Defeasible Reasoning in SROEL: from Rational Entailment to Rational Closure

In this work we study a rational extension $SROEL^R T$ of the low complexity description logic SROEL, which underlies the OWL EL ontology language. The extension involves a typicality operator T, whose semantics is based on Lehmann and Magidor's ranked models and allows for the definition of defeasible inclusions. We consider both rational entailment and minimal entailment. We show that deciding instance checking under minimal entailment is in general $\Pi^P_2$-hard, while, under rational entailment, instance checking can be computed in polynomial time. We develop a Datalog calculus for instance checking under rational entailment and exploit it, with stratified negation, for computing the rational closure of simple KBs in polynomial time.