Polylogarithmic Independence Can Fool DNF Formulas

We show that any k-wise independent probability measure on {0, 1}n can O(m2ldr2ldr2-radick/10)-fool any boolean function computable by an rn-clauses DNF (or CNF) formula on n variables. Thus, for each constant c > 0. there is a constant e > 0 such that any boolean function computable by an m-clauses DNF (or CNF) formula can be in m-e-fooled by any clog in-wise probability measure. This resolves, asymptotically and up to a logm factor, the depth-2 circuits case of a conjecture due to Linial and Nisan (1990). The result is equivalent to a new characterization of DNF (or CNF) formulas by low degree polynomials. It implies a similar statement for probability measures 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 PRG's ofO(log2 m log n)-seed length for m-clauses DNF (or CNF) formulas on n variables, improving previously known seed lengths.

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