发表机构
Facultad de Ingeniería y Ciencias Universidad Adolfo Ibáñez（阿道夫·伊巴涅斯大学工程与科学学院）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本研究证明同步二维多数规则的预测问题是P完全的，通过时间编码实现信息交叉，解决了近三十年悬而未决的问题。
AI 中文摘要
我们证明在logspace多一归约下，同步二维多数规则的预测问题是P完全的，解决了近三十年来悬而未决的问题。1997年，Moore确立了三维及以上维度的P完全性，并猜想二维情形存在高效的并行算法。我们考虑一个$n\ imes n$的环面，每个单元遵循其四个最近邻居的多数状态，并在平局时保持当前状态。给定一个明确指定的初始构型和时间$T$，预测问题询问指定单元在时间$T$是否处于状态$+1$。核心挑战是在均匀、单调、扩散的局部规则下，使独立的信息流在平面中交叉。我们通过布尔值的时间编码克服了这一障碍：两种值都产生活动，但通过信号到达时间加以区分。这种编码产生了一个交叉，它保留两种值，并与导线、复制、AND门和OR门组合，以模拟任意单调布尔电路。因此，一个倾向于一致的局部规则仍然可以在二维中传输、组合和交叉独立信息。我们还证明了在没有规定时间范围的情况下，判定指定单元是否曾达到$+1$的问题也是P完全的。预测结果扩展到同一邻域上的每个均匀对称有符号多数规则，包括少数规则。
英文摘要
We prove that prediction for the synchronous two-dimensional majority rule is $\mathrm{P}$-complete under logspace many-one reductions, resolving a problem open for almost three decades. In 1997, Moore established $\mathrm{P}$-completeness in dimension three and higher and conjectured that the two-dimensional case admits an efficient parallel algorithm. We consider an $n\times n$ torus on which each cell follows the majority of its four nearest neighbors and retains its current state in a tie. Given an explicitly specified initial configuration and a time $T$, prediction asks whether a designated cell is in state $+1$ at time $T$. The central challenge is to make independent information streams cross in the plane under a homogeneous, monotone, diffusive local rule. We overcome this obstacle through a temporal encoding of Boolean values: both values generate activity, but are distinguished by signal arrival times. This encoding yields a crossover that preserves both values and composes with wires, duplication, and AND and OR gates to simulate arbitrary monotone Boolean circuits. Thus a local rule that favors agreement can nevertheless transport, combine, and cross independent information in two dimensions. We also prove $\mathrm{P}$-completeness for deciding whether a designated cell ever reaches $+1$, without a prescribed time horizon. The prediction result extends to every uniform symmetric signed majority rule on the same neighborhood, including the minority rule.
