ICALP 2007: Automata, Languages and Programming pp 195-206

Approximation by DNF: Examples and Counterexamples

• Ryan O’Donnell
• Karl Wimmer
Conference paper

DOI: 10.1007/978-3-540-73420-8_19

Part of the Lecture Notes in Computer Science book series (LNCS, volume 4596)
Cite this paper as:
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

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]).