Robustness of Constraint Automata for Description Logics with Concrete Domains

TL;DR

The paper addresses the decidability and optimal complexity of the knowledge-base consistency problem for description logics with concrete domains, formulated as $\mathcal{ALCO}(\mathcal{D})$. It introduces constraint automata parameterized by $\mathcal{D}$ that operate on infinite data trees and encode CD-restrictions as symbolic constraints in transitions, enabling a reduction to the nonemptiness problem for $\text{Büchi}$ tree automata. Under conditions such as the completion property, amalgamation property, $\omega$-compactness, bounded arity, and $k$-EXPTIME CSP, the nonemptiness problem lies in $EXPTIME$, yielding $EXPTIME$-membership for ontology consistency and extending to inverse roles, functional roles and constraint assertions. Overall, the approach demonstrates robustness and improves on prior automata-based and tableaux-based results, broadening applicability to more expressive concrete domains and CD-restrictions.

Abstract

Decidability or complexity issues about the consistency problem for description logics with concrete domains have already been analysed with tableaux-based or type elimination methods. Concrete domains in ontologies are essential to consider concrete objects and predefined relations. In this work, we expose an automata-based approach leading to the optimal upper bound EXPTIME, that is designed by enriching the transitions with symbolic constraints. We show that the nonemptiness problem for such automata belongs to EXPTIME if the concrete domains satisfy a few simple properties. Then, we provide a reduction from the consistency problem for ontologies, yielding EXPTIME-membership. Thanks to the expressivity of constraint automata, the results are extended to additional ingredients such as inverse roles, functional role names and constraint assertions, while maintaining EXPTIME-membership, which illustrates the robustness of the approach