The Closure of Monadic NP

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 (7 1 1 ), 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 7 1 1 , i.e., 7 1 1 with the restriction 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 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 first-order quantification. We define closed monadic NP to be the closure of monadic NP under first-order quantification and existential unary second-order quantification. Thus, closed monadic NP differs from monadic NP in that we allow the possibility of arbitrary interleavings of firstorder quantifiers 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 contains an undirected graph property not in the closure of monadic NP under first-order quantification and Boolean operations. Our lower-bound proofs require a number of new game-theoretic techniques. Academic Press

Partially Ordered Connectives and ∑1 1 on Finite Models

“…For instance, the complexity theorist's headache caused by the NP = coNP-problem can now be shared by the logician working on the Σ 1 1 = Π 1 1 -problem. 3 Indeed, logicians took up the challenge and nowadays separating logics related to complexity classes is one of their main occupations. By and large they go about by mapping out fragments of various relevant logics.…”

“…A point in case is Fagin's [10] study of the monadic fragments of Σ 1 1 and Π 1 1 , showing that they do not coincide. The results in [10] did arouse a lot of interest in monadic languages [2,3,20], but somewhat disappointingly, we are still waiting for methods to separate binary, existential, second-order logic from 3-ary, existential, second-order logic, see [5], or even from binary, universal, second-order logic.…”

Fragments of Monadic Second-Order Logics Over Word Structures

“…Ajtai and al, and Matz, in [1,18], have introduced and studied some fragments of MSO that contains ∃MSO. One of these fragments is the first-order closure of ∃MSO.…”

First order quantifiers in monadic second order logic

“…They posed the problem of whether the corresponding hierarchy is strict. Marcinkowski [Mar99] showed that Directed Reachability is not in FO(Σ 1 ), answering a question in [AFS98]. The tools of [AFS00] and [Mar99] were put in an abstract form by Janin and Marcinkowski in [JM01], to study the expressive power of fragments of MSO defined by prefix classes.…”

“…Ajtai, Fagin, and Stockmeyer in [AFS98] and [AFS00] proposed closed monadic NP, in which first order quantifiers are freely mixed with monadic second order existential quantifiers, as the "right" monadic version of NP. They posed the problem of whether the corresponding hierarchy is strict.…”