Polylogarithmic Independence Can Fool DNF Formulas

Abstract

We show that any k-wise independent probability distribution on $\{0,1\}^n$ $O(m^{2.2}$ $2^{-\sqrt{k}/10})$-fools any boolean function computable by an m-clause disjunctive normal form (DNF) (or conjunctive normal form (CNF)) formula on n variables. Thus, for each constant $e>0$, there is a constant $c>0$ such that any boolean function computable by an m-clause DNF (or CNF) formula is $m^{-e}$-fooled by any $c\log^2m$-wise probability distribution. This resolves up to an $O(\log m)$ factor the depth-2 circuit case of a conjecture due to Linial and Nisan [Combinatorica, 10 (1990), pp. 349–365]. The result is equivalent to a new characterization of DNF (or CNF) formulas by low degree polynomials. It implies a similar statement for probability distributions with the small bias property. Using known explicit constructions of small probability spaces having the limited independence property or the small bias property, we directly obtain a large class of explicit pseudorandom generators of $O(\log^2m\log n)$-seed length for m-clause DNF (or CNF) formulas on n variables, improving previously known seed lengths.