A resolution principle for constrained logics

Abstract

We introduce a constrained logic scheme with a resolution principle for clauses whose variables are constrained by a constraint theory. Constraints can be seen as quantifier restrictions filtering out the values that any interpretation of the underlying constraint theory can assign to the variables of a formula with such restricted quantifiers. We present a resolution principle for constrained clauses, where unification is replaced by testing constraints for satisfiability over the constraint theory. We show that constrained resolution is sound and complete in that a set of constrained clauses is unsatisfiable over the constraint theory if and only if for each model of the constraint theory we can deduce a constrained empty clause whose constraint is satisfiable in that model. We demonstrate that we cannot require a better result in general. But we give conditions, under which at most finitely many such empty clauses are needed or even better only one empty clause as in classical resolution, sorted resolution or resolution with theory unification.