The Latent Space of Equational Theories

TL;DR

This work constructs a latent, geometry-based representation of equational theories by embedding the 4694 theories from the Equational Theories project into a Stone-paired feature space derived from finite magmas. It connects model-theoretic structure (Stone pairing, Stone spectrum) with proof-theoretic structure (implication graph and its atomic steps) to reveal organized, directionally oriented flows of reasoning in a three-dimensional latent space obtained via PCA. Key contributions include demonstrating that the latent space clusters theories by signature, quantifies how reversible (provably equivalent) theories cluster tightly, and exposes radial and parallel patterns in implication flows, with a Z-axis indication of conjugacy. The findings suggest a deep, statistics-based correspondence between model-theoretic and proof-theoretic notions, offering new tools for analyzing logical deduction and guiding future research in the intersection of finite model theory, universal algebra, and machine-assisted reasoning.

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.

Figures (10)

• Figure 1: How each equation $\mathsf{Eqn}{[k]}$ is assigned a position in the hypercube $\mathbb{H}_{n}$ of dimension $n=\#\mathsf{Mags}$ the number of sampled finite magmas $A_1,\dots,A_n$.
• Figure 2: Stone spectra of the commutative law (top-left) and of the associative law (bottom-left) for a sampling of $n=20\,000$ magmas of size 4 (red curve) and of $n=24\, 000$ magmas of size 8 (blue curve). The Stone spectra of the conjugate theories $\mathsf{Eqn}{[63]}$ and $\mathsf{Eqn}{[271]}$ are identical (top and bottom, right).
• Figure 3: The Stone interference spectrum of $\mathsf{Eqn}{[4400]}$ and $\mathsf{Eqn}{[4533]}$ detects a degree of statistical dependence between the two conjugate equational theories (left) while the Stone interference spectrum of $\mathsf{Eqn}{[1092]}$ and $\mathsf{Eqn}{[4092]}$ shows full statistical independence between the two equational theories (right).
• Figure 4: PCA Explainability statistics for a random sample of $n=20\,000$ magmas of size 8. The values of the first ten singular values (left) and the explained variance ratio (right) for the first ten principal components.
• Figure 5: Latent space of equational theories where every theory is colored according to its signature $(a,b)$, where $a$ and $b$ are the numbers of diamond operations on the lefthand and righthand side of the theory.