The closure of Monadic NP (extended abstract)

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 (El), which allows sentences consisting of a string of existential second-order quantifiers followed by a first-order formula. Monadic NP is the class of problems expressible in monadic Cl, i.e., Xi with therestriction that the second-order quantifiers 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 3colorable belongs to monadic NP, because 3colorability 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, monadicNPis notarobustclass, inthatitisnotclosed under first-order quantification. We define closed monadic NP to be the closure of monadic NP under first-order quanthlcation and existential unary second-order quantification. Thus, closed monadic NP differs from monadic NP in that we allow the possibility of arbitrary interleavings of tirstorder qunntifiers among the existential unary second-order quantifiers, We show that closed monadic NP is a natural, rich, and robust subclass of NP. As evidence for its richness, we show that not only is it a proper extension of monadic NP, but that it contains properties not in various other extensions of monadic NP. In particular, we show that closed monadic NP contnins an undirected graph property not in the closure of monadic NP under first-order quantification and Boolean operations, Our lower-boundproofsrequire a number of new game-theoretic techniques. *The full vcrnlon of this pnpcr, including proofs, can be obtaioed from: hltp://v/wv~,nlmadcn,ibm.comlcs/people/