Michel Rigo

Formal Languages, Automata and Numeration Systems 1

Introduction to Combinatorics on Words

• Gebundenes Buch

Produktbeschreibung

Formal Languages, Automaton and Numeration Systems presents readers with a review of research related to formal language theory, combinatorics on words or numeration systems, such as Words, DLT (Developments in Language Theory), ICALP, MFCS (Mathematical Foundation of Computer Science), Mons Theoretical Computer Science Days, Numeration, CANT (Combinatorics, Automata and Number Theory).
Combinatorics on words deals with problems that can be stated in a non-commutative monoid, such as subword complexity of finite or infinite words, construction and properties of infinite words, unavoidable regularities or patterns. When considering some numeration systems, any integer can be represented as a finite word over an alphabet of digits. This simple observation leads to the study of the relationship between the arithmetical properties of the integers and the syntactical properties of the corresponding representations. One of the most profound results in this direction is given by the celebrated theorem by Cobham. Surprisingly, a recent extension of this result to complex numbers led to the famous Four Exponentials Conjecture. This is just one example of the fruitful relationship between formal language theory (including the theory of automata) and number theory.

Contents to include: algebraic structures, homomorphisms, relations, free monoid finite words, prefixes, suffixes, factors, palindromes
periodicity and Fine-Wilf theorem
infinite words are sequences over a finite alphabet
properties of an ultrametric distance, example of the p-adic norm
topology of the set of infinite words
converging sequences of infinite and finite words, compactness argument
iterated morphism, coding, substitutive or morphic words
the typical example of the Thue-Morse word
the Fibonacci word, the Mex operator, the n-bonacci words
wordscomingfromnumbertheory(baseexpansions,continuedfractions,...) the taxonomy of Lindenmayer systems
S-adic sequences, Kolakoski word
repetition in words, avoiding repetition, repetition threshold
(complete) de Bruijn graphs
concepts from computability theory and decidability issues
Post correspondence problem and application to mortality of matrices
origins of combinatorics on words
bibliographic notes
languages of finite words, regular languages
factorial, prefix/suffix closed languages, trees and codes
unambiguous and deterministic automata, Kleene's theorem
growth function of regular languages
non-deterministic automata and determinization
radix order, first word of each length and decimation of a regular language
the theory of the minimal automata
an introduction to algebraic automata theory, the syntactic monoid and the
syntactic complexity
star-free languages and a theorem of Schu tzenberger
rational formal series and weighted automata
context-free languages, pushdown automata and grammars
growth function of context-free languages, Parikh's theorem
some decidable and undecidable problems in formal language theory
bibliographic notes
factor complexity, Morse-Hedlund theorem
arithmetic complexity, Van Der Waerden theorem, pattern complexity recurrence, uniform recurrence, return words
Sturmian words, coding of rotations, Kronecker's theorem
frequencies of letters, factors and primitive morphism
critical exponent
factor complexity of automatic sequences
factor complexity of purely morphic sequences
primitive words, conjugacy, Lyndon word
abelianisation and abelian complexity
bibliographic notes
automatic sequences, equivalent definitions
a theorem of Cobham, equivalence of automatic sequences with constant
length morphic sequences
a few examples of well-known automatic sequences
about Derksen's theorem
some morphic sequences are not automatic
abstract numeration system and S-automatic sequences
k . infinity -automatic sequences
bibliographic notes
numeration systems, greedy algorithm
positional numeration systems, reco

Produktdetails

• ISTE .1
• Verlag: Wiley & Sons
• 1. Auflage
• Seitenzahl: 338
• Erscheinungstermin: 17. November 2014
• Englisch
• Abmessung: 240mm x 161mm x 23mm
• Gewicht: 674g
• ISBN-13: 9781848216150
• ISBN-10: 1848216157
• Artikelnr.: 40546087

Autorenporträt

Michel RIGO, Full professor, University of Liège, Department of Math., Belgium.

Inhaltsangabe

FOREWORD ix INTRODUCTION xiii CHAPTER 1. WORDS AND SEQUENCES FROM SCRATCH 1
1.1. Mathematical background and notation 2 1.1.1. About asymptotics 4
1.1.2. Algebraic number theory 5 1.2. Structures, words and languages 11
1.2.1. Distance and topology 16 1.2.2. Formal series 24 1.2.3. Language,
factor and frequency 28 1.2.4. Period and factor complexity 33 1.3.
Examples of infinite words 36 1.3.1. About cellular automata 43 1.3.2.
Links with symbolic dynamical systems 46 1.3.3. Shift and orbit closure 59
1.3.4. First encounter with ²-expansions 62 1.3.5. Continued fractions 69
1.3.6. Direct product, block coding and exercises 70 1.4. Bibliographic
notes and comments 77 CHAPTER 2. MORPHIC WORDS 85 2.1. Formal definitions
89 2.2. Parikh vectors and matrices associated with a morphism 96 2.2.1.
The matrix associated with a morphism 98 2.2.2. The tribonacci word 99 2.3.
Constant-length morphisms 107 2.3.1. Closure properties 117 2.3.2. Kernel
of a sequence 119 2.3.3. Connections with cellular automata 120 2.4.
Primitive morphisms 122 2.4.1. Asymptotic behavior 127 2.4.2. Frequencies
and occurrences of factors 127 2.5. Arbitrary morphisms 133 2.5.1.
Irreducible matrices 134 2.5.2. Cyclic structure of irreducible matrices
144 2.5.3. Proof of theorem 2.35 150 2.6. Factor complexity and Sturmian
words 153 2.7. Exercises 159 2.8. Bibliographic notes and comments 163
CHAPTER 3. MORE MATERIAL ON INFINITE WORDS 173 3.1. Getting rid of erasing
morphisms 174 3.2. Recurrence 185 3.3. More examples of infinite words 191
3.4. Factor Graphs and special factors 202 3.4.1. de Bruijn graphs 202
3.4.2. Rauzy graphs 206 3.5. From the Thue-Morse word to pattern avoidance
219 3.6. Other combinatorial complexity measures 228 3.6.1. Abelian
complexity 228 3.6.2. k-Abelian complexity 237 3.6.3. k-Binomial complexity
245 3.6.4. Arithmetical complexity 249 3.6.5. Pattern complexity 251 3.7.
Bibliographic notes and comments 252 BIBLIOGRAPHY 257 INDEX 295 SUMMARY OF
VOLUME 2 303