Improved XOR Lemmas for Communication Complexity

Abstract

We give communication lower bounds for computing the $n$-fold XOR of a given Boolean function $f$, denoted $f^{\oplus n}(x,y) := f(x_1,y_1)\oplus\ldots\oplus f(x_n,y_n)$, in both the deterministic and the randomized setting. In addition, we also give deterministic communication lower bounds on computing the composition of 2 functions, $g\circ f(x,y) := g(f(x_1,y_1),\ldots,f(x_n,y_n))$. Below for some absolute constant $C_0 > 0$ and all $C > C_0$ we show the following:\begin{enumerate} \item \textbf{Randomized XOR Lemma.} If $f$ requires $C$ bits to be computed with some constant success probability then, computing $f^{\oplus n}$ with probability at least $1/2 + \exp(-\Omega(n))$ requires $\tilde\Omega(C\sqrt{n})$ bits. \item \textbf{Deterministic XOR Lemma.} If $f$ requires $C$ bits to be computed deterministically then, computing $f^{\oplus n}$ deterministically requires $\Omega(n\sqrt C)$ bits. \item \textbf{Lifting Theorem.} For any function $g$, having sensitivity $s$ and degree $d$, and any $f$ requiring $C$ bits to be computed deterministically, computing $g\circ f$ deterministically requires $\Omega(\min\{s,d\}\cdot\sqrt C)$ bits. \end{enumerate} We prove the above results using information theory.In particular, the randomized XOR lemma is proved using a new notion of information that we call marginal information.