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Ternary Gamma Semirings: From Neural Implementation to Categorical Foundations

Ruoqi Sun

Abstract

This paper establishes a theoretical framework connecting neural network learning with abstract algebraic structures. We first present a minimal counterexample demonstrating that standard neural networks completely fail on compositional generalization tasks (0% accuracy). By introducing a logical constraint -- the Ternary Gamma Semiring -- the same architecture learns a perfectly structured feature space, achieving 100% accuracy on novel combinations. We prove that this learned feature space constitutes a finite commutative ternary $Γ$-semiring, whose ternary operation implements the majority vote rule. Comparing with the recently established classification of Gokavarapu et al., we show that this structure corresponds precisely to the Boolean-type ternary $Γ$-semiring with $|T|=4$, $|Γ|=1$}, which is unique up to isomorphism in their enumeration. Our findings reveal three profound conclusions: (i) the success of neural networks can be understood as an approximation of mathematically ``natural'' structures; (ii) learned representations generalize because they internalize algebraic axioms (symmetry, idempotence, majority property); (iii) logical constraints guide networks to converge to these canonical forms. This work provides a rigorous mathematical framework for understanding neural network generalization and inaugurates the new interdisciplinary direction of Computational $Γ$-Algebra.

Ternary Gamma Semirings: From Neural Implementation to Categorical Foundations

Abstract

This paper establishes a theoretical framework connecting neural network learning with abstract algebraic structures. We first present a minimal counterexample demonstrating that standard neural networks completely fail on compositional generalization tasks (0% accuracy). By introducing a logical constraint -- the Ternary Gamma Semiring -- the same architecture learns a perfectly structured feature space, achieving 100% accuracy on novel combinations. We prove that this learned feature space constitutes a finite commutative ternary -semiring, whose ternary operation implements the majority vote rule. Comparing with the recently established classification of Gokavarapu et al., we show that this structure corresponds precisely to the Boolean-type ternary -semiring with , }, which is unique up to isomorphism in their enumeration. Our findings reveal three profound conclusions: (i) the success of neural networks can be understood as an approximation of mathematically ``natural'' structures; (ii) learned representations generalize because they internalize algebraic axioms (symmetry, idempotence, majority property); (iii) logical constraints guide networks to converge to these canonical forms. This work provides a rigorous mathematical framework for understanding neural network generalization and inaugurates the new interdisciplinary direction of Computational -Algebra.
Paper Structure (24 sections, 2 theorems, 2 equations, 6 tables)

This paper contains 24 sections, 2 theorems, 2 equations, 6 tables.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Definition of Ternary $\Gamma$-Semirings
  4. The Majority Vote Operation
  5. Categorical Perspective
  6. Experimental Design: A Minimal Counterexample
  7. The XOR Task
  8. Standard Neural Network Performance
  9. Implementation of the Ternary Gamma Semiring
  10. Results
  11. Algebraic Structure Analysis
  12. Truth Table Enumeration
  13. Verification of Algebraic Properties
  14. Correspondence with Gokavarapu's Classification
  15. Categorical Interpretation
  16. ...and 9 more sections

Key Result

Theorem 1

The feature space learned by our Ternary Gamma Semiring, together with the ternary operation $\phi$ defined by the majority vote rule, constitutes a finite commutative ternary $\Gamma$-semiring. This structure is isomorphic to the Boolean-type ternary $\Gamma$-semiring with $|T|=4$, $|\Gamma|=1$ in

Theorems & Definitions (2)

  • Theorem 1: Correspondence Theorem
  • Proposition 2