Mechanical Proof Systems for Logic II, Consensus Programs and Their Processing

We continue the investigations of [Ra90, Ra91, RM89] and study the automated theorem proving for reasoning about perception of reasoning agents and their consensus reaching. Using the techniques of [Ra91] and of Logic programming ([Ap90, NS93]) we develop the processing techniques for consensus programs.

Representation Theory for Default Logic

Default logic can be regarded as a mechanism to represent families of belief sets of a reasoning agent. As such, it is inherently second-order. In this paper, we study the problem of representability of a family of theories as the set of extensions of a default theory. We give a complete solution to the representability by means of normal default theories. We obtain partial results on representability by arbitrary default theories. In particular, we construct examples of denumerable families of nonincluding theories that are not representable. We also study the concept of equivalence between default theories. We show that for every normal default theory there exists a normal prerequisitefree theory with the same set of extensions. We derive a representation result connecting normal default logic with a version of CWA.

More on modal aspects of default logic

Abstract. 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.