- Published 2017 in ArXiv

Given a multiset F of points in the Euclidean plane and a set R of robots such that |R| = |F |, the Pattern Formation (PF ) 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. Here we design a new deterministic distributed algorithm that solves PF for any pattern when asynchronous robots start from asymmetric configurations, without chirality. The focus on asymmetric configurations might be perceived as an over-simplification of the subject due to the common feeling with the PF problem by the scientific community. However, we demonstrate that this is not the case. 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. Our new methodology 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 the obtained results, the new techniques might help in revisiting previous results in order to design new algorithms. In fact, it comes out that well-established results for PF like [Fujinaga et al., SIAM J. Comp. 44(3) 2015] or more recent approaches like [Bramas et al., Brief Announcement PODC 2016] revealed to be not correct. Our claim is not just based on some marginal counter-examples but we show how fundamental properties have been completely ignored, hence affecting the rationale behind the proposed strategies.

@article{Cicerone2017AsynchronousPF,
title={Asynchronous Pattern Formation: the effects of a rigorous approach},
author={Serafino Cicerone and Gabriele Di Stefano and Alfredo Navarra},
journal={CoRR},
year={2017},
volume={abs/1706.02474}
}