Üresin and Dubois’ paper “Parallel Asynchronous Algorithms for Discrete Data” shows how a class of synchronous iterative algorithms may be transformed into asynchronous iterative algorithms. They then prove that the correctness of the resulting asynchronous algorithm can be guaranteed by reasoning about the synchronous algorithm alone. These results have been used to prove the correctness of various distributed algorithms, including in the fields of routing, numerical analysis and peer-to-peer protocols. In this paper we demonstrate several ways in which the assumptions that underlie this theory may be relaxed. Amongst others, we (i) expand the set of schedules for which the asynchronous iterative algorithm is known to converge and (ii) weaken the conditions that users must prove to hold to guarantee convergence. Furthermore, we demonstrate that two of the auxiliary results in the original paper are incorrect, and explicitly construct a counter-example. Finally, we also relax the alternative convergence conditions proposed by Gurney based on ultrametrics. Many of these relaxations and errors were uncovered after formalising the work in the proof assistant Agda. This paper describes the Agda code and the library that has resulted from this work. It is hoped that the library will be of use to others wishing to formally verify the correctness of asynchronous iterative algorithms.

中文翻译：

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Agda 中 Üresin 和 Dubois 异步定点理论的放松

Üresin 和 Dubois 的论文“离散数据的并行异步算法”展示了如何将一类同步迭代算法转换为异步迭代算法。然后他们证明可以通过单独对同步算法进行推理来保证最终异步算法的正确性。这些结果已被用于证明各种分布式算法的正确性，包括在路由、数值分析和对等协议领域。在本文中，我们展示了几种可以放松作为该理论基础的假设的方法。其中，我们 (i) 扩展了已知异步迭代算法收敛的调度集，以及 (ii) 削弱用户必须证明的条件以保证收敛。此外，我们证明原始论文中的两个辅助结果是不正确的，并明确构造了一个反例。最后，我们还放宽了 Gurney 基于 Ultrametrics 提出的替代收敛条件。在证明助手 Agda 将工作正式化后，发现了许多这些松弛和错误。本文描述了 Agda 代码和由此工作产生的库。希望该库对希望正式验证异步迭代算法正确性的其他人有用。在证明助手 Agda 将工作正式化后，发现了许多这些松弛和错误。本文描述了 Agda 代码和由此工作产生的库。希望该库对希望正式验证异步迭代算法正确性的其他人有用。在证明助手 Agda 将工作正式化后，发现了许多这些松弛和错误。本文描述了 Agda 代码和由此工作产生的库。希望该库对希望正式验证异步迭代算法正确性的其他人有用。