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Formal Languages, Automata and Numeration Systems, Volume 2

ISBN: 978-1-84821-788-1
246 pages
November 2014, Wiley-ISTE
Formal Languages, Automata and Numeration Systems, Volume 2 (1848217889) cover image

Description

The interplay between words, computability, algebra and arithmetic has now proved its relevance and fruitfulness. Indeed, the cross-fertilization between formal logic and finite automata (such as that initiated by J.R. Büchi) or between combinatorics on words and number theory has paved the way to recent dramatic developments, for example, the 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. Objects are considered with a perspective that comes both from theoretical computer science and mathematics. Theoretical computer science offers here topics such as decision problems and recognizability issues, whereas mathematics offers concepts such as discrete dynamical systems.

The main goal is to give a quick access, for students and researchers in mathematics or computer science, to actual research topics at the intersection between automata and formal language theory, number theory and combinatorics on words.

The second of two volumes on this subject, this book covers regular languages, numeration systems, formal methods applied to decidability issues about infinite words and sets of numbers.

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Table of 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

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

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

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Author Information

Michel Rigo is Professor at the Department of Mathematics at the University of Liège, Belgium.
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Reviews

"This book follows [Formal languages, automata and numeration systems. Vol. 1. Introduction to combinatorics on words. Hoboken, NJ: John Wiley& Sons; London: ISTE (2014;Zbl 6384340)]. It contains essentially two parts that are quite interesting." (Zentralblatt MATH 2016)
This book follows [Formal languages, automata and numeration systems. Vol. 1. Introduc-
tion to combinatorics on words. Hoboken, NJ: John Wiley& Sons; London: ISTE (2014;
Zbl 6384340)]. It contains essentially two parts that are quite interesting.
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