It is known that there exist multi-prover interactive protocols ($\mathsf{MIP}$ protocols) for the complexity class $\mathsf{NEXP}$, succinct $\mathsf{MIP}$ protocols for $\mathsf{NP}$ and multi-prover interactive protocols with shared entanglement ($\mathsf{MIP}^\ast$ protocols) for $\mathsf{RE}$. This extraordinary power of multi-prover interactive proof systems comes from the assumption that provers do not communicate with each other during the protocols. If they are allowed to communicate freely, the setting is the same as in the single-prover case, and the computational power of the system becomes significantly weaker. In this paper, we investigate for the first time the setting where communication (i.e., leakage of information) between provers is allowed but bounded. We introduce two techniques to approach this question and show that multi-prover interactive proof systems are robust against some amount of leakage. Our first technique is based on parallel repetition theorems. We apply it to show that for any polynomial $p$, we can construct two-prover one-round $\mathsf{MIP}$ and $\mathsf{MIP}^\ast$ protocols for $\mathsf{NEXP}$ and $\mathsf{RE}$, respectively, that are robust against $p(n)$ bits of leakage. We further derive our second technique to convert any low-soundness PCP construction to a two-prover one-round $\mathsf{MIP}$ protocol for $\mathsf{NP}$ robust against leakage. We also discuss the relation between robustness against leakage in multi-prover interactive proof systems and the Sliding Scale Conjecture in the PCP literature.

翻译：已知存在针对复杂性类$\mathsf{NEXP}$的多证明者交互式协议（$\mathsf{MIP}$协议）、针对$\mathsf{NP}$的简洁$\mathsf{MIP}$协议以及针对$\mathsf{RE}$的共享纠缠多证明者交互式协议（$\mathsf{MIP}^\ast$协议）。多证明者交互式证明系统的异常强大源于协议过程中证明者之间不允许通信的假设。若允许自由通信，则归约为单证明者情形，系统的计算能力将显著减弱。本文首次研究允许证明者间存在有限通信（即信息泄露）的场景。我们引入两种技术方法探索该问题，并证明多证明者交互式证明系统对一定量的泄露具有鲁棒性。第一种技术基于并行重复定理。我们应用该技术表明：对任意多项式$p$，可分别构造针对$\mathsf{NEXP}$和$\mathsf{RE}$的双证明者单轮$\mathsf{MIP}$协议与$\mathsf{MIP}^\ast$协议，使其能抵御$p(n)$比特的泄露。进一步推导出第二种技术，可将任意低声音性PCP构造转化为针对$\mathsf{NP}$且对泄露鲁棒的双证明者单轮$\mathsf{MIP}$协议。文中还讨论了多证明者交互式证明系统的泄露鲁棒性与PCP文献中的滑动尺度猜想之间的关系。