Reasoning with power defaults

This paper introduces power default reasoning (PDR), a framework for non-monotonic reasoning based on the domain-theoretic idea of modeling default rules with partial-information in a higher-order setting. PDR lifts a non-monotonic operator at the base (syntactic) level to a well-behaved, almost monotonic operator in the higher-order space of the Smyth power-domain-effectively a space of sets of models. Working in the model space allows us to prove the dichotomy theorem and the extension splitting theorem, leading to a more well-behaved logic and (modulo the usual complexity conjectures) a less complex logic than standard default logic. Specifically, we prove that skeptical normal default inference is a problem complete for co-NP(3) in the Boolean hierarchy for strict propositional logic and NP(4)-complete in general. These results (by changing the underlying semantics) contrasts favorably with similar results of Gottlob (J. Logic Comput. 2(3) (1992) 397-425), who proves that standard skeptical default reasoning is Pi(2)(P)-complete. Furthermore, we show that the skeptical non-monotonic consequence relation, defined using our domain-theoretic semantics, obeys all of the laws for preferential consequence relations defined by Kraus, Lehmann, and Magidor. In particular, we get the property of being able to reason by cases, and the so-called law of cautious monotony. Both of these laws fail for the standard propositional default logic of Reiter (Artificial Intelligence 13 (1980) 81-132), but hold in PDR as a consequence of the dichotomy theorem and the extension splitting theorem. (C) 2004 Elsevier B.V. All rights reserved.