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TR05-020 | 22nd November 2004 00:00
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#### 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. 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.