Britz, Katarina, Casini, Giovanni, Meyer, Thomas, Moodley, Kody, Sattler, Uli, Varzinczak, Ivan

We extend description logics (DLs) with non-monotonic reasoning features. We start by investigating a notion of defeasible subsumption in the spirit of defeasible conditionals as studied by Kraus, Lehmann and Magidor in the propositional case. In particular, we consider a natural and intuitive semantics for defeasible subsumption, and investigate KLM-style syntactic properties for both preferential and rational subsumption. Our contribution includes two representation results linking our semantic constructions to the set of preferential and rational properties considered. Besides showing that our semantics is appropriate, these results pave the way for more effective decision procedures for defeasible reasoning in DLs. Indeed, we also analyse the problem of non-monotonic reasoning in DLs at the level of entailment and present an algorithm for the computation of rational closure of a defeasible ontology. Importantly, our algorithm relies completely on classical entailment and shows that the computational complexity of reasoning over defeasible ontologies is no worse than that of reasoning in the underlying classical DL ALC.

Casini, Giovanni (University of Luxembourg) | Fermé, Eduardo (Universidade da Madeira) | Meyer, Thomas (University of Cape Town) | Varzinczak, Ivan (University of Artois)

Belief change and non-monotonic reasoning are usually viewed as two sides of the same coin, with results showing that one can formally be defined in terms of the other. In this paper we investigate the integration of the two formalisms by studying belief change for a (preferential) non-monotonic framework. We show that the standard AGM approach to belief change can be transferred to a preferential non-monotonic framework in the sense that change operations can be defined on conditional knowledge bases. We take as a point of departure the results presented by Casini and Meyer (2017), and we develop and extend such results with characterisations based on semantics and entrenchment relations, showing how some of the constructions defined for propositional logic can be lifted to our preferential non-monotonic framework.

Booth, Richard, Casini, Giovanni, Meyer, Thomas, Varzinczak, Ivan

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.

Casini, Giovanni, Straccia, Umberto, Meyer, Thomas

Description Logics (DLs) under Rational Closure (RC) is a well-known framework for non-monotonic reasoning in DLs. In this paper, we address the concept subsumption decision problem under RC for nominal safe $\mathcal{ELO}_\bot$, a notable and practically important DL representative of the OWL 2 profile OWL 2 EL. Our contribution here is to define a polynomial time subsumption procedure for nominal safe $\mathcal{ELO}_\bot$ under RC that relies entirely on a series of classical, monotonic $\mathcal{EL}_\bot$ subsumption tests. Therefore, any existing classical monotonic $\mathcal{EL}_\bot$ reasoner can be used as a black box to implement our method. We then also adapt the method to one of the known extensions of RC for DLs, namely Defeasible Inheritance-based DLs without losing the computational tractability.

Casini, Giovanni (University of Luxembourg) | Meyer, Thomas (University of Cape Town)

We consider the problem of obtaining coherence in a propositional knowledge base using techniques from Belief Change. Our motivation comes from the field of formal ontologies where coherence is interpreted to mean that a concept name has to be satisfiable. In the propositional case we consider here, this translates to a propositional formula being satisfiable. We define belief change operators in a framework of nonmonotonic preferential reasoning.We show how the introduction of defeasible information using contraction operators can be an effective means for obtaining coherence.

Booth, Richard (Mahasarakham University) | Casini, Giovanni (University of Pretoria and University of Luxembourg) | Meyer, Thomas Andreas (University of Cape Town) | Varzinczak, Ivan José (Universidade Federal de Rio de Janeiro)

Propositional Typicality Logic (PTL) is a recently proposed logic, obtained by enriching classical propositional logic with a typicality operator. 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. We investigate different (semantic) versions of entailment for PTL, based on the notion of Rational Closure as defined by Lehmann and Magidor for KLM-style conditionals, and constructed using minimality. 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 satis- fied 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 define two primary forms of entailment for PTL and discuss their advantages and disadvantages

Booth, Richard (Mahasarakham University) | Casini, Giovanni (University of Pretoria) | Meyer, Thomas (University of Pretoria) | Varzinczak, Ivan José (Universidade Federal do Rio de Janeiro)

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.

Straccia, Umberto (ISTI - CNR) | Casini, Giovanni (Scuola Normale Superiore)

Defeasible inheritance networks are a non-monotonic framework that deals with hierarchical knowledge. On the other hand, rational closure is acknowledged as a landmark of the preferential approach. We will combine these two approaches and define a new non-monotonic closure operation for propositional knowledge bases that combines the advantages of both. Then we redefine such a procedure for Description Logics, a family of logics well-suited to model structured information. In both cases we will provide a simple reasoning method that is build on top of the classical entailment relation.