Lower bounds on the state complexity of population protocols

Abstract

AbstractPopulation 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 $$x \ge \eta $$ x ≥ η have state complexity $$\mathcal {O}(\log \eta )$$ O ( log η ) for leaderless protocols and $$\mathcal {O}(\log \log \eta )$$ O ( log log η ) for protocols with leaders. We obtain the first non-trivial lower bounds: the state complexity of $$x \ge \eta $$ x ≥ η is $$\Omega (\log \log \eta )$$ Ω ( log log η ) for leaderless protocols, and the inverse of a non-elementary function for protocols with leaders.