Using model theory to find w-admissible concrete domains

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. One contribution of this paper is to strengthen the existing undecidability results further by showing that concrete domains even weaker than the ones considered in the previous proofs may cause undecidability. To regain decidability in the presence of GCIs, quite strong restrictions, in sum called w-admissiblity, need to be imposed on the concrete domain. On the one hand, we generalize the notion of w-admissiblity from concrete domains with only binary predicates to concrete domains with predicates of arbitrary arity. On the other hand, we relate w-admissiblity to well-known notions from model theory. In particular, we show that finitely bounded, homogeneous structures yield w-admissible concrete domains. This allows us to show w-admissibility of concrete domains using existing results from model theory