Simplifying analyses of chemical reaction networks for approximate majority

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 (formula presented) time efficiency with high probability, given an initial gap of size (formula presented) when the total molecular count in the mixture is n. Motivated by their intriguing but complex proof, we provide simpler, and more intuitive proofs of correctness and efficiency for two bi-molecular CRNs for Approximate Majority, including that of Angluin et al. Key to our approach is to show how the bi-molecular CRNs essentially emulate a tri-molecular CRN with just two reactions and two species. Our results improve on those of Angluin et al. in that they hold even with an initial gap of (formula presented). Our analysis approach, which leverages the simplicity of a tri-molecular CRN to ultimately reason about bi-molecular CRNs, may be useful in analyzing other CRNs too.