Stable Model Semantics for Description Logic Terminologies

This paper studies a stable model semantics for Description Logic (DL) knowledge bases (KBs) and for (possibly cyclic) terminologies, ultimately showing that terminologies under the proposed semantics can be equipped with effective reasoning algorithms. The semantics is derived using Quantified Equilibrium Logic, and---in contrast to the usual semantics of DLs based on classical logic---supports default negation and allows to combine the open-world and the closed-world assumptions in a natural way. Towards understanding the computational properties of this and related formalisms, we show a strong undecidability result that applies not only to KBs under the stable model semantics, but also to the more basic setting of minimal model reasoning. Specifically, we show that concept satisfiability in minimal models of an ALCIO KB is undecidable. We then turn our attention to (possibly cyclic) DL terminologies, where ontological axioms are limited to definitions of concept names in terms of complex concepts. This restriction still yields a very rich setting. We show that standard reasoning problems, like concept satisfiability and subsumption, are ExpTime-complete for terminologies expressed in ALCI under the stable model semantics.