思想的syzygy：通过最小自由分解改进LLM的CoT
Syzygy of Thoughts: Improving LLM CoT with the Minimal Free Resolution
AI总结：
Syzygy of Thoughts通过引入最小自由分解方法，改进LLM的链式思维推理，提升复杂问题的解决能力和推理准确性。
摘要：
链式思维（CoT）提示通过将问题分解为连续步骤来增强大型语言模型（LLM）的推理能力，模仿人类逻辑并减少错误。然而，复杂的任务具有庞大的解空间和模糊的约束，往往超出单一推理链的能力。受交换代数和代数几何中的最小自由分解（MFR）启发，我们提出思想的syzygy（SoT）——一种扩展CoT的新型框架，通过引入辅助且相互关联的推理路径。SoT捕捉更深层次的逻辑依赖性，使问题解决更加稳健和结构化。MFR将模块分解为一系列自由模块的序列，具有最小的秩，为复杂系统提供结构化的分析方法。该方法引入了“模块”、“贝蒂数”、“自由性”、“映射”、“精确性”和“最小性”等概念，使原始复杂问题能够系统地分解为逻辑完整的最小子问题，同时保留关键问题特征并减少推理长度。我们测试了SoT在多样化的数据集（如GSM8K、MATH）和模型（如GPT-4o-mini、Qwen2.5）上的表现，实现了推理准确率与主流CoT标准相匹配或超越。此外，通过将采样过程与代数约束对齐，我们的方法增强了LLM推理时间的可扩展性，确保了透明的推理和高性能。我们的代码将在https://github.com/dlMARiA/Syzygy-of-thoughts公开。
英文摘要：
Chain-of-Thought (CoT) prompting enhances the reasoning of large language models (LLMs) by decomposing problems into sequential steps, mimicking human logic and reducing errors. However, complex tasks with vast solution spaces and vague constraints often exceed the capacity of a single reasoning chain. Inspired by Minimal Free Resolution (MFR) in commutative algebra and algebraic geometry, we propose Syzygy of Thoughts (SoT)-a novel framework that extends CoT by introducing auxiliary, interrelated reasoning paths. SoT captures deeper logical dependencies, enabling more robust and structured problem-solving. MFR decomposes a module into a sequence of free modules with minimal rank, providing a structured analytical approach to complex systems. This method introduces the concepts of "Module", "Betti numbers","Freeness", "Mapping", "Exactness" and "Minimality", enabling the systematic decomposition of the original complex problem into logically complete minimal subproblems while preserving key problem features and reducing reasoning length. We tested SoT across diverse datasets (e.g., GSM8K, MATH) and models (e.g., GPT-4o-mini, Qwen2.5), achieving inference accuracy that matches or surpasses mainstream CoTs standards. Additionally, by aligning the sampling process with algebraic constraints, our approach enhances the scalability of inference time in LLMs, ensuring both transparent reasoning and high performance. Our code will be publicly available at https://github.com/dlMARiA/Syzygy-of-thoughts.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
