Asynchronous Arbitrary Pattern Formation: the effects of a rigorous approach

Given any multiset F of points in the Euclidean plane and a set R of robots such that |R| = |F|, the Arbitrary Pattern Formation (APF) problem asks for a distributed algorithm that moves robots so as to reach a configuration similar to F. Similarity means that robots must be disposed as F regardless of translations, rotations, reflections, uniform scalings. Initially, each robot occupies a distinct position. When active, a robot operates in standard Look–Compute–Move cycles. Robots are asynchronous, oblivious, anonymous, silent and execute the same distributed algorithm. So far, the problem has been mainly addressed by assuming chirality, that is robots share a common left–right orientation. We are interested in removing such a restriction. While working on the subject, we faced several issues that required close attention. We deeply investigated how such difficulties were overcome in the literature, revealing that crucial arguments for the correctness proof of the existing algorithms have been neglected. The systematic lack of rigorous arguments with respect to necessary conditions required for providing correctness proofs deeply affects the validity as well as the relevance of strategies proposed in the literature. Here we design a new deterministic distributed algorithm that fully characterizes APF showing its equivalence with the well-known Leader Election problem in the asynchronous model without chirality. Our approach is characterized by the use of logical predicates in order to formally describe our algorithm as well as its correctness. In addition to the relevance of our achievements, our techniques might help in revising previous results. In fact, it comes out that well-established results like (Fujinaga et al. in SIAM J Comput 44(3):740–785, 2015), more recent approaches like (Bramas and Tixeuil, in: Proceedings of the 35th ACM SIGACT-SIGOPS symposium on principles of distributed computing (PODC), 2016; Bramas and Tixeuil, in: Proceedings of the 18th international symposium on stabilization, safety, and security of distributed systems (SSS), 2016) and ‘unofficial’ results like (Dieudonné et al., in: CoRR arXiv:0902.2851, 2009) revealed to be not correct.