标题： 细胞自动机在二维和一维中的响应曲线 - 严格计算的一个例子 显示英文标题

标题： Response curves for cellular automata in one and two dimensions - an example of rigorous calculations

摘要： 我们考虑计算二进制细胞自动机的响应曲线的问题——即描述经过多次规则迭代后1的密度与初始1的密度之间关系的曲线。 我们展示如何以规则130为例来解决这个问题。 对于此规则，有限字符串的预像集表现出可识别的模式，因此可以计算某些有限字符串的预像基数以及这些字符串在通过迭代随机初始配置$n$次得到的配置中出现的概率。 响应曲线可以在CA规则130的一维和二维版本中严格计算。 我们还讨论了一种特殊情况，即完全无序的初始配置，也就是1和0的密度都等于1/2的随机配置。 显示英文摘要

摘要： We consider the problem of computing a response curve for binary cellular automata -- that is, the curve describing the dependence of the density of ones after many iterations of the rule on the initial density of ones. We demonstrate how this problem could be approached using rule 130 as an example. For this rule, preimage sets of finite strings exhibit recognizable patterns, and it is therefore possible to compute both cardinalities of preimages of certain finite strings and probabilities of occurrence of these strings in a configuration obtained by iterating a random initial configuration $n$ times. Response curves can be rigorously calculated in both one- and two-dimensional versions of CA rule 130. We also discuss a special case of totally disordered initial configurations, that is, random configurations where the density of ones and zeros are equal to 1/2.