Population protocols (Angluin et al. in PODC, 2004) are a model of distributed computation in which indistinguishable, finite-state agents interact in pairs to decide if their initial configuration, i.e., the initial number of agents in each state, satisfies a given property. In a seminal paper Angluin et al. classified population protocols according to their communication mechanism, and conducted an exhaustive study of the expressive power of each class, that is, of the properties they can decide (Angluin et al. in Distrib Comput 20(4):279–304, 2007). In this paper we study the correctness problem for population protocols, i.e., whether a given protocol decides a given property. A previous paper (Esparza et al. in Acta Inform 54(2):191–215, 2017) has shown that the problem is decidable for the main population protocol model, but at least as hard as the reachability problem for Petri nets, which has recently been proved to have non-elementary complexity. Motivated by this result, we study the computational complexity of the correctness problem for all other classes introduced by Angluin et al., some of which are less powerful than the main model. Our main results show that for the class of observation models the complexity of the problem is much lower, ranging from \(\varPi _2^p\) to PSPACE.

种群协议（Angluin et al. in PODC, 2004）是一种分布式计算模型，其中不可区分的有限状态代理成对交互以确定它们的初始配置（即每个状态中的初始代理数量）是否满足给定的财产。在一篇开创性的论文中，Angluin 等人。根据通信机制对种群协议进行分类，并对每个类的表达能力进行详尽的研究，即它们可以决定的属性（Angluin 等人在 Distrib Comput 20(4):279–304, 2007） . 在本文中，我们研究人口协议的正确性问题，即给定协议是否决定给定属性。之前的一篇论文（Esparza et al. in Acta Inform 54(2):191–215, 2017）表明，对于主要人口协议模型，该问题是可判定的，但至少与 Petri 网的可达性问题一样困难，最近已证明它具有非基本复杂性。受此结果的启发，我们研究了 Angluin 等人引入的所有其他类的正确性问题的计算复杂性，其中一些类不如主模型强大。我们的主要结果表明，对于观察模型类，问题的复杂性要低得多，范围从\(\varPi _2^p\)到PSPACE。