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The Garden of Eden theorem for cellular automata on group sets

机译:集团套装蜂窝自动机的伊甸园定理园

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We prove the Garden of Eden theorem for big-cellular automata with finite set of states and finite neighbourhood on right amenable left homogeneous spaces with finite stabilisers. It states that the global transition function of such an automaton is surjective if and only if it is pre-injective. Pre-lnjectivity means that two global configurations that differ at most on a finite subset and have the same image under the global transition function must be identical. The theorem is proven by showing that the global transition function of an automaton as above is surjective if and only if its image has maximal entropy and that its image has maximal entropy if and only if it is pre-injective. Entropy of a subset of global configurations measures the asymptotic growth rate of the number of finite patterns with growing domains that occur in the subset.
机译:我们向大型蜂窝自动机的伊甸园定理的花园提供了有限的状态,有限邻域与有限稳定剂有限的左右均匀空间。它指出这种自动机的全局转换功能是且仅当它是预注射的时样子。预注射器意味着两个全局配置在有限子集上最多不同,并且在全局转换函数下具有相同的图像必须是相同的。通过显示自动机的全局转换功能是且仅当其图像具有最大熵并且仅当它才有最大熵而且仅当它是预制件时的最大熵并且才有最大的熵,而且才能才能预制熵。全局配置子集的熵测量了子集中发生的生长域的有限模式数量的渐近生长速率。

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