The idea of circumscription can be explained on a simple example. We would like to represent information about the locations of blocks in a blocks world, using the "default":

This paper exhibits some problematic cases of defeasible or nonmonotonic reasoning that tend to be handled incorrectly by all of the theories of defeasible and nonmonotonic reasoning in the current literature. The paper focuses particularly on default logic, circumscription, and the author's own argument-based approach to defeasible reasoning. A proposal is made for how to deal with these problematic cases. The paper closes with a demonstration that the proposed solution is able to differentiate, in a congenial way, between cases having the structure of the lottery paradox and cases having the structure of the paradox of the preface. The algorithm proposed for computing justificational status has been implemented in the automated defeasible reasoner OSCAR. 1. Introduction The purpose of this paper is to exhibit some problematic cases of defeasible or nonmonotonic reasoning that tend to be handled incorrectly by all of the theories of defeasible and nonmonotonic reasoning in the curr...

This paper describes a uniform formalization of much of the current work in AI on inference systems. We show that many of these systems, including first-order theorem provers, assumption-based truth maintenance systems (atms's) and unimplemented formal systems such as default logic or circumscription can be subsumed under a single general framework. We begin by defining this framework, which is based on a mathematical structure known as a bilattice. We present a formal definition of inference using this structure, and show that this definition generalizes work involving atms's and some simple nonmonotonic logics. Following the theoretical description, we describe a constructive approach to inference in this setting; the resulting generalization of both conventional inference and atms's is achieved without incurring any substantial computational overhead. We show that our approach can also be used to implement a default reasoner, and discuss a combination of default and atms methods th...

We propose a new approach to the use of circumscription for representing knowledge. Nested abnormality theories are similar to simple abnormality theories introduced by McCarthy, except that their axioms may have a nested structure, with each level corresponding to another application of the circumscription operator. The new style of applying circumscription sometimes leads to more economical and elegant formalizations. Mathematical properties of nested abnormality theories may be easier to investigate. These advantages are demonstrated by recasting several familiar applications of circumscription in the new format, including some examples of inheritance hierarchies, the domain closure assumption and causal minimization. Nested abnormality theories provide also a convenient representation for the explanation closure approach to the frame problem developed by Schubert.

We describe a new approach to default reasoning, based on a principle of indifference among possible worlds. We interpret default rules as extreme statistical statements, thus obtaining a knowledge base KB comprised of statistical and first-order statements. We then assign equal probability to all worlds consistent with KB in order to assign a degree of belief to a statement '. The degree of belief can be used to decide whether to defeasibly conclude '. Various natural patterns of reasoning, such as a preference for more specific defaults, indifference to irrelevant information, and the ability to combine independent pieces of evidence, turn out to follow naturally from this technique. Furthermore, our approach is not restricted to default reasoning; it supports a spectrum of reasoning, from quantitative to qualitative. It is also related to other systems for default reasoning. In particular, we show that the work of [ Goldszmidt et al., 1990 ] , which applies maximum entropy ideas t...

In a recent paper we have proposed terminological default logic as a formalism which combines both means for structured representation of classes and objects, and for default inheritance of properties. The major drawback which terminological default logic inherits from general default logic is that it does not take precedence of more specific defaults over more general ones into account. This behaviour has already been criticized in the general context of default logic, but it is all the more problematic in the terminological case where the emphasis lies on the hierarchical organization of concepts. The present paper addresses the problem of modifying terminological default logic such that more specific defaults are preferred. It turns out that the existing approaches for expressing priorities between defaults do not seem to be appropriate for this purpose. Therefore we shall consider an alternative approach for dealing with prioritization in the framework of Reiter's def...

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We present a new approach to deal with default information based on the theory of belief functions. Our semantic structures, inspired by Adams' epsilon semantics, are epsilon-belief assignments, where mass values are either close to 0 or close to 1. In the first part of this paper, we show that these structures can be used to give a uniform semantics to several popular non-monotonic systems, including Kraus, Lehmann and Magidor's system P, Pearl's system Z, Brewka's preferred sub-theories, Geffner's conditional entailment, Pinkas' penalty logic, possibilistic logic and the lexicographic approach. In the second part, we use epsilon-belief assignments to build a new system, called LCD, and show that this system correctly addresses the well-known problems of specificity, irrelevance, blocking of inheritance, ambiguity, and redundancy.

In this paper I argue that we do not understand the process of default reasoning. A number of examples are given which serve to distinguish different default reasoning systems. It is shown that if we do not make our assumptions explicit we get into trouble with disjunctive knowledge, and if we make our assumptions explicit, we run foul of the lottery paradox. None of the current popular default reasoning systems work on all of the examples. It is argued that the lottery paradox does arise in default reasoning and can cause problems. It is also shown that some of the intuitively plausible requirements for default reasoning are incompatible. How different systems cope with this is discussed. 1 Introduction Default reasoning is the ability to jump to a conclusion based on the lack of evidence to the contrary. Deduction in standard logic does not allow such reasoning; if some proposition follows from a set of axioms, it follows from a superset of the axioms. There have been many proposal...