AI system solves open algebraic combinatorics problems and raises questions
AI Discovery in Combinatorics: From Problems to Solutions and Back to Bigger Problems
Artificial IntelligenceMachine Learning
Summary
Algebraic combinatorics involves finding simple rules that explain complex math formulas. The authors created a set of 29 challenging problems that ask for such simple rules. They worked with OpenAI to see how well AI could find solutions, and the AI managed to propose answers to several problems. This also led to bigger questions about how to share AI-made discoveries ethically and how to check if the proofs are correct.
What this means in practice
- •For mathematics software engineers: Incorporate AI-driven tools to generate and test combinatorial algorithms for complex math problems, enhancing automated theorem proving workflows.
- •For mathematical data analysts: Use structured benchmarks and interactive scaffolds to validate and explore new combinatorial algorithm candidates generated by AI systems.
Authors
Eugenio Cainelli, Lorenzo Luccioli, Alessandro Iraci, Giovanni Paolini
Abstract
We introduce $qt$Bench, a benchmark of 29 open problems in algebraic combinatorics. These problems ask for a simple combinatorial algorithm that explains given polynomial coefficients or identities. To help AI agents search for such algorithms, we provide a scaffold for playing with combinatorial objects, countering cheating attempts, and testing candidate solutions. To explore how far frontier AI systems can go on these problems, we collaborated with OpenAI, whose researchers obtained proposed solutions to several of them. This highlights bigger meta-problems: is AI discovery at scale ethical? How should these results be released? How should proof digestion be organized? We discuss these questions and give our views.