PP is as Hard as the Polynomial-Time Hierarchy

In this paper, two interesting complexity classes, PP and ⊕ P, are compared with PH, the polynomial-time hierarchy. It is shown that every set in PH is polynomial-time Turing reducible to a set in PP, and PH is included in BP · ⊕ P. As a consequence of the results, it follows that PP ⊆ PH (or ⊕ P ⊆ PH) implies a collapse of PH. A stronger result is also shown: every set in PP(PH) is polynomial-time Turing reducible to a set in PP.