First order quantifiers in monadic second order logic

Abstract

Abstract. This paper studies the expressive power that an extra first order quantifier adds to a fragment of monadic second order logic, extending the toolkit of Janin and Marcinkowski [JM01].We introduce an operation existsn (S) on properties S that says "there are n components having S". We use this operation to show that under natural strictness conditions, adding a first order quantifier word u to the beginning of a prefix class V increases the expressive power monotonically in u. As a corollary, if the first order quantifiers are not already absorbed in V , then both the quantifier alternation hierarchy and the existential quantifier hierarchy in the positive first order closure of V are strict.We generalize and simplify methods from Marcinkowski [Mar99] to uncover limitations of the expressive power of an additional first order quantifier, and show that for a wide class of properties S, S cannot belong to the positive first order closure of a monadic prefix class W unless it already belongs to W .We introduce another operation alt(S) on properties which has the same relationship with the Circuit Value Problem as reach(S) (defined in These hierarchy problems have been hard to attack. Fagin suggested studying monadic second order logic (MSO), a simplified fragment of full second order logic, in which second order quantifiers are only allowed over unary relations, i.e., subsets of the underlying universe. MSO was indeed tractable. In [Fag75] Fagin himself used Ehrenfeucht-Fraïssé games to show that existential MSO (called monadic NP) is not closed under negation, thus separating monadic NP from monadic co-NP. Matz and Thomas [MT97] showed that the monadic quantifier alternation hierarchy is strict. In particular, they showed that Σ n ⊂ B(Σ n ) ⊂ ∆ n+1 ⊂ Σ n+1 , where B denotes Boolean closure. Their argument was based on growth rates of definable functions.[

Rank Hierarchies for Generalized Quantifiers

“…Intuitively, G(A, B) is the union of n + 1 copies of A and n + 1 copies of B, which is also equipped with an "edge" relation E(x, y), that connects each c i ∈ C to each element of the corresponding copy of A, and similarly for D and B. In the terminology of [JM01] and [KL04], c i is the root of a cone over A i , and d i is the root of a cone over B i , in the model G(A, B).…”