Publications of Michel Rigo     Results 21-40 of 120.   1 2 3 4 5 6   Use of the wavelet theory as a tool to investigate the l-abelian complexity of a sequenceKleyntssens, Thomas ; Nicolay, Samuel ; Vandomme, Elise et alPoster (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 ▲]Detailed reference viewed: 31 (11 ULiège) Jouer avec les mots, pourquoi et comment ?Rigo, Michel 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 ▲]Detailed reference viewed: 85 (10 ULiège) Is Büchi's theorem useful for you?Rigo, Michel 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 ▲]Detailed reference viewed: 71 (5 ULiège) Invariant games and non-homogeneous Beatty sequencesRigo, Michel 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 ▲]Detailed reference viewed: 26 (5 ULiège) Another Generalization of Abelian Equivalence: Binomial Complexity of Infinite Words (long version)Rigo, Michel ; Salimov, Pavelin Theoretical Computer Science (2015), 601The 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 ▲]Detailed reference viewed: 39 (3 ULiège) An analogue of Cobham's theorem for graph directed iterated function systemsCharlier, Emilie ; Leroy, Julien ; Rigo, Michel in Advances in Mathematics (2015), 280Feng 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 ▲]Detailed reference viewed: 59 (23 ULiège) A New Approach to the 2-Regularity of the ℓ-Abelian Complexity of 2-Automatic SequencesParreau, Aline; Rigo, Michel ; Rowland, Eric et alin The Electronic Journal of Combinatorics (2015), 22(1), 127We 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 ▲]Detailed reference viewed: 63 (18 ULiège) Avoiding 2-binomial squares and cubesRao, Michaël; Rigo, Michel ; Salimov, Pavelin Theoretical Computer Science (2015), 572Two 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 ▲]Detailed reference viewed: 22 (7 ULiège) FaceBook aime les maths !Rigo, Michel Conference (2014, November 06)Nos recherches réalisées sur le Net, les sites que nous visitons, nos habitudes, notre profil Facebook, le contenu de nos chats ou tweets sont autant de données collectées en très grand nombre. On estime ... [more ▼]Nos recherches réalisées sur le Net, les sites que nous visitons, nos habitudes, notre profil Facebook, le contenu de nos chats ou tweets sont autant de données collectées en très grand nombre. On estime que le marché des données pèsera près de 1000 milliards de dollars en 2020. Les applications sont nombreuses : publicité ciblée, développement de nouveaux services et produits, etc. Ainsi, à l'heure du "big-data", Google prédit la propagation de la grippe en fonction des requêtes reçues, FICO anticipe le fait que vous preniez vos médicaments à l'heure prescrite en se basant sur des données démographiques, la chaîne américaine de grands magasins TARGET détecte la grossesse de ses clientes avant même leur famille, Netflix propose des recommandations de films adaptées aux goût du spectateur, etc. Dans cet exposé, on présentera quelques modélisations simples, en termes de graphes, et on verra comment des outils mathématiques apportent des solutions efficaces à des problèmes complexes comme la détection de groupes dans un réseau ou la désignation de personnes influentes. [less ▲]Detailed reference viewed: 76 (8 ULiège) Formal languages, automata and numeration systems, volume 1: Introduction to combinatorics on wordsRigo, Michel 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 ▲]Detailed reference viewed: 37 (4 ULiège) A new approach to the 2-regularity of the ℓ-abelian complexity of 2-automatic sequences (extended abstract)Parreau, Aline; Rigo, Michel ; Rowland, Eric et alConference (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 ▲]Detailed reference viewed: 30 (8 ULiège) A conjecture on the 2-abelian complexity of the Thue-Morse wordRigo, Michel ; Parreau, Aline ; Vandomme, Elise 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 ▲]Detailed reference viewed: 83 (22 ULiège) Formal languages, automata and numeration systems, volume 2: Applications to recognizability and decidabilityRigo, Michel Book published by ISTE-Wiley (2014)The interplay between words, computability, algebra and arithmetic has now proved its relevance and fruitfulness. Indeed, the cross-fertilization between formal logic and finite automata (such as that ... [more ▼]The interplay between words, computability, algebra and arithmetic has now proved its relevance and fruitfulness. Indeed, the cross-fertilization between formal logic and finite automata (such as that initiated by J.R. Büchi) or between combinatorics on words and number theory has paved the way to recent dramatic developments, for example, the transcendence results for the real numbers having a “simple” binary expansion, by B. Adamczewski and Y. Bugeaud. This book is at the heart of this interplay through a unified exposition. Objects are considered with a perspective that comes both from theoretical computer science and mathematics. Theoretical computer science offers here topics such as decision problems and recognizability issues, whereas mathematics offers concepts such as discrete dynamical systems. The main goal is to give a quick access, for students and researchers in mathematics or computer science, to actual research topics at the intersection between automata and formal language theory, number theory and combinatorics on words. The second of two volumes on this subject, this book covers regular languages, numeration systems, formal methods applied to decidability issues about infinite words and sets of numbers. [less ▲]Detailed reference viewed: 71 (2 ULiège) A note on abelian returns in rotation wordsRampersad, Narad; Rigo, Michel ; Salimov, Pavel in Theoretical Computer Science (2014), 528Pursuing the study started by Rigo, Salimov and Vandomme, we use elementary number-theoretic techniques to characterize rotation words having a finite set of abelian returns to all prefixes. We also make ... [more ▼]Pursuing the study started by Rigo, Salimov and Vandomme, we use elementary number-theoretic techniques to characterize rotation words having a finite set of abelian returns to all prefixes. We also make the connection between the three gap theorem and the number of semi-abelian returns for Sturmian words, simplifying some arguments developed by Puzynina and Zamboni. [less ▲]Detailed reference viewed: 155 (19 ULiège) Special issue dedicated to the 14th "Journées montoises d'informatique théorique"Bruyère, Véronique; Jungers, Raphaël; Hollanders et alin RAIRO : Informatique Théorique et Applications = Theoretical Informatics and Applications (2014), 48Detailed reference viewed: 18 (1 ULiège) On the number of abelian bordered words (with an example of automatic theorem-proving)Goc, Daniel; Rampersad, Narad; Rigo, Michel et alin International Journal of Foundations of Computer Science (2014), 8In the literature, many bijections between (labeled) Motzkin paths and various other combinatorial objects are studied. We consider abelian (un)bordered words and show the connection with irreducible ... [more ▼]In the literature, many bijections between (labeled) Motzkin paths and various other combinatorial objects are studied. We consider abelian (un)bordered words and show the connection with irreducible symmetric Motzkin paths and paths in $\mathbb{Z}$ not returning to the origin. This study can be extended to abelian unbordered words over an arbitrary alphabet and we derive expressions to compute the number of these words. In particular, over a $3$-letter alphabet, the connection with paths in the triangular lattice is made. Finally, we characterize the lengths of the abelian unbordered factors occurring in the Thue--Morse word using some kind of automatic theorem-proving provided by a logical characterization of the $k$-automatic sequences. [less ▲]Detailed reference viewed: 22 (3 ULiège) Ces mathématiques que l'on dit pures, et leurs applications : des objets mathématiques aux objets industriels, technologiques, informatiquesRigo, Michel Conference (2013, November 14)Detailed reference viewed: 46 (7 ULiège) Mathémagie IIIRigo, Michel Speech/Talk (2013)Cet exposé est dans la continuité de mes précédentes prestations comme apprenti-magicien. Il s'adresse au plus grand nombre et ne nécessite pas de prérequis particulier. Je réaliserai 6 ou 7 tours de ... [more ▼]Cet exposé est dans la continuité de mes précédentes prestations comme apprenti-magicien. Il s'adresse au plus grand nombre et ne nécessite pas de prérequis particulier. Je réaliserai 6 ou 7 tours de "mathémagie" (tours de cartes, divination, mentalisme,...). Pour chaque tour, le scénario sera identique : réalisation du tour, explication, mise en évidence des structures et résultats mathématiques sous-jacents et enfin, illustration de ces concepts mathématiques mis en oeuvre dans d'autres contextes (informatique, théorie de l'information, ...) [less ▲]Detailed reference viewed: 297 (16 ULiège) Another Generalization of Abelian Equivalence: Binomial Complexity of Infinite WordsRigo, Michel ; Salimov, Pavel in Lecture Notes in Computer Science (2013), 8079The 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 ... [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. 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 the Sturmian words and of the Thue-Morse word. We also mention the possible avoidance of 2-binomial squares. [less ▲]Detailed reference viewed: 100 (24 ULiège) On the Number of Abelian Bordered WordsRampersad, Narad; Rigo, Michel ; Salimov, Pavel in Lecture Notes in Computer Science (2013), 7907In the literature, many bijections between (labeled) Motzkin paths and various other combinatorial objects are studied. We consider abelian (un)bordered words and show the connection with irreducible ... [more ▼]In the literature, many bijections between (labeled) Motzkin paths and various other combinatorial objects are studied. We consider abelian (un)bordered words and show the connection with irreducible symmetric Motzkin paths and paths in Z not returning to the origin. This study can be extended to abelian unbordered words over an arbitrary alphabet and we derive expressions to compute the number of these words. In particular, over a 3-letter alphabet, the connection with paths in the triangular lattice is made. Finally, we study the lengths of the abelian unbordered factors occurring in the Thue--Morse word. [less ▲]Detailed reference viewed: 86 (14 ULiège)