On the distribution of sensitivities of symmetric Boolean functions

06/26/2023

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by   Jon T. Butler, et al.

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A Boolean function f(x⃗) is sensitive to bit x_i if there is at least one input vector x⃗ and one bit x_i in x⃗, 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.