Automation of Reasoning pp 55-65 | Cite as

# Automatic Theorem Proving With Renamable and Semantic Resolution

## Abstract

The theory of J. A. Robinson’s resolution principle, an inference rule for first-order predicate calculus, is unified and extended. A theorem-proving computer program based on the new theory is proposed and the proposed semantic resolution program is compared with hyper-resolution and set-of-support resolution programs. Renamable and semantic resolution are defined and shown to be identical. Given a model *M*, semantic resolution is the resolution of a latent clash in which each “electron” is at least sometimes false under *M;* the nucleus is at least sometimes true under *M.*

The completeness theorem for semantic resolution and all previous completeness theorems for resolution (including ordinary, hyper-, and set-of-support resolution) can be derived from a slightly more general form of the following theorem. If *U* is a finite, truth-functionally un-satisfiable set of nonempty clauses and if *M* is a ground model, then there exists an unresolved maximal semantic clash*E* _{ 1 },
*E* _{ 2 }, ..., *E* _{ q },
*C* with nucleus *C* such that any set containing *C* and one or more of the electrons E_{1}, E_{2}, ..., Eq is an unresolved semantic clash in *U.*

### Keywords

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