Domain and Formula Circumscription
An earlier formulation of circumscription by McCarthy is based on minimizing the domain of first-order models, rather than the extension of predicates. Namely, a model is considered less than another if it has a smaller domain, and the two models coincide on the evaluation of the common tuples of values. This version of circumscription can be reduced to predicate circumscription.
Formula circumscription was a later formalism introduced by McCarthy. This is a generalization of circumscription in which the extension of a formula is minimized, rather than the extension of a predicate. In other words, a formula can be specified so that the set of tuples of values of the domain that satisfy the formula is made as small as possible.
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