Simulating Compositional Meaning in Natural Language Using Cartesian Decomposition and Tensor Product Operations

Abstract

This paper develops a mathematically grounded, operator-theoretic framework for modeling compositional meaning in natural language by applying Cartesian decomposition and tensor product operations. This paper argues that separating linear operators into Hermitian and skew-Hermitian components allows for finer modeling of literal vs. contextual semantic elements. By elaborating the theory and presenting new illustrative examples, including novel constructions such as 'ambitious engineers evaluate prototypes' and 'curious children solve puzzles', we demonstrate how this framework captures the interactions between syntactic structure and semantic content. We further explore applications in natural language inference, interpretable AI, and sustainable language technologies. The result is a comprehensive model that is transparent, extensible, and aligns with both cognitive principles and formal linguistic theory. The frame work aims to bridge symbolic and sub-symbolic approaches to semantics, enabling not only more nuanced understanding of language, but also robust and interpretable NLP systems.