Population protocols are a model of computation in which an arbitrary number of indistinguishable finite-state agents interact in pairs. The goal of the agents is to decide by stable consensus whether their initial global configuration satisfies a given property, specified as a predicate on the set of configurations. The state complexity of a predicate is the number of states of a smallest protocol that computes it. Previous work by Blondin \textit{et al.} has shown that the counting predicates x≥η have state complexity O(logη) for leaderless protocols and O(loglogη) for protocols with leaders. We obtain the first non-trivial lower bounds: the state complexity of x≥η is Ω(loglogη) for leaderless protocols, and the inverse of a non-elementary function for protocols with leaders.
@article{arxiv.2102.11619,
title = {Lower Bounds on the State Complexity of Population Protocols},
author = {Philipp Czerner and Javier Esparza and Jérôme Leroux},
journal= {arXiv preprint arXiv:2102.11619},
year = {2022}
}