Approximate majority analyses using tri-molecular chemical reaction networks

Abstract

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.