Using model theory to find w-admissible 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. 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.

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

“…To the best of our knowledge, the only article besides [35] where concrete domains satisfying the EHD property are studied in the context of ALC with GCIs is [60,61]. 7 There, the authors consider specific concrete domains based on integers equipped with a linear order and provide an exponential upper bound for reasoning using an automata-theoretic algorithm. Interestingly, their upper bound holds not only for constraints along paths, but also for the traditional integration of concrete domain into DLs.…”

“…In the setting considered in the present paper, where concrete domain restrictions always have access to equality, JD is actually needed to ensure decidability. If the equality predicate is dropped from concrete domain restrictions, then the decidability results in [7,65] do not depend on JD. However, all examples of ω-admissible concrete domains presented in [65] satisfy JD since equality is contained in the signature.…”

“…The following theorem is shown in [6,7] by extending the tableau-based decision procedure given in [65] to our more general definition of ω-admissibility. The main motivation for the definition of ω-admissible concrete domains in [65] was that they can capture qualitative calculi of time and space.…”