The emergence of complex behaviors in cellular automata is an area that has been widely developed in recent years with the intention to generate and analyze automata that produce space-moving patterns or gliders that interact in a periodic background. Frequently, this type of automata has been found through either an exhaustive search or a meticulous construction of the evolution rule. In this study, the specification of cellular automata with complex behaviors was obtained by utilizing randomly generated specimens. In particular, it proposed that a cellular automaton of $n$ states should be specified at random and then extended to another automaton with a higher number of states so that the original automaton operates as a periodic background where the additional states serve to define the gliders. Moreover, this study presented an explanation of this method. Furthermore, the random way of defining complex cellular automata was studied by using mean-field approximations for various states and local entropy measures. This specification was refined with a genetic algorithm to obtain specimens with a higher degree of complexity. With this methodology, it was possible to generate complex automata with hundreds of states, demonstrating that randomly defined local interactions with multiple states can construct complexity.

中文翻译：

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生成复杂元胞自动机的随机扩展方法

元胞自动机中复杂行为的出现是近年来得到广泛发展的一个领域，旨在生成和分析自动机，这些自动机会产生空间移动模式或在周期性背景中相互作用的滑翔机。通常，这种类型的自动机是通过详尽的搜索或对进化规则的细致构建找到的。在这项研究中，通过利用随机生成的样本获得了具有复杂行为的元胞自动机的规范。特别是，它提出应该随机指定一个 $n$ 状态的元胞自动机，然后扩展到另一个具有更多状态的自动机，以便原始自动机作为周期性背景运行，其中附加状态用于定义滑翔机. 而且，这项研究对这种方法进行了解释。此外，通过对各种状态和局部熵测量使用平均场近似，研究了定义复杂元胞自动机的随机方式。该规范通过遗传算法进行了细化，以获得具有更高复杂度的样本。使用这种方法，可以生成具有数百个状态的复杂自动机，证明随机定义的具有多个状态的局部交互可以构建复杂性。