A Calculus for Generating Ground Explanations (Technical Report)

TL;DR

This paper introduces a modified superposition calculus designed to generate ground explanations for clause satisfiability, with proofs of correctness, completeness, and conditions for termination, aiding human understanding in automated reasoning.

Contribution

It presents a novel calculus for generating explanations in satisfiability problems, extending abductive reasoning with formal guarantees and practical procedures.

Findings

Proves correctness and completeness of the calculus.

Develops a termination condition for explanation generation.

Provides a procedure for generating human-useful explanations.

Abstract

We present a modification of the superposition calculus that is meant to generate explanations why a set of clauses is satisfiable. This process is related to abductive reasoning, and the explanations generated are clauses constructed over so-called abductive constants. We prove the correctness and completeness of the calculus in the presence of redundancy elimination rules, and develop a sufficient condition guaranteeing its termination; this sufficient condition is then used to prove that all possible explanations can be generated infinite time for several classes of clause sets, including many of interest to the SMT community. We propose a procedure that generates a set of explanations that should be useful to a human user and conclude by suggesting several extensions to this novel approach.