Shape irregularity of Life-Like Network Automaton rules as an indicator of classification performance

Previous work has found a gap between the scale of neural networks that reliably learn Conway's Game of Life, and minimal networks capable of representing the classic cellular automaton with hard-coded parameter values. Viewing neural network learning as a search process suggests a dependence on networks large enough to contain sub-networks with lucky initializations (sometimes known as 'winning tickets') that actually learn the task. In this work, we reorient our perspective from discovering Life rules as a search problem back to a learning problem, and reason that with fitting inductive biases, the problem should be much more amenable to minimal networks. We find that network variants with several alternative activation functions meaningfully outperform the default choice of Rectified Linear Units, and in particular, that a 2nd degree polynomial activation function consistently learns Life dynamics with or without the benefit of learning neural weights. Our results provide an informative demonstration of the benefits of matching learning to the task at hand and challenge the easy default choice of scale for all problems. In particular, we advocate for the use of cellular automata as simple test domains for developing strategies that can benefit machine learning for science, physics-based deep learning, and interpretable machine learning.

Stephen Wolfram proclaimed in his 2003 seminal work "A New Kind Of Science" that simple recursive programs in the form of Cellular Automata (CA) are a promising approach to replace currently used mathematical formalizations, e.g. differential equations, to improve the modeling of complex systems. Over two decades later, while Cellular Automata have still been waiting for a substantial breakthrough in scientific applications, recent research showed new and promising approaches which combine Wolfram's ideas with learnable Artificial Neural Networks: So-called Neural Cellular Automata (NCA) are able to learn the complex update rules of CA from data samples, allowing them to model complex, self-organizing generative systems. The aim of this paper is to review the existing work on NCA and provide a unified modular framework and notation, as well as a reference implementation in the open-source library NCAtorch. Supplementary materials, videos, and code are available at the project website: https://www.neural-cellular-automata.org/

As the input dimension $n$ grows, rule-based machine learning, such as Learning Classifier Systems (LCSs), faces a fundamental scalability bottleneck for function approximation: both rule count and parameter count grow exponentially with $n$. Traditional LCSs partition the $n$-dimensional input space directly, requiring $\mathcal{O}(m^n)$ rules for adequate coverage, where $m$ is the per-variable resolution. This article breaks from this paradigm by reorganizing rules dimension-wise, guided by the Kolmogorov-Arnold representation theorem: any continuous $n$-dimensional function can be expressed as a finite superposition of one-dimensional functions. The proposed Kolmogorov-Arnold Classifier System (KACS) decomposes the target function into one-dimensional subproblems and assigns a dedicated ruleset to each, reducing the worst-case rule count from $\mathcal{O}(m^n)$ to $\mathcal{O}(mn^2)$ and replacing $n$-dimensional local models with one-dimensional models requiring only two parameters per rule, independent of $n$. We also provide the first constructive proof that an LCS, namely KACS, is a universal approximator for continuous functions on compact domains. Evaluated against a direct $n$-dimensional input space partitioning approach under otherwise identical conditions, KACS achieves competitive accuracy in many settings while using only 2% to 40% of the parameters. Our implementation is available at https://github.com/YNU-NakataLab/KACS.