On the Sensitivity of Cyclically-Invariant Boolean Functions

TL;DR

This paper constructs a cyclically invariant Boolean function with sensitivity (n^{1/3}), answering two open questions about the lower bounds of sensitivity for transitive-invariant functions and their product of sensitivities.

Contribution

It provides a counterexample to previous conjectures by constructing a cyclically invariant Boolean function with lower sensitivity than expected and proves a polynomial relation between sensitivity and block sensitivity for minterm-transitive functions.

Findings

Constructed a cyclically invariant Boolean function with sensitivity (n^{1/3})

Answered two open questions negatively about sensitivity bounds

Established polynomial relation between sensitivity and block sensitivity for minterm-transitive 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.