Deciding regularity of hairpin completions of regular languages in polynomial time

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.

Two-Sided Derivatives for Regular Expressions and for Hairpin Expressions

“…Given an integer k > 0 and an involution H over an alphabet Γ, the hairpin k-completion of two languages L 1 and L 2 over Γ is the language H k (L 1 , L 2 ) = {αβγH(β)H(α) | α, β, γ ∈ Γ * ∧ (αβγH(β) ∈ L 1 ∨ βγH(β)H(α) ∈ L 2 ) ∧ |β| = k} (see Figure 1). Hairpin completion has been deeply studied [2,6,9,10,11,12,13,14,16,18,19,20,21,22,23]. The hairpin completion of formal languages has been introduced in [9] by reason of its application to biochemistry.…”

“…It aroused numerous studies that investigate theoretical and algorithmic properties of hairpin completions or related operations (see for example [14,18,21]). One of the most recent result concerns the problem of deciding regularity of hairpin completions of regular languages; it can be found in [11] as well as a complete bibliography about hairpin completion. Hairpin completions of regular languages are proved to be linear context-free from [9].…”