Living Without Beth and Craig: Definitions and Interpolants in Description and Modal Logics with Nominals and Role Inclusions

The Craig interpolation property (CIP) states that an interpolant for implication exists iff it is valid. projective Beth definability (PBDP) explicit definition a formula stating implicit Thus, the CIP and PBDP reduce potentially hard existence problems to entailment in underlying logic. Description (and modal) logics with nominals and/or role inclusions do not enjoy nor PBDP, but interpolants definitions have many applications, particular concept learning, ontology engineering, ontology-based data management. In this article we show that, even without Craig, of decidable description such as \(\mathcal {ALCO} \) , {ALCH} {ALCHOI} corresponding hybrid modal logics. However, living makes these harder than entailment: become 2ExpTime-complete presence or universal modality, coNExpTime-complete otherwise. We also analyze if all symbols (except one defined) are admitted definition. case complexity depends on whether considers individual names. Finally, consider problem computing they exist turn upper bound proof into algorithm them, at least inclusions.