A note on a problem in communication complexity

TL;DR

This paper proves a version of Tarui's Theorem in communication complexity, establishing relationships between classes like PH^{cc}, PP^{cc}, and M^{cc}, and clarifies the limitations of polynomial space within these classes.

Contribution

It extends Tarui's Theorem to communication complexity, linking classes and showing polynomial space is not contained in M^{cc}.

Findings

PH^{cc} is contained in BP·PP^{cc}.

High mc-rigidity problems lie outside the polynomial hierarchy.

Polynomial space is not contained in M^{cc}.

Abstract

In this note, we prove a version of Tarui's Theorem in communication complexity, namely $PH^{cc} \subseteq BP\cdot PP^{cc}$. Consequently, every measure for $PP^{cc}$ leads to a measure for $PH^{cc}$, subsuming a result of Linial and Shraibman that problems with high mc-rigidity lie outside the polynomial hierarchy. By slightly changing the definition of mc-rigidity (arbitrary instead of uniform distribution), it is then evident that the class $M^{cc}$ of problems with low mc-rigidity equals $BP\cdot PP^{cc}$. As $BP\cdot PP^{cc} \subseteq PSPACE^{cc}$, this rules out the possibility, that had been left open, that even polynomial space is contained in $M^{cc}$.