A Sequent-Type Calculus for Three-Valued Default Logic, Or: Tweety Meets Quartum Non Datur

More on Modal Aspects of Default Logic1

Fundamenta Informaticae ◽

10.3233/fi-1992-171-207 ◽

1992 ◽

Vol 17 (1-2) ◽

pp. 99-116

Author(s):

V. Wiktor Marek ◽

Miroslaw Truszczynski

Keyword(s):

Modal Logic ◽

Default Logic ◽

Nonmonotonic Logic ◽

Default Reasoning ◽

Default Theory ◽

Autoepistemic Logic ◽

The Relationship ◽

Natural Classes ◽

Weak Extension

Investigations of default logic have been so far mostly concerned with the notion of an extension of a default theory. It turns out, however, that default logic is much richer. Namely, there are other natural classes of objects that might be associated with default reasoning. We study two such classes of objects with emphasis on their relations with modal nonmonotonic formalisms. First, we introduce the concept of a weak extension and study its properties. It has long been suspected that there are close connections between default and autoepistemic logics. The notion of weak extension allows us to precisely describe the relationship between these two formalisms. In particular, we show that default logic with weak extensions is essentially equivalent to autoepistemic logic, that is, nonmonotonic logic KD45. In the paper we also study the notion of a set of formulas closed under a default theory. These objects are shown to correspond to stable theories and to modal logic S5. In particular, we show that skeptical reasoning with sets closed under default theories is closely related with provability in S5. As an application of our results we determine the complexity of reasoning with weak extensions and sets closed under default theories.

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Default Logic Models of Certain Inheritance Systems with Exceptions

Fundamenta Informaticae ◽

10.3233/fi-1993-182-404 ◽

1993 ◽

Vol 18 (2-4) ◽

pp. 129-149

Author(s):

Serge Garlatti

Keyword(s):

Hierarchical Structure ◽

Formal Semantics ◽

Sufficient Conditions ◽

Nonmonotonic Reasoning ◽

Default Logic ◽

Representation Systems ◽

The Hierarchical Structure ◽

Necessary And Sufficient ◽

Structure Of Knowledge ◽

Made In

Representation systems based on inheritance networks are founded on the hierarchical structure of knowledge. Such representation is composed of a set of objects and a set of is-a links between nodes. Objects are generally defined by means of a set of properties. An inheritance mechanism enables us to share properties across the hierarchy, called an inheritance graph. It is often difficult, even impossible to define classes by means of a set of necessary and sufficient conditions. For this reason, exceptions must be allowed and they induce nonmonotonic reasoning. Many researchers have used default logic to give them formal semantics and to define sound inferences. In this paper, we propose a survey of the different models of nonmonotonic inheritance systems by means of default logic. A comparison between default theories and inheritance mechanisms is made. In conclusion, the ability of default logic to take some inheritance mechanisms into account is discussed.

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