The local structure theory for cellular automata (CA) can be viewed as an finite-dimensional approximation of infinitely dimensional system. While it is well known that this approximation works surprisingly well for some CA, it is still not clear why it is the case, and which CA rules have this property. In order to shed some light on this problem, we present an example of a four input CA for which probabilities of occurrence of short blocks of symbols can be computed exactly. This rule is number conserving and possesses a blocking word. Its local structure approximation correctly predicts steady-state probabilities of small length blocks, and we present a rigorous proof of this fact, without resorting to numerical simulations. We conjecture that the number-conserving property together with the existence of the blocking word are responsible for the observed perfect agreement between the finite-dimensional approximation and the actual infinite-dimensional dynamical system.

中文翻译：

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用有限维动力学系统逼近数守恒元胞自动机的动力学

元胞自动机（CA）的局部结构理论可以看作是无限维系统的有限维逼近。虽然众所周知，这种近似对于某些 CA 非常有效，但仍不清楚为什么会出现这种情况，以及哪些 CA 规则具有此属性。为了阐明这个问题，我们提出了一个四输入 CA 的示例，可以精确计算短符号块的出现概率。这条规则是数字守恒的，并且有一个阻塞词。它的局部结构近似正确地预测了小长度块的稳态概率，我们对此事实提出了严格的证明，而无需借助数值模拟。