Population protocols are a model of computation in which an arbitrary number of anonymous finite-state agents are interacting in order to decide by stable consensus if an initial configuration extended with some extra agents called leaders satisfies some property. In this paper, we focus on $n$-counting predicates that ask, given an initial configuration, if the number of agents in a given state is at least $n$. In a recent work it was exhibited for infinitely many $n$, a population protocol with at most $O(\log\log(n))$ states that decides the $n$-counting predicate. We prove that this bound is almost optimal, by observing that any population protocol deciding such a predicate requires at least $\Omega((\log\log(n))^{\frac{1}{3}})$ states.

翻译：人口协议是一种计算模型,其中任意数目的匿名限定国家代理人相互作用,以便以稳定协商一致的方式决定,如果最初配置扩大后,一些额外的代理人称为领导人,能够满足某些财产。在本文中,我们侧重于以美元计数的上游,如果某个州内的代理人数量至少为美元,则根据初始配置,我们要求计算该上游。在最近一项工作中,它展示了无限多的美元,而人口协议最多为O(log\log(n))美元,确定以美元计数的上游。我们通过观察任何决定这种上游的人口协议至少需要$\Omega(log\log)(n)\\\\\\\frac{{%3}州,证明这一约束几乎是最佳的,我们发现任何确定这种上游的人口协议都要求至少需要$\Omega(log\log)(n)\\frac{{{{{{{{{{{{{{{{{{{{{{{{{{{{})$(n)美元。