Consider a set of $n$ mobile entities, called robots, located and operating on a continuous circle, i.e., all robots are initially in distinct locations on a circle. The \textit{gathering} problem asks to design a distributed algorithm that allows the robots to assemble at a point on the circle. Robots are anonymous, identical, and homogeneous. Robots operate in a deterministic Look-Compute-Move cycle within the circular path. Robots agree on the clockwise direction. The robot's movement is rigid and they have limited visibility $\pi$, i.e., each robot can only see the points of the circle which is at an angular distance strictly less than $\pi$ from the robot. Di Luna \textit{et al}. [DISC'2020] provided a deterministic gathering algorithm of oblivious and silent robots on a circle in semi-synchronous (\textsc{SSync}) scheduler. Buchin \textit{et al}. [IPDPS(W)'2021] showed that, under full visibility, $\mathcal{OBLOT}$ robot model with \textsc{SSync} scheduler is incomparable to $\mathcal{FSTA}$ robot (robots are silent but have finite persistent memory) model with asynchronous (\textsc{ASync}) scheduler. Under limited visibility, this comparison is still unanswered. So, this work extends the work of Di Luna \textit{et al}. [DISC'2020] under \textsc{ASync} scheduler for $\mathcal{FSTA}$ robot model.

翻译：考虑一组称为机器人的$n$个移动实体，它们位于并运行在一个连续圆上，即所有机器人初始时分布在圆的不同位置。\textit{聚集}问题要求设计一种分布式算法，使得机器人能够汇聚到圆上的某一点。机器人是匿名的、相同的且均质的。机器人在圆形路径内以确定性的"观察-计算-移动"循环运作。机器人约定顺时针方向为正向。机器人的移动是刚性的，且具有有限可视性$\pi$，即每个机器人只能看到与其角距离严格小于$\pi$的圆上点。Di Luna等人[DISC'2020]在半同步（\textsc{SSync}）调度器下，给出了圆上无记忆且静默机器人的确定性聚集算法。Buchin等人[IPDPS(W)'2021]证明，在全可视性条件下，采用\textsc{SSync}调度器的$\mathcal{OBLOT}$机器人模型与采用异步（\textsc{ASync}）调度器的$\mathcal{FSTA}$机器人模型（机器人静默但具有有限持久记忆）不可比较。在有限可视性条件下，这种比较仍未得到解答。因此，本文在\textsc{ASync}调度器下，将Di Luna等人[DISC'2020]的工作扩展至$\mathcal{FSTA}$机器人模型。