Extending Defeasible Reasoning Beyond Rational Closure

Datalog is a declarative logic programming language that uses classical logical reasoning as its basic form of reasoning. Defeasible reasoning is a form of non-classical reasoning that is able to deal with exceptions to general assertions in a formal manner. The KLM approach to defeasible reasoning is an axiomatic approach based on the concept of plausible inference. Since Datalog uses classical reasoning, it is currently not able to handle defeasible implications and exceptions. We aim to extend the expressivity of Datalog by incorporating KLM-style defeasible reasoning into classical Datalog. We present a systematic approach for extending the KLM properties and a well-known form of defeasible entailment: Rational Closure. We conclude by exploring Datalog extensions of less conservative forms of defeasible entailment: Relevant and Lexicographic Closure. We provide algorithmic definitions for these forms of defeasible entailment and prove that the definitions are LM-rational.

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.

Description logics have been extended in a number of ways to support defeasible reasoning in the KLM tradition. Such features include preferential or rational defeasible concept subsumption, and defeasible roles in complex concept descriptions. Semantically, defeasible subsumption is obtained by means of a preference order on objects, while defeasible roles are obtained by adding a preference order to role interpretations. In this paper, we address an important limitation in defeasible extensions of description logics, namely the restriction in the semantics of defeasible concept subsumption to a single preference order on objects. We do this by inducing a modular preference order on objects from each preference order on roles, and use these to relativise defeasible subsumption. This yields a notion of contextualised rational defeasible subsumption, with contexts described by roles. We also provide a semantic construction for and a method for the computation of contextual rational closure, and present a correspondence result between the two.