References of "Waxweiler, Laurent"
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See detailThe minimal automaton recognizing mN in a linear numeration system
Charlier, Emilie ULg; Rampersad, Narad ULg; Rigo, Michel ULg et al

in Integers: Electronic Journal of Combinatorial Number Theory (2011), 11B(A4), 1-24

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. Under some mild assumptions, we also study the state complexity of the trim minimal automaton accepting the greedy representations of the multiples of m>1 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 detailState complexity of testing divisibility
Charlier, Emilie ULg; Rampersad, Narad ULg; Rigo, Michel ULg et al

in McQuillan, Ian, Pighizzini, Giovanni (Ed.) Proceedings Twelfth Annual Workshop on Descriptional Complexity of Formal Systems (2010, August)

Under some mild assumptions, we 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 ... [more ▼]

Under some mild assumptions, we 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 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 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 detailCaractère reconnaissable d'ensembles de polynômes à coefficients dans un corps fini
Waxweiler, Laurent ULg

Doctoral thesis (2009)

Nous nous plaçons dans le cadre de l'anneau des polynômes sur un corps fini. Si P est un polynôme de degré au moins 1, tout polynôme Q se décompose de manière unique sous la forme d'une combinaison ... [more ▼]

Nous nous plaçons dans le cadre de l'anneau des polynômes sur un corps fini. Si P est un polynôme de degré au moins 1, tout polynôme Q se décompose de manière unique sous la forme d'une combinaison linéaire de puissances de P, dont les coefficients sont des polynômes dont le degré est strictement inférieur à celui de P. À une telle décomposition, nous associons un mot que nous appelons la P-représentation du polynôme Q. Un ensemble de polynômes est alors qualifié de P-reconnaissable si il existe un automate fini déterministe qui accepte l'ensemble des P-représentations de ses éléments. Dans cette thèse, nous montrons que les ensembles P-reconnaissables sont exactement ceux qui sont définissables par une formule du premier ordre dans une certaine structure S(P) basée sur un prédicat dépendant du polynôme P. Nous donnons aussi une caractérisation des ensembles P-reconnaissables en terme de suites P-automatiques. Nous apportons également une réponse partielle à la question de savoir quels sont les ensembles reconnaissables simultanément dans toutes les bases de degré au moins 1. Finalement, nous montrons que si P et Q sont deux polynômes de degré au moins 1 et multiplicativement indépendants, alors la multiplication est définissable dans la réunion des structures S(P) et S(Q). [less ▲]

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See detailA note on syndeticity, recognizable sets and Cobham's theorem
Rigo, Michel ULg; Waxweiler, Laurent ULg

in Bulletin of the European Association for Theoretical Computer Science (2006), 88

In this note, we give an alternative proof of the following result. Let p,q>=2 be two multiplicatively independent integers. If an infinite set of integers is both p- and q-recognizable, then it is ... [more ▼]

In this note, we give an alternative proof of the following result. Let p,q>=2 be two multiplicatively independent integers. If an infinite set of integers is both p- and q-recognizable, then it is syndetic. Notice that this result is needed in the classical proof of the celebrated Cobham’s theorem. Therefore the aim of this paper is to complete [13] and [1] to obtain an accessible proof of Cobham’s theorem. [less ▲]

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