8,558 research outputs found
Variations of the Morse-Hedlund Theorem for k-Abelian Equivalence
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
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
In this paper we prove the decidability of the HD0L ultimate periodicity
problem
On a generalization of Abelian equivalence and complexity of infinite words
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 be a finite non-empty set. Two finite words and
in are said to be -Abelian equivalent if for all of length
less than or equal to the number of occurrences of in is equal to
the number of occurrences of in This defines a family of equivalence
relations on bridging the gap between the usual notion of
Abelian equivalence (when ) and equality (when We show that
the number of -Abelian equivalence classes of words of length grows
polynomially, although the degree is exponential in 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
-Abelian equivalence classes of factors of of length We show
that the complexity function is intimately linked with
periodicity. More precisely we define an auxiliary function q^k: \nats
\rightarrow \nats and show that if for
some k \in \ints ^+ \cup {+\infty} and the is ultimately
periodic. Moreover if is aperiodic, then if and only if is Sturmian. We also
study -Abelian complexity in connection with repetitions in words. Using
Szemer\'edi's theorem, we show that if has bounded -Abelian
complexity, then for every D\subset \nats with positive upper density and for
every positive integer there exists a -Abelian power occurring in
at some position $j\in D.
Open and closed complexity of infinite words
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 of an infinite word with
each belonging to a fixed finite set we say is closed
if either or if is a complete first return to some factor
of Otherwise is said to be open. We show that for an aperiodic
word the complexity functions (resp.
that count the number of closed (resp. open) factors of of each
given length are both unbounded. More precisely, we show that if is
aperiodic then and for any syndetic subset of However,
there exist aperiodic infinite words verifying
Keywords: word complexity, periodicity, return words
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