Reasoning with Axioms: Theory and Pratice

TL;DR

This paper provides a formal framework for reasoning with axioms in description logics, analyzing optimization techniques like lazy unfolding and absorption to improve efficiency while ensuring correctness.

Contribution

It introduces a formal framework for these techniques, establishes conditions for their safe application, and demonstrates how to relax safety conditions for better efficiency.

Findings

Formal framework for reasoning with axioms

Conditions for safe application of optimization techniques

Improved efficiency in subsumption testing algorithms

Abstract

When reasoning in description, modal or temporal logics it is often useful to consider axioms representing universal truths in the domain of discourse. Reasoning with respect to an arbitrary set of axioms is hard, even for relatively inexpressive logics, and it is essential to deal with such axioms in an efficient manner if implemented systems are to be effective in real applications. This is particularly relevant to Description Logics, where subsumption reasoning with respect to a terminology is a fundamental problem. Two optimisation techniques that have proved to be particularly effective in dealing with terminologies are lazy unfolding and absorption. In this paper we seek to improve our theoretical understanding of these important techniques. We define a formal framework that allows the techniques to be precisely described, establish conditions under which they can be safely applied, and prove that, provided these conditions are respected, subsumption testing algorithms will still function correctly. These results are used to show that the procedures used in the FaCT system are correct and, moreover, to show how efficiency can be significantly improved, while still retaining the guarantee of correctness, by relaxing the safety conditions for absorption.