General default theories

Abstract

In this chapter we introduce default logic. Default logic is concerned with proof systems which are obtained from propositional logic by adding some nonstandard inference rules called defaults. A default differs from an inference rule, as defined in the previous chapter, in that its premises are of two types. Premises of the first type, called prerequisites, act in the same way as the premises of an inference rule. In order to apply a default it is necessary that each of its prerequisites be proved. Premises of the second type, called justifications, do not need a proof. It is only required that each of them be possible in order for a default to be applicable. In this chapter we will formally define defaults and default logic and make precise what is meant by a proof and by being possible. To this end, we will introduce the notion of an extension for a default theory. Extensions can be thought of as descriptions of all possible knowledge (belief) sets that are represented by a default theory. They play a similar role to that of the set of all consequences of a theory in propositional logic. In this chapter we study the notion of an extension and provide several characterizations of extensions as well as algorithms to compute them.