More on modal aspects of default logic

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.

[1]  Paolo Mancarella,et al.  Abductive Logic Programming , 1992, LPNMR.

[2]  Victor W. Marek,et al.  Nonmonotonic logic - context-dependent reasoning , 1997, Artificial intelligence.

[3]  Raymond Reiter,et al.  A Logic for Default Reasoning , 1987, Artif. Intell..

[4]  Miroslaw Truszczynski,et al.  Embedding Default Logic into Modal Nonmonotonic Logics , 1991, LPNMR.

[5]  Robert C. Moore.  Semantical Considerations on Nonmonotonic Logic , 1985, IJCAI.

[6]  Christine Froidevaux,et al.  Negation by Default and Unstratifiable Logic Programs , 1991, Theor. Comput. Sci..

[7]  Victor W. Marek,et al.  Autoepistemic logic , 1991, JACM.

[8]  Kurt Konolige,et al.  On the Relation Between Default and Autoepistemic Logic , 1987, Artif. Intell..

[9]  Richard E. Ladner,et al.  The Computational Complexity of Provability in Systems of Modal Propositional Logic , 1977, SIAM J. Comput..

[10]  Georg Gottlob,et al.  Complexity Results for Nonmonotonic Logics , 1992, J. Log. Comput..

[11]  Victor W. Marek,et al.  The Relationship Between Logic Program Semantics and Non-Monotonic Reasoning , 1989, ICLP.

[12]  David S. Johnson,et al.  Computers and Intractability: A Guide to the Theory of NP-Completeness , 1978 .

[13]  Victor W. Marek,et al.  Stable Semantics for Logic Programs and Default Theories , 1989, NACLP.

[14]  Drew McDermott,et al.  Nonmonotonic Logic II: Nonmonotonic Modal Theories , 1982, JACM.

[15]  Ilkka Niemelä.  On the decidability and complexity of autoepistemic reasoning , 1992, Fundam. Informaticae.

[16]  Gerhard Brewka.  Cumulative Default Logic: In Defense of Nonmonotonic Inference Rules , 1991, Artif. Intell..

[17]  Max J. Cresswell,et al.  A companion to modal logic , 1984 .

[18]  Grigori F. Shvarts.  Autoepistemic Modal Logics , 1990, TARK.

[19]  Drew McDermott,et al.  Default Reasoning, Nonmonotonic Logics, and the Frame Problem , 1986, AAAI.

[20]  Tomasz Imielinski,et al.  Results on Translating Defaults to Circumscription , 1985, IJCAI.