On the distribution of sensitivities of symmetric Boolean functions

TL;DR

This paper investigates the maximum sensitivity of symmetric Boolean functions, revealing that most such functions tend to have the highest possible sensitivity, which impacts their complexity analysis.

Contribution

It provides a count of symmetric Boolean functions with maximum sensitivity and shows that most have sensitivity equal to the number of variables, $n$, highlighting limitations of sensitivity as a complexity measure.

Findings

Most symmetric Boolean functions have maximum sensitivity $n$.

Sensitivity is limited as a complexity measure for symmetric functions.

The paper counts symmetric functions with maximum sensitivity.

Abstract

A Boolean function $f({\vec x})$ is sensitive to bit $x_i$ if there is at least one input vector $\vec x$ and one bit $x_i$ in $\vec x$, such that changing $x_i$ changes $f$. A function has sensitivity $s$ if among all input vectors, the largest number of bits to which $f$ is sensitive is $s$. We count the $n$-variable symmetric Boolean functions that have maximum sensitivity. We show that most such functions have the largest possible sensitivity, $n$. This suggests sensitivity is limited as a complexity measure for symmetric Boolean functions.