# Solving the Conjugacy Decision Problem via Machine Learning

## TL;DR

This paper applies machine learning techniques to solve the conjugacy decision problem in non-free groups, demonstrating high accuracy and potential underlying mathematical relationships.

## Contribution

It extends machine learning methods to finitely presented non-free groups, including polycyclic and metabelian groups, for conjugacy decision problems.

## Key findings

- Decision tree classifiers achieved high accuracy.
- Random forests and neural networks also performed well.
- Results suggest underlying mathematical patterns in conjugacy.

## Abstract

Machine learning and pattern recognition techniques have been successfully applied to algorithmic problems in free groups. In this paper, we seek to extend these techniques to finitely presented non-free groups, with a particular emphasis on polycyclic and metabelian groups that are of interest to non-commutative cryptography. As a prototypical example, we utilize supervised learning methods to construct classifiers that can solve the conjugacy decision problem, i.e., determine whether or not a pair of elements from a specified group are conjugate. The accuracies of classifiers created using decision trees, random forests, and N-tuple neural network models are evaluated for several non-free groups. The very high accuracy of these classifiers suggests an underlying mathematical relationship with respect to conjugacy in the tested groups.