The Latent Space of Equational Theories

TL;DR

This paper introduces a latent space for equational theories, constructed using machine learning and finite model theory, revealing structured logical implications and reasoning patterns among theories.

Contribution

It combines ideas from machine learning and finite model theory to create a novel latent space for equational theories, enabling analysis of their logical structure.

Findings

The latent space reveals structured chains of logical implications.

Reasoning flows are oriented and well-structured within the space.

The approach provides new insights into the organization of equational theories.

Abstract

Building on the collaborative Equational Theories project initiated by Terence Tao fifteen months ago, and combining it with ideas coming from machine learning and finite model theory, we construct a latent space of equational theories where each equational theory is located at a specific location, determined by its statistical behavior with respect to a large sample of finite magmas. This experiment enables us to observe for the first time how reasoning flows and produces surprisingly oriented and well-structured chains of logical implications in the latent space of equational theories.