Pointwise Circumscription
Pointwise circumscription is a variant of first-order circumscription that has been introduced by Vladimir Lifschitz. In the propositional case, pointwise and predicate circumscription coincide. The rationale of pointwise circumscription it minimize the value of a predicate for each tuple of values separately, rather than minimizing the extension of the predicate. For example, there are two models of with domain, one setting and the other setting . Since the extension of in the first model is while the extension for the second is, circumscription only selects the first model.
In pointwise circumscription, each tuple of values is considered separately. For example, in the formula one would consider the value of separately from . A model is minimal only it is not possible to turn any such value from true to false while still satisfying the formula. As a result, the model in which is selected by pointwise circumscription because turning only into false does not satisfy the formula, and the same happens for .
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