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

翻译：人口协议是一种计算模型,任意数量无法区分的有限国家代理物在其中进行双对互动。 代理物的目标是以稳定一致的方式决定其初始全球配置是否满足一个特定属性, 指定该属性为所有初始配置集的前提。 上游的复杂性是计算该属性的最小协议的状态。 布隆丁等人先前的工作显示, 计算无领导者协议的美元值( log\ log\ log\ eta) 和无领导者协议的美元值( log\ log\ eta) 和 $\ mathcal{ O} (\ log\ log\ log\ eta) 。 我们获得了第一个非三维的下界限: $x\ geq\ eta$ 的状态复杂性是 $\ Omega( log\ log\\ eta), 与领导者协议的不具有非元素功能 。