Formal language theory of hairpin formations

The (bounded) hairpin completion, the hairpin lengthening, and their iterated versions are operations on formal languages which have been inspired by the hairpin formation in DNA biochemistry. In this paper we discuss the hairpin formations from a language theoretic point of view. The hairpin completion of a formal language has been defined in 2006. In the first paper on this topic it has been shown that the hairpin completion of a regular language is not necessarily regular but always linear context-free. In the same paper the open problem was stated, if it is decidable, whether the hairpin completion of a regular language is regular. We solved this problem positively in 2009. Here, we prove that the problem is NL-complete and we present an algorithm whose time complexity is bounded by a polynomial of degree 8. As a byproduct of our technique to prove the complexities, we obtain that the hairpin completion of a regular language is actually an unambiguous linear context-free language. In addition, we provide results concerning language classes within the regular languages. We show that if the hairpin completion of an aperiodic (i. e., star-free) language is regular, then the hairpin completion is aperiodic, too. The same is true for the language class induced by the variety LDA. The hairpin lengthening is a variant of the hairpin completion which has been investigated first in 2010. The hairpin lengthening of a regular language seems to behave quite similar to the hairpin completion: It is not necessarily regular but linear context-free. We prove, however, that the hairpin lengthening of a regular language may be an inherent ambiguous linear language. We also consider the problem if it is decidable, whether the hairpin lengthening of a regular language is regular, but we are only able to prove decidability for the one-sided case of hairpin lengthening. (The one-sided case is closer to biochemistry, yet the two-sided case is more interesting from a theoretic point of view). The bounded hairpin completion can be seen as a weaker variant of the hairpin completion. It is well-known that all language classes in the Chomsky hierarchy are closed under bounded hairpin completion and that context-free, context-sensitive, and recursively enumerable languages are closed under iterated bounded hairpin completion. In 2009 it was asked in literature, whether the regular languages are closed under iterated bounded hairpin completion as well. We solve this question by presenting a more general result. We give an effective representation for the iterated bounded hairpin completion which uses union, intersection with regular sets, and concatenation with regular. Thus, all language classes which are (effectively) closed under these basic operations are also (effectively) closed under iterated bounded hairpin completion. This applies to all classes in the Chomsky hierarchy and to all usual complexity classes. Furthermore, we give an exponential lower and up-