Probing the ability of automata networks to solve decision problems has received a continuous attention in the literature, and specially with the automata reaching the answer by distributed consensus, i.e., their all taking on a same state, out of two. In the case of binary automata networks, regardless of the kind of update employed, the networks should display only two possible attractors, the fixed points $0^L$ and $1^L$, for all cyclic configurations of size $L$. A previous investigation into the space of one-dimensional, binary, radius-2 cellular automata identified a restricted subset of rules as potential solvers of decision problems, but the reported results were incomplete and lacked sufficient detail for replication. To address this gap, we conducted a comprehensive reevaluation of the entire radius-2 rule space, by filtering it with all configuration sizes from 5 to 20, according to their basins of attraction being formed by only the two expected fixed points. A set of over fifty-four thousand potential decision problem solvers were then obtained. Among these, more than forty-five thousand were associated with 3 well-defined decision problems, and precise formal explanations were provided for over forty thousand of them. The remaining candidate rules suggest additional problem classes yet to be fully characterised. Overall, this work substantially extends the understanding of radius-2 cellular automata, offering a more complete picture of their capacity to solve decision problems by consensus.

翻译：探究自动机网络解决决策问题的能力在文献中持续受到关注，特别是通过分布式共识机制得出答案的自动机——即所有单元从两种状态中收敛至同一状态。对于二元自动机网络而言，无论采用何种更新方式，在所有规模为L的循环构型中，网络应仅呈现两个可能的吸引子：不动点$0^L$与$1^L$。先前对一维二元半径-2元胞自动机规则空间的研究曾识别出一个受限规则子集作为决策问题的潜在求解器，但已报道的结果存在不完整性且缺乏足够的可复现细节。为填补这一空白，我们通过对整个半径-2规则空间进行系统性重评估，依据其吸引盆仅由两个预期不动点构成的标准，使用规模从5到20的所有构型进行筛选。最终获得超过五万四千条潜在的决策问题求解规则。其中超过四万五千条规则与三个明确定义的决策问题相关联，并对其中四万余条规则提供了精确的形式化解释。其余候选规则暗示了尚未完全表征的其他问题类别。总体而言，本研究显著拓展了对半径-2元胞自动机的理解，为其通过共识机制解决决策问题的能力提供了更完整的图景。