Evolving Structures in Complex Systems

In this paper we propose an approach for measuring growth of complexity of emerging patterns in complex systems such as cellular automata. We discuss several ways how a metric for measuring the complexity growth can be defined. This includes approaches based on compression algorithms and artificial neural networks. We believe such a metric can be useful for designing systems that could exhibit open-ended evolution, which itself might be a prerequisite for development of general artificial intelligence. We conduct experiments on 1D and 2D grid worlds and demonstrate that using the proposed metric we can automatically construct computational models with emerging properties similar to those found in the Conway’s Game of Life, as well as many other emergent phenomena. Interestingly, some of the patterns we observe resemble forms of artificial life. Our metric of structural complexity growth can be applied to a wide range of complex systems, as it is not limited to cellular automata.

💡 Research Summary

The paper tackles the long‑standing problem of quantifying structural evolution and complexity growth in dynamical systems, using cellular automata (CA) as a testbed. The authors argue that open‑ended evolution—a prerequisite for artificial general intelligence—requires two observable properties: self‑preservation and a measurable increase in complexity over time. To capture these properties they propose two complementary metrics based on data compression.

The first metric is a simple compressed length (C). A CA’s global state at a given time step is serialized into a string (one symbol per cell state) and fed to a lossless compressor (DEFLATE via zlib, or PAQ). The resulting byte length serves as a proxy for Kolmogorov‑type complexity: highly regular configurations compress to short strings, while random‑looking configurations yield longer outputs. Applying this to all 256 elementary 1‑D CA (ECA) over 512 steps, the authors plot compressed lengths and cluster them with K‑means. The clusters align with Wolfram’s four classes: the longest compressed lengths correspond to Class 3 (chaotic), the shortest to Class 1 (homogeneous), and intermediate lengths map to Class 4 (complex, localized structures). This reproduces earlier findings that compression can separate CA behavior classes and validates the metric as a quick, implementation‑light alternative to entropy or Lyapunov analyses.

The second metric, joint compression score (µ), measures how much structure persists across time. For a delay τ, the states at times t and t‑τ are concatenated and compressed together; µ is defined as (C(S_t)+C(S_{t‑τ}))/C(S_t‖S_{t‑τ}). If a pattern survives from t‑τ to t (e.g., a glider or a self‑replicating seed), the joint compression will be smaller than the sum of the individual compressed lengths, pushing µ toward 1. The authors demonstrate that µ alone tends to select trivial rules that either copy the input unchanged or merely shift it, because those maximize overlap without generating new complexity. To filter out such degenerate cases they add a second condition: a positive increase in compressed length between the initial and final states (ΔC > 0). The combined filter (high µ + ΔC) successfully isolates rules that both preserve structures and exhibit growing disorder—exactly the kind of dynamics expected from open‑ended evolution.

The paper also discusses the challenges of extending these ideas to two‑dimensional CA. Linear compressors have dictionary size limits and are not invariant to spatial transformations (rotations, flips, translations). The authors propose a count‑based predictor that tallies pattern frequencies, and suggest using image‑specific compressors (e.g., PNG) for 2‑D grids, but acknowledge that current implementations are limited and that more sophisticated n‑dimensional compressors or neural‑network based predictors would be needed for scalability.

Experimental results cover both 1‑D and 2‑D grid worlds. In 1‑D, the combined metric discovers several ECA rules that generate glider‑like moving structures and occasional self‑replication, reminiscent of Conway’s Game of Life dynamics. In 2‑D experiments, the same methodology uncovers patterns that grow, maintain, and replicate themselves, with visual similarity to artificial life forms. The authors highlight that these emergent behaviors arise without hand‑crafted rule design, demonstrating the utility of the metric for automated search in large rule spaces.

In the discussion, the authors emphasize that compression‑based metrics are computationally cheap, conceptually clear, and applicable beyond CA to any system where a symbolic representation of the state can be constructed. They also note limitations: dependence on the chosen compressor (different compressors yield different absolute lengths), sensitivity to boundary conditions, and the lack of true invariance to geometric transformations in higher dimensions. Future work is outlined as integrating modern neural compressors (e.g., variational autoencoders), extending the framework to physical or biological simulations, and exploring theoretical connections between compression‑based complexity and algorithmic information theory.

Overall, the paper contributes a practical, dual‑metric approach for detecting and quantifying structural complexity growth in discrete dynamical systems, offering a promising tool for the automated discovery of open‑ended evolutionary dynamics that could inform the design of more adaptable, general AI systems.