CircuitBuilder: From Polynomials to Circuits via Reinforcement Learning
This addresses the challenge of auto-proof generation and theoretical complexity for researchers in computational algebra, though it is incremental as it applies existing RL methods to a new domain.
The paper tackles the problem of discovering efficient arithmetic circuits to compute polynomials by formulating it as a reinforcement learning game, where SAC achieves high success rates on two-variable targets and PPO+MCTS scales to three variables with steady improvement.
Motivated by auto-proof generation and Valiant's VP vs. VNP conjecture, we study the problem of discovering efficient arithmetic circuits to compute polynomials, using addition and multiplication gates. We formulate this problem as a single-player game, where an RL agent attempts to build the circuit within a fixed number of operations. We implement an AlphaZero-style training loop and compare two approaches: Proximal Policy Optimization with Monte Carlo Tree Search (PPO+MCTS) and Soft Actor-Critic (SAC). SAC achieves the highest success rates on two-variable targets, while PPO+MCTS scales to three variables and demonstrates steady improvement on harder instances. These results suggest that polynomial circuit synthesis is a compact, verifiable setting for studying self-improving search policies.