Description Logic (DL) based ontologies and non-monotonic rules provide complementary features whose combination is crucial in many applications. In hybrid knowledge bases (KBs), which combine both formalisms, for large real-world applications, often integrating knowledge originating from different sources, inconsistencies can easily occur. These commonly trivialize standard reasoning and prevent us from drawing any meaningful conclusions. When restoring consistency by changing the KB is not possible, paraconsistent reasoning offers an alternative by allowing us to obtain meaningful conclusions from its consistent part. In this paper, we address the problem of efficiently obtaining meaningful conclusions from (possibly inconsistent) hybrid KBs. To this end, we define two paraconsistent semantics for hybrid KBs which, beyond their differentiating properties, are faithful to well-known paraconsistent semantics as well as the non-paraconsistent logic they extend, and tractable if reasoning in the DL component is.

Description Logic (DL) based ontologies and non-monotonic rules provide complementary features whose combination is crucial in many applications. In hybrid knowledge bases (KBs), which combine both formalisms, for large real-world applications, often integrating knowledge originating from different sources, inconsistencies can easily occur. These commonly trivialize standard reasoning and prevent us from drawing any meaningful conclusions. When restoring consistency by changing the KB is not possible, paraconsistent reasoning offers an alternative by allowing us to obtain meaningful conclusions from its consistent part. In this paper, we address the problem of efficiently obtaining meaningful conclusions from (possibly inconsistent) hybrid KBs. To this end, we define two paraconsistent semantics for hybrid KBs which, beyond their differentiating properties, are faithful to well-known paraconsistent semantics as well as the non-paraconsistent logic they extend, and tractable if reasoning in the DL component is.

We consider the problem of reasoning from inconsistent hybrid theories, i.e., combinations of a structural part given by a classical first order theory (e.g., an ontology) and a rules part as a set of declarative logic program rules (under answer-set semantics). Paraconsistent reasoning is achieved by defining an appropriate semantics, so-called paraconsistent semi-equilibrium model semantics for such hybrid theories. Appropriateness of the semantics is established with respect to desirable properties attesting design objectives, such us to generalize the underlying semantics in case of consistency, as well as to generalize existing paraconsistent semantics for the individual parts. A complexity analysis of corresponding reasoning tasks complements these results.