Ordering Finite Variable Types with Generalized Quanti ers

Let Q be a nite set of generalized quantiiers. By L k (Q) we denote the k-variable fragment of FO(Q), rst order logic extended with Q. We show that for each k, there is a PFP(Q)-deenable linear pre-order whose equivalence classes in any nite structure A are the L k (Q)-types in A. For some special classes of generalized quantiiers Q, we show that such an ordering of L k (Q)-types is already deenable in IFP(Q). As applications of the above results, we prove some generalizations of the Abiteboul-Vianu theorem. For instance, we show that for any nite set Q of modular counting quantiiers, P = PSPACE if, and only if, IFP(Q) = PFP(Q) over nite structures. On the other hand, we show that an ordering of L k (Q)-types is not always deenable in IFP(Q). Indeed , we construct a single, polynomial time computable quantiier P such that the equivalence relation k;P , and hence ordering on L k (P)-types, is not deen-able in IFP(P).