Cellular automata are topological dynamical systems. We consider the problem of deciding whether two cellular automata are conjugate or not. We also consider deciding strong conjugacy, that is, conjugacy by a map that commutes with the shift maps. We show that the following two sets of pairs of one-dimensional one-sided cellular automata are recursively inseparable:

(i)

pairs where the first cellular automaton has strictly higher entropy than the second one, and

(ii)

pairs that are strongly conjugate and both have zero topological entropies.

This implies that the following decision problems are undecidable: Given two one-dimensional one-sided cellular automata F and G: Are F and G conjugate? Is F a factor of G? Is F a subsystem of G? All of these are undecidable in both strong and weak variants (whether the homomorphism is required to commute with the shift or not, respectively).

We also prove the same results for reversible two-dimensional cellular automata.

中文翻译：

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关于细胞自动机的结合问题

元胞自动机是拓扑动力学系统。我们考虑决定两个细胞自动机是否共轭的问题。我们还考虑确定强大的共轭性，即与换挡图换向的图的共轭性。我们证明以下两对一维单面细胞自动机对是递归不可分割的：

（一世）

对，其中第一个细胞自动机的熵严格高于第二个细胞自动机的熵

（ii）

对是强共轭的，并且都具有零拓扑熵。

这意味着以下决策问题无法确定：给定两个一维单面细胞自动机F和G：F和G共轭吗？是˚F的因素摹？是˚F的一个子系统摹？在强变体和弱变体中，所有这些都是不确定的（是否需要同态分别与移位相通）。

我们还证明了可逆二维细胞自动机的相同结果。