On Winning Strategies with Unary Quantifiers

A combinatorial argument for two nite structures to agree on all sentences with bounded quantiier rank in rst-order logic with any set of unary generalized quantiiers, is given. It is known that connectivity of nite structures is neither in monadic 1 1 nor in L !! (Q u), where Q u is the set of all unary generalized quantiiers. Using this combinatorial argument and a combination of second-order Ehrenfeucht-Fra ss e games developed by Ajtai and Fagin, we prove that connectivity of nite structures is not in monadic 1 1 with any set of unary quantiiers, even if sentences are allowed to contain built-in relations of moderate degree. The combinatorial argument is also used to show that no class (if it is not in some sense trivial) of nite graphs deened by forbidden minors, is in L !! (Q u). Especially, the class of planar graphs is not in L !! (Q u).