CircuitBuilder: From Polynomials to Circuits via Reinforcement Learning

TL;DR

This paper introduces a reinforcement learning framework for synthesizing arithmetic circuits that compute polynomials, demonstrating the effectiveness of RL algorithms like SAC and PPO+MCTS in circuit discovery tasks.

Contribution

It formulates polynomial circuit synthesis as a reinforcement learning problem and compares RL algorithms, showing their potential for efficient circuit construction.

Findings

SAC outperforms other methods on two-variable targets.

PPO+MCTS scales to three variables and improves on complex instances.

Reinforcement learning effectively guides circuit synthesis for polynomials.

Abstract

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.