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

TL;DR

The paper targets a core bottleneck in LLM mathematical reasoning: logical relationship understanding (LRO), which accounts for the majority of errors and is not significantly mitigated by Chain-of-Thought Fine-Tuning (CoT-SFT). It introduces First-Step Logical Reasoning (FSLR), a lightweight, explicit supervision framework that trains models to predict the initial planning step (which variables to use and which operation to apply), thereby isolating and strengthening LRO. Across multiple models and datasets, FSLR achieves consistent in-distribution and out-of-distribution gains over CoT-SFT, while reducing training tokens by over 80% and speeding up training by 4–6×. The approach also demonstrates strong generalization, robustness to teacher quality, and improved performance on GSM-Symbolic, highlighting its potential to yield more reliable and transferable mathematical reasoning in LLMs.

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\%.

Figures (7)

• Figure 1: Error analysis on Qwen2.5-7B comparing Base, CoT-SFT, and FSLR(Ours) models. Each bar shows the breakdown of correct predictions, logical relationship understanding errors, and other errors.
• Figure 2: Overview of the FSLR framework, consisting of two stages: data generation and model training. The framework leverages a teacher LLM to generate first planning step guidance, which is then used to fine-tune the target LLM via supervised fine-tuning.
• Figure 3: Radar chart showing model accuracy stratified by problem complexity (number of reasoning steps). Results shown are for LLaMA-3.1-8B with LLaMA-3.1-70B as the teacher model. FSLR consistently outperforms both Base and CoT-SFT methods, with particularly notable advantages on more complex problems requiring 6-8 reasoning steps.
• Figure 4: Training time (in minutes) for CoT-SFT and FSLR across different models and data sources. FSLR achieves substantial speedup over CoT-SFT on both GSM8K (left) and SVAMP (right), reducing training time by approximately 4-6$\times$ while maintaining competitive performance.
• Figure 5: Zero-shot performance on GSM-Symbolic benchmark. All models are trained using LLaMA-3.1-70B as the teacher model. FSLR achieves the best performance across all three models, demonstrating superior generalization.