Fixed-Point Logics, Generalized Quantifiers, and Oracles

The monotone circuit problem Q MC is shown to be complete for xed point logic IFP under quantiier free reductions. Enhancing the circuits with an oracle Q leads to a problem complete for IFP(Q). By contrast, if L is any extension of FO with generalized quantiiers, one can always nd a Q such that IFP(Q) is not contained in L(Q). For L = FO(Q MC), we have L IFP but L(Q) < IFP(Q). The adjunction of Q reveals the diierence between these two representations of the class of IFP{deenable queries. Also for partial xed point logic, PFP a complete problem based on circuits is given, and, concerning the adjunction of further quantiiers, similar results as for IFP are proved. On ordered structures, where our results still hold, this reads as follows: For any given oracle Q, the complexity class PTIME Q (or PSPACE Q in a bounded oracle model) can be characterized by an extension of FO with a uniform sequence of quantiiers. However, there is no such logic L that satisses L(Q) PTIME Q (or PSPACE Q) for all Q. We also study second order logic. Each level i of the polynomial hierarchy has a complete circuit problem. Closing FO under this problem does not exceed i+1. Hence, the polynomial hierarchy collapses to a certain level if and only if there is a class Q such that SO FO(Q) holds on nite structures.