Automata Theory and its Applications

441 Pages·2001·8.76 MB·English

by  Bakhadyr Khoussainov

Progress in Computer Science and Applied Logic
Volume 21
Editor
John C. Cherniavsky, National Science Foundation
Associate Editors
Robert Constable, Cornell University
Jean Gallier, University of Pennsylvania
Richard Platek, Cornell University
Richard Statman, Carnegie-Mellon University
Bakhadyr Khoussainov
Anil Nerode
Automata Theory
and its Applications
Springer Science+Business Media, LLC
Bakhadyr Khoussainov Anii Nerode
Department of Computer Science Department of Mathematics
The University of Auckland Cornell University
Auckland Ithaca, NY 14850
New Zealand U.S.A.
Library of Congress Cataloging-in-Publication Data
Khoussainov, Bakhadyr, 1961-
Automata theory and its applications / Bakhadyr Khoussainov, Anii Nerode.
p. cm. - (Progress in computer science and applied logic; v. 21)
Includes bibliographical references and index.
ISBN 978-1-4612-6645-7 ISBN 978-1-4612-0171-7 (eBook)
DOI 10.1007/978-1-4612-0171-7
1. Machine theory. 1. Nerode, Anii, 1932- II. Title. III. Series.
QA267.K56 2001
511.3-dc21 2001025819
CIP
AMS Subject Classifications: O3D05, 68Q05, 68Q1O, 68Q45, 68Q70, 68RI5
Printed on acid-free paper
© 2001 Springer Science+Business Media New York
Originally published by Birkhauser Boston in 2001
Softcover reprint ofthe hardcover lst edition 2001
AII rights reserved. This work may not be translated or copied in whole Of in par! without the written
permission of the publisher. except for brief excerpts in connection with reviews or scholarlyanalysis.
Use in connection with any form of information storage and retrieval, electronic adaptation, computer
software, or by similar or dis similar methodology now known or hereafter developed is forbidden.
The use of general descriptive names, trade names, trademarks, etc., in this publication, even if the
former are not especially identified, is not to be taken as a sign that such names, as understood by the
Trade Marks and Merchandise Marks Act, may accordingly be used freely by anyone.
SPIN 10776946
ISBN 978-1-4612-6645-7
Reformatted from author's files in L:\TEX 2e by John Spiegelman, Philadelphia, PA
987 6 543 2 1
Contents
Preface xi
1 Basic Notions 1
1.1 Sets ... 1
1.2 Sequences and Tuples · . · . 5
1.3 Functions, Relations, Operations 6
1.4 Equivalence Relations. 11
1.5 Linearly Ordered Sets 14
1.6 Partially Ordered Sets 17
1.7 Graphs. 19
1.8 Induction. 24
1.9 Trees and Konig's Lemma. 27
1.10 Countable and Uncountable Sets 30
1.10.1 Countable Sets. . . · . · . 32
1.10.2 Diagonalization and Uncountable Sets. 35
1.11 Algorithms. · . · .... . . 36
2 Finite Automata 39
2.1 Two Examples . .. · . . .... 40
2.1.1 The Consumer-Producer Problem .. 40
2.1.2 A Monkey and Banana Problem. 42
2.2 Finite Automata .. · ...... 46
vi Contents
2.2.1 Definition of Finite Automata and Languages .. 46
2.2.2 Runs (Computations) of Finite Automata. 50
2.2.3 Accessibility and Recognizability. 56
2.3 Closure Properties . . . . . . . . . . . . . . . . 58
2.3.1 Union and Intersection. . . . . . . . . . . 58
2.3.2 Complementation and Nondeterminism .. 62
2.3.3 On the Exponential Blow-Up of Complementation. 66
2.3.4 Some Other Operations. 67
2.3.5 Projections of Languages. 70
2.4 The Myhill-Nerode Theorem 72
2.5 The Kleene Theorem ... 76
2.5.1 Regular Languages.. . 76
2.5.2 Regular Expressions. . 79
2.5.3 The Kleene Theorem .. 81
2.6 Generalized Finite Automata 85
2.7 The Pumping Lemma and Decidability 93
2.7.1 Basic Problems. ... 93
2.7.2 The Pumping Lemma. 93
2.7.3 Decidability. ...... 95
2.8 Relations and Finite Automata 97
2.9 Finite Automata with Equations . 101
2.9.1 Preliminaries. . . . . . . . 101
2.9.2 Properties of E-Languages .. 103
2.10 Monadic Second Order Logic of Strings 105
2.10.1 Finite Chains ............ 105
2.10.2 The Monadic Second Order Logic of Strings. 106
2.10.3 Satisfiability. .......... 109
2.10.4 Discussion and Plan About SFS. 111
2.10.5 From Automata to Formulas. . 112
2.10.6 From Formulas to Automata. . 117
3 Biichi Automata 121
3.1 Two Examples . ................. 122
3.1.1 The Dining Philosophers Problem. . . . . 123
3.1.2 Consumer-Producer Problem Revisited .. 125
3.2 Btichi Automata .............. 127
3.2.1 Basic Notions . . . . . . . . . . . . 128
3.2.2 Union, Intersection, and Projection. 139
3.3 The Btichi Theorem ....... 143
3.3.1 Auxiliary Results.. . . . . . . . 143
3.3.2 Btichi's Characterization. . . . . 147
3.4 Complementation for Btichi Automata 150
3.4.1 Basic Notations. ........ 150
Contents vii
3.4.2 Congruence~ ...... . 151
3.5 The Complementation Theorem. 154
3.6 Determinism .......... . 160
3.7 Miiller Automata ....... . 162
3.7.1 Motivation and Definition. 163
3.7.2 Properties of Miiller Automata .. 167
3.7.3 Sequential Rabin Automata. 170
3.8 The McNaughton Theorem 175
3.8.1 Flag Points .. . 175
3.8.2 The Theorem. ... . 179
3.9 Decidability ....... . 184
3.10 Biichi Automata and the Successor Function 187
3.10.1 w-Strings as Structures.. . . . . . . 187
3.10.2 Monadic Second Order Formalism. 189
3.10.3 Satisfiability. . .......... . 191
3.10.4 From Biichi Automata to Formulas. 194
3.10.5 From Formulas to Biichi Automata. 198
3.10.6 Decidability and Definability in SIS. 201
3.11 An Application of the McNaughton Theorem . 202
4 Games Played on Finite Graphs 209
4.1 Introduction........ 209
4.2 Finite Games ...... . 210
4.2.1 Informal Description .. 210
4.2.2 Definition of Finite Games and Examples .. 212
4.2.3 Finding The Winners.. . . . . . . . . . 215
4.3 Infinite Games . . . . . . . . . . . . . . . . . 218
4.3.1 Informal Description and an Example .. 218
4.3.2 Formal Definition of Games .. 220
4.3.3 Strategies ........... . 222
4.4 Update Games and Update Networks 224
4.4.1 Update Games and Examples. 225
4.4.2 Deciding Update Networks. 226
4.5 Solving Games .......... . 231
4.5.1 Forgetful Strategies. . ... . 231
4.5.2 Constructing Forgetful Strategies. 236
4.5.3 No-Memory Forcing Strategies. 239
4.5.4 Finding Winning Forgetful Strategies. 243
5 Rabin Automata 249
5.1 Rabin Automata . . . . . . . . . . . . . . 250
5.1.1 Union, Intersection, and Projection. 259
5.2 Special Automata ............ . 262
viii Contents
5.2.1 Basic Properties of Special Automata ... 263
5.2.2 A Counterexample to Complementation .. 265
5.3 Game Automata .... 270
5.3.1 What Is a Game? 270
5.3.2 Game Automata. 272
5.3.3 Strategies ..... 274
5.4 Equivalence of Rabin and Game Automata . 276
5.5 Terminology: Arenas, Games, and Strategies 281
5.6 The Notion of Rank . 287
5.7 Open Games . . . . . 290
5.8 Congruence Relations 292
5.9 Sewing Theorem ... 295
5.10 Can Mr. (E) Visit C Infinitely Often? 300
5.10.1 Determinacy Theorem for Games (n, [C), E) .. 301
5.10.2 An Example of More Complex Games. 304
5.11 The Determinacy Theorem ............. . 306
5.11.1 GH-Games and Last Visitation Record. . .. . 306
5.11.2 The Restricted Memory Determinacy Theorem. 308
5.12 Complementation and Decidability . . . . . . . . 318
5.12.1 Forgetful Determinacy Theorem.. . . . . . 318
5.12.2 Solution of the Complementation Problem. 319
5.12.3 Decidability. . .............. . 327
6 Applications of Rabin Automata 329
6.1 Structures and Types. . . . . . . . . . 330
6.2 The Monadic Second Order Language 333
6.3 Satisfiability and Theories . . . . . . . 336
6.4 Isomorphisms ............ . 338
6.5 Definability in T and Decidability of S2S 339
6.5.1 ~-Valued Trees as Structures ... . 340
6.5.2 Definable Relations. . ...... . 341
6.5.3 From Rabin Automata to Formulas. 343
6.5.4 From Formulas to Rabin Automata. 346
6.5.5 Definability and Decidability. 349
6.6 The Structure with Successors . . . 350
(J)
6.7 Applications to Linearly Ordered Sets 354
6.7.1 Two Algebraic Facts .. . 354
6.7.2 Decidability. ..... . 358
6.8 Application to Unary Algebras 361
6.8.1 Unary Structures. . . . 361
6.8.2 Enveloping Algebras ... 363
6.8.3 Decidability. ... . . . 365
6.9 Applications to Cantor's Discontinuum . 369
Contents ix
6.9.1 A Brief Excursion to Cantor's Discontinuum. . 369
6.9.2 Cantor's Discontinuum as a Topological Space. 372
6.9.3 Expressing Subsets of CD in S2S. 374
6.9.4 Decidability Results. . . . . . . . . . . . 378
6.10 Application to Boolean Algebras . . . . . . . . 382
6.10.1 A Brief Excursion into Boolean Algebras. 382
6.10.2 Ideals, Factors, and Subalgebras of Boolean Algebras. 385
6.10.3 Maximal Ideals of Boolean Algebras. . 388
6.10.4 The Stone Representation Theorem. . . 390
6.10.5 Homomorphisms of Boolean Algebras. 392
6.10.6 Decidability Results. . . . . . . . . . . 397
Bibliography 403
Index 423
Preface
The theory of finite automata on finite stings, infinite strings, and trees has had a dis
tinguished history. First, automata were introduced to represent idealized switching
circuits augmented by unit delays. This was the period of Shannon, McCullouch
and Pitts, and Howard Aiken, ending about 1950. Then in the 1950s there was
the work of Kleene on representable events, of Myhill and Nerode on finite coset
congruence relations on strings, of Rabin and Scott on power set automata. In the
1960s, there was the work of Btichi on automata on infinite strings and the second
order theory of one successor, then Rabin's 1968 result on automata on infinite
trees and the second order theory of two successors. The latter was a mystery until
the introduction of forgetful determinacy games by Gurevich and Harrington in
1982. Each of these developments has successful and prospective applications in
computer science. They should all be part of every computer scientist's toolbox.
Suppose that we take a computer scientist's point of view. One can think
of finite automata as the mathematical representation of programs that run us
ing fixed finite resources. Then Btichi's SIS can be thought of as a theory of
programs which run forever (like operating systems or banking systems) and are
deterministic. Finally, Rabin's S2S is a theory of programs which run forever and
are nondeterministic. Indeed many questions of verification can be decided in the
decidable theories of these automata. These automata also arise in other languages
such as temporal logic and the fL-calculus.
Suppose we take a mathematical logician 's point of view. Each of the classes
of automata discussed has a corresponding natural decidable theory. As Rabin
showed, many theories can be proved decidable by coding them into S2S. Even