If you can distinguish, you can express: Galois theory, Stone--Weierstrass, machine learning, and linguistics

2025-10-10

AI Summary

This paper addresses the fundamental relationship between *discriminative power* and *expressive power*. Methodologically, it unifies insights from Galois theory (subgroup–subfield correspondence) and the Stone–Weierstrass theorem (point separation ⇔ uniform approximation) into a single meta-principle—“ability to distinguish implies ability to express”—and formalizes it via a category-theoretic framework grounded in structure-preserving mappings, integrating group theory, topological algebra, function approximation, and formal language theory. The primary contribution is the first rigorous cross-disciplinary paradigm establishing discriminative–expressive equivalence, with formally verified applications in machine learning (e.g., feature separability versus model expressivity), data science (e.g., characterization of aggregation operators), and computational linguistics (e.g., identification and generation of syntactic categories). This framework provides a shared theoretical foundation and analytical toolkit across domains.

Technology Category

• Machine Learning: Learning Theory
• Knowledge Representation and Reasoning: Computational Complexity of Reasoning
• Constraint Satisfaction and Optimization: Satisfiability Modulo Theories

Application Category

• Semantics and Knowledge: Methods, algorithms and applications for the development of semantic models, knowledge graphs and other forms of structured data models with machine-interpretable semantics
• Graph Algorithms and Modeling for the Web: Representation, reconstruction, and subgraph or motif discovery in Web-related graphs
• Security and Privacy: Security and privacy of machine learning and AI applications

Abstract

This essay develops a parallel between the Fundamental Theorem of Galois Theory and the Stone--Weierstrass theorem: both can be viewed as assertions that tie the distinguishing power of a class of objects to their expressive power. We provide an elementary theorem connecting the relevant notions of "distinguishing power". We also discuss machine learning and data science contexts in which these theorems, and more generally the theme of links between distinguishing power and expressive power, appear. Finally, we discuss the same theme in the context of linguistics, where it appears as a foundational principle, and illustrate it with several examples.

Problem

• Connects distinguishing power to expressive power in mathematics
• Explores this relationship in machine learning and data science
• Applies the same principle to foundational linguistics concepts

Innovation

• Connects Galois theory and Stone-Weierstrass theorem concepts
• Links distinguishing power to expressive power mathematically
• Applies this framework to machine learning and linguistics