Population protocols are a model of distributed computation in which a collection of indistinguishable finite-state agents interact randomly in pairs to decide a predicate of their initial configuration. The agents decide by achieving a stable consensus on whether the predicate holds or not. It is known that population protocols can decide exactly the predicates expressible in Presburger arithmetic. Recently, Lossin et al. have introduced a notion of protocol robustness against adversarial crash failures. They show that all atomic Presburger predicates can be decided by robust protocols, and ask whether the same holds for every Presburger predicate. We make progress towards settling this question by proving that all predicates expressible in monadic Presburger arithmetic have robust protocols. In addition, we analyze the cost of robustness in terms of state complexity. We study the ratio between the number of states of the smallest robust protocol for a given predicate and the smallest protocol for it. We show that the cost of robustness is at least double exponential in the size of the predicate, and prove that the robust protocols by Lossin et al. for threshold predicates x >= k have optimal state complexity.

翻译：群体协议是一种分布式计算模型，其中一组不可区分的有限状态智能体通过随机配对交互，判定其初始配置是否满足某个谓词。智能体通过达成稳定共识来判定谓词是否成立。已知群体协议恰好能判定Presburger算术中可表达的谓词。近期，Lossin等人引入了针对对抗性崩溃故障的协议鲁棒性概念。他们证明所有原子Presburger谓词均可由鲁棒协议判定，并询问是否每个Presburger谓词都具有相同性质。我们通过证明所有一元Presburger算术可表达的谓词均存在鲁棒协议，推动了对该问题的解决。此外，我们分析了鲁棒性在状态复杂度方面的代价。研究了给定谓词的最小鲁棒协议状态数与最小协议状态数之比。我们证明鲁棒性的代价至少为谓词大小的双指数级，并表明Lossin等人针对阈值谓词x ≥ k设计的鲁棒协议具有最优状态复杂度。