Sharper Results on the Expressive Power of Generalized Quantifiers

In this paper we improve on some results of 3] and extend them to the setting of implicit deenability. We show a strong necessary condition on classes of structures on which PSPACE can be captured by extending PFP with a nite set of generalized quantiiers. For IFP and PTIME the limitation of expressive power of generalized quantiiers is shown only on some speciic nontrivial classes. These results easily extend to implicit closure of these logics. In fact, we obtain a nearly complete characterization of classes of structures on which IMP(PFP) can capture PSPACE if nitely many generalized quantiiers are also allowed. We give a new proof of one of the main results of 3], characterizing the classes of structures on which L ! 1;! (Q) collapses to FO(Q), where Q is a set of nitely many generalized quantiiers. This proof easily generalizes to the case of implicit deenability, unlike the quantiier elimination argument of 3] which does not easily get adapted to implicit deenabil-ity setting. This result is then used to show the limitation of expressive power of implicit closure of L ! 1;! (Q). Finally, we adapt the technique of quantiier elimination due to Scott Weinstein, used in 3], to show that IMP(L k (Q))-types can be isolated in the same logic.