8,558 research outputs found

    Variations of the Morse-Hedlund Theorem for k-Abelian Equivalence

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    In this paper we investigate local-to-global phenomena for a new family of complexity functions of infinite words indexed by k >= 0. Two finite words u and v are said to be k-abelian equivalent if for all words x of length less than or equal to k, the number of occurrences of x in u is equal to the number of occurrences of x in v. This defines a family of equivalence relations, bridging the gap between the usual notion of abelian equivalence (when k = 1) and equality (when k = infinity). Given an infinite word w, we consider the associated complexity function which counts the number of k-abelian equivalence classes of factors of w of length n. As a whole, these complexity functions have a number of common features: Each gives a characterization of periodicity in the context of bi-infinite words, and each can be used to characterize Sturmian words in the framework of aperiodic one-sided infinite words. Nevertheless, they also exhibit a number of striking differences, the study of which is one of the main topics of our paper

    Variations of the Morse-Hedlund theorem for k-abelian equivalence

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    In this paper we investigate local-to-global phenomena for a new family of complexity functions of infinite words indexed by k ≥ 0. Two finite words u and v are said to be k-abelian equivalent if for all words x of length less than or equal to k, the number of occurrences of x in u is equal to the number of occurrences of x in v. This defines a family of equivalence relations, bridging the gap between the usual notion of abelian equivalence (when k = 1) and equality (when k = ∞). Given an infinite word w, we consider the associated complexity function which counts the number of k-abelian equivalence classes of factors of w of length n. As a whole, these complexity functions have a number of common features: each gives a characterization of periodicity in the context of bi-infinite words, and each can be used to characterize Sturmian words in the framework of aperiodic one-sided infinite words. Nevertheless, they also exhibit a number of striking differences, the study of which is one of the main topics of our paper

    Decidability of the HD0L ultimate periodicity problem

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    In this paper we prove the decidability of the HD0L ultimate periodicity problem

    On a generalization of Abelian equivalence and complexity of infinite words

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    In this paper we introduce and study a family of complexity functions of infinite words indexed by k \in \ints ^+ \cup {+\infty}. Let k \in \ints ^+ \cup {+\infty} and AA be a finite non-empty set. Two finite words uu and vv in AA^* are said to be kk-Abelian equivalent if for all xAx\in A^* of length less than or equal to k,k, the number of occurrences of xx in uu is equal to the number of occurrences of xx in v.v. This defines a family of equivalence relations k\thicksim_k on A,A^*, bridging the gap between the usual notion of Abelian equivalence (when k=1k=1) and equality (when k=+).k=+\infty). We show that the number of kk-Abelian equivalence classes of words of length nn grows polynomially, although the degree is exponential in k.k. Given an infinite word \omega \in A^\nats, we consider the associated complexity function \mathcal {P}^{(k)}_\omega :\nats \rightarrow \nats which counts the number of kk-Abelian equivalence classes of factors of ω\omega of length n.n. We show that the complexity function P(k)\mathcal {P}^{(k)} is intimately linked with periodicity. More precisely we define an auxiliary function q^k: \nats \rightarrow \nats and show that if Pω(k)(n)<qk(n)\mathcal {P}^{(k)}_{\omega}(n)<q^k(n) for some k \in \ints ^+ \cup {+\infty} and n0,n\geq 0, the ω\omega is ultimately periodic. Moreover if ω\omega is aperiodic, then Pω(k)(n)=qk(n)\mathcal {P}^{(k)}_{\omega}(n)=q^k(n) if and only if ω\omega is Sturmian. We also study kk-Abelian complexity in connection with repetitions in words. Using Szemer\'edi's theorem, we show that if ω\omega has bounded kk-Abelian complexity, then for every D\subset \nats with positive upper density and for every positive integer N,N, there exists a kk-Abelian NN power occurring in ω\omega at some position $j\in D.

    Open and closed complexity of infinite words

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    In this paper we study the asymptotic behaviour of two relatively new complexity functions defined on infinite words and their relationship to periodicity. Given a factor ww of an infinite word x=x1x2x3x=x_1x_2x_3\cdots with each xix_i belonging to a fixed finite set A,\mathbb{A}, we say ww is closed if either wAw\in \mathbb{A} or if ww is a complete first return to some factor vv of x.x. Otherwise ww is said to be open. We show that for an aperiodic word xAN,x\in \mathbb{A}^\mathbb{N}, the complexity functions ClxCl_x (resp. Opx)Op_x) that count the number of closed (resp. open) factors of xx of each given length are both unbounded. More precisely, we show that if xx is aperiodic then lim infnNOpx(n)=+\liminf_{n\in \mathbb{N}} Op_x(n)=+\infty and lim supnSClx(n)=+\limsup_{n\in S} Cl_x(n)=+\infty for any syndetic subset SS of N.\mathbb{N}. However, there exist aperiodic infinite words xx verifying lim infnNClx(n)<+.\liminf_{n\in \mathbb{N}}Cl_x(n)<+\infty. Keywords: word complexity, periodicity, return words
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