On the Hairpin Incompletion

Hairpin completion and its variant called bounded hairpin completion are operations on formal languages, inspired by a hairpin formation in molecular biology. Another variant called hairpin lengthening has been recently introduced and studied on the closure properties and algorithmic problems concerning several families of languages. In this paper, we introduce a new operation of this kind, called hairpin incompletion which is not only an extension of bounded hairpin completion, but also a restricted (bounded) variant of hairpin lengthening. Further, the hairpin incompletion operation provides a formal language theoretic framework that models a bio-molecular technique nowadays known as Whiplash PCR. We study the closure properties of language families under both the operation and its iterated version. We show that a family of languages closed under intersection with regular sets, concatenation with regular sets, and finite union is closed under one-sided iterated hairpin incompletion, and that a family of languages containing all linear languages and closed under circular permutation, left derivative and substitution is also closed under iterated hairpin incompletion.