On the Sensitivity of Cyclically-Invariant Boolean Functions

Details On the Sensitivity of Cyclically-Invariant Boolean Functions On the Sensitivity of Cyclically-Invariant Boolean Functions

2005-01-13

arxiv.org/abs/cs/0501026v1

Computer Science

Computational Complexity

Scientific paper

In this paper we construct a cyclically invariant Boolean function whose sensitivity is $\Theta(n^{1/3})$. This result answers two previously published questions. Tur\'an (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.

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