Living Without Beth and Craig - Explicit Definitions and Interpolants in Description Logics with Nominals (Extended Abstract).

The Craig interpolation property (CIP) states that an interpolant for an implication exists iff it is valid. The projective Beth definability property (BDP) states that an explicit definition exists iff a formula stating implicit definability is valid. Thus, they transform potentially hard existence problems into deduction problems in the underlying logic. Description Logics with nominals do not have the CIP nor BDP, but in particular, deciding and computing explicit definitions of concepts or individuals has many potential applications in ontology engineering and ontology-based data management. In this article we show two main results: even without Craig and Beth, the existence of interpolants and explicit definitions is decidable in the description logics with nominals ALCO and ALCIO. However, living without Craig and Beth makes this problem harder than deduction: we prove that the existence problems become 2ExpTime-complete, thus one exponential harder than validity.