An extension of the Lyndon--Schützenberger result to pseudoperiodic words

  • Authors:

  • Elena Czeizler;Eugen Czeizler;Lila Kari;Shinnosuke Seki

  • Affiliations:

  • Department of Computer Science, The University of Western Ontario, London, Ontario, Canada N6A 5B7;Department of Computer Science, The University of Western Ontario, London, Ontario, Canada N6A 5B7;Department of Computer Science, The University of Western Ontario, London, Ontario, Canada N6A 5B7;Department of Computer Science, The University of Western Ontario, London, Ontario, Canada N6A 5B7

  • Venue:
  • Information and Computation
  • Year:
  • 2011

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Abstract

One of the particularities of information encoded as DNA strands is that a string u contains basically the same information as its Watson-Crick complement, denoted here as @q(u). Thus, any expression consisting of repetitions of u and @q(u) can be considered in some sense periodic. In this paper, we give a generalization of Lyndon and Schutzenberger's classical result about equations of the form u^l=v^nw^m, to cases where both sides involve repetitions of words as well as their complements. Our main results show that, for such extended equations, if l=5,n,m=3, then all three words involved can be expressed in terms of a common word t and its complement @q(t). Moreover, if l=5, then n=m=3 is an optimal bound. These results are established based on a complete characterization of all possible overlaps between two expressions that involve only some word u and its complement @q(u), which is also obtained in this paper.