The Power of the Middle Bit of a #P Function

We introduce the class MP of languages L which can be solved in polynomial time with the additional information of one bit from a #P function f . We prove that the polynomial hierarchy and the classes ModkP, k ≥ 2, are low for this class. We show that the middle bit of f(x) is as powerful as any other bit, and that a wide range of bits around the middle have the same power. By contrast, the O(log n) many least significant bits are equivalent to ⊕P [BeGiHe 90], and by a simple corollary to the result that PP is closed under intersection [BeReSp 91], the O(log n) many most significant bits are equivalent to PP; hence these bits are probably weaker. We study also the subclass AmpMP of languages whose MP representations can be “amplified,” showing that BPP⊕P ⊆ AmpMP, and that important subclasses of AmpMP are low for MP. We translate some of these results to the area of circuit complexity using MidBit (middle bit) gates. A MidBit gate over w inputs x1, . . . , xw is a gate which outputs the value of the blog(w)/2cth bit in the binary representation of the number ∑w i=1 xi. We show that every language in ACC can be computed by a family of depth-2 deterministic circuits of size 2(logn) O(1) with a MidBit gate at the root and AND-gates of fan-in (log n)O(1) at the leaves. This result improves the known upper bounds for the class ACC.