The Boolean hierarchy I: structural properties
SIAM Journal on Computing
Nonmonotonic reasoning, preferential models and cumulative logics
Artificial Intelligence
Hard problems for simple default logics
Proceedings of the first international conference on Principles of knowledge representation and reasoning
Handbook of theoretical computer science (vol. B)
Functional programming and databases
Research topics in functional programming
Logic of domains
The formal semantics of programming languages: an introduction
The formal semantics of programming languages: an introduction
The complexity of default reasoning under the stationary fixed point semantics
Information and Computation
Aspects of partial information in databases
Aspects of partial information in databases
Information flow: the logic of distributed systems
Information flow: the logic of distributed systems
Experimenting with power default reasoning
AAAI '98/IAAI '98 Proceedings of the fifteenth national/tenth conference on Artificial intelligence/Innovative applications of artificial intelligence
Domains and lambda-calculi
Clausal logic and logic programming in algebraic domains
Information and Computation
LPNMR '97 Proceedings of the 4th International Conference on Logic Programming and Nonmonotonic Reasoning
Complexity of Power Default Reasoning
LICS '97 Proceedings of the 12th Annual IEEE Symposium on Logic in Computer Science
A lattice theoretic approach to computation based on a calculus of partially ordered type structures (property inheritance, semantic nets, graph unification)
The semantics of grammar formalisms seen as computer languages
ACL '84 Proceedings of the 10th International Conference on Computational Linguistics and 22nd annual meeting on Association for Computational Linguistics
A Categorical View on Algebraic Lattices in Formal Concept Analysis
Fundamenta Informaticae
A Categorical View on Algebraic Lattices in Formal Concept Analysis
Fundamenta Informaticae
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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 Π2P-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.