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\bar{α}, and outputs w' = xαy\bar{α}\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 α and \bar{α} 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 α and \bar{α}, we prove that the iterated hairpin completion of w is regular. For other classes with higher numbers of occurrences of α and \bar{α}, we prove a necessary and sufficient condition for the iterated hairpin completion of a word in these classes to be regular.