On the sensitivity conjecture for disjunctive normal forms

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) ≤ 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.