Pith. sign in

REVIEW 3 major objections 6 minor 36 references

Learning Elementary Cellular Automata with Transformers

T0 review · 3 major / 6 minor · reviewed 2026-08-12 · deepseek-v4-flash

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

Pith's one-line read This paper claims that a Transformer trained on random Elementary Cellular Automaton orbits abstracts the underlying Boolean rule and that the ability to plan multiple steps ahead is controlled by the number of layers.

desk verdict Clean toy-domain task comparison, but the depth-scaling result is not supported by the reported single-run experiments. read the letter →

arxiv 2412.01417 v1 pith:K5W2QN33 submitted 2024-12-02 cs.NE cs.AIcs.FL

Mikhail Burtsev This is my paper · ORCID
classification cs.NEcs.AIcs.FL MSC 37B1568T07
keywords elementarycellularautomatatransformersruleabstractionmulti-stepplanningdepthscalingautoregressivegenerationchain-of-thoughtBooleanfunctions
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

This paper asks whether a Transformer can infer the local rule of an Elementary Cellular Automaton from raw orbit data and then use that rule to plan ahead. It claims that a Transformer trained on random orbits with held-out rules generalizes across Boolean functions of fixed arity, so the network is abstracting a rule class rather than memorizing examples. The paper also claims that adding future-state or rule prediction to the training loss helps the model form better internal rule representations, which improves longer-horizon prediction and autoregressive generation. Finally, it reports a depth-scaling effect: predicting each additional step into the future requires more Transformer layers, suggesting that fixed-depth models are the bottleneck for multi-step planning.

What carries the argument

The central object is the Elementary Cellular Automaton (ECA), a one-dimensional lattice of binary cells updated synchronously by a local rule $\rho: \{0,1\}^{2r+1}\to\{0,1\}$; with $W=20$ and $r=2$ there are $2^{32}\approx4.3\times10^9$ possible rules. The machinery is a Transformer encoder with masked tokens standing for unknown future bits or the unknown rule, trained under four task variants: O-S (orbit to next state), O-O (orbit to several future states), O-SR (orbit to next state plus rule), and RO-S (given orbit and rule, predict state). The rule-recovery estimate $T_{\min}=2^{2r+1}(\ln 2^{2r+1}+\gamma)/W\approx6.47$ from the coupon collector's problem tells how many orbit steps are needed, and the depth sweep on the O-O task is what carries the claim that each planning step needs more layers.

What would settle it

Repeat the O-O depth experiment with multiple random seeds and learning-rate schedules at layer counts 2, 4, 7, and 10, and record accuracy distributions. If the apparent thresholds shift by more than a layer or disappear when seeds vary, the claim that each planning step requires additional layers would not be supported; alternatively, a wider 4-layer model that predicts step 3 as well as the 7-layer model would falsify the depth-specific conclusion.

Watch

Extended reading notes

Core claim

On the paper's own terms, the central discovery is that a Transformer encoder with full self-attention can learn to predict the next state of an Elementary Cellular Automaton from a ten-step orbit and, more strongly, can do so for local rules never seen in training, meaning it has abstracted the underlying Boolean function rather than memorized a specific rule. When asked to jump several steps ahead without intermediate context, accuracy falls sharply, from about 0.96 for next-state prediction to below 0.75 for two or three steps ahead. Training with an orbit-to-orbit objective or with simultaneous rule prediction both improve planning, with the rule-prediction variant (O-SR) generalizing better at longer horizons, while explicitly supplying the rule (RO-S) does not rescue multi-step jumps. A depth experiment shows that four-step planning requires about ten layers, leading the author to conclude that sequential computation in Transformers is bounded by depth and that recurrence or adaptive computation time are promising remedies.

Load-bearing premise

The depth-scaling conclusion assumes that the exact layer counts at which accuracy first rises are real signals rather than random variation between training runs; no repeated runs or error bars are reported.

Editorial extensions

If this is right

  • Transformers trained on random ECA orbits can predict the next state of unseen local rules with high per-bit accuracy, so rule abstraction from data alone is possible for Boolean functions of fixed arity.
  • Adding future-state or rule prediction to the loss improves both next-state accuracy and autoregressive rollout quality; the rule-prediction variant generalizes best at look-ahead steps.
  • Explicitly giving the rule (RO-S) does not fix multi-step planning beyond one or two steps, indicating the difficulty is in propagating intermediate state information, not in knowing the rule.
  • Autoregressive generation outperforms direct multi-step look-ahead, so models are better at short state-by-state prediction than at jumping ahead without intermediate context.
  • The number of layers required for accurate O-O prediction grows with planning horizon (2 layers for step 1, 4 for step 2, 7 for step 3, 10 for step 4), suggesting depth is the key architectural resource for sequential computation.

Reading between the lines

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

  • If the depth-to-horizon scaling holds beyond this toy setup, fixed-depth models should systematically fail on planning problems whose solution requires more sequential rule applications than the network has layers; that is a testable prediction for arithmetic or multi-hop reasoning benchmarks.
  • The O-SR result suggests rule-identification losses could act as a regularizer that compresses the hidden state into a causal rule; one could test this by probing the model's hidden vectors for the rule and correlating probe accuracy with planning accuracy.
  • The coupon-collector estimate implies that the number of orbit steps needed to identify a rule grows exponentially in neighborhood radius; scaling ECAs to larger $r$ should make rule abstraction and planning degrade in a predictable way.
  • Because RO-S fails at multi-step jumps despite having the rule, the bottleneck is not rule knowledge but the propagation of intermediate states; architectures with explicit memory or recurrent state updates should be tested on the same task.
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

3 major / 6 minor

Summary. The manuscript studies whether Transformer encoders can learn the local rules of Elementary Cellular Automata (ECA) from observed orbits and use them for multi-step planning. Four training objectives are compared: next-state prediction (O-S), multi-step prediction (O-O), joint state and rule prediction (O-SR), and state prediction given the rule (RO-S). The authors report that the model reaches about 0.96 per-bit accuracy for next-state prediction on held-out rules, that look-ahead accuracy drops for k≥2, that O-SR improves longer look-ahead over O-O, that autoregressive rollouts are more accurate than direct look-ahead, and that deeper models are needed to predict more steps ahead, with step 1 needing 2 layers, step 2 needing 4, step 3 needing 7, and step 4 needing 10 layers.

Significance. If the claims hold, the paper offers a clean testbed for studying how Transformers abstract discrete rules and a concrete demonstration that multi-step planning scales with depth, with implications for training objectives (longer horizons, rule prediction) and architecture (recurrence, adaptive computation) in LLMs. The setup is well chosen: test rules are disjoint from training rules, so next-state accuracy is evidence of generalization rather than memorization, and the authors provide code and data (GitHub link). The main limitation is that the experimental evidence is currently too thin—single runs, no error bars, and an undefined accuracy threshold for the depth scaling—so the central claims are plausible but not yet established.

major comments (3)
  1. [3 Results and Discussion (Figure 3)] The depth-scaling conclusion that "each additional planning step requires more computational layers" is not supported as presented. The thresholds (step 2 at 4 layers, step 3 at 7, step 4 at 10 layers) are read from single runs with no stated accuracy criterion for "predicting a step," and no error bars or seeds are reported. Additionally, because d_model=512 and 8 heads are held fixed while the layer count changes from 1 to 11, the experiment conflates depth with parameter count and training dynamics; a shallow-wide control is not reported. This point is load-bearing because the abstract and conclusions use this result to motivate recurrence and adaptive computation time.
  2. [3 Results and Discussion (Figures 2 and 3)] All accuracy figures appear to come from a single training run per condition; no random seeds, error bars, or significance tests are reported. Consequently, comparative statements such as "the O-SR model outperformed the O-O model" (0.85 vs 0.75 for k=2,3) and the "near-perfect accuracy" of RO-S for k=0,1 are not quantitatively supported. Please report at least a small number of seeds with mean and standard deviation, or otherwise justify the stability of the observed differences.
  3. [2 Methods] The training configuration is underspecified: the paper does not state the optimizer, learning rate and schedule, batch size, number of training steps, or compute budget, and for the O-SR objective it does not specify how the state and rule losses are weighted. Without these details the experiments are not reproducible, and the claim that rule prediction in the loss "enhances the ability to form internal representations" cannot be separated from arbitrary training choices. Please add full training details and, ideally, a sensitivity check on the loss weight.
minor comments (6)
  1. [3 Results and Discussion] The text refers to "Figure 3A," but the figure has no panel A; the caption reads "Figure 3: Adding layers improves prediction of ECA orbit." Please correct the cross-reference.
  2. [2 Methods] The coupon-collector estimate Tmin treats each cell-time observation as an independent draw, but adjacent cells in the same time step have overlapping neighborhoods, so the draws are correlated. The agreement with Figure 2A is therefore heuristic; please state this caveat or provide a more careful estimate.
  3. [2 Methods / Figure 1] The definition of "look-ahead steps" could be clearer: Figure 2B includes k=0 while the text defines k∈{1,2,3}; please state explicitly what k=0 denotes (presumably next-state prediction) and how "per-bit accuracy averaged over free runs" is computed.
  4. [3 Results and Discussion] The O-SR model's rule prediction accuracy is never reported, so the claim that it forms an internal representation of the rule is only indirectly evidenced. Reporting the rule-prediction accuracy (and how rule outputs are decoded) would strengthen the mechanistic interpretation.
  5. [3 Results and Discussion] In Figure 2D, the comparison between autoregressive (AR) and look-ahead (LA) predictions needs a precise definition of both modes; in particular, for AR, specify whether the model consumes its own previous output tokens as context and whether teacher forcing is used at any stage.
  6. [4 Conclusions] The phrase "we also confirmed that the model's depth plays a crucial role" overstates the evidence given the single-run, threshold-free analysis in Figure 3; please soften the wording to match the strength of the results.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the paper is an empirical training and evaluation study whose results are measured on held-out ECA rules, with no fitted input renamed as a prediction.

full rationale

The paper's central claims are empirical measurements, not derivations from fitted constants. The test set uses rules 'exclusive and not present in the training set', so the reported generalization across Boolean functions is measured on held-out data rather than forced by construction. The auxiliary rule-prediction loss in O-SR is an additional training objective, and its effect on planning accuracy is measured, not assumed. The Tmin coupon-collector estimate is an external calculation that is subsequently compared with the observed accuracy plateau in Fig. 2A; no parameter is fitted to the plateau and then reused to produce the plateau. The depth-scaling conclusion in Fig. 3 is an observed accuracy-versus-layers relationship; while the absence of error bars and an explicit accuracy threshold is a legitimate correctness concern, that is a statistical inference issue, not circularity. The paper contains no load-bearing self-citation: the only author-linked artifact is a public repository for dataset and code, and the cited prior work on cellular automata, transformers, and reasoning is external. No step in the derivation reduces, by the paper's own equations or by citation, to a restatement of its inputs. Therefore the appropriate finding is no significant circularity.

Assumptions & free parameters 2 free parameters · 3 assumptions · 0 invented entities

The paper introduces no new particles, forces, or theoretical entities. Its central claims rest on the training/evaluation setup, the rule-disjoint split, the Tmin approximation, and the accuracy metric, none of which is fitted to produce the claimed results.

free parameters (2)
  • Input orbit length k = 10 time steps
    Chosen because accuracy plateaus after 8 steps (Fig. 2A); all planning and depth experiments use 10 input states, so the reported accuracies and layer thresholds are specific to this fixed context length.
  • Minimum layer count thresholds for planning steps = 2, 4, 7, 10 layers for steps 1, 2, 3, 4
    Read from Figure 3 without confidence intervals; these thresholds are the operational definition of the depth-scaling claim.
assumptions (3)
  • domain assumption Training and test sets use disjoint random local rules, so good test performance indicates generalization to unseen rules.
    The claim of rule abstraction rests on this split; the paper states test rules are exclusive and not present in the training set.
  • standard math The coupon-collector estimate Tmin = 2^5 (ln 2^5 + gamma) / 20 approximates the orbit length needed to identify a rule.
    This is used to justify 10 input steps; it assumes each observed local neighborhood is drawn roughly uniformly at random and independent across time steps, which is not guaranteed for correlated ECA orbits.
  • domain assumption Per-bit accuracy computed on free runs is a valid proxy for rule learning and planning ability.
    The entire evaluation uses average per-bit accuracy; small per-bit differences are interpreted as meaningful gains, yet no confidence intervals accompany the numbers.

how reviews work

0 comments
Cite this review

Pith. "Pith review of Learning Elementary Cellular Automata with Transformers." pith.science (2026). https://pith.science/paper/K5W2QN33

@misc{pith2026241201417,
  author       = {Pith},
  title        = {Pith review of: Learning Elementary Cellular Automata with Transformers},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/K5W2QN33}},
  note         = {Machine review of arXiv:2412.01417}
}
read the original abstract

Large Language Models demonstrate remarkable mathematical capabilities but at the same time struggle with abstract reasoning and planning. In this study, we explore whether Transformers can learn to abstract and generalize the rules governing Elementary Cellular Automata. By training Transformers on state sequences generated with random initial conditions and local rules, we show that they can generalize across different Boolean functions of fixed arity, effectively abstracting the underlying rules. While the models achieve high accuracy in next-state prediction, their performance declines sharply in multi-step planning tasks without intermediate context. Our analysis reveals that including future states or rule prediction in the training loss enhances the models' ability to form internal representations of the rules, leading to improved performance in longer planning horizons and autoregressive generation. Furthermore, we confirm that increasing the model's depth plays a crucial role in extended sequential computations required for complex reasoning tasks. This highlights the potential to improve LLM with inclusion of longer horizons in loss function, as well as incorporating recurrence and adaptive computation time for dynamic control of model depth.

Figures

Figures reproduced from arXiv: 2412.01417 by the authors.

Figure 1
Figure 1. Learning Elementary Cellular Automata (ECA) with Transformers. A. Examples of training samples. Orbit of ECA is a sequence of binary strings of size W = 20. First k = 10 states marked by red rectangle encode Transformer input. B. Given a part of the orbit Transformer with full attention learns to predict the next state (O-S), the next few steps (O-O), the next state and a rule (O-SR), or predict the next state given… view at source ↗
Figure 2
Figure 2. Transformer learns to predict the next state of ECA but struggles to plan ahead. A. Accuracy of the next state prediction for ECA orbit with size 20 generated by Boolean function of 5 arguments for different input state lengths. B. Planning accuracy for different training settings (see the main text for details). C. Accuracy for autoregressive generation of ECA orbit. D. Comparison of autoregressive (AR) and look ah… view at source ↗
Figure 3
Figure 3. Adding layers improves prediction of ECA orbit. Accuracy of O-O training for different number of layers. Additionally, we evaluated the performance of the four models — trained under the O-S, O-O, O-SR, and RO-S tasks — when used to generate continuations of the input orbit O10 up to O20 by predicting each subsequent state autoregressively (see Figure 2C). As expected, the success of these models in this task correl… view at source ↗

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

36 extracted references · 19 canonical work pages

  1. [1]

    Learning to reason with llms

    OpenAI. Learning to reason with llms. https://openai.com/index/ learning-to-reason-with-llms/ , 2024. Accessed: 2024-09-23

    2024

  2. [2]

    Faith and fate: Limits of transformers on compositionality

    Nouha Dziri, Ximing Lu, Melanie Sclar, Xiang Lorraine Li, Liwei Jiang, Bill Yuchen Lin, Sean Welleck, Peter West, Chandra Bhagavatula, Ronan Le Bras, et al. Faith and fate: Limits of transformers on compositionality. Advances in Neural Information Processing Systems, 36, 2024

    2024

  3. [3]

    A & b== b & a: Triggering logical reasoning failures in large language models

    Yuxuan Wan, Wenxuan Wang, Yiliu Yang, Youliang Yuan, Jen-tse Huang, Pinjia He, Wenxiang Jiao, and Michael R Lyu. A & b== b & a: Triggering logical reasoning failures in large language models. arXiv preprint arXiv:2401.00757, 2024

    arXiv 2024

  4. [4]

    Conditional and modal reasoning in large language models

    Wesley H Holliday and Matthew Mandelkern. Conditional and modal reasoning in large language models. arXiv preprint arXiv:2401.17169, 2024

    arXiv 2024

  5. [5]

    Inductive learning of logical theories with llms: A complexity-graded analysis

    João Pedro Gandarela, Danilo S Carvalho, and André Freitas. Inductive learning of logical theories with llms: A complexity-graded analysis. arXiv preprint arXiv:2408.16779, 2024

    arXiv 2024

  6. [6]

    Liar, liar, logical mire: A benchmark for suppositional reasoning in large language models

    Philipp Mondorf and Barbara Plank. Liar, liar, logical mire: A benchmark for suppositional reasoning in large language models. arXiv preprint arXiv:2406.12546, 2024

    arXiv 2024

  7. [7]

    Llms still can’t plan; can lrms? a preliminary evaluation of openai’s o1 on planbench, 2024

    Karthik Valmeekam, Kaya Stechly, and Subbarao Kambhampati. Llms still can’t plan; can lrms? a preliminary evaluation of openai’s o1 on planbench, 2024

    work page 2024

  8. [8]

    Attention is All you Need

    Ashish Vaswani, Noam Shazeer, Niki Parmar, Jakob Uszkoreit, Llion Jones, Aidan N Gomez, Łukasz Kaiser, and Illia Polosukhin. Attention is All you Need. In Advances in neural information processing systems, pages 5998–6008, 2017. URL http://papers.nips.cc/ paper/7181-attention-is-all-you-need . 5

    work page 2017

Show all 36 references
  1. [9]

    Approximations by superpositions of a sigmoidal function

    George Cybenko. Approximations by superpositions of a sigmoidal function. Mathematics of Control, Signals and Systems, 2:183–192, 1989

    1989

  2. [10]

    Multilayer feedforward networks are universal approximators

    Kurt Hornik, Maxwell Stinchcombe, and Halbert White. Multilayer feedforward networks are universal approximators. Neural networks, 2(5):359–366, 1989

    1989

  3. [11]

    Are transformers universal approximators of sequence-to-sequence functions? arXiv preprint arXiv:1912.10077, 2019

    Chulhee Yun, Srinadh Bhojanapalli, Ankit Singh Rawat, Sashank J Reddi, and Sanjiv Kumar. Are transformers universal approximators of sequence-to-sequence functions? arXiv preprint arXiv:1912.10077, 2019

    1912 arXiv

  4. [12]

    Representational strengths and limitations of transformers

    Clayton Sanford, Daniel J Hsu, and Matus Telgarsky. Representational strengths and limitations of transformers. Advances in Neural Information Processing Systems, 36, 2024

    2024

  5. [13]

    Uni- versal transformers

    Mostafa Dehghani, Stephan Gouws, Oriol Vinyals, Jakob Uszkoreit, and Lukasz Kaiser. Uni- versal transformers. In International Conference on Learning Representations, 2019. URL https://openreview.net/forum?id=HyzdRiR9Y7

    2019

  6. [14]

    On the computational power of transformers and its implications in sequence modeling

    Satwik Bhattamishra, Arkil Patel, and Navin Goyal. On the computational power of transformers and its implications in sequence modeling. arXiv preprint arXiv:2006.09286, 2020

    2006 arXiv

  7. [15]

    Attention is turing-complete

    Jorge Pérez, Pablo Barceló, and Javier Marinkovic. Attention is turing-complete. Journal of Machine Learning Research, 22(75):1–35, 2021

    2021

  8. [16]

    Transformers as recognizers of formal languages: A survey on expressivity

    Lena Strobl, William Merrill, Gail Weiss, David Chiang, and Dana Angluin. Transformers as recognizers of formal languages: A survey on expressivity. arXiv preprint arXiv:2311.00208, 2023

    2023 arXiv

  9. [17]

    Deep learning for symbolic mathematics

    Guillaume Lample and François Charton. Deep learning for symbolic mathematics. arXiv preprint arXiv:1912.01412, 2019

    1912 arXiv

  10. [18]

    End-to-end symbolic regression with transformers

    Pierre-Alexandre Kamienny, Stéphane d’Ascoli, Guillaume Lample, and François Charton. End-to-end symbolic regression with transformers. Advances in Neural Information Processing Systems, 35:10269–10281, 2022

    2022

  11. [19]

    Deep symbolic regression for recurrent sequences

    Stéphane d’Ascoli, Pierre-Alexandre Kamienny, Guillaume Lample, and François Charton. Deep symbolic regression for recurrent sequences. arXiv preprint arXiv:2201.04600, 2022

    2022 arXiv

  12. [20]

    Luis M. Antunes. Cellpylib: A python library for working with cellular automata. Journal of Open Source Software, 6(67):3608, 2021. doi: 10.21105/joss.03608. URL https://doi.org/ 10.21105/joss.03608

    2021 doi

  13. [21]

    Learning cellular automaton dynamics with neural networks

    N Wulff and J A Hertz. Learning cellular automaton dynamics with neural networks. Advances in Neural Information Processing Systems, 5, 1992

    1992

  14. [22]

    Cellular automata as convolutional neural networks

    William Gilpin. Cellular automata as convolutional neural networks. Physical Review E, 100 (3):032402, 2019

    2019

  15. [23]

    Generalization over different cellular automata rules learned by a deep feed-forward neural network, 2021

    Marcel Aach, Jens Henrik Goebbert, and Jenia Jitsev. Generalization over different cellular automata rules learned by a deep feed-forward neural network, 2021

    2021

  16. [24]

    Growing neural cellular automata

    Alexander Mordvintsev, Ettore Randazzo, Eyvind Niklasson, and Michael Levin. Growing neural cellular automata. Distill, 5(2):e23, 2020

    2020

  17. [25]

    Hierarchical neural cellular automata

    Ritu Pande and Daniele Grattarola. Hierarchical neural cellular automata. In Artificial Life Conference Proceedings 35, volume 2023, page 20. MIT Press One Rogers Street, Cambridge, MA 02142-1209, USA journals-info . . . , 2023

    2023

  18. [26]

    Cellular automata, many-valued logic, and deep neural networks

    Yani Zhang and Helmut Bölcskei. Cellular automata, many-valued logic, and deep neural networks. arXiv preprint arXiv:2404.05259, 2024

    2024 arXiv

  19. [27]

    E (n)-equivariant graph neural cellular automata

    Gennaro Gala, Daniele Grattarola, and Erik Quaeghebeur. E (n)-equivariant graph neural cellular automata. arXiv preprint arXiv:2301.10497, 2023

    2023 arXiv

  20. [28]

    Attention-based neural cellular au- tomata

    Mattie Tesfaldet, Derek Nowrouzezahrai, and Chris Pal. Attention-based neural cellular au- tomata. Advances in Neural Information Processing Systems, 35:8174–8186, 2022. 6

    2022

  21. [29]

    Learning graph cellular automata

    Daniele Grattarola, Lorenzo Livi, and Cesare Alippi. Learning graph cellular automata. Ad- vances in Neural Information Processing Systems, 34:20983–20994, 2021

    2021

  22. [30]

    Learning spatio- temporal patterns with neural cellular automata

    Alex D Richardson, Tibor Antal, Richard A Blythe, and Linus J Schumacher. Learning spatio- temporal patterns with neural cellular automata. PLOS Computational Biology, 20(4):e1011589, 2024

    2024

  23. [31]

    Learning locally interacting discrete dynamical systems: Towards data-efficient and scalable prediction

    Beomseok Kang, Harshit Kumar, Minah Lee, Biswadeep Chakraborty, and Saibal Mukhopad- hyay. Learning locally interacting discrete dynamical systems: Towards data-efficient and scalable prediction. In Alessandro Abate, Mark Cannon, Kostas Margellos, and Antonis Pa- pachristodoul...

    2024

  24. [32]

    It’s hard for neural networks to learn the game of life

    Jacob M Springer and Garrett T Kenyon. It’s hard for neural networks to learn the game of life. In 2021 International Joint Conference on Neural Networks (IJCNN), pages 1–8. IEEE, 2021

    2021

  25. [33]

    Data-centric approach to constrained machine learning: A case study on conway’s game of life

    Anton Bibin and Anton Dereventsov. Data-centric approach to constrained machine learning: A case study on conway’s game of life. arXiv preprint arXiv:2408.12778, 2024

    2024 arXiv

  26. [34]

    Reconstructing cellular automata rules from observations at nonconsecutive times

    Veit Elser. Reconstructing cellular automata rules from observations at nonconsecutive times. Physical Review E, 104(3):034301, 2021

    2021

  27. [35]

    Lifegpt: Topology-agnostic generative pretrained transformer model for cellular automata

    Jaime A Berkovich and Markus J Buehler. Lifegpt: Topology-agnostic generative pretrained transformer model for cellular automata. arXiv preprint arXiv:2409.12182, 2024

    2024 arXiv

  28. [36]

    Chain-of-thought prompting elicits reasoning in large language models

    Jason Wei, Xuezhi Wang, Dale Schuurmans, Maarten Bosma, Fei Xia, Ed Chi, Quoc V Le, Denny Zhou, et al. Chain-of-thought prompting elicits reasoning in large language models. Advances in neural information processing systems, 35:24824–24837, 2022. 7

    2022

Pith tools

Reviewed August 12, 2026 · model on record in the stance chip above.