We present a family of automata networks that solve the k-parity problem when run in parallel. These solutions are constructed by connecting cliques in a non-cyclical fashion. The size of the local neighbourhood is linear in the size of the alphabet, and the convergence time is proven to always be the diameter of the interaction graph. We show that this family of solutions can be slightly altered to obtain an equivalent family of solutions to the k-synchronisation problem, which means that these solutions converge from any initial configuration to the cycle which contains all the uniform configurations over the alphabet, in order.

中文翻译：

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使用并行自动机网络快速解决 k 奇偶校验和 k 同步问题

我们提出了一系列自动机网络，可以解决并行运行时的 k 奇偶校验问题。这些解决方案是通过以非循环方式连接派系来构建的。局部邻域的大小与字母表的大小成线性关系，并且收敛时间被证明始终是交互图的直径。我们证明，这个解决方案族可以稍微改变以获得 k 同步问题的等效解决方案族，这意味着这些解决方案从任何初始配置收敛到包含字母表上所有统一配置的循环，按顺序。