Equations on partial words

¹ Department of Computer Science, University of North Carolina, P.O. Box 26170, Greensboro, North Carolina 27402–6170, USA; blanchet@uncg.edu A research assignment from the University of North Carolina at Greensboro is gratefully acknowledged. Some of this assignment was spent at the LIAFA: Laboratoire d'Informatique Algorithmique: Fondements et Applications of Université Paris 7, Paris, France, and at the University of Debrecen, Debrecen, Hungary.
² Department of Mathematics, Texas A&M University, College Station, TX 77843–3368, USA.
³ Department of Mathematics, Tennessee Tech University, Box 5054, Cookeville, TN 38505–0001, USA.

Received: 8 August 2006
Accepted: 23 October 2007

Abstract

It is well-known that some of the most basic properties of words, like the commutativity (xy = yx) and the conjugacy (xz = zy), can be expressed as solutions of word equations. An important problem is to decide whether or not a given equation on words has a solution. For instance, the equation x^My^N = z^P has only periodic solutions in a free monoid, that is, if x^My^N = z^P holds with integers m,n,p ≥ 2, then there exists a word w such that x, y, z are powers of w. This result, which received a lot of attention, was first proved by Lyndon and Schützenberger for free groups. In this paper, we investigate equations on partial words. Partial words are sequences over a finite alphabet that may contain a number of “do not know” symbols. When we speak about equations on partial words, we replace the notion of equality (=) with compatibility (↑). Among other equations, we solve xy ↑ yx, xz ↑ zy, and special cases of x^myⁿ ↑ z^p for integers m,n,p ≥ 2.