Approximation by DNF: Examples and Counterexamples

  • Ryan O’Donnell
  • Karl Wimmer
Part of the Lecture Notes in Computer Science book series (LNCS, volume 4596)

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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Copyright information

© Springer-Verlag Berlin Heidelberg 2007

Authors and Affiliations

  • Ryan O’Donnell
    • 1
  • Karl Wimmer
    • 1
  1. 1.Carnegie Mellon University, Pittsburgh PA 15213USA

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