On kernels, defaults and even graphs

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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.

This revised version was published online in June 2006 with corrections to the Cover Date.