Normal form results for default logic V. Wiktor Marek Miroslaw Truszczy´ nski Department of Computer Science University of Kentucky Lexington, KY 40506–0027 marek@ms.uky.edu, mirek@ms.uky.edu Abstract In this paper we continue investigations of proof theory of default logic. It turns out that, similarly to classical logic, default theories can be represented in normal forms. 1 Introduction and preliminaries In this paper we develop a representation theory for default logic of Reiter ([Rei80]). The question is whether one can find “normal forms” for default theories, that is, if there are syntactical constraints which can be imposed on default theories without changing extensions. In this section we intro- duce basic definitions and recall fundamental concepts of default logic. We introduce the notion of representability of default theories in Section 2 and prove a number of results of both positive and negative nature. A weaker notion, semi-representability, is studied in Section 3. We prove that with this weaker notion we can represent every default theory by a default the- ory with all rules either monotonic (that is justification-free) or semi-normal ([Eth88]). In Section 4 we discuss another structure associated with default logic, a weak extension. We show that every finite family of finitely gener- ated theories can be represented as a family of weak extensions of a suitably constructed default theory. A result on autoepistemic expansions is given as a corollary. We use standard logical notation. The reader is referred to [Fit90] and [Men64] for the unexplained concepts. The presentation of default logic follows one we gave in [MT93]. 1