The syntactic monoid of hairpin-free languages

Abstract

The study of hairpin-free words has been initiated in the context of DNA computing. DNA strands that, theoretically speaking, are finite strings over the alphabet {A, G, C, T} are used in DNA computing to encode information. Due to the fact that A is complementary to T and G to C, DNA single strands that are 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 continues the theoretical study of hairpin-free languages. We study algebraic properties of hairpin-free words and hairpins. We also give a complete characterization of the syntactic monoid of the language consisting of all hairpin-free words over a given alphabet and illustrate it with an example using the DNA alphabet.

DNA Computing — Foundations and Implications

“…Several other studies have been published, focusing on formal-language aspects of the hairpin formation. A complete characterization of the syntactic monoid of the language consisting of all hairpin-free words over a given alphabet was given in [71]. The reference [83] described formal language operations of hairpin completion and reduction and studied their closure and other mathematical properties.…”

Watson-Crick Bordered Words and Their Syntactic Monoid

“…More recently, in [10] we have discussed the syntactic monoid properties of the set of all hairpin-free words. In this paper we use these methods to study the algebraic properties of the set of all involution-bordered words.…”

Deciding regularity of hairpin completions of regular languages in polynomial time

“…Referring to [27], α should consist of at least 9 bases, otherwise the bond between α and α is too weak. Hairpin completions are often seen as undesirable byproducts that occur during DNA computations and, therefore, sets of DNA strands have been investigated that do not tend to form hairpins or other undesired hybridizations, see e. g., [4,7,8,13,14] and the references within. On the other hand, DNA algorithms have been designed that make good use of hairpins and hairpin completions.…”