Syzygy of Thoughts: Improving LLM CoT with the Minimal Free Resolution
Authors: name: Chenghao Li, name: Chaoning Zhang, name: Yi Lu, name: Jiaquan Zhang, name: Qigan Sun, name: Xudong Wang, name: Jiwei Wei, name: Guoqing Wang, name: Yang Yang, name: Heng Tao Shen

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.