Disjunctivity and other properties of sets of pseudo-bordered words

Abstract

The concepts of pseudo-bordered and pseudo-unbordered words are in large part motivated by research in theoretical DNA computing, wherein the Watson–Crick complementarity of DNA strands is modelled as an antimorphic involution, that is, a function $\theta$ which is an antimorphism, $\theta (uv)=\theta (v) \theta (u)$, and an involution, $\theta (\theta (u))=u$, for all words u, v over the DNA alphabet. In particular, a word w is said to be $\theta$-bordered (or pseudo-bordered) if there exists a word $v \in \Sigma ^{+}$ that is a proper prefix of w, while $\theta (v)$ is a proper suffix of w. A word which is not $\theta$-bordered is $\theta$-unbordered. This paper continues the exploration of properties (for the case where $\theta$ is a morphic involution) of the set of $\theta$-unbordered words, $D_{\theta }(1)$, and the sets of words that have exactly i $\theta$-borders, $D_{\theta }(i)$, $i \ge 2$. We prove that, under some conditions, the set $D_{\theta }(i)$ is disjunctive for all $i \ge 1$, and that the set $D_{\theta }^i(1){\setminus }D(i)$ is disjunctive for all $i\ge 2$, where D(i) denotes the set of words with exactly i borders. We also discuss conditions for catenations of languages of $\theta$-unbordered words to remain $\theta$-unbordered, and anticipate further generalizations by showing that the set of all $\theta$-bordered words is not context-free for all morphisms $\theta$ over an alphabet $\Sigma$ with $|\Sigma | \ge 3$ such that $\theta (a) \ne a$ for all $a \in \Sigma$ and $\theta ^{3}$ equals the identity function on $\Sigma$.