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

Abstract: 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.

• The paper presents a novel integration of differentiable logic gate networks and neural cellular automata to establish robust local update rules in discrete systems.
• The paper demonstrates successful learning of Conway’s Game of Life and the generation of diverse patterns such as checkerboards, lizard shapes, and multi-color stripes.
• The paper highlights DiffLogic CA’s resilience to noise and scalability, suggesting its potential for programmable computing substrates and advanced hardware applications.

An Overview of Differentiable Logic Cellular Automata: Integration of NCA and DLGNs for Pattern Generation

Differentiable Logic Cellular Automata (DiffLogic CA) introduces a novel computational model integrating Neural Cellular Automata (NCA) and Differentiable Logic Gate Networks (DLGNs). This paper delineates the process by which DiffLogic CA leverages gradient-based training resulting in robust local update rules within cellular automata, operating in discrete state spaces. The studies conducted illustrate the capability of this model, particularly in understanding the rules of Conway's Game of Life and generating multi-color patterns, among other complex tasks.

Key Contributions

The integration of DLGNs with NCA stands out as a distinct approach to overcome inherent limitations regarding interpretability and computational efficiency in standard neural networks. The paper outlines several contributions:

• Development of DiffLogic CA combining principles of NCA with the discrete structures of DLGNs.
• Implementation of DLGNs in recurrent spatial and temporal settings for effective pattern generation.
• Elucidation of the robustness of these systems to noise and perturbations, affirming their utility in distributed computing frameworks.

Experimental Insights

The paper articulates four experiments highlighting the versatility of DiffLogic CA:

1. Learning Conway's Game of Life: The model accurately demonstrated the rules of the Game of Life using binary state representations. A pivotal observation was the accurate reproduction of dynamic patterns on larger scales utilizing encoded circuits.
2. Pattern Generation: DiffLogic CA was tasked with generating a checkerboard pattern, showcasing directional propagation without intrinsic directional bias. Insights into grid size invariance and pattern resilience against perturbations were obtained, suggesting robustness akin to biological systems.
3. Growing Complex Shapes: DiffLogic CA learned to grow a lizard shape, extending its application beyond simple patterns. Notably, scaling experiments indicated potential scalability without reliance on boundary conditions.
4. Multi-Color Pattern Production: This involved generating diagonal stripes in multiple colors, highlighting the capability of DiffLogic CA in coordinating multichannel outputs. The model emerged successful in producing requisite color patterns efficiently.

Implications and Future Directions

The DiffLogic CA framework promises substantial advancements in discrete self-organizing systems, facilitating the generation of robust computational structures. The implications are noteworthy:

• It presents a route towards programmable computing substrates capable of operating efficiently in discrete settings, potentially mapping to FPGA or specialized hardware for accelerated inference.
• The model's demonstrated resilience and fault tolerance indicate applicability in robust computing environments.

Future work might explore hierarchical architectures for layered self-organization, dynamic gating for operational flexibility, and potential hardware implementations where discrete logic circuits like DiffLogic CA could be optimized for rapid computations.

Conclusion

In summation, Differentiable Logic Cellular Automata deliver an innovative pathway towards constructing robust programmable systems grounded in discrete, interpretable logic networks alongside adaptive neural computation. This work furnishes a foundational framework with ample potential for further exploration in distributed computing architecture.