Watson-Crick Powers of a Word

Abstract

AbstractIn this paper we define and investigate the binary word operation of strong-\(\theta \)-catenation (denoted by \(\otimes \)) where \(\theta \) is an antimorphic involution modelling the Watson-Crick complementarity of DNA single strands. When iteratively applied to a word u, this operation generates all the strong-\(\theta \)-powers of u (defined as any word in \(\{u, \theta (u)\}^+\)), which amount to all the Watson-Crick powers of u when \(\theta = \theta _{DNA}\) (the antimorphic involution on the DNA alphabet \(\varDelta = \{A, C, G, T\}\) that maps A to T and C to G). In turn, the Watson-Crick powers of u represent DNA strands usually undesirable in DNA computing, since they attach to themselves via intramolecular Watson-Crick complementarity that binds u to \(\theta _{DNA}(u)\), and thus become unavailable for other computational interactions. We find necessary and sufficient conditions for two words u and v to commute with respect to the operation of strong-\(\theta \)-catenation. We also define the concept of \(\otimes \)-primitive root pair of a word, and prove that it always exists and is unique.KeywordsDNA computingMolecular computingBinary word operationsAlgebraic properties