Probabilistic quantifiers, adversaries, and complexity classes : An overview

Abstract

We consider inclusion relations among a multitude of classical complexity classes and classes with probabilistic components. A key tool is a method for characterizing such classes in terms of the ordinary quantifiers ∃ and ∀ together with a quantifier ∃+, which means roughly “for most”, applied to polynomial-time predicates. This approach yields a uniform treatment which leads to easier proofs for class-inclusion and hierarchy-collapse results. Furthermore the method captures some recently introduced game classes and game hierarchies. This survey also includes a charting of class-inclusion and oracle-based separation results.