ASP for Minimal Entailment in a Rational Extension

This work exploits Answer Set Programming (ASP) for reasoning in a rational extension of SROEL(⊓,×) [5], the low complexity description logic which underlies the OWL EL ontology language. It is based on a preferential approach to defeasible reasoning in description logics (DLs) [2, 3], which has been developed along the lines of the preferential semantics introduced by Kraus, Lehmann and Magidor [4, 6]. Following [3], we have considered an extension of SROEL(⊓,×) with a typicality operator T, which allows the definition of defeasible inclusions T(C) ⊑ D (“the typical C elements are Ds”). In this extension, SROEL(⊓,×)T, instance checking under rational entailment has polynomial complexity. We observe that the notion of minimal canonical model introduced in [3] as a semantic characterization of the rational closure for ALC is not adequate to capture many knowledge bases (KBs) in SROEL(⊓,×)T. In particular, when nominals or the universal role are used, a KB may have no canonical model at all. The T-minimal model semantics is introduced as an alternative to the minimal canonical model semantics. It weakens the canonical model condition in [3], by requiring that only for the concepts C such that T(C) occurs in the KB (or in the query), an instance of C has to exist in the model, when C is satisfiable wrt the KB. For the KBs having minimal canonical models with the same rank assignment to concepts as in the rational closure, we show thatT-minimal models capture the same defeasible inferences as minimal canonical models. We prove that, for arbitrary SROEL(⊓,×)T KBs, instance checking under Tminimal entailment is Π 2 -complete. Based on a Small Model result, where models correspond to answer sets of a suitable ASP encoding, we exploit Answer Set Preferences and the asprin framework [1] for reasoning underT-minimal entailment.