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

• URL: http://arxiv.org/abs/2601.03682v1
• Date: Wed, 07 Jan 2026 08:15:01 GMT
• Title: From Implicit to Explicit: Token-Efficient Logical Supervision for Mathematical Reasoning in LLMs
• Authors: Shaojie Wang, Liang Zhang,
• Abstract summary: Large language models (LLMs) exhibit limited logical reasoning abilities in mathematical problem-solving.<n>Errors related to logical relationship understanding account for over 90% of incorrect predictions.<n>We propose First-Step Logical Reasoning, a lightweight training framework targeting logical relationship understanding.
• Score: 5.703029996279753
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: Recent studies reveal that large language models (LLMs) exhibit limited logical reasoning abilities in mathematical problem-solving, instead often relying on pattern-matching and memorization. We systematically analyze this limitation, focusing on logical relationship understanding, which is a core capability underlying genuine logical reasoning, and reveal that errors related to this capability account for over 90\% of incorrect predictions, with Chain-of-Thought Supervised Fine-Tuning (CoT-SFT) failing to substantially reduce these errors. To address this bottleneck, we propose First-Step Logical Reasoning (FSLR), a lightweight training framework targeting logical relationship understanding. Our key insight is that the first planning step-identifying which variables to use and which operation to apply-encourages the model to derive logical relationships directly from the problem statement. By training models on this isolated step, FSLR provides explicit supervision for logical relationship understanding, unlike CoT-SFT which implicitly embeds such relationships within complete solution trajectories. Extensive experiments across multiple models and datasets demonstrate that FSLR consistently outperforms CoT-SFT under both in-distribution and out-of-distribution settings, with average improvements of 3.2\% and 4.6\%, respectively. Moreover, FSLR achieves 4-6x faster training and reduces training token consumption by over 80\%.

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