Population protocols are a model of distributed computation in which an arbitrary number of indistinguishable finite-state agents interact in pairs to decide some property of their initial configuration. We investigate the behaviour of population protocols under adversarial faults that cause agents to silently crash and no longer interact with other agents. As a starting point, we consider the property ``the number of agents exceeds a given threshold $t$'', represented by the predicate $x \geq t$, and show that the standard protocol for $x \geq t$ is very fragile: one single crash in a computation with $x:=2t-1$ agents can already cause the protocol to answer incorrectly that $x \geq t$ does not hold. However, a slightly less known protocol is robust: for any number $t' \geq t$ of agents, at least $t' - t+1$ crashes must occur for the protocol to answer that the property does not hold. We formally define robustness for arbitrary population protocols, and investigate the question whether every predicate computable by population protocols has a robust protocol. Angluin et al. proved in 2007 that population protocols decide exactly the Presburger predicates, which can be represented as Boolean combinations of threshold predicates of the form $\sum_{i=1}^n a_i \cdot x_i \geq t$ for $a_1,...,a_n, t \in \mathbb{Z}$ and modulo prdicates of the form $\sum_{i=1}^n a_i \cdot x_i \bmod m \geq t $ for $a_1, \ldots, a_n, m, t \in \mathbb{N}$. We design robust protocols for all threshold and modulo predicates. We also show that, unfortunately, the techniques in the literature that construct a protocol for a Boolean combination of predicates given protocols for the conjuncts do not preserve robustness. So the question remains open.

翻译：群体协议是一种分布式计算模型，其中任意数量的不可区分有限状态智能体通过成对交互来决定其初始配置的某些性质。我们研究了在对抗性故障下群体协议的行为，这些故障会导致智能体静默崩溃并不再与其他智能体交互。作为起点，我们考虑性质"智能体数量超过给定阈值$t$"，由谓词$x \geq t$表示，并证明$x \geq t$的标准协议非常脆弱：在$x:=2t-1$个智能体的计算中，仅需一次崩溃就可能导致协议错误地判定$x \geq t$不成立。然而，一个稍鲜为人知的协议具有鲁棒性：对于任意$t' \geq t$个智能体，至少需要发生$t' - t+1$次崩溃才会使协议判定该性质不成立。我们正式定义了任意群体协议的鲁棒性，并研究每个可由群体协议计算的谓词是否都存在鲁棒协议的问题。Angluin等人于2007年证明群体协议恰好能判定Presburger谓词，这些谓词可表示为形式为$\sum_{i=1}^n a_i \cdot x_i \geq t$（其中$a_1,...,a_n, t \in \mathbb{Z}$）的阈值谓词与形式为$\sum_{i=1}^n a_i \cdot x_i \bmod m \geq t$（其中$a_1, \ldots, a_n, m, t \in \mathbb{N}$）的模谓词的布尔组合。我们为所有阈值谓词和模谓词设计了鲁棒协议。同时证明，现有文献中根据子谓词协议构造布尔组合谓词协议的技术无法保持鲁棒性。因此该问题仍然悬而未决。