Deductive Logic in Language Models: Horizontal vs. Vertical Reasoning

Round 1

Reviewer 1 Report

Comments and Suggestions for Authors

The paper studies how small language models learn logical reasoning, comparing step-by-step reasoning with implicit reasoning inside model layers. It finds that Chain-of-Thought training helps models learn rule-based inference instead of relying on simple shortcuts, but that it plays different roles in different reasoning settings.

The paper is generally well structured and technically interesting; however, the following comments were identified:

Comment 1:
The paper would benefit from an overview figure showing the complete workflow of the study, including data generation, model training, evaluation, and interpretability analysis. This would make the methodology easier to follow for readers.

Comment 2:
The introduction provides sufficient general background, but the research question could be stated more explicitly. The authors should add a short paragraph at the end of the introduction clearly explaining the main research question, objectives, and contribution. The connection with Chain-of-Thought and curriculum-learning literature could also be strengthened.

Comment 3:
Figures 2–5 are informative but visually very similar, which makes it difficult for the reader to immediately understand the differences between the illustrated mechanisms. The paper would benefit from clearer figure titles, stronger visual highlighting, or combining these figures into one multi-panel figure that explicitly contrasts rule completion, rule chaining, and final decision making.

Comment 4:
In Figure 7, the caption states that each group contains 15 bars, but visually only 14 bars seem to be shown. The authors should clarify what each bar represents and check whether one distance category is missing, hidden, or whether the caption should be corrected.

Comment 5:
A similar issue appears in Figure 8: the caption states that each group contains 15 bars, corresponding to examples of length 1–15 Ops, but visually only 14 bars seem to be shown. The authors should clarify whether one category is missing, hidden, or whether the caption should be corrected. Adding clearer x-axis labels or a legend would improve readability.

Comment 6:
Figures 7 and 8 seem more appropriate for the results/analysis section rather than the conclusion, because they present experimental evidence related to CoT and curriculum learning. Placing them closer to the corresponding discussion would improve the logical flow and readability of the paper.

Comment 7:
The methods are mostly well described, but some technical details could be clearer. In particular, the authors should better explain how thresholds are selected in the truncated pseudoinverse method and provide more information about robustness across different random seeds and training runs.

Comment 8:
The model architecture is described sufficiently, but the computational setup is not. The authors should add basic information about the hardware used and the approximate training time to improve reproducibility.

Comment 9:
Some important training results and model details are placed in the appendix, although they are relevant for understanding the main findings. The paper would be clearer if the key appendix results, especially convergence behavior, model settings, and robustness across runs, were summarized in the main Results section.

 

Comments on the Quality of English Language

Line 123, explaination → explanation

Author Response

Please see the attachment.

Author Response File: Author Response.pdf

Reviewer 2 Report

Comments and Suggestions for Authors

Summary of the manuscript

This manuscript investigates how small transformer-based language models implement deductive reasoning at the mechanistic level. Rather than evaluating reasoning performance alone, the study aims to explain the internal computational mechanisms that emerge when transformer models are trained from scratch on symbolic multi-step deductive reasoning tasks. The authors distinguish between two reasoning paradigms. Horizontal reasoning refers to autoregressive reasoning in which intermediate inference steps are explicitly generated and reused through Chain-of-Thought (CoT) supervision. Vertical reasoning refers to implicit reasoning performed internally across transformer layers before the first output token is generated.

To study these paradigms, the paper introduces two synthetic reasoning tasks. The first is a logical consequence task requiring models to determine whether a queried implication follows from a chain of symbolic implications. The second is a binary tree navigation task in which the model must generate the root-to-leaf path corresponding to a target node. Both tasks are designed to minimize linguistic complexity and dataset biases, allowing the analysis to focus on algorithmic reasoning rather than language understanding.

The experimental analysis employs small GPT-2-style transformer models trained from scratch under both CoT and non-CoT supervision. Using mechanistic interpretability techniques—including attention visualization, residual stream decoding via LogitLens, circuit analysis, and a newly proposed truncated pseudoinverse method for decoding query, key, and value representations—the authors investigate how the learned computations are implemented internally.

For the logical consequence task, the study reports that a shallow attention-only transformer develops interpretable circuits corresponding to rule completion, rule chaining, and final decision making, with induction-head-like mechanisms playing a central role. For the binary tree navigation task, the authors find that successful reasoning requires a deeper model in which computation unfolds vertically across transformer layers before output generation.

The manuscript further compares training with and without Chain-of-Thought supervision. The experiments suggest that CoT facilitates horizontal reasoning by explicitly decomposing inference into intermediate steps, whereas for vertical reasoning it appears to function more as a curriculum-learning mechanism that enables the gradual acquisition of increasingly complex reasoning patterns. When CoT supervision is removed, the models generally fail to learn the intended reasoning procedure and instead either memorize the training data or exploit unintended dataset biases.

The authors conclude that mechanistic interpretability can reveal explicit computational circuits underlying deductive reasoning in small transformer models and that the role of Chain-of-Thought supervision differs depending on the computational structure of the reasoning task

Comments to the Authors

The manuscript's main contribution is the circuit-level explanation of how transformer models implement symbolic deductive reasoning under two different computational settings. Making this contribution more explicit at the end of the introduction would help readers better understand the novelty of the work and provide a clearer roadmap for the remainder of the paper.

The distinction between horizontal and vertical reasoning provides a useful conceptual framework for organizing the manuscript. However, γiving more solid definitions of these concepts the first time they appear would improve the article’s readability.

The related work is comprehensive and appropriately discusses recent studies on mechanistic interpretability and reasoning in transformer models. Nevertheless, the research gap could be articulated more explicitly. While the review summarizes previous contributions effectively, it is less clear what specific aspect of transformer reasoning remained unresolved prior to this work and how the present study advances existing knowledge. Adding text at the end of the literature review explicitly contrasting previous findings with the contributions of the present manuscript would improve the positioning of the paper.

The experimental methodology is a strοng point of the manuscript. The synthetic datasets are carefully constructed, considerable attention is given to preventing shortcut learning, and the authors transparently discuss the discovery and removal of dataset biases. The inclusion of unsuccessful experiments and analyses without Chain-of-Thought supervision also increases confidence in the validity of the reported conclusions. The statistical validation, appropriately complements the qualitative mechanistic analyses and provides additional quantitative evidence supporting the proposed interpretation of the learned circuits.

The proposed truncated pseudoinverse decoding approach is an interesting addition to the interpretability methodology employed throughout the paper. Although the technique is demonstrated through several examples, its motivation could be explained more clearly.

The conclusions are generally supported by the presented experiments. Nevertheless, a few statements describing the role of Chain-of-Thought supervision could be expressed more cautiously. The experiments convincingly demonstrate that CoT plays different roles in the two symbolic reasoning tasks investigated in this study. However, some formulations may be interpreted as making broader claims about transformer reasoning in general. Since the paper deliberately focuses on carefully controlled symbolic tasks, slightly qualifying these statements—for example by explicitly referring to "the tasks considered in this study"—would better align the conclusions with the scope of the experimental evidence. The discussion section already acknowledges several limitations, and extending this careful wording consistently throughout the manuscript would further strengthen the paper.

The figures illustrating the learned computational circuits are central to the manuscript and generally of high quality. However, Figures 3 through 6 require frequent cross-referencing between the figures and the main text to follow the reasoning process in order to make the figures easier to interpret.

Recommendation

This manuscript presents a thoughtful and technically rigorous investigation of deductive reasoning in transformer language models from the perspective of mechanistic interpretability. The experimental design is sound, the analyses are thorough, and the conclusions are largely supported by the reported evidence. The requested revisions concern clarification of the conceptual framing, sharper positioning of the work within the existing literature, improved presentation of the proposed methodology, and more careful calibration of several conclusions. These are relatively minor revisions that would improve the clarity and impact of an already strong contribution. I therefore recommend acceptance after minor revision.

Author Response

Please see the attachment.

Author Response File: Author Response.pdf