A logical characterization of the counting hierarchy

Abstract

In this article we give a logical characterization of the counting hierarchy. The counting hierarchy is the analogue of the polynomial hierarchy, the building block being Probabilistic polynomial time PP instead of NP. We show that the extension of first-order logic by second-order majority quantifiers of all arities describes exactly the problems in the counting hierarchy. We also consider extending the characterization to general proportional quantifiers Q k r interpreted as “more than an r -fraction of k -ary relations”. We show that the result holds for rational numbers of the form s /2 m but for any other 0 < r < 1 the corresponding logic satisfies the 0-1 law.