Efficient solving of the word equations in one variable

Abstract

A word equation in n variables x1,…, xn over an alphabet C is a pair E = (φ(x1,…,xn),Ψ(x1,…,xn)) of words over the alphabet C ∪ {x1,…, xn}. A solution of E is any n-tuple (X1,…, Xn) of words over C such that φ(X1,…,Xn)=Ψ(X1,…,Xn). The existence of a solution for any given equation E is decidable, as shown by Yu. I. Khmelevskiǐ [3] for up to four variables and by G. S. Makanin [6] for any number of variables. However, as shown by A. Kow and L. Pacholskl [4], these impressive decidability results can unfortunately not be matched by efficient algorithms of resolution, except for some restricted classes of equations. In this vein, W. Charatonik and L. Pacholski [1] give a polynomial algorithm, in terms of the equation length |E| - |φ| + |Ψ| for the equations in two variables and very roughly estimate at O(|E|5) the time complexity for solving those in one variable. For the latter, using rather fine combinatorial methods, we give an O(|E| log |E|) algorithm, the best one so far known.