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. 0)P. As a consequence of the results, it follows that PP PH (or 03P___ 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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In this paper, it is shown that many natural counting classes, such as PP, C=P, and MOD#, are at least as computationally hard as PH (the polynomial-time hierarchy) in the following sense: for each K of the counting classes above, every set in K(PH) is polynomial-time randomized many-one reducible to a set in K with two-sided exponentially small error probability. As a consequence of the result, we see that all the counting classes above are computationally harder than PH unless PH collapses to a finite level. Some other consequences are also shown. 'The research reported here was done while the authors visited the

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