The expressive power of finitely many generalized quantifiers

We consider extensions of first order logic (FO) and fixed [Bpoint logic (FP) by means of generalized quantifiers in the sense of P. Lindstrom (1966). We show that adding a finite set of such quantifiers to FP fails to capture PTIME, even over a fixed signature. We also prove a stronger version of this result for PSPACE, which enables us to establish a weak version of a conjecture formulated previously by Ph.G. Kolaitis and M.Y. Vardi (1992). These results are obtained by defining a notion of element type for bounded variable logics with finitely many generalized quantifiers. Using these, we characterize the classes of finite structures over which the infinitary logic L_∞w ^w extended by a finite set of generalized quantifiers Q and is no more expressive than first order logic extended by the quantifiers in Q