Catalan Numbers and Power Laws in Cellular Automaton Rule 14
Catalan Numbers and Power Laws in Cellular Automaton Rule 14
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元胞自动机中的加泰罗尼亚数和幂律规则 14
DOI：
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发表时间：
2007
影响因子：
0.3
通讯作者：
J. Haroutunian
中科院分区：
文献类型：
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作者：
H. Fuks;J. Haroutunian
我们讨论一个基本的细胞自动机的密度衰减到其极限值作为迭代次数的幂n$的例子。使用的事实，这条规则保存的块10的数量和一些其他块的原像表现出密切相关的模式，在规则184中观察到的模式，我们推导出表达式的所有块的长度为3的$n$步原像的数量。这些表达式涉及加泰罗尼亚数，并与迭代概率测度的基本性质一起，它们允许我们计算$n$迭代后的密度，以及长度小于或等于3的任意块的发生概率。
We discuss example of an elementary cellular automaton for which the density of ones decays toward its limiting value as a power of the number of iterations $n$. Using the fact that this rule conserves the number of blocks 10 and that preimages of some other blocks exhibit patterns closely related to patterns observed in rule 184, we derive expressions for the number of $n$-step preimages of all blocks of length 3. These expressions involve Catalan numbers, and together with basic properties of iterated probability measures they allow us to to compute the density of ones after $n$ iterations, as well as probabilities of occurrence of arbitrary block of length smaller or equal to 3.