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
