The standard population protocol model assumes that when two agents interact, each observes the entire state of the other agent. We initiate the study of $\textit{message complexity}$ for population protocols, where the state of an agent is divided into an externally-visible $\textit{message}$ and an internal component, where only the message can be observed by the other agent in an interaction. We consider the case of $O(1)$ message complexity. When time is unrestricted, we obtain an exact characterization of the stably computable predicates based on the number of internal states $s(n)$: If $s(n) = o(n)$ then the protocol computes semilinear predicates (unlike the original model, which can compute non-semilinear predicates with $s(n) = O(\log n)$), and otherwise it computes a predicate decidable by a nondeterministic $O(n \log s(n))$-space-bounded Turing machine. We then introduce novel $O(\mathrm{polylog}(n))$ expected time protocols for junta/leader election and general purpose broadcast correct with high probability, and approximate and exact population size counting correct with probability 1. Finally, we show that the main constraint on the power of bounded-message-size protocols is the size of the internal states: with unbounded internal states, any computable function can be computed with probability 1 in the limit by a protocol that uses only $\textit{1-bit}$ messages.

中文翻译：

---

填充协议的消息复杂度

标准人口协议模型假设当两个代理交互时，每个代理都会观察另一个代理的整个状态。我们开始研究群体协议的 $\textit{message 复杂性}$，其中代理的状态分为外部可见的 $\textit{message}$ 和内部组件，其中只有消息可以通过交互中的另一个代理。我们考虑 $O(1)$ 消息复杂度的情况。当时间不受限制时，我们根据内部状态 $s(n)$ 的数量获得稳定可计算谓词的精确表征：如果 $s(n) = o(n)$ 那么协议计算半线性谓词（与原始模型，可以用 $s(n) = O(\log n)$) 计算非半线性谓词，否则它计算一个可由非确定性 $O(n \log s(n))$-space-bounded Turing machine 判定的谓词。然后，我们引入了新的 $O(\mathrm{polylog}(n))$ 预期时间协议，用于军政府/领导人选举和通用广播的概率很高，近似和精确的人口规模计数正确的概率为 1。最后，我们展示对有界消息大小协议的威力的主要约束是内部状态的大小：对于无界内部状态，任何可计算函数都可以通过仅使用 $\textit{1 的协议在限制中以概率 1 计算-bit}$ 消息。