Self-Supervised Theorem Discovery in a Formal Axiomatic System

2026-06-27

AI Summary

This work investigates how an agent can autonomously discover mathematically useful theorems starting solely from axioms and inference rules, without relying on human-provided prior knowledge such as mathematical texts, code, or existing theorem libraries. To this end, the authors propose a self-supervised theorem discovery algorithm that iteratively alternates between proof search and extraction of useful theorems within a formal axiomatic system, progressively constructing a reusable theorem library to enhance subsequent reasoning. This approach achieves, for the first time, fully autonomous theorem discovery without any human-supplied theorems. The resulting collection of tens of thousands of theorems exhibits both formal correctness and mathematical relevance, enabling the independent solution of human-designed benchmark problems and significantly improving the theorem-proving performance of large language models when used as prompting lemmas.

Abstract

Recent artificial intelligence (AI) systems have shown remarkable progress in mathematical reasoning. Many existing approaches, including large language models (LLMs), draw on human prior knowledge in the form of mathematical text, code, or theorem libraries. Although these approaches are highly effective in practice, it remains an open question whether an agent can autonomously discover useful theorems without such human priors. We study this question in a formal axiomatic system by developing an agent that starts from axioms and inference rules alone and gradually grows a library of useful theorems. Concretely, we propose a self-supervised theorem-discovery algorithm that alternates between proof search and useful-theorem extraction, building a theorem library whose entries are reused as lemmas for subsequent proof search. Experiments show that the agent discovers tens of thousands of theorems and finds proofs for human-written benchmark problems, suggesting that its discoveries include theorems meaningful from a human mathematical perspective. Furthermore, the discovered theorems improve LLM proof performance when provided as prompt lemmas, indicating that they can serve as external knowledge for LLM reasoning. Our results provide evidence that useful theorems can emerge from proof search without relying on human-provided theorem libraries. More broadly, they suggest a path toward self-evolving AI systems for mathematics whose discoveries remain formally verifiable.

Problem

• self-supervised
• theorem discovery
• formal axiomatic system
• mathematical reasoning
• autonomous discovery

Innovation

• self-supervised theorem discovery
• formal axiomatic system
• proof search
• lemma extraction
• AI for mathematics