title: Syzygy of Thoughts: Improving LLM CoT with the Minimal Free Resolution
summary: 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.
paper_authors._id: 6800e4285fba288bf14ea9b1
paper_authors.name: Chenghao Li
paper_authors._id: 6800e4285fba288bf14ea9b2
paper_authors.name: Chaoning Zhang
paper_authors._id: 6800e4285fba288bf14ea9b3
paper_authors.name: Yi Lu
paper_authors._id: 6800e4285fba288bf14ea9b4
paper_authors.name: Jiaquan Zhang
paper_authors._id: 6800e4285fba288bf14ea9b5
paper_authors.name: Qigan Sun
paper_authors._id: 6800e4285fba288bf14ea9b6
paper_authors.name: Xudong Wang
paper_authors._id: 6800e4285fba288bf14ea9b7
paper_authors.name: Jiwei Wei
paper_authors._id: 6800e4285fba288bf14ea9b8
paper_authors.name: Guoqing Wang
paper_authors._id: 6800e4285fba288bf14ea9b9
paper_authors.name: Yang Yang
paper_authors._id: 6800e4285fba288bf14ea9ba
paper_authors.name: Heng Tao Shen
paper_title: Syzygy of Thoughts: Improving LLM CoT with the Minimal Free Resolution
paper_summary: 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.
paper_ai_summary: Syzygy of Thoughts (SoT) enhances large language models by introducing auxiliary reasoning paths inspired by Minimal Free Resolution from algebra, resulting in more robust problem-solving and improved performance.
paper_ai_keywords: Chain-of-Thoughts Minimal Free Resolution Syzygy of Thoughts Module Betti numbers Freeness Mapping Exactness Minimality GSM8K MATH GPT-4o-mini Qwen2.5