# Differentiable Logic Cellular Automata: From Game of Life to Pattern Generation

arXiv: https://arxiv.org/abs/2506.04912 · HTML: https://mlsanity.com/paper/2506.04912/

## Key facts

- Cited by 9 papers in the map (data through 14 Oct 2026).
- Builds on 5 of its 22 references that are in the map.
- First posted on arXiv 5 Jun 2025.

## Builds on (most cited first, 5 of 5)

- [Deep Differentiable Logic Gate Networks](https://mlsanity.com/paper/2210.08277/) (2022)
- [Convolutional Differentiable Logic Gate Networks](https://mlsanity.com/paper/2411.04732/) (2024)
- [Attention-based Neural Cellular Automata](https://mlsanity.com/paper/2211.01233/) (2022)
- [Image segmentation via Cellular Automata](https://mlsanity.com/paper/2008.04965/) (2020)
- [Physical Neural Cellular Automata for 2D Shape Classification](https://mlsanity.com/paper/2203.07548/) (2022)

## Built on by (most cited first, 9 of 9)

- [Neural Cellular Automata for ARC-AGI](https://mlsanity.com/paper/2506.15746/) (2025)
- [Neural cellular automata: applications to biology and beyond classical AI](https://mlsanity.com/paper/2509.11131/) (2025)
- [From MNIST to ImageNet: Understanding the Scalability Boundaries of Differentiable Logic Gate Networks](https://mlsanity.com/paper/2509.25933/) (2025)
- [Differentiable Weightless Controllers: Learning Logic Circuits for Continuous Control](https://mlsanity.com/paper/2512.01467/) (2025)
- [Self-Organising Digital Circuits](https://mlsanity.com/paper/2608.02606/) (2026)
- [Logic-Gated Time-Shared Feedforward Networks for Alternating Finite Automata: Exact Simulation and Learnability](https://mlsanity.com/paper/2604.01228/) (2026)
- [Von Neumann Networks](https://mlsanity.com/paper/2605.05780/) (2026)
- [The Artificial Experimentalist: Discovery and Control of Self-Organizing Phenomena with Autotelic Reinforcement Learning](https://mlsanity.com/paper/2608.26116/) (2026)
- [Applications of Neural Cellular Automata: State of the Art, Challenges and Opportunities](https://mlsanity.com/paper/2609.24595/) (2026)

## Abstract (arXiv)

This paper introduces Differentiable Logic Cellular Automata (DiffLogic CA), a novel combination of Neural Cellular Automata (NCA) and Differentiable Logic Gates Networks (DLGNs). The fundamental computation units of the model are differentiable logic gates, combined into a circuit. During training, the model is fully end-to-end differentiable allowing gradient-based training, and at inference time it operates in a fully discrete state space. This enables learning local update rules for cellular automata while preserving their inherent discrete nature. We demonstrate the versatility of our approach through a series of milestones: (1) fully learning the rules of Conway's Game of Life, (2) generating checkerboard patterns that exhibit resilience to noise and damage, (3) growing a lizard shape, and (4) multi-color pattern generation. Our model successfully learns recurrent circuits capable of generating desired target patterns. For simpler patterns, we observe success with both synchronous and asynchronous updates, demonstrating significant generalization capabilities and robustness to perturbations. We make the case that this combination of DLGNs and NCA represents a step toward programmable matter and robust computing systems that combine binary logic, neural network adaptability, and localized processing. This work, to the best of our knowledge, is the first successful application of differentiable logic gate networks in recurrent architectures.