Involutively Bordered Words

In this paper we study a generalization of the classical notions of bordered and unbordered words, motivated by DNA computing. DNA strands can be viewed as finite strings over the alphabet {A, G, C, T}, and are used in DNA computing to encode information. Due to the fact that A is Watson-Crick complementary to T and G to C, DNA single strands that are Watson-Crick complementary can bind to each other or to themselves in either intended or unintended ways. One of the structures that is usually undesirable for biocomputation, since it makes the affected DNA string unavailable for future interactions, is the hairpin: If some subsequences of a DNA single string are complementary to each other, the string will bind to itself forming a hairpin-like structure. This paper studies a mathematical formalization of a particular case of hairpins, the Watson-Crick bordered words. A Watson-Crick bordered word is a word with the property that it has a prefix that is Watson-Crick complementary to its suffix. More generally, we investigate the notion of θ-bordered words, where θ is a morphic or antimorphic involution. We show that the set of all θ-bordered words is regular, when θ is an antimorphic involution and the set of all θ-bordered words is context-sensitive when θ is a morphic involution. We study the properties of θ-bordered and θ-unbordered words and also the relation between θ-bordered and θ-unbordered words and certain type of involution codes.