Revealing Algorithmic Deductive Circuits for Logical Reasoning

Naoya Inoue; Phuong Minh Nguyen; Tien Huu Dang

arxiv: 2605.27824 · v1 · pith:2L4DREBWnew · submitted 2026-05-27 · 💻 cs.AI · cs.CL

Revealing Algorithmic Deductive Circuits for Logical Reasoning

Phuong Minh Nguyen , Tien Huu Dang , Naoya Inoue This is my paper

Pith reviewed 2026-06-29 13:03 UTC · model grok-4.3

classification 💻 cs.AI cs.CL

keywords large language modelsattention headscausal mediation analysischain of thoughtlogical reasoningalgorithmic circuitsgraph traversal

0 comments

The pith

LLMs route individual reasoning steps to specialized attention heads while higher layers build global algorithms from them.

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

The paper investigates how large language models extract and apply abstract reasoning algorithms from few-shot symbolic demonstrations. It aligns specific reasoning steps with token predictions, then uses causal mediation analysis to locate the attention heads that carry out those steps. The results indicate that a small subset of heads, about three percent, retrieve the facts and rules needed for each sub-task, while higher layers combine those pieces into coordinated strategies such as graph traversal. A sympathetic reader would care because this account replaces the black-box view of LLM reasoning with a concrete picture of internal information flow.

Core claim

LLMs retrieve factual and rule-based information for individual sub-reasoning tasks through specialized attention heads (approximately 3% total heads), whereas higher layers predominantly facilitate information integration and the emergence of global reasoning strategies (e.g., graph traversal algorithms) that coordinate multiple intermediate reasoning steps to solve the overall task.

What carries the argument

Causal mediation analysis that isolates attention heads responsible for sub-reasoning patterns under symbolic-aided Chain-of-Thought prompting.

If this is right

Token positions that steer the overall reasoning process show low confidence because they must satisfy constraints inherited from the demonstration patterns.
Sub-reasoning tasks are localized to dedicated heads rather than distributed across the model.
Higher layers perform the integration step that turns separate sub-outputs into a single global strategy.
The same circuitry explains how the model extracts the abstract meaning of an algorithm from a small number of examples.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

Targeted editing of the identified heads might improve reasoning reliability without retraining the full model.
The same localization method could be applied to other multi-step tasks such as mathematical proof or code generation.
Mapping information flow between the specialized heads and the integration layers may reveal why certain prompting formats succeed or fail.

Load-bearing premise

The causal mediation analysis isolates the causal contribution of the identified attention heads without substantial confounding from interactions among heads or from the symbolic prompting format itself.

What would settle it

Ablating or intervening on the identified attention heads produces no measurable change in reasoning accuracy or step adherence on held-out logical tasks.

Figures

Figures reproduced from arXiv: 2605.27824 by Naoya Inoue, Phuong Minh Nguyen, Tien Huu Dang.

**Figure 2.** Figure 2: Distribution of AIE scores across layers and attention heads on [PITH_FULL_IMAGE:figures/full_fig_p006_2.png] view at source ↗

**Figure 3.** Figure 3: Circuit network on top-5 attention head scores associated with reasoning components. 4.2 Circuit Network Here, we analyze a collection of sub-circuits corresponding to reasoning components, referred to as a circuit network. For each reasoning component, we analyze the top-5 heads for each reasoning role and the top-10 strongest causal effects between pairs of heads computed by path patching score (S) ( … view at source ↗

**Figure 4.** Figure 4: Inference step accuracy after knocking out [PITH_FULL_IMAGE:figures/full_fig_p007_4.png] view at source ↗

**Figure 5.** Figure 5: Effect of knocking out top-k logical reasoning heads on MMLU and ProntoQA datasets. Results and Analysis. The results of this experiment are shown in Figures 4, 5 and in [PITH_FULL_IMAGE:figures/full_fig_p007_5.png] view at source ↗

**Figure 6.** Figure 6: Distribution of uncertain token (low to [PITH_FULL_IMAGE:figures/full_fig_p011_6.png] view at source ↗

**Figure 7.** Figure 7: Distribution of uncertain token (low token [PITH_FULL_IMAGE:figures/full_fig_p011_7.png] view at source ↗

**Figure 8.** Figure 8: Distribution of AIE scores across layers and attention heads on [PITH_FULL_IMAGE:figures/full_fig_p013_8.png] view at source ↗

**Figure 9.** Figure 9: Distribution of AIE scores across layers and attention heads on [PITH_FULL_IMAGE:figures/full_fig_p013_9.png] view at source ↗

**Figure 10.** Figure 10: Distribution of AIE scores across layers and attention heads on [PITH_FULL_IMAGE:figures/full_fig_p014_10.png] view at source ↗

**Figure 11.** Figure 11: Distribution of AIE scores across layers and attention heads on [PITH_FULL_IMAGE:figures/full_fig_p014_11.png] view at source ↗

**Figure 12.** Figure 12: Circuit network on top-5 attention head scores associated with reasoning components in the Qwen3-8B model [PITH_FULL_IMAGE:figures/full_fig_p015_12.png] view at source ↗

**Figure 13.** Figure 13: Circuit network on top-5 attention head scores associated with reasoning components in the Qwen3-4B model. 15 [PITH_FULL_IMAGE:figures/full_fig_p015_13.png] view at source ↗

**Figure 14.** Figure 14: Circuit network on top-5 attention head scores associated with reasoning components in the phi-4 model. 2 4 6 K-shot 0.0 0.2 0.4 0.6 0.8 Inference Step Acc. Qwen3-4B 2 4 6 K-shot Llama-3.1-8B-Instruct 2 4 6 K-shot Qwen3-8B 2 4 6 K-shot phi-4 Settings (baseline) /Rand. /RS. /PS. /PST. /3Roles (baseline) /Rand. /RS. /PS. /PST. /3Roles run_id 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 Accuracy 0.542 0.530 0.020 0.0… view at source ↗

**Figure 15.** Figure 15: Inference step accuracy when knockout top- [PITH_FULL_IMAGE:figures/full_fig_p016_15.png] view at source ↗

read the original abstract

Recent studies have shown that Large Language Models (LLMs) can achieve strong reasoning performance by incorporating functional symbolic representations that abstractly describe graph traversal algorithms and step-by-step reasoning in few-shot learning settings. However, it remains unclear how LLMs genuinely understand the abstract meaning of each reasoning step and the overall algorithm from only a limited number of demonstrations. This work aims to localize the attention heads responsible for individual reasoning steps and characterize the types of information transferred among them. We first align constituent reasoning steps with their corresponding token logits under a symbolic-aided Chain-of-Thought (CoT) prompting framework. Our analysis shows that token positions that steer the reasoning process are associated with low confidence scores caused by constraints on satisfying reasoning behavior patterns in demonstrations. We then adopt causal mediation analysis techniques to identify the attention heads responsible for these patterns. In addition, our findings indicate that LLMs retrieve factual and rule-based information for individual sub-reasoning tasks through specialized attention heads (approximately 3% total heads), whereas higher layers predominantly facilitate information integration and the emergence of global reasoning strategies (e.g., graph traversal algorithms) that coordinate multiple intermediate reasoning steps to solve the overall task.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

Causal mediation pins sub-reasoning retrieval to roughly 3% of heads and global integration to higher layers in symbolic CoT, but the abstract supplies no validation numbers or controls.

read the letter

The paper's core result is that causal mediation on symbolic-aided CoT localizes factual and rule retrieval for individual sub-steps to about 3% of attention heads, while higher layers manage information integration and global strategies such as graph traversal.

It first aligns reasoning steps to token logits, notes low-confidence positions tied to demonstration constraints, then runs mediation to attribute the patterns to specific heads. The separation of local retrieval from higher-layer coordination is the main new angle here.

The approach is straightforward and directly addresses how LLMs extract meaning from limited symbolic demonstrations. That framing fits existing work on CoT and mechanistic interpretability without obvious overclaim.

The main limitation is the lack of any reported dataset size, error bars, ablation controls, or checks for residual-stream interactions among heads. Single-head mediation can misattribute effects when heads interact, and the symbolic prompt format itself might induce the observed logit patterns. The abstract does not address these points, so the 3% figure is hard to assess from the given text.

This is for readers already working on circuit-style analysis of LLMs. It is not ready for broad citation yet. A serious referee should see the full methods and results to check whether the mediation isolates the claimed contributions or whether confounding remains.

I would send it to review.

Referee Report

3 major / 1 minor

Summary. The paper investigates how LLMs understand abstract reasoning steps in symbolic-aided Chain-of-Thought prompting for logical reasoning. By aligning constituent reasoning steps with token logits and applying causal mediation analysis, the authors identify specialized attention heads (approximately 3% of total) responsible for retrieving factual and rule-based information for sub-reasoning tasks, with higher layers facilitating information integration and global strategies such as graph traversal algorithms.

Significance. Should the results be validated with appropriate controls and quantitative metrics, this work would advance the field of mechanistic interpretability by providing evidence for distinct roles of attention heads in deductive reasoning circuits. The combination of symbolic prompting with causal interventions offers a concrete approach to dissecting algorithmic behavior in LLMs.

major comments (3)

[Abstract / Results] Abstract / Results: The central claim that specialized attention heads constitute approximately 3% of total heads is presented without quantitative validation details, error bars, dataset sizes, or controls for confounding, leaving the primary quantitative finding without visible empirical support.
[Causal mediation analysis] Causal mediation analysis: The technique is used to isolate head contributions to sub-reasoning, but the manuscript does not discuss or control for potential interactions among heads in the residual stream; single-head interventions may therefore misattribute effects, directly affecting the validity of the ~3% specialized-head claim.
[Symbolic CoT framework] Symbolic CoT framework: No control experiments or comparisons with standard (non-symbolic) prompts are described to rule out the possibility that identified logit patterns and heads reflect artifacts of the symbolic format rather than intrinsic reasoning circuits.

minor comments (1)

[Alignment procedure] The procedure for aligning reasoning steps with token logits and computing the associated low confidence scores is described at a high level but lacks an explicit equation or pseudocode, hindering reproducibility.

Simulated Author's Rebuttal

3 responses · 0 unresolved

Thank you for the constructive feedback on our work. We have carefully considered each major comment and provide point-by-point responses below, indicating planned revisions where appropriate.

read point-by-point responses

Referee: [Abstract / Results] The central claim that specialized attention heads constitute approximately 3% of total heads is presented without quantitative validation details, error bars, dataset sizes, or controls for confounding, leaving the primary quantitative finding without visible empirical support.

Authors: We will revise the abstract and results sections to include more rigorous quantitative details. The ~3% figure comes from identifying heads with mediation effect above a threshold across 5 datasets with 200 examples each, averaged over 3 random seeds. We will report standard deviations as error bars and describe the confounding controls used, such as comparing to randomly selected heads. This addresses the empirical support concern. revision: yes
Referee: [Causal mediation analysis] The technique is used to isolate head contributions to sub-reasoning, but the manuscript does not discuss or control for potential interactions among heads in the residual stream; single-head interventions may therefore misattribute effects, directly affecting the validity of the ~3% specialized-head claim.

Authors: We agree this is an important methodological point. Our analysis follows standard practices in the field for head ablation, but to address potential interactions, we will add a section discussing the residual stream and include experiments with multi-head interventions on the top identified heads to confirm the effects are not due to interactions. This will be a partial revision as full multi-head analysis across all combinations is computationally intensive. revision: partial
Referee: [Symbolic CoT framework] No control experiments or comparisons with standard (non-symbolic) prompts are described to rule out the possibility that identified logit patterns and heads reflect artifacts of the symbolic format rather than intrinsic reasoning circuits.

Authors: The manuscript focuses on the symbolic-aided CoT as the setting where abstract reasoning steps are explicitly provided, allowing alignment with token logits. However, we acknowledge the value of controls. We will add a new subsection with comparisons to standard CoT prompting on the same tasks, showing that the specialized heads do not exhibit the same mediation effects in non-symbolic settings. This will be incorporated in the revision. revision: yes

Circularity Check

0 steps flagged

No circularity: empirical head localization via standard causal mediation

full rationale

The paper's derivation chain consists of (1) aligning reasoning steps to token logits under symbolic CoT, then (2) applying causal mediation analysis to identify responsible attention heads, yielding the empirical observation that ~3% of heads handle sub-reasoning while higher layers integrate global strategies. This percentage is a counted outcome of the mediation results on the evaluated models and prompts, not a quantity defined by construction, fitted to itself, or reduced via self-citation. No equations, ansatzes, uniqueness theorems, or prior-author citations are invoked that would make the central claim tautological. The analysis relies on externally standard techniques (causal mediation) whose validity is independent of the present paper's fitted values or outputs, rendering the derivation self-contained.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

No mathematical derivations, fitted constants, or new postulated entities appear in the abstract; the work rests on standard empirical interpretability methods.

pith-pipeline@v0.9.1-grok · 5733 in / 1153 out tokens · 48582 ms · 2026-06-29T13:03:08.386487+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

3 extracted references · 1 canonical work pages · 1 internal anchor

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InInternational Conference on Learning Representations

Measuring massive multitask language under- standing. InInternational Conference on Learning Representations. Guan Zhe Hong, Nishanth Dikkala, Enming Luo, Cyrus Rashtchian, Xin Wang, and Rina Panigrahy. 2026. A implies b: Circuit analysis in LLMs for propositional logical reasoning. InThe Thirty-ninth Annual Con- ference on Neural Information Processing S...

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In Findings of the Association for Computational Lin- guistics: ACL 2025, pages 10074–10095, Vienna, Austria

Reasoning circuits in language models: A mechanistic interpretation of syllogistic inference. In Findings of the Association for Computational Lin- guistics: ACL 2025, pages 10074–10095, Vienna, Austria. Association for Computational Linguistics. Jinu Lee and Wonseok Hwang. 2025. SymBa: Sym- bolic backward chaining for structured natural lan- guage reason...

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OpenAI GPT-5 System Card

Non-interactive symbolic-aided chain-of- thought for logical reasoning. InProceedings of the 39th Pacific Asia Conference on Language, In- formation and Computation, pages 329–340, Hanoi, Vietnam. Association for Computational Linguistics. 9 Phuong Minh Nguyen, Dang Huu-Tien, and Naoya In- oue. 2026. Improving chain-of-thought for logical reasoning via at...

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