Approximation by DNF: Examples and Counterexamples

Say that f : {0, 1}n → {0, 1} 2-approximates g : {0, 1}n → {0, 1} if the functions disagree on at most an 2 fraction of points. This paper contains two results about approximation by DNF and other small-depth circuits: (1) For every constant 0 < 2 < 1/2 there is a DNF of size 2 √ n) that 2-approximates the Majority function on n bits, and this is optimal up to the constant in the exponent. (2) There is a monotone function F : {0, 1}n → {0, 1} with total influence (AKA average sensitivity) I(F) ≤ O(log n) such that any DNF or CNF that .01-approximates F requires size 2 log n) and such that any unbounded fan-in AND-OR-NOT circuit that .01-approximates F requires size Ω(n/ log n). This disproves a conjecture of Benjamini, Kalai, and Schramm (appearing in [BKS99, Kal00, KS05]).