Syzygy of Thoughts: Improving LLM CoT with the Minimal Free Resolution

TL;DR

SoT addresses limitations of Chain-of-Thought in high-dimensional reasoning by introducing a Minimal Free Resolution–inspired framework that constructs interdependent syzygy reasoning paths. It maps algebraic notions—Module, Freeness, Mapping, Exactness, Minimality, Betti numbers—to prompting and search procedures, enabling structured decomposition, redundancy pruning, and logical closure verification. Across nine benchmarks and multiple LLM backbones, SoT consistently surpasses CoT, CoT-SC, GoT, and AoT while maintaining stable variance and a favorable accuracy–cost balance, with public code available. This work provides a theoretically grounded, scalable approach to robust LLM reasoning with potential extensions to multimodal and iterative frameworks.

Abstract

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.

Figures (6)

• Figure 1: SoT (Ours) achieved performance improvements compared to CoT and CoT-SC on two models across nine datasets. The inner circle shows three methods of Qwen2.5, while the outer circle shows three methods of 4o-mini.
• Figure 2: The mathematical, abstract analogy, and CoT analogy of Module, Freeness, Mapping, Exactness, Minimality, and Betti Number.
• Figure 3: SoT Overview. Through the six modules: Module Freeness, Mapping, Exactness, Minimality, and Betti, MFR can decompose and deconstruct a complex reasoning problem, aiding LLMs in generating more accurate answers.
• Figure 4: Cost–accuracy trade-off across different reasoning frameworks. Each bubble corresponds to one configuration of CoT, CoT-SC, GoT, AoT, or the proposed SoT, and the bubble area is proportional to the average number of tokens consumed per problem.
• Figure 5: Overall analysis: (left) Betti number sensitivity, (middle) stability under different temperatures, and (right) accuracy distribution across tasks.