References of "Rigo, Michel"
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See detailOn Cobham's theorem
Durand, Fabien; Rigo, Michel ULg

in Handbook of Automata (in press)

Let k >= 2 be an integer. A set X of integers is k-recognizable if the language of k-ary representations of the elements in X is accepted by a finite automaton. The celebrated theorem of Cobham from 1969 ... [more ▼]

Let k >= 2 be an integer. A set X of integers is k-recognizable if the language of k-ary representations of the elements in X is accepted by a finite automaton. The celebrated theorem of Cobham from 1969 states that if a set of integers is both k-recognizable and ℓ-recognizable, then it is a finite union of arithmetic progressions. We present several extensions of this result to nonstandard numeration systems, we describe the relationships with substitutive and automatic words and list Cobham-type results in various contexts. [less ▲]

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See detailAsymptotic properties of free monoid morphisms
Charlier, Emilie ULg; Leroy, Julien ULg; Rigo, Michel ULg

in Linear Algebra & its Applications (2016), 500

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See detailGeneralized Pascal triangle for binomial coefficients of finite words
Stipulanti, Manon ULg; Leroy, Julien ULg; Rigo, Michel ULg

Poster (2016, April 05)

Abstract. We introduce a generalization of Pascal triangle based on bino- mial coefficients of finite words. These coefficients count the number of times a word appears as a subsequence of another finite ... [more ▼]

Abstract. We introduce a generalization of Pascal triangle based on bino- mial coefficients of finite words. These coefficients count the number of times a word appears as a subsequence of another finite word. Similarly to the Sierpinski gasket that can be built as the limit set, for the Hausdorff distance, of a convergent sequence of normalized compact blocks extracted from Pascal triangle modulo 2, we describe and study the first properties of the subset of [0, 1] × [0, 1] associated with this extended Pascal triangle modulo a prime p. [less ▲]

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See detailUne généralisation du triangle de Pascal
Stipulanti, Manon ULg; Rigo, Michel ULg; Leroy, Julien ULg

Conference (2016, March 22)

We introduce a generalization of Pascal triangle based on binomial coefficients of finite words. These coefficients count the number of times a word appears as a subsequence of another finite word ... [more ▼]

We introduce a generalization of Pascal triangle based on binomial coefficients of finite words. These coefficients count the number of times a word appears as a subsequence of another finite word. Similarly to the Sierpinski gasket that can be built as the limit set, for the Hausdorff distance, of a convergent sequence of normalized compact blocks extracted from Pascal triangle modulo 2, we describe and study the first properties of the subset of [0, 1]×[0, 1] associated with this extended Pascal triangle modulo a prime p. [less ▲]

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See detailGeneralized Pascal triangle for binomial coefficients of words
Leroy, Julien ULg; Rigo, Michel ULg; Stipulanti, Manon ULg

in Advances in Applied Mathematics (2016), 80

We introduce a generalization of Pascal triangle based on binomial coefficients of finite words. These coefficients count the number of times a word appears as a subsequence of another finite word ... [more ▼]

We introduce a generalization of Pascal triangle based on binomial coefficients of finite words. These coefficients count the number of times a word appears as a subsequence of another finite word. Similarly to the Sierpiński gasket that can be built as the limit set, for the Hausdorff distance, of a convergent sequence of normalized compact blocks extracted from Pascal triangle modulo 2, we describe and study the first properties of the subset of [0, 1] × [0, 1] associated with this extended Pascal triangle modulo a prime p. [less ▲]

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See detailDefining multiplication in some additive expansions of polynomial rings
Point, Françoise; Rigo, Michel ULg; Waxweiler, Laurent

in Communications in Algebra (2016), 44

Adapting a result of R. Villemaire on expansions of Presburger arithmetic, we show how to define multiplication in some expansions of the additive reduct of certain Euclidean rings. In particular, this ... [more ▼]

Adapting a result of R. Villemaire on expansions of Presburger arithmetic, we show how to define multiplication in some expansions of the additive reduct of certain Euclidean rings. In particular, this applies to polynomial rings over a finite field. [less ▲]

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See detailCombinatorics, Words and Symbolic Dynamics
Berthé, Valérie; Rigo, Michel ULg

Book published by Cambridge University Press (2016)

Internationally recognised researchers look at developing trends in combinatorics with applications in the study of words and in symbolic dynamics. They explain the important concepts, providing a clear ... [more ▼]

Internationally recognised researchers look at developing trends in combinatorics with applications in the study of words and in symbolic dynamics. They explain the important concepts, providing a clear exposition of some recent results, and emphasise the emerging connections between these different fields. Topics include combinatorics on words, pattern avoidance, graph theory, tilings and theory of computation, multidimensional subshifts, discrete dynamical systems, ergodic theory, numeration systems, dynamical arithmetics, automata theory and synchronised words, analytic combinatorics, continued fractions and probabilistic models. Each topic is presented in a way that links it to the main themes, but then they are also extended to repetitions in words, similarity relations, cellular automata, friezes and Dynkin diagrams. The book will appeal to graduate students, research mathematicians and computer scientists working in combinatorics, theory of computation, number theory, symbolic dynamics, tilings and stringology. It will also interest biologists using text algorithms. [less ▲]

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See detailJouer avec les mots, pourquoi et comment ?
Rigo, Michel ULg

Learning material (2016)

Ce texte reprend l'essentiel de ma présentation à la Brussels Math. Summer School du 4 août 2015. Il s'agit d'une courte introduction à la combinatoire des mots. A l'instar de Raymond Queneau et ses cent ... [more ▼]

Ce texte reprend l'essentiel de ma présentation à la Brussels Math. Summer School du 4 août 2015. Il s'agit d'une courte introduction à la combinatoire des mots. A l'instar de Raymond Queneau et ses cent mille milliards de poèmes, nous construisons des suites aux propriétés surprenantes. Pour ne pas allonger le texte, nous avons décidé d'éviter l'emploi d'automates finis. Les premiers résultats en combinatoire des mots remontent au début du siècle précédent, avec les travaux du mathématicien norvégien Axel Thue. Cette branche des mathématiques étudie la structure et les arrangements apparaissant au sein de suites finies, ou infinies, de symboles appartenant à un ensemble fini. Un carré est la juxtaposition de deux répétitions d'un même mot. On dira qu'un mot comme `taratata' contient un carré. Il est aisé de vérifier que, si on dispose uniquement de deux symboles, alors tout mot de longueur au moins 4 contient un carré. Cette observation amène de nombreuses questions simples à formuler : Avec trois symboles, peut-on construire un mot arbitrairement long ne contenant pas de carré ? Si on se limite à deux symboles, peut-on construire un mot arbitrairement long sans cube, i.e., évitant la juxtaposition de trois répétitions d'un même mot ? En fonction de la taille de l'alphabet, quels motifs doivent nécessairement apparaître et quels sont ceux qui sont évitables ? Que se passe-t-il si on autorise certaines permutations ? [less ▲]

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See detailComputing k-binomial equivalence and avoiding binomial repetitions
Rigo, Michel ULg

Conference (2015, October 28)

In this talk, I will first recall basic results on binomial coefficients of words, then review the connections and differences with Parikh matrices. As a generalization of abelian equivalence, two words u ... [more ▼]

In this talk, I will first recall basic results on binomial coefficients of words, then review the connections and differences with Parikh matrices. As a generalization of abelian equivalence, two words u and v are k-binomially equivalent if every word of length at most k appears as a subword of u exactly as many times as it appears as a subword of v. So a k-binomial square is a word uv where u and v are k-binomially equivalent. We will discuss avoidance of squares and cubes in infinite words (this is a joint word with M. Rao). Finally, I will consider the question of deciding whether or not two finite words are k-binomially equivalent. This problem has recently been shown to be decidable in polynomial time by Freydenberger, Gawrychowski et al. [less ▲]

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See detailUse of the wavelet theory as a tool to investigate the l-abelian complexity of a sequence
Kleyntssens, Thomas ULg; Nicolay, Samuel ULg; Vandomme, Elise ULg et al

Poster (2015, September 23)

The concept of k-automatic sequences is at the intersection of number theory and formal language theory. It has been generalized by the notion of k-regularity that allows to study sequences with values in ... [more ▼]

The concept of k-automatic sequences is at the intersection of number theory and formal language theory. It has been generalized by the notion of k-regularity that allows to study sequences with values in a (possibly infinite) ring. This concept provides us with structural information about how the different terms of the sequence are related to each other. They are many different notions related to the measure of complexity of an infinite sequence w. A classical approach is its factor complexity. In an abelian context, the analogue to the factor complexity is the abelian complexity where the number of distinct factors of length n is counted up to abelian equivalence. The notion of abelian complexity was extended to that of l-abelian complexity. In this talk, I propose to use tools from the wavelet theory to analyze the l-abelian complexity. For the numerical simulations, I apply the wavelet leaders method that allows to study the pointwise regularity of signals. [less ▲]

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See detailJouer avec les mots, pourquoi et comment ?
Rigo, Michel ULg

Scientific conference (2015, August 04)

A l'instar de Raymond Queneau et ses cent mille milliards de poèmes, cet exposé a pour but de compter et de construire des mots aux propriétés parfois surprenantes. Les premiers résultats en combinatoire ... [more ▼]

A l'instar de Raymond Queneau et ses cent mille milliards de poèmes, cet exposé a pour but de compter et de construire des mots aux propriétés parfois surprenantes. Les premiers résultats en combinatoire des mots remontent au début du siècle précédent, avec les travaux du mathématicien norvégien Axel Thue. Cette branche des mathématiques étudie la structure et les arrangements apparaissant au sein de suites finies, ou infinies, de symboles appartenant à un ensemble fini. Donnons un exemple rudimentaire. Un carré est la juxtaposition de deux répétitions d'un mot, ainsi "coco" ou "bonbon" sont des carrés. On dira alors qu'un mot comme "taratata" contient un carré. Il est aisé de vérifier que, si on dispose uniquement de deux symboles "a" et "b", alors tout mot de longueur au moins 4 contient un des carrés "aa", "bb", "abab" ou encore "baba". On dira donc que, sur deux symboles, les carrés sont inévitables. Cette observation pose des questions intéressantes et simples à formuler : Avec trois symboles, peut-on construire un mot arbitrairement long ne contenant pas de carré ? Si on se limite à deux symboles, peut-on construire un mot arbitrairement long sans cube, i.e., évitant la juxtaposition de trois répétitions d'un même mot ? En fonction de la taille de l'alphabet, quels motifs doivent nécessairement apparaître et quels sont ceux qui sont évitables ? Que se passe-t-il si on autorise certaines permutations ? etc. Dans cet exposé, on passera en revue quelques constructions simples de mots finis ou infinis : mot de Thue-Morse, mot de Fibonacci, mots Sturmiens. Nous montrerons aussi que les applications sont nombreuses : arithmétique, transcendance en théorie des nombres, informatique mathématique et théorie des automates, pavages du plan, dynamique symbolique et codage de rotations, infographie, géométrie discrète et représentation de segment de droites à l'écran, bio-informatique, ... [less ▲]

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See detailIs Büchi's theorem useful for you?
Rigo, Michel ULg

Conference (2015, May 28)

Almost a century ago, Presburger showed that the first order theory of the natural numbers with addition is decidable. Following the work of Büchi in 1960, this result still holds when adding a function ... [more ▼]

Almost a century ago, Presburger showed that the first order theory of the natural numbers with addition is decidable. Following the work of Büchi in 1960, this result still holds when adding a function $V_k$ to the structure, where $V_k(n)$ is the largest power of $k\ge 2$ diving $n$. In particular, this leads to a logical characterization of the $k$-automatic sequences. During the last few years, many applications of this result have been considered in combinatorics on words, mostly by J. Shallit and his coauthors. In this talk, we will present this theorem of Büchi where decidability relies on finite automata.Then we will review some results about automatic sequences or morphic words that can be proved automatically (i.e., the proof is carried on by an algorithm). Finally, we will sketch the limitation of this technique. With a single line formula, one can prove automatically that the Thue-Morse word has no overlap but, hopefully, not all the combinatorial properties of morphic words can be derived in this way. [less ▲]

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See detailInvariant games and non-homogeneous Beatty sequences
Rigo, Michel ULg

Scientific conference (2015, January 08)

The aim of this talk is to introduce some notions arising in combinatorial game theory and make the connection with combinatorics on words. We characterize all pairs of complementary non-homogenous Beatty ... [more ▼]

The aim of this talk is to introduce some notions arising in combinatorial game theory and make the connection with combinatorics on words. We characterize all pairs of complementary non-homogenous Beatty sequences (A_n)n≥0 and (B_n)n≥0 for which there exists an invariant game having exactly {(A_n,B_n)∣n≥0}∪{(B_n,A_n)∣n≥0} as set of P-positions. Using the notion of Sturmian word and tools arising in symbolic dynamics and combinatorics on words, this characterization can be translated to a decision procedure relying only on a few algebraic tests about algebraicity or rational independence. Given any four real numbers defining the two sequences, up to these tests, we can therefore decide whether or not such an invariant game exists. [less ▲]

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See detailAnother Generalization of Abelian Equivalence: Binomial Complexity of Infinite Words (long version)
Rigo, Michel ULg; Salimov, Pavel

in Theoretical Computer Science (2015), 601

The binomial coefficient of two words $u$ and $v$ is the number of times $v$ occurs as a subsequence of $u$. Based on this classical notion, we introduce the $m$-binomial equivalence of two words refining ... [more ▼]

The binomial coefficient of two words $u$ and $v$ is the number of times $v$ occurs as a subsequence of $u$. Based on this classical notion, we introduce the $m$-binomial equivalence of two words refining the abelian equivalence. Two words $x$ and $y$ are $m$-binomially equivalent, if, for all words $v$ of length at most $m$, the binomial coefficients of $x$ and $v$ and respectively, $y$ and $v$ are equal. The $m$-binomial complexity of an infinite word $x$ maps an integer $n$ to the number of $m$-binomial equivalence classes of factors of length $n$ occurring in $x$. We study the first properties of $m$-binomial equivalence. We compute the $m$-binomial complexity of two classes of words: Sturmian words and (pure) morphic words that are fixed points of Parikh-constant morphisms like the Thue--Morse word, i.e., images by the morphism of all the letters have the same Parikh vector. We prove that the frequency of each symbol of an infinite recurrent word with bounded $2$-binomial complexity is rational. [less ▲]

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See detailAn analogue of Cobham's theorem for graph directed iterated function systems
Charlier, Emilie ULg; Leroy, Julien ULg; Rigo, Michel ULg

in Advances in Mathematics (2015), 280

Feng and Wang showed that two homogeneous iterated function systems in $\mathbb{R}$ with multiplicatively independent contraction ratios necessarily have different attractors. In this paper, we extend ... [more ▼]

Feng and Wang showed that two homogeneous iterated function systems in $\mathbb{R}$ with multiplicatively independent contraction ratios necessarily have different attractors. In this paper, we extend this result to graph directed iterated function systems in $\mathbb{R}^n$ with contraction ratios that are of the form $\frac{1}{\beta}$, for integers $\beta$. By using a result of Boigelot {\em et al.}, this allows us to give a proof of a conjecture of Adamczewski and Bell. In doing so, we link the graph directed iterated function systems to Büchi automata. In particular, this link extends to real numbers $\beta$. We introduce a logical formalism that permits to characterize sets of $\mathbb{R}^n$ whose representations in base $\beta$ are recognized by some Büchi automata. This result depends on the algebraic properties of the base: $\beta$ being a Pisot or a Parry number. The main motivation of this work is to draw a general picture representing the different frameworks where an analogue of Cobham's theorem is known. [less ▲]

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See detailAvoiding 2-binomial squares and cubes
Rao, Michaël; Rigo, Michel ULg; Salimov, Pavel

in Theoretical Computer Science (2015), 572

Two finite words $u,v$ are $2$-binomially equivalent if, for all words $x$ of length at most $2$, the number of occurrences of $x$ as a (scattered) subword of $u$ is equal to the number of occurrences of ... [more ▼]

Two finite words $u,v$ are $2$-binomially equivalent if, for all words $x$ of length at most $2$, the number of occurrences of $x$ as a (scattered) subword of $u$ is equal to the number of occurrences of $x$ in $v$. This notion is a refinement of the usual abelian equivalence. A $2$-binomial square is a word $uv$ where $u$ and $v$ are $2$-binomially equivalent. In this paper, considering pure morphic words, we prove that $2$-binomial squares (resp. cubes) are avoidable over a $3$-letter (resp. $2$-letter) alphabet. The sizes of the alphabets are optimal. [less ▲]

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See detailA New Approach to the 2-Regularity of the ℓ-Abelian Complexity of 2-Automatic Sequences
Parreau, Aline; Rigo, Michel ULg; Rowland, Eric ULg et al

in The Electronic Journal of Combinatorics (2015), 22(1), 127

We prove that a sequence satisfying a certain symmetry property is 2-regular in the sense of Allouche and Shallit, i.e., the Z-module generated by its 2-kernel is finitely generated. We apply this theorem ... [more ▼]

We prove that a sequence satisfying a certain symmetry property is 2-regular in the sense of Allouche and Shallit, i.e., the Z-module generated by its 2-kernel is finitely generated. We apply this theorem to develop a general approach for studying the l-abelian complexity of 2-automatic sequences. In particular, we prove that the period-doubling word and the Thue--Morse word have 2-abelian complexity sequences that are 2-regular. Along the way, we also prove that the 2-block codings of these two words have 1-abelian complexity sequences that are 2-regular. [less ▲]

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See detailFormal languages, automata and numeration systems, volume 1: Introduction to combinatorics on words
Rigo, Michel ULg

Book published by ISTE-Wiley (2014)

The goal is not to have an encyclopedic presentation of the subject, but to familiarize the reader with a series of selected selected topics on words (words, morphisms, factor complexity, Sturmian words ... [more ▼]

The goal is not to have an encyclopedic presentation of the subject, but to familiarize the reader with a series of selected selected topics on words (words, morphisms, factor complexity, Sturmian words, ...). The philosophy is to rigorously present the concepts being illustrated with many examples (particularly in relations to numeration systems or symbolic dynamics). The reader should be able to quickly gain access to current research problems or attend a conference on the subject. Interactions between combinatorics, arithmetic and automata theory are also highlighted. The book requires little (or no) prerequisites and thus should be accessible to a wide audience (computer scientists/mathematicians, at Master/graduate level). The first volume can be used for a course in one semester in combinatorics of words (e.g. I give regularly the first two chapters to read to my students, the last one serving as complement for the 'advanced' students). [less ▲]

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See detailA new approach to the 2-regularity of the ℓ-abelian complexity of 2-automatic sequences (extended abstract)
Parreau, Aline; Rigo, Michel ULg; Rowland, Eric ULg et al

Conference (2014, September)

We show that a sequence satisfying a certain symmetry property is 2-regular in the sense of Allouche and Shallit. We apply this theorem to develop a general approach for studying the ℓ-abelian complexity ... [more ▼]

We show that a sequence satisfying a certain symmetry property is 2-regular in the sense of Allouche and Shallit. We apply this theorem to develop a general approach for studying the ℓ-abelian complexity of 2-automatic sequences. In particular, we prove that the period-doubling word and the Thue–Morse word have 2-abelian complexity sequences that are 2-regular. Along the way, we also prove that the 2-block codings of these two words have 1-abelian complexity sequences that are 2-regular. [less ▲]

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See detailA conjecture on the 2-abelian complexity of the Thue-Morse word
Rigo, Michel ULg; Parreau, Aline ULg; Vandomme, Elise ULg

Conference (2014, January 20)

The Thue-Morse word is a well-known and extensively studied 2-automatic sequence. For example, it is trivially abelian periodic and its abelian complexity takes only two values. For an integer k, the k ... [more ▼]

The Thue-Morse word is a well-known and extensively studied 2-automatic sequence. For example, it is trivially abelian periodic and its abelian complexity takes only two values. For an integer k, the k-abelian complexity is a generalization of the abelian complexity, corresponding to the case where k=1. Formally, two words u and v of the same length are k-abelian equivalent if they have the same prefix (resp. suffix) of length k-1 and if, for all words x of length k, the numbers of occurrences of x in u and v are the same. This notion has received some recent interest, see the works of Karhumäki et al. The k-abelian complexity of an infinite word x maps an integer n to the number of k-abelian classes partitioning the set of factors of length n occurring in x. The aim of this talk is to study the 2-abelian complexity a(n) of the Thue-Morse word. We conjecture that a(n) is 2-regular in the sense of Allouche and Shallit. This question can be related to a work of Madill and Rampersad (2012) where the (1)-abelian complexity of the paper folding word is shown to be 2-regular. We will present some arguments supporting our conjecture. They are based on functions counting some subword of length 2 occuring in prefixes of the Thue-Morse word. [less ▲]

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