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We survey here various approaches which were proposed to incorporate negation in logic programs. We concentrate on the proof-theoretic and model-theoretic issues and the relationships between them.

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Logical models of argument formalize commonsense reasoning while taking process and computation seriously. This survey discusses the main ideas which characterize different logical models of argument. It presents the formal features of a few main approaches to the modeling of argumentation. We trace the

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The idea of ordering plays a basic role in commonsense reasoning for addressing three interrelated tasks: inconsistency handling, belief revision and plausible inference. We study the behavior of non-monotonic inferences induced by various methods for priority-based handling of inconsistent sets of classical formulas. One of them is based on a lexicographic ordering of maximal consistent subsets, and refines Brewka's preferred sub-theories. This new approach leads to a non-monotonic inference which satisfies the "rationality " property while solving the problem of blocking of property inheritance. It differs from and improves previous equivalent approaches such as Gardenfors and Makinson's expectation-based inference, Pearl's System Z and possibilistic logic. 1

Ontologies play a crucial role in the development of the Semantic Web as a means for defining shared terms in web resources. They are formulated in web ontology languages, which are based on expressive description logics. Significant research efforts in the semantic web community are recently directed towards representing and reasoning with uncertainty and vagueness in ontologies for the Semantic Web. In this paper, we give an overview of approaches in this context to managing probabilistic uncertainty, possibilistic uncertainty, and vagueness in expressive description logics for the Semantic Web.

A key issue when reasoning with default rules is how to order them so as to derive plausible conclusions according to the more specific rules applicable to the situation under concern, to make sure that default rules are not systematically inhibited by more general rules, and to cope with the problem of irrelevance of facts with respect to exceptions. Pearl's system Z enables us to rank-order default rules. In this paper we show how to encode such a rank-ordered set of defaults in possibilistic logic. We can thus take advantage of the deductive machinery available in possibilistic logic. We point out that the notion of inconsistency tolerant inference in possibilistic logic corresponds to the bold inference ; 1 in system Z. We also show how to express defaults by means of qualitative possibility relations. Improvements to the ordering provided by system Z are also proposed.

In this paper we argue that for realistic applications involving default reasoning it is necessary to reason about the priorities of defaults. Existing approaches require the knowledge engineer to explicitly state all relevant priorities which are then handled in an extra-logical manner, or they are restricted to priorities ba-sed on specificity, neglecting other relevant criteria. We present an approach where priority information can be represented within the logical language. Our approach is based on PDL, a prioritized exten-sion of Reiter’s Default Logic recently proposed by the same author. In PDL the generation of extensions is controlled by an ordering of the defaults. This pro-perty is used here in the following way: we first build Reiter extensions of a given default theory. These ex-tensions contain explicit information about the prio-rities of defaults. We then eliminate every extension E that cannot be reconstructed as a PDL extension based on a default ordering that is compatible with the priority information in E. An example from legal reasoning illustrates the power of our approach. 1.

The lexicographic closure of any given finite set D of normal defaults is defined. A conditional assertion a b is in this lexicographic closure if, given the defaults D and the fact a, one would conclude b. The lexicographic closure is essentially a rational extension of D, and of its rational closure, defined in a previous paper. It provides a logic of normal defaults that is different from the one proposed by R. Reiter and that is rich enough not to require the consideration of non-normal defaults. A large number of examples are provided to show that the lexicographic closure corresponds to the basic intuitions behind Reiter's logic of defaults. 1 Plan of this paper Section 2 is a general introduction, describing the goal of this paper, in relation with Reiter's Default Logic and the program proposed in [12] by Lehmann and Magidor. Section 3 first discusses at length some general principles of the logic of defaults, with many examples, and, then, puts this paper in perspe...