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See detailSyntactical complexity of periodic sets
Charlier, Emilie ULg

Conference (2012, June)

We are interested in the links between numbers and their representations. In the decimal numeration system, a number is even if its representation ends with 0, 2, 4, 6, or 8, which give rise to a simple ... [more ▼]

We are interested in the links between numbers and their representations. In the decimal numeration system, a number is even if its representation ends with 0, 2, 4, 6, or 8, which give rise to a simple representation set, that is, accepted by a finite automaton. More generally, in an integer base numeration system, any divisibility criterion is recognized by a finite automaton. The state complexity of a regular language is the number of states of the minimal automaton accepting this language. The syntactic complexity is the size of the associated syntactical monoid. The first one only takes into account the context on the right side, whereas the second considers both left and right contexts. Given a set of integers, we want to answer the following questions. What is the associated state complexity? What is the associated syntactic complexity? These questions extend to more general numeration systems, as linear numeration systems or abstract numeration systems. Such problems are motivated by decidability problems. For example : given an automaton recognizing a set of integers, decide if this set is ultimately periodic? [less ▲]

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See detailMonoïde syntaxique et numérations
Charlier, Emilie ULg

Conference (2012, June)

La complexité en nombre d'états (state complexity) d'un langage régulier est le nombre d'états de son automate minimal. La complexité syntaxique est le cardinal du monoïde syntaxique associé. La première ... [more ▼]

La complexité en nombre d'états (state complexity) d'un langage régulier est le nombre d'états de son automate minimal. La complexité syntaxique est le cardinal du monoïde syntaxique associé. La première ne prend en compte que le contexte à droite alors la seconde considère les contextes à gauche et à droite. Nous nous intéressons aux liens entre les nombres et leurs représentations. Dans le système décimal, un nombre est pair si sa représentation se termine par 0, 2, 4, 6 ou 8, ce qui donne lieu à ensemble de représentations simple, c'est-à-dire reconnu par automate. Plus généralement, dans une base entière, tout critère de divisibilité se traduit par un automate. Etant donné un ensemble d'entiers, nous voulons répondre aux questions suivantes. Quel est le nombre d'états de l'automate minimal associé. Quelle est la complexité syntaxique associée ? Ces questions s'étendent à des numérations plus générales. Ces problèmes sont motivés par des problèmes de décidabilité. Par exemple : étant donné un automate reconnaissant un ensemble d'entiers, décider si cet ensemble est ou non ultimement périodique. [less ▲]

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See detailAn introduction to abstract numeration systems
Charlier, Emilie ULg

Scientific conference (2012, May)

Abstract numeration systems were introduced in 2001 by P. Lecomte and M. Rigo. This new way to represent numbers generalizes that of usual positional numeration systems such as integer base numeration ... [more ▼]

Abstract numeration systems were introduced in 2001 by P. Lecomte and M. Rigo. This new way to represent numbers generalizes that of usual positional numeration systems such as integer base numeration systems and linear numeration systems. Some standard properties are preserved in this wider framework though some others are not. Yet, the advantages of these systems stem from their great generality: current research on this subject strives to highlight the properties that are independent of the target numeration system, such as properties related to the complexity of the numeration language. In this talk I will introduce this topic. In particular, I will present many open questions in the area and highlight the connections with combinatorics on words. [less ▲]

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See detailAbstract numeration systems
Charlier, Emilie ULg

Scientific conference (2012, May)

Abstract numeration systems were introduced in 2001 by P. Lecomte and M. Rigo. This new way to represent numbers generalizes that of usual positional numeration systems such as integer base numeration ... [more ▼]

Abstract numeration systems were introduced in 2001 by P. Lecomte and M. Rigo. This new way to represent numbers generalizes that of usual positional numeration systems such as integer base numeration systems and linear numeration systems. Some standard properties are preserved in this wider framework though some others are not. Yet, the advantages of these systems stem from their great generality: current research on this subject strives to highlight the properties that are independent of the target numeration system, such as properties related to the complexity of the numeration language. In this talk I will introduce this topic. In particular, I will present many open questions in the area and highlight the connections with combinatorics on words. [less ▲]

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See detailEnsembles reconnaissables de rationnels
Charlier, Emilie ULg

Scientific conference (2012, April)

In this talk I will introduce the recognizable sets of rationals, following the formalism of Jeffrey Shallit. I will point out several open questions regarding this topic.

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See detailEnumeration and decidable properties of automatic sequences
Charlier, Emilie ULg; Rampersad, Narad; Shallit, Jeffrey

in International Journal of Foundations of Computer Science (2012), 23(5), 1035-1066

We show that various aspects of k-automatic sequences — such as having an unbordered factor of length n — are both decidable and effectively enumerable. As a consequence it follows that many related ... [more ▼]

We show that various aspects of k-automatic sequences — such as having an unbordered factor of length n — are both decidable and effectively enumerable. As a consequence it follows that many related sequences are either k-automatic or k-regular. These include many sequences previously studied in the literature, such as the recurrence function, the appearance function, and the repetitivity index. We also give some new characterizations of the class of k-regular sequences. Many results extend to other sequences defined in terms of Pisot numeration systems. [less ▲]

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See detailMulti-dimensional sets recognizable in all abstract numeration systems
Charlier, Emilie ULg; Lacroix, Anne ULg; Rampersad, Narad ULg

in RAIRO : Informatique Théorique et Applications = Theoretical Informatics and Applications (2012), 46(1), 51-65

We prove that the subsets of N^d that are S-recognizable for all abstract numeration systems S are exactly the 1-recognizable sets. This generalizes a result of Lecomte and Rigo in the one-dimensional ... [more ▼]

We prove that the subsets of N^d that are S-recognizable for all abstract numeration systems S are exactly the 1-recognizable sets. This generalizes a result of Lecomte and Rigo in the one-dimensional setting. [less ▲]

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See detailOrbits of language operations: finiteness and upper bounds
Charlier, Emilie ULg

Scientific conference (2011, August)

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See detailEnumeration and decidable properties of automatic sequences
Charlier, Emilie ULg

Conference (2011, July)

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See detailEnumeration and decidable properties of automatic sequences
Charlier, Emilie ULg; Rampersad, Narad; Shallit, Jeffrey

in Actes de Numération 2011 (2011, June)

We show that various aspects of k-automatic sequences such as having an unbordered factor of length n are both decidable and e ectively enumerable. As a consequence it follows that many related sequences ... [more ▼]

We show that various aspects of k-automatic sequences such as having an unbordered factor of length n are both decidable and e ectively enumerable. As a consequence it follows that many related sequences are either k-automatic or k-regular. These include many sequences previously studied in the literature, such as the recurrence function, the appearance function, and the repetitivity index. We also give a new characterization of the class of k-regular sequences. Many results extend to other sequences de ned in terms of Pisot numeration systems. [less ▲]

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See detailFinite orbits of language operations
Charlier, Emilie ULg

Scientific conference (2011, May)

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See detailAbstract numeration systems or decimation of languages
Charlier, Emilie ULg

Scientific conference (2011, April)

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See detailA numeration point of view on the HD0L periodicity problem
Charlier, Emilie ULg

Scientific conference (2011, January)

A HD0L system is a 5-tuple G = (∆, Γ, f, g, w) where • ∆ and Γ are alphabet; • f : ∆^∗ → ∆^∗ is a morphism; • g : ∆^∗ → Γ^∗ is a morphism; • w is a finite word over ∆. If w is a prefix of f(w) and if g(f ... [more ▼]

A HD0L system is a 5-tuple G = (∆, Γ, f, g, w) where • ∆ and Γ are alphabet; • f : ∆^∗ → ∆^∗ is a morphism; • g : ∆^∗ → Γ^∗ is a morphism; • w is a finite word over ∆. If w is a prefix of f(w) and if g(f^ω(w)) is an infinite word over Γ, where f^ω(w) denotes the limit lim_{n→+∞}f^n(w), then we define the infinite word generated by G to be ω(G) = g(f^ω(w)). The question is to decide whether the infinite word ω(G) is ultimately periodic. This open problem is called the HD0L periodicity problem. It is not hard to see that we may assume that w is a letter. Furthermore, it is well known that we can assume that f is a non-erasing morphism and g is a coding. Therefore we will always consider that all these additional hypotheses hold. On the one hand, if f is uniform of length b, then ω(G) is b-automatic. In that particular case the problem is known to be decidable. Various proofs of this result have been given by several authors. On the other hand, in the general case, when f is not necessarily uniform, ω(G) is S-automatic for some abstract numeration system S. Therefore the HD0L periodicity problem is equivalent to the following problem involving numeration systems. Given an abstract numeration system S, is it decidable whether an S-recognizable set X ⊆ N is ultimately periodic? The numeration language L and the set X are given through DFAs accepting L and rep_S(X) respectively. Thanks to this numeration point of view, we can give decision procedures for large classes of numeration systems. In this talk, I will discuss some techniques used to provide such decision procedures. [less ▲]

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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 detailThe growth function of S-recognizable sets
Charlier, Emilie ULg; Rampersad, Narad

in Theoretical Computer Science (2011), 412(39), 5400-5408

A set X ⊆ N is S-recognizable for an abstract numeration system S, if the set rep_S (X) of its representations is accepted by a finite automaton. We show that the growth function of an S-recognizable set ... [more ▼]

A set X ⊆ N is S-recognizable for an abstract numeration system S, if the set rep_S (X) of its representations is accepted by a finite automaton. We show that the growth function of an S-recognizable set is always either Θ((log(n))^(c−df) n^f ) where c, d ∈ N and f ≥ 1, or Θ(n^r θ^(Θ(n^q))), where r, q ∈ Q with q ≤ 1. If the number of words of length n in the numeration language is bounded by a polynomial, then the growth function of an S-recognizable set is Θ(nr ), where r ∈ Q with r ≥ 1. Furthermore, for every r ∈ Q with r ≥ 1, we can provide an abstract numeration system S built on a polynomial language and an S-recognizable set such that the growth function of X is Θ(n^r ). For all positive integers k and ℓ, we can also provide an abstract numeration system S built on an exponential language and an S-recognizable set such that the growth function of X is Θ((log(n))^k n^ℓ). [less ▲]

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See detailEnumeration and decidable properties of automatic sequences
Charlier, Emilie ULg; Rampersad, Narad; Shallit, Jeffrey

in Lecture Notes in Computer Science (2011), 6795

We show that various aspects of k-automatic sequences — such as having an unbordered factor of length n — are both decidable and effectively enumerable. As a consequence it follows that many related ... [more ▼]

We show that various aspects of k-automatic sequences — such as having an unbordered factor of length n — are both decidable and effectively enumerable. As a consequence it follows that many related sequences are either k-automatic or k-regular. These include many sequences previously studied in the literature, such as the recurrence function, the appearance function, and the repetitivity index. We also give a new characterization of the class of k-regular sequences. Many results extend to other sequences defined in terms of Pisot numeration systems. [less ▲]

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See detailFinite orbits of language operations
Charlier, Emilie ULg; Domaratzski, Michael; Harju, Tero et al

in Lecture Notes in Computer Science (2011), 6638

We consider a set of natural operations on languages, and prove that the orbit of any language L under the monoid generated by this set is finite and bounded, independently of L. This generalizes previous ... [more ▼]

We consider a set of natural operations on languages, and prove that the orbit of any language L under the monoid generated by this set is finite and bounded, independently of L. This generalizes previous results about complement, Kleene closure, and positive closure. [less ▲]

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See detailRepresenting real numbers in a generalized numeration system
Charlier, Emilie ULg; Le Gonidec, Marion; Rigo, Michel ULg

in Journal of Computer & System Sciences (2011), 77

We show how to represent an interval of real numbers in an abstract numeration system built on a language that is not necessarily regular. As an application, we consider representations of real numbers ... [more ▼]

We show how to represent an interval of real numbers in an abstract numeration system built on a language that is not necessarily regular. As an application, we consider representations of real numbers using the Dyck language. We also show that our framework can be applied to the rational base numeration systems. [less ▲]

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See detailS-automatic sets
Charlier, Emilie ULg

Scientific conference (2010, December)

In this talk, I will discuss some new properties of S-automatic sets. First I will show that a multidimensional set is S-automatic for all abstract numeration systems S if and only if it is 1-automatic ... [more ▼]

In this talk, I will discuss some new properties of S-automatic sets. First I will show that a multidimensional set is S-automatic for all abstract numeration systems S if and only if it is 1-automatic. This result is surprising in the following sense: the class of multidimensional 1-automatic sets is a strict subclass of that of semi-linear sets. Hence, this result is not a generalization of the well-known result in integer base numeration systems: a multidimensional set is b-automatic for all integer bases b ≥ 1 if and only if it is semi-linear. Second I will describe the possible behaviors of the nth -term of an S-automatic set, depending on the growth function (i.e., the number of words of length n) of the numeration language. [less ▲]

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See detailCriteria for recognizability in abstract numeration systems
Charlier, Emilie ULg

Scientific conference (2010, October)

In this talk, I will first shortly introduce abstract numeration systems. Then I will prove some first results I have regarding the growth function of recognizable sets. I will conjecture which functions ... [more ▼]

In this talk, I will first shortly introduce abstract numeration systems. Then I will prove some first results I have regarding the growth function of recognizable sets. I will conjecture which functions can be the growth functions of recognizable sets in general, depending on the growth functions of the numeration language and of the sublanguage under consideration. [less ▲]

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