# Logical Structure as Knowledge: Enhancing LLM Reasoning via Structured Logical Knowledge Density Estimation

arXiv: https://arxiv.org/abs/2509.24836 · HTML: https://mlsanity.com/paper/2509.24836/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- Placed in Reasoning benchmarks and evaluation, one of 17 directions in LLM reasoning.
- Builds on 20 of its 50 references that are in the map.
- First posted on arXiv 29 Sep 2025.

## Placed in

- LLM reasoning / [Reasoning benchmarks and evaluation](https://mlsanity.com/lr/reasoning-benchmarks-and-evaluation/)

## Builds on (most cited first, 20 of 20)

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- [SemDeDup: Data-efficient learning at web-scale through semantic deduplication](https://mlsanity.com/paper/2303.09540/) (2023)
- [LLMs Can Easily Learn to Reason from Demonstrations Structure, not content, is what matters!](https://mlsanity.com/paper/2502.07374/) (2025)
- [A Survey of Efficient Reasoning for Large Reasoning Models: Language, Multimodality, and Beyond](https://mlsanity.com/paper/2503.21614/) (2025)
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- [Unlocking the Potential of Difficulty Prior in RL-based Multimodal Reasoning](https://mlsanity.com/paper/2505.13261/) (2025)
- [AttentionInfluence: Adopting Attention Head Influence for Weak-to-Strong Pretraining Data Selection](https://mlsanity.com/paper/2505.07293/) (2025)
- [Finding the Minimal Parameter Budget for Implicit Reasoning: A Data Complexity Driven Scaling Law for Language Models](https://mlsanity.com/paper/2504.03635/) (2025)
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## Built on by (most cited first, 1 of 1)

- [Make LLM Learn to Synthesize from Streaming Experiences through Feedback](https://mlsanity.com/paper/2605.29940/) (2026)

## Abstract (arXiv)

The reasoning capabilities of Large Language Models (LLMs) are increasingly attributed to training data quality rather than mere parameter scaling. However, existing data-centric paradigms often equate quality with factuality or diversity and ignore the internal logical complexity of training samples. In this work, we propose that natural language harbors Structured Logical Knowledge manifested through entailment relationships and logical topologies. To quantify this, we introduce Structured Logical Knowledge Density (SLKD), a novel metric that measures logical information content by decomposing natural language into executable predicates and logical primitives. Our analysis reveals a significant logical disparity in current datasets where sparse logical signals predominate. Consequently, we propose a density aware re-cognizing optimization strategy that prioritizes high-density logical samples to enhance with the LLM's reasoning ability. Extensive experiments demonstrate that our approach enhances reasoning performance and generalization without increasing total data volume. These results, further validated within a reinforcement learning framework, suggest that elevating logical density is more critical than expanding data scale for realizing the full cognitive potential of LLMs. The released code is available in the Appendix C.