Population protocols are a class of algorithms for modeling distributed computation in networks of finite-state agents communicating through pairwise interactions. Their suitability for analyzing numerous chemical processes has motivated the adaptation of the original population protocol framework to better model these chemical systems. In this paper, we further the study of two such adaptations in the context of solving approximate majority: persistent-state agents (or catalysts) and spontaneous state changes (or leaks). Based on models considered in recent protocols for populations with persistent-state agents, we assume a population with $n$ catalytic input agents and $m$ worker agents, and the goal of the worker agents is to compute some predicate over the states of the catalytic inputs. We call this model the Catalytic Input (CI) model. For $m = \Theta(n)$, we show that computing the exact majority of the input population with high probability requires at least $\Omega(n^2)$ total interactions, demonstrating a strong separation between the CI model and the standard population protocol model. On the other hand, we show that the simple third-state dynamics of Angluin et al. for approximate majority in the standard model can be naturally adapted to the CI model: we present such a constant-state protocol for the CI model that solves approximate majority in $O(n \log n)$ total steps w.h.p. when the input margin is $\Omega(\sqrt{n \log n})$. We then show the robustness of third-state dynamics protocols to the transient leaks events introduced by Alistarh et al. In both the original and CI models, these protocols successfully compute approximate majority with high probability in the presence of leaks occurring at each step with probability $\beta \leq O\left(\sqrt{n \log n}/n\right)$, exhibiting a resilience to leaks similar to that of Byzantine agents in previous works.

翻译：种群协议是一类用于建模有限状态代理通过成对交互进行分布式计算的算法。其适用于分析多种化学过程的特性促使研究者对原始种群协议框架进行调整，以更准确地描述这些化学系统。本文在求解近似多数问题的背景下，进一步研究了两类改进方案：持久状态代理（即催化剂）与自发状态变化（即泄漏）。基于近期含持久状态代理种群协议研究中的模型，我们假设种群包含$n$个催化输入代理和$m$个工作代理，工作代理的目标是计算关于催化输入状态的某种谓词。我们将该模型称为催化输入（CI）模型。对于$m = \Theta(n)$的情形，我们证明以高概率计算输入种群的精确多数至少需要$\Omega(n^2)$次总交互，这表明CI模型与标准种群协议模型之间存在显著差异。另一方面，我们证明Angluin等人针对标准模型中近似多数问题提出的简单三态动力学可自然适配至CI模型：我们提出了一个面向CI模型的常状态协议，当输入裕度为$\Omega(\sqrt{n \log n})$时，该协议可在$O(n \log n)$步内以高概率求解近似多数。进一步，我们证明了这类三态动力学协议对Alistarh等人提出的瞬态泄漏事件的鲁棒性：在原始模型和CI模型中，当每步发生泄漏的概率$\beta \leq O\left(\sqrt{n \log n}/n\right)$时，这些协议均能以高概率成功计算近似多数，展现出与先前工作中拜占庭代理类似的抗泄漏能力。