On the iterated hairpin completion

Abstract

The hairpin completion is a natural operation on formal languages which has been inspired by biochemistry and DNA-computing. In this paper we solve two problems which were posed first in 2008 and 2009, respectively, and still left open: 1.) It is known that the iterated hairpin completion of a regular language is not context-free in general, but it was open whether the iterated hairpin completion of a singleton or finite language is regular or at least context-free. We will show that it can be non-context-free. (It is of course context-sensitive.) 2.) A restricted but also very natural variant of the hairpin completion is the bounded hairpin completion. It was unknown whether the iterated bounded hairpin completion of a regular language remains regular. We prove that this is indeed the case. Actually we derive a more general result. We will present a general representation of the iterated bounded hairpin completion for any language using basic operations. Thus, each language class closed under these basic operations is also closed under iterated bounded hairpin completion.