Learning to Discover Efficient Mathematical Identities

TL;DR

This paper presents machine learning methods, including an n-gram model and a recursive neural network, to efficiently discover mathematical identities that reduce computational complexity, surpassing brute-force and human capabilities.

Contribution

It introduces a novel attribute grammar framework and two learning approaches to guide symbolic expression tree search for discovering efficient mathematical identities.

Findings

Learned identities beyond brute-force reach

Guided tree search improves discovery efficiency

Neural network approach outperforms simple models

Abstract

In this paper we explore how machine learning techniques can be applied to the discovery of efficient mathematical identities. We introduce an attribute grammar framework for representing symbolic expressions. Given a set of grammar rules we build trees that combine different rules, looking for branches which yield compositions that are analytically equivalent to a target expression, but of lower computational complexity. However, as the size of the trees grows exponentially with the complexity of the target expression, brute force search is impractical for all but the simplest of expressions. Consequently, we introduce two novel learning approaches that are able to learn from simpler expressions to guide the tree search. The first of these is a simple n-gram model, the other being a recursive neural-network. We show how these approaches enable us to derive complex identities, beyond reach of brute-force search, or human derivation.