7 - Resolution

Summary

In this chapter we study another form of inference, which forms the keystone of logic programming and certain theorem-proving systems. We do not aim at giving a complete introduction to the theory of logic programming; rather, we want to show how resolution is connected with other formalisms and to provide a proof-theoretic road to the completeness theorem for SLD-resolution.

The first three sections deal with propositional resolution, unification and resolution in predicate logic. The last two sections illustrate for Cp and Ip how deductions in a suitably chosen variant of the Gentzen systems can be directly translated into deductions based on resolution, which often permits us to lift strategies for proof search in Gentzen systems to resolution-based systems. The extension of these methods to predicate logic is more or less straightforward.

Introduction to resolution

Propositional linear resolution is a “baby example” of resolution methods, which is not of much interest in itself, but may serve as an introduction to the subject.

We consider programs consisting of finitely many sequents (clauses) of the form Γ ⇒ P, P a propositional variable and Γ a finite multiset of propositional variables (“definite clauses”, “Horn clauses” or “Horn sequents”). A goal or query Γ is a finite (possibly empty) set of propositional variables, and may be identified with the sequent Γ ⇒. [] is the empty goal.