Formal Languages, Automata and Numeration Systems 2
Applications to Recognizability and Decidability
Published on 10. September 2014
246 pages
978-1-119-04295-2 (ISBN)
Description
The interplay between words, computability, algebra andarithmetic 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 betweencombinatorics on words and number theory has paved the way torecent dramatic developments, for example, the transcendenceresults for the real numbers having a "simple" binaryexpansion, by B. Adamczewski and Y. Bugeaud.
This book is at the heart of this interplay through a unifiedexposition. Objects are considered with a perspective that comesboth from theoretical computer science and mathematics. Theoreticalcomputer science offers here topics such as decision problems andrecognizability issues, whereas mathematics offers concepts such asdiscrete dynamical systems.
The main goal is to give a quick access, for students andresearchers in mathematics or computer science, to actual researchtopics at the intersection between automata and formal languagetheory, number theory and combinatorics on words.
The second of two volumes on this subject, this book coversregular languages, numeration systems, formal methods applied todecidability issues about infinite words and sets of numbers.
Reviews / Votes
"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.More details
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Michel Rigo
Formal Languages, Automata and Numeration Systems 2
Applications to Recognizability and Decidability
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Formal Languages, Automata and Numeration Systems 2
Applications to Recognizability and Decidability
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Michel Rigo is Professor at the Department of Mathematics at the University of Liège, Belgium.
Content
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 polynomialgrowth 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 automaticsequences 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 beta-expansions 107
2.5.1. Representation of real numbers 107
2.5.2. Link between representations of integers and real numbers112
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 ISSUES129
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 elimination146
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 theorem159
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
