Bispecial factors in circular non-pushy D0L languages

We study bispecial factors in fixed points of morphisms. In particular, we propose a simple method of how to find all bispecial words of non-pushy circular D0L-systems. This method can be formulated as an algorithm. Moreover, we prove that non-pushy circular D0L-systems are exactly those with finite critical exponent.Comment: 18 pages, 5 figure

On almost cylindrical languages and the decidability of the D0L and PWD0L primitivity problems

AbstractPrimitive words and their properties have always been of fundamental importance in the study of formal language theory. Head and Lando in Periodic D0L Languages proposed the idea of deciding whether or not a given D0L language has the property that every word in it is a primitive word. After reducing the general problem to the case in which h is injective, it will be shown that primitivity is decidable when ((A)h)∗ is an almost cylindrical set. Moreover, in this case, it is shown that the set of words which generate primitive sequences (given a particular D0L scheme) is an algorithmically constructible context-sensitive language. An undecidability result for the PWD0L primitivity problem and decidability results for cases of the RWD0L primitivity problem are also given

On images of D0L and DT0L power series

AbstractThe D0L and DT0L power series are generalizations of D0L and DT0L languages. We continue the study of these series by investigating various decidability questions concerning the images of D0L and DT0L power series

Decidability of the HD0L ultimate periodicity problem

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

A Note on Symmetries in the Rauzy Graph and Factor Frequencies

We focus on infinite words with languages closed under reversal. If frequencies of all factors are well defined, we show that the number of different frequencies of factors of length n+1 does not exceed 2C(n+1)-2C(n)+1.Comment: 7 page

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 $A$ be a finite non-empty set. Two finite words $u$ and $v$ in $A^*$ are said to be $k$-Abelian equivalent if for all $x\in A^*$ 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 $\thicksim_k$ on $A^*,$ bridging the gap between the usual notion of Abelian equivalence (when $k=1$) and equality (when $k=+\infty).$ We show that the number of $k$-Abelian equivalence classes of words of length $n$ grows polynomially, although the degree is exponential in $k.$ 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 $k$-Abelian equivalence classes of factors of $\omega$ of length $n.$ We show that the complexity function $\mathcal {P}^{(k)}$ is intimately linked with periodicity. More precisely we define an auxiliary function q^k: \nats \rightarrow \nats and show that if $\mathcal {P}^{(k)}_{\omega}(n)<q^k(n)$ for some k \in \ints ^+ \cup {+\infty} and $n\geq 0,$ the $\omega$ is ultimately periodic. Moreover if $\omega$ is aperiodic, then $\mathcal {P}^{(k)}_{\omega}(n)=q^k(n)$ if and only if $\omega$ is Sturmian. We also study $k$-Abelian complexity in connection with repetitions in words. Using Szemer\'edi's theorem, we show that if $\omega$ has bounded $k$-Abelian complexity, then for every D\subset \nats with positive upper density and for every positive integer $N,$ there exists a $k$-Abelian $N$ power occurring in $\omega$ at some position $j\in D.