Polylogarithmic independence fools AC0 circuits
Journal of the ACM (JACM)
BQP and the polynomial hierarchy
Proceedings of the forty-second ACM symposium on Theory of computing
Improved pseudorandom generators for depth 2 circuits
APPROX/RANDOM'10 Proceedings of the 13th international conference on Approximation, and 14 the International conference on Randomization, and combinatorial optimization: algorithms and techniques
Bounded Independence Fools Halfspaces
SIAM Journal on Computing
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We show that any k-wise independent probability measure on {0, 1}^n can {\text{O}}(\sqrt {m^{2.2} 2^{ - \sqrt {k/10} } } )-fool any boolean function computable by an m-clauses DNF (or CNF) formula on n variables. Thus, for each constante 0, there is a constantc 0 such that any boolean function computable by anm-clauses DNF (or CNF) formula can be {\text{m}}^{ - e} e-fooled by any c\log ^2 m-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 of {\text{O}}(\log ^2 m\log n) -seed length for m-clauses DNF (or CNF) formulas on n variables, improving previously known seed lengths.