On the Sensitivity of Cyclically-Invariant Boolean Functions

Sourav Chakraborty

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On the Sensitivity of Cyclically-Invariant Boolean Functions

Sourav Chakraborty

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Abstract

In this paper we construct a cyclically invariant Boolean function whose sensitivity is $Θ(n^{1/3})$. This result answers two previously published questions. Turán (1984) asked if any Boolean function, invariant under some transitive group of permutations, has sensitivity $Ω(\sqrt{n})$. Kenyon and Kutin (2004) asked whether for a ``nice'' function the product of 0-sensitivity and 1-sensitivity is $Ω(n)$. Our function answers both questions in the negative. We also prove that for minterm-transitive functions (a natural class of Boolean functions including our example) the sensitivity is $Ω(n^{1/3})$. Hence for this class of functions sensitivity and block sensitivity are polynomially related.

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Sourav Chakraborty. 2005-01-13. On the Sensitivity of Cyclically-Invariant Boolean Functions. https://arxiv.org/abs/cs/0501026

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