On the Sensitivity of Cyclically-Invariant Boolean Functions

Abstract

In this paper we construct a cyclically invariant Boolean function whose sensitivity is $\Theta (n^{1/3})$. This result answers two previously published questions. Turán (1984) asked if any Boolean function, invariant under some transitive group of permutations, has sensitivity $\Omega (\sqrt{n})$. Kenyon and Kutin (2004) asked whether for a ``nice'' function the product of 0-sensitivity and 1-sensitivity is $\Omega (n)$. 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 $\Omega (n^{1/3})$. Hence for this class of functions sensitivity and block sensitivity are polynomially related.