On the regularity of iterated hairpin completion of a single word

Abstract

Hairpin completion is an abstract operation modeling a DNA bio-operation which receives as input a DNA strand $w = xαy \calpha$, and outputs $w^{\prime }=x\alpha y\bar{\alpha }\bar{x}$, where $\bar{x}$ denotes the Watson-Crick complement of $x$. In this paper, we focus on the problem of finding conditions under which the iterated hairpin completion of a given word is regular. According to the numbers of words $\alpha$ and $\calpha$ that initiate hairpin completion and how they are scattered, we classify the set of all words $w$. For some basic classes of words $w$ containing small numbers of occurrences of $\alpha$ and $\calpha$, we prove that the iterated hairpin completion of $w$ is regular. For other classes with higher numbers of occurrences of $\alpha$ and $\calpha$, we prove a necessary and sufficient condition for the iterated hairpin completion of a word in these classes to be regular.

17 pages, 1 figure, submitted to Fundamenta Informaticae