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Looking for Complexity at Phase Boundaries in Continuous Cellular Automata

Vassilis Papadopoulos, Guilhem Doat, Arthur Renard, Clément Hongler

TL;DR

The paper addresses the difficulty of finding complex, emergent behaviors in high-dimensional continuous Artificial Life systems by introducing the Phase Transition Finder (PTF), a lightweight, non-ML search that targets parameter-space boundaries between defined phases. PTF defines simple, quantitative phase criteria, samples transition regions, and uses a dichotomy along a parameter path to locate phase interfaces, leveraging continuity in parameter space. When applied to multi-channel Lenia, PTF markedly increases the incidence of interesting dynamics (e.g., solitons, self-reproduction) while maintaining scalability, achieving roughly a two- to three-fold improvement over random sampling depending on priors. This approach provides a complementary tool for rapid, large-scale exploration of continuous ALife models, with potential synergy when combined with learning-based methods and extensions to other phase definitions and systems.

Abstract

One key challenge in Artificial Life is designing systems that display an emergence of complex behaviors. Many such systems depend on a high-dimensional parameter space, only a small subset of which displays interesting dynamics. Focusing on the case of continuous systems, we introduce the 'Phase Transition Finder'(PTF) algorithm, which can be used to efficiently generate parameters lying at the border between two phases. We argue that such points are more likely to display complex behaviors, and confirm this by applying PTF to Lenia showing it can increase the frequency of interesting behaviors more than two-fold, while remaining efficient enough for large-scale searches.

Looking for Complexity at Phase Boundaries in Continuous Cellular Automata

TL;DR

The paper addresses the difficulty of finding complex, emergent behaviors in high-dimensional continuous Artificial Life systems by introducing the Phase Transition Finder (PTF), a lightweight, non-ML search that targets parameter-space boundaries between defined phases. PTF defines simple, quantitative phase criteria, samples transition regions, and uses a dichotomy along a parameter path to locate phase interfaces, leveraging continuity in parameter space. When applied to multi-channel Lenia, PTF markedly increases the incidence of interesting dynamics (e.g., solitons, self-reproduction) while maintaining scalability, achieving roughly a two- to three-fold improvement over random sampling depending on priors. This approach provides a complementary tool for rapid, large-scale exploration of continuous ALife models, with potential synergy when combined with learning-based methods and extensions to other phase definitions and systems.

Abstract

One key challenge in Artificial Life is designing systems that display an emergence of complex behaviors. Many such systems depend on a high-dimensional parameter space, only a small subset of which displays interesting dynamics. Focusing on the case of continuous systems, we introduce the 'Phase Transition Finder'(PTF) algorithm, which can be used to efficiently generate parameters lying at the border between two phases. We argue that such points are more likely to display complex behaviors, and confirm this by applying PTF to Lenia showing it can increase the frequency of interesting behaviors more than two-fold, while remaining efficient enough for large-scale searches.
Paper Structure (8 sections, 1 equation, 3 figures, 1 table)

This paper contains 8 sections, 1 equation, 3 figures, 1 table.

Table of Contents

  1. Introduction
  2. Idea and algorithm
  3. Phase Transition Finder Algorithm
  4. An illustrative example: multi-channel Lenia
  5. Results on Lenia
  6. Discussion and research directions
  7. Acknowledgements
  8. Code and simulations

Figures (3)

  • Figure 1: 2-dimensional parameter space for an Alife model. Two points in different phases are located, and then by dichotomy, we rapidly approach the phase transition region.
  • Figure 2: Top: Example kernels $K_{ji}$. Each kernel is convoluted with the corresponding labeled channel on top. The colors of the circles indicates to which channel the convolution contributes (i.e., the '$i$' in $K_{ji}$). Bottom : Cross-section of one kernel $K_{ji}$, displaying the 8 free parameters ($\sum_j \beta_j=1$).
  • Figure 3: Final snapshot for a simulation of 1000 steps in Lenia. Left of the red line: parameters obtained with PTF. Right : Parameters obtained with prior.