Abstract

Hairpin formations arise in biochemical processes and play an important role in DNA-computing. We study language theoretical properties of hairpin formations and our new results concern the hairpin completion Hκ(L1,L2) of two regular languages L1 and L2 and the iterated hairpin lengthening HLκ∗(L) of any language L.Assume that L1 and L2 belong to a certain variety of regular languages which satisfies a mild closure property (being closed by a restricted concatenation), then either Hκ(L1,L2) is not regular or it belongs to the same variety as L1 and L2. This result applies, in particular, to the class of first-order definable languages (which is the class of aperiodic or star-free languages) and it applies to the class of first-order definable languages in two variables with predicates < and successor +1 (i.e., languages in the variety LDA).Furthermore, we solve an open question from Manea et al. [15]: we prove that regular languages are closed under iterated hairpin lengthening. (This has been independently shown in Manea et al. [16].) However, the result here is more precise: if L belongs to a class of languages which satisfies a certain closure property with respect to locally testable languages, then HLκ∗(L) belongs to the same class. This applies to the class of aperiodic languages and many other classes. We also show that the variety DA is not closed under the operation L↦HLκ∗(L). For LDA the situation remains open.

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