References of "Rigo, Michel"
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See detailStructure of the minimal automaton of a numeration language
Charlier, Emilie ULg; Rampersad, Narad ULg; Rigo, Michel ULg et al

in Actes de LaCIM 2010 (2010, August)

We study the structure of automata accepting the greedy representations of N in a wide class of numeration systems. We describe the conditions under which such automata can have more than one strongly ... [more ▼]

We study the structure of automata accepting the greedy representations of N in a wide class of numeration systems. We describe the conditions under which such automata can have more than one strongly connected component and the form of any such additional components. Our characterization applies, in particular, to any automaton arising from a Bertrand numeration system. Furthermore, we show that for any automaton A arising from a system with a dominant root beta>1, there is a morphism mapping A onto the automaton arising from the Bertrand system associated with the number beta. [less ▲]

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See detailInvariant games
Duchêne, Eric; Rigo, Michel ULg

in Theoretical Computer Science (2010), 411

In the context of 2-player removal games, we define the notion of invariant game for which each allowed move is independent of the position it is played from. We present a family of invariant games which ... [more ▼]

In the context of 2-player removal games, we define the notion of invariant game for which each allowed move is independent of the position it is played from. We present a family of invariant games which are variations of Wythoff's game. The set of P-positions of these games are given by a pair of complementary Beatty sequences related to the irrational quadratic number $\alpha_k = (1; \overline{1, k})$. We also provide a recursive characterization of this set. [less ▲]

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See detailSystèmes de numération abstraits et combinatoire des mots (habilitation à diriger des recherches)
Rigo, Michel ULg

Post doctoral thesis (2010)

We summary the main properties of abstract numeration systems and their links to combinatorics on words and combinatorial game theory.

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See detailExtensions and restrictions of Wythoff's game preserving its P positions
Duchêne, Eric; Fraenkel, Aviezri; Nowakowski, Richard et al

in Journal of Combinatorial Theory - Series A (2010), 117

We consider extensions and restrictions of Wythoff's game having exactly the same set of P positions as the original game. No strict subset of rules give the same set of P positions. On the other hand, we ... [more ▼]

We consider extensions and restrictions of Wythoff's game having exactly the same set of P positions as the original game. No strict subset of rules give the same set of P positions. On the other hand, we characterize all moves that can be adjoined while preserving the original set of P positions. Testing if a move belongs to such an extended set of rules is shown to be doable in polynomial time. Many arguments rely on the infinite Fibonacci word, automatic sequences and the corresponding number system. With these tools, we provide new two-dimensional morphisms generating an infinite picture encoding respectively P positions of Wythoff's game and moves that can be adjoined. [less ▲]

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See detailSpecial issue dedicated to the twelfth "Journées montoises d'informatique théorique"
Bruyère, Véronique; Rigo, Michel ULg

in RAIRO : Informatique Théorique et Applications = Theoretical Informatics and Applications (2010), 44(1), 1-192

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See detailSpecial issue dedicated to the second "AutoMathA conference"
Bruyère, Véronique; Pin, Jean-Eric; Restivo, Antonio et al

in Discrete Mathematics & Theoretical Computer Science (2010), 12

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See detailStructure of the minimal automaton of a numeration language and applications to state complexity
Charlier, Emilie ULg; Rampersad, Narad ULg; Rigo, Michel ULg et al

in Actes des Journées Montoises d'Informatique Théorique (2010)

We study the structure of automata accepting the greedy representations of N in a wide class of numeration systems. We describe the conditions under which such automata can have more than one strongly ... [more ▼]

We study the structure of automata accepting the greedy representations of N in a wide class of numeration systems. We describe the conditions under which such automata can have more than one strongly connected component and the form of any such additional components. Our characterization applies, in particular, to any automaton arising from a Bertrand numeration system. Furthermore, we show that for any automaton A arising from a system with a dominant root > 1, there is a morphism mapping A onto the automaton arising from the Bertrand system associated with the number . Under some mild assumptions, we also study the state complexity of the trim minimal automaton accepting the greedy representations of the multiples of m>=2 for a wide class of linear numeration systems. As an example, the number of states of the trim minimal automaton accepting the greedy representations of mN in the Fibonacci system is exactly 2m^2. [less ▲]

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

in Discrete Mathematics (2010), 310

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 numeration system S. In this paper, we consider an ... [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 numeration system S. In this paper, we consider an analogous definition in a multidimensional setting and study its relation to the shapesymmetric infinite words introduced by Arnaud Maes. More precisely, for d>1, we show that a multidimensional infinite word x over a finite alphabet 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; Lacroix, Anne ULg; Rigo, Michel ULg

in Journal of Integer Sequences (2010), 13

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 detailMathémagie et au-delà
Rigo, Michel ULg

Learning material (2010)

Nous présentons ici 5 tours de magie ne nécessitant aucune habileté particulière de la part de l'apprenti magicien : des tours de cartes, des tours de divination et le célèbre tour du ``barman aveugle ... [more ▼]

Nous présentons ici 5 tours de magie ne nécessitant aucune habileté particulière de la part de l'apprenti magicien : des tours de cartes, des tours de divination et le célèbre tour du ``barman aveugle avec des gants de boxe''. Contrairement au magicien qui ne dévoile jamais ses secrets, ici, nous expliquons que ces tours reposent sur diverses propriétés et constructions mathématiques. Ces dernières débouchent sur de véritables questions de recherche actuelle en théorie des graphes ou en combinatoire des mots et même sur de possibles applications en robotique et automatisation ! Il y en aura donc pour tous les goûts... et tous les niveaux (suivant l'auditoire, l'exposé sera adapté au niveau des élèves de la 4ième à la 6ième secondaire, voire même aux étudiants universitaires). Ce texte présente donc un matériel qui dépassera souvent (et de loin) le ``spectacle''. [less ▲]

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See detailIndex and References
Berthé, Valérie; Rigo, Michel ULg

in Combinatorics, Automata and Number Theory (2010)

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See detailAbstract numeration systems (Chapter 3)
Lecomte, Pierre ULg; Rigo, Michel ULg

in Berthé, Valérie; Rigo, Michel (Eds.) Combinatorics, Automata and Number Theory (2010)

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See detailPreliminaries (Chapter 1)
Berthé, Valérie; Rigo, Michel ULg

in Berthé, Valérie; Rigo, Michel (Eds.) Combinatorics, Automata and Number Theory (2010)

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See detailIntroduction
Berthé, Valérie; Rigo, Michel ULg

in Rigo, Michel; Berthé, Valérie (Eds.) Combinatorics, Automata and Number Theory (2010)

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See detailOn the Periodicity of Morphic Words
Halava, Vesa; Harju, Tero; Kärki, Tomi et al

in Lecture Notes in Computer Science (2010), 6224

Given a morphism h prolongable on a and an integer p, we present an algorithm that calculates which letters occur infinitely often in congruent positions modulo p in the infinite word hω(a). As a ... [more ▼]

Given a morphism h prolongable on a and an integer p, we present an algorithm that calculates which letters occur infinitely often in congruent positions modulo p in the infinite word hω(a). As a corollary, we show that it is decidable whether a morphic word is ultimately p-periodic. Moreover, using our algorithm we can find the smallest similarity relation such that the morphic word is ultimately relationally p-periodic. The problem of deciding whether an automatic sequence is ultimately weakly R-periodic is also shown to be decidable. [less ▲]

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See detailNumeration Systems: a Link between Number Theory and Formal Language Theory
Rigo, Michel ULg

in Lecture Notes in Computer Science (2010), 6224

We survey facts mostly emerging from the seminal results of Alan Cobham obtained in the late sixties and early seventies. We do not attempt to be exhaustive but try instead to give some personal ... [more ▼]

We survey facts mostly emerging from the seminal results of Alan Cobham obtained in the late sixties and early seventies. We do not attempt to be exhaustive but try instead to give some personal interpretations and some research directions. We discuss the notion of numeration systems, recognizable sets of integers and automatic sequences. We brie y sketch some results about transcendence related to the representation of real numbers. We conclude with some applications to combinatorial game theory and veri cation of in nite-state systems and present a list of open problems. [less ▲]

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See detailInvariant games
Duchêne, Eric; Rigo, Michel ULg

Conference (2009, September)

In the context of 2-player removal games, we define the notion of invariant game for which each allowed move is independent of the position it is played from. We present a family of invariant games which ... [more ▼]

In the context of 2-player removal games, we define the notion of invariant game for which each allowed move is independent of the position it is played from. We present a family of invariant games which are variations of Wythoff's game. The set of P-positions of these games are given by a pair of complementary Beatty sequences related to the irrational quadratic number $\alpha_k = (1; \overline{1, k})$. We also provide a recursive characterization of this set. [less ▲]

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See detailA characterization of multidimensional S-automatic sequences
Charlier, Emilie ULg; Kärki, Tomi ULg; Rigo, Michel ULg

in Actes des rencontres du CIRM, 1 (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 extended ... [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 extended abstract, we consider an analogous definition in a multidimensional setting and present the connection to the shape-symmetric infinite words introduced by Arnaud Maes. More precisely, for d ≥ 2, we state that a multidimensional infinite word x : N^d → \Sigma over a finite alphabet \Sigma 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 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 ▲]

Detailed reference viewed: 56 (17 ULg)