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From Implicit to Explicit: Token-Efficient Logical Supervision for Mathematical Reasoning in LLMs

Paper · 2026 · Shaojie Wang and Liang Zhang

Summary

LLMs solving math word problems often lean on memorized patterns rather than working out how the quantities in a problem relate. The authors had GPT-4o classify the wrong answers of several models, such as Qwen2.5-7B on GSM8K. Misreading these "logical relationships" explained more than 90% of failures, and standard CoT-SFT barely reduced that share. They attribute this to full-trajectory training, which supervises such relationships only indirectly.

Their remedy is First-Step Logical Reasoning (FSLR). A teacher model, guided by a restrictive prompt, writes only the opening planning step for each problem: which quantities to combine and with which operation, with no arithmetic. The target model is then fine-tuned on these short targets alone.

Across several models and benchmarks, FSLR beats CoT-SFT on average by 3.2% in-distribution and 4.6% out-of-distribution. It also trains 4–6× faster on far fewer tokens. The work suggests that cheap, targeted supervision of problem understanding can outperform imitation of complete solutions.

Key points

  • Models:
    • Teachers: LLaMA-3.1-70B-Instruct, Qwen2.5-72B-Instruct, or the target model itself (self-generated).
    • Targets: LLaMA-3.1-8B-Instruct, Qwen2.5-7B-Instruct and Qwen3-4B-Instruct.
  • Training cost: FSLR uses roughly 81–87% fewer training tokens than CoT-SFT and trains about 4–6× faster.
  • GSM-Symbolic results (FSLR vs CoT-SFT):
    • LLaMA-3.1-8B: 79.6% vs 70.7% (the base model scores 52.9%).
    • Qwen2.5-7B: 78.4% vs 68.7%.
    • Qwen3-4B: 78.4% vs 70.5%.
  • Stability of first planning steps (on GSM8K training problems, judged by GPT-4o):
    • Within one teacher across temperatures 0, 0.5 and 1.0, they are semantically consistent for 84.7% of problems.
    • Across the three teacher sources, they agree for 75.4% of problems.
  • Limitations: the method is tested only on mathematics and relies on teacher-annotated SFT. The authors leave reinforcement learning for future work.

Our summary of the paper, in our own words — not its text. The work belongs to its authors; read it at the links above.

 Institutions

🎓The Hong Kong University of Science and Technology (Guangzhou)Guangzhou · China53 works · 1,367 citations

 Fields, topics and concepts

📚 Shelves 6The field of science this work belongs to (OpenAlex fields).Computer Science · 16,667 worksArtificial Intelligence · 16,194 worksArts and Humanities · 16,039 worksLanguage and Linguistics · 15,891 worksMathematics · 177 worksApplied Mathematics · 46 works

🧭 Topics 2Research subjects and languages the work is about.Language models & pretraining · 3,320 worksReasoning · 1,426 works

Cite

Cite as Shaojie Wang and Liang Zhang. 2026. From Implicit to Explicit: Token-Efficient Logical Supervision for Mathematical Reasoning in LLMs. Findings of the Association for Computational Linguistics: ACL 2026.

The paper itself, not this page. BibTeX from the library's record of the work. DOI 10.18653/v1/2026.findings-acl.1420

@inproceedings{wang2026implicit,
  title = {{From Implicit to Explicit: Token-Efficient Logical Supervision for Mathematical Reasoning in LLMs}},
  author = {Shaojie Wang and Liang Zhang},
  booktitle = {Findings of the Association for Computational Linguistics: ACL 2026},
  year = {2026},
  doi = {10.18653/v1/2026.findings-acl.1420},
  url = {https://doi.org/10.18653/v1/2026.findings-acl.1420}
}

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