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...cription for well-founded theories. 1 Introduction Circumscription is proposed by McCarthy [6] as a possible solution for the qualification problem in AI. He later introduced predicate circumscription=-=[7]-=- as a generalization of previous version which he called domain circumscription. Given a theory T and a set of predicates P, predicate circumscription minimizes the extensions of P that satisfy T. The...

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...r discussion is restricted to single predicate circumscription, we give the two definitions below for a single predicate whereas they are originally for multiple predicates. Definition 1.3 (Lifschitz =-=[5]-=-) A theory T is well-founded w.r.t (P; Z) if for each model A of T, there exists a (P; Z)-minimal model B such that BsT P; Z A. However, there are perfectly reasonable theories, such as the simple the...

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...uces sensible results. Therefore, these new definitions also coincide with the original definition of circumscription for well-founded theories. 1 Introduction Circumscription is proposed by McCarthy =-=[6]-=- as a possible solution for the qualification problem in AI. He later introduced predicate circumscription[7] as a generalization of previous version which he called domain circumscription. Given a th...

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... A of T, there exists a (P; Z)-minimal model B such that BsT P; Z A. However, there are perfectly reasonable theories, such as the simple theory of natural successor (the example given in Etherington =-=[3]-=-), that are not well-founded. Another approach to the same problem is to weaken circumscription by generalizing the minimality criterion which will accordingly increase the class of theories that can ...

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...Predicate Circumscription The original definition of circumscription is based on minimal models. However, if a theory has no minimal models, then circumscribing it gives an inconsistent theory. Davis =-=[1]-=- and Etherington [3] provide examples of such theories for domain and predicate circumscription, respectively. Liau et. al. [4] presents Etherington's example and shows how their definition of abstrac...

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...same problem is to weaken circumscription by generalizing the minimality criterion which will accordingly increase the class of theories that can be consistently handled. Definition 1.4 (Liau et. al. =-=[4]-=-) Given a theory T, a model A of T is abstractly (P; Z)-minimal if for each model BsT P; Z A, B = A. Since a theory can have abstractly minimal models without having minimal models (i.e., theory of su...