Formal Languages, Automata and Numeration Systems : Introduction to Combinatorics on Words.
Title:
Formal Languages, Automata and Numeration Systems : Introduction to Combinatorics on Words.
Author:
Rigo, Michel.
ISBN:
9781119008217
Personal Author:
Edition:
1st ed.
Physical Description:
1 online resource (338 pages)
Series:
Networks and Telecommunications Series
Contents:
Cover page -- Half-Title page -- Title page -- Copyright page -- Contents -- Foreword -- Introduction -- I.1. What this book is or is not about -- I.2. A few words about what you will find -- I.3. How to read this book -- I.4. Acknowledgments -- 1: Words and Sequences from Scratch -- 1.1. Mathematical background and notation -- 1.1.1. About asymptotics -- 1.1.2. Algebraic number theory -- 1.2. Structures, words and languages -- 1.2.1. Distance and topology -- 1.2.2. Formal series -- 1.2.3. Language, factor and frequency -- 1.2.4. Period and factor complexity -- 1.3. Examples of infinite words -- 1.3.1. About cellular automata -- 1.3.2. Links with symbolic dynamical systems -- 1.3.3. Shift and orbit closure -- 1.3.4. First encounter with β-expansions -- 1.3.5. Continued fractions -- 1.3.6. Direct product, block coding and exercises -- 1.4. Bibliographic notes and comments -- 2: Morphic Words -- 2.1. Formal definitions -- 2.2. Parikh vectors and matrices associated with a morphism -- 2.2.1. The matrix associated with a morphism -- 2.2.2. The tribonacci word -- 2.3. Constant-length morphisms -- 2.3.1. Closure properties -- 2.3.2. Kernel of a sequence -- 2.3.3. Connections with cellular automata -- 2.4. Primitive morphisms -- 2.4.1. Asymptotic behavior -- 2.4.2. Frequencies and occurrences of factors -- 2.5. Arbitrary morphisms -- 2.5.1. Irreducible matrices -- 2.5.2. Cyclic structure of irreducible matrices -- 2.5.3. Proof of theorem 2.35 -- 2.6. Factor complexity and Sturmian words -- 2.7. Exercises -- 2.8. Bibliographic notes and comments -- 3: More Material on Infinite Words -- 3.1. Getting rid of erasing morphisms -- 3.2. Recurrence -- 3.3. More examples of infinite words -- 3.4. Factor Graphs and special factors -- 3.4.1. de Bruijn graphs -- 3.4.2. Rauzy graphs -- 3.5. From the Thue-Morse word to pattern avoidance.
3.6. Other combinatorial complexity measures -- 3.6.1. Abelian complexity -- 3.6.2. k-Abelian complexity -- 3.6.3. k-Binomial complexity -- 3.6.4. Arithmetical complexity -- 3.6.5. Pattern complexity -- 3.7. Bibliographic notes and comments -- Bibliography -- Index -- Volume 2 - Contents -- Volume 2 - Index.
Abstract:
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 se.
Local Note:
Electronic reproduction. Ann Arbor, Michigan : ProQuest Ebook Central, 2017. Available via World Wide Web. Access may be limited to ProQuest Ebook Central affiliated libraries.
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Electronic Access:
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