FactorLibrary: From Polynomials to Circuits via Recursive Subgoals

2026-06-24

AI Summary

This work addresses the fundamental problem in algebraic complexity theory of finding minimal arithmetic circuits for polynomials over finite fields. The authors formulate this task as a reinforcement learning problem and introduce a FactorLibrary mechanism that stores and reuses factorizable subexpressions as recursive subgoals, effectively mitigating search space explosion and enhancing generalization. By integrating bottom-up and top-down strategies, they optimize the approach using variants of Gumbel-PPO-MCTS, PPO+MCTS, and SAC algorithms. Among these, the top-down PPO+MCTS agent achieves the best performance, successfully synthesizing verified optimal circuits with a success rate of 91.8% on polynomials of complexity at most eight.

Abstract

Finding minimal arithmetic circuits for polynomials over finite fields is a combinatorially hard problem central to algebraic complexity theory. We formulate it as a reinforcement learning problem in two directions, bottom-up and top-down. To address the challenge of a fast-growing combinatorial search space, we introduce FactorLibrary, which stores factorizable subexpressions that serve as reusable subgoals across training episodes. We trained a bottom-up agent with Gumbel-PPO-MCTS and two top-down agents with PPO+MCTS and SAC. The PPO+MCTS top-down agent exhibited the most stable performance, finding certified optimal circuits up to complexity $8$ with a success rate of $91.8\%$.

Problem

• arithmetic circuits
• polynomials
• finite fields
• algebraic complexity
• combinatorial optimization

Innovation

• FactorLibrary
• arithmetic circuits
• reinforcement learning
• subgoal reuse
• polynomial factorization