Small-Bias is Not Enough to Hit Read-Once CNF

Abstract

Small-bias probability spaces have wide applications in pseudorandomness which naturally leads to the study of their limitations. Constructing a polynomial complexity hitting set for read-once CNF formulas is a basic open problem in pseudorandomness. We show in this paper that this goal is not achievable using small-bias spaces. Namely, we show that for each read-once CNF formula F with probability of acceptance p and with m clauses each of size c, there exists a δ-biased distribution μ on {0, 1}n such that δ = 2−Ω(logm log(1/p)) and no element in the support of μ satisfies F, where n = mc (assuming that e−m≤p≤p0, where p0 > 0 is an absolute constant). In particular if p = n−Θ(1), the needed bias is 2−Ω(log 2 n), which requires a hitting set of size 2Ω(log2n). Our lower bound on the needed bias is asymptotically tight. The dual version of our result asserts that if flow: { 0 , 1 } n→ ℝ is such that and E[flow] > 0 and flow(x) ≤ 0 for each x ∈ {0, 1}n such that F(x) = 0, then the L1-norm of the Fourier transform of flow is at least E[flow]2Ω(logm log(1/p)). Our result extends a result due to De, Etesami, Trevisan, and Tulsiani (APPROX-RANDOM 2010) who proved that the small-bias property is not enough to obtain a polynomial complexity PRG for a family of read-once formulas of Θ(1) probability of acceptance. © 2016, Springer Science+Business Media New York.