Almost-natural proofs

Abstract: 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. For those who hope to separate P from NP using random function properties in some sense, discrimination is interesting, because it is constructive, yet may be thought of as a minor alteration of a property of a random function. The proof relies heavily on the self-defeating character of natural proofs. Our proof technique also yields an unconditional result, namely that there exist almost-large and useful properties that are constructive, if we are allowed to call non-uniform low-complexity classes "constructive." We note, though, that this unconditional result can also be proved by a more conventional counting argument. Finally, we give an alternative proof, communicated to us by Salil Vadhan at FOCS 2008, of one of our theorems, and we make some speculative remarks on the future prospects for proving strong circuit lower bounds.