The Magnitude of Categories of Texts Enriched by Language Models (2501.06662v1)

Abstract: The purpose of this article is twofold. Firstly, we use the next-token probabilities given by a LLM to explicitly define a $[0,1]$-enrichment of a category of texts in natural language, in the sense of Bradley, Terilla, and Vlassopoulos. We consider explicitly the terminating conditions for text generation and determine when the enrichment itself can be interpreted as a probability over texts. Secondly, we compute the M\"obius function and the magnitude of an associated generalized metric space $\mathcal{M}$ of texts using a combinatorial version of these quantities recently introduced by Vigneaux. The magnitude function $f(t)$ of $\mathcal{M}$ is a sum over texts $x$ (prompts) of the Tsallis $t$-entropies of the next-token probability distributions $p(-|x)$ plus the cardinality of the model's possible outputs. The derivative of $f$ at $t=1$ recovers a sum of Shannon entropies, which justifies seeing magnitude as a partition function. Following Leinster and Schulman, we also express the magnitude function of $\mathcal M$ as an Euler characteristic of magnitude homology and provide an explicit description of the zeroeth and first magnitude homology groups.

• The paper introduces a novel framework connecting LLM probabilities with enriched category theory to model linguistic texts.
• It constructs a [0,1]-enriched category and a generalized metric space by transforming next-token probabilities using -ln, linking language with geometry.
• The work applies Möbius inversion to compute magnitude, relating combinatorial properties to Tsallis and Shannon entropies for language analysis.

Analyzing the Magnitude of Categories of Texts Enriched by LLMs

This paper primarily introduces a novel framework that connects enriched category theory and LLMs through LLMs. It explores how semantic and statistical properties of natural languages can be encapsulated using mathematical structures derived from LLM probabilities. This work is a continuation of previous efforts by Bradley, Terilla, and Vlassopoulos to exploit category theory as a means to elucidate the inherent structure in linguistic corpora.

The paper's strategy is two-fold; it first employs next-token probabilities from an LLM to create a $[0,1]$-enriched category of strings. Then, the same probabilities are leveraged to form a generalized metric space using the $([0,\infty],\geq,+,0)$ framework, providing insights into the magnitude and geometric properties of these categories.

To construct this $[0,1]$-category, objects are defined as strings ranging over a finite token alphabet, and the enrichment over the unit interval is achieved by assigning to each pair of strings a value that reflects the probability of one being an extension of the other. This definition is adjusted for sentence initiators ($\bot$) and terminators ($\dagger$) to account for constraints imposed by LLMs. As opposed to explaining semantic content through interpretation, this model uses probabilistic information directly generated by LLMs. Fascinatingly, probabilities for strings are shown to be well-defined over the set of terminating states for a given input.

Switching perspectives, by transforming probabilities to distances using $-\ln:[0,1]\to[0,\infty]$, a generalized metric space, $\mathcal{M}$, is introduced. This provides a novel geometric interpretation, where higher probabilities correspond to shorter distances, thus facilitating a bridge between LLMs and geometric measure theory in novel ways.

The paper makes significant technical contributions in computing the magnitude of this enriched category. Magnitude, which extends classical notions of cardinality and Euler characteristics to enriched categories, is employed to derive combinatorial quantities corresponding to the categories of texts. Using the method of M\"obius inversion adapted for generalized metric spaces, the magnitude function $f(t)$ is meticulously calculated. The paper builds on Vigneaux's work to offer a precise combinatorial approach to deriving the magnitude of these spaces, associating it with Tsallis entropies and model output cardinalities. Specifically, the derivative of the magnitude at $t=1$ connects to Shannon entropies, offering a partition function interpretation that is valuable for understanding how information is distributed over the outputs of texts.

Additionally, the paper delves deeper into magnitude homology, a topological invariant that provides further insights into the complexity and connectedness of the language structures modeled by LLMs. By exploring these homological aspects, the work opens pathways to link algebraic topology with informational measures in language.

Practical and theoretical implications of this research are substantial. By pioneering a quantitative framework to understand natural language processing from a category-theoretic and geometrical standpoint, the paper sets the stage for advancements in how we might structure computational models for understanding and generating human languages. The research encourages future work to explore diverse paths, such as semantically enriching LLMs with additional categorical constructs or investigating further topological and geometric applications in AI.

In conclusion, this paper advances our understanding of LLMs by framing language through enriched category theory, offering both a sophisticated theoretical analysis and a robust computational methodology. This intersection of probabilistic modeling, geometric thinking, and categorical methods holds promising potential for the future development of AI and natural language interfaces.