论文标题
一维多数蜂窝自动机中配置的结构：从单元稳定到配置周期性
The Structure of Configurations in One-Dimensional Majority Cellular Automata: From Cell Stability to Configuration Periodicity
论文作者
论文摘要
我们研究（同步）一维细胞自动机的动力学，具有周期性边界条件，该动力学根据半径$ r $的多数规则而发展。我们介绍了一个概念，即我们将单元稳定性表达出可能在这种情况下出现的可能出现的配置的结构。我们的主要发现是，除了表格$（0^{r+1} 0^*+1^{r+1} 1^*）^* $的配置外，它们始终是固定点，它们可能会收敛到其他配置，这些配置可能会收敛到，这些配置被称为固定点或2-cycles或2-cycles，具有特定的Spating Spatecatienceclect。也就是说，这些配置中的每一种都是$ s^* $的形式，其中$ s $由$ o（r^2）$连续具有相同状态的单元格组成，每个这样的序列最多为$ r $，$ s $的总长度为$ o（r^2）$。我们表明，类似的结果也适用于少数族裔规则。
We study the dynamics of (synchronous) one-dimensional cellular automata with cyclical boundary conditions that evolve according to the majority rule with radius $ r $. We introduce a notion that we term cell stability with which we express the structure of the possible configurations that could emerge in this setting. Our main finding is that apart from the configurations of the form $ (0^{r+1}0^* + 1^{r+1}1^*)^* $, which are always fixed-points, the other configurations that the automata could possibly converge to, which are known to be either fixed-points or 2-cycles, have a particular spatially periodic structure. Namely, each of these configurations is of the form $ s^* $ where $ s $ consists of $ O(r^2) $ consecutive sequences of cells with the same state, each such sequence is of length at most $ r $, and the total length of $ s $ is $ O(r^2) $ as well. We show that an analogous result also holds for the minority rule.
