Formal Languages, Automata and Numeration Systems 1
Introduction to Combinatorics on Words
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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 regul...
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
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
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