Lecture 14 : The Complexity of Counting II

Let M be the polynomial-time TM and p be the polynomial bound such that f(x) = #{y ∈ {0, 1}p(|x|) | M(x, y) = 1}. Define M ′(x,N, y′) for N a length p(|x|) binary string viewed as an integer and |y′| = p(|x|) + 1 by M ′(x,N, 0y) = M(x, y) and M ′(x,N, 1y) = 1 iff y ≥ N . Note that there will be precisely 2p(|x|) − N strings y′ = 1y such that M ′(x,N, y′) accepts. Therefore f(x) > N if and only if there are > N strings y′ = 0y such that M ′(x,N, y′) for a total of > 2p(|x|) witness strings y′ such thatM ′(x,N, y′) accepts, which is> 1/2 of all strings of the witness length. The PP oracle will be for the language defined by the machine M ′ with input (x,N) and length p(n) + 1. It will accept (x,N) precisely when f(x) > N . Definition 1.2. A function g is #P-complete iff