Lower Bounds on the State Complexity of Population Protocols

02/23/2021

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by   Philipp Czerner, et al.

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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 all initial 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(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(logloglogη) for leaderless protocols, and the inverse of a non-elementary function for protocols with leaders.