An Algorithm for Distributive Unification

We consider unification of terms under the equational theory of two-sided distributivity D with the axioms x*(y+z) = x*y + x*z and (x+y)*z = x*z + y*z. The main result of this paper is that Dunification is decidable by giving a non-deterministic transformation algorithm. The generated unification are: an AC1-problem with linear constant restrictions and a second-order unification problem that can be transformed into a word-unification problem that can be decided using Makanin's algorithm. This solves an open problem in the field of unification. Furthermore it is shown that the word-problem can be decided in polynomial time, hence D-matching is NP-complete. Introduction Unification (solving equations) in equationally defined theories has several applications in computer science, an overview and further references can be found in [Si89], [Ki89], [BS94]. Since 1982, there was interest in appying methods of unification also if distributive axioms were present. These Distributive axioms are very common in algebraic structures and every-day mathematics, for example in solving Diophantine equations, but solving equations in these structures is a hard task, for example there is no algorithm for solving Diophantine equations. P. Szabó [Sz82] has considered unification in several equational theories where some axioms are dropped (not the distributive ones), proved undecidability results for several combinations, a minimal set of axioms being two-sided distributivity plus associativity of addition. Peter Szabó left open the question of a unification algorithm for the theory D, where all axioms but the two distributive ones are dropped. Tidén and Arnborg [TA87] investigated other combinations motivated by the occurrence of distributive axioms in a modelling of communicating processes by Bergstra and Klop [BK86]. In [TA87] there is a unification algorithm for one distributive axiom and complexity results for unification in combinations of one distributive axiom with a multiplicative unit element. In particular they showed that there is another small set of axioms, one-sided distributivity, associativity of addition and a multiplicative unit, for which unification is undecidable. The most recent results on Dunification are in [Con93], where Contejean gave a unification algorithm for the subset of product terms. The author gave an algorithm for the theory of one distributive axiom with a multiplicative unit ([Sch93]). It turned out that the unification problem for this theory has a complexity that is comparable with the complexity of associative unification.