On the distribution of sensitivities of symmetric Boolean functions

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.

arXiv comment5 pages, 0 figures The submitted paper is a journal version of "Enumeration of Symmetric Boolean Functions By Sensitivity" by J. Butler, T. Sasao, and S. Nagayama presented at the Reed-Muller Workshop, Matsue, Japan on May 24, 2023. Paper was presented, but not distributed. Authors retained copyright