On the Iterated Hairpin Completion

TL;DR

This paper investigates the properties of the iterated hairpin completion operation on formal languages, proving closure for broad language classes and providing counterexamples for regularity of singleton completions.

Contribution

It establishes that all classes closed under certain operations are also closed under iterated bounded hairpin completion and shows that singleton languages can have non-regular iterated hairpin completions.

Findings

Regular languages are closed under iterated bounded hairpin completion.

All standard language classes are closed under this operation.

Some singleton languages have non-regular, even non-context-free, iterated hairpin completions.

Abstract

The (bounded) hairpin completion and its iterated versions are operations on formal lan- guages which have been inspired by the hairpin formation in DNA-biochemistry. The paper answers two questions asked in the literature about the iterated hairpin completion. The first question is whether the class of regular languages is closed under iterated bounded hairpin completion. Here we show that this is true by providing a more general result which applies to all the classes of languages which are closed under finite union, intersection with regular sets, and concatenation with regular sets. In particular, all Chomsky classes and all standard complexity classes are closed under iterated bounded hairpin completion. In the second part of the paper we address the question whether the iterated hairpin completion of a singleton is always regular. In contrast to the first question, this one has a negative answer. We exhibit an example of a singleton language whose iterated hairpin completion is not regular, actually it is not context-free, but context-sensitive.