First order quantifiers in~monadic second order logic

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. Wegeneralize and simplifymethods fromMarcinkowski [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 classW unless it already belongs toW . We introduce another operation alt(S) on properties which has the same relationship with the Circuit Value Problem as reach(S) (defined in [JM01]) has with the Directed Reachability Problem. We use alt(S) to show that Πn 6⊆ FO(Σn), Σn 6⊆ FO(∆n), and ∆n+1 6⊆ FOB(Σn), solving some open problems raised