Equations on Partial Words
- Cite this paper as:
- Blanchet-Sadri F., Blair D.D., Lewis R.V. (2006) Equations on Partial Words. In: Královič R., Urzyczyn P. (eds) Mathematical Foundations of Computer Science 2006. MFCS 2006. Lecture Notes in Computer Science, vol 4162. Springer, Berlin, Heidelberg
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 xmyn = zp has only periodic solutions in a free monoid, that is, if xmyn = zp 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 xmyn ↑ zp for integers m, n, p ≥2. ...
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