The Closure of Monadic NP

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It is a well-known result of Fagin that the complexity class NP coincides with the class of problems expressible in existential second-order logic ( 1 1), which allows sentences consisting of a string of existential second-order quanti ers followed by a rst-order formula. Monadic NP is the class of problems expressible in monadic 1 1, i.e., 1 1 with the restriction that the second-order quanti ers are all unary, and hence range only over sets (as opposed to ranging over, say, binary relations). For example, the property of a graph being 3-colorable belongs to monadic NP, because 3-colorability can be expressed by saying that there exists three sets of vertices such that each vertex is in exactly one of the sets and no two vertices in the same set are connected by an edge. Unfortunately, monadic NP is not a robust class, in that it is not closed under rst-order quanti cation. We de ne closed monadic NP to be the closure of monadic NP under rst-order quanti cation and existential unary ...

Miklós Ajtai, Ronald Fagin, Larry J. Stockm

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