Relationships between the number of inputs and other complexity measures of Boolean functions

TL;DR

This paper establishes new bounds on relevant variables of Boolean functions using various complexity measures, unifies previous results, and enhances key inequalities, thereby strengthening the theoretical understanding of Boolean function complexity.

Contribution

It generalizes and refines existing bounds, unifies multiple complexity measures, and improves the block sensitivity versus degree inequality, impacting the sensitivity conjecture proof.

Findings

New bounds on relevant variables in terms of complexity measures

Unified framework for previous bounds

Improved block sensitivity vs. degree inequality

Abstract

We generalize and extend the ideas in a recent paper of Chiarelli, Hatami and Saks to prove new bounds on the number of relevant variables for boolean functions in terms of a variety of complexity measures. Our approach unifies and refines all previously known bounds of this type. We also improve Nisan and Szegedy's well-known block sensitivity vs. degree inequality by a constant factor, thereby improving Huang's recent proof of the sensitivity conjecture by the same constant.