Some of the applications of OWL and RDF (e.g. biomedical knowledge representation and semantic policy formulation) call for extensions of these languages with nonmonotonic constructs such as inheritance with overriding. Nonmonotonic description logics have been studied for many years, however no practical such knowledge representation languages exist, due to a combination of semantic difficulties and high computational complexity. Independently, low-complexity description logics such as DL-lite and EL have been introduced and incorporated in the OWL standard. Therefore, it is interesting to see whether the syntactic restrictions characterizing DL-lite and EL bring computational benefits to their nonmonotonic versions, too. In this paper we extensively investigate the computational complexity of Circumscription when knowledge bases are formulated in DL-lite_R, EL, and fragments thereof. We identify fragments whose complexity ranges from P to the second level of the polynomial hierarchy, as well as fragments whose complexity raises to PSPACE and beyond.

The research on low-complexity nonmonotonic description logics recently identified a fragment of EL with bottom, supporting defeasible inheritance with overriding, where reasoning can be carried out in polynomial time. We contribute to that framework by supporting more axiom schemata and all the concept constructors of EL++ without increasing asymptotic complexity. Moreover, we show that all the syntactic restrictions we adopt are necessary by proving several coNP-hardness results.

We analyze the complexity of reasoning in EL with defeasible inclusions, and extend previous results by tightening lower and upper complexity bounds and by relaxing some syntactic restrictions. We further extend the old framework by supporting arbitrary priority relations over defeasible inclusions.

We analyze the complexity of reasoning with circumscribed low-complexity DLs such as DL-lite and the EL family, under suitable restrictions on the use of abnormality predicates. We prove that in circumscribed DL-lite R complexity drops from NExp NP to the second level of the polynomial hierarchy. In EL, reasoning remains ExpTime-hard, in general. However, by restricting the possible occurrences of existential restrictions, we obtain membership in Sigma p 2 and Pi p 2 for an extension of EL.