A Dichotomy Result for Ramsey Quantifiers

Abstract. Ramsey quantifiers are a natural object of study not only for logic and computer science, but also for formal semantics of natu-ral language. Restricting attention to finite models leads to the natural question whether all Ramsey quantifiers are either polynomial-time com-putable or NP-hard, and whether we can give a natural characterization of the polynomial-time computable quantifiers. In this paper, we first show that there exist intermediate Ramsey quantifiers and then we prove a dichotomy result for a large and natural class of Ramsey quantifiers, based on a reasonable and widely-believed complexity assumption. We show that the polynomial-time computable quantifiers in this class are exactly the constant-log-bounded Ramsey quantifiers.

Enhancing Fixed Point Logic with Cardinality Quantifiers

Let Q IPP be any quantifier such that FO(QIFP), first-order logic enhanced with Q IPP and its vectorizations, equals inductive fixed point logic, IFP in expressive power. It is known that for certain quantifiers Q, the equivalence FO(QIFP) ≡ IFP is no longer true if Q is added on both sides. Rather, we have FO (QIFP, Q) < IFP(Q) in such cases. We extend these results to a great variety of quantifiers, namely all unbounded simple cardinality quantifiers. Our argument also applies to partial fixed point logic, PFP. In order to establish an analogous result for least fixed point logic, LFP, we exhibit a general method to pass from arbitrary quantifiers to monotone quantifiers. Our proof shows that the three isomorphism problem is not definable in, infinitary logic extended with all monadic quantifiers and their vectorizations, where a finite bound is imposed to the number of variables as well as to the number of nested quantifiers in Q1. This strengthens a result of Etessami and Immerman by which tree isomorphism is not definable in TC + COUNTIN