A proof procedure is an algorithm (technically, a semi-decision procedure) which identifies a formula as valid (or unsatisfiable) when appropriate, and may not terminate when the formula is invalid (satisfiable). Since a proof procedure concerns a logic the procedure takes a special form, superimposing a search strategy on an inference calculus. We will consider a certain collection of proof procedures in the light of an inference calculus format that abstracts the concept of logic programming. This formulation allows us to look beyond SLD-resolution, the proof procedure that underlies Prolog, to generalizations and extensions that retain an essence of logic programming structure. The inference structure used in the formulation of the logic programming concept and first realization, Prolog, evolved from the work done in the subdiscipline called automated theorem proving. While many proof procedures have been developed within this subdiscipline, some of which appear in Volume 1 of this handbook, we will present a narrow selection, namely the proof procedures which are clearly ancestors of the first proof procedure associated with logic programming, SLD-resolution. Extensive treatment of proof procedures for automated theorem proving appear in Bibel [Bibel, 1982], Chang and Lee [Chang and Lee, 1973] and Loveland [Loveland, 1978]. Although the consideration of proof procedures for automated theorem proving began about 1958 we begin our overview with the introduction of the resolution proof procedure by Robinson in 1965. We then review the linear resolution procedures, model elimination and SL-resolution procedures. Our exclusion of other proof procedures from consideration here is due to our focus, not because other procedures are less important historically or for general use within automated or semi-automated theorem process. After a review of the general resolution proof procedure, we consider the linear refinement for resolution and then further restrict the procedure format to linear input resolution. Here we are no longer capable of treating full first-order logic, but have forced ourselves to address a smaller domain, in essence the renameable Horn clause formulas. By leaving the resolution format, indeed leaving traditional formula representation, we see there exists a linear input procedure for all of first-order logic.
Category Theoretic Semantics for Theorem Proving in Logic Programming: Embracing the Laxness
