Superposition for Fixed Domains

TL;DR

This paper introduces a novel superposition calculus that explicitly handles existential variables and fixed domains, enabling more precise reasoning about models in first-order logic theories.

Contribution

It presents the first superposition calculus capable of explicitly representing existential variables and computing within a fixed domain, enhancing reasoning capabilities.

Findings

The calculus is sound and refutationally complete for fixed domain semantics.

It can prove properties of minimal models for saturated Horn theories.

Enables reasoning beyond traditional superposition approaches.

Abstract

Superposition is an established decision procedure for a variety of first-order logic theories represented by sets of clauses. A satisfiable theory, saturated by superposition, implicitly defines a minimal term-generated model for the theory. Proving universal properties with respect to a saturated theory directly leads to a modification of the minimal model's term-generated domain, as new Skolem functions are introduced. For many applications, this is not desired. Therefore, we propose the first superposition calculus that can explicitly represent existentially quantified variables and can thus compute with respect to a given domain. This calculus is sound and refutationally complete in the limit for a first-order fixed domain semantics. For saturated Horn theories and classes of positive formulas, we can even employ the calculus to prove properties of the minimal model itself, going beyond the scope of known superposition-based approaches.