Deciding Regularity of Hairpin Completions of Regular Languages in Polynomial Time

TL;DR

This paper proves that determining whether the hairpin completion of regular languages remains regular is decidable in polynomial time, providing an efficient algorithm and analyzing growth properties.

Contribution

It introduces a polynomial time algorithm for deciding the regularity of hairpin completions of regular languages and analyzes their growth behavior.

Findings

Decidability of regularity is polynomial time

Hairpin completions are unambiguous linear context-free languages

Growth of hairpin completion matches that of the original languages

Abstract

The hairpin completion is an operation on formal languages that has been inspired by the hairpin formation in DNA biochemistry and by DNA computing. In this paper we investigate the hairpin completion of regular languages. It is well known that hairpin completions of regular languages are linear context-free and not necessarily regular. As regularity of a (linear) context-free language is not decidable, the question arose whether regularity of a hairpin completion of regular languages is decidable. We prove that this problem is decidable and we provide a polynomial time algorithm. Furthermore, we prove that the hairpin completion of regular languages is an unambiguous linear context-free language and, as such, it has an effectively computable growth function. Moreover, we show that the growth of the hairpin completion is exponential if and only if the growth of the underlying languages is exponential and, in case the hairpin completion is regular, then the hairpin completion and the underlying languages have the same growth indicator.