On the satisfiability of circumscription

doi:10.1016/0004-3702(86)90028-7

Artificial Intelligence

Volume 28, Issue 1, February 1986, Pages 17-27

https://doi.org/10.1016/0004-3702(86)90028-7 Get rights and content

Abstract

Etherington, Mercer and Reiter showed, on the basis of ideas of Bossu and Siegel, that circumscription cannot lead to inconsistency for universal formulas. We extend this result in three directions: to formulas of a more general syntactic form, to circumscription with some predicate symbols allowed to vary, and to prioritized circumscription.

References (6)

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Circumscription—a form of non-monotonic reasoning
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J. McCarthy
Applications of circumscription to formalizing common-sense knowledge
Artificial Intelligence
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G. Bossu et al.
Saturation, nonmonotonic reasoning and the closed-world assumption
Artificial Intelligence
(1985)

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Representing and reasoning with priorities are important in commonsense reasoning. This paper introduces a framework of prioritized logic programming (PLP), which has a mechanism of explicit representation of priority information in a program. When a program contains incomplete or indefinite information, PLP is useful for specifying preference to reduce non-determinism in logic programming. Moreover, PLP can realize various forms of commonsense reasoning in AI such as abduction, default reasoning, circumscription, and their prioritized variants. The proposed framework increases the expressive power of logic programming and exploits new applications in knowledge representation.
An extension of pointwise circumscription
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In this paper we generalize Lifschitz's pointwise circumscription ander the first-order framework. The generalized version has the ability for simultaneously minimizing several predicates at finitely many pinpoints in a pointwise manner. We show that if an anderlying first-order theory is almost existential, then the extended pointwise circumscription is complete with respect to minimal model semantics. Almost existential formulas are in the dual form of almost universal formulas, which was proposed by Lifschitz to investigate the satisfiability of circumscription. This completeness result is a generalization of the result by Kolaitis and Papadimitriou, who regarded the case of existential formulas. We also give a partial answer to the question for exponential growth of the size of first-order formulas equivalent to circumscription. Moreover we clarify that Lifschitz's pointwise circumscription is complete in a slightly wider class of positive formulas.
Circumscription. Completeness reviewed
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In this paper we demonstrate that some results on the completeness of P-defining theories published earlier are incorrect. We point out that by restricting the original propositions to well-founded theories results somewhat weaker than the original ones can be retained. We also present a theorem that provides some insight into the relation between completeness and reducibility and helps to identify the theories whose minimal models can be adequately handled with circumscription.
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In this paper, we present an alternative approach to the generalization of circumscription. Traditionally, the generalization of circumscription involves the change of ordering among models, while in the present study we only try to generalize the minimality criteria of models. We define the notion of abstractly minimal (or (P,Z)-minimal) models by isomorphism. Under this generalization we come up with the fact that some theories which are unsatisfiable in the original circumscription will be satisfiable now. Moreover, we prove that this generalization is completely coincident with the original circumscription in the case of well-founded theories.
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Research noteOn the satisfiability of circumscription

Abstract

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Research note
On the satisfiability of circumscription