Byzantine-tolerant circle formation by oblivious mobile robots

Consider a system that consists of a group of mobile robots roaming in the two-dimensional plane. Each robot occupies a point in the plane, and is equipped with sensors to observe the positions of the other robots. Each robot proceeds by repeatedly (1) observing the environment, (2) computing a destination based on the observed positions of robots, and (3) moving toward the computed destination. Robots are unable to communicate directly, and can only interact by observing each others' positions. In addition, all robots execute the same deterministic algorithm, however some of them can be Byzantine, and exhibit arbitrary behaviors that are not in accordance with their local algorithms. Finally, robots are oblivious (i.e., stateless), meaning that they can not remember their previous observations and actions. In this model, we address the problem of coordination between these robots from a computational viewpoint, and we focus on a basic coordination problem, namely the Byzantine-tolerant circle formation problem. In other words, we study the feasibility of positioning a group of mobile robots into forming a circle in Byzantine fault model. The contribution of this paper is as follows: Let N be the number of robots in the system, and let f be the number of Byzantine robots. We first show that there exists no algorithm that solves the Byzantine-tolerant circle formation when N ≤ 2f + 2 in a fully synchronous system. On the other hand, when N >; 2f + 2, we provide an algorithm in the fully synchronous model. Then, we show that when N ≤ 3f + 2, the problem of Byzantine circle formation has no solution in the semi-synchronous model when robots' activations follow a k-bounded scheduler for k ≥ 2, that is while the slowest robot is activated once, the fastest robot is activated at most k times (k-bounded). Finally, we conjecture that the problem is impossible in general for any N when k ≥ 2 in the semi-synchronous model even for f = 1.