Intelligence at the Edge of Chaos

Thank you for your detailed review and constructive feedback. We are glad that you found our approach novel and appreciated our use of elementary cellular automata (ECA) as a framework for exploring the connection between complexity and emergent intelligence. In response to your feedback, we have addressed both major and minor points to enhance the robustness of our findings and the clarity of our presentation. Specifically:

• We conducted additional experiments to validate the role of spatiotemporal sampling in our results.
• Baseline comparisons with randomly initialized transformers were added to evaluate the impact of ECA pretraining.
• Additional experiments with smaller models were included to assess the role of model capacity in our findings.

Below, we provide detailed responses to your comments and questions.

Weaknesses
Comment 1:
The method of generating training data introduces randomness due to spatial windowing, which may impact the results. The attention patterns observed might arise from missing information outside the spatial window, rather than intrinsic model behavior.
Response:
Great observation! While it is true that information outside the spatial window could diffuse into the observed space, we suspect that the effect may be minimal for the following reason: at any given state, only two cells (out of 100) at most could be influenced by this “missing” information, leaving the majority of the state unaffected.
To validate this, we reran our experiments with models trained on the entire 1000-dimensional spatial width to eliminate any randomness. The data was generated with symmetric boundary conditions to ensure there were no boundary effects. We recreated the attention plot from Figure 4 and observed the same patterns as in our original results, as seen here. This suggests that the greater attention observed in complex models is not due to the missing information but rather reflects an intrinsic feature of the models’ behavior.
Due to resource constraints, we performed this experiment with the four rules shown in Figure 4. We hope to extend it to all rules for an updated version of the manuscript.

Comment 2:
The models should not be referred to as “large language models” (LLMs), as they are neither large nor trained on natural language.
Response:
We acknowledge this perspective. However, as we are using a GPT-2 architecture (which is traditionally referred to as an LLM), we thought it is fair to use this term prior to pretraining. However, to address your concern, we have updated the manuscript to refer to the trained models as “transformer models” when discussing their post-pretraining behavior.

Comment 3:
Please clarify the definition of “binary vector” (line 158).
Response:
The term “binary vector” was defined in Section 3.1 as representing the spatial pattern of the automaton at a given time. To improve clarity, we have reiterated this definition on line 161.

Comment 4:
Why isn’t Rule 110 included in ARC Easy or Chess Move evaluations (line 232)?
Response:
At the time of the initial submission, some of our models had not completed training. We have since included Rule 110 in Figure 2 and updated the corresponding analyses.

Comment 5:
Would complex ECAs outperform a randomly initialized transformer?
Response:
To address this, we added baseline results for a randomly initialized transformer. The results show that transformers pretrained on complex ECAs consistently outperform their randomly initialized counterparts, highlighting the value of structured complexity in pretraining. These results have been incorporated into the revised manuscript (Figure 2).

Question 1:
How does this relate to Reservoir Computing (RC)?
Response:
Thank you for this insightful question. While reservoir computing (RC) leverages the fixed dynamics of randomly initialized reservoirs to solve tasks, our study fully trains transformer model parameters on specific ECA rules. This allows our models to adapt to the complexity of the data, whereas RC relies on the reservoir’s inherent dynamics.
Our findings highlight that models trained on complex rules develop richer internal representations, as evidenced by their attention patterns. This suggests that improvements in performance are driven by the training data complexity, not fixed dynamics, distinguishing our work from traditional RC approaches. We have added a section to the related work highlighting these comparisons.

Question 2:
Beyond the trivial explanation mentioned above, what other reasons might explain why the transformer learns to use attention? Have you experimented with varying model sizes?
Response:
Beyond external randomness, the transformer’s use of attention likely reflects its capacity to model long-range dependencies in complex data. To investigate this further, we conducted an additional experiment with a small, one-layer transformer model (Appendix A). Our findings show that smaller models require more data to achieve comparable performance, suggesting that attention use is influenced by both the complexity of the data and the model’s capacity. We plan to explore scaling laws with larger models in future work.

Question 3:
Does this relate to the principle of computational equivalence?
Response:
Yes, this concept is highly relevant to our work! The principle of computational equivalence suggests that processes that are not trivially simple are capable of similar levels of computation. ECAs are a canonical example of this, as Wolfram used them to illustrate the principle.
However, our findings suggest an important nuance: while chaotic and complex systems may both be computationally equivalent, models trained on chaotic data may require more data or compute to learn effective representations compared to models trained on complex data. This aligns with the idea that the efficiency of computation differs across systems.
We have added a discussion of this principle to the Related Work section of the manuscript.

We hope the revisions address your concerns and improve the overall quality of the manuscript.

Thank you for your review and thoughtful feedback. We are pleased that you found our work innovative and appreciated our focus on the “Edge of Chaos” concept and its connection to emergent intelligence. In response to your feedback, we have taken steps to address the identified weaknesses and expand the scope of our evaluation.

Question 1:
Could you include more downstream tasks, like Nim games or cylinder chess, to expand the evaluation?

Response:
Great suggestion! To expand the evaluation, we included a downstream task based on the Nim game. The results show a trend consistent with our other downstream tasks, where models pretrained on higher-complexity CA data perform better on the Nim game task. This further supports our hypothesis that structured complexity fosters better generalization. We have incorporated the details and results of this experiment into Figure 8 in Appendix C in the revised manuscript.

Question 2:
Could you test on smaller models, such as ResNet-50, BERT, or Mamba-170m, to validate the results?

Response:
To evaluate the impact of model size, we conducted scaling law experiments with different model sizes, including the GPT-2 small (85M parameters) model used in our other experiments and also a new one-layer transformer (67k parameters) model. These experiments confirmed our findings, which are included in Appendix A:

• Models trained on higher-complexity CA data require more training data to converge compared to simpler data (Figure 6).
• Larger models pretrained on the same CA rule achieve equivalent validation performance with less data, highlighting a data efficiency advantage for larger architectures.

These results suggest that both the complexity of the training data and the capacity of the model influence the trade-off between data volume and performance. We plan to explore this scaling behavior further in future work.

We hope these additions address your concerns and strengthen the overall contribution of our work.

Thank you for your thoughtful and detailed review. We are delighted that you found our paper well-motivated and enjoyed the use of elementary cellular automata (ECA) as a framework for controlling and analyzing complexity. In response to your feedback, we have included the baseline performance of untrained models in the relevant figures (e.g., Figure 2) to provide a clearer comparison. Below, we provide detailed responses to your questions and comments.

Weakness 1:
The presented quantitative results are rather weak in terms of significance and correlation values (e.g., Figure 2 shows low r-values).
Response:
We acknowledge that the correlation coefficient in Figure 2 is not particularly high. However, we hypothesized that this would occur because the relationship between complexity and performance is non-monotonic, and this is in fact what we observe: performance peaks at intermediate complexity levels (the “edge of chaos”) and decreases at both lower and higher complexities. This non-linear trend inherently limits the strength of the linear correlation coefficient as a descriptor.
In order to quantify the nonlinear association, we computed the distance correlation coefficient, a measure of nonlinear correlation, and found that it is statistically significant for each of our downstream tasks as well. The distance correlation coefficients were 0.7163 for ARC Easy, 0.4444 for ARC Hard, and 0.4329 for Chess. We will add these to the manuscript.

To avoid misunderstanding, we clarified in the manuscript that the correlation coefficient is provided to quantify the overall relationship, but the key takeaway is the qualitative insight: performance peaks at intermediate complexity and deteriorates with excessive complexity.

Weakness 2:
For Wolfram’s complexity class categorization, does the p-test show significance? The absolute accuracy difference between categories seems low.
Response:
We performed significance tests to analyze the significance of differences between Wolfram’s complexity classes.

• For the ARC-E and ARC-H tasks, differences between all categories are statistically significant except for Class III vs. Class IV, which aligns with expectations, as both classes produce highly complex data and only sufficiently chaotic rules deteriorate in performance.
• For the chess task, both the Pearson correlation and the distance correlation coefficient were statistically significant. However, we observed that differences between Wolfram classes were not significant, likely due to the smaller number of rules sampled from each class (owing to the high computational cost of training chess models).

We will add these significance tests to the manuscript.

Weakness 3:
Did the authors run experiments with multiple seeds?
Response:
Due to resource constraints, we were unable to repeat the full training procedure for all models across multiple seeds. However, we conducted runs for several models with different seeds and observed nearly identical training dynamics, suggesting that the results are consistent across seeds. The training runs can be seen here. We plan to add additional replicates in an updated version of the manuscript.

Question 1:
Could you comment on the differences between “complexity” and “diversity” in the ECA model? Do any of the complexity measures capture diversity as well? Are complex data always more diverse and vice versa?
Response:
Complexity and diversity, while interrelated, capture different characteristics of data generated by ECA models. Complexity refers to the intricacy of the patterns and structures observed over time, reflecting how challenging the underlying system is to predict or describe. It often emerges from the interplay between order and chaos, as seen in systems operating at the "Edge of Chaos." Diversity, on the other hand, pertains to the richness or variety of states and transitions that a system can exhibit. It focuses on the breadth of possibilities rather than the difficulty of understanding or predicting them.

Complex systems are not necessarily diverse, and diverse systems are not always complex. For example:

• A highly chaotic ECA rule may produce a narrow range of unpredictable states (high complexity, low diversity).
• Conversely, a periodic ECA rule may exhibit many unique but easily predictable states (low complexity, high diversity).

Some measures of complexity, like Krylov complexity, partially overlap with diversity by capturing the number of unique states observed. However, the relationship is not direct or universal across all complexity measures, as many focus on structural or temporal patterns rather than state variety. We believe this distinction is critical to understanding how ECA-generated data influences model performance.

Question 2:
In Figure 2 (violin plot), it seems like variance is also correlated with complexity and downstream tasks. Could the authors comment on this?
Response:
We agree that the variance in performance is an interesting observation. The variance is mostly explained with the number of distinct rules in each complexity class. For example, periodic rules (Class II) encompass a broader range of distinct rules than other classes, leading to greater variability in the generated data and model performance.

Question 3:
I see a different tendency for Krylov complexity compared to the other measures in Figure 3. High variance is observed for both low and high Krylov complexity. Why is this?
Response:
Krylov complexity measures the number of linearly independent basis vectors generated by a dynamical system’s evolution operator. Its behavior is distinct from other measures because:

1. It is bounded by the dimensionality of the system.
2. For simple or periodic states, Krylov complexity remains low as these systems evolve in small subspaces.
3. For complex and chaotic systems in finite-dimensional space, Krylov complexity saturates quickly as they span the full space $1$ .

This explains the observed behavior in Figure 3: rules with simple or periodic dynamics exhibit low Krylov complexity, while chaotic and complex rules show high Krylov complexity with little middle ground.
This distinction underscores why we report results with five different complexity measures (Lempel-Ziv, Krylov, compression complexity, Lyapunov exponent, and Wolfram class), as each captures unique aspects of the system’s dynamics.

Question 4:
Baseline performance before pretraining on ECA data is not provided.
Response:
We have now added baseline performance using randomly initialized transformer models in the relevant figures (e.g., Figure 2). These results clearly demonstrate the improvements achieved through ECA pretraining.

We hope the revisions address your concerns and enhance the overall quality of the manuscript.

References

$1$ Hashimoto, Koji, et al. "Krylov complexity and chaos in quantum mechanics." Journal of High Energy Physics 2023.11 (2023): 1-41.

I appreciate the authors response and additional experiments and analysis. I still think the evaluation and analysis on downstream tasks are rather limited, though I think it is a interesting setup. I will keep my score as it is, looking forward for the future work.

Thank you for your thoughtful review and constructive feedback. We are glad that the reviewer found our work interesting and appreciated the simplicity of our approach to generating structured data with varying degrees of complexity. We also appreciate your recognition of the novel use of cellular automata (CA) data to pretrain large language models (LLMs).

We agree that improving the precision of our language and addressing ambiguities would enhance the clarity and impact of our work. In response to your comments, we have made the following high-level changes:

• We have limited the use of the term “intelligence” and instead emphasized the development of structured and generalizable representations throughout the manuscript.
• We have clarified that all CA rules are simple in their definitions, but their outputs differ in complexity. Where applicable, we explicitly refer to the complexity of the generated data rather than the rules themselves.
• Added experiments: We incorporated new results to explore the role of temporal structure in learning representations, as well as the relationship between data volume and model performance as suggested.

Below, we address each of your specific questions in detail.

Question 1:
What role does the temporal structure of CA sequences play in learning generalizable representations?

Response:
To investigate this, we conducted the suggested experiment by randomly shuffling the temporal order of elementary cellular automata (ECA) states during training while preserving their spatial organization. This augmentation thus removes the sequential information of the data, isolating the spatial patterns. Our findings show that this operation resulted in a significant drop in downstream performance (ARC Easy). This demonstrates that temporal structure is critical for learning representations that generalize effectively. Spatial complexity alone is insufficient to account for the observed improvements; temporal information plays a key role in enabling models to learn more sophisticated representations. We have incorporated the results of this experiment into Figure 7 in Appendix B in the revised manuscript.

Questions 2 and 3:
Is the amount of data required from CA pretraining comparable to the amount of data required using natural language (or similar complex data) to reach similar generalization effects? What is the trade-off between complexity and volume of data?

Response:
Interesting questions! Comparing data requirements between cellular automata (CA) and natural language is inherently challenging. While CA-generated data can exhibit high structural complexity, natural language is likely more information-dense, embedding semantic relationships, contextual nuances, and real-world grounding. These attributes likely contribute to natural language’s capacity to foster generalization across diverse tasks. However, quantifying this information density is challenging due to the lack of universal complexity metrics for natural language, unlike the well-defined measures available for CAs. To address this rigorously, we envision an experiment where models are pretrained on a fixed amount of natural language data and evaluated in the same manner as the ECA-pretrained models, e.g. on ARC, chess, etc. By controlling the number of tokens seen during pretraining, we could compare relative performance. However, carrying out such an experiment as well as measuring the complexity of natural language is beyond the scope of this study.

As an alternative approach to answer these questions that is within the scope of our work, we performed a new experiment to investigate the trade-off between complexity and data volume by analyzing model performance as a function of the number of tokens seen during ECA pre-training (see Appendix A). We found that models trained on higher-complexity ECA data require more training data to converge compared to simpler data (Figure 6). Additionally, larger models pre-trained on the same ECA rule achieve equivalent validation performance with less data (and converge to better performance overall), highlighting a data efficiency advantage for larger models. These results suggest that both the complexity of the training data and the capacity of the model significantly influence the trade-off between data volume and performance. We have included these findings in Figure 6 and plan to explore the scaling behavior in more detail in future work.

This is a great paper, but it's missing a discussion of the existing literature on the pretraining of transformers on procedural data.

Some examples below, covering language/reasoning data [1-11] and vision data [12].

1. Insights into Pre-training via Simpler Synthetic Tasks (2022)
2. Pretraining with Artificial Language: Studying Transferable Knowledge in LMs (2022)
3. On the Transferability of Pre-trained Language Models: A Study from Artificial Datasets (2022)
4. Injecting Structural Hints: Using Language Models to Study Inductive Biases in Language Learning (2023)
5. Modeling Rapid Language Learning by Distilling Bayesian Priors into ANNs (2023)
6. Synthetic Pre-Training Tasks for Neural Machine Translation (2023)
7. Exploring Model Depth and Data Complexity Through the Lens of Cellular Automata (2024)
8. Learning Universal Predictors (2024)
9. Pre-training with Synthetic Data Helps Offline Reinforcement Learning (2024)
10. SIP: Injecting a Structural Inductive Bias into a Seq2Seq Model by Simulation (2024)
11. Between Circuits and Chomsky: Pre-pretraining on Formal Languages Imparts Linguistic Biases (2025)
12. Scaling Backwards: Minimal Synthetic Pre-training? (2024)

For future readers of this paper, it may also be worth highlighting that the "improvements" when pretraining on ECA data are truly marginal. On the chess move prediction, things improve, at best, from 19.5 to 20.5% accuracy. On the other tailor-made tasks ("ARC easy/hard"), there's no improvement in accuracy, only a faster convergence.

BTW the summary given in the meta review above completely misses the point of the paper.