The Expressive Power of Finitely Many Generalized Quantifiers

Abstract

We consider extensions of first order logic (FO) and fixed point logic (FP) by means of generalized quantifiers in the sense of Lindström. We show that adding a finite set of such quantifiers to FP fails to capture PTIME, even over a fixed signature, strengthening earlier results. We also prove a stronger version of this result for PSPACE, which enables us to establish that PSPACE ≠ FO on any infinite class of ordered structures, a weak version of the ordered conjecture formulated by Ph. G. Kolaitis and M. Y. Vardi (Fixpoint logic vs. infinitary logic in finite-model theory, in "Proceedings, 7th IEEE Symposium on Logic in Computer Science, 1992," pp. 46-57). 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 Lω∞ω extended by a finite aw set of generalized quantifiers Q is no more expressive than first order logic extended by the quantifiers in Q.