Flow-Lenia.png: Evolving Multi-Scale Complexity by Means of Compression
Tadashi Adachi, Solvi Arnold, Takafumi Mochizuki, Kimitoshi Yamazaki
TL;DR
This work addresses how to quantify and steer multi-scale complexity in Flow Lenia using a compression-based proxy grounded in Kolmogorov complexity to facilitate open-ended evolution in cellular automata. It formalizes a fitness function, $Fitness = (1/(S+1)) \sum_{s=0}^S | C(x, s) - T |$, with $C$ as the inverse compression after downsampling to scale $2^{-s}$, and mitigates symmetry artifacts via polar re-rasterization; a genetic algorithm evolves Flow Lenia patterns under a fixed hyperparameter configuration. Experiments show that patterns evolve toward prescribed complexity targets and reveal the extrema of the accessible complexity range (roughly $0.21$ to $0.62$), with higher targets producing more intricate, multi-scale structures when using multi-scale analysis. The results demonstrate the viability of compression-based complexity control for guiding open-ended exploration and suggest directions to combine complexity objectives with novelty and functional selection in future work.
Abstract
We propose a fitness measure quantifying multi-scale complexity for cellular automaton states, using compressibility as a proxy for complexity. The use of compressibility is grounded in the concept of Kolmogorov complexity, which defines the complexity of an object by the size of its smallest representation. With this fitness function, we explore the complexity range accessible to the well-known Flow Lenia cellular automaton, using image compression algorithms to assess state compressibility. Using a Genetic Algorithm to evolve Flow Lenia patterns, we conduct experiments with two primary objectives: 1) generating patterns of specific complexity levels, and 2) exploring the extrema of Flow Lenia's complexity domain. Evolved patterns reflect the complexity targets, with higher complexity targets yielding more intricate patterns, consistent with human perceptions of complexity. This demonstrates that our fitness function can effectively evolve patterns that match specific complexity objectives within the bounds of the complexity range accessible to Flow Lenia under a given hyperparameter configuration.