Schematic Saturation for Decision and Unification Problems

Abstract

We show that if a set of clauses S is finitely saturated by the Resolution inference system, and if S ∪ N can be finitely saturated by Resolution, then S ∪ Nθ can be saturated in polynomial time, for any ground substitution θ. If N contains only unit clauses consisting of predicate symbols with distinct variables as arguments, then this implies the polynomial time decidability of ground unit clauses modulo S. But for a given N, this implies decidability modulo S of clauses of a particular structure. In addition, if S is a set of Horn clauses, and if a negative literal is selected in each clause containing one, then this implies the polynomial time solvability of the unification problem for the theory Nθ modulo S. We also consider the unification problem, where only certain positions of a clause are allowed to contain variables. We mark those positions, and show that if S ∪ N can be finitely saturated, under a modification of the duplicate removal rule, then S ∪ Nθ, can be saturated in exponential time for any ground substitution θ, and therefore the unification problem modulo S is solvable in exponential time for such goals. However, if we further restrict the conditions, then unification is solvable in polynomial time for such goals. The theories of membership, append and addition fall into this last class. Therefore, unifiability in the theory of membership, append, and addition is decidable in polynomial time for unit clauses if the last position of the predicate is ground.