On the regularity of iterated hairpin completion of a single word

TL;DR

This paper investigates the conditions under which the iterated hairpin completion of a DNA-inspired word operation results in a regular language, providing classifications and necessary and sufficient conditions for regularity.

Contribution

It offers a classification of words based on occurrences of specific substrings and establishes criteria for the regularity of their iterated hairpin completions.

Findings

Regularity for words with few occurrences of α and ᾱ

Necessary and sufficient conditions for regularity in complex cases

Classification of words based on initiation and scattering of hairpin completion

Abstract

Hairpin completion is an abstract operation modeling a DNA bio-operation which receives as input a DNA strand $w = x\alpha y \calpha$, and outputs $w' = x \alpha y \bar{\alpha} \bar{x}$, 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.