Probabilistic Quantifiers and Games

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 3 and V together with a quantifier 3+, 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. @?I 1988 Academic Press. Inc.