The Precise Complexity of Reasoning in $\mathcal{ALC}$ with $ω$-Admissible Concrete Domains (Extended Version)
Stefan Borgwardt, Filippo De Bortoli, Patrick Koopmann
TL;DR
The paper addresses the precise complexity of reasoning in $\\mathcal{ALC}(\\mathfrak{D})$ when the concrete domain $\\mathfrak{D}$ is $\\omega$-admissible. It introduces a type-elimination algorithm built on augmented types, local systems, and ABox types to decide ontology consistency, proving $ExpTime$-completeness provided the CSP$$(\\mathfrak{D})$ is decidable in exponential time. It further extends the framework to predicate and feature assertions, showing preservation of complexity for homogeneous $\\omega$-admissible domains and reducing certain assertions to standard $ALC(\\mathfrak{D})$ reasoning. The results apply to canonical examples like Allen's interval algebra, RCC8, and $\\mathbb{Q}$, and pave the way for extensions to $ALCI(\\mathfrak{D})$ and $ALCQ(\\mathfrak{D})$ as well as definability and abduction questions in this setting.
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 $ω$-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 $ω$-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 $ω$-admissible domains, we can add feature assertions without affecting the complexity.