Population Protocols Revisited: Parity and Beyond

For nearly two decades, population protocols have been extensively studied, yielding efficient solutions for central problems in distributed computing, including leader election, and majority computation, a predicate type in Presburger Arithmetic closely tied to population protocols. Surprisingly, no protocols have achieved both time- and space-efficiency for congruency predicates, such as parity computation, which are complementary in this arithmetic framework. This gap highlights a significant challenge in the field. To address this gap, we explore the parity problem, where agents are tasked with computing the parity of the given sub-population size. Then we extend the solution for parity to compute congruences modulo an arbitrary $m$. Previous research on efficient population protocols has focused on protocols that minimise both stabilisation time and state utilisation for specific problems. In contrast, this work slightly relaxes this expectation, permitting protocols to place less emphasis on full optimisation and more on universality, robustness, and probabilistic guarantees. This allows us to propose a novel computing paradigm that integrates population weights (or simply weights), a robust clocking mechanism, and efficient anomaly detection coupled with a switching mechanism (which ensures slow but always correct solutions). This paradigm facilitates universal design of efficient multistage stable population protocols. Specifically, the first efficient parity and congruence protocols introduced here use both $O(\log^3 n)$ states and achieve silent stabilisation in $O(\log^3 n)$ time. We conclude by discussing the impact of implicit conversion between unary and binary representations enabled by the weight system, with applications to other problems, including the computation and representation of (sub-)population sizes.