ASP for Minimal Entailment in a Rational Extension
            of SROEL (Extended Abstract)⋆

                       Laura Giordano and Daniele Theseider Dupré

                        DISIT, Università del Piemonte Orientale, Italy

    This work exploits Answer Set Programming (ASP) for reasoning in a rational ex-
tension 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 typical-
ity operator T, which allows the definition of defeasible inclusions T(C) ⊑ D (“the
typical C elements are Ds”). In this extension, SROEL(⊓, ×)R T, instance check-
ing 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(⊓, ×)R 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) oc-
curs 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 that T-minimal models
capture the same defeasible inferences as minimal canonical models.
    We prove that, for arbitrary SROEL(⊓, ×)R T KBs, instance checking under T-
minimal entailment is Π2P -complete. Based on a Small Model result, where models
correspond to answer sets of a suitable ASP encoding, we exploit Answer Set Prefer-
ences and the asprin framework [1] for reasoning under T-minimal entailment.
References
1. G. Brewka, J. P. Delgrande, J. Romero, and T. Schaub. asprin: Customizing answer set pref-
   erences without a headache. In Proc. AAAI 2015, pages 1467–1474, 2015.
2. K. Britz, J. Heidema, and T. Meyer. Semantic preferential subsumption. In G. Brewka and
   J. Lang, editors, Proc. KR 2008, pages 476–484, 2008.
3. L. Giordano, V. Gliozzi, N. Olivetti, and G. L. Pozzato. Semantic characterization of rational
   closure: From propositional logic to description logics. Artif. Intell., 226:1–33, 2015.
4. S. Kraus, D. Lehmann, and M. Magidor. Nonmonotonic reasoning, preferential models and
   cumulative logics. Artif. Intell., 44(1-2):167–207, 1990.
5. Krötzsch, M.: Efficient Inferencing for OWL EL, Proc. JELIA 2010, 2010.
6. D. Lehmann and M. Magidor. What does a conditional knowledge base entail? Artificial
   Intelligence, 55(1):1–60, 1992.

⋆
    This extended abstract is based on the the paper ”ASP for minimal entailment in a rational
    extension of SROEL”, published in TPLP, 16(5-6):738–754, 2016. The research has been par-
    tially supported by INDAM - GNCS Project 2016 Defeasible Reasoning in Description Logics.