An enriched category theory of language: from syntax to semantics

Abstract: State of the art LLMs return a natural language text continuation from any piece of input text. This ability to generate coherent text extensions implies significant sophistication, including a knowledge of grammar and semantics. In this paper, we propose a mathematical framework for passing from probability distributions on extensions of given texts, such as the ones learned by today's LLMs, to an enriched category containing semantic information. Roughly speaking, we model probability distributions on texts as a category enriched over the unit interval. Objects of this category are expressions in language, and hom objects are conditional probabilities that one expression is an extension of another. This category is syntactical -- it describes what goes with what. Then, via the Yoneda embedding, we pass to the enriched category of unit interval-valued copresheaves on this syntactical category. This category of enriched copresheaves is semantic -- it is where we find meaning, logical operations such as entailment, and the building blocks for more elaborate semantic concepts.

• The paper's main contribution is a unified framework that integrates syntax and semantics using enriched category theory over probability measures.
• It models language expressions as morphisms with conditional probabilities, enabling operations like conjunction, disjunction, and implication in semantic space.
• The approach connects to generalized metric spaces and tropical geometry, offering innovative insights for the development of interpretable AI and NLP systems.

An Enriched Category Theory of Language: From Syntax to Semantics

Introduction

The paper "An enriched category theory of language: from syntax to semantics" (2106.07890) introduces a sophisticated mathematical framework that connects the probabilistic nature of LLMs, specifically those used for text continuation, with an enriched category theory. This novel approach models language as a collection of enriched categories to incorporate both the syntactical arrangement of language and the distributional probabilities inherent in LLMs. By leveraging these categories, the authors propose a transition from syntactic structures to meaningful semantic entities, thus encapsulating both syntax and semantics in a unified theoretical construct.

Enriched Categories in Language

At the core of the paper's contribution is the development of a language syntax category $\mathcal{L}$, enriched over the unit interval $[0,1]$, which represents conditional probabilities as morphisms between language expressions. This enrichment allows the probability that a text extends another to be seamlessly integrated into the category structure, maintaining both compositional and distributional linguistic information. The paper argues that this probabilistically-enriched view naturally aligns with the empirical capabilities of current LLMs, which seem to intuitively grasp semantic concepts through unsupervised learning.

The Semantic Framework

From the syntax category, the authors use the Yoneda embedding to define a semantic category of $[0,1]$-valued copresheaves on the syntax category. This enriched copresheaf category, $\widehat{\mathcal{L}}$, is where semantic structures are formally defined and manipulated. The authors propose that these copresheaves represent the meaning potential of texts, capturing the contexts in which texts appear through a dynamic semantics lens. Operations on these copresheaves, such as conjunction, disjunction, and implication, are mathematically formalized using enriched categorical limits and colimits.

Operations on Semantic Entities

The paper provides a rigorous definition of operations on the semantic entities represented in copresheaves, including conjunction and disjunction via weighted products and coproducts, and a form of implication through an enriched internal hom. These operations are essential for modeling semantic relationships and entailments between language entities and are derived from the enriched categorical framework that extends the classical set-based approach to functional reasoning over probabilistic contexts.

Generalized Metric Spaces and Tropical Geometry

Beyond categorical constructs, the paper introduces a geometric interpretation by transforming the syntax category into a generalized metric space, where the notion of 'distance' between language texts informs the semantic copresheaves. This approach allows for a tropical geometry perspective, offering a new potential for visualizing and reasoning about language structures. In this view, semantic operations gain additional interpretation in terms of tropical polynomial division and metric tree structures, hinting at deep connections with advanced mathematical fields like tropical convexity and module theory.

Implications and Future Work

The implications of this theoretical framework are significant for the future development of AI and NLP systems. The mathematical rigor introduced could guide the development of new LLM architectures capable of explicit semantic reasoning without sacrificing the flexibility of current generative models. The underlying enriched categorical structure offers potential pathways for constructing interpretable and manipulable semantic layers within neural networks. Additionally, this work suggests further exploration into tropical geometry's role in understanding neural activation patterns and decision boundaries in LLMs.

Conclusion

This paper advances the foundational understanding of semantics in artificial intelligence through the innovative application of enriched category theory to language modeling. It sets the stage for a new line of inquiry that marries advanced mathematical concepts with practical AI systems, paving the way for future research into both theoretical and applied aspects of language understanding. The proposed framework has the potential to significantly influence the design and interpretation of AI LLMs, presenting a compelling avenue for exploring the intersection of mathematics and linguistics.