Circle Formation of Weak Robots and Lyndon Words

A Lyndon word is a non-empty word strictly smaller in the lexicographic order than any of its suffixes, except itself and the empty word. In this paper, we show how Lyndon words can be used in the distributed control of a set of n weak mobile robots. By weak, we mean that the robots are anonymous, memoryless, without any common sense of direction, and unable to communicate in an other way than observation. An efficient and simple deterministic protocol to form a regular n-gon is presented and proven for n prime.