Introduction

A seminal result of Makanin 1977 states that the existential theory of equations over free monoids is decidable. Makanin achieved this result by presenting an algorithm which solves the satisfiability problem for word equations with constants. The satisfiability problem is usually stated for a single equation, but this is no loss of generality.

This chapter provides a self-contained presentation of Makanin's result. The presentation has been inspired by Schulz 1992a. In particular, we show the result of Makanin in a more general setting, due to Schulz, by allowing that the problem instance is given by a word equation L = R together with a list of rational languages Lx ⊆ A*, where x ∈ Ω denotes an unknown and A is the alphabet of constants. We will see that it is decidable whether or not there exists a solution σ:Ω → A* which, in addition to σ(L) = σ(R), satisfies the rational constraints σ(x)∈ Lx for all x ∈ Ω. Using an algebraic viewpoint, rational constraints mean to work over some finite semigroup, but we do not need any deep result from the theory of finite semigroups. The presence of rational constraints does not make the proof of Makanin's result much harder; however, the more general form is attractive for various applications.

In the following we explain the outline of the chapter; for some background information and more comments on recent developments we refer to the Notes.