# 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 \(O(\log n)\) 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 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 \(\varOmega (\sqrt{n \log n})\). 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.

## Keywords

Approximate Majority Chemical reaction networks Population protocols## References

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