SCL with Theory Constraints

TL;DR

This paper extends the SCL calculus to SCL(T), enabling model-guided resolution in first-order logic with background theories, improving efficiency by avoiding redundancy and potentially becoming a decision procedure under certain conditions.

Contribution

It introduces the SCL(T) calculus, a novel model-guided resolution method for first-order logic with background theories, eliminating redundancy and enabling decision procedures.

Findings

SCL(T) avoids tautology and subsumption tests due to non-redundancy.

SCL(T) functions as a semi-decision procedure for certain clause sets.

Under finite model property, SCL(T) can be a decision procedure.

Abstract

We lift the SCL calculus for first-order logic without equality to the SCL(T) calculus for first-order logic without equality modulo a background theory. In a nutshell, the SCL(T) calculus describes a new way to guide hierarchic resolution inferences by a partial model assumption instead of an a priori fixed order as done for instance in hierarchic superposition. The model representation consists of ground background theory literals and ground foreground first-order literals. One major advantage of the model guided approach is that clauses generated by SCL(T) enjoy a non-redundancy property that makes expensive testing for tautologies and forward subsumption completely obsolete. SCL(T) is a semi-decision procedure for pure clause sets that are clause sets without first-order function symbols ranging into the background theory sorts. Moreover, SCL(T) can be turned into a decision procedure if the considered combination of a first-order logic modulo a background theory enjoys an abstract finite model property.