How to Use Deep Learning to Identify Sufficient Conditions: A Case Study on Stanley's $e$-Positivity

TL;DR

This work presents an AI-guided framework for identifying sufficient conditions $P$ that imply a target mathematical statement $Q$, by modifying loss functions to prioritize $100\%$ precision and using attribution methods to steer conjecture formation. It combines a precision-focused two-stage workflow with exploratory data analysis to derive conjectures of the form $\forall z:\ X_{i_1}(z)\in S_1,\ldots, X_{i_q}(z)\in S_q \Rightarrow Y(z)\in B$. Demonstrated on Stanley's $e$-positivity problem, the approach rediscovers that co-triangle-free graphs are $e$-positive and identifies the number of claws as a key invariant, while suggesting that continuous graph invariants drive classification. The study delivers a practical, reproducible pipeline with public code for generating and testing conjectures in pure mathematics, highlighting AI-assisted conjecture generation as a valuable tool for mathematical insight and discovery.

Abstract

In a study, published in \emph{Nature}, researchers from DeepMind and mathematicians demonstrated a general framework using machine learning to make conjectures in pure mathematics. Their work uses neural networks and attribution techniques to guide human intuition towards making provable conjectures. Here, we build upon this framework to develop a method for identifying sufficient conditions that imply a given mathematical statement. Our approach trains neural networks with a custom loss function that prioritizes high precision. Then uses attribution techniques and exploratory data analysis to make conjectures. As a demonstration, we apply this process to Stanley's problem of $e$-positivity of graphs--a problem that has been at the center of algebraic combinatorics for the past three decades. Guided by AI, we rediscover that one sufficient condition for a graph to be $e$-positive is that it is co-triangle-free, and that the number of claws is the most important factor for $e$-positivity. Based on the most important factors in Saliency Map analysis of neural networks, we suggest that the classification of $e$-positive graphs is more related to continuous graph invariants rather than the discrete ones. Furthermore, using neural networks and exploratory data analysis, we show that the claw-free and claw-contractible-free graphs with $10$ and $11$ vertices are $e$-positive, resolving a conjecture by Dahlberg, Foley, and van Willigenburg.

Figures (5)

• Figure 1: Flowchart of the framework for guiding mathematicians' intuition by using machine learning models in Nature21
• Figure 2: The framework for finding sufficient conditions
• Figure 3: Comparing the number of triangle-free graphs and $(3+1)$-free posets
• Figure 4: Saliency Map visualizations
• Figure 5: $e$-positivity and the four most important graph invariants

Theorems & Definitions (3)

• Remark 1
• Theorem 1
• Theorem 2