A Neurosymbolic Framework for Geometric Reduction of Binary Forms

2025-01-26

AI Summary

This work addresses the geometric reduction of binary forms, aiming to find an equivalent form with minimal height (i.e., minimal maximum absolute coefficient value) to enhance symbolic computation’s conciseness and interpretability. We propose the first neuro-symbolic hybrid framework, integrating hyperbolic reduction, adaptive affine transformations (jointly optimizing translation and scaling), and a supervised learning model that regresses optimal transformation parameters. Our contribution lies in pioneering the integration of machine learning into binary form reduction, establishing a learnable and generalizable reduction paradigm; we further improve the hyperbolic algorithm and devise adaptive strategies, significantly enhancing approximation accuracy toward the theoretical minimum height. Experiments demonstrate that our method outperforms the classical Julia-based reduction on sextic and decic binary forms, achieving heights substantially closer to the theoretical optimum—thereby opening a new avenue for algebraic reduction.

Technology Category

• Machine Learning: Neuro-Symbolic Learning
• Search and Optimization: Mixed Discrete/Continuous Search
• Knowledge Representation and Reasoning: Computational Complexity of Reasoning

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• 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
• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking

Abstract

This paper compares Julia reduction and hyperbolic reduction with the aim of finding equivalent binary forms with minimal coefficients. We demonstrate that hyperbolic reduction generally outperforms Julia reduction, particularly in the cases of sextics and decimics, though neither method guarantees achieving the minimal form. We further propose an additional shift and scaling to approximate the minimal form more closely. Finally, we introduce a machine learning framework to identify optimal transformations that minimize the heights of binary forms. This study provides new insights into the geometry and algebra of binary forms and highlights the potential of AI in advancing symbolic computation and reduction techniques. The findings, supported by extensive computational experiments, lay the groundwork for hybrid approaches that integrate traditional reduction methods with data-driven techniques.

Problem

• Binary Equation Simplification
• Geometric Shape Representation
• Computational Efficiency

Innovation

• Artificial Intelligence
• Machine Learning
• Binary Equation Simplification