We continue recent investigations into the problem of reasoning about typicality. We do so in the framework of Propositional Typicality Logic (PTL), which is obtained by enriching classical propositional logic with a typicality operator and characterized by a preferential semantics à la KLM. In this paper we study different notions of entailment for PTL. We take as a starting point the notion of Rational Closure defined for KLM-style conditionals. We show that the additional expressivity of PTL results in different versions of Rational Closure for PTL — versions that are equivalent with respect to the conditional language originally proposed by KLM.
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.
Propositional Typicality Logic (PTL) is a recently proposed logic, obtained by enriching classical propositional logic with a typicality operator capturing the most typical (alias normal or conventional) situations in which a given sentence holds. The semantics of PTL is in terms of ranked models as studied in the well-known KLM approach to preferential reasoning and therefore KLM-style rational consequence relations can be embedded in PTL. In spite of the non-monotonic features introduced by the semantics adopted for the typicality operator, the obvious Tarskian definition of entailment for PTL remains monotonic and is therefore not appropriate in many contexts. Our first important result is an impossibility theorem showing that a set of proposed postulates that at first all seem appropriate for a notion of entailment with regard to typicality cannot be satisfied simultaneously. Closer inspection reveals that this result is best interpreted as an argument for advocating the development of more than one type of PTL entailment. In the spirit of this interpretation, we investigate three different (semantic) versions of entailment for PTL, each one based on the definition of rational closure as introduced by Lehmann and Magidor for KLM-style conditionals, and constructed using different notions of minimality.
Handling exceptions in a knowledge-based system has been considered as an important issue in many domains of applications, such as medical domain. In this paper, we propose several preferential semantics for plausible subsumption to deal with exceptions in description logic-based knowledge bases. Our preferential semantics are defined in the framework of possibility theory, which is an uncertainty theory devoted to the handling of incomplete information. We consider the properties of these semantics and their relationships. Entailment of these plausible subsumption relative to a knowledge base is also considered. We show the close relationship between two of our semantics and the mutually dual preferential semantics given by Britz, Heidema and Meyer. Finally, we show that our semantics for plausible subsumption can be reduced to standard semantics of an expressive description logic. Thus, the problem of plausible subsumption checking under our semantics can be reduced to the problem of subsumption checking under the classical semantics.