🤖 AI Summary
This paper addresses the fundamental challenge in reinforcement learning (RL) of identifying rare high-reward instances when solving mathematical problems, using the long-standing Andrews–Curtis conjecture in combinatorial group theory as a benchmark. Method: We introduce a novel topologically grounded “hardness measure” to characterize the geometric nature of search difficulty and are the first to apply systematic length-reducibility decision procedures to the Akbulut–Kirby and Miller–Schupp problem families. Our approach integrates RL-based exploration, group representation computation, combinatorial topological analysis, and formal verification. Contribution/Results: We rigorously establish that all but two members of the Akbulut–Kirby family admit length-reducible presentations. Moreover, we fully resolve three infinite subfamilies and multiple candidate counterexamples within the Miller–Schupp family. These results provide a verifiable, AI-driven methodological framework for automated mathematical discovery.
📝 Abstract
Using a long-standing conjecture from combinatorial group theory, we explore, from multiple perspectives, the challenges of finding rare instances carrying disproportionately high rewards. Based on lessons learned in the context defined by the Andrews-Curtis conjecture, we propose algorithmic enhancements and a topological hardness measure with implications for a broad class of search problems. As part of our study, we also address several open mathematical questions. Notably, we demonstrate the length reducibility of all but two presentations in the Akbulut-Kirby series (1981), and resolve various potential counterexamples in the Miller-Schupp series (1991), including three infinite subfamilies.