Darlington, J.

Some theorem-proving strategies based on the resolution principle


These programs have generated proofs of some interesting propositions of number theory, in addition to theorems of first-order functional logic and group theory. Quantifiers do not occur in these formulae, since existentially quantified variables have been replaced by functions of universally quantified ones, and the remaining variables may therefore be taken as universally quantified. Starting with a finite number of individual constants, all possible substitutions were made in the input clauses, and the resulting conjunction of substitution instances was tested for truth-functional consistency. Davis in fact proved that any substitution instance Lo/L2 v vL„ that contains at least one unmated literal may be erased without affecting consistency, and that the test for consistency may be confined to'linked conjuncts', i.e., conjuncts (or clauses) wherein each literal Li is negated by a mate This'theorem on linked conjuncts' provides a necessary, though not a sufficient, condition of relevance: any substitution instance containing an unmated literal is demonstrably irrelevant to consistency and may be deleted forthwith, but the remaining substitution instances are not necessarily all relevant.