First-Order Logic Theorem Proving and Model Building via Approximation and Instantiation

TL;DR

This paper introduces a novel method for first-order logic theorem proving that uses approximation and instantiation to simplify clause sets, enabling decidable satisfiability checks while preserving unsatisfiability, and guiding refutations.

Contribution

It presents a new approximation technique that extends signatures and preserves unsatisfiability, improving refutational completeness in first-order logic theorem proving.

Findings

The approach is refutationally complete.

It preserves unsatisfiability during approximation.

Refutations can be lifted or refined based on simplified clause sets.

Abstract

In this paper we consider first-order logic theorem proving and model building via approximation and instantiation. Given a clause set we propose its approximation into a simplified clause set where satisfiability is decidable. The approximation extends the signature and preserves unsatisfiability: if the simplified clause set is satisfiable in some model, so is the original clause set in the same model interpreted in the original signature. A refutation generated by a decision procedure on the simplified clause set can then either be lifted to a refutation in the original clause set, or it guides a refinement excluding the previously found unliftable refutation. This way the approach is refutationally complete. We do not step-wise lift refutations but conflicting cores, finite unsatisfiable clause sets representing at least one refutation. The approach is dual to many existing approaches in the literature because our approximation preserves unsatisfiability.