Computing Real Numbers with Large-Population Protocols Having a Continuum of Equilibria

TL;DR

This paper extends the Large-Population Protocol model to include systems with a continuum of equilibria, enabling the computation of a broader class of real numbers, including transcendental ones, using population protocols.

Contribution

It removes the finitary equilibrium restriction in LPPs, allowing the computation of all numbers computable by GPACs or CRNs, including transcendental numbers.

Findings

LPPs with continuum of equilibria can compute all GPAC/CRN computable numbers.

The algorithm converts GPACs/CRNs into LPPs, broadening the class of computable numbers.

Fixes a gap in previous LPP constructions for algebraic numbers.

Abstract

Bournez, Fraigniaud, and Koegler defined a number in [0,1] as computable by their Large-Population Protocol (LPP) model, if the proportion of agents in a set of marked states converges to said number over time as the population grows to infinity. The notion, however, restricts the ordinary differential equations (ODEs) associated with an LPP to have only finitely many equilibria. This restriction places an intrinsic limitation on the model. As a result, a number is computable by an LPP if and only if it is algebraic, namely, not a single transcendental number can be computed under this notion. In this paper, we lift the finitary requirement on equilibria. That is, we consider systems with a continuum of equilibria. We show that essentially all numbers in [0,1] that are computable by bounded general-purpose analog computers (GPACs) or chemical reaction networks (CRNs) can also be computed by LPPs under this new definition. This implies a rich series of numbers (e.g., the reciprocal of Euler's constant, $\pi/4$, Euler's $\gamma$, Catalan's constant, and Dottie number) are all computable by LPPs. Our proof is constructive: We develop an algorithm that transfers bounded GPACs/CRNs into LPPs. Our algorithm also fixes a gap in Bournez et al.'s construction of LPPs designed to compute any arbitrary algebraic number in [0,1].