Embedded pattern formation by asynchronous robots without chirality

We consider the Embedded Pattern Formation (epf) problem introduced in Fujinaga et al. (SIAM J Comput 44(3):740–785, 2015). Given a set F of distinct points in the Euclidean plane (called here fixed-points) and a set R of robots such that (Formula presented.), the problem asks for a distributed algorithm that moves robots so as to occupy all points in F. Initially, each robot occupies a distinct position. When active, a robot operates in standard Look-Compute-Move cycles. In one cycle, a robot perceives the current configuration in terms of the robots’ positions and the fixed-points (Look) according to its own coordinate system, decides whether to move toward some direction (Compute), and in the positive case it moves (Move). Cycles are performed asynchronously for each robot. Robots are oblivious, anonymous, silent and execute the same deterministic algorithm. In the mentioned paper, the problem has been investigated by endowing robots with chirality, that is they share a common left-right orientation. Here we consider epf without chirality, and we fully characterize when it can be solved by designing a deterministic distributed algorithm that works for all configurations but those identified as unsolvable. The algorithm has been designed according to a rigorous approach, characterized by the use of logical predicates associated to each move used by the robots. This induces a greater level of detail that provides us rigorous bases to state the correctness of the algorithm.