ar X iv : 0 80 5 . 13 85 v 1 [ cs . C C ] 9 M ay 2 00 8 ALMOST - NATURAL PROOFS

Razborov and Rudich have famously shown that so-called natural proofs are not useful for separating P from NP unless hard pseudorandom number generators do not exist. Their 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. We show unconditionally 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. Moreover, if we assume that hard pseudorandom number generators exist, then a simple, explicit property that we call discrimination suffices to separate P from NP . For those who hope to separate P from NP using “random function properties” in some sense, discrimination is interesting, because it is almost natural and may be thought of as a “minor alteration” of a property of a random function. In more detail, one quantitative corollary of our results is that if γ(n) = n and λ(n) = n for constants k > 1 and ǫ > 0, then we can show unconditionally that there exists a SIZE(γ)-natural property of density 2−q(n) containing no SIZE(λ) Boolean functions of n variables, where q is a quasipolynomial function. We also show that, if 2 ǫ -hard pseudorandom number generators exist, then there is a nearly linear-time computable natural property of density 2−q(n) that separates NP from P/poly, where q grows slightly faster than a quasi-polynomial function. Compare this with Razborov and Rudich’s result that under the same hypothesis, q cannot be replaced by a polynomial function.