  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. We 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. We 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/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.
