APPROXIMATING THE AND-OR TREE

Alexander A Sherstov

doi:10.4086/toc.2013.v009a020

Abstract

The approximate degree of a Boolean function f is the least degree of a real polynomial that approximates f within 1=3 at every point. We prove that the function V n i=1 W n j=1 xi j, known as the AND-OR tree, has approximate degree W(n). This lower bound is tight and closes a line of research on the problem, the best previous bound being W(n 0:75 ). More generally, we prove that the function V m=1 W n j=1 xi j has approximate degree W( p mn); which is tight. The same lower bound was obtained independently by Bun and Thaler (2013) using related techniques.

Highlights

Over the past two decades, representations of Boolean functions by real polynomials have played an important role in theoretical computer science
| f (x) − p(x)| ≤ 1 3 for every x ∈ {0, 1}n, In other words, we are interested in the pointwise approximation of Boolean functions by real polynomials
The approximate degree has been used to obtain the fastest known algorithms for PAC-learning DNF formulas [42, 19] and read-once formulas [4] and the fastest known algorithm for agnostically learning disjunctions [17]. Another well-known use of the approximate degree is an algorithm for approximating the inclusion-exclusion formula based on its initial terms [22, 16, 33, 43]

Summary

Introduction

Over the past two decades, representations of Boolean functions by real polynomials have played an important role in theoretical computer science. The approximate degree has been used many times to prove tight lower bounds on quantum query complexity, e. The approximate degree has been used to obtain the fastest known algorithms for PAC-learning DNF formulas [42, 19] and read-once formulas [4] and the fastest known algorithm for agnostically learning disjunctions [17] Another well-known use of the approximate degree is an algorithm for approximating the inclusion-exclusion formula based on its initial terms [22, 16, 33, 43]. These applications motivate the study of the approximate degree as a complexity measure in its own right. This lower bound is tight for all m and n, by the results of Høyer et al [15]

Proof overview

Independent work by Bun and Thaler

Preliminaries

Main Result