Even with AI, Bijection Discovery is Still Hard: The Opportunities and Challenges of OpenEvolve for Novel Bijection Construction

TL;DR

This paper investigates the use of OpenEvolve, an AI-assisted program synthesis system, for discovering combinatorial bijections, highlighting its potential and current limitations in aiding mathematical research.

Contribution

The paper demonstrates the application of OpenEvolve to bijection problems, including an open problem, and discusses lessons learned about its capabilities and challenges.

Findings

OpenEvolve shows promise for combinatorial problem solving.

Current systems struggle with discovering novel, research-level bijections.

Human mathematicians remain essential in complex bijection discovery.

Abstract

Evolutionary program synthesis systems such as AlphaEvolve, OpenEvolve, and ShinkaEvolve offer a new approach to AI-assisted mathematical discovery. These systems utilize teams of large language models (LLMs) to generate candidate solutions to a problem as human readable code. These candidate solutions are then 'evolved' with the goal of improving them beyond what an LLM can produce in a single shot. While existing mathematical applications have mostly focused on problems of establishing bounds (e.g., sphere packing), the program synthesis approach is well suited to any problem where the solution takes the form of an explicit construction. With this in mind, in this paper we explore the use of OpenEvolve for combinatorial bijection discovery. We describe the results of applying OpenEvolve to three bijection construction problems involving Dyck paths, two of which are known and one of which is open. We find that while systems like OpenEvolve show promise as a valuable tool for combinatorialists, the problem of finding novel, research-level bijections remains a challenging task for current frontier systems, reinforcing the need for human mathematicians in the loop. We describe some lessons learned for others in the field interested in exploring the use of these systems.