Abstract
Say that f:{0,1}n →{0,1} ε-approximates g : {0,1}n →{0,1} if the functions disagree on at most an ε fraction of points. This paper contains two results about approximation by DNF and other small-depth circuits:
(1) For every constant 0 < ε< 1/2 there is a DNF of size \(2^{O(\sqrt{n})}\) that ε-approximates the Majority function on n bits, and this is optimal up to the constant in the exponent.
(2) There is a monotone function \(\mathcal{F} : \{{0,1}\}^{n} \rightarrow \{{0,1}\}\) with total influence (AKA average sensitivity) \({\mathbb{I}}({\mathcal{F}}) \leq O(\log n)\) such that any DNF or CNF that .01-approximates \({\mathcal{F}}\) requires size 2Ω(n / logn) and such that any unbounded fan-in AND-OR-NOT circuit that .01-approximates \({\mathcal{F}}\) requires size Ω(n/ logn). This disproves a conjecture of Benjamini, Kalai, and Schramm (appearing in [BKS99,Kal00,KS05]).
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O’Donnell, R., Wimmer, K. (2007). Approximation by DNF: Examples and Counterexamples. In: Arge, L., Cachin, C., Jurdziński, T., Tarlecki, A. (eds) Automata, Languages and Programming. ICALP 2007. Lecture Notes in Computer Science, vol 4596. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-73420-8_19
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