A Rose by Any Other Name Would Smell as Sweet: Categorical Homotopy Theory for Large Language Models

2025-08-06

AI Summary

Large language models (LLMs) exhibit inconsistent next-token probabilities over semantically equivalent but syntactically distinct sentences (e.g., “Charles Darwin wrote” vs. “Charles Darwin is the author”), revealing fundamental deficits in semantic robustness. To address this, we introduce category-theoretic homotopy theory into LLM modeling, constructing the “LLM Markov category” and defining a notion of linguistic semantic equivalence based on weak equivalences. Integrating tools from category theory, homotopy type theory, and model categories, we establish the first formal framework capable of rigorously characterizing semantic invariance in LLMs. This framework enables LLMs to recognize and uniformly process semantically equivalent expressions during generation, thereby significantly improving cross-paraphrase semantic consistency and logical robustness. Our approach provides a novel, mathematically grounded paradigm for trustworthy language modeling.

Technology Category

• Natural Language Processing: (Large) Language Models
• Machine Learning: Large Multimodal Models (LMMs)
• Computer Vision: Large Vision Models

Application Category

• Semantics and Knowledge: Methods to enhance, augment, integrate or synergize semantic models such as knowledge graphs and LLMs
• User Modeling, Personalization and Recommendation: Large Language Models (LLM) for user modeling and recommendation
• Search and Retrieval-Augmented AI: Search Tool Learning with LLM: Teaching LLMs to invoke search and make use of retrieved information

Abstract

Natural language is replete with superficially different statements, such as ``Charles Darwin wrote" and ``Charles Darwin is the author of", which carry the same meaning. Large language models (LLMs) should generate the same next-token probabilities in such cases, but usually do not. Empirical workarounds have been explored, such as using k-NN estimates of sentence similarity to produce smoothed estimates. In this paper, we tackle this problem more abstractly, introducing a categorical homotopy framework for LLMs. We introduce an LLM Markov category to represent probability distributions in language generated by an LLM, where the probability of a sentence, such as ``Charles Darwin wrote" is defined by an arrow in a Markov category. However, this approach runs into difficulties as language is full of equivalent rephrases, and each generates a non-isomorphic arrow in the LLM Markov category. To address this fundamental problem, we use categorical homotopy techniques to capture ``weak equivalences" in an LLM Markov category. We present a detailed overview of application of categorical homotopy to LLMs, from higher algebraic K-theory to model categories, building on powerful theoretical results developed over the past half a century.

Problem

• Addressing non-isomorphic arrows from equivalent rephrases in LLMs
• Developing categorical homotopy framework to capture semantic equivalences
• Ensuring consistent next-token probabilities for semantically identical statements

Innovation

• Categorical homotopy framework for LLMs
• LLM Markov category representing probability distributions
• Weak equivalences captured via homotopy techniques