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

2025-11-25

AI Summary

Constructing explicit bijections—central yet challenging in enumerative combinatorics—remains largely manual and intuition-driven, especially for open problems such as certain bijections on Dyck paths. Method: This work introduces OpenEvolve, the first evolutionary program synthesis system applied to bijection discovery, instantiated on three Dyck-path bijection problems (two known, one open). It integrates large language models for generating human-readable candidate code, formal verification–driven iterative refinement, and mathematician-in-the-loop collaboration. Contribution/Results: The system successfully reconstructs and improves two known bijections and produces a promising candidate for the open problem, demonstrating the feasibility of AI-assisted bijection discovery. However, it also exposes limitations in deep semantic understanding and mathematical intuition modeling, underscoring the necessity of human–AI synergy. This work establishes the first application paradigm of evolutionary program synthesis to combinatorial bijection research and opens a new avenue for AI-augmented theoretical mathematics exploration.

Technology Category

• Humans and AI: Other Foundations of Human Computation & AI
• Search and Optimization: Evolutionary Computation
• Machine Learning: Evolutionary Learning

Application Category

• Economics, Online Markets and Human Computation: Research challenges in human and human-AI computation
• Web Mining and Content Analysis: Models for Web evolution
• Graph Algorithms and Modeling for the Web: Representation, reconstruction, and subgraph or motif discovery in Web-related graphs

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.

Problem

• Exploring evolutionary AI systems for combinatorial bijection discovery
• Applying OpenEvolve to known and open Dyck path bijection problems
• Assessing challenges in finding novel bijections with current AI systems

Innovation

• Uses teams of LLMs to generate candidate solutions
• Evolves solutions beyond single LLM generation capabilities
• Applies evolutionary synthesis to combinatorial bijection discovery