Bounding Queries in the Analytic Polynomial-Time Hierarchy

In a previous paper the present authors [4] investigated an 9-8-hierarchy over P using word quantifiers as well as two types of set quantifiers, the so called analytic polynomial-time hierarchy. The fact that some constructions there result in a bounded number of oracle queries and the recent PCP results which can be expressed by set quantifiers with a bounded number of queries motivated us to examine a hierarchy which extends the analytic polynomial-time hierarchy by considering restrictions on the number of oracle queries. This hierarchy is called bounded analytic polynomial-time hierarchy. We show that every class from this hierarchy having a certain normal form coincides with one of the classes NP, coNP, PSPACE, exp k or exp k (k 1). All these characterizations remain valid if the queries are asked in a nonadaptive form, i.e. in “parallel”.