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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]).

Keywords

Boolean Function Monotone Function Random Graph Graph Property Input Gate 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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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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