Given any multiset F of points in the Euclidean plane and a set R of robots such that $$|R|=|F|$$|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.

中文翻译：

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异步任意模式形成：严格方法的影响

给定欧几里德平面中点的任何多重集 F 和机器人集 R，使得 $$|R|=|F|$$|R|=|F|，任意模式形成 (APF) 问题需要分布式算法移动机器人以达到类似于 F 的配置。 相似性意味着机器人必须被设置为 F，而不管平移、旋转、反射、统一缩放。最初，每个机器人占据不同的位置。当处于活动状态时，机器人以标准的“观察-计算-移动”循环运行。机器人是异步的、不经意的、匿名的、沉默的并且执行相同的分布式算法。到目前为止，这个问题主要是通过假设手性来解决的，即机器人共享一个共同的左右方向。我们有兴趣取消此类限制。在研究这个主题时，我们遇到了几个需要密切关注的问题。我们深入研究了文献中如何克服这些困难，揭示了现有算法正确性证明的关键论点被忽略了。在提供正确性证明所需的必要条件方面，系统缺乏严格的论证，这深深影响了文献中提出的策略的有效性和相关性。在这里，我们设计了一种新的确定性分布式算法，它充分表征了 APF，显示了它与无手性异步模型中众所周知的领导者选举问题的等价性。我们的方法的特点是使用逻辑谓词来正式描述我们的算法及其正确性。除了我们成就的相关性之外，我们的技术可能有助于修改以前的结果。实际上，