The Game of Life (Life), a well known algorithm within the broader class of cellular automata (CA), exhibits complex emergent dynamics, with extreme sensitivity to initial conditions. Modeling and predicting such intricate behavior without explicit knowledge of the system's underlying topology presents a significant challenge, motivating the development of algorithms that can generalize across various grid configurations and boundary conditions. We develop a decoder-only generative pretrained transformer model to solve this problem, showing that our model can simulate Life on a toroidal grid with no prior knowledge on the size of the grid, or its periodic boundary conditions (LifeGPT). LifeGPT is topology-agnostic with respect to its training data and our results show that a GPT model is capable of capturing the deterministic rules of a Turing-complete system with near-perfect accuracy, given sufficiently diverse training data. We also introduce the idea of an'autoregressive autoregressor' to recursively implement Life using LifeGPT. Our results pave the path towards true universal computation within a large language model (LLM) framework, synthesizing of mathematical analysis with natural language processing, and probing AI systems for situational awareness about the evolution of such algorithms without ever having to compute them. Similar GPTs could potentially solve inverse problems in multicellular self-assembly by extracting CA-compatible rulesets from realworld biological systems to create new predictive models, which would have significant consequences for the fields of bioinspired materials, tissue engineering, and architected materials design. Cellular automata (CA) have long been a subject of profound interest within the fields of computer science and mathematics, owing to their intricate and emergent behaviors. CA algorithms are uniquely characterized by their combination of computational simplicity--evolving solely by local state-transition rules--and broad dynamical behavior, encompassing static, periodic, chaotic, and complex patterns, depending on the ruleset and initial condition (IC) being used. These properties render CA algorithms particularly valuable for simulating a wide array of natural phenomena, such as the propagation of forest fires[1], traffic flow dynamics[2], chemical reactions[3], and recrystallization[4]. The inherent behavioral unpredictability of CA (following human inspection of their rulesets) has hindered advancements in these subfields, confining CA to the realm of phenomenological modeling and subsequently preventing their evolution into mature, predictive tools for systems or phenomena where do not yet know a closed-form ruleset.

Many graph algorithms can be viewed as sets of rules that are iteratively applied, with the number of iterations dependent on the size and complexity of the input graph. Existing machine learning architectures often struggle to represent these algorithmic decisions as discrete state transitions. Therefore, we propose a novel framework: GraphFSA (Graph Finite State Automaton). GraphFSA is designed to learn a finite state automaton that runs on each node of a given graph. We test GraphFSA on cellular automata problems, showcasing its abilities in a straightforward algorithmic setting. For a comprehensive empirical evaluation of our framework, we create a diverse range of synthetic problems. As our main application, we then focus on learning more elaborate graph algorithms. Our findings suggest that GraphFSA exhibits strong generalization and extrapolation abilities, presenting an alternative approach to represent these algorithms.

The disparity in access to machine learning tools for medical imaging across different regions significantly limits the potential for universal healthcare innovation, particularly in remote areas. Our research addresses this issue by implementing Neural Cellular Automata (NCA) training directly on smartphones for accessible X-ray lung segmentation. We confirm the practicality and feasibility of deploying and training these advanced models on five Android devices, improving medical diagnostics accessibility and bridging the tech divide to extend machine learning benefits in medical imaging to low- and middle-income countries (LMICs). We further enhance this approach with an unsupervised adaptation method using the novel Variance-Weighted Segmentation Loss (VWSL), which efficiently learns from unlabeled data by minimizing the variance from multiple NCA predictions. This strategy notably improves model adaptability and performance across diverse medical imaging contexts without the need for extensive computational resources or labeled datasets, effectively lowering the participation threshold. Our methodology, tested on three multisite X-ray datasets -- Padchest, ChestX-ray8, and MIMIC-III -- demonstrates improvements in segmentation Dice accuracy by 0.7 to 2.8%, compared to the classic Med-NCA. Additionally, in extreme cases where no digital copy is available and images must be captured by a phone from an X-ray lightbox or monitor, VWSL enhances Dice accuracy by 5-20%, demonstrating the method's robustness even with suboptimal image sources.

Elementary Cellular Automata (ECA) are a well-studied computational universe that is, despite its simple configurations, capable of impressive computational variety. Harvesting this computation in a useful way has historically shown itself to be difficult, but if combined with reservoir computing (RC), this becomes much more feasible. Furthermore, RC and ECA enable energy-efficient AI, making the combination a promising concept for Edge AI. In this work, we contrast ECA to substrates of Partially-Local CA (PLCA) and Homogeneous Homogeneous Random Boolean Networks (HHRBN). They are, in comparison, the topological heterogeneous counterparts of ECA. This represents a step from ECA towards more biological-plausible substrates. We analyse these substrates by testing on an RC benchmark (5-bit memory), using Temporal Derrida plots to estimate the sensitivity and assess the defect collapse rate. We find that, counterintuitively, disordered topology does not necessarily mean disordered computation. There are countering computational "forces" of topology imperfections leading to a higher collapse rate (order) and yet, if accounted for, an increased sensitivity to the initial condition. These observations together suggest a shrinking critical range.

Neural Cellular Automata (NCA) have proven to be effective in a variety of fields, with numerous biologically inspired applications. One of the fields, in which NCAs perform well is the generation of textures, modelling global patterns from local interactions governed by uniform and coherent rules. This paper aims to enhance the usability of NCAs in texture synthesis by addressing a shortcoming of current NCA architectures for texture generation, which requires separately trained NCA for each individual texture. In this work, we train a single NCA for the evolution of multiple textures, based on individual examples. Our solution provides texture information in the state of each cell, in the form of an internally coded genomic signal, which enables the NCA to generate the expected texture. Such a neural cellular automaton not only maintains its regenerative capability but also allows for interpolation between learned textures and supports grafting techniques. This demonstrates the ability to edit generated textures and the potential for them to merge and coexist within the same automaton. We also address questions related to the influence of the genomic information and the cost function on the evolution of the NCA.

Building effective imitation learning methods that enable robots to learn from limited data and still generalize across diverse real-world environments is a long-standing problem in robot learning. We propose EquiBot, a robust, data-efficient, and generalizable approach for robot manipulation task learning. Our approach combines SIM(3)-equivariant neural network architectures with diffusion models. This ensures that our learned policies are invariant to changes in scale, rotation, and translation, enhancing their applicability to unseen environments while retaining the benefits of diffusion-based policy learning such as multi-modality and robustness. We show in a suite of 6 simulation tasks that our proposed method reduces the data requirements and improves generalization to novel scenarios. In the real world, we show with in total 10 variations of 6 mobile manipulation tasks that our method can easily generalize to novel objects and scenes after learning from just 5 minutes of human demonstrations in each task.

After decades of attention, emergence continues to lack a centralized mathematical definition that leads to a rigorous emergence test applicable to physical flocks and swarms, particularly those containing both deterministic elements (eg, interactions) and stochastic perturbations like measurement noise. This study develops a heuristic test based on singular value curve analysis of data matrices containing deterministic and Gaussian noise signals. The minimum detection criteria are identified, and statistical and matrix space analysis developed to determine upper and lower bounds. This study applies the analysis to representative examples by using recorded trajectories of mixed deterministic and stochastic trajectories for multi-agent, cellular automata, and biological video. Examples include Cucker Smale and Vicsek flocking, Gaussian noise and its integration, recorded observations of bird flocking, and 1D cellular automata. Ensemble simulations including measurement noise are performed to compute statistical variation and discussed relative to random matrix theory noise bounds. The results indicate singular knee analysis of recorded trajectories can detect gradated levels on a continuum of structure and noise. Across the eight singular value decay metrics considered, the angle subtended at the singular value knee emerges with the most potential for supporting cross-embodiment emergence detection, the size of noise bounds is used as an indication of required sample size, and the presence of a large fraction of singular values inside noise bounds as an indication of noise.

Neural Cellular Automata (NCA) is a class of Cellular Automata where the update rule is parameterized by a neural network that can be trained using gradient descent. In this paper, we focus on NCA models used for texture synthesis, where the update rule is inspired by partial differential equations (PDEs) describing reaction-diffusion systems. To train the NCA model, the spatio-temporal domain is discretized, and Euler integration is used to numerically simulate the PDE. However, whether a trained NCA truly learns the continuous dynamic described by the corresponding PDE or merely overfits the discretization used in training remains an open question. We study NCA models at the limit where space-time discretization approaches continuity. We find that existing NCA models tend to overfit the training discretization, especially in the proximity of the initial condition, also called "seed". To address this, we propose a solution that utilizes uniform noise as the initial condition. We demonstrate the effectiveness of our approach in preserving the consistency of NCA dynamics across a wide range of spatio-temporal granularities. Our improved NCA model enables two new test-time interactions by allowing continuous control over the speed of pattern formation and the scale of the synthesized patterns. We demonstrate this new NCA feature in our interactive online demo. Our work reveals that NCA models can learn continuous dynamics and opens new venues for NCA research from a dynamical system's perspective.

Reservoir Computing with Cellular Automata (ReCA) is a relatively novel and promising approach. It consists of 3 steps: an encoding scheme to inject the problem into the CA, the CA iterations step itself and a simple classifying step, typically a linear classifier. This paper demonstrates that the ReCA concept is effective even in arguably the simplest implementation of a ReCA system. However, we also report a failed attempt on the UCR Time Series Classification Archive where ReCA seems to work, but only because of the encoding scheme itself, not in any part due to the CA. This highlights the need for ablation testing, i.e., comparing internally with sub-parts of one model, but also raises an open question on what kind of tasks ReCA is best suited for.

This paper presents Coralai, a framework for exploring diverse ecosystems of Neural Cellular Automata (NCA). Organisms in Coralai utilize modular, GPU-accelerated Taichi kernels to interact, enact environmental changes, and evolve through local survival, merging, and mutation operations implemented with HyperNEAT and PyTorch. We provide an exploratory experiment implementing physics inspired by slime mold behavior showcasing the emergence of competition between sessile and mobile organisms, cycles of resource depletion and recovery, and symbiosis between diverse organisms. We conclude by outlining future work to discover simulation parameters through measures of multi-scale complexity and diversity. Code for Coralai is available at https: //github.com/aidanbx/coralai,