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.

翻译：我们提出了一类能够在并行运行中解决k-奇偶性问题的自动机网络族。这些解通过以非循环方式连接团结构来构建。局部邻域规模与字母表大小呈线性关系，且收敛时间被证明始终等于交互图直径。我们证明该解族经轻微调整后可获得等价于k-同步问题的解族，这意味着这些解能从任意初始构型收敛至包含字母表上所有均匀构型的有序循环。