Formal verification of a proof procedure for the description logic ALC

TL;DR

This paper presents a formally verified proof procedure for the description logic ALC, ensuring soundness, completeness, and termination through Isabelle proof assistant, enhancing reliability of DL reasoners.

Contribution

It introduces a semantic tableau-based proof procedure for ALC and formally verifies its properties using Isabelle, which was not previously done for such reasoners.

Findings

Proved soundness, completeness, and termination of the ALC proof procedure.

Validated the proof procedure through formal verification in Isabelle.

Provided a certified basis for DL reasoners like RACER and FaCT++.

Abstract

Description Logics (DLs) are a family of languages used for the representation and reasoning on the knowledge of an application domain, in a structured and formal manner. In order to achieve this objective, several provers, such as RACER and FaCT++, have been implemented, but these provers themselves have not been yet certified. In order to ensure the soundness of derivations in these DLs, it is necessary to formally verify the deductions applied by these reasoners. Formal methods offer powerful tools for the specification and verification of proof procedures, among them there are methods for proving properties such as soundness, completeness and termination of a proof procedure. In this paper, we present the definition of a proof procedure for the Description Logic ALC, based on a semantic tableau method. We ensure validity of our prover by proving its soundness, completeness and termination properties using Isabelle proof assistant. The proof proceeds in two phases, first by establishing these properties on an abstract level, and then by instantiating them for an implementation based on lists.