We define an extension to the standard population protocol that provides each agent with a clock signal that indicates when the agent has waited long enough for the protocol to have converged. We represent “long enough” as an infinite time interval, and treat the protocol as operating over transfinite time. Over finite time intervals, the protocol behaves as in the standard model. At nonzero limit ordinals, corresponding to clock ticks, the protocol switches to a limit of previous configurations supplemented by a clock signal appearing as an extra component in some of the agents' states. Fairness generalizes to transfinite executions straightforwardly. We show that a clocked population protocol running in less than ${\omega }^k$ time for any fixed $k\ge 2$ is equivalent in power to a nondeterministic Turing machine with space complexity logarithmic in the population size, and can be simulated using only finitely many clock ticks.

我们定义了标准填充协议的扩展，该扩展为每个代理提供了时钟信号，该时钟信号指示代理何时等待足够长的时间以使协议收敛。我们将“足够长的时间”表示为无限的时间间隔，并将该协议视为在超限时间内运行。在有限的时间间隔内，协议的行为与标准模型中的行为相同。在与时钟滴答相对应的非零限制标准下，协议会切换到先前配置的限制，并以时钟信号作为代理的某些状态中的额外组成部分来补充。公平性直接概括为超限执行。我们证明了时钟填充协议的运行时间少于${\omega }^{ķ}$ 任何固定的时间 $ķ\ge \mathrm{2个}$ 在功率上等同于不确定性的图灵机，该图灵机的空间复杂度在种群大小上为对数，并且只能使用有限的多个时钟滴答进行模拟。