Loop formulas for circumscription

Clark's completion is a simple nonmonotonic formalism and a special case of several nonmonotonic logics. Recently there has been work on extending completion with “loop formulas” so that general cases of nonmonotonic logics such as logic programs (under the answer set semantics) and McCain–Turner causal logic can be characterized by propositional logic in the form of “completion + loop formulas”. In this paper, we show that the idea is applicable to McCarthy's circumscription in the propositional case, with Lifschitz's pointwise circumscription playing the role of completion. We also show how to embed propositional circumscription in logic programs and in causal logic, inspired by the uniform characterization of “completion + loop formulas”.