An Algebraic View on p-Admissible Concrete Domains for Lightweight Description Logics

Abstract

Concrete domains have been introduced in Description Logics (DLs) to enable reference to concrete objects (such as numbers) and predefined predicates on these objects (such as numerical comparisons) when defining concepts. To retain decidability when integrating a concrete domain into a decidable DL, the domain must satisfy quite strong restrictions. In previous work, we have analyzed the most prominent such condition, called \(\omega \)-admissibility, from an algebraic point of view. This provided us with useful algebraic tools for proving \(\omega \)-admissibility, which allowed us to find new examples for concrete domains whose integration leaves the prototypical expressive DL \(\mathcal {ALC}\) decidable.When integrating concrete domains into lightweight DLs of the \(\mathcal {EL}\) family, achieving decidability is not enough. One wants reasoning in the resulting DL to be tractable. This can be achieved by using so-called p-admissible concrete domains and restricting the interaction between the DL and the concrete domain. In the present paper, we investigate p-admissibility from an algebraic point of view. Again, this yields strong algebraic tools for demonstrating p-admissibility. In particular, we obtain an expressive numerical p-admissible concrete domain based on the rational numbers. Although \(\omega \)-admissibility and p-admissibility are orthogonal conditions that are almost exclusive, our algebraic characterizations of these two properties allow us to locate an infinite class of p-admissible concrete domains whose integration into \(\mathcal {ALC}\) yields decidable DLs.