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Complexity of propositional nested circumscription and nested abnormality theories

Complexity of propositional nested circumscription and nested abnormality theories Circumscription has been recognized as an important principle for knowledge representation and common-sense reasoning. The need for a circumscriptive formalism that allows for simple yet elegant modular problem representation has led Lifschitz (AIJ, 1995) to introduce nested abnormality theories (NATs) as a tool for modular knowledge representation, tailored for applying circumscription to minimize exceptional circumstances. Abstracting from this particular objective, we propose L CIRC , which is an extension of generic propositional circumscription by allowing propositional combinations and nesting of circumscriptive theories. As shown, NATs are naturally embedded into this language, and are in fact of equal expressive capability. We then analyze the complexity of L CIRC and NATs, and in particular the effect of nesting. The latter is found to be a source of complexity, which climbs the Polynomial Hierarchy as the nesting depth increases and reaches PSPACE-completeness in the general case. We also identify meaningful syntactic fragments of NATs which have lower complexity. In particular, we show that the generalization of Horn circumscription in the NAT framework remains coNP-complete, and that Horn NATs without fixed letters can be efficiently transformed into an equivalent Horn CNF, which implies polynomial solvability of principal reasoning tasks. Finally, we also study extensions of NATs and briefly address the complexity in the first-order case. Our results give insight into the “cost” of using L CIRC (respectively, NATs) as a host language for expressing other formalisms such as action theories, narratives, or spatial theories. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png ACM Transactions on Computational Logic (TOCL) Association for Computing Machinery

Complexity of propositional nested circumscription and nested abnormality theories

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References (57)

Publisher
Association for Computing Machinery
Copyright
Copyright © 2005 by ACM Inc.
ISSN
1529-3785
DOI
10.1145/1055686.1055688
Publisher site
See Article on Publisher Site

Abstract

Circumscription has been recognized as an important principle for knowledge representation and common-sense reasoning. The need for a circumscriptive formalism that allows for simple yet elegant modular problem representation has led Lifschitz (AIJ, 1995) to introduce nested abnormality theories (NATs) as a tool for modular knowledge representation, tailored for applying circumscription to minimize exceptional circumstances. Abstracting from this particular objective, we propose L CIRC , which is an extension of generic propositional circumscription by allowing propositional combinations and nesting of circumscriptive theories. As shown, NATs are naturally embedded into this language, and are in fact of equal expressive capability. We then analyze the complexity of L CIRC and NATs, and in particular the effect of nesting. The latter is found to be a source of complexity, which climbs the Polynomial Hierarchy as the nesting depth increases and reaches PSPACE-completeness in the general case. We also identify meaningful syntactic fragments of NATs which have lower complexity. In particular, we show that the generalization of Horn circumscription in the NAT framework remains coNP-complete, and that Horn NATs without fixed letters can be efficiently transformed into an equivalent Horn CNF, which implies polynomial solvability of principal reasoning tasks. Finally, we also study extensions of NATs and briefly address the complexity in the first-order case. Our results give insight into the “cost” of using L CIRC (respectively, NATs) as a host language for expressing other formalisms such as action theories, narratives, or spatial theories.

Journal

ACM Transactions on Computational Logic (TOCL)Association for Computing Machinery

Published: Apr 1, 2005

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