quiescent state, 5. # !$ A is the end-of-input symbol, 6. S : S3 Si is the deterministic local transition function for non-communication cells satisfying &(so,S O ,S O ) = SO, 7. 6,d : S3 x ( A U {#}) + 2’ is the nondeterministic local transition function for the communication cell satisfying VSI, s2, s3 E S, a E A U { #} : b n d ~ ls,2 , s 3 , a> # 0. Let M be a G-IA (G for guessing). A configuration of M at some time t 2 0 is a description of its global state which is actually a pair (wt, ct), where wt E A* is the remaining input sequence and ct : Z 3 S is a mapping that maps the single cells to their current states. During its course of computation a G-IA steps nondeterministically through a sequence of configurations. The configuration (WO, CO) at time 0 is defined by the input word 200 and the mapping co(i) = SO, i € Z,while subsequent configurations are chosen according t o the global transition function And: Let (wt ,ct), t 2 0, be a configuration then the possible successor configurations (wt+l, ct+l) are as follows:

where i E Z \ {0}, and a = it, wt+l = E if wt = E , and a = a l , wt+l = a2 . . . a, if wt = a1 . . . a,. Thus, the global transition function And is induced by 6

76

and

6nd.

The i-fold composition of

And is defined as follows:

If the state set is a Cartesian product of some smaller sets S = Sl x . . . x S k , we will use the notion register for the single parts of a state. The concatenation of one of the registers of all cells respectively forms a track. A G-IA is deterministic if 6 n d ( S 1 , ~ 2 , s g , u )is a singleton for all states S ~ , S Z , S QE S and all input symbols u E A U {#}. Deterministic iterative arrays are denoted by IA.

Definition 2 Let M = ( S ,6, bnd, S O , #,A, F) be a G-IA. 1. A word w E A* is accepted by M if there exists a time step i E N such that ci(0) E F for some (wi,ci) E n i i ( ( w , c 0 ) ) . 2. L ( M ) = { w E A* I w is accepted by M } is the language accepted by

M.

+

3. Let t : NO + N, t(n) 2 n 1, be a mapping and iw be the minimal time step a t which M accepts a w E L ( M ) in some computation. If all w E L ( M ) are accepted within i, 5 t(lw1) time steps, then L is said to be of time complexity t. The family of all languages which can be accepted by a G-IA with time complexity t is denoted by -Yt(G-IA). In the sequel we will use a corresponding notion for other types of acceptors. If t equals the function n 1 acceptance is said t o be in real-time and we write -YTt(G-IA). The linear-time languages -Ylt(G-IA) are defined according t o -Yit(G-IA) = UkEQ,k,lLk.n(G-IA). There is a natural way t o restrict the nondeterminism of the arrays. One can limit the number of allowed nondeterministic state transitions of the communication cell. For this reason a deterministic local transformation 6d : S3 x ( AU {#}) -+ S for the communication cell is provided and the global transformation induced by 6 and bd is denoted by Ad. Let g : No + NObe a mapping that gives the number of allowed nondeterministic transitions dependent on the length of the input. The resulting system ( S ,6, dnd, 6 d , S O , #, A,F ) is a gG-IA (g guess IA) if starting with the initial configuration (w0,co) the possible configurations at some time i are given by the global transformations

+

77

as follows: ifi=O

U

A$-g(lwl)l ((w', c'))

otherwise

( W ' , C ' ) ~ A ~ ~((w0,co)) ''"')~

Observe that in this definition the nondeterministic transitions have to be applied before the deterministic ones. This is not a serious restriction since nondeterministic transitions for later time steps can be guessed and stored in advance (cf. second part of the proof of Theorem 3). Up to now we have g not required t o be effective. Of course, for almost all applications we will have to do so but some of our general results can be developed without such requirement.

3

Guess Reduction

This section is devoted to the reduction of the number of nondeterministic transformations. In the sequel we will make extensively use of the ability of IAs t o simulate a pushdown storage [8,2] or a queue [4] on some track in real-time. The communication cell contains the symbol at the top of the stack or the queue. The left-to-right inclusion in the following theorem is not immediate since there might be computation paths of the kgG-IA that cannot appear for the gG-IA. Therefore, the kgG-IA must be able t o verify whether or not its communication cell has performed g(n) nondeterministic transitions.

Theorem 3 Let g : NO -+ No be a mapping and k E N be a constant. If t : No -+ N, t(n)2 n + 1, is a mapping such that t(n)2 k .g(n) for almost all

n E N then 2t(gG-IA) = 2 t ( l C . gG-IA) Proof. The crucial point in proving the inclusion 2t(gG-IA) C Zt(kgG-IA) is that a kgG-IA M' which is designated to simulate a given gG-IA M with the same time complexity must not simulate too many nondeterministic transitions of M . Therefore, the communication cell of M' is equipped with a pushdown storage. During its nondeterministic transitions M ' either can simulate a nondeterministic step of M whereby k - 1 specific symbols are pushed or can simulate a deterministic step of M whereby one symbol is popped.

78

Once M' decided t o simulate a deterministic transition it has to do so for its remaining nondeterministic steps, whereby again one symbol is popped respectively. In order t o accept the input M' has to pop the last symbol from the stack exactly a t time step k . g(n) which is its last nondeterministic one. Let m be the number of time steps at which symbols are pushed. Then we have m . ( k - 1) = k . g(n) - m m = g(n). To see the other inclusion D49,(kgG-IA) _49,(gG-IA)we use again a pushdown storage. The communication cell of a gG-IA M' simulating a kgG-IA M without any loss of time pushes k - 1 nondeterministically determined functions d : S3 x ( AU {#}) --t S satisfying d(s1,s2] s3, a) E bnd(Sl1s 2 , s3, a) (here bnd denotes the nondeterministic transition function for the communication cell of M ) during each of its nondeterministic transitions. Additionally] it simulates a nondeterministic transition of M . During the first deterministic transitions such a function is popped and applied to the states of the communication cell and its neighbors and the current input symbol which yields the next state of the communication cell. Hence a nondeterministic transition in M is simulated deterministically. Altogether M ' performs g(n) + (k- 1).g(n) = k.g(n) nondeterministic transitions and accepts exactly 0 the same language as M . A constant number of nondeterministic transitions does not increase the power of IAs. The principle of the proof is t o simulate all finitely many choices on different tracks.

Theorem 4 Let t : constant then

0.10

+ N,

t(n) 2 n + 1 be a mapping. If k E N is a

9t(kG-IA) zz L&(IA) The next corollary extends the previous results.

Corollary 5 Let g : NO + NO be a mapping and q E Q, 0 < q rational number such that g(n) = Lqn] for almost all n 6 N, then

5 1, be a

2Tt(gG-IA)= TTt(idG-1A) 4

Nondeterministic Hierarchy

Definition 6 Let L g A* be a language over an alphabet A and 1 E NObe a constant. 1. Two words w and w' are 1-equivalent with respect to L if wwl E L

u w'wl E L for all w1 E A'

79

2. N ( n ,1, L ) denotes the number of 1-equivalence classes of words of length n with respect to L (i.e. Iww1I = n).

+

Lemma 7 Let g : No -+ No, g(n) 5 n 1, be a mapping. If L E L%(gG-IA) then there exist constants p , q E N such that

Proof. Let M = ( S ,6,dnd,6 d , S O , #, A,F ) be a real-time gG-IA which accepts L. We define 4 = max { Ibnd(S1,s2, s 3 , .)I Is1,S 2 , 5 3 E

s A a E A}

In order t o determine an upper bound to the number of 1-equivalence classes we consider the possible configurations of M after reading all but I input symbols. The remaining computation depends on the last 1 input symbols and the states of the cells -1 - 1,.. . , 0, . . . , I 1. For the 21 3 states there are 1S12'+3 different possibilities. Let p = ISI5 then due t o lS12'+3 = IS121.1S13 = (IS12)'.1S/3 5 (lS12)'.(IS13)' = (IS12.1S13)'5 p' we haveat most p' different possibilities for at most qg(n)different computation paths. Since the number of equivalence classes is not affected by the last 1 input symbols

+

*dn)

in total there are at most ( p ' )

= p'.qg(")classes.

+

0

The following result does not follow for structural reasons since there might be accepting computation paths of the fG-IA that cannot appear for the gG-IA. Therefore, the fG-IA must be able to verify whether or not its communication cell has performed g(n) nondeterministic transitions.

Theorem 8 Let f : NO + NO,f(n) 5 ,: and g : NO -+ NO,g(n) 5 f(n), be two increasing mappings such that V m, n, E N : f(m) = f ( n )==+ g(m) = g(n). If L, = ( ~ g ( ~ ) b f ( ~ ) - gI n ( ~ E) N} belongs to the family &(IA) then 2 T t

(SG-IA)

C -%t (fG-IA)

Proof. Let M be a real-time gG-IA that accepts the language L. A real-time f G-IA M' which simulates M works as follows. Since f 2 g M' can guess the time step g(n) and therefore simulate M directly. Additionally, M' has to verify that its guess was correct. Otherwise the computation must not be accepting. It is known that deterministic linear-time IAs can be sped-up to (2 . i d ) time [14]. Thus, L, belongs to 22id(IA). Now M' simulates such an acceptor M" on an additional track. During the first g(n) time steps M' simulates M" under the assumption that M" fetches input symbols a. From the guessed

80

time step g ( n ) up t o the last nondeterministic step f(n) M ' simulates M" under the assumption that M" fetches input symbols b, respectively, and during the last n - f ( n )time steps M' simulates M" without input. altogether M' simulates at least 2 . i d Due to the condition f ( n ) 5 time steps of M " . If M ' guessed g ( n ) correctly it simulates M" for the input ~ g ( ~ ) b f ( ~ ) - and, g ( ~ )hence, an accepting computation. On the other hand, if M' simulates an accepting computation then it guessed a time step t such that the input ~ ~ b f ( " belongs )-~ to L,. It follows t E { g ( m ) I f ( m ) = f ( n ) } and due to the assumption V m, n, E N : f ( m )= f ( n )==+ g ( m ) = g ( n ) it holds t = g ( n ) . Therefore, M' can verify whether its guess was correct and, thus, accepts L in real-time. The following situation may clarify the necessity of the condition V m, n, E N : f ( m ) = f ( n ) ==+ g(m) = g(n). Let m < n and f ( m ) = f ( n ) and g(m) < g ( n ) . Since c ~ g ( ~ ) b f ( " ) - g ( ~belongs ) t o L, the word ~ g ( ~ ) b f ( ~ ) - g ( ~ ) does. Consequently, for an input of length m the word c ~ g ( ~ ) b f ( ~ ) - would g ( ~ ) lead t o an accepting computation but since g(m) < g ( n ) the time step g might be guessed wrong. Now we are going to extend the previous result to a hierarchy of properly included language families. Theorem 9 Let f : No -+ No and g : NOt NObe two mappings which meet the conditions of Theorem 8. If additionally f E o(1og) and g E o(f) then

-Z-t(gG-IA)c -%t(fG-IA) Proof. We define a mapping h : NO t N by h(n) = 2 f ( n ) . h is increasing since f is. Moreover, since f E o(1og) for all k E Q, k 2 0, it .holds 2f(n) limn-,m = limn+m -= 0 and therefore h E o ( n k ) . Especially for Ic = $ it follows that themappingm(n) = max{n' E NOI ( h ( n ) + l ) . ( n ' + l5 ) n } is unbounded, and for large n we obtain m(n) > h(n). Now we define a language L that belongs to T,t(fG-IA) but does not belong to z,t(gG-IA).

L= { $ T ~ ~ $ ~ 2 $ . . . $ ~ j ~ y ~ ) 3 n E D J : j = E {h O( ,nl )} rAn ~( ni ) , l ~ i ~ j , A T = ~7, - ( h ( n ) 1 ) .(m(n)+ 1) A 3 1 5 i' 5 j : W ~ = I yR}

+

It follows that L is not empty (cf. Example 10). Assume now L E zTt(gG-IA). Then by Lemma 7 there exist constants p , q E N such that ~(n,m(n +)1,L) 5 p(rn(n)+l).q."'"'. Since g E o ( f ) for all IC E 9, IC 2 0, it k.g n 2E.d") = lim n+m = limn+m 2 f o = 0. Thus, we obtain holds limn+m

#

81

2"g E o ( 2 f ) = o(h). Therefore, for large n the number of equivalence classes is bounded as follows: N ( n ,m(n)+ 1,L ) 5 p ( m ( n ) + l ) . P ' " ' < -p 2 4 n - 210g ( p ).2. m ( n ).2'"9(4).9(" ) -

) 4 " )

Let k be log(q), then 21°g(q)'g(") = 2"g(") E o ( h ) . Now we can find a constant no such that for all n 2 no: 2 . log(p) . 2"g(") < ah(n). It follows

< 2m(n),h(n).$

210g(p).2.m(n).2'0g(P)'9(")

On the other hand, let for all n E N and for every subset U = ( w l , . . . ,w ~ ( of (0, l } m ( na) word u be defined according to u = $'w~$. . . $ w ~ ( ~where )$ T = n - ( h ( n ) 1) . (m(n) 1). Then for all y E (0, l } m ( n ) :

+

+

yE

u

-

uyRe E L

(2h:G))

Since there exist at least 2m(n) different words wi there are different subsets U . For every pair U , V of subsets one can find a wi belonging t o U \ V or t o V \ U . It follows UW?$ E L w VW?$ fj L and, hence,

): ;(

N ( n ,m(n)+ 1,L ) 2

=

2 4 7 4 . ( 2 4 7 4 - 1) . . . . . ( 2 4 7 4 - h(n)+ 1) h(n)!

( 2 4 4 - qn))h(") h(n)W From m(n) > h(n) for large n it follows 2m(n) - h(n) > - 2m(n).5. Thus

> - 2m(n).h(").; From the contradiction we obtain L fj -Y't(gG-IA). It remains to show L E -Y.t(fG-IA). An fG-IA M which accepts L has t o check whether j = h ( n ) ,whether all the wi are of the same length, whether T < h(n) (from which now follows that lwil = m ( n ) ) and , whether there exists an i' such that wit = yR. Accordingly M performs four tasks in parallel.

82

For the first task M simulates a stack and pushes a symbol 1 at every nondeterministic transformation. After the last nondeterministic transformation the pushed string is handled as a counter which is decremented every time step a new wi appears in the input. The decrementation starts for w2. The number of wis is accepted if the counter is 0 after reading the input because is the binary number 2 f ( n )- 1 = h ( n )- 1. For the second task M uses two more stacks. The subword w1 is pushed onto one of them. When M fetches w2 it pushes w2 to the second stack and pops w1 from the first stack whereby their lengths are compared symbol by symbol. This task is repeated up to wj. The third task uses another stack on which the first T symbols $ of the input are pushed. Subsequently for each subword ‘uli one of them is popped. The last task is to find an i’ such that W ~ = I yR. Here the nondeterminism is used. During the first f(n)nondeterministic steps a binary string is guessed bit by bit and pushed onto a stack. From time f ( n ) on it is handled as a counter which is decremented for every subword wi. If it is 0 the next word is pushed onto another stack. It will be popped and compared symbol by symbol when the word y appears in the input. Thus, the i’ is guessed during the nondeterministic transformations. 0

At first glance the witness L for the proper inclusion seems to be rather complicated. But here is a natural example for a hierarchy: Example 10 Let i > 1 be a constant and f(n) = log[il(n) and g(n) = l ~ g [ ~ + l I ( nThen ). by Theorem 9 we have Trt(gG-IA) c 9?t(fG-IA). Since 9Lt(IA) is identical to the linear-time cellular automata languages [22] and {anb2n-n I n E N} is acceptable by such devices { a g ( n ) b f ( n ) - g ( n ) I n E N} E Tlt(IA) holds. Moreover, from g E log(f) follows V m , n E N : f ( m ) = f ( n ) ==+ g(m) = g(n). Thus, the conditions of Theorem 8 are met. Trivially, g is of order o(f), E.g. for i = 2 we obtain m(4) = 0, m(8) = 1, m(l6) = 2, E L. m ( 3 2 ) = 4, and $01$11$10$00~11~ 5

Closure Properties

Besides that closure properties are interesting of their own they are a powerful tool for relating families of languages. Our first results in this sections deal with Boolean operations. Lemma 11 Let g : No -+ No and t : No -+ N, t(n) 2 n + 1, be two mappings. Then the family 9t(gG-IA) is closed under union and intersection and trivially contains 9t(IA).

83

Proof. Using the same two channel technique of [9] and [22] the assertion can easily be seen. Each cell consists of two registers in which acceptors for both languages are simulated in parallel. Now we turn to more language specific closure properties. For some functions g the families 5Yrt(gG-IA) are closed under concatenation and for some others are not. At first we consider the closure under marked concatenation. Lemma 12 Let g : N o + No be an increasing mapping such that the language { a g ( m ) b m - g ( m ) I m E N} belongs to 5YTi(IA). Then the family -YTt(gG-IA) is closed under marked concatenation. Proof. Let L1 resp. L2 be formal languages over the alphabets A1 resp. A2 which are acceptable in real-time by the gG-IAs MI resp. M z . Let L denote the marked concatenation of L1 and Lz: i.e.,

L = { W I C W Z I w1 E L1 and

202

E Lz}

where c 6 A1 U A2 is a marking symbol. A gG-IA M that accepts L in real-time works as follows. ATcA; is a regular language and, therefore, belongs trivially to 5YTt(gG-IA). Since 2Zrt(gG-IA) is closed under intersection (cf. Lemma 11) it is sufficient to consider inputs of the form ATcA; only. Let w = wlcw2 with w1 E A;, w2 E A;, and n1 = 1w11, n2 = 12021. Now the idea is as follows: On input w the array M simulates the behavior of MI (on input wl)until reading the marking symbol c and subsequently the behavior of Ma (on input 2 0 2 ) . M accepts w iff both simulations are accepting. The simulation of M1 can be performed directly since g is monotonically increasing and therefore g(n) 2 g ( n 1 ) . But the time step g ( n l ) has to be guessed and verified. In order t o perform this task an acceptor for the language L' = { a g ( m ) b m - g ( m ) I m E N} is simulated on an additional track in parallel. Thereby an input symbol a is assumed for each nondeterministic step (up to the guessed time g(n1)) and an input symbol b for each deterministic step (up t o the end of simulation at time n l ) . So the number 2 resp. y of simulated nondeterministic resp. deterministic transitions corresponds to a word azbY belonging to L' iff there exists an m E N such that 2 = g(m) and y = m - g(m). Thus, iff n1 = z y = g(m) m - g(m)= m. The simulation of M Zis performed similarly. However, a problem would arise with the nondeterministic transitions if g(n) < n1 + 1 g(nz). Therefore, during its nondeterministic transitions M uses a queue into which it

+

+

+

84

pipes nondeterministically chosen local transition functions corresponing to a possible nondeterministic transition of Ma (cf. the proof of Theorem 3). During the simulation of the nondeterministic transitions of M 2 these functions are successively extracted from the queue and applied to the communication cell. 0 The assertions of the lemma can essentially be weakened. Let h be a homomorphism such that h(z) = a for z # b and h(b) = b. Then instead of requiring L = {a9(")bm-g(") I m E N} to be acceptable in real-time by some iterative array it is sufficient to require that some language L' with h(L') = L belongs to T T t (IA). By $4' we denote the set of functions g : N + No, g(n) 5 n, such that there exists a language L' E Trt(IA) whose image under h is {ag(m)bm-g(m)I m E N}. So in fact any family TTt(gG-IA) where g E $4' is closed under marked concatenation. The usage of a marking symbol can be omitted if the limiting function g allows a gG-IA to determine a possible concatenation point by its own (for instance nondeterministically by using a b-ary counter). Hence, we obtain the following corollary.

Corollary 13 Let g : No + NObe a mapping with g E R(1og). If.&(gG-IA) is closed under marked concatenation then it is closed under concatenation. On the other hand, there exist functions g for which gG-IA is not closed under concatenation. The proof follows essentially an idea presented in [7] to show that the family A$t(IA) is not closed under concatenation. Theorem 14 Let g : NO + No, g E o(loglog), be a mapping. -Y,.t(gG-IA) is not closed under concatenation.

Then

Proof. Let A be the alphabet consisting of the four symbols 0 , 1,a, and b. Further let L1 = A* and denote by L2 the language of palindromes over A, i.e. the set of all words w over A which are identical to their reversals wR.As it has been shown in [7] L1 as well as L2 are belonging to TTt(IA) and thus to TTt(gG-IA). Consider now the concatenation L = L1 L2 and assume contrarily that L belongs to A!Tt(gG-IA), too. Then let W , = {Owl I w E {a,b},} for n E OJ and define for each subset U = {wl, . . . wk} of W , the word u as

u={

E thk

if U = @ otherwise

85

where the

u1,.

. . , U k are recursively defined by uo = E ,

ui+l

15 i = wi+lwRwi, R

5 m - 1.

One easily sees that lul = n(2k - 1) and that for all w E W, it holds w E U iff u w E L. Therefore (choosing k = n ) there are especially at least different n-equivalence classes with respect to L in the set of words of length n2n over A. Hence using the assumption on g we can work out a contradiction to Lemma 7 for a sufficiently large n. So L is not acceptable in real-time by a gG-IA, i.e. -YTt(gG-IA)is not closed under concatenation. 0

(z)

Note that one can additionally show that for g E o(log1og) the corresponding family Z T t (gG-IA) is not closed under marked iteration although it might be closed under marked concatenation. Theorem 15 Let g : No -+ NO,g E o(log), be a mapping. Then the family ZTt(gG-IA)is not closed under reversal. Proof. Consider the language L consisting of all marked concatenations of binary sequences of equal length where the first sequence occurs at least twice, i.e

L = {W1$. . . w k $ I k 2 2 A 3 m E N : ‘Wi E ( 0 , ,}I A325j5k:wl

15 i

5 k,

=wj}.

We are going to show that L belongs to -YTt(IA) 2 -YTt(gG-IA), but LR 6 -YTt(gG-IA). An iterative array M that accepts L in real-time works as follows. The communication cell is equipped with a queue through which symbols can be piped in first-in-first-out manner. At the beginning of the computation M stores its input symbols to the queue until the first symbol $ appears. Afterwards at every time step one symbol is extracted from the queue and compared to the current input symbol. At the same time step it is stored in the queue again. Thus, the symbols of w1 circulate through the queue and w1 is compared with all the wi, 2 5 i 5 k , serially. It remains to show that LR does not belong to 2’Tt(gG-IA). Let us assume that LR is acceptable by some gG-IA in real-time. Let us consider the equivalence classes N ( ( m 1)2, ( m l ) ,L R ) . For every pair of different subsets { X I , . . . ,x,} and {yl, . . . ,ym} of the set (0, I}, there are words $21. . . $x, and $ y l . . . $ym which belong to different such (rn 1)-equivalence classes. W.1.o.g. let 21 $! { y l , . . . ,ym}. Then $21 ... $ x m $ x l belongs to LR

+

+

+

86

(2)

+

whereas $y1. . . $y,$z1 does not. Hence, there are at least such (m 1)equivalence classes. Since f E o(1og) we obtain a contradiction t o Lemma 7 0 for a. sufficiently large m which concludes the proof. The last two results deal with the closure under homomorphisms. Theorem 16 Let g : NO+ NObe a mapping. If 9',t(gG-IA) C 2ZTt(idG-IA) then 9',t(gG-IA) is closed under &-free homomorphism iff z',t(gG-IA) = .YTt (idG-IA) .

Proof. One can show that the family 2',t(idG-IA) coincides with the closure of 9',t(IA) under &-freehomomorphisms and forms an AFL which is closed under intersection and reversal. Consequently 2',t (idG-IA) is closed under &-freehomomorphisms, too, implying the closure of 9 T t(gG-IA) under &-free homomorphism if 2+ (gG-IA) = 2',t(idG-IA) holds. On the other hand, since 9',t(IA) C .Y',t(gG-IA) it follows that the closure of &(IA) under €-free homomorphisms (which is dP,t(idG-IA)) is contained in the closure of 9',t(gG-IA). If the latter family is T',t(gG-IA) itself then it follows 2,.t(idG-IA) C 2',L(gG-IA) C 9',t(idG-IA), i.e T',t(gG-IA) = 9',t(idG-IA) 0 Corollary 17 Let g : No + No be a mapping. If 9',t(gG-IA) c ZTt(idG-1A) then 9',t(gG-IA) is not closed under &-freehomomorphism, homomorphism and &-freesubstitution and substitution. By Theorem 9 such functions exist. References

1. Beyer, W. T . Recognition of topological invariants by iterative arrays. Technical Report TR-66, MIT, Cambridge, Proj. MAC, 1969. 2. Buchholz, Th. and Kutrib, M. Some relations between massively parallel arrays. Parallel Comput. 23 (1997), 1643-1662. 3. Buchholz, Th., Klein, A., and Kutrib, M. One guess one-way cellular arrays. In: Proc. Int. Sym. on Mathematical Foundations of Computer Science (MFCS). LNCS 1450, Springer, 1998, 807-815. 4. Buchholz, Th., Klein, A., and Kutrib, M. Iterative arrays with limited nondeterministic communication cell. IFIG Research Report 9901, Institute of Informatics, University of Giessen, Giessen, 1999. 5. Buss, J. and Goldsmith, J. Nondeterminism within P . SIAM J. Comput. 22 (1993), 560-572.

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6. Chang, J. H., Ibarra, 0. H., and Palis, M. A. Parallel parsing on a oneway array of finite-state machines. IEEE Trans. Comput. C-36 (1987), 64-75. 7. Cole, S. N. Real-time computation b y n-dimensional iterative arrays of finite-state machines. IEEE Trans. Comput. C-18 (1969), 349-365. 8. Culik 11, K. and Yu, S. Iterative tree automata. Theoret. Comput. Sci. 32 (1984), 227-247. 9. Dyer, C. R. One-way bounded cellular automata. Inform. Control 44 (1980), 261-281. 10. Fischer, P. C. Generation of primes b y a one-dimensional real-time iterative array. J . Assoc. Comput. Mach. 12 (1965), 388-394. 11. Fischer, P. C. and Kintala, C. M. R. Real-time computations with restricted nondeterminism. Math. Systems Theory 12 (1979), 219-231. 12. Hromkovic, J. et al. Measures of nondeterminism in finite automata. In: Proc. Int. Conf. on Automata, Languages, and Programming (ICALP). LNCS 1853, Springer, 2000, 199-210. 13. Ibarra, 0. H. and Jiang, T. On one-way cellular arrays. SIAM J. Comput. 16 (1987), 1135-1154. 14. Ibarra, 0. H. and Palis, M. A. Some results concerning linear iterative (systolic) arrays. J. Parallel and Distributed Comput. 2 (1985), 182-218. 15. Ibarra, 0. H. and Palis, M. A. Two-dimensional iterative arrays: Characterizations and applications. Theoret. Comput. Sci. 57 (1988), 47-86. 16. Kintala, C. M. and Fischer, P. C. Refining nondeterminism in relativized complexity classes. SIAM J. Comput. 13 (1984), 329-337. 17. Kintala, C. M. and Wotschke, D. Amounts of nondeterminism in finite automata. Acta Inf. 13 (1980), 199-204. 18. Salomaa, K. and Yu, S. Limited nondeterminism for pushdown automata. Bulletin of the EATCS 50 (1993), 186-193. 19. Salomaa, K. and Yu, S. Measures of nondeterminism for pushdown automata. J. Comput. System Sci. 49 (1994), 362-374. 20. Seiferas, J . I. Iterative arrays with direct central control. Acta Inf. 8 (1977), 177-192. 21. Seiferas, J. I. Linear-time computation b y nondeterministic multidimensional iterative arrays. SIAM J . Comput. 6 (1977), 487-504. 22. Smith 111, A. R. Real-time language recognition b y one-dimensional cellular automata. J. Comput. System Sci. 6 (1972), 233-253. 23. Terrier, V. On real time one-way cellular array. Theoret. Comput. Sci. 141 (1995), 331-335.

88

R-TRIVIAL LANGUAGES OF WORDS ON COUNTABLE ORDINALS OLIVIER CARTON Imtitut Gaspard Monge Universite' d e Marile-la- Valle'e, F-77454 Marne-la- Valle'e C e d e x 2, Prance, Email: 01 i v i e r . C a r t o n h n i v - m l v . f r , Url: h t t p :/ / w w w - i gm .un i v-m 1v .f r / - c a r t on/ Following the recently proved variet,y theorem for transfinite words we give, in this paper, three instances of correspondence between varieties of finit,e WI -semigroups and varieties of wl-languages. We first characterize the class of languages which are recognized by automata in which overlapping limit transit,ions end in t,he same state. I t turns out. that. the corresponding variety of w~-semigroupsis defined by an equation which has a topological interpretation in the case of infinite words. It characterizes languages of infinite words in the class A2 = l I z n C z of t,he Bore1 hierarchy. This result is used t,o prove that an wl-latiguage is recognized by an extensive automaton if and only if its syntacric wl-semigroup is R-uivial and satisfies the Az-equation. This result extends Eilenberg's result, concerning Rtrivial semigroups and extensive automata. We finally characterize wl-languages recognized by extensive automata whose limit transitions are trivial.

1

Introduction

Finite semigroups are the algebraic counterpart of automata. The first deep result using semigroup recognition is due to Schtitzenberger 14. He proved that the syntactic semigroup of a recognizable language L is finite and aperiodic (i.e. group-free) if and only if L is star-free, i.e., it belongs to the s~nallestclass of languages containing the letters and closed under product a i d finite boolean operations. The idea of using algebraic properties of syntactic seinigroups to classify recognizable languages was developed by Eilenberg *, wlro showed that there exists a one-to-one correspondence between varieties of finite semigroups (class of semigroups closed under taking sub-semigroups, quotients and finite direct products) and certain classes of languages, the varieties of Iaiiguages. This theorem is known as the variety theorem. Since that time the tlieory of varieties of recognizable languages has been widely developed (see and '). For instance, it has been shown by Eilenberg that a language is recognized by an extensive automaton if and only if its syntactic semigroup is R-trivial (see Chap. 10 in *). Furthermore, such languages can he described by very special rational expressions. Automata on infinite words were introduced by Biiclii 6. A few years later, Buchi extended this notion to ordinals '. The challeirge was tlieii to

89

extend the algebraic approach to infinite words in a first step a i d to orcliiials in a second step. For infinite words, there is now a rather satisfying theory The couiiterpart of culminating in the works of Wilke 1 5 , Perrin and Pin this theory for ordinal words was a bit slower to develop. Wojciechowski l 6 defined rational expressions and proved that they are equivalent to automata. The algebraic theory was first settled for ordinals less than w" and later extended t o countable ordinals in 5 . The key algebraic notioii is tliat of an wl-semigroup which extends the notion of an w-semigroup introduced in 15,'. Roughly speaking, an wl-semigroup is a structure in which the product of any sequence of a countable number of elements is possible. The variety theorem is also extended to words on countable ordinals in 5 . In this paper, we give three instances of correspondence betweeii varieties of finite wl-semigroups and varieties of wl-languages. We first, cliaixterize the class of languages which are recognized by automata in which overlappilig limit transitions end in the same state. It turns out that tlie corresponding variety of wl-semigroups is defined by an equation which has a topological interpretation in the case of infinite words. This equation characterizes languages of infinite words in the class A2 = rIgnC.2 of the Bore1 hierarcliy 15, We use this result to characterize wl-languages recognized by extensive automata. An wl-language is indeed recognized by an extensive automaton if and only if its syntactic wl-semigroup is R-trivial and satisfies the A.L-equtitioit. This result extends Eilenberg's result concerning R-trivial semigroups and extensive automata. We finally characterize wl-languages recogiiized by extensive automata whose limit transitions are trivial. The paper is organized as follows. Basic definitions of words, autoinata, rational expressions and wl-semigroups are recalled iii Sectioii 2. The tliree instances of correspondence are given in Section 3. ' 9 " .

2

Notation and Basic Definitions

This section is devoted to basic notation and definitions on ordinals, words, rational expressions, automata and wl-semigroups. 2.1

Ordinals

We refer the reader to l 3 for a complete introduction to the theory of orcliiials. An ordinal is a class for isomorphism of well-founded linear orderings. 11-1 this paper, ordinals are usually denoted by lower Greek letters like u, 8, y. An ordinal a: is said to be a successor if a = p+ 1 for some ordinal 8. An ordinal is either 0, a successor ordinal or a limit ordinal. As usual, we identify the

90

linear order on ordinals with the membership. A n ordiiial CY is ttieii identified with the set of ordinals srrialler than a. In this paper, we maiiily use ordiiials to index sequences. Let n be an ordinal. A sequence r of lengtll u (or ail a-sequence) of elements from a set E is a function which maps any ordinal y smaller that a to an element of E. A sequence r is usually denoted by x = ( z ~ ) ~ In < ~this . paper, we only use countable ordinals, except for w1 which denotes the first uncountable ordinal.

2.2

Words

Let A be a finite set called the alphabet whose elements are called k t t e r s . For an ordinal a , an a-sequence of letters is also called a word of length n or an a - w o r d over A . The sequence of length 0 which has no element is called t,he e m p t y word and it is denoted by A. The length of a word r is denoted by 111 '. For an ordinal a , Aa denotes the set of all words of' length a. The set of' all words of countable length over A is denoted by Ah. A subset of Ah is called a language or an w1 -language. Let ( z ~ ) - ,
nyCcr

2.3

R a t i o n a l expressions

Rational expression for transfinite words have been defined by Wojciechowski 16. They extend the usual rational expressions for finite words with two other iterations. Rational expressions for transfinite words are well-formed expressions build from the constants ( U ) ~ ~which A denote the letters of the alphabet and the constant X which denotes the empty word. The operators used in the expressions are the binary operators + and . which respectively denote the union and the concatenation and the unary operators *, w aiid b which respectively denotes the finite, the omega and the ordiiial iterations. The omega and ordinal iterations are defined as follows. =

I

( ~ 0 x 1. ~ . .zi

xy I a

E X > and ~b = {

< w1 aiid x?

E X>

7< a

As usual, the braces and the dot are usually omitted in rational expressioiis. Thus, the expression ( { u } { u } . { b } O ) b is written ( u ubw)b.

+

+

91

2.4

Automata

Buchi automata on transfinite words are a generalization of usual (Kleene) automata on finite words, with a second transition function for limit ordiiials. States reached at limit points depend only on states reached Lefore.

Definition 1 An automaton A is a 5-tu,ple (&, .4, E , I ,F ) Uhe7.e (2 is the finite set of states, A the finite alphabet, E (0x A x Q)u ( P ( Q )x 0)the set of transitions, I Q the set of initial states and b7 the set Of firlal states.

c

c

c

Transitions are either of the form ( q , u,q') or of the form ( P ,q ) where P is a subset of Q. A transition of the former case is called a su.cce.s.sor trunsition and it is denoted by p % q. A transition of the latter case is called a limit transition and it is denoted by P -+q . We now explain how these automata are used to define languages. Before describing a path in an automaton, we define the cofinal set of a sequence at some limit point.

Definition 2 Let c be ct-sequence of states and let ,8 5 Q be (I Limit orrlznal. W e denote by cof,j(c), the set of states q such that for (my S < 0 , tlrew exists 6 < y < p such, that cr = q . This definition allows us t o define the notion of a path in an automaton.

Definition 3 Let A = ( Q ,A , E , I , F ) be an automaton. A path c lebeled b y a word x = (uy)7
2. for any p < a, qp

n!3 -+

qp+1 is

a szuccessor transition of A;

3. for any limit ordinal p, cofp(c)

+ q/j Zs a limit

trwrisitiorl of A.

The word z = ( C L ~ ) ~is
92

Figure 1. An automaton recognizing X = (aA')h(uA*

+ A)

Example 4 Consider the automaton pictured in Figure 1. The successor transitions are pictured like a labeled graph. This autoinaton recognizes the wl-language of words having an a at the first position and at all limit positions. This wl-language is denoted by the rational expression ( u A ~ ) ~ ( u-t AA). * The wl-language recognized by the autoinaton of Figure 1 can be denoted by a rational expression. It turns out that automata and rational expressions have been proved to be equivalent 16. Any wl-language recognized by an automaton can be denoted by a rational expression and conversely, ally w1language denoted by a rational expression can be recognized by an automaton. Such an wl-language is said to be rational. An automaton is deterministic if and only if it satisfies the following three conditions. It has only one initial state i. For any state p E Q aiid for any letter u E A there is at most one state q such that p 3 q is a successor transition of A. For any subset P C Q there is at most one state y such that P + q is a limit transition of A. The last two conditions implies that for any state q, any word labels at most one path starting at y. The autoiriat,on of Figure 1 is deterministic. 2.5

w1 -semigroups

The notion of wl-semigroups have been introduced in '. They are a generalizaRoughly speaking, an w1-semigroup tion of w-semigroups introduced in is a set S equipped with a product which maps any sequence of countable length over S to an element of S. This product satisfies a generalizatioii of the associativity called w1 -associativity. l5l9.

Definition 5 An wl-semigroup is a set S equipped with a function cp : Sb -+ S , called an w1 -product, which satisfies the following properties I . For any element s E S , p(s) = s.

93

An wl-semigroup is not an algebra in the usual sense since the product does not have a finite arity. Even if the wl-semigroup is finite, the description of the product is not finite since the product of any sequence of countable length must be given. However, we will see later that the product of a finite wl-semigroup can be described in a finite way. U p to this question of iiifiiiite arity, the notion of wl-semigroups fits into the general framework of universal algebra and the following notions are self understanding: morphism of w1semigroups, quotient of w1 -semigroups, sub-wl -semigroup, congruence of w1semigroups. An wl-language X is recognized by a morpliism $ tioin ,4Qto S if and only if there is a subset P of S such that .Y = ( P ) . It was proved in ti that an wl-language is rational if and only if it is recogiiizecl by a fiiiite wl-semigroup. The wl-product ‘p of an wl-semigroup, naturally induces an inner product defined by z . y = ‘p(zy) and a function z c) z7 defined by xT = ‘ p ( x ” ) . The wl-associativity satisfied by ‘p implies that the inner product is associative and that the function r satisfies

s ( t s ) r = (st)‘

and

( s ” ) ~= sr for any n

> 0.

A function satisfying tliese two identities is said to be coniputitrle with the semigroup structure of S given by the inner product. Wlieii tlre w1semigroup S is finite, the inner product and the functioii r I-+ T‘ coiripletely describe the wl-product ‘p. Therefore, a finite wl-semigroup can be viewed as a finite semigroup equipped with an additional compatible functioii 7. In the sequel, we identify a sequence over an wl-semigroup a i d the product of its elements. As in semigroup theory, the product of a sequence of elements is denoted by mere concatenation aiid the applicatioii of cp is omitted. Following this convention, the application of r is then denoted by w since xr = ‘p(zc“). From now on, we will not distinguish between a finite wl-semigroup and a finite semigroup with an additional compatible function w. We point out that both notions do not coincide anymore when the w1 -semigroups considered are not finite. In a finite wl-semigroup, there is an integer K such tliat sn is aii idempotent for any element s of S. This integer is generally denoted by w in the literature but w already denotes the first limit ordinal in our context.

94

3

Instances of correspondences

The variety theorem for finite words is due to Eilenherg *. Ile showed that there exists a one-to-one correspondence between varieties of finite seinigroups (classes of semigroups closed under taking sub-semigroups, quotients and finite direct products) and certain classes of languages called the varieties of languages. A classical example is the theorem of Schiitzenberger l4 which states that a set of finite words is star-free if and only if its syntactic semigroup is aperiodic. The variety theorem was extended to words on countable ordinals in 5 . The result of Schiitzenberger was first extended to ordinal words of length less than ww in and then to all words of courit,i\hle leiigtll i n '. In the following two subsections, we give three instalices of correspol~dences between varieties of fiitite wl-semigroups and varieties of W I -1alrg11ages. The first example is a characterization of w1 -languages recogiiized by automata in which overlapping limit transitions end iii the sainr state. This class of wl-languages is characterized by an equation which has already been used in the case of languages of infinite words. This equation will be needed in the second example which is a characterization of w1 -languages recognized by extensive automata. The third example is a refinement of the second one. It is a characterization of wl-languages recognized by extensive autoiriata in which limit transitions have a very special form.

3.1 A, -languages Our first example characterizes wl-languages recognized by autmlata in which overlapping limit transitions end in the same state. An wl-language can be recognized by such an automaton iff its syntactic semigroup satisfies an equation which, in the framework of infinite words, characterizes languages in the class A, of the Bore1 hierarchy (see Thm 6.2 in 15). This justifies the terminology. Theorem 6 For a n w1 -language X , the following conditions are equivalent.

1. X is recognized by a deterministic and complete automaton such th,at i f P + q and Pi -+ qi are two limit transitions with P n P i # 0 , t h e n q = 4'.

2. T h e syntactic w1 -serniyro,up of X .satisfies the ecputdori ( ~ g ~ ) ~ = y'(xuT)". The automaton pictured in Figure 1 fulfills the property of t,lie theorein. Note that the determinism and the completeness of the autoinatoil are really necessary. In Thm 6.2 of 15, the A2-languages are characterized by the equation

95 (z"y")"x" = (x"yx)"yw. The following Iernina sliows that, tliis equation is equivalent to the equation given in the previous tlieorein. Jt also provides a useful property of wl-semigroups satisfying this equation.

Lemma 7 For any finite wl-semigroup S , the follo*wing propositions are equivalent. 1. S satisfies the equ.ation (xy")"y" = (xy")"

I

2. S satisfies the equation (x"y")"z" = ( ~ " y " ) ~ v " . 3. A n y elements s, s',x,x' E S sucl~,that s R s', s z = s (md 5'3.' = s' sutisfy sxw

= s'xI"

The third statement of the leirirria means that if the w1 -semigroup S satisfies the equation (zyr)"yW = (zy")", the value of sx" for s and 3: in S sucli that sx = s only depends on the R-class of s and not on a:. Proof We first prove that (1) and (2) are equivalent. We aim to show that (~"y")"x" = (x"y")"y'. Denote by e aiid f the two idempotents z" and y". Since ( e f ) " e = ( e f e ) " arid ee" = e", one has in the first hand that (ef)"ew = (efe)",". We are in position to apply equation (xy")"yw = (xy")" with e f in the role of z and e in tJie role of y". This yields (ef)"e" = ( e f e ) " . Observing that ( e f e ) w = e ( j e ) d = ( e l ) " , one has the equality (ef)"e" = (ef)"'. Applying again (qj")"y" = (zy")" with e in the role of x and f in the role of y", one has i n the seconcl hand that (ef)"f" = ( e f ) " . Combining the two equalities, one gets tlie cquality (z"yK)"xw= (xxyx)"yw as required. We aim to show that (a:y")"y" = (xy")". Denote by e a i d f the two idempotents (xy")" and y". One has ef = e and therefore (ef)" = e. Applying equation (xry")"xw = (xryn)*yw with e in the role of a:" and f in tlie role y", one gets eew = e f" which is (xyn)"yW = (zy")". We now prove that (1) and (3) are equivalent. We suppose that s R s', sx = s and s'z' = s' and we aim to show that szw= s'z'". Since s R s', there are two elements y a,nd y' sucli that, sy = s' and s'y' = s. Define the two idempotents e and e' by e = (@"y'a:")" and e' = (y'z"yx'")". Applying equation (zy")"yw = (ry")"with yx'"y' in the role of x and x" in the role y", one obtains ex" = e". Symmetrically. one also has e'x'" = e'". By definition of e and e', one has se = s, s'e' = s'. This yields sz" = sew and s ' d " = s'e'". The equality sew = s'e'" follows finally from the definition of e and e'. We aim to show that (x:y")"yW = ( ~ 9 " ) " . Let f be tlie ideiripotent y K . Then, one has (xf)"f = (zf)"and (xf)"(xf)" = ( a : f ) " . We are ill position

96

t o apply (3) with f in the role of 2: and (zf)"in the role of s, s' aiicl yields (xf)"f" = (xf)"(xf)" = (zf)"as required. 0

3.'.

This

We now come to the proof of the previous theorem. From any complete deterministic automaton with no transitions P -+ q and P' + (I'sucli that P n P' # 0 and q # q', we define an wl-semigroup satisfying the equation (zy")"y" = (xy")" and recognizing the same wl-language. Corivcrsely, frorn any wl-semigroup satisfying the equation (zy")"y'" = (zg")", we construct an automaton with no transitions P 3 q and P' -+ q' such that, P n P' # 0 and q # 9'. Proof Let A be a deterministic and complete ailtoinaton. For m y state (I and any word x, there is a unique path q 3 q' labeled by x. Tliis defines a right action of Ah on Q defined by q . x = 4'. This right action defines a semigroup S and a morphism 7r of semigroup frorn Ah into S. If for any limit transitions P + p and P' -+ p' of A such that P n P' # 0 one has p = p', this semigroup S can naturally be endowed with a structure of w1 -semigroup. The compatible operation w is defined as follows. Let q be a state aiid Y be an element of S . Define the sequence (qn)nlo of states by q, = q . s'l. Let P be the set of states which appear infinitely often in the sequence ( q n ) r r l O , that is P = { q I Vrz 2 0 3k 2 12 q = q a } . Define q . sd = p where P' -+ p is a limit transition of A such that P C P'. Note that the stsate p does not depend on the choice of the transition P' + p . Indeed, if P' -+ 7) and P" -+ p' are two limit transitions such that P c P' and P c P", then P' f~ P" # 0 and p = p'. It is pure routine to check that (sn)" = s'" and ~ ( t s= ) ~( ~ tand ) ~that the wl-semigroup satisfies the equation (zyK)"yw = (zy")". The morphism K from Ah to S is then a morphism of wl-seinigroup. By construction, it recognizes the set X. Conversely, let $ : Ab -+ S be a morphism into a finite wl-semigroup satisfying the equation (xy")"~'" = (zyn)'". We construct a cleteririiiiistic aiid complete automaton whose state set is S' , that is S with a neutral eleirierit 1. The initial state is neutral element 1. The successor transitions are the transitions of the form s -% s $ ( ~ ) for s E S1 and u E A. The limit trailsitions are the transitions of the form P + s where P is a subset ( ~ 1 , .. . , s , ~ }such that sz R s j for any 1 5 i , j 5 a. By the third statement of the previous lemma, there is a unique element r in S such that for any xz satisfying s z x , = s,,one has sixy = r . We define then the limit transition P -+ r . By constructioil, for any word z, there is a path 1 3 s where s = $(r) and the automaton recognizes the set X . 0

97

3.2 R-trivial languages An extensive automaton is an automaton in which any cyclilig pat11 is trivial. If there is a paths from p to q and from q to p in such a11 ; \ L I ~ O I I ~ A ~ O Itlien I, the states p and q are equal, These automata capture the iiotioii o f I J ~ O C W Swliicli cannot return to a previously visited state. The progress of sucli a process is always increasing. I t has been shown by Eilenberg that a language of fiiiite words is recognized by an extensive automaton if and only if its syntactic semigroup is R-trivial (see Chap. 10 in *). We extend here this result. We first define extensive automata for ordinal words and characterize the wl-languages recognized by these automata. We also characterize the w1 -languages recognized by a subclass of these automata. We first extend the notion of extensive automata (see p. 93 in lo).

Definition 8 An automaton A = ( Q ,A , E , I , F ) is suid to be exteiisive if its set of states is equipped with a totul ordering 5 su.ch that: I . for any successor transition p

2. for any limit transition P

-+ q

3 q, it holds p 5 q, and any p E P , it h,olds p 5 q .

I t follows directly from the definition that any path in an extensive i\Utomaton is an increasing sequence of states. Therefore any cycling path inust be trivial. Conversely, if any cycling path is trivial, the accessibility relation defines a partial ordering on the states. This partial ordering ca.n he embedded into a total ordering which obviously meets the requirernents of the previous definition. Since any cycling path is trivial in an extensive automaton, the cofinal set of a path at some Iii-Iiit point is reduced to a single state. I’lierefore. we can always assume that all the liinit transitions are of the form P + (1 where P = { p } is a singleton. If the automaton is deterministic and extensive, two transitions P -+ y and PI -+ q‘ such that P n PI # 0 satisfy q = q’. Since the autoiIiaton is extensive, i t can be assumed that P = { p } and P’ = { p ’ } . If P n P’ # 0 , one has p = p’ and P = P‘. If furthermore the automaton is deterministic, one has then q = q’. This shows that any wl-language recognized by a deterministic extensive automaton is a 4 2 wl-language. In 8, it is proved that a. language of finite words is recognized by an extensive automaton if and only if its syntactic semigroup is R-trivial. Is is also proved that such a language can be denoted by a rational expression of a very special form. More precisely, it is shown that it is ecplill t,o a fillitre

98

union of products B,*uoB;al ., .B;-Iu,-l B; where Bk C A aid u p 4 BA.(see Prop 3.2 and Thm 3.3 in lo). The following theorem extends thcse results to words on countable ordinals Theorem 9 Let dent.

x be an w1-languuye.

The fo16owiny propositions are equz'ti-

1. X is recognized b y a deternvinistic eztewiue uutoniaton.

2. X is recognized by an R-triuial w1 -semigroGp whach sotasfies the equation (zy")Ty" = (2yT)".

3. X is equal t o a finite union of products L1 . . . L,, wh,ere euch LA.f o r Ic < '12 is either B;uk with B k g A a n d uk $ BA. or BZ with B k C A or witlit B~ A and ak $ B~

and L , is either B;, BZ or B! with B,,

A.

We have alrexty mentioned that all limit traiisitioiis of an extensive automaton are of the form { p } + q. If a pa.tlr uses a transition { p } + q with p # q , there is a gap in the path. The path reinains in the state p and then jumps t o the state q at some limit point. The following theorem chara.cterizes wl-languages recognized by extensive automata in which transitioiis of the form { p } -+ q with p # q are not allowed. All limit trarisitions are of the form { p } -+ p. This theorem is therefore a refinement of the previous oiie. Theorem 10 Let X be w1-lunyuaye. The follo'winy lent.

ure ec~uiva-

prO~JOSitl~ns

1. X is recognized by u deterministic extensive automuton d l of ruhictr limit transitions ure of the forrri { p } + p . 2. X is recognized by a n w1 -sernigroeup wh.lch satisfies the equation xb'= 9 .

3.

x

is equal to a finite union ofproducts B $ ~ ~ B $. x. .,~ f - ~ a , , -B:~ ~ iu/i,ere BI, C A and a k $ Bk.

Note first that an wl-semigroup satisfying the equation 2:" = zT is R-trivial and satisfies the Az-equation ( ~ y ~ ) ~=y (zf)". '" Indeed, one has ( ~ : y ) ~=x z(yz)T = z(y2)" = (xu)" = (1:yy. The statements of the previous two theorems are very close. We will prove them simultaneously. The proof is split in severa.1 lemmas. This first, lenllrlii

99

shows that a R-trivial seinigroup can tie naturally eiirlowed with a structure of Rirthermore, this w1 -semigroup satisfies the required equatioii. Note that in general there is 110 caiioiiical way tcJ elidow a fiiiite st.rriigroup with a structure of wl-sernigroup. I t caii even be sliown that soirie scinigroups like the finite groups caiinot be eiidowed with a structure of wl-seiriigroup. w1 -semigroup.

Lemma 11 Let S be a R-trivial semigroup and let SO a fixed element of S . T h e function s c+ s"sg endows S with, u structure of w1 -.semi.qroup sutisfying the equation (se)*ew = ( s e ) W .Furthermore, if SO is a n e u t r d element, then S satisfies the equation sw = sn. Proof Suppose that the semigroup S is R-trivial. It satisfies tlie equation (zy)"z = (zg)". One 1ia.s ( S ' ~ ) * S = ~ sxso and s(ts)*so = (stjk.sso = (st)"sO and the function s c+ s"s0 is compatible. One verifies that (se)"ed = (se)*ensO = (se)"so = ( ~ e ) 'The ~ . last statement is obvious. 0 If 5 is an ordering on a set &, the set S of all increasiiig fuiictioiis froin G.1 t o Q endowed with the coiripositioii as product is a R-trivial semigroup. Therefore, the previous leinina can be applied to any subseinigroup of tlie semigroup of all increasing functions. I t is not required in Theorems 9 and 10 that the autoinaton is complete. It is just supposed that it is deterininistic arid extensive. Without loss of generality, it can be always assumed that the automaton is complete as stated in the following lemma. Lemma 12 A detenninistzc and eztenszve automaton cun be

T I L U complete. ~ ~

Proof Let A be a deteririinistic aiid extensive autoiriaton aiicl let 5 tw tlie ordering on Q. If A is not coinplete, a new state qo sucli that (I 5 qo is added with the following transitions. If there is no transition labeled by a letter CL outgoing from a state q , the transition q 3 qo is added. If for a statc p , there is no limit transition of the foriri { p } + q , a traiisition { p } + qu is added. Moreover, the transitioiis qo 4qo for u in A and the transitioii { q o ) + qO are added. The new autoinatoil is complete and it is still deteririiitistic aitd extensive. 0 The following shows that if the wl-language X is recogiiized hy a deterministic and extensive autoiriaton, then is recognized by a i l R-trivial w1semigroup which satisfies the &-equatioii.

100

Lemma 13 Let X be an wl-language recognized by a deteniiinistic extensive automaton A. T h e n X is recognized by a R-triviul w1 -semigrosup satisfying the . if all the liirvit transitions of the equation ( ~ g " ) = ~ y( ~~ y " ) ~Furthernore automaton are of the form { p } -+ p , the wl-semigroup satisfies the cyuation %w - 9 . Proof By the previous lemma, it rnay be assumed that the autolnaton is complete. Let 5 be the ordering on Q. Let 5' be the increasing functioils from Q to Q a.nd let SO be an element of 5' such that if {p} -+ q is a limit transition of d,then p . SO = q . We claim that the wl-semigroup S with the operation w defined by sw = s"s0 recognizes X . Let .si/ : .4Q -+ S he the morphism defined by +(u) = s, where ea.cl-1s, is a fhct,ion of S such that p-s, = q whenever p -% q is a successor transition of A. We clairn that for any word z and any state q , p . + ( x ) = q iff p ' z in A. The proof is by illduction on the length of z. If 1x1 is a successor ordinal, the result follows directly from the definition of the functions s., Otherwise, let a be the limit ordinal 1x1. Let 2 = zn be a factorization such that s = +(xo) and e = $ ( x ? ~for ) n 2 1. Let yn = 5 0 . . .zn. Since the sequence qn = p ' p n is increasing, there is an integer 1x0 such that qIL= qnc for 'n 2 'no. Then q is the unique sta.te such that { q n o } -+ q is a limit transition of A. By the induction hypothesis, one has qo = p ' g n = psse" for 72 2 n o . Since +(x) = xu = se"s0 and q = qno ' s o , one has p z = p . +(z). Furthermore, if all limit tra.nsitions of A are of tlie form {p} -+ p , the element SO can he chosen equal to the neutral elemelit. Tlien, the wl-semigroup satisfies the equation zw = 2". 0

nn<# +

Lemma 14 Let X be a n wl-Iangu.age recogruized b y a detenninistic extensive automaton A. Then, at can be denoted by a reyular expression a s (Lescribed in Theorems 9 and 10.

Proof For any states p and q , denote by Ap,q the subalpliabet { a I 1) 3 ( I } and by ?; the unique state such that { p } -+ ?; is a limit transitioii of A. Let S be the set of increasing sequences of states defined by S = { ( q l , . . . ,Qn) I (11 E I and

(11

<

< . < qTLa ~ q,t d * .

E

f}.

The set X recognized by A is then equal to to the finite unioii

For any sequence s = (q1,. . . ,qn) of S , the set X s = {z I q1 3 qIL} is tlie set S of words which labels a path from ql to qn going through the states of s. Since

101

the automaton A is extensive, the states rnust be visited in the orclciirig of s. I t suffices to prove that each X, is a finite union of product of tlLe required form. We define the sets Lk as follows '!h

,qk

if

,Vk+1

';k,qkAqk,q,+l '~k,Qk"(/k?qk+l

Atk,ak

&. = q k

t/t' = q k + l otherwise if

and L, by

L,, =

if 4 1 = (111 L,, = L,, = ,4;,,qn otherwise

I t is clear that X , is equal to the product L1 . . . L , L . Siiice t h > \ \ ~ t , o l l \ iA ~t~)~~ is deterministic, the subalphabets L4qk , q k and A,, , c , r + l are disjoint. 7'licrefore, each Lk is equal to a finite union of wl-languages of the required form. 0

Acknowledgment The author would like to thank Jean-Eric Pin for his highly relcv a n t coinments on a preliminary version of this paper. He also would like to tJhNIJkt,he anonymous referee for his very helpful suggestions.

References 1. J. Ahneida. Finite Semigroups und Universal Algebru, voluirie 3 of Series in algebra. World Scientific, 1994. 2. N. Bedon. Automata, semigroups and recognizability of words on ordinals. Int. J . Aly. Cornput., 8(1):1-21, 1998. 3. N. Bedon. Logic over words on denumerable ordinals. Techiiical R.eport 2000-04, Institut Gaspard Monge, 2000. 4. N. Bedon. Star-free sets of words on ordinals. Inform. Coniput., 166:93111,2001. 5 . N. Bedon and 0. Carton. An Eilenherg theorem for words 011 countable ordinals. In C. L. Lucchesi and A. V. Moura, editors, LA7'1N.98, vdutne 1380 of Lect. Notes 171 Cornput. Sci., pages 53-64. Spriiigei,, 1998. 6. J. R. Biichi. O n a decision method i n the restricted second-orcler itritkmetic. In Proc. Int. Congress Loyz'c, Meth,odoloTy urid Philosophy of science, Berkeley 1960, pages 1-11. Stanford IJniversity I'ress, 1962. 7. J . R. Biichi. Tra.iisfinite autoinata. recursions a.nd weak second order theory of ordinals. In Proc. Int. Conyress Logac, Jvlethorloloyy, u71d Philosophy of Science, Jerusalem 1964, pages 2-23. North Holland, 1965.

102

8. S. Eileriberg. Automutu, La.rtgzla!ges und Machines, volume B. .4ci\delriic Press, 1976. 9. D. Perrin a.nd J.-I~.pin. ~emigroupsand a.utoi-riataon infiiiite words. 11.1 J. Fountain and V. A. R. Gould, editors, N A T O Aduanced S t u d y Fnstitute Semigroups, Formal Languages U71d Groups, pages 49-72. Kluwer academic publishers, 1995. 10. J.-8. Pin. Varieties of Forinal Languages. North Oxford, London a.nd Plenum, New-York, 1986. Traduction de Vurie'te's de L a n ! p ~ ~ e~. sO ~ ~ I 11. J.-E. Pin. Handbook of formal languages. volui-tie 1 , CllilIJteI' Syntactic semigroups, pages 679-746. Springer, 1997. 12. J.-E. Pin. Positive varieties and infiiiite worc~s. 111 C. L. imxlresi a n c ~ A. V. Moura., editors, LATIN'98, volume 1380 of Lect. Notes 571 Gomput. Sci., pages 76-87. Springer, 1998. 13. J. G. Rosenstein. Linear Ordering. Academic Press, New York, 1982. 14. M. P. Schutzenberger. On finite inonoids having only trivial subgroups. Inform. Coritrol, 8:190-194, 1965. 15. T. Wilke. An algebraic theory for regular languages of finite and infinite words. hit. J . Aly. Comput., 3(4):447-489, 1993. 16. J. Wojciechowski. Finite automata. on transfiiiite sequences and regu1a.r expressions. Fundamento infor~naticce,8 (3-4):379-396, I 985.

103

THETHEORY

OF RATIONAL RELATIONS ON TRANSFINITE STRINGS

CHRISTIAN CHOFFRUT

SERGEGRIGORIEFF

LIAFA, UniversitB Paris 7

LIAFA, UniversitB Paris 7

2, pl. Jussieu 75251 Paris Cedex 05 France

2, pl. Jussieu 75251 Paris Cedex 05

ccQliafa.jussieu.fr

segQliafa.jussieu.fr

France

1 Introduction In order to explain the purpose of this paper, we recall briefly how the theory of “rationality” developed in the last fourty years. The theory of sets of finite strings recognized by finite automata, also known as regular or rational sets was developed in the fifties. It rapidly extended in two directions. Indeed, by solving the decidability problem of the second order monadic theory of one successor, Buchi was led naturally to introduce the notion of finite automata working on infinite strings. He further extended this result to the monadic theory of all denumerable ordinals, and by doing so he again modified the original notion of finite automata to suit his new purpose, [9]. At this point, the equivalence between the notions of recognizability (by “finite automata”), rationality (by “rational expressions”) and definability (by “monadic second order logics”) was achieved as far as strings of denumerable lengths were concerned. In the late sixties, Elgot and Mezei wrote an historical paper on rational relations [15] which was a successful attempt to construct the theory of relations between free monoids that could be recognized by so-called n-tape automata. Though hard to read, it contained the basic results of the theory. In the mid eighties Gire and Nivat showed that the theory of rational relations on finite strings carries over to

104

infinite strings, [16]. More recently, Wilke made a breakthrough by giving an algebraic characterization of the rational infinite strings via the notion of “right binoids” (now known as “Wilke algebras”), [25]. This construct happens to be the natural extension of finite monoids for infinite strings. Elaborating on this notion, Bedon [2] showed that rational transfinite strings of countable length recognized by finite automata can also be recognized by finite algebras (the wl-algebras). In the present paper we draw the theory further by showing that almost everything carries over to transfinite strings. The proof of most elementary properties are mere paraphrase of those for finite strings, so we state them here for self-containment but we leave the simple verifications to the reader. For a few results completely new proof techniques are required, like the equivalent of the “second factorization theorem” (in Eilenberg’s terminology [14, p. 248]), cf. Thm. 27. Finally, some properties no longer hold beyond the length ww like the uniformization problem, cf. Thm. 33 and 55.4. Observe finally that our properties do not cover all those of [16] since the concatenation of strings is only partially defined. In some sense our general framework seems to be more natural. For example, when dealing with infinite strings the property that the family of relations recognized by automata is closed under concatenation needs some intricate case analysis ([16, pp. 107-1101) while a purely formal proof works for transfinite strings. Also, the notion of direct product of rational subsets (the “recognizable relations”) is completely clarified now with the wl-algebras of Wilke.

2

Preliminaries

Here we present the basic facts on ordinals, transfinite strings and finite automata for transfinite strings, extending thus the elementary definitions usual for finite strings.

2.1

Ordinals

In this paper we shall deal with ordinals less than the first nondenumerable ordinal w1. We refer the interested reader to the standard

105

textbooks, e.g., [24] and [23], for a thorough exposition of the material on ordinals. We recall an ordinal is prime if it cannot be expressed as the sum of two smaller ordinals; these ordinals are exactly the powers of w. The (unique) Cantor’s normal form of an ordinal 0 < a < w1 is the sum

where 0 5 n < w,A, > . . . > XI > Xo, 0 5 ai < w for all i 5 n and 0 < a,. The ordinal A, is the degree and the ordinal XO is the tgpe of a. The degree and the type of 0 are equal to 0. A few elementary definitions on sequences indexed by ordinals will be used in the sequel. We review them here. DEFINITION1. Let a be a limit ordinal. An increasing sequence (/?,),
< +

2.2

+

Transfinite strings

Given a finite alphabet A , a string is a mapping u from some a < w1 into A . Equivalently, u is a sequence of elements of A indexed by an ordinal Q. We denote up the element indexed by /? < a in this sequence. The ordinal a is the length of u, denoted by 1u(. The collection of all

106

strings is denoted by A<("' and we will call them improperly transfinite though they might also be of finite length. The empty string, denoted by 1, is the string of length 0 and is the unit of A<W1as a monoid. By extension, the degree of a string u is the degree of its length. For a E A , Iu(,denotes the length in the letter a of the string u,i.e., the ordinal of the subsequence indexed by the set of positions ,O < a for which u p = a. The set of strings is partially ordered by the "prefix relation" : u 5 w if there exists w such that 21 = uw.

2.3

Continuous mappings

FYom now on and unless otherwise stated, the term "increasing" when applied to strings, is to be understood relative to the prefix ordering. The string w is the limit of the increasing sequence (u,),<~ of prefixes of w if for all prefixes w of w there exists an index 7 such that w is a prefix of u,. E.g., limi,,(ab)i = limi,,(ab)Za = (ab)". Let h : A<"' + be increasing with respect to the prefix ordering. We say that h is continuous if it commutes with limits of strings, i.e., if for all limit ordinals X and all string u = lim,,x u, E A<W1,equality h(u) = lim,,x h(u,) holds. can be extended in a unique Any mapping h from A into way as an increasing (relative to the prefix ordering) and continuous (in the above sense) function of into Thus, all morphisms will be determined by the images of the letters. This can be proved by transfinite induction by setting h(ua) = h(u)h(a)for all u E and a E A and h(u) = sup{h(w) I w is a prefix of u} if the length of u is a limit ordinal. This latter string is indeed well determined, it is the string of length a = sup{lh(w)l I w is a prefix of u} whose prefix of length ,O < a is the prefix of length p of every string h(v) where w a prefix of u and ,O 5 lh(u)I < a. Similarly, every mapping h of A into the direct product B,<,' x El,<"' can be extended to a mapping of A<,' into B:"' x B,<"' by posing h(u) = ((wh)(u),(.2h)(u)).

107

1. If

p < Q then gp+1

satisfies ( g p , u p , g p + l ) E EA.

2. If p 5 Q is a limit ordinal then ( P , g p ) E Ee where P set of persistent elements in the run (q^l)y
C Q is the

+

Observe that a run with label u has length Iu( 1. The run is successful if go E Q- and qa E &+. DEFINITION 5. 1) A subset of is Buchi recognizable if it is the set of labels of successful runs of some Buchi automaton. We denote Buchi(A<"l) the family of Buchi recognizable subsets of A<Wi 2) We let Biichi(A
I

X

E

Buchi(A<"l)}.

A word of caution. The way we introduced the relation E is a slight modification of Buchi's original treatment of limit cases (for which the set of persistent states itself is considered as the limit state). However, this does not change the family of recognized subsets of A<W1as the reader may easily verify.

REMARK 6. The family of Buchi recognizable languages is easily seen to be closed by union and intersection. Closure by complementation is a difficult result (Buchi, 1965, [8],cf. also [9]). The next result insures that given an automaton, the set of strings that label some path visiting a specified subset of states is rational. It will be used later in Theorem 27. LEMMA7. Let A be a Buchi automaton, Q its set of states and V C Q a fixed subset. For all p , g E Q , the set Xp,v,q of strings that label some run from p to g which visits exactly the states in V i s Buchi recognizable. Proof. If A = (Q, A, Q - , Q+, EA,Ee) then X,,v,, is recognized by the automaton I3 = (Q x 2Q, A, { ( p ,0)}, ( ( 4 , V ) } FA, , Fe) where the transition sets FA, Fe satisfy the following conditions 1. for all q l , q 2 E Q, W C Q, a E A we have ((gl,W),a,(qZ,WU{g:!})) E FA if and only i f ( q l , a , q 2 ) E EA

108

3

Finite automata on transfinite strings

There are two different ways of defining a finite automaton on transfinite strings. Both are due to Buchi [9]. The first one was extensively studied by Choueka and considers strings of length less than wnfl for a given n, see [13]. The second one deals with strings of arbitrary countable lengths and was investigated (and actually extended to uncountable ordinals) by Wojciechowski [27]. Both formalisms are equivalent when restricted to strings of length less than wn+’ for some n < w, (cf. Theorem 17), so we shall use which one is more amenable depending on the question under investigation.

3.1

Buchi automata on transfinite strings

Here we recall precisely the above mentioned definitions of finite automata for transfinite strings starting with what is nowadays known as Buchi’s automaton. The main new point is about the definition of limit transitions and relies on the notion of cofinal sequences, as defined in paragraph 2.1.

3. Let Q be a finite set, let a be a limit ordinal and let DEFINITION (qp)p
109

2. for all (41,. . . ,q'} C Q, W C Q, r E Q we have ( S ,(r,W U { r } ) )E F' if and only if ( ( 4 1 , . . . ,q k } , r ) E Et where = ((41, W ) ,. . . , ( Q k , W ) )

s

3.2

Choueka automata on transfinite strings

Buchi's notion of transition at limit step a involves all possible cofinal sequences to a. In case a < wn+l is a limit ordinal with type k (such that 0 < k 5 n) then a = ,O w' for some ,O and there is a canonical sequence cofinal to a , namely ( p ~ ' - ' m ) ~ This < ~ is. the one considered at limit steps for the so-called Choueka automata which deal with strings having length less than wn+' for some n < w . They require a special notion of limit, known as "Choueka-continuity" in the literature, see [13]. Given a set Q, we let [Qlo = Q , [&I - 2[&Ik-' \ (0) if k > 0 , [Q]; = U,,5k,n[Q]' .

+

+

+

DEFINITION8. A transfinite sequence s of length a 1 < w"' is Choueka-continuous over the set Q if - sp+1 E Q for all ,8 < a , - sp = { e E [Q]z I 3"Ok, e = sy+wl,k}for all p 5 a of the form p = y wz+' (i.e. ,O is limit of type i I). In particular, s maps a 1 into [Q]; (and not into Q) where T is the type of a.

+

+

+

DEFINITION9. A Choueka automaton is a quintuple A = (Q, A , E , Q-, Q+) where Q is the (finite) set of states, A is the input alphabet, Q- C Q is the set of initial states, Q+ C: [Q]; is the set of final states and E C [Q]; x A x Q is the set of transitions. The notion of run with label u = (up)p<, as interpreted for Buchi's automata, extends naturally to Choueka's automata. Indeed, it is a Choueka-continuous sequence (qp)p<(y E [Q]; satisfying (qp, up, qp+1) E E for all p < a. Observe that it is clear from Definitions 8 and 9 that qp E [Q]' if ,O is of type k .

110

A run is successful if its first state 40 is an initial state and its last state qa is a final state. A subset of A<wn+'is Choueka recognizable if it is the set of labels of successful runs of some Choueka automaton.

REMARK 10. The family of Choueka recognizable languages is easily seen t o be closed by union and intersection. Closure by complementation is also true (Choueka, [13]).

3.3

Rational sets of transfinite strings

A fundamental result is that Kleene's theorem can be extended to transfinite strings for both notions of automata. To that purpose, one has to consider two new operations on sets of strings. DEFINITION11. 1) The w-power of a set X, denoted X", is the set of strings obtained by concatenating w-sequences of strings in X . 2) The wl-iteration of a set X is the set X<"l = Ua<"' X", where X" is the set of strings obtained by concatenating &-sequences of strings in X (in particular, Xo = {I)). 3) The n-trace-w-power of a set X is n-trace(X") = X" n A<wn+'.

REMARK 12. 1) X<"l is the closure of XU(1) under w-power, i.e. it is the smallest set Y which contains XU { 1) and is closed under w-power: X U { 1) Y = Y". It is also the closure of X U { 1) under Kleene-star and w-power, i.e. the smallest set Y such that XU{ 1) Y = Y* = Y". In fact, since (X U (1))" 2 X*, such a Y necessarily satisfies Y = Y2 and an easy induction over ordinals less than w1 (using equalities Y = Y 2 and Y = Y" for the respective successor and limit cases) shows that Y = Y" for all Q < w1,hence Y = Y<"l. 2) Observe that there is no need for a similar closure operation relative to the n-trace-w-power. In fact, an easy induction over i shows that xa>n A < w ~ + * the i-th iterate of n-trace-w-power of XU { 1) is (U,<,.

+

+

All these iterates coincide for i 2 n 1, therefore the (n 1)-th iterate of n-trace-w-power of X U (1) is closed under n-trace-w-power. DEFINITION 13. 1) [Wojciechowski, 1985 [27]]Rat(A<"') denotes the least family of subsets of A<W1that contains the empty set, the singleton sets consisting of a letter and is closed under set union, concatenation, Kleene-star, w-power and wl-iteration.

111

2) [Choueka, 1978 [13]]Rat(A<W"fl)denotes the least family of subsets of A<wn+' that contains the empty set, tbe singleton sets consisting of a letter and is closed under set union, concatenation, Kleene-star, and the n-trace-w-power.

The following can be proven with the same structural induction technique a s that for rational subsets of free monoids. Recall that a rational substitution of C into A<W1is a mapping o that assigns a rational subset of AcW1to each c E C. Given an ordinal a , one can o(up). extend o to A<" by setting o((up)pca)=

np<,

PROPOSITION 14. I f a : C Rat(ACW1)is a rational substitution and X E Rat(Ccw1) then o ( X ) = U{a(z) I II: E X } is in Rat(A<"l). An analog property holds with wn+' in place of w1. --f

3.4 Kleene type results As for finite strings, there is an equivalence for subsets of transfinite strings between recognizability via some type of finite automaton and expressability via some type of operations. We state these results for transfinite lengths less than for some n < w and for arbitrary lengths less than w1.

THEOREM 15 (Choueka, 1974,1131, cf. also Bedon, 1996, [l]). . Rat(A<Wnf')i s exactly the family of Chouelca recognizable-subsets of A<8+1 THEOREM 16 (Wojciechowski, 1985,[27]). Rat(A<W1)is exactly the family of Buchi recognizable subsets of A<W1. (considered as a subset of A<'"') is It is easy to check that Buchi recognizable. Now, as a corollary of the above theorems, we see that Choueka recognizability is a special case of Buchi's recognizability. This is proven in [l]but it can more easily be seen by arguing on the lengths of the strings.

THEOREM 17 ([13],[1]). Let X C . The following conditions are equivalent: 1) X is Buchi recognizable 2) X is Chouelca recognizable 3) x = Y n ~ < w ~ f+o rl some Buchi recognizable set Y C A<W1

112

4

Rational relations on transfinite strings

4.1 Two-tape Buchi and Choueka automata The idea of Buchi and Choueka automata described above extends to two-tape automata. DEFINITION 18. A two-tape Buchi automaton on transfinite strings is where Q,Q-,Q+ are a construct A = ( Q , A , B ,Q-,Q+,EA,EB,E~) as in Definition 4, A, B are finite alphabet and

The elements of (2), ( 3 ) and (4),are respectively called the A , B and limit transitions. The notions of run, label, successful run are straightforward extensions of the corresponding notions in finite automaton, the only difference being that labels are pairs of strings rather than single strings. The relation in A<W1x B<"'l defined by the two-tape Buchi automaton is the set of pairs (u, v) that are the labels of some successful run and is said to be Buchi recognizable. DEFINITION 19. A two-tape Choueka automaton on transfinite strings is a construct A = ( & , A ,B , Q-, Q+, E ) where Q,Q-,Q+ are as in Definition 9, A , B is a finite alphabet and where E C [Ql; x ( ( A x (1)) u ((1) x BN x Q x F w n + l defined by the two-tape Choueka The relation in automaton is said to be Choueka recognizable.

4.2

Rational relations

The operations needed for defining rational relations are those introduced for subsets extended in the usual way to pairs of strings. For instance Definition 11 obviously extends from strings to pairs of strings which leads to the following extension of Definition 13.

113

DEFINITION 20. 1) Rat(A<"l x W W 1 denotes ) the least family of subsets of A<W1x that contains the empty set, the singleton sets { ( a , l)}, ((1,b ) ) for a E A, b E B and is closed under set union, concatenation, Kleene-star, w-power and w1-iteration. 2) Rat (A<Wnf1 x BCwn+l) denotes the least family of subsets of x B<Wn+ that contains the empty set, the singleton sets { ( a ,l)},{ (1,b ) } for a E A, b E B and is closed under set union, concatenation, Kleenestar, and the n-trace of w-power. The closure under rational substitution seen in Proposition 14 extends easily to relations.

PROPOSITION 21. I ) If g : C Rat(A<Wl)and T : C --+ Rat(BCW1) are rational substitutions and X E Rat(C<wl)then (0, T ) ( X )= UsEXD(.) x T ( . ) is in Rat(A<"l x B<"'l). 2) If o : C Rat (ACwn+') and T : C -+ Rat ) are rational substitutions and X E Rat(C<WP+l)then (a,T ) ( X )= UzEXa ( z ) x ).( is in Rat(A<WnPC1 B < W n p f l >. ---f

--f

4.3

The first factorization theorem

We recall that a morphism 4 : A* --+ B* is alphabetic whenever + ( A ) G B U (1) holds and that it is strictly alphabetic whenever $ ( A ) G B holds. These notions make sense when applied to morphisms from A<W1into B<W1. Proposition 21 admits a reciprocal which is given some normalized forms in Proposition 22 below and Theorems 27,28. The first form we consider is the transfinite extension of a result on finite strings first observed by Nivat, 1968, [22], which is called the first factorization theorem in Eilenberg [14] p. 240. Its proof is a paraphrase of the proof for finite strings, cf., e.g., [6, Thm. 111.4.1.1 and consists in considering pairs (a, l),(1,b) as letters of a new alphabet C and introducing the projections from C onto A U (1) and B U (1).

PROPOSITION 22. The following conditions are equivalent 1) R E Rat(A<"l x B<W1) 2) there exist a finite alphabet C , a rational subset K of C<W1and A U {l}, T B : C B U {I} such two alphabetic morphisms T A : C --f

--+

114

that R = { ( T A ( ~ ) , T B ( z )I)z E K } . Moreover, one can suppose

(*)

b'c E

C(TA(C) is empty

A n analog equivalence holds with wnfl

7 r ~ ( c is )

not empty)

in place of w1.

As a corollary, we get the relational version of Theorem 17.

THEOREM 23. Let R C_ A<"'"+' x B<wn+l . The following conditions are equivalent: 1) R is Buchi recognizable 2) R is Choueka recognizable 3) R = Sfl(A<""+' x BiW"+l)for some Buchi recognizable relation S. Proof. 1) + 2). Let K , T A , T B be as in 2) of Proposition 22 with condition (*). Clearly, I T A ( z ) ~ I~rg(z)l 21 . 1 for all z E C<"l. Since R c - A<wn+' B < W n + ' we see that K 2 C<"n+' . We conclude by Theorem 17 and Proposition 22 applied to wn+'. 2) 3 3). If R is Choueka recognizable then by Proposition 22 with condition (*) applied to wn+l and Theorem 17 there exists K = K' n C<w"+lwhere K' is Buchi recognizable such that R = { ( n ~ ( z n) ,~ ( x )1 ) z E K } . Then R = Sn(A<""+l xB<""+l ) WhereS = { ( T A ( x ) , T B ( z ) )I x E K'}. 3) + 1).Let S = { ( T A ( z ) , T B ( z ) )15 E K } for some K E Rat C<"l with condition (*). Then R = { ( ~ ~ ( z ) , 7 r r g1 z ( zE) )KnC<""+ }. 0

+

0

Another corollary of the first factorization theorem is the closure under composition property. The proof is also a paraphrase of the same property for finite strings, [6, Thm 111.4.4.1. We reproduce it here for the sake of completeness.

PROPOSITION 24 (Closure under composition). If R E Rat(A<"1 x BcW1) and S E Rat(B<"" x C<"l) then R o S E B<""+', C<""+l f o r Rat(A<"1 x C<"l). The same holds with all 0 5 n < w . Proof. The proof goes exactly as in the finite length case, see [6]. Denote by r j B the projection of AUB onto A and by the projection

r2ic

115

I

of A U B U C onto A U B. Let R = {(7riB(z),7riB(z))2 E K } , S = { ( T ~ ~ ( ~ ) , T ~ I~ z( ZE) L ) } , where K C ( A U B)<W1and L 2 ( B U C)<W1 are rational languages. Now, the composition R o S can be written R o S = { ( rABC A (z),7rgBC(z)) I z E M } , where M = ( T ~ ; ~ ) - ~ (f K l (7rggc)-'(L). ) The inclusion from right to left is straightforward. As for the left to right inclusion, observe that if A , B , C are disjoint, u E ( A U B)<W1 and v E ( B U C)<W1 are such that 7riB(u)= 7r,""(v) then there exists w E ( A U B U C)<Wlsuch B (w) = u and 7 r ~ ~ " ( w= ) v. Finally, observe that M is rathat T AABC tional due to the commutation of 7r-' with set union, concatenation, Kleene-star, w-power and wl-iteration and second to the closure under intersection of the family of rational sets of transfinite strings. 0

4.4

Rational Biichi transducers

Here we modify the notion of finite transducer in such a way as transforming it into a Buchi automaton on A where each transition is equipped with an output in Rat(B<"l). In other words, as for finite and infinite strings, there is an alternative definition where the third component of the transitions are rational subsets of

DEFINITION25. A Biichi transducer on transfinite strings is a construct 7 = (&, A , B , &-, EA,Ee, F ) where Q, A , B , Q- are as in Definition 4, EA is a finite subset of (Q x A x R a t ( W w l ) x Q), Ee is a subset of 2Q x Q and F is a mapping from Q to Rat(BCW1). A run of 7 on a transfinite input string (u,),<~ E A<W1is a pair of sequences ((q,)TQ, (X7JVSA) where 1) qo E Q- and for all 7 < X we have (qv,uq,X,,q,+l) E EA 2) if 5 X is limit then (P,qq) E Ee where P is the set of persistent states in (qc)c-,. 3) XA = F ( q x ) The output of the run is the concatenation product X = X,). Transducer 7 defines the relation R G A<W1x B<W1 which associates with every transfinite input string u the union set of all outputs of runs on u.

(n,,,

116

4.5

Eilenberg’s second factorization theorem: the case

< w1 The following technical result happens to be the crux for the transfinite extension of Eilenberg’s second factorization theorem.

LEMMA26. Let Q be a finite set, let a and X be limit ordinals and let (gp)pca be a sequence of elements of Q. Let (&,),
Proof. Denote by {Q(’),. . . Q @ ) }the collection of persistent elements in the sequence (&,),
+

2) Conversely, let g E Q ( j ) ,with 1 5 j 5 Ic. Then Q ( j )= QVi for some increasing sequence ( ~ i ) i cofinal < ~ to A. Choose an arbitrary element ei E [&, with gEi = q. Then the sequence ( e i ) i C w is cofinal to cy 0 and thus g is persistent in (gp)p
117

THEOREM 27. Given a relation R A<"'1 x B<"'', the following properties are equivalent 1) R is rational 2) R is defined by some 2-tape Buchi automaton 3) R is defined b y some Buchi transducer C<"'lD, 4) there exist finite alphabets C, D , a rational subset K a strictly alphabetic morphism 4 : C --+ A and a rational substitution a : (cU 0) --+ Rat(B<"'') such that R = U d E D , z d E K { 4x( ~ a(@ )} Proof. With no loss of generality we can suppose that A and B are disjoint. 1) % 2). As in the finite string case, there is a one-to-one correspondence between 2-tape Buchi automata on alphabets A, B and one tape Buchi automata working on alphabet ( A x (1)) U ((1) x B ) . The relation associated to the 2-tape Buchi automaton being defined as in condition 2 in Proposition 22. Thus, 1) w 2) is a mere reformulation of that last proposition. 3) + 4). Given the Buchi transducer 7 = (Q, A, B , Q-, EA,El, F ) , let C, D be disjoint finite sets of new symbols such that C is in one-to-one correspondence with the set of quadruples (4, a , X , p ) of EA,and D is in one-to-one correspondence with the set of pairs ( q , X ) E Q xRat(B<"'l) such that F ( q ) = X . Denote by [ q , a , X , p ]or [q,X ] whichever, the element of C , D in this correspondence. Define a strictly alphabetic morphism 4 : C + A and a rational substitution : (CUD) -+ Rat(B<"'l) as follows: $ ( [ q , a, X , p ] ) = a , a([q,a , X , p ] ) = a([q, X I ) = X . Transform 7 into an one-tape Buchi automaton A = (QU {4+},CUD,Q-,{q+},EL"D,Ee) by defining - ((I,[4,a,X,P],P)E E&D if and only if ( Q , a , X , P )E EA - (q, [ q , X ] , q + E ) ELuD if and only if F ( q ) = X . Clearly, A recognizes a subset K of C<"'lD such that R = {(4(x),a ( z d ) )1 d E D,zd E K } . 4) + 1). This is a direct consequence of Proposition 21. 2) + 3). This is the last implication that remains to be proved. The rest of this paragraph is devoted to its proof. Using condition 2 in Proposition 22, let+ R = { ( T A ( z ) , T B ( z ) ) I z E K } where K E R a t ( ( AU B)<"l) is recognized by the (l-tape) automaton A =

118

(Q, A U B, Q - , Q+, E A U B , El). We shall associate to A a Buchi transducer 7 which, in a first approach, follows the same