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arxiv: 2605.24597 · v1 · pith:POZVR6BInew · submitted 2026-05-23 · 💻 cs.AI · cs.CL· cs.LG

Learning to Reason Efficiently with A* Post-Training

Andreas Opedal , Francesco Ignazio Re , Abulhair Saparov , Mrinmaya Sachan , Bernhard Sch\"olkopf , Ryan Cotterell This is my paper

Pith reviewed 2026-06-30 13:51 UTC · model grok-4.3

classification 💻 cs.AI cs.CLcs.LG
keywords A* searchnatural language inferenceLLM reasoningpost-trainingsupervised fine-tuningreinforcement learningprocess reward modelsdeductive reasoning
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The pith

Training small LLMs on A* search traces enables them to outperform much larger models on natural language inference.

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

The paper investigates whether LLMs can learn to produce correct and efficient reasoning steps by using A* search to generate training data for natural language inference tasks. Small Llama-3.2 models are post-trained either through supervised fine-tuning on A* execution traces or reinforcement learning with A*-informed reward models. This leads to substantial gains, allowing 1B to 3B parameter models to surpass the performance of the much larger DeepSeek-V3.2 model. The work highlights a trade-off where A* signals improve both accuracy and efficiency compared to simple correctness rewards.

Core claim

LLMs can learn to generate correct and efficient proofs when trained on execution traces produced by A* search, with small models achieving higher accuracy than larger models after this post-training.

What carries the argument

A* search used to produce optimal execution traces of correct inference steps for training LLMs via fine-tuning or RL.

If this is right

  • Small models achieve near-zero to high accuracy on deductive reasoning after A* post-training.
  • A*-informed process rewards balance accuracy and efficiency better than pure correctness signals.
  • Models trained with imperfect heuristics perform better on larger search spaces.
  • Both supervised fine-tuning on traces and RL with A* rewards are effective training techniques.

Where Pith is reading between the lines

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

  • Classical search algorithms like A* could be integrated more broadly into LLM training pipelines for reasoning tasks.
  • This method might allow smaller, more efficient models to handle complex inference without scaling up parameters.
  • The approach could be tested on other search-based problems beyond natural language inference.

Load-bearing premise

The execution traces generated by A* search contain patterns of correct and efficient reasoning that small LLMs can learn and apply to their own outputs.

What would settle it

Testing the trained small models on a new set of natural language inference problems and finding they do not reach high accuracy or do not outperform larger models would falsify the main claim.

Figures

Figures reproduced from arXiv: 2605.24597 by Abulhair Saparov, Andreas Opedal, Bernhard Sch\"olkopf, Francesco Ignazio Re, Mrinmaya Sachan, Ryan Cotterell.

Figure 1
Figure 1. Figure 1: We study whether LLMs can learn to reason correctly and efficiently in natural language [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Performance of LLMs finetuned with SFT across different search orders, as compared to an [PITH_FULL_IMAGE:figures/full_fig_p008_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: Results on ProofWriter for GRPO-finetuned models, starting from SFT checkpoints that were trained on A* traces under the true cost-to-go. Points on the Pareto front are highlighted. We observe that all reward models are placed on at least one of the two accuracy-efficiency Pareto fronts. the problems that are solved are done so efficiently. Dijkstra’s, on the other hand, generalizes best to the test set in… view at source ↗
Figure 4
Figure 4. Figure 4: Results on DeepRD for GRPO-finetuned models, starting from SFT checkpoints that were trained on A* traces under the dependency heuristic. Points on the Pareto front are highlighted. The accuracy-efficiency trade-off is shown even more starkly than in [PITH_FULL_IMAGE:figures/full_fig_p010_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: Comparison of number of proof steps between different search strategies. The histograms [PITH_FULL_IMAGE:figures/full_fig_p021_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: Comparison of number of proof steps between different search strategies. The histograms [PITH_FULL_IMAGE:figures/full_fig_p021_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: Hypergraph induced by the logic program from Fig. 1, showing all 12 atoms in the minimal [PITH_FULL_IMAGE:figures/full_fig_p022_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: Results on ProofWriter for Llama-3.2-1B-Instruct finetuned with GRPO starting from the SFT checkpoint that was trained under the dependency heuristic. (Note that [PITH_FULL_IMAGE:figures/full_fig_p023_8.png] view at source ↗
Figure 9
Figure 9. Figure 9: Results on DeepRD for Llama-3.2-1B-Instruct finetuned with GRPO starting from the SFT checkpoint that was trained under the true cost-to-go. We note that RL is not effective in this case, in which the SFT reference model is weak on the task. B.3 Training Details For the ProofWriter data we restrict the number of generated tokens to 1024, whereas for the DeepRD data we restrict it to 2048. These maximum len… view at source ↗
read the original abstract

Many applications of large language models (LLMs) require deductive reasoning, yet models frequently produce incorrect or redundant inference steps. We frame natural language inference as a search problem where the final answer is the valid proof itself, requiring a reasoning procedure in which intermediate inferences are correct. Specifically, we investigate whether LLMs can learn to generate correct and efficient proofs with guidance from A* search -- an algorithm that guarantees an optimally efficient path to a goal. We explore two training techniques: supervised fine-tuning on execution traces from A* and reinforcement learning with A*-informed process reward models. Empirically, we find that Llama-3.2 models in the 1B--3B range benefit substantially from A* post training, going from near-zero accuracy to outperforming DeepSeek-V3.2 -- a much larger model. Our analysis uncovers a trade-off: while simple correctness rewards maximize accuracy, A*-informed signals strike a balance between accuracy and efficiency. Furthermore, we find that on larger search spaces, models trained with imperfect heuristics exhibit superior accuracy. Our results demonstrate a promising direction towards reasoning guided by principles derived from classical search algorithms.

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.

Referee Report

2 major / 1 minor

Summary. The paper frames natural language inference as a search problem in which the goal is a valid proof, and investigates whether small Llama-3.2 models (1B–3B) can learn correct and efficient reasoning by supervised fine-tuning on A* execution traces and by reinforcement learning that uses A*-informed process reward models. It reports that this post-training raises accuracy from near zero to levels that exceed DeepSeek-V3.2, while also documenting an accuracy–efficiency trade-off and better performance of imperfect-heuristic models on larger search spaces.

Significance. If the reported gains are robust, the work supplies concrete evidence that classical optimal-search algorithms can supply training signals that improve both correctness and efficiency in small language models, offering a route to reasoning capability that does not rely solely on scale.

major comments (2)
  1. [Abstract] Abstract, paragraph 2: the claim that A* traces enable the model to internalize correct inference steps that transfer to standalone generation is load-bearing for the central empirical result, yet the abstract provides no description of how node validity or the heuristic is computed during trace collection; if these steps rely on an external verifier or larger model unavailable at inference time, the observed accuracy jump would not demonstrate internalization of the reasoning procedure itself.
  2. The manuscript does not report dataset sizes, number of A* traces, or ablation controls that isolate the contribution of the A* heuristic versus simple correctness rewards; without these, it is impossible to assess whether the reported outperformance of DeepSeek-V3.2 is robust or sensitive to post-hoc choices in trace generation.
minor comments (1)
  1. [Abstract] The abstract refers to “A*-informed process reward models” without defining how the A* cost or heuristic is converted into a scalar reward signal.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the constructive feedback. We address the major comments point by point below and will revise the manuscript to improve clarity and completeness.

read point-by-point responses

  1. Referee: [Abstract] Abstract, paragraph 2: the claim that A* traces enable the model to internalize correct inference steps that transfer to standalone generation is load-bearing for the central empirical result, yet the abstract provides no description of how node validity or the heuristic is computed during trace collection; if these steps rely on an external verifier or larger model unavailable at inference time, the observed accuracy jump would not demonstrate internalization of the reasoning procedure itself.

    Authors: We agree the abstract should specify the trace-generation mechanics. Node validity is determined by an external symbolic verifier that checks logical entailment (available only during A* trace collection), while the heuristic is a simple estimate of remaining unresolved premises. These signals are absent at inference; the reported accuracy gains on standalone generation therefore reflect internalization. We will add a concise clause to the abstract describing the verifier and heuristic roles. revision: yes

  2. Referee: [—] The manuscript does not report dataset sizes, number of A* traces, or ablation controls that isolate the contribution of the A* heuristic versus simple correctness rewards; without these, it is impossible to assess whether the reported outperformance of DeepSeek-V3.2 is robust or sensitive to post-hoc choices in trace generation.

    Authors: We acknowledge that explicit counts and targeted ablations were omitted from the initial submission. The revised version will report the precise number of A* traces, dataset sizes, and include new ablation results comparing A*-informed process rewards against pure correctness rewards, allowing readers to evaluate the heuristic's incremental contribution. revision: yes

Circularity Check

0 steps flagged

No circularity: empirical results rest on external comparisons, not self-referential definitions or fitted inputs.

full rationale

The paper reports an empirical study training Llama-3.2 models on A* execution traces via SFT and RL with process reward models, then measuring accuracy and efficiency gains against external baselines including DeepSeek-V3.2. No equations, fitted parameters, or derivations are presented that would make the accuracy numbers reduce to the training inputs by construction. The central claims rely on observable performance differences on held-out problems, which are falsifiable independently of the trace-generation procedure. No self-citation chains, uniqueness theorems, or ansatzes are invoked in the provided text to support the results. This is the standard case of a self-contained empirical paper.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Abstract-only review yields no explicit free parameters, axioms, or invented entities; the central claim rests on the unstated premise that A* traces are transferable training signals.

pith-pipeline@v0.9.1-grok · 5756 in / 1096 out tokens · 26449 ms · 2026-06-30T13:51:31.774675+00:00 · methodology

discussion (0)

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Reference graph

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6 extracted references · 6 canonical work pages · 1 internal anchor

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