Unification problems with one-sided distributivity

Abstract

This work is a study of unification in some equational theories that have a one-sided distributivity axiom: x x (y + z) = x x y + x x z. First one-sided distributivity, the theory which has only this axiom, is studied. It is shown that, although one-sided distributivity is a simple theory in many ways, its unification problem is not trivial, and known universal unification procedures fail to provide a decision procedure for it. We give a unification procedure based on a process of decomposition combined with a generalized occurs check, which may be applied in any permutative theory, and another test. These tests together ensure termination. Next, we show that unification is undecidable if the laws of associativity x + (y + z) = (x + y) + z and a one-sided unit element x x 1 = x are added to one-sided distributivity. Unification under one-sided distributivity with (one-sided) unit element is shown to be as hard as Markov's problem (associative unification), whereas unification under two-sided distributivity, with or without unit element, is NP-hard. The study of these problems is motivated by possible applications in circuit synthesis and by the need for gaining insight in the problem of combining theories with overlapping sets of operator symbols.