On the distribution of sensitivities of symmetric Boolean functions

Jon T. Butler; Tsutomu Sasao; Shinobu Nagayama

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arXiv · 2306.14401

On the distribution of sensitivities of symmetric Boolean functions

Jon T. Butler·Tsutomu Sasao·Shinobu Nagayama

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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.

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BibTeX RIS

Jon T. Butler, Tsutomu Sasao, Shinobu Nagayama. 2023-06-26. On the distribution of sensitivities of symmetric Boolean functions. https://arxiv.org/abs/2306.14401

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