Using Model Theory to Find Decidable and Tractable Description Logics with Concrete Domains

Abstract

Concrete domains have been introduced in the area of Description Logic to enable reference to concrete objects (such as numbers) and predefined predicates on these objects (such as numerical comparisons) when defining concepts. Unfortunately, in the presence of general concept inclusions (GCIs), which are supported by all modern DL systems, adding concrete domains may easily lead to undecidability. To regain decidability of the DL mathcal {ALC} in the presence of GCIs, quite strong restrictions, in sum called omega -admissibility, were imposed on the concrete domain. On the one hand, we generalize the notion of omega -admissibility from concrete domains with only binary predicates to concrete domains with predicates of arbitrary arity. On the other hand, we relate omega -admissibility to well-known notions from model theory. In particular, we show that finitely bounded homogeneous structures yield omega -admissible concrete domains. This allows us to show omega -admissibility of concrete domains using existing results from model theory. 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. 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.