Revised: March 24, 2017

Published: December 28, 2018

**Keywords:**lower bounds, separation of complexity classes, multiparty communication complexity, unbounded-error communication complexity, PP, UPP

**Categories:**complexity theory, lower bounds, communication complexity, multiparty communication complexity, complexity classes

**ACM Classification:**F.1.3, F.2.3

**AMS Classification:**68Q17, 68Q15

**Abstract:**
[Plain Text Version]

The *unbounded-error* communication complexity of
a Boolean function $F$ is the limit of the
$\epsilon$-error randomized complexity of $F$ as
$\epsilon\to1/2.$ Communication complexity with
*weakly unbounded error* is defined similarly but
with an additive penalty term that depends on
$1/2-\epsilon$. Explicit functions are known whose
two-party communication complexity with unbounded error
is logarithmic compared to their complexity
with weakly unbounded
error. Chattopadhyay and Mande (ECCC TR16-095, Theory of
Computing 2018)
recently generalized this exponential separation to the
number-on-the-forehead multiparty model.
We show how to derive such an exponential separation
from known two-party work, achieving a quantitative
improvement along the way. We present several proofs
here, some as short as half a page.

In more detail, we construct a $k$-party communication problem $F\colon(\{0,1\}^{n})^{k}\to\{0,1\}$ that has complexity $O(\log n)$ with unbounded error and $\Omega(\sqrt n\,/\,4^{k})$ with weakly unbounded error, reproducing the bounds of Chattopadhyay and Mande. In addition, we prove a quadratically stronger separation of $O(\log n)$ versus $\Omega(n\,/\,4^k)$ using a nonconstructive argument.

A preliminary version of this paper appeared in ECCC, Report TR16-138, 2016.