A swarm of anonymous oblivious mobile robots, operating in deterministic Look-Compute-Move cycles, is confined within a circular track. All robots agree on the clockwise direction (chirality), they are activated by an adversarial semi-synchronous scheduler (SSYNCH), and an active robot always reaches the destination point it computes (rigidity). Robots have limited visibility: each robot can see only the points on the circle that have an angular distance strictly smaller than a constant $\vartheta$ from the robot's current location, where $0<\vartheta\leq\pi$ (angles are expressed in radians). We study the Gathering problem for such a swarm of robots: that is, all robots are initially in distinct locations on the circle, and their task is to reach the same point on the circle in a finite number of turns, regardless of the way they are activated by the scheduler. Note that, due to the anonymity of the robots, this task is impossible if the initial configuration is rotationally symmetric; hence, we have to make the assumption that the initial configuration be rotationally asymmetric. We prove that, if $\vartheta=\pi$ (i.e., each robot can see the entire circle except its antipodal point), there is a distributed algorithm that solves the Gathering problem for swarms of any size. By contrast, we also prove that, if $\vartheta\leq \pi/2$, no distributed algorithm solves the Gathering problem, regardless of the size of the swarm, even under the assumption that the initial configuration is rotationally asymmetric and the visibility graph of the robots is connected. The latter impossibility result relies on a probabilistic technique based on random perturbations, which is novel in the context of anonymous mobile robots. Such a technique is of independent interest, and immediately applies to other Pattern-Formation problems.

翻译：一群匿名无意识移动机器人在确定性“看-计算-移动”周期中运行，受限于圆形轨道。所有机器人对顺时针方向（手性）达成一致，它们由对抗性半同步调度器（SSYNCH）激活，且激活的机器人始终能到达其计算的目标点（刚性）。机器人具有有限可见性：每个机器人仅能看到圆周上与自身当前位置角距离严格小于常数$\vartheta$的点，其中$0<\vartheta\leq\pi$（角度以弧度表示）。我们研究此类机器人群体的聚集问题：即所有机器人初始位于圆周上的不同位置，任务是在有限步数内到达圆周上同一点，无论调度器如何激活它们。注意，由于机器人的匿名性，若初始配置具有旋转对称性则此任务不可行；因此我们必须假设初始配置是旋转非对称的。我们证明：若$\vartheta=\pi$（即每个机器人能看到除对映点外的整个圆周），存在分布式算法可解决任意规模群体的聚集问题。相反，我们还证明：若$\vartheta\leq \pi/2$，无论群体规模大小，即使假设初始配置旋转非对称且机器人可见图连通，也不存在能解决聚集问题的分布式算法。后者不可行性结果基于随机扰动的概率技术，这在匿名移动机器人领域具有新颖性。该技术具有独立研究价值，并可立即应用于其他模式形成问题。