We present an analysis of an additive cellular automaton (CA) under asynchronous dynamics. The asynchronous scheme is maxmin-\(\omega\), a deterministic system, introduced in our previous work with a binary alphabet. Extending this work, we study the impact of a larger alphabet, which also allows a meaningful inference of the behaviour of the resultant CA from the asymptotic behaviour of the maxmin-\(\omega\) update system. Far from being a straightforward positive correlation between complexity and alphabet size, we show that there is a region of \(\omega\) and alphabet size where complexity of CA is maximal. Thus, despite employing a fixed CA rule, the complexity of this CA can be controlled by \(\omega\) and alphabet size. The main message is that the effect of maxmin-\(\omega\) updating on the state of a network can be well understood, especially if the state alphabet is counter-intuitively large.

中文翻译：

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字母大小对maxmin-$$ \ omega $$ω元胞自动机的模式复杂度的影响

我们介绍了异步动力学下的加性细胞自动机（CA）的分析。异步方案是maxmin- \（\ omega \），这是一种确定性系统，在我们之前的工作中使用二进制字母引入。扩展这项工作，我们研究了较大字母的影响，这也允许根据maxmin- （\ omega \）更新系统的渐近行为来有意义地推断所得CA的行为。远不是复杂度和字母大小之间的直接正相关，而是表明存在一个\（\ omega \）和字母大小的区域，其中CA的复杂度最大。因此，尽管采用了固定的CA规则，但该CA的复杂度仍可以由\（\ omega \）来控制和字母大小。主要信息是，可以很好地理解maxmin- （\ omega \）更新对网络状态的影响，尤其是在状态字母过大地违反直觉的情况下。