Word unification and transformation of generalized equations

Abstract

Makanin's algorithm shows that it is decidable whether a word equation has a solution. The original description was hard to understand and not designed for implementation. Since words represent a fundamental data type, various authors have given improved descriptions. In this paper we present a version of the algorithm that probably cannot be further simplified without fundamentally new insights that exceed Makanin's original ideas. We give a transformation that is efficient and conceptually simple and applies to arbitrary generalized equations. No further subprocedure is needed for the generation of the search tree. Particular attention is then given to the proof that proper generalized equations are transformed into proper generalized equations. This point, which is important for the termination argument, was treated erroneously in other papers. We also show that a combination of the basic algorithm for string-unification and Makanin's algorithm offers a simple solution to the problem of terminating minimal and complete word unification.