Word Equations with Two Variables

The problem whether the set of all equations that are satisfiable in some free semigroup or, equivalently, in an algebra of words with concatenation is recursive (usually called the satisfiability problem for semigroup equations) was first formulated by A.A. Markov in early sixties (see [3]). Special cases of the problem were solved affirmatively by A.A. Markov (see [3]), Yu.I. Khmelevskiı̆ [8], [7], G. Plotkin, [14] and A. Lentin [11]. The full positive solution, was given by G.S. Makanin in a paper [12], which is long and very technical. Makanin’s decision procedure for equational satisfiability in semigroups has received a lot of attention in the literature. Undoubtedly, this is because the notion of an algebra of words (or strings) with the operation of concatenation is of fundamental importance in computer science: many algorithms and data structures refer to words. Thus, several improvements of Makanin’s algorithm have been given by H. Abdulrab, J.-P. Pecuchet, K. Schulz, A. Kościelski and L. Pacholski (see [2], [13], [15], [9], and [10]), and attempts have even been made to implement the algorithm (see [1]). Moreover, related unification problems have been studied. In particular, J. Jaffar, in [6], basing on the Makanin’s decision procedure, described an algorithm which, when an equation has a solution, generates all its solutions and halts if the set of solutions is finite. An important fact used in the Makanin’s algorithm and in the unification algorithms based on it, is that the periodicity exponent of a minimal solution of a word equation can be bounded by a recursive function of the length of the equation. In fact, V.K.Bulitko, in [4], proved that if d is the length of an equation, then the index of periodicity of its minimal solution (see below) does not exceed (6d)2 2d4 + 2. Kościelski and Pacholski ([9], [10]) forced this bound down to 21.07d . They also prove a lower bound of 20.29d for the exponent of periodicity of minimal solutions of a word equation of length d. Although the bound on the exponent of periodicity given by Kościelski and Pacholski gave an overexponential improvement of the algorithm its complexity is still so high that it prohibits any applications in practice. Moreover, Kościelski and Pacholski [10] proved that the problem of the solvability of word equations is N P-hard, even if a linear bound is put on the length of possible solutions. Thus, for a given constant c > 2 the problem of the existence of a solution of length cd for an equation of length d is N P-complete. This implies, that there does not exist any fast algorithm, which decides solvability of all word equations and suggests that good algorithms can be found only for restricted classes of equations. This paper contains the first report on our research project with the aim to describe classes of word equations for which fast algorithms, deciding solvability or giving actual solutions, exist. In this paper by ”fast” we mean ”deterministic polynomial time”. Of course for many actual applications it would be better to consider more restricted classes like linear time or DTIME(nlog(n)). This problem will be considered in subsequent papers. We consider equations which have at most two distinct variables. For such equations we give a deterministic polynomial time algorithm deciding their solvability. Our technique and algorithm is based on the notion of an ”equation in exponent” which has been introduced by Yu.I. Khmelevskiı̆ [7].