On the Power of Many One-Bit Provers

TL;DR

This paper explores the complexity of multi-prover interactive proof systems where each prover sends only one bit, revealing a rich structure that relates various complexity classes depending on soundness parameters.

Contribution

It characterizes the complexity classes of 1-bit k-prover games across different soundness regimes, extending understanding beyond the single-prover case.

Findings

For low soundness, the class equals BPP.

For intermediate soundness, the class equals SZK.

For high soundness, the class includes NEXP.

Abstract

We study the class of languages, denoted by $\MIP[k, 1-\epsilon, s]$, which have $k$-prover games where each prover just sends a \emph{single} bit, with completeness $1-\epsilon$ and soundness error $s$. For the case that $k=1$ (i.e., for the case of interactive proofs), Goldreich, Vadhan and Wigderson ({\em Computational Complexity'02}) demonstrate that $\SZK$ exactly characterizes languages having 1-bit proof systems with"non-trivial" soundness (i.e., $1/2 < s \leq 1-2\epsilon$). We demonstrate that for the case that $k\geq 2$, 1-bit $k$-prover games exhibit a significantly richer structure: + (Folklore) When $s \leq \frac{1}{2^k} - \epsilon$, $\MIP[k, 1-\epsilon, s] = \BPP$; + When $\frac{1}{2^k} + \epsilon \leq s < \frac{2}{2^k}-\epsilon$, $\MIP[k, 1-\epsilon, s] = \SZK$; + When $s \ge \frac{2}{2^k} + \epsilon$, $\AM \subseteq \MIP[k, 1-\epsilon, s]$; + For $s \le 0.62 k/2^k$ and sufficiently large $k$, $\MIP[k, 1-\epsilon, s] \subseteq \EXP$; + For $s \ge 2k/2^{k}$, $\MIP[k, 1, 1-\epsilon, s] = \NEXP$. As such, 1-bit $k$-prover games yield a natural "quantitative" approach to relating complexity classes such as $\BPP$,$\SZK$,$\AM$, $\EXP$, and $\NEXP$. We leave open the question of whether a more fine-grained hierarchy (between $\AM$ and $\NEXP$) can be established for the case when $s \geq \frac{2}{2^k} + \epsilon$.