Pith. sign in

REVIEW 2 major objections 5 minor 72 references

AutomataGPT: Forecasting and Ruleset Inference for Two-Dimensional Cellular Automata

T0 review · 2 major / 5 minor · reviewed 2026-08-06 · deepseek-v4-flash

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗

Pith's one-line read AutomataGPT: a transformer pretrained on 100 isotropic 2D cellular-automaton rules forecasts unseen rules at 98.5% and recovers working rules at up to 96%.

desk verdict Forward CA forecasting with a small transformer is solid and worth reading; the inverse-problem headline numbers overstate what is measured, and the 'without hand-crafted priors' claim needs a caveat. read the letter →

arxiv 2506.17333 v1 pith:FZ3RXV2V submitted 2025-06-19 cs.LG cond-mat.dis-nncond-mat.mtrl-sciq-bio.QM

Jaime A. Berkovich , Noah S. David , Markus J. Buehler This is my paper · ORCID
classification cs.LGcond-mat.dis-nncond-mat.mtrl-sciq-bio.QM MSC 37B1568Q8068T07
keywords cellularautomatatwo-dimensionalbinarygenerativepretrainedtransformersrulespacesforecastingrulesetinferencedeeplearning
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

AutomataGPT is a decoder-only transformer pretrained on about one million simulated trajectories spanning 100 distinct rules of a restricted family of two-dimensional binary cellular automata. The paper claims that widening the rule set in pretraining, from 2 to 10 to 100 rules, makes the same architecture progressively better at both the forward task (predicting the next grid state from an initial state and a rule matrix) and the inverse task (recovering a rule matrix from one initial-state/next-state pair). On previously unseen rules from the same family, the model reaches 98.5% perfectly correct one-step forecasts, 96% functional rule-application accuracy, and 82% rule-matrix recovery under the paper's metric. The authors read this as evidence that broad rule-space pretraining yields a rules-agnostic 'CA computer' and a route toward inferring rules of real-world coarse-grained dynamics from data alone.

What carries the argument

The load-bearing object is the rules matrix (RM), a $2\times18$ binary adjacency matrix whose columns are the 18 metastates of an isotropic binary Moore-neighborhood cellular automaton: the center cell's state (0 or 1) combined with the number of its eight neighbors in state 1, with orientation ignored. This count-based representation compresses the rule space from $2^{512}$ oriented $3\times3$ kernels to $2^{18}=262{,}144$ rules, and both tasks are posed as autoregressive token completion over flattened strings of RM, initial grid, and next grid using a 22-token vocabulary. The metastate compression is what makes the problem tractable for a small transformer, and it is also the representational prior that bounds the scope of the generalization claim.

What would settle it

Generate initial-state/next-state pairs from an orientation-dependent binary Moore-neighborhood rule (same center state and same neighbor count, but different next states depending on which neighbor is 1) and ask the inverse model to infer the rule. Because the 18-metastate representation cannot encode orientation, the inferred rule should fail to reproduce the next state when applied; success would falsify the claim that the representation is the limiting factor.

Watch

Extended reading notes

Core claim

The central discovery claimed is that a single transformer, trained only on synthetic CA trajectories, can internalize a general and programmable way to execute 2D binary deterministic cellular automata, and can invert one time step to recover a working rule. The evidence is monotonic scaling: moving from 2 to 10 to 100 distinct rules in pretraining raises forward accuracy, the fraction of perfectly correct forecasts, and both inverse accuracy metrics, while the correlation between test accuracy and the distance to the closest training rules — measured by Hamming distance, Jaccard distance, and Jensen–Shannon divergence — drops toward zero, with $R^2 = 0.00$ for functional rule-application accuracy at $N_{\mathrm{RM}} = 100$. The paper interprets the disappearing correlation as reduced inductive bias: for rules far from anything seen in training, the model offers 'creative' degenerate rules that still reproduce the observed dynamics, and that creativity does not cost accuracy.

Load-bearing premise

The load-bearing premise is that orientation can be ignored and only the number of 1-neighbors matters; if a target rule depends on which neighbor is where, the model's 18-metastate representation cannot express it, so the reported generalization does not apply.

Editorial extensions

If this is right

  • If the scaling result is correct, one small transformer can replace a rule-by-rule simulator for the entire isotropic 2D binary cellular-automaton family, including Game-of-Life-like rules.
  • Rule-space breadth in pretraining becomes a concrete, model-agnostic axis for improving both forecasting and inverse inference.
  • The near-zero $R^2$ at $N_{\mathrm{RM}}=100$ implies that, within this family, the model's inferences are effectively independent of which rules appeared in training.
  • Because a single time-step pair already yields working rules, feeding longer orbits should shrink the set of degenerate rules that all reproduce the observed state.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • The 'without hand-crafted priors' claim sits alongside the paper's own permutation-invariance assumption: the 18-metastate count-based representation is itself a hand-crafted prior, so the demonstrated generalization holds within that isotropic family, not for general 2D cellular automata.
  • The 82% 'exact rule-matrix match' figure uses a metric that counts any rule matrix reproducing the observed next state as perfect, so it may include degenerate rules different from the ground truth; the exact ground-truth identity rate is not reported separately.
  • A direct scaling test is to pretrain on all $2^{18}$ possible rules; if accuracy does not reach 100%, the ceiling is representational or architectural rather than a matter of rule-space coverage.
  • Lifting the permutation-invariance assumption would give an oriented-kernel version of the model, and testing whether the same scaling benefits appear for directional rules (for example, traffic or chemical-wave automata) is a natural next step.
Share X Bluesky LinkedIn Reddit HN

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

2 major / 5 minor

Summary. The paper introduces AutomataGPT, a decoder-only transformer pretrained on synthetic trajectories from two-dimensional binary deterministic cellular automata with r=1 Moore neighborhoods on 16x16 toroidal grids. Three versions are trained on 2, 10, and 100 distinct rules for each of two tasks: a forward task (given a rules matrix and initial condition, predict the next state) and an inverse task (given an initial condition and next state, infer the rules matrix). For the forward task, the NRM=100 model achieves 98.5% perfect one-step predictions. For the inverse task, the authors report up to 96% functional application accuracy and 82% rule-matrix inference accuracy, and interpret decreasing correlation with training-rule distance as evidence of reduced inductive bias and increased 'creativity.' The central claims are that scaling rule-space breadth improves both forward and inverse performance without hand-crafted priors.

Significance. If the claims are taken at face value, the forward result is a clean demonstration that a small transformer can learn to execute arbitrary rules from a restricted isotropic 2D CA family when trained on broad rule coverage, and the inverse result would suggest a path toward interpretable rule discovery. The paper's strengths include public code and data, an independent ground-truth evaluation for the forward problem, and a systematic comparison across rule-space sizes. However, the significance is substantially reduced by two issues: the rule representation itself is a hand-crafted isotropic prior that limits all conclusions to 2^18 of the 2^512 general 2D CA rule space, and the headline inverse metric ARMI counts degenerate solutions as perfect, so '82% exact rule-matrix match' is not an exact-match rate. The forward result remains sound, but the broader generalization claims and the inverse headline need re-scoping.

major comments (2)
  1. [Section 4.4.1 step 4 and Eq. (3)] The claim in the Abstract and Section 1 that AutomataGPT generalizes over '2D binary deterministic CA' 'without hand-crafted priors' is contradicted by the paper's own rule representation. In Section 4.3.3, metastates are defined as permutation-invariant: only the center state and the number of neighbors in state 1 matter, yielding 18 count-based metastates and a 2^18 rule space (Eq. 8). This excludes oriented 3x3 kernels, which would give 2^512 rules. The model is trained and tested exclusively on this isotropic, count-based family, so the results demonstrate generalization only within that specific hand-chosen family, not over general 2D binary deterministic CA. The text should explicitly scope all claims to the isotropic family and remove or qualify the 'without hand-crafted priors' phrasing.
  2. [Section 4.4.1 step 4 and Eq. (3)] The forward result is independent of this issue because it is evaluated directly against ground-truth GS2, but the inverse headline is load-bearing and currently overstates what is measured.
minor comments (5)
  1. [Section 2.1, Fig. 3c] The text states that 'for NRM = 200, 100% of samples tested yielded AGS2I ≥ 0.9,' but no NRM=200 model was trained; this should be NRM=100.
  2. [Section 2.2] The sentence 'the error fraction initially increased from NRM = 2 to (See Fig. 1.)' is incomplete and should read 'from NRM = 2 to NRM = 10' or be reworded.
  3. [Equations (1) and (2)] Both the per-sample accuracy and the averaged accuracy are denoted AGS2I, which is confusing; one of these should be renamed, for example AGS2I_i for the per-sample value.
  4. [Figure 3 and Figure 4 captions] The captions use 'affect' where 'effect' is intended in 'the affect of NRM.'
  5. [Section 4.3.3] The phrase 'losslessly represented as a binary adjacency matrix' should be qualified: the representation is lossless only with respect to the chosen permutation-invariant, count-based metastate set, not for arbitrary 3x3 oriented neighborhoods.

Circularity Check

2 steps flagged · score 6.0 of 10

The '82% exact rule-matrix match' is defined by construction as reproducing GS2, not as recovering the ground-truth RM; the 'without hand-crafted priors' claim is contradicted by the paper's own permutation-invariant 2x18 RM formalism.

  1. self definitional [Abstract; Sec. 2.2 Eq. (3); Sec. 4.4.1 inverse step 3]
    "reconstructs the governing update rule with up to 96 % functional (application) accuracy and 82 % exact rule-matrix match. ... ARMI = 1, if the inferred RM yields the exact ground truth GS2 ... ARMI was set to 1 if AIRMA was also found to be 1 (indicating a degenerate solution)."

    ARMI, the source of the 82% 'exact rule-matrix match', assigns a perfect score to any inferred RM that reproduces the sample GS2, regardless of whether the matrix equals the ground-truth RM. Section 4.4.1 explicitly implements this as ARMI = 1 whenever AIRMA = 1, so functional equivalence on one observed transition is the entire criterion. The claimed exact-rule reconstruction is therefore defined in terms of the forward forecast rather than the inverse target: a model that overfits the single IC/GS2 pair is scored as perfect without recovering the true rule. The 82% figure is a behavioral-fit rate renamed as an exact match, so this part of the inverse claim reduces by construction to evaluating whether the inferred RM reproduces GS2.

  2. other [Abstract; Sec. 4.3.3 RM Formalization; Sec. 4.3.6 GS2 Generation]
    "These results demonstrate ... without hand-crafted priors. ... We assume metastates are permutation invariant, meaning the orientation of neighboring states does not change the identity of the metastate ... we defined RMs as 2 x 18 2D arrays. ... This operation was count-based (isotropic)."

    The paper's own rule formalism is a hand-crafted prior: each orientation-dependent 3x3 neighborhood is compressed into an 18-metastate count-based representation (center state plus number of 1-neighbors), reducing the oriented rule space from 2^512 to 2^18. All datasets, prompts, and inverse targets use this 2x18 format, so every reported accuracy is by construction about this isotropic family. The abstract's 'without hand-crafted priors' claim is therefore not supported by the paper's setup; the results demonstrate generalization within a specially chosen low-dimensional rule encoding, not over general 2D binary deterministic CA.

full rationale

The forward forecasting result is largely self-contained: held-out RM/IC pairs are evaluated against independently computed ground-truth GS2, so the 98.5% one-step forecast accuracy is not a circular claim. The inverse AIRMA functional-application metric is also meaningful as a measure of behavioral fidelity. However, the '82% exact rule-matrix match' is circular by construction: Eq. (3) defines perfect inverse inference as any RM that yields the exact GS2, and Sec. 4.4.1 equates this with AIRMA = 1, which explicitly includes degenerate RMs. Thus the exact-match statistic is not an exact-recovery statistic; it is a re-labeling of forward-consistency on a single transition. Separately, the paper's central 'without hand-crafted priors' framing is contradicted by Sec. 4.3.3's permutation-invariant, count-based 2x18 metastate assumption, which restricts the entire study to a 2^18-rule isotropic family. This is a scope limitation and an overclaim rather than a derivation-circularity, but it weakens the general-CA interpretation. These issues are partial: the forward numbers and the functional-application numbers retain independent content, so the overall circularity score is moderate rather than extreme.

Assumptions & free parameters 1 free parameters · 4 assumptions · 0 invented entities

The central claims rest on a heavily structured rule encoding (count-based isotropic 2x18 RMs) and metric definitions that credit degenerate solutions. No free scientific parameters are fitted to external data; the only hand-set values are standard ML hyperparameters. No invented entities are introduced.

free parameters (1)
  • Transformer hyperparameters (dim, depth, heads, learning rate, mask probability) = 256, 6, 4, 1e-4, 0.15
    Selected heuristically (Section 4.2.1) for the 16GB GPU; the paper does not test sensitivity, so their specific values are hand-chosen inputs to the result.
assumptions (4)
  • domain assumption The rule space is restricted to count-based isotropic 2D binary rules with 18 metastates (2x18 RM).
    Invoked in Section 4.3.3 to define RMs; this is a hand-crafted prior that excludes oriented rules and reduces rule space from 2^512 to 2^18.
  • domain assumption Metastates are permutation invariant with respect to neighbor orientation.
    Section 4.3.3 states the orientation of neighbors does not change a metastate; without this, the RM encoding would need 2^9 columns.
  • domain assumption Evaluation is limited to one-step (GS2) transitions on 16x16 toroidal grids.
    Section 4.3.6 defines GS2 as a single update; the paper does not evaluate multi-step forecasting or larger grids.
  • ad hoc to paper ARMI treats any RM that reproduces the exact GS2 as a perfect inference, including degenerate solutions.
    Eq. (3) in Section 2.2; this definition drives the '82% exact rule-matrix match' headline while not requiring ground-truth RM recovery.

how reviews work

0 comments
Cite this review

Pith. "Pith review of AutomataGPT: Forecasting and Ruleset Inference for Two-Dimensional Cellular Automata." pith.science (2026). https://pith.science/paper/FZ3RXV2V

@misc{pith2026250617333,
  author       = {Pith},
  title        = {Pith review of: AutomataGPT: Forecasting and Ruleset Inference for Two-Dimensional Cellular Automata},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/FZ3RXV2V}},
  note         = {Machine review of arXiv:2506.17333}
}
read the original abstract

Cellular automata (CA) provide a minimal formalism for investigating how simple local interactions generate rich spatiotemporal behavior in domains as diverse as traffic flow, ecology, tissue morphogenesis and crystal growth. However, automatically discovering the local update rules for a given phenomenon and using them for quantitative prediction remains challenging. Here we present AutomataGPT, a decoder-only transformer pretrained on around 1 million simulated trajectories that span 100 distinct two-dimensional binary deterministic CA rules on toroidal grids. When evaluated on previously unseen rules drawn from the same CA family, AutomataGPT attains 98.5% perfect one-step forecasts and reconstructs the governing update rule with up to 96% functional (application) accuracy and 82% exact rule-matrix match. These results demonstrate that large-scale pretraining over wider regions of rule space yields substantial generalization in both the forward (state forecasting) and inverse (rule inference) problems, without hand-crafted priors. By showing that transformer models can faithfully infer and execute CA dynamics from data alone, our work lays the groundwork for abstracting real-world dynamical phenomena into data-efficient CA surrogates, opening avenues in biology, tissue engineering, physics and AI-driven scientific discovery.

Figures

Figures reproduced from arXiv: 2506.17333 by the authors.

Figure 1
Figure 1. Illustration of the process of computing one future global state (or ‘game state’) for a 2D binary deterministic r = 1 cellular automaton, based on an inputted initial 16 × 16 binary grid representing an initial global state/initial condition (IC) and 2D binary array representing local state transition rules, referred to here as a ‘rules matrix’ (RM). Local states are updated based on a 3 × 3 kernel (moving window) … view at source ↗
Figure 2
Figure 2. a) Illustration of the forward and inverse problems investigated in this study, with respect to the ground truth. The ground truth shows the standard relationship between the IC, RM, and GS2 for a given CA system sample: the rules defined by the RM are applied to the IC to compute GS2. Question marks represent the unknowns (GS2 or RM) that AutomataGPT is tasked with inferring for each problem. b) Dataset generation … view at source ↗
Figure 3
Figure 3. Accuracy increases significantly with more extensive training on rulesets, demonstrating AutomataGPT’s generalization from 2 to 100 rules. a) Bar chart illustrating the change in AutomataGPT’s accuracy on the forward problem averaged over the entire testing set (AGS2I) versus the number of RMs used for training (NRM). b) Bar chart depicting the affect of NRM on the fraction of perfect GS2 inferences across the testi… view at source ↗
Figures from the paper (3 more)
Figure 4
Figure 4. Figure 4: a) Bar chart illustrating the change in inverse problem inference accuracy averaged over the testing set (A) for two distinct accuracy metrics (Rules Matrix Inference Accuracy – ARMI – and Inferred Rules Matrix Application Accuracy – AIRMA) versus the number of RMs use…
Figure 5
Figure 5. Figure 5: Plots depicting the capabilities of varying versions of AutomataGPT to infer RMs (the inverse problem) for a testing set with samples generated using RMs not present in any training data. Specifically, model accuracy vs. RM difference averaged across the closest two RM…
Figure 6
Figure 6. Figure 6: Transformation of a graph representation of state transition in 2D binary CA to a 2D binary adjacency matrix. We refer to the latter mathematical object as a ‘rules matrix.’ As shown in [PITH_FULL_IMAGE:figures/full_fig_p012_6.png]

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

72 extracted references · 58 canonical work pages

  1. [1]

    Theory of cellular automata: A survey.Theoretical Computer Science, 334(1):3–33, April 2005

    Jarkko Kari. Theory of cellular automata: A survey.Theoretical Computer Science, 334(1):3–33, April 2005

    work page 2005

  2. [2]

    Computation in Cellular Automata: A Selected Review

    Melanie Mitchell. Computation in Cellular Automata: A Selected Review. In Non- Standard Computation, pages 95–140. John Wiley & Sons, Ltd, 1998. Section: 4 _eprint: https://onlinelibrary.wiley.com/doi/pdf/10.1002/3527602968.ch4

    work page doi:10.1002/3527602968.ch4 1998

  3. [3]

    Cellular automata model for Urban Road traffic flow Considering Internet of Vehicles and emergency vehicles.Journal of Computational Science, 47:101221, November 2020

    Han-Tao Zhao, Xin Zhao, Lu Jian-cheng, and Liu-yan Xin. Cellular automata model for Urban Road traffic flow Considering Internet of Vehicles and emergency vehicles.Journal of Computational Science, 47:101221, November 2020

    work page 2020

  4. [4]

    Hernández Encinas, L

    A. Hernández Encinas, L. Hernández Encinas, S. Hoya White, A. Martín del Rey, and G. Ro- dríguez Sánchez. Simulation of forest fire fronts using cellular automata.Advances in Engineering Software, 38(6):372–378, June 2007

    work page 2007

  5. [5]

    Kier, Paul G

    Lemont B. Kier, Paul G. Seybold, and Chao-Kun Cheng.Modeling Chemical Systems Using Cellular Automata. Springer Science & Business Media, October 2005. Google-Books-ID: eHVAQKbdxEcC

    work page 2005

  6. [6]

    Modeling and simulation of highway traffic using a cellular automaton approach

    Ding Ding. Modeling and simulation of highway traffic using a cellular automaton approach. U.U.D.M. Project Report 2011:25, Department of Mathematics, Uppsala University, Uppsala, Sweden, December 2011

    work page 2011

  7. [7]

    Wolfram Media, 2002

    Stephen Wolfram.A new kind of science. Wolfram Media, 2002

    work page 2002

  8. [8]

    Unpredictability and Computational Irreducibility

    Hervé Zwirn and Jean-Paul Delahaye. Unpredictability and Computational Irreducibility. In Hector Zenil, editor,Irreducibility and Computational Equivalence: 10 Years After Wolfram’s A New Kind of Science, pages 273–295. Springer, Berlin, Heidelberg, 2013

    work page 2013

Show all 72 references
  1. [9]

    Computational Irreducibility and Computational Analogy.Complex Systems, 24(2):149–174, June 2015

    UFR de Physique (LIED, Université Paris 7) CMLA (ENS Cachan, France) and IHPST (CNRS, France) and Hervé Zwirn. Computational Irreducibility and Computational Analogy.Complex Systems, 24(2):149–174, June 2015

    2015

  2. [10]

    Springer and Garrett T

    Jacob M. Springer and Garrett T. Kenyon. It’s Hard for Neural Networks To Learn the Game of Life, September 2020. arXiv:2009.01398 [cs]

    2020 arXiv

  3. [11]

    Mathematical Games.Scientific American, 223(4):120–123, 1970

    Martin Gardner. Mathematical Games.Scientific American, 223(4):120–123, 1970. Publisher: Scientific American, a division of Nature America, Inc

    1970

  4. [12]

    Growing Neural Cellular Automata

    Alexander Mordvintsev, Ettore Randazzo, Eyvind Niklasson, and Michael Levin. Growing Neural Cellular Automata. Distill, 5(2):e23, February 2020

    2020

  5. [13]

    Richardson, Tibor Antal, Richard A

    Alex D. Richardson, Tibor Antal, Richard A. Blythe, and Linus J. Schumacher. Learning spatio-temporal patterns with Neural Cellular Automata, April 2024. arXiv:2310.14809

    2024 arXiv

  6. [14]

    Attention-based Neural Cellular Automata

    Mattie Tesfaldet, Derek Nowrouzezahrai, and Chris Pal. Attention-based Neural Cellular Automata. Advances in Neural Information Processing Systems, 35:8174–8186, December 2022

    2022

  7. [15]

    What’s Really Going On in Machine Learning? Some Minimal Models, August 2024

    Stephen Wolfram. What’s Really Going On in Machine Learning? Some Minimal Models, August 2024

    2024

  8. [16]

    Attention is All you Need

    Ashish Vaswani, Noam Shazeer, Niki Parmar, Jakob Uszkoreit, Llion Jones, Aidan N Gomez, Ł ukasz Kaiser, and Illia Polosukhin. Attention is All you Need. InAdvances in Neural Information Processing Systems, volume 30. Curran Associates, Inc., 2017

    2017

  9. [17]

    Brown, Benjamin Chess, Rewon Child, Scott Gray, Alec Radford, Jeffrey Wu, and Dario Amodei

    Jared Kaplan, Sam McCandlish, Tom Henighan, Tom B. Brown, Benjamin Chess, Rewon Child, Scott Gray, Alec Radford, Jeffrey Wu, and Dario Amodei. Scaling Laws for Neural Language Models, January

  10. [18]

    METAGENE-1: Metagenomic Foundation Model for Pandemic Monitoring, January

    Ollie Liu, Sami Jaghouar, Johannes Hagemann, Shangshang Wang, Jason Wiemels, Jeff Kaufman, and Willie Neiswanger. METAGENE-1: Metagenomic Foundation Model for Pandemic Monitoring, January

  11. [19]

    Scaling transformer neural networks for skillful and reliable medium-range weather forecasting, December 2023

    Tung Nguyen, Rohan Shah, Hritik Bansal, Troy Arcomano, Sandeep Madireddy, Romit Maulik, Veerab- hadra Kotamarthi, Ian Foster, and Aditya Grover. Scaling transformer neural networks for skillful and reliable medium-range weather forecasting, December 2023. arXiv:2312.03876 [physics]

    2023 arXiv

  12. [20]

    A transformer neural network for predicting near-surface temperature

    Emy Alerskans, Joachim Nyborg, Morten Birk, and Eigil Kaas. A transformer neural network for predicting near-surface temperature. Meteorological Applications, 29(5):e2098, 2022. _eprint: https://onlinelibrary.wiley.com/doi/pdf/10.1002/met.2098

    2022 doi

  13. [21]

    STC-ViT:SpatioTemporalContinuousVisionTransformer for Weather Forecasting, May 2024

    HiraSaleem, FloraSalim, andCormacPurcell. STC-ViT:SpatioTemporalContinuousVisionTransformer for Weather Forecasting, May 2024. arXiv:2402.17966 [cs]

    2024

  14. [22]

    Bernard Moussad, Rahmatullah Roche, and Debswapna Bhattacharya. The transformative power of transformers in protein structure prediction.Proceedings of the National Academy of Sciences of the United States of America, 120(32):e2303499120, August 2023

    2023

  15. [23]

    Language Agents with Reinforcement Learning for Strategic Play in the Werewolf Game, February 2024

    Zelai Xu, Chao Yu, Fei Fang, Yu Wang, and Yi Wu. Language Agents with Reinforcement Learning for Strategic Play in the Werewolf Game, February 2024. arXiv:2310.18940 [cs] version: 3

    2024 arXiv

  16. [24]

    A Survey on Large Language Model-Based Game Agents, April 2024

    Sihao Hu, Tiansheng Huang, Fatih Ilhan, Selim Tekin, Gaowen Liu, Ramana Kompella, and Ling Liu. A Survey on Large Language Model-Based Game Agents, April 2024. arXiv:2404.02039 [cs] version: 1

    2024 arXiv

  17. [25]

    Large Language Models as Agents in Two-Player Games, February

    Yang Liu, Peng Sun, and Hang Li. Large Language Models as Agents in Two-Player Games, February

  18. [26]

    Overview of the Transformer- based Models for NLP Tasks

    Anthony Gillioz, Jacky Casas, Elena Mugellini, and Omar Abou Khaled. Overview of the Transformer- based Models for NLP Tasks. In2020 15th Conference on Computer Science and Information Systems (FedCSIS), pages 179–183, September 2020

    2020

  19. [27]

    Transformers in Speech Processing: A Survey, March 2023

    Siddique Latif, Aun Zaidi, Heriberto Cuayahuitl, Fahad Shamshad, Moazzam Shoukat, and Junaid Qadir. Transformers in Speech Processing: A Survey, March 2023. arXiv:2303.11607 [cs, eess]

    2023 arXiv

  20. [28]

    N. S. Mamatov, N. A. Niyozmatova, Sh. Sh. Abdullaev, A. N. Samijonov, and K. K. Erejepov. Speech Recognition Based On Transformer Neural Networks. In2021 International Conference on Information Science and Communications Technologies (ICISCT), pages 1–5, November 2021

    2021

  21. [29]

    Neural speech synthesis with transformer network

    Naihan Li, Shujie Liu, Yanqing Liu, Sheng Zhao, and Ming Liu. Neural speech synthesis with transformer network. InProceedings of the Thirty-Third AAAI Conference on Artificial Intelligence and Thirty-First Innovative Applications of Artificial Intelligence Conference and Ninth...

    2019

  22. [30]

    Vision Transformers: A Review of Architecture, Applications, and Future Directions

    Abdelhafid Berroukham, Khalid Housni, and Mohammed Lahraichi. Vision Transformers: A Review of Architecture, Applications, and Future Directions. In2023 7th IEEE Congress on Information Science and Technology (CiSt), pages 205–210, December 2023. ISSN: 2327-1884

    2023

  23. [31]

    Markus J. Buehler. FieldPerceiver: Domain agnostic transformer model to predict multiscale physical fields and nonlinear material properties through neural ologs.Materials Today, 57:9–25, July 2022

    2022

  24. [32]

    Markus J. Buehler. MechGPT, a Language-Based Strategy for Mechanics and Materials Modeling That Connects Knowledge Across Scales, Disciplines, and Modalities.Applied Mechanics Reviews, 76(021001), January 2024

    2024

  25. [33]

    Markus J. Buehler. PRefLexOR: preference-based recursive language modeling for exploratory optimiza- tion of reasoning and agentic thinking.npj Artificial Intelligence, 1:4, 2025. Published: 14 May 2025; Accepted: 22 March 2025; Received: 01 November 2024

    2025

  26. [34]

    Markus J. Buehler. In situ graph reasoning and knowledge expansion using Graph-PRefLexOR.Advanced Intelligent Discovery, n/a(n/a):202500006

  27. [35]

    Berkovich and Markus J

    Jaime A. Berkovich and Markus J. Buehler. LifeGPT: Topology-Agnostic Generative Pretrained Transformer Model for Cellular Automata, October 2024. arXiv:2409.12182 version: 2

    2024 arXiv

  28. [36]

    Learning Elementary Cellular Automata with Transformers

    Mikhail Burtsev. Learning Elementary Cellular Automata with Transformers. InThe 4th Workshop on Mathematical Reasoning and AI at NeurIPS’24, October 2024

    2024

  29. [37]

    World Models.Zenodo, March 2018

    David Ha and Jürgen Schmidhuber. World Models.Zenodo, March 2018. 17 A preprint - August 13, 2025

    2018

  30. [38]

    Stanley, Phillip Isola, and David Ha

    Akarsh Kumar, Chris Lu, Louis Kirsch, Yujin Tang, Kenneth O. Stanley, Phillip Isola, and David Ha. Automating the Search for Artificial Life with Foundation Models, December 2024. arXiv:2412.17799 [cs]

    2024 arXiv

  31. [39]

    Reynolds

    Craig W. Reynolds. Flocks, herds and schools: A distributed behavioral model. InProceedings of the 14th annual conference on Computer graphics and interactive techniques, SIGGRAPH ’87, pages 25–34, New York, NY, USA, August 1987. Association for Computing Machinery

    1987

  32. [40]

    How Particle Life emerges from simplicity, December 2022

    Tom Mohr. How Particle Life emerges from simplicity, December 2022

    2022

  33. [41]

    Clusters

    Jeffrey Ventrella. Clusters

  34. [42]

    Life Worth Mentioning: Complexity in Life-Like Cellular Automata

    Eric Peña and Hiroki Sayama. Life Worth Mentioning: Complexity in Life-Like Cellular Automata. Artificial Life, 27(2):105–112, May 2021

    2021

  35. [43]

    Cellular Automata rules lexicon - Life

    Mirek Wojtowicz. Cellular Automata rules lexicon - Life

  36. [44]

    Lenia: Biology of Artificial Life.Complex Systems, 28(3):251–286, October 2019

    Hong Kong and Bert Wang-Chak Chan. Lenia: Biology of Artificial Life.Complex Systems, 28(3):251–286, October 2019

    2019

  37. [45]

    Toward Artificial Open-Ended Evolution within Lenia using Quality-Diversity, June 2024

    Maxence Faldor and Antoine Cully. Toward Artificial Open-Ended Evolution within Lenia using Quality-Diversity, June 2024. arXiv:2406.04235 [cs]

    2024 arXiv

  38. [46]

    Nature’s hierarchical materials.Progress in Materials Science, 52(8):1263–1334, November 2007

    Peter Fratzl and Richard Weinkamer. Nature’s hierarchical materials.Progress in Materials Science, 52(8):1263–1334, November 2007

    2007

  39. [47]

    From single cells to tissue self-organization

    Aline Xavier da Silveira dos Santos and Prisca Liberali. From single cells to tissue self-organization. The FEBS Journal , 286(8):1495–1513, 2019. _eprint: https://onlinelibrary.wiley.com/doi/pdf/10.1111/febs.14694

    2019 doi

  40. [48]

    Olayioye, Markus J

    Haiqian Yang, Florian Meyer, Shaoxun Huang, Liu Yang, Cristiana Lungu, Monilola A. Olayioye, Markus J. Buehler, and Ming Guo. Learning Collective Cell Migratory Dynamics from a Static Snapshot with Graph Neural Networks.PRX Life, 2(4):043010, November 2024. Publisher: American...

    2024

  41. [49]

    Application of Artificial Intelligence in Tissue Engineering.Tissue Engineering Part B: Reviews, April 2024

    Reza Bagherpour, Ghasem Bagherpour, and Parvin Mohammadi. Application of Artificial Intelligence in Tissue Engineering.Tissue Engineering Part B: Reviews, April 2024. Publisher: Mary Ann Liebert, Inc., publishers

    2024

  42. [50]

    PhD thesis, University of Dayton, 2010

    Mary Elizabeth Kundrat.A Comprehensive Series for Predicting Bone Dynamics: Forecasting Osseous Tissue Formation using the Molecular Structure of a Biomaterial. PhD thesis, University of Dayton, 2010

    2010

  43. [51]

    Ko, Reza Mousavi, and Daniel Lobo

    Jason M. Ko, Reza Mousavi, and Daniel Lobo. Computational Systems Biology of Morphogenesis. In Sonia Cortassa and Miguel A. Aon, editors,Computational Systems Biology in Medicine and Biotechnology: Methods and Protocols, pages 343–365. Springer US, New York, NY, 2022

    2022

  44. [52]

    Fletcher and James M

    Alexander G. Fletcher and James M. Osborne. Seven challenges in the multiscale model- ing of multicellular tissues. WIREs Mechanisms of Disease , 14(1):e1527, 2022. _eprint: https://onlinelibrary.wiley.com/doi/pdf/10.1002/wsbm.1527

    2022 doi

  45. [53]

    Computer modeling in developmental biology: growing today, essential tomorrow

    James Sharpe. Computer modeling in developmental biology: growing today, essential tomorrow. Development, 144(23):4214–4225, December 2017

    2017

  46. [54]

    Kivelson

    Sophia Kivelson and Steven A. Kivelson. Defining emergence in physics.npj Quantum Materials, 1(1):1–2, November 2016. Publisher: Nature Publishing Group

    2016

  47. [55]

    Emergent properties from organisms to ecosystems: towards a realistic approach

    Jean-François Ponge. Emergent properties from organisms to ecosystems: towards a realistic approach. Biological Reviews of the Cambridge Philosophical Society, 80(3):403–411, August 2005

    2005

  48. [56]

    How a life-like system emerges from a simplistic particle motion law.Scientific Reports, 6(1):37969, November 2016

    Thomas Schmickl, Martin Stefanec, and Karl Crailsheim. How a life-like system emerges from a simplistic particle motion law.Scientific Reports, 6(1):37969, November 2016. Publisher: Nature Publishing Group

    2016

  49. [57]

    Systems biology, emergence and antireductionism.Saudi Journal of Biological Sciences, 23(5):584–591, September 2016

    Srdjan Kesić. Systems biology, emergence and antireductionism.Saudi Journal of Biological Sciences, 23(5):584–591, September 2016

    2016

  50. [58]

    Weisstein

    Eric W. Weisstein. Moore Neighborhood. Publisher: Wolfram Research, Inc

  51. [59]

    R. W. Hamming. Error Detecting and Error Correcting Codes.Bell System Technical Journal, 29(2):147– 160, 1950. _eprint: https://onlinelibrary.wiley.com/doi/pdf/10.1002/j.1538-7305.1950.tb00463.x

    1950

  52. [60]

    The Distribution of the Flora in the Alpine Zone.New Phytologist, 11(2):37–50, 1912

    Paul Jaccard. The Distribution of the Flora in the Alpine Zone.New Phytologist, 11(2):37–50, 1912. _eprint: https://onlinelibrary.wiley.com/doi/pdf/10.1111/j.1469-8137.1912.tb05611.x. 18 A preprint - August 13, 2025

    1912

  53. [61]

    T. T. Tanimoto. An Elementary Mathematical Theory of Classification and Prediction. Technical Report PB167360, International Business Machines Corp., New York., 1958. Num Pages: 12

    1958

  54. [62]

    On the Jensen–Shannon Symmetrization of Distances Relying on Abstract Means

    Frank Nielsen. On the Jensen–Shannon Symmetrization of Distances Relying on Abstract Means. Entropy, 21(5):485, May 2019

    2019

  55. [63]

    Modelling cellular automata with partial differential equations

    Stephen Omohundro. Modelling cellular automata with partial differential equations. Physica D: Nonlinear Phenomena, 10(1):128–134, January 1984

    1984

  56. [64]

    Xin-She Yang and Y. Young. Cellular Automata, PDEs, and Pattern Formation, March 2010. arXiv:1003.1983 [nlin]

    2010 arXiv

  57. [65]

    OpenAI, Josh Achiam, Steven Adler, Sandhini Agarwal, Lama Ahmad, Ilge Akkaya, Florencia Leoni Ale- man, Diogo Almeida, Janko Altenschmidt, Sam Altman, Shyamal Anadkat, Red Avila, Igor Babuschkin, Suchir Balaji, Valerie Balcom, Paul Baltescu, Haiming Bao, Mohammad Bavarian, Jef...

    2024 arXiv

  58. [66]

    lucidrains/x-transformers, October 2024

    Phil Wang. lucidrains/x-transformers, October 2024. original-date: 2020-10-24T22:13:25Z

    2024

  59. [67]

    Harris, K

    Charles R. Harris, K. Jarrod Millman, Stéfan J. van der Walt, Ralf Gommers, Pauli Virtanen, David Cournapeau, Eric Wieser, Julian Taylor, Sebastian Berg, Nathaniel J. Smith, Robert Kern, Matti Picus, Stephan Hoyer, Marten H. van Kerkwijk, Matthew Brett, Allan Haldane, Jaime Fe...

    2025

  60. [68]

    Weisstein

    Eric W. Weisstein. Adjacency Matrix. Publisher: Wolfram Research, Inc

  61. [69]

    HuggingFace’s Tokenizers, April 2023

    Anthony Moi and Nicolas Patry. HuggingFace’s Tokenizers, April 2023. original-date: 2019-11- 01T17:52:20Z. 20

    2023

  62. [2020]

    16 A preprint - August 13, 2025

    arXiv:2001.08361 [cs]. 16 A preprint - August 13, 2025

    2001 arXiv

  63. [2024]

    arXiv:2402.08078 [cs]

    arXiv

  64. [2025]

    arXiv:2501.02045 [q-bio]

    arXiv

Pith tools

Reviewed August 6, 2026 · model on record in the stance chip above.