Open-ended evolution requires unbounded possibilities that evolving entities can explore. The cardinality of a set of those possibilities thus has a significant implication for the open-endedness of evolution. I propose that facilitating formation of higher-order entities is a generalizable, effective way to cause a cardinality leap in the set of possibilities that promotes open-endedness. I demonstrate this idea with a simple, proof-of-concept toy model called Hash Chemistry that uses a hash function as a fitness evaluator of evolving entities of any size or order. Simulation results showed that the cumulative number of unique replicating entities that appeared in evolution increased almost linearly along time without an apparent bound, demonstrating the effectiveness of the proposed cardinality leap. It was also observed that the number of individual entities involved in a single replication event gradually increased over time, indicating evolutionary appearance of higher-order entities. Moreover, these behaviors were not observed in control experiments in which fitness evaluators were replaced by random number generators. This strongly suggests that the dynamics observed in Hash Chemistry were indeed evolutionary behaviors driven by selection and adaptation taking place at multiple scales.

中文翻译：

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开放式进化的基数飞跃：哈希化学的理论思考和论证

开放式进化需要进化实体可以探索的无限可能性。因此，一组这些可能性的基数对进化的开放性具有重要意义。我建议促进高阶实体的形成是一种可推广的有效方法，可以在促进开放性的一组可能性中引起基数飞跃。我用一个简单的、概念验证的玩具模型来证明这个想法，称为哈希化学，它使用哈希函数作为任何大小或顺序的进化实体的适应度评估器。模拟结果表明，进化中出现的独特复制实体的累积数量几乎随时间线性增加，没有明显的界限，证明了所提出的基数跳跃的有效性。还观察到参与单个复制事件的个体实体的数量随着时间的推移逐渐增加，表明更高阶实体的进化外观。此外，在将适应度评估器替换为随机数生成器的对照实验中，没有观察到这些行为。这有力地表明，在哈希化学中观察到的动态确实是由在多个尺度上发生的选择和适应驱动的进化行为。