CrystleLLM

Quotient Space Reasoning for Large Language Models

CrystleLLM applies quotient space theory from abstract algebra to analyze and reduce semantic redundancy in LLM reasoning chains. The key insight is that many tokens in a reasoning trace are semantically equivalent under task-relevant equivalence relations -- they convey the same logical step in different surface forms. By identifying and collapsing these equivalence classes, we can dramatically compress reasoning without losing accuracy.

The analysis reveals striking redundancy: 64-72% of tokens in typical chain-of-thought reasoning traces are semantically redundant when projected onto the quotient space defined by logical equivalence. CrystleLLM exploits this by constructing a compressed reasoning representation that operates in quotient space, reducing token consumption by 26.8% on average while maintaining or improving accuracy.

Experiments span three major benchmarks (GSM8K for math, MMLU for knowledge, ARC for science reasoning) across three model families (Mistral-7B, Qwen2.5, Llama-70B), demonstrating that the redundancy pattern is architecture-independent. Larger models show higher redundancy, suggesting that scaling laws create increasing opportunities for quotient space compression.

The framework provides both analytical tools (measuring redundancy in existing reasoning traces) and generation tools (producing compressed reasoning directly), making it applicable to both post-hoc analysis and real-time inference optimization.

• 64-72% semantic redundancy identified in LLM reasoning chains via quotient space analysis
• 26.8% token reduction while maintaining or improving accuracy
• Evaluated on GSM8K, MMLU, ARC across Mistral-7B, Qwen2.5, and Llama-70B
• Architecture-independent redundancy pattern: larger models show higher redundancy
• Both analytical and generative tools for quotient space reasoning