Alternation, Sparsity and Sensitivity : Bounds and Exponential Gaps

Abstract

$\newcommand{\sp}{\mathsf{sparsity}}\newcommand{\s}{\mathsf{s}}\newcommand{\al}{\mathsf{alt}}$ The well-known Sensitivity Conjecture states that for any Boolean function $f$, block sensitivity of $f$ is at most polynomial in sensitivity of $f$ (denoted by \s(f)). The XOR Log-Rank Conjecture states that for any $n$ bit Boolean function, $f$ the communication complexity of a related function $f^{\oplus}$ on $2n$ bits, (defined as $f^{\oplus}(x,y)=f(x\oplus y)$) is at most polynomial in logarithm of the sparsity of $f$ (denoted by \sp(f)). A recent result of Lin and Zhang (2017) implies that to confirm the above conjectures it suffices to upper bound alternation of $f$ (denoted \al(f)) for all Boolean functions $f$ by polynomial in \s(f) and logarithm of \sp(f), respectively. In this context, we show the following : * There exists a family of Boolean functions for which \al(f) is at least exponential in \s(f) and \al(f) is at least exponential in \log \sp(f). En route to the proof, we also show an exponential gap between \al(f) and the decision tree complexity of $f$, which might be of independent interest. * As our main result, we show that, despite the above gap between \al(f) and \log \sp(f), the XOR Log-Rank Conjecture is true for functions with the alternation upper bounded by $poly(\log n)$. It is easy to observe that the Sensitivity Conjecture is also true for this class of functions. * The starting point for the above result is the observation (derived from Lin and Zhang (2017)) that for any Boolean function $f$ and $m \ge 2$, deg(f)\le \al(f)deg_2(f)deg_m(f) where $deg(f)$, $deg_2(f)$ and $deg_m(f)$ are the degrees of $f$ over $\mathbb{R}$, $\mathbb{F}_2$ and $\mathbb{Z}_m$ respectively. We also show three further applications of this observation.Comment: 19 pages, 1 figure, Journal versio