On kernels, defaults and even graphs

Abstract

Extensions in prerequisite‐free, disjunction‐free default theories have been shown to be in direct correspondence with kernels of directed graphs; hence default theories without odd cycles always have a “standard” kind of an extension. We show that, although all “standard” extensions can be enumerated explicitly, several other problems remain intractable for such theories: Telling whether a non‐standard extension exists, enumerating all extensions, and finding the minimal standard extension. We also present a new graph‐theoretic algorithm, based on vertex feedback sets, for enumerating all extensions of a general prerequisite‐free, disjunction‐free default theory (possibly with odd cycles). The algorithm empirically performs well for quite large theories.