Representation theory for default logic

Abstract

In propositional and predicate logics we often represent formulas and theories by means of formulas or theories of a simpler syntactic form but with the same logical properties. For instance, it is well known that for every propositional formula φ there exists a formula ψ in the conjunctive normal form such that φ and ψ have the same models (are logically equivalent). In fact, there is a simple algorithm to construct ψ out of φ. In this chapter we study the representability issue for default logic. We provide several results showing that default theories can be represented by syntactically simpler theories in such a way that extensions are preserved. We show that, in some cases, these “normal form” default theories can easily be constructed. One of the highlights of this chapter is the result showing that, at the cost of adding new atoms to the language, every default theory can be assigned an equivalent default theory with all rules either monotonic, that is, justification-free, or semi-normal. We also show that prerequisite-free normal default theories can be represented within the framework of CWA. Finally, we discuss the representability of families of sets as collections of weak extensions of default theories.