References of "Rigo, Michel"
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See detailMultidimensional generalized automatic sequences and shape-symmetric morphic words
Charlier, Emilie ULg; Kärki, Tomi; Rigo, Michel ULg

in Proceedings of AutoMathA (2009)

An infinite word is S-automatic if, for all n ≥ 0, its (n + 1)st letter is the output of a deterministic automaton fed with the representation of n in the considered numeration system S. In this paper, we ... [more ▼]

An infinite word is S-automatic if, for all n ≥ 0, its (n + 1)st letter is the output of a deterministic automaton fed with the representation of n in the considered numeration system S. In this paper, we consider an analogous definition in a multidimensional setting and study the relationship with the shape-symmetric infinite words as introduced by Arnaud Maes. Precisely, for d ≥ 2, we show that a multidimensional infinite word x : N^d → Σ over a finite alphabet Σ is S-automatic for some abstract numeration system S built on a regular language containing the empty word if and only if x is the image by a coding of a shape-symmetric infinite word. [less ▲]

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See detailOn the Recognizability of Self-Generating Sets
Kärki, Tomi ULg; Lacroix, Anne ULg; Rigo, Michel ULg

in Lecture Notes in Computer Science (2009), 5734

Let I be a finite set of integers and F be a finite set of maps of the form n->k_i n + l_i with integer coefficients. For an integer base k>=2, we study the k-recognizability of the minimal set X of ... [more ▼]

Let I be a finite set of integers and F be a finite set of maps of the form n->k_i n + l_i with integer coefficients. For an integer base k>=2, we study the k-recognizability of the minimal set X of integers containing I and satisfying f(X)\subseteq X for all f in F. In particular, solving a conjecture of Allouche, Shallit and Skordev, we show under some technical conditions that if two of the constants k_i are multiplicatively independent, then X is not k-recognizable for any k>=2. [less ▲]

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See detailA Decision Problem for Ultimately Periodic Sets in Non-standard Numeration Systems
Bell, Jason; Charlier, Emilie ULg; Fraenkel, Aviezri et al

in International Journal of Algebra & Computation (2009), 19

Consider a non-standard numeration system like the one built over the Fibonacci sequence where nonnegative integers are represented by words over {0,1} without two consecutive 1. Given a set X of integers ... [more ▼]

Consider a non-standard numeration system like the one built over the Fibonacci sequence where nonnegative integers are represented by words over {0,1} without two consecutive 1. Given a set X of integers such that the language of their greedy representations in this system is accepted by a finite automaton, we consider the problem of deciding whether or not X is a finite union of arithmetic progressions. We obtain a decision procedure for this problem, under some hypothesis about the considered numeration system. In a second part, we obtain an analogous decision result for a particular class of abstract numeration systems built on an infinite regular language. [less ▲]

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See detailSyndeticity and independent substitutions
Durand, Fabien; Rigo, Michel ULg

in Advances in Applied Mathematics (2009), 42

We associate in a canonical way a substitution to any abstract numeration system built on a regular language. In relationship with the growth order of the letters, we de ne the notion of two independent ... [more ▼]

We associate in a canonical way a substitution to any abstract numeration system built on a regular language. In relationship with the growth order of the letters, we de ne the notion of two independent substitutions. Our main result is the following. If a sequence x is generated by two independent substitutions, at least one being of exponential growth, then the factors of x appearing in nitely often in x appear with bounded gaps. As an application, we derive an analogue of Cobham's theorem for two independent substitutions (or abstract numeration systems) one with polynomial growth, the other being exponential. [less ▲]

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See detailLes codes correcteurs
Rigo, Michel ULg

Learning material (2009)

Qu'on le veuille ou non, nous vivons dans un monde où les informations et les moyens de communication sont omniprésents. Les exemples où se mêlent informatique et télécommunications sont nombreux. Que ce ... [more ▼]

Qu'on le veuille ou non, nous vivons dans un monde où les informations et les moyens de communication sont omniprésents. Les exemples où se mêlent informatique et télécommunications sont nombreux. Que ce soit l'Internet, la musique achetée en ligne ou gravée sur CD, les baladeurs de type iPod et autres appareils portables, les supports mémoire de tout type, la photographie numérique, la présence d'ordinateurs de bord sophistiqués dans nos voitures, etc. Sans code correcteur, le CD (implémentant un code correcteur de type Reed-Solomon) et de nombreux autres produits n'auraient probablement jamais connu le succès ! [less ▲]

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See detailAbstract numeration systems on bounded languages and multiplication by a constant
Charlier, Emilie ULg; Rigo, Michel ULg; Steiner, Wolfgang

in INTEGERS: Electronic Journal of Combinatorial Number Theory (2008), 8(1-A35), 1-19

A set of integers is $S$-recognizable in an abstract numeration system $S$ if the language made up of the representations of its elements is accepted by a finite automaton. For abstract numeration systems ... [more ▼]

A set of integers is $S$-recognizable in an abstract numeration system $S$ if the language made up of the representations of its elements is accepted by a finite automaton. For abstract numeration systems built over bounded languages with at least three letters, we show that multiplication by an integer $\lambda\ge2$ does not preserve $S$-recognizability, meaning that there always exists a $S$-recognizable set $X$ such that $\lambda X$ is not $S$-recognizable. The main tool is a bijection between the representation of an integer over a bounded language and its decomposition as a sum of binomial coefficients with certain properties, the so-called combinatorial numeration system. [less ▲]

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See detailA decision problem for ultimately periodic sets in non-standard numeration systems
Charlier, Emilie ULg; Rigo, Michel ULg

in Actes des Journées Montoises d'Informatique Théorique (2008)

Consider a non-standard numeration system like the one built over the Fibonacci sequence where nonnegative integers are represented by words over {0, 1} without two consecutive 1. Given a set X of ... [more ▼]

Consider a non-standard numeration system like the one built over the Fibonacci sequence where nonnegative integers are represented by words over {0, 1} without two consecutive 1. Given a set X of integers such that the language of their greedy representations in this system is accepted by a finite automaton, we consider the problem of deciding whether or not X is a finite union of arithmetic progressions. We obtain a decision procedure under some hypothesis about the considered numeration system. In a second part, we obtain an analogous decision result for a particular class of abstract numeration systems built on an infinite regular language. [less ▲]

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See detailCombinatorics, Automata and Number Theory 2006
Berthé, Valérie; Lecomte, Pierre ULg; Rigo, Michel ULg

in Theoretical Computer Science (2008), 391

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See detailCubic Pisot Unit Combinatorial Games
Duchêne, Eric; Rigo, Michel ULg

in Monatshefte für Mathematik (2008), 155

Generalized Tribonacci morphisms are de ned on a three letters alphabet and generate the so-called generalized Tribonacci words. We present a family of combinatorial removal games on three piles of tokens ... [more ▼]

Generalized Tribonacci morphisms are de ned on a three letters alphabet and generate the so-called generalized Tribonacci words. We present a family of combinatorial removal games on three piles of tokens whose set of P-positions is coded exactly by these generalized Tribonacci words. To obtain this result, we study combinatorial properties of these words like gaps between consecutive identical letters or recursive de nitions of these words. Beta-numeration systems are then used to show that these games are tractable, i.e., deciding whether a position is a P-position can be checked in polynomial time. [less ▲]

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See detailA morphic approach to combinatorial games : the Tribonacci case
Duchêne, Eric; Rigo, Michel ULg

in Theoretical Informatics and Applications. Informatique Théorique et Applications (2008), 42

We propose a variation of Wythoff's game on three piles of tokens, in the sense that the losing positions can be derived from the Tribonacci word instead of the Fibonacci word for the two piles game ... [more ▼]

We propose a variation of Wythoff's game on three piles of tokens, in the sense that the losing positions can be derived from the Tribonacci word instead of the Fibonacci word for the two piles game. Thanks to the corresponding exotic numeration system built on the Tribonacci sequence, deciding whether a game position is losing or not can be computed in polynomial time. [less ▲]

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See detailSyntactictal and automatic properties of sets of polynomials over finite fields
Rigo, Michel ULg

in Finite Fields & Their Applications (2008), 42

Syntactical properties of representations of integers in various number systems are well-known and have been extensively studied. In this paper, we transpose the notion of recognizable set of integers ... [more ▼]

Syntactical properties of representations of integers in various number systems are well-known and have been extensively studied. In this paper, we transpose the notion of recognizable set of integers into the framework of the polynomial ring over a finite field F. We define B-recognizable sets of polynomials over F and consider their first properties. [less ▲]

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See detailA Decision Problem for Ultimately Periodic Sets in Non-standard Numeration Systems
Charlier, Emilie ULg; Rigo, Michel ULg

in Lecture Notes in Computer Science (2008), 5162

Consider a non-standard numeration system like the one built over the Fibonacci sequence where nonnegative integers are represented by words over {0, 1} without two consecutive 1. Given a set X of ... [more ▼]

Consider a non-standard numeration system like the one built over the Fibonacci sequence where nonnegative integers are represented by words over {0, 1} without two consecutive 1. Given a set X of integers such that the language of their greedy representations in this system is accepted by a finite automaton, we consider the problem of deciding whether or not X is a finite union of arithmetic progressions. We obtain a decision procedure under some hypothesis about the considered numeration system. In a second part, we obtain an analogous decision result for a particular class of abstract numeration systems built on an infinite regular language. [less ▲]

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See detailLa matrice cachée de Google
Rigo, Michel ULg

Learning material (2008)

Inutile de le présenter : Google est le moteur de recherche le plus connu et le plus utilisé au monde. Mais comment les concepteurs de Google font-ils pour classer les milliers de pages se rapportant à un ... [more ▼]

Inutile de le présenter : Google est le moteur de recherche le plus connu et le plus utilisé au monde. Mais comment les concepteurs de Google font-ils pour classer les milliers de pages se rapportant à un mot-clé donné, de façon telle que les pages les plus représentatives occupent toujours les premières positions du classement ? Ce tour de force repose sur de véritables résultats mathématiques combinant théorie des graphes et algèbre linéaire. [less ▲]

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See detailAbout frequencies of letters in generalized automatic sequences
Nicolay, Samuel ULg; Rigo, Michel ULg

in Theoretical Computer Science (2007), 374(1-3), 25-40

We present some asymptotic results about the frequency of a letter appearing in a generalized unidimensional automatic sequence. Next, we study multidimensional generalized automatic sequences and the ... [more ▼]

We present some asymptotic results about the frequency of a letter appearing in a generalized unidimensional automatic sequence. Next, we study multidimensional generalized automatic sequences and the corresponding frequencies. (C) 2006 Elsevier B.V. All rights reserved. [less ▲]

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See detailDistribution of additive functions with respect to numeration systems on regular languages
Grabner, Peter J.; Rigo, Michel ULg

in Theory of Computing Systems (2007), 40(3), 205-223

We study the distribution of values of additive functions related to numeration systems defined via regular languages.

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See detailOdometers on regular languages
Berthé; Rigo, Michel ULg

in Theory of Computing Systems (2007), 40(1), 1-31

Odometers or "adding machines" are usually introduced in the context of positional numeration systems built on a strictly increasing sequence of integers. We generalize this notion to systems defined on ... [more ▼]

Odometers or "adding machines" are usually introduced in the context of positional numeration systems built on a strictly increasing sequence of integers. We generalize this notion to systems defined on an arbitrary infinite regular language. In this latter situation, if (A, <) is a totally ordered alphabet, then enumerating the words of a regular language L over A with respect to the induced genealogical ordering gives a one-to-one correspondence between N and L. In this general setting the odometer is not defined on a set of sequences of digits but on a set of pairs of sequences where the first (resp. the second) component of the pair is an infinite word over A (resp. an infinite sequence of states of the minimal automaton of L). We study some properties of the odometer such as continuity, injectivity, surjectivity, minimality,... We then study some particular cases: we show the equivalence of this new function with the classical odometer built upon a sequence of integers whenever the set of greedy representations of all the integers is a regular language; we also consider substitution numeration systems as well as the connection with beta-numerations. [less ▲]

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See detailspecial issue dedicated to the tenth ``Journées montoises d'informatique théorique''
Bruyère, Véronique; Rigo, Michel ULg

in Discrete Mathematics & Theoretical Computer Science (2007), 9(2), 1

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See detailPirates informatiques et mathématique modulaire
Rigo, Michel ULg

Learning material (2007)

De nos jours, l'envoi de messages secrets requiert la manipulation de nombres ayant plus de cent chiffres décimaux. Nous illustrons une technique cryptographique standard (le RSA) dont la sécurité réside ... [more ▼]

De nos jours, l'envoi de messages secrets requiert la manipulation de nombres ayant plus de cent chiffres décimaux. Nous illustrons une technique cryptographique standard (le RSA) dont la sécurité réside dans le fait qu'il est "rapide", sur le plus banal des ordinateurs personnels, de calculer le produit de deux "grands" nombres (cela se compte au pire en secondes), alors que le temps nécessaire pour effectuer l'opération inverse de factorisation prend, dans l'état actuel des connaissances mathématiques, énormément plus de temps (que l'on pourrait estimer en milliards d'années même pour un super-calculateur !). [less ▲]

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See detailOn the cost and complexity of the successor function
Berthé, Valérie; Frougny, Christiane; Rigo, Michel ULg et al

in Arnoux, P.; Bédaride, N. (Eds.) Proceedings of WORDS 2007 (2007)

For a given numeration system, the successor function maps the representation of an integer n onto the representation of its successor n+1. In a general setting, the successor function maps the n-th word ... [more ▼]

For a given numeration system, the successor function maps the representation of an integer n onto the representation of its successor n+1. In a general setting, the successor function maps the n-th word of a genealogically ordered language L onto the (n+1)-th word of L. We show that, if the ratio of the number of elements of length n + 1 over the number of elements of length n of the language has a limit b> 1, then the amortized cost of the successor function is equal to b/(b − 1). From this, we deduce the value of the amortized cost for several classes of numeration systems (integer base systems, canonical numeration systems associated with a Parry number, abstract numeration systems built on a rational language, and rational base numeration systems). [less ▲]

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See detailSyntactictal and automatic properties of sets of polynomials over finite fields
Rigo, Michel ULg

Conference (2006, October)

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