Unification problems with one-sided distributivity

Abstract

We show that unification in the equational theory defined by the one-sided distributivity law x × (y+z)=x×y+x×z is decidable and that unification is undecidable if the laws of associativity x+(y+z)=(x+y)+z and unit element 1×x=x× 1=x are added. Unification under one-sided distributivity with unit element is shown to be as hard as Markov's problem, whereas unification under two-sided distributivity, with or without unit element, is NP-hard. A quadratic time unification algorithm for one-sided distributivity, which may prove interesting since available universal unification procedures fail to provide a decision procedure for this theory, is outlined. 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.