The Precise Complexity of Reasoning in $\mathcal{ALC}$ with $\omega$-Admissible Concrete Domains (Extended Version)

TL;DR

This paper establishes that reasoning in $ ext{ALC}$ extended with $ ext{omega}$-admissible concrete domains is ExpTime-complete, providing an algorithm and analyzing the impact of feature assertions on complexity.

Contribution

It introduces a type elimination algorithm for $ ext{ALC}$ with $ ext{omega}$-admissible concrete domains and shows that reasoning remains ExpTime-complete even with feature assertions.

Findings

Deciding consistency is ExpTime-complete for $ ext{ALC}( ext{D})$ with $ ext{omega}$-admissible domains.

The proposed algorithm effectively handles concept satisfiability with concrete domains.

Feature assertions can be added without increasing complexity under certain conditions.

Abstract

Concrete domains have been introduced in the context of Description Logics to allow references to qualitative and quantitative values. In particular, the class of $\omega$-admissible concrete domains, which includes Allen's interval algebra, the region connection calculus (RCC8), and the rational numbers with ordering and equality, has been shown to yield extensions of $\mathcal{ALC}$ for which concept satisfiability w.r.t. a general TBox is decidable. In this paper, we present an algorithm based on type elimination and use it to show that deciding the consistency of an $\mathcal{ALC}(\mathfrak{D})$ ontology is ExpTime-complete if the concrete domain $\mathfrak{D}$ is $\omega$-admissible and its constraint satisfaction problem is decidable in exponential time. While this allows us to reason with concept and role assertions, we also investigate feature assertions $f(a,c)$ that can specify a constant $c$ as the value of a feature $f$ for an individual $a$. We show that, under conditions satisfied by all known $\omega$-admissible domains, we can add feature assertions without affecting the complexity.