Defeasible Entailment: from Rational Closure to Lexicographic Closure and Beyond

In this paper we present what we believe to be the first systematic approach for extending the framework for de- feasible entailment first presented by Kraus, Lehmann, and Magidor—the so-called KLM approach. Drawing on the properties for KLM, we first propose a class of basic defea- sible entailment relations. We characterise this basic frame- work in three ways: (i) semantically, (ii) in terms of a class of properties, and (iii) in terms of ranks on statements in a knowlege base. We also provide an algorithm for computing the basic framework. These results are proved through vari- ous representation results. We then refine this framework by defining the class of rational defeasible entailment relations. This refined framework is also characterised in thee ways: se- mantically, in terms of a class of properties, and in terms of ranks on statements. We also provide an algorithm for com- puting the refined framework. Again, these results are proved through various representation results. We argue that the class of rational defeasible entail- ment relations—a strengthening of basic defeasible entail- ment which is itself a strengthening of the original KLM proposal—is worthy of the term rational in the sense that all of them can be viewed as appropriate forms of defeasi- ble entailment. We show that the two well-known forms of defeasible entailment, rational closure and lexicographic clo- sure, fall within our rational defeasible framework. We show that rational closure is the most conservative of the defeasi- ble entailment relations within the framework (with respect to subset inclusion), but that there are forms of defeasible en- tailment within our framework that are more “adventurous” than lexicographic closure

A Semantic Perspective on Belief Change in a Preferential Non-Monotonic Framework

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 be- lief 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 depar- ture 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

On the Entailment Problem for a Logic of Typicality

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 appro- priate. We investigate different (semantic) versions of entailment for PTL, based on the notion of Ra- tional Closure as defined by Lehmann and Magidor for KLM-style conditionals, and constructed using minimality. Our first important result is an impossi- bility theorem showing that a set of proposed postu- lates 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 ad- vocating 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

What Does Entailment for PTL Mean?

We continue recent investigations into the problem of reason- ing 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 a 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

Introducing Defeasibility into OWL Ontologies

In recent years, various approaches have been developed for repre- senting and reasoning with exceptions in OWL. The price one pays for such ca- pabilities, in terms of practical performance, is an important factor that is yet to be quantified comprehensively. A major barrier is the lack of naturally oc- curring ontologies with defeasible features - the ideal candidates for evaluation. Such data is unavailable due to absence of tool support for representing defea- sible features. In the past, defeasible reasoning implementations have favoured automated generation of defeasible ontologies. While this suffices as a prelimi- nary approach, we posit that a method somewhere in between these two would yield more meaningful results. In this work, we describe a systematic approach to modify real-world OWL ontologies to include defeasible features, and we ap- ply this to the Manchester OWL Repository to generate defeasible ontologies for evaluating our reasoner DIP (Defeasible-Inference Platform). The results of this evaluation are provided together with some insights into where the performance bottle-necks lie for this kind of reasoning. We found that reasoning was feasible on the whole, with surprisingly few bottle-necks in our evaluation

Rational Defeasible Reasoning for Description Logics

In this paper, we extend description logics (DLs) with non-monotonic reasoning fea- tures. We start by investigating a notion of defeasible subsumption in the spirit of defeasible conditionals as studied by Kraus and colleagues in the propositional case. In particular, we consider a natural and intuitive semantics for defeasible subsumption, and we investi- gate syntactic properties (à la Gentzen) for both preferential and rational subsumptions and prove representation results for the description logic ALC. Such representation results pave the way for more effective decision procedures for defeasible reasoning in DLs. We analyse the problem of non-monotonic reasoning in DL at the level of entailment for both TBox and ABox reasoning, and present an adaptation of rational closure for the DL en- vironment. Importantly, we also show that computing it can be reduced to classical ALC entailment. One of the stumbling blocks to evaluating performance scalability of rational closure is the absence of naturally occurring DL-based ontologies with defeasible features. We overcome this barrier by devising an approach to introduce defeasible subsumption into classical real-world ontologies. Such semi-natural defeasible ontologies, together with a purely artificial set, are used to test our rational closure algorithms. We found that performance is scalable on the whole with no major bottlenecks