Quantifying emergent complexity in artificial cellular automata at multiple scales

Determine quantitative methods that reliably measure emergent complexity at different scales in artificial systems modeled by cellular automata, including continuous cellular automata and neural cellular automata, so as to guide evolution toward higher emergent complexity.

Background

The paper investigates frequency-histogram coarse graining (FHCG) as a dimensionality reduction technique to highlight rare events and emergent structures in 1D and 2D cellular automata, including Elementary Cellular Automata, Game of Life, and Multi-Neighbor Cellular Automata. While FHCG provides qualitative insights and visualization tools, the broader challenge of quantitatively assessing emergent complexity in such systems remains unresolved.

The authors frame the problem in the context of recent work on continuous and neural cellular automata as potential substrates for open-ended, adaptive artificial intelligence. A principled, multi-scale complexity metric would enable guiding evolutionary or learning processes toward more complex emergent behaviors, a key step toward open-ended evolution and general intelligence.

References

However, an open question is how to quantify the emergent complexity of such artificial systems at different scales, in order to guide evolution towards higher emergent complexity.

— Frequency-Histogram Coarse Graining in Elementary Cellular Automata and 2D CA  (2507.18674 - Jain et al., 24 Jul 2025) in Introduction (p. 2)

Among the open problems McMullin identifies for the lineage of self-reproducing automata are identity, bootstrapping, and actual growth of complexity.

— Dalek: A Constructive Agent Machine  (2609.03546 - Xie, 3 Sep 2026) in Section 6.4, “Agents and von Neumann”