Automated Proof of Polynomial Inequalities via Reinforcement Learning

2025-03-09

AI Summary

Traditional algebraic methods for automated polynomial inequality proving are constrained by degree truncation, limiting their efficacy on high-degree or sparse polynomials. Method: This paper proposes a reinforcement learning (RL)-based framework that reformulates inequality proving as a linear programming problem, equivalently cast as a basis-selection task over the Krivine–Stengle nonnegative basis representation. It introduces Proximal Policy Optimization (PPO) to efficiently search the basis space for feasible representations and incorporates a fast Fourier transform (FFT)-accelerated multivariate polynomial multiplication to drastically speed up action evaluation. Contribution/Results: We implement the open-source tool APPIRL, which outperforms state-of-the-art algebraic provers on standard benchmarks. Moreover, APPIRL successfully computes tight semidefinite relaxations for the maximum stable set problem, demonstrating both empirical effectiveness and generalization capability beyond synthetic benchmarks.

Technology Category

• Constraint Satisfaction and Optimization: Satisfiability Modulo Theories
• Knowledge Representation and Reasoning: Automated Reasoning and Theorem Proving
• Search and Optimization: Learning to Search

Application Category

• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking
• Graph Algorithms and Modeling for the Web: Foundation models and LLMs for Web-related graphs
• Systems and Infrastructure for Web, Mobile and WoT: Applied ML and AI for Web-based mobile applications

Abstract

Polynomial inequality proving is fundamental to many mathematical disciplines and finds wide applications in diverse fields. Current traditional algebraic methods are based on searching for a polynomial positive definite representation over a set of basis. However, these methods are limited by truncation degree. To address this issue, this paper proposes an approach based on reinforcement learning to find a {Krivine-basis} representation for proving polynomial inequalities. Specifically, we formulate the inequality proving problem as a linear programming (LP) problem and encode it as a basis selection problem using reinforcement learning (RL), achieving a non-negative {Krivine basis}. Moreover, a fast multivariate polynomial multiplication method based on Fast Fourier Transform (FFT) is employed to enhance the efficiency of action space search. Furthermore, we have implemented a tool called {APPIRL} (Automated Proof of Polynomial Inequalities via Reinforcement Learning). Experimental evaluation on benchmark problems demonstrates the feasibility and effectiveness of our approach. In addition, {APPIRL} has been successfully applied to solve the maximum stable set problem.

Problem

• Proves polynomial inequalities using reinforcement learning.
• Overcomes truncation degree limitations in traditional methods.
• Enhances efficiency with FFT-based polynomial multiplication.

Innovation

• Reinforcement learning for polynomial inequalities
• Fast Fourier Transform enhances search efficiency
• APPIRL tool automates inequality proving process