On Solving Equations and Disequations

Abstract

We are interested in the problem of solving a system 〈s_i= t_i: 1≤i≤n,p_j≠q_j:1≤j≤m〉 of equations and disequations, also known as disunification. Solutions to disunification problems are substitutions for the variables of the problem that make the two terms of each equation equal, but leave those of the disequations different. We investigate this in both algebraic and logical contexts where equality is defined by an equational theory and more generally by a definite clause equality theory E. We show how E-disunification can be reduced to E-unification, that is solving equations only, and give a disunification algorithm for theories given a unification algorithm. In fact this result shows that for theories where the solutions of all unification problems can be represented by finitely many substitutions, the solutions of all disunifieation problems can also be represented finitely. We apply disunifieation to handle negation in logic programming with equality in a similar style to Colmerauer‘s logic programming with rational trees, and to represent many solutions to AC-unification problems by a few solutions to AC1-disunification problems.