Article

Annals of Mathematics and Artificial Intelligence

, Volume 20, Issue 1, pp 1-12

On kernels, defaults and even graphs

  • Yannis DimopoulosAffiliated withMax‐Planck‐Institut für Informatik
  • , Vangelis MagirouAffiliated withAthens University of Economics
  • , Christos H. PapadimitriouAffiliated withCS&EE Department, University of California at San Diego

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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.