Iterated Hairpin Completions of Non-crossing Words

TL;DR

This paper investigates the properties of iterated hairpin completions of non-crossing words, proving that regularity is decidable and that non-regular cases are also not context-free, advancing understanding in formal language theory.

Contribution

It establishes decidability of regularity for iterated hairpin completions of non-crossing words and shows non-regular, non-context-free cases.

Findings

Regularity of iterated hairpin completion is decidable for non-crossing words.

If not regular, the iterated hairpin completion is also not context-free.

Provides answers to open questions in formal language theory regarding hairpin operations.

Abstract

Iterated hairpin completion is an operation on formal languages that is inspired by the hairpin formation in DNA biochemistry. Iterated hairpin completion of a word (or more precisely a singleton language) is always a context-sensitive language and for some words it is known to be non-context-free. However, it is unknown whether regularity of iterated hairpin completion of a given word is decidable. Also the question whether iterated hairpin completion of a word can be context-free but not regular was asked in literature. In this paper we investigate iterated hairpin completions of non-crossing words and, within this setting, we are able to answer both questions. For non-crossing words we prove that the regularity of iterated hairpin completions is decidable and that if iterated hairpin completion of a non-crossing word is not regular, then it is not context-free either.