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Pseudopower Avoidance

Chiniforooshan E., Kari L., Xu Z.

Fundamenta Informaticae

2012

Vol. 114, nr 1

55-72

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 theWatson-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 }, whereinA 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 &Sigma: 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 . 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.