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Communication Dans Un Congrès Année : 2017

Cyclic Complexity of Words

Résumé

We introduce and study a complexity function on words $c_x(n),$ called \emph{cyclic complexity}, which counts the number of conjugacy classes of factors of length $n$ of an infinite word $x.$ We extend the well-known Morse-Hedlund theorem to the setting of cyclic complexity by showing that a word is ultimately periodic if and only if it has bounded cyclic complexity. Unlike most complexity functions, cyclic complexity distinguishes between Sturmian words of different slopes. We prove that if $x$ is a Sturmian word and $y$ is a word having the same cyclic complexity of $x,$ then up to renaming letters, $x$ and $y$ have the same set of factors. In particular, $y$ is also Sturmian of slope equal to that of $x.$ Since $c_x(n)=1$ for some $n\geq 1$ implies $x$ is periodic, it is natural to consider the quantity $\liminf_{n\rightarrow \infty} c_x(n).$ We show that if $x$ is a Sturmian word, then $\liminf_{n\rightarrow \infty} c_x(n)=2.$ We prove however that this is not a characterization of Sturmian words by exhibiting a restricted class of Toeplitz words, including the period-doubling word, which also verify this same condition on the limit infimum. In contrast we show that, for the Thue-Morse word $t$, $\liminf_{n\rightarrow \infty} c_t(n)=+\infty.$

Dates et versions

hal-01024885 , version 1 (16-07-2014)

Identifiants

Citer

Julien Cassaigne, Gabriele Fici, Marinella Sciortino, Luca Q. Zamboni. Cyclic Complexity of Words. MFCS 2014 (39th International Symposium), Faculty of Informatics, Eötvös Loránd University, Budapest, and the Department of Foundations of Computer Science, Faculty of Science and Informatics, University of Szeged, in cooperation with EATCS., Aug 2014, Budapest, Hungary. pp.36 - 56, ⟨10.1016/j.jcta.2016.07.002⟩. ⟨hal-01024885⟩
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