Almost-Natural Proofs

Razborov and Rudich have shown that so-called natural proofs are not useful for separating P from NP unless hard pseudorandom number generators do not exist. This famous result is widely regarded as a serious barrier to proving strong lower bounds in circuit complexity theory. By definition, a natural combinatorial property satisfies two conditions, constructivity and largeness. Our main result is that if the largeness condition is weakened slightly, then not only does the Razborov–Rudich proof break down, but such “almost-natural” (and useful) properties provably exist. Specifically, under the same pseudorandomness assumption that Razborov and Rudich make, a simple, explicit property that we call discrimination suffices to separate P/poly from NP; discrimination is nearly linear-time computable and almost large, having density 2−q(n) where q is a quasi-polynomial function. (This is a slightly stronger result than the one announced in the FOCS 2008 extended of this paper.) ...

• Type: Conference
• Year: 2008
• Where: FOCS
• Authors: Timothy Y. Chow
• Keywords: Circuit Lower Bound, FOCS 2008, Lower Bound, Natural Proof, Theoretical Computer Science