Asynchronous Separation of Unconscious Colored Robots

We consider the recently introduced model of autonomous computational mobile entities called unconscious colored robots. The entities are the traditional oblivious silent mobile robots operating in the Euclidean plane in Look-Compute-Move cycles. However, each robot has a permanent external mark (or color) from a finite set, visible by the other robots, but not by the robot itself. The basic problem for these robots is separation, requiring all the robots with the same color to separate from the other robots, each group forming a recognizable geometric shape (e.g., circle, point, line); this task must be performed in finite time, in spite of the robots being unconscious of their own color, unable to communicate, and oblivious. This problem has been studied and solved in the synchronous setting (SSS 2023). In this paper we show that the problem is solvable also under the more difficult asynchronous adversary, provided the robots agree on the orientation of one axis, and no robot is uniquely colored. The proof is constructive: we present a distributed algorithm that allows unconscious colored robots with one-axis agreement to separate into parallel lines under the asynchronous scheduler.