Symbolic Equation Solving via Reinforcement Learning

Abstract

Machine-learning methods are gradually being adopted in a wide variety of social, economic, and scientific contexts, yet they are notorious for struggling with exact mathematics. A typical example is computer algebra, which includes tasks like simplifying mathematical terms, calculating formal derivatives, or finding exact solutions of algebraic equations. Traditional software packages for these purposes are commonly based on a huge database of rules for how a specific operation (e.g., differentiation) transforms a certain term (e.g., sine function) into another one (e.g., cosine function). These rules have usually needed to be discovered and subsequently programmed by humans. Efforts to automate this process by machine-learning approaches are faced with challenges like the singular nature of solutions to mathematical problems, when approximations are unacceptable, as well as hallucination effects leading to flawed reasoning. We propose a novel deep-learning interface involving a reinforcement-learning agent that operates a symbolic stack calculator to explore mathematical relations. By construction, this system is capable of exact transformations and immune to hallucination. Using the paradigmatic example of solving linear equations in symbolic form, we demonstrate how our reinforcement-learning agent autonomously discovers elementary transformation rules and step-by-step solutions.