On the Sensitivity Conjecture for Disjunctive Normal Forms

TL;DR

This paper investigates the sensitivity conjecture for Boolean functions within the class of functions with the Normalized Block property, establishing a quadratic upper bound on block sensitivity in terms of sensitivity and extending known structural results.

Contribution

It proves that functions with the Normalized Block property have block sensitivity at most quadratically related to sensitivity, and extends structural characterizations of Boolean functions.

Findings

For functions with the Normalized Block property, bs(f) ≤ 4s(f)^2.

Almost all functions with quadratic sensitivity-block sensitivity separation have the Normalized Block property.

Boolean functions are determined by their values on a Hamming ball of radius 2s(f), with matching lower bounds constructed.

Abstract

The sensitivity conjecture of Nisan and Szegedy [CC '94] asks whether for any Boolean function $f$, the maximum sensitivity $s(f)$, is polynomially related to its block sensitivity $bs(f)$, and hence to other major complexity measures. Despite major advances in the analysis of Boolean functions over the last decade, the problem remains widely open. In this paper, we consider a restriction on the class of Boolean functions through a model of computation (DNF), and refer to the functions adhering to this restriction as admitting the Normalized Block property. We prove that for any function $f$ admitting the Normalized Block property, $bs(f) \leq 4s(f)^2$. We note that (almost) all the functions mentioned in literature that achieve a quadratic separation between sensitivity and block sensitivity admit the Normalized Block property. Recently, Gopalan et al. [ITCS '16] showed that every Boolean function $f$ is uniquely specified by its values on a Hamming ball of radius at most $2s(f)$. We extend this result and also construct examples of Boolean functions which provide the matching lower bounds.