Cardinality leap for open-ended evolution: Theoretical consideration and demonstration by hash chemistry

Hiroki Sayama*

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review

3 Citations (Scopus)


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.

Original languageEnglish
Pages (from-to)104-116
Number of pages13
JournalArtificial Life
Issue number2
Publication statusPublished - 2019


  • Cardinality leap
  • Hash Chemistry
  • Higher-order entities
  • Open-ended evolution
  • Set of possibilities
  • Universal fitness evaluator

ASJC Scopus subject areas

  • Biochemistry, Genetics and Molecular Biology(all)
  • Artificial Intelligence


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