This collaborative volume presents recent trends arising from the fruitful interaction between the themes of combinatorics on words, automata and formal language theory, and number theory. Presenting several important tools and concepts, the authors also reveal some of the exciting and important relationships that exist between these different fields. Topics include numeration systems, word complexity function, morphic words, Rauzy tilings and substitutive dynamical systems, Bratelli diagrams, frequencies and ergodicity, Diophantine approximation and transcendence, asymptotic properties of digital functions, decidability issues for D0L systems, matrix products and joint spectral radius. Topics are presented in a way that links them to the three main themes, but also extends them to dynamical systems and ergodic theory, fractals, tilings and spectral properties of matrices. Graduate students, research mathematicians and computer scientists working in combinatorics, theory of computation, number theory, symbolic dynamics, fractals, tilings and stringology will find much of interest in this book.

/pdf/combinatorics-automata-and-number-theory-54vcmxgyks.pdf

Combinatorics, Automata and Number Theory

Finite presentations of infinite structures The model theory of finite structures is intimately connected to various fields in computer science, including complexity theory, databases, and verification. In particular, there is a close relationship between complexity classes and the expressive power of logical languages, as witnessed by the fundamental theorems of descriptive complexity theory, such as Fagin's Theorem and the Immerman-Vardi Theorem (see [78, Chapter 3] for a survey). However, for many applications, the strict limitation to finite structures has turned out to be too restrictive, and there have been considerable efforts to extend the relevant logical and algorithmic methodologies from finite structures to suitable classes of infinite ones. In particular this is the case for databases and verification where infinite structures are of crucial importance [130]. Algorithmic model theory aims to extend in a systematic fashion the approach and methods of finite model theory, and its interactions with computer science, from finite structures to finitely-presentable infinite ones. There are many possibilities to present infinite structures in a finite manner. A classical approach in model theory concerns the class of computable structures ; these are countable structures, on the domain of natural numbers, say, with a finite collection of computable functions and relations. Such structures can be finitely presented by a collection of algorithms, and they have been intensively studied in model theory since the 1960s. However, from the point of view of algorithmic model theory the class of computable structures is problematic.

/pdf/automata-based-presentations-of-infinite-structures-hbdebtpnuq.pdf

Automata-based presentations of infinite structures

We prove that the uniform recurrence of morphic sequences is decidable. For this we show that the number of derived sequences of uniformly recurrent morphic sequences is bounded. As a corollary we obtain that uniformly recurrent morphic sequences are primitive substitutive sequences.

Decidability of uniform recurrence of morphic sequences

https://hal.archives-ouvertes.fr/hal-00407667/document

Syndeticity and independent substitutions

Counting the number of non-zero coefficients in rows of generalized Pascal triangles

We give some generalizations of k-automatic sequences replacing the k-ary system by an abstract numeration system on a regular language. We study some of the closure properties of these sequences and the possible extension to the multidimensional case or to infinite alphabets. The equivalence of these sequences and morphic predicates is given and the relationship to recognizability is also investigated.

More on generalized automatic sequences

An Automata Theoretic Decidability Proof for First-Order Theory of ⟨ℕ,<,P⟩ with Morphic Predicate P.

Morphisms and almost-periodicity

We construct an infinite family of brunnian links whose projections give the family of Venn diagrams for many sets constructed by Edwards.

A family of brunnian links based on Edwards' construction of Venn diagrams

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More on generalized automatic sequences

An Automata Theoretic Decidability Proof for First-Order Theory of ⟨ℕ,<,P⟩ with Morphic Predicate P.

Morphisms and almost-periodicity

A family of brunnian links based on Edwards' construction of Venn diagrams