Formal Languages, Automata and Numeration Systems Volume 2

274 Pages·2014·2.916 MB·

by  Michel Rigo

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The interplay between words, computability, algebra and arithmetic has M
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now proved its relevance and fruitfulness. Indeed, the cross-fertilization c
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between formal logic and finite automata (such as that initiated by J.R. e
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Büchi) or between combinatorics on words and number theory has R
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paved the way to recent dramatic developments, for example, the g
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transcendence results for the real numbers having a “simple” binary
expansion, by B. Adamczewski and Y. Bugeaud.
This book is at the heart of this interplay through a unified exposition. F
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Objects are considered with a perspective that comes both from
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theoretical computer science and mathematics. Theoretical computer m
science offers here topics such as decision problems and recognizability a
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issues, whereas mathematics offers concepts such as discrete
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dynamical systems. a
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The main goal is to give a quick access, for students and researchers in g
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mathematics or computer science, to actual research topics at the
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intersection between automata and formal language theory, number g
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theory and combinatorics on words.
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The second of two volumes on this subject, this book covers regular
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languages, numeration systems, formal methods applied to decidability u
Formal Languages,
issues about infinite words and sets of numbers. t
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Automata and
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Numeration Systems 2
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Michel Rigo is Professor at the Department of Mathematics at the
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University of Liège, Belgium. u
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Applications to Recognizability
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s Michel Rigo

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www.iste.co.uk

Formal Languages, Automata and Numeration Systems 2
To Christelle, Aurore and Maxime.
Series Editor
Valérie Berthé
Formal Languages, Automata
and Numeration Systems 2

Applications to Recognizability and
Decidability

Michel Rigo
First published 2014 in Great Britain and the United States by ISTE Ltd and John Wiley & Sons, Inc.
Apart from any fair dealing for the purposes of research or private study, or criticism or review, as
permitted under the Copyright, Designs and Patents Act 1988, this publication may only be reproduced,
stored or transmitted, in any form or by any means, with the prior permission in writing of the publishers,
or in the case of reprographic reproduction in accordance with the terms and licenses issued by the
CLA. Enquiries concerning reproduction outside these terms should be sent to the publishers at the
undermentioned address:
ISTE Ltd John Wiley & Sons, Inc.
27-37 St George’s Road 111 River Street
London SW19 4EU Hoboken, NJ 07030
UK USA
www.iste.co.uk www.wiley.com

© ISTE Ltd 2014
The rights of Michel Rigo to be identified as the author of this work have been asserted by him in
accordance with the Copyright, Designs and Patents Act 1988.
Library of Congress Control Number: 2014945517

British Library Cataloguing-in-Publication Data
A CIP record for this book is available from the British Library
ISBN 978-1-84821-788-1
Contents
FOREWORD . . . . . . . . . . . . . . . . . . . . . . . . . . . ix
INTRODUCTION . . . . . . . . . . . . . . . . . . . . . . . . xiii
CHAPTER 1. CRASH COURSE ON REGULAR
LANGUAGES . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.1. Automata and regular languages . . . . . . . . . . 2
1.2. Adjacency matrix. . . . . . . . . . . . . . . . . . . . 14
1.3. Multidimensional alphabet . . . . . . . . . . . . . 17
1.4. Two pumping lemmas. . . . . . . . . . . . . . . . . 19
1.5. The minimal automaton . . . . . . . . . . . . . . . 23
1.6. Some operations preserving regularity . . . . . . 29
1.7. Links with automatic sequences and
recognizable sets . . . . . . . . . . . . . . . . . . . . 32
1.8. Polynomial regular languages . . . . . . . . . . . . 37
1.8.1. Tiered words . . . . . . . . . . . . . . . . . . . . 40
1.8.2. Characterization of regular languages
of polynomial growth. . . . . . . . . . . . . . . . 43
1.8.3. Growing letters in morphic words . . . . . . . 49
1.9. Bibliographic notes and comments . . . . . . . . . 51
CHAPTER 2. A RANGE OF NUMERATION
SYSTEMS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55
2.1. Substitutive systems . . . . . . . . . . . . . . . . . 58
vi FormalLanguages,AutomataandNumerationSystems2
2.2. Abstract numeration systems . . . . . . . . . . . . 67
2.2.1. Generalization of Cobham’s theorem on
automatic sequences . . . . . . . . . . . . . . . . 74
2.2.2. Some properties of abstract numeration
systems . . . . . . . . . . . . . . . . . . . . . . . . 86
2.3. Positional numeration systems . . . . . . . . . . . 89
2.4. Pisot numeration systems . . . . . . . . . . . . . . 98
2.5. Back to β-expansions . . . . . . . . . . . . . . . . . 107
2.5.1. Representation of real numbers . . . . . . . . 107
2.5.2. Link between representations of
integers and real numbers . . . . . . . . . . . . 112
2.5.3. Ito–Sadahiro negative base systems . . . . . . 114
2.6. Miscellaneous systems . . . . . . . . . . . . . . . . 117
2.7. Bibliographical notes and comments . . . . . . . 123
CHAPTER 3. LOGICAL FRAMEWORK AND
DECIDABILITY ISSUES . . . . . . . . . . . . . . . . . . . . 129
3.1. A glimpse at mathematical logic . . . . . . . . . . 132
3.1.1. Syntax . . . . . . . . . . . . . . . . . . . . . . . . 132
3.1.2. Semantics . . . . . . . . . . . . . . . . . . . . . . 136
3.2. Decision problems and decidability . . . . . . . . 140
3.3. Quantifier elimination in Presburger
arithmetic . . . . . . . . . . . . . . . . . . . . . . . . 143
3.3.1. Equivalent structures . . . . . . . . . . . . . . . 143
3.3.2. Presburger’s theorem and quantifier
elimination . . . . . . . . . . . . . . . . . . . . . . 146
3.3.3. Some consequences of Presburger’s
theorem . . . . . . . . . . . . . . . . . . . . . . . . 150
3.4. Büchi’s theorem . . . . . . . . . . . . . . . . . . . . 156
3.4.1. Definable sets . . . . . . . . . . . . . . . . . . . . 157
3.4.2. A constructive proof of Büchi’s theorem . . . . 159
3.4.3. Extension to Pisot numeration systems . . . . 168
3.5. Some applications . . . . . . . . . . . . . . . . . . . 170
3.5.1. Properties about automatic sequences . . . . 170
3.5.2. Overlap-freeness . . . . . . . . . . . . . . . . . . 172
3.5.3. Abelian unbordered factors . . . . . . . . . . . 173
3.5.4. Periodicity . . . . . . . . . . . . . . . . . . . . . . 177
Contents vii
3.5.5. Factors . . . . . . . . . . . . . . . . . . . . . . . . 178
3.5.6. Applications to Pisot numeration
systems . . . . . . . . . . . . . . . . . . . . . . . . 180
3.6. Bibliographic notes and comments . . . . . . . . . 183
CHAPTER 4. LIST OF SEQUENCES . . . . . . . . . . . . 187
BIBLIOGRAPHY . . . . . . . . . . . . . . . . . . . . . . . . 193
INDEX . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 231
SUMMARY OF VOLUME 1 . . . . . . . . . . . . . . . . . . 235