PP is as Hard as the Polynomial-Time Hierarchy

In this paper, two interesting complexity classes, PP and $ \oplus {\text{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 ${\text{BP}} \cdot \oplus {\text{P}}$. As a consequence of the results, it follows that ${\text{PP}} \subseteq {\text{PH}}$ (or $\oplus {\text{P}} \subseteq {\text{PH}}$) 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.

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