Sharper results on the expressive power of generalized quantifiers

Abstract

In this paper we improve on some results of [3] and extend them to the setting of implicit definability. We show a strong necessary condition on classes of structures on which PSPACE can be captured by extending PFP with a finite set of generalized quantifiers. For IFP and PTIME the limitation of expressive power of generalized quantifiers is shown only on some specific 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 finitely many generalized quantifiers are also allowed.

We give a new proof of one of the main results of [3], characterizing the classes of structures on which L ^ω_∞,ω (Q) collapses to FO(Q), where Q is a set of finitely many generalized quantifiers. This proof easily generalizes to the case of implicit definability, unlike the quantifier elimination argument of [3] which does not easily get adapted to implicit definability setting. This result is then used to show the limitation of expressive power of implicit closure of L ^ω_∞,ω (Q).

Finally, we adapt the technique of quantifier elimination due to Scott Weinstein, used in [3], to show that IMP(Lk(Q))-types can be isolated in the same logic.