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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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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 detailRepresenting real numbers in a generalized numeration system
Charlier, Emilie ULg

Conference (2010, June)

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 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 detailAbstract Numeration Systems : Recognizability, Decidability, Multidimensional S-Automatic Words, and Real Numbers
Charlier, Emilie ULg

Doctoral thesis (2009)

In this dissertation we study and we solve several questions regarding abstract numeration systems. Each particular problem is the focus of a chapter. The first problem concerns the study of the ... [more ▼]

In this dissertation we study and we solve several questions regarding abstract numeration systems. Each particular problem is the focus of a chapter. The first problem concerns the study of the preservation of recognizability under multiplication by a constant in abstract numeration systems built on polynomial regular languages. The second is a decidability problem, which has been already studied notably by J. Honkala and A. Muchnik and which is studied here for two new cases: the linear positional numeration systems and the abstract numeration systems. Next, we focus on the extension to the multidimensional setting of a result of A. Maes and M. Rigo regarding S-automatic infinite words. Finally, we propose a formalism to represent real numbers in the general framework of abstract numeration systems built on languages that are not necessarily regular. We end by a list of open questions in the continuation of the present work. [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 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 detailA decision problem for ultimately periodic sets in non-standard numeration systems
Charlier, Emilie ULg

Scientific conference (2008, December)

Given a linear numeration system U and a set X (include in N) such that repU(X) is recognized by a (deterministic) finite automaton. Is it decidable whether or not X is ultimately periodic, i.e., whether ... [more ▼]

Given a linear numeration system U and a set X (include in N) such that repU(X) is recognized by a (deterministic) finite automaton. Is it decidable whether or not X is ultimately periodic, i.e., whether or not X is a finite union of arithmetic progressions? Honkala showed that this problem turns out to be decidable for the usual b-ary numeration system (b greater than 2) defined by U_n = bU_{n-1} for n greater than 1 and U_0 = 1. In this work, we give a decision procedure for this problem for a large class of linear numeration systems. [less ▲]

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

Conference (2008, May)

We consider the following decidability problem: Given a linear numeration system U and a set X ⊆ N such that rep_U(X) is recognized by a (deterministic) finite automaton. Is it decidable whether or not X ... [more ▼]

We consider the following decidability problem: Given a linear numeration system U and a set X ⊆ N such that rep_U(X) is recognized by a (deterministic) finite automaton. Is it decidable whether or not X is ultimately periodic, i.e., whether or not X is a finite union of arithmetic progressions? In this work, we give a decision procedure for this problem whenever U is a linear numeration system such that N is U -recognizable and satisfying a relation of the form U_{i+k} = a_1 U_{i+k−1} + · · · + a_k U_i with a_k = ±1 (the main reason for this assumption is that 1 and −1 are the only two integers invertible modulo n for all n ≥ 2). [less ▲]

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