We consider several questions on the computational power of PP, the class of sets accepted by polynomial time-bounded probabilistic Turing machines with two-sided unbounded error probability. In particular, we consider the questions whether PP is as powerful as PSPACE and whether PP is more powerful than PH (the polynomial-time hierarchy). These questions have remained open. In fact, only a few results have been shown even in the relativized case.
In the present paper we deal with the above questions in a manner similar to Long (1985), Balcázar (1986), and Long and Selman (1986). We introduce a quantitative restriction on access to a given oracle for probabilistic oracle machines. The restriction, denoted as PRR(·), allows polynomial time-bounded probabilistic oracle machines to query only polynomially many positive strings but exponentially many negative strings in its computation tree. We show that for every oracle A that PPR(NP)(A) is included in PP(A). This strengthens the result shown recently by Biegel (1989). In fact, we have their result as an immediate consequence.
We next show some positive relativization results on the above questions. We show that seprating PP from PSPACE with some sparse oracle implies the separation of PP from PSPACE in the nonrelativized case. We also show that separating PP from PH with some sparse oracle implies the separation in the nonrelativized case. We also compare D # P and with PH. In particular, we show that D # P = PH iff D # (S) = PH (S) for some sparse oracle set S. Thus, the inequality D # P ≠ PH can also be relativized for all sparse oracle sets.
