Lower bounds on the state complexity of population protocols

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 et al. has shown that the counting predicates x≥ η have state complexity O(log η) for leaderless protocols and O(log log η) for protocols with leaders. We obtain the first non-trivial lower bounds: the state complexity of x≥ η is Ω (log log η) for leaderless protocols, and the inverse of a non-elementary function for protocols with leaders.