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 exists(n) (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 [JM01]) has with the Directed Reachability Problem. We use alt(S) to show that Pi(n) not subset of or equal to FO(Sigma(n)), Sigma(n) not subset of or equal to FO(Delta(n)), and Delta(n+1) not subset of or equal to FOB(Sigma(n)), solving some open problems raised in [Mat98].