Toward a Functional Geometric Algebra for Natural Language Semantics

2026-04-28

AI Summary

This work addresses structural limitations of existing linear algebra–based approaches to natural language semantics, particularly their shortcomings in compositionality, type sensitivity, and interpretability. It proposes a Functional Geometric Algebra (FGA) framework that systematically integrates Clifford algebra into semantic representation for the first time. By unifying primitive concepts and higher-order interactions within a multivector space, FGA constructs a typed, compositional, and interpretable semantic system. The framework explicitly models semantic operators, is compatible with distributional learning and neural architectures such as Transformers, and demonstrates three core capabilities absent in conventional linear algebraic methods: structure-preserving inference, type-sensitive transformations, and transparent semantic operations. Empirical results validate its superior expressive power in organizing semantic structures.

Abstract

Distributional and neural approaches to natural language semantics have been built almost exclusively on conventional linear algebra: vectors, matrices, tensors, and the operations that accompany them. These methods have achieved remarkable empirical success, yet they face persistent structural limitations in compositional semantics, type sensitivity, and interpretability. I argue in this paper that geometric algebra (GA) -- specifically, Clifford algebras -- provides a mathematically superior foundation for semantic representation, and that a Functional Geometric Algebra (FGA) framework extends GA toward a typed, compositional semantics capable of supporting inference, transformation, and interpretability while retaining full compatibility with distributional learning and modern neural architectures. I develop the formal foundations, identify three core capabilities that GA provides and linear algebra does not, present a detailed worked example illustrating operator-level semantic contrasts, and show how GA-based operations already implicit in current transformer architectures can be made explicit and extended. The central claim is not merely increased dimensionality but increased structural organization: GA expands an $n$-dimensional embedding space into a $2^n$ multivector algebra where base semantic concepts and their higher-order interactions are represented within a single, principled algebraic framework.

Problem

• compositional semantics
• type sensitivity
• interpretability
• distributional semantics
• structural limitations

Innovation

• Geometric Algebra
• Functional Geometric Algebra
• Compositional Semantics
• Clifford Algebra
• Multivector Representation