Pseudopower Avoidance

Repetition avoidance has been intensely studied since Thue's work in the early 1900's. In this paper, we consider another type of repetition, called pseudopower, inspired by the Watson-Crick complementarity property of DNA sequences. A DNA single strand can be viewed as a string over the four-letter alphabet {A,C,G, T}, wherein A is the complement of T, while C is the complement of G. Such a DNA single strand will bind to a reverse complement DNA single strand, called its Watson-Crick complement, to form a helical double-stranded DNA molecule. The Watson-Crick complement of a DNA strand is deducible from, and thus informationally equivalent to, the original strand. We use this fact to generalize the notion of the power of a word by relaxing the meaning of “sameness” to include the image through an antimorphic involution, the model of DNA Watson-Crick complementarity. Given a finite alphabet Σ, an antimorphic involution is a function θ : Σ* → Σ* which is an involution, i.e., θ2 equals the identity, and an antimorphism, i.e., θ(uv) = θ(v)θ(u), for all u ∈ Σ*. For a positive integer k, we call a word w a pseudo-kth-power with respect to θ if it can be written as w = u1 ... uk, where for 1 ≤ i, j ≤ k we have either ui = uj or ui = θ(uj). The classical kth-power of a word is a special case of a pseudo-kth-power, where all the repeating units are identical. We first classify the alphabets Σ and the antimorphic involutions θ for which there exist arbitrarily long pseudo-kth-power-free words. Then we present efficient algorithms to test whether a finite word w is pseudo-kth-power-free.