On the Power of One Bit of a #P Function

We introduce the class MP of languages L which can be solved in polynomial time with an oracle for one selected bit of the value f (y) of a #P function f on a selected argument y. This extends the much-studied language classes P and PP, which correspond to the power of the least and most signiicant bits, respectively. We show that MP is captured by the power of the middle bit; namely: a language L is in MP ii for some #P function f 0 and all x, x 2 L () the middle bit of f 0 (x) in binary notation is a `1'. Also, S. Toda's proof Tod89, Tod91] that the polynomial hierarchy (PH) is contained in P #P actually gives: PH BPPP] CP MP. The class MP has complete problems, and is closed under complements and under polynomial-time many-one reducibility. We show that MP is closed under intersection ii, for any xed k > 0, k bits of a #P function are no more powerful than one. Moreover, if there is a polynomial-time construction for the closure under intersection, then O(log n)-many bits have the same power as one. We examine the subclass AmpMP of languages whose MP representations can be \ampliied," showing that BPPP] AmpMP, and that for any p m-closed subclass C of AmpMP, MP C = MP. Hence BPPP] is low for MP, and if C=P AmpMP, then PP PP = MP. Finally, our work leads to a purely mathematical question about the size of integer-valued polynomials p(x; y) which satisfy certain congruence relations, one which also matters to the theory of bounded-depth circuits.