Leaderless Population Protocols Decide Double-exponential Thresholds
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Population protocols are a model of distributed computation in which finite-state agents interact randomly in pairs. A protocol decides for any initial configuration whether it satisfies a fixed property, specified as a predicate on the set of configurations. The state complexity of a predicate is smallest number of states of any protocol deciding that predicate. For threshold predicates of the form x≥ k, with k constant, prior work has shown that they have state complexity Θ(loglog k) if the protocol is extended with leaders. For ordinary protocols it is only known to be in Ω(loglog k)∩𝒪(log k). We close this remaining gap by showing that it is Θ(loglog k) as well, i.e. we construct protocols with 𝒪(n) states deciding x≥ k with k≥2^2^n.
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