Approximate Majority is a well-studied problem in the context of chemical reaction networks (CRNs) and their close relatives, population protocols: Given a mixture of two types of species with an initial gap between their counts, a CRN computation must reach consensus on the majority species. Angluin, Aspnes, and Eisenstat proposed a simple population protocol for Approximate Majority and proved correctness and \(O(\log n)\) time efficiency with high probability, given an initial gap of size \(\omega (\sqrt{n}\log n)\) when the total molecular count in the mixture is n. Motivated by their intriguing but complex proof, we provide a new analysis of several CRNs for Approximate Majority, starting with a very simple tri-molecular protocol with just two reactions and two species. We obtain simple analyses of three bi-molecular protocols, including that of Angluin et al., by showing how they emulate the tri-molecular protocol. Our results improve on those of Angluin et al. in that they hold even with an initial gap of \(\varOmega (\sqrt{n \log n})\). We prove that our tri-molecular CRN is robust even when there is some uncertainty in the reaction rates, when some molecules are Byzantine (i.e., adversarial), or when activation of molecules is triggered by epidemic. We also analyse a natural variant of our tri-molecular protocol for the more general problem of multi-valued consensus. Our analysis approach, which leverages the simplicity of a tri-molecular CRN to ultimately reason about these many variants, may be useful in analysing other CRNs too.

中文翻译：

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使用三分子化学反应网络的近似多数分析

在化学反应网络（CRN）及其近亲，种群协议的背景下，近似多数是一个经过充分研究的问题：给定两种生物的混合物，其计数之间存在初始差距，因此CRN计算必须达成共识。多数种。Angluin，Aspnes和Eisenstat提出了一个用于近似多数的简单人口协议，并在给定初始大小为\（\ omega（\ sqrt {n} ）的情况下，以很高的概率证明了正确性和\（O（\ log n）\）时间效率。\ log n）\）当混合物中的总分子数为n。受它们有趣但复杂的证明的激励，我们提供了一种针对大多数近似CRN的新分析，从一个非常简单的只有两个反应和两个物种的三分子协议开始。通过显示它们如何模拟三分子协议，我们获得了对三种双分子协议（包括Angluin等人）的简单分析。我们的结果优于Angluin等人的结果。因为它们甚至以\（\ varOmega（\ sqrt {n \ log n}）\）的初始间隙保持。我们证明，即使反应速率存在不确定性，某些分子是拜占庭分子（即对抗分子）或流行病引发分子激活时，我们的三分子CRN仍然是可靠的。对于多值共识的更普遍问题，我们还分析了三分子协议的自然变体。我们的分析方法利用了三分子CRN的简单性来最终推断出许多这样的变异，在分析其他CRN时也可能有用。