30 research outputs found
Easy cases of the D0L sequence equivalence problem
AbstractTo test the equivalence of two binary D0L sequences it suffices to compare the first four terms of the sequences. We introduce a larger class of D0L systems for which sequence equivalence can be decided by considering the first ten initial terms
Marked D0L systems and the 2n-conjecture
AbstractWe show that to test the equivalence of two D0L sequences over an n-letter alphabet generated by marked morphisms it suffices to compare the first 2n+1 initial terms of the sequences. Under an additional condition it is enough to consider the 2n initial terms
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.
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