Deep Reinforcement Learning for Synthesizing Functions in Higher-Order Logic

TL;DR

This paper introduces a deep reinforcement learning framework using tree neural networks and Monte Carlo Tree Search within HOL4 to synthesize functions in higher-order logic, demonstrating significant success in combinator and Diophantine problems.

Contribution

It presents a novel integration of TNNs and MCTS for function synthesis in HOL4, enabling recursive improvement and outperforming existing ATPs in certain tasks.

Findings

65% success rate on combinator synthesis tasks

78.5% success rate on Diophantine equation problems

Demonstrates effective self-supervised learning in proof assistant environment

Abstract

The paper describes a deep reinforcement learning framework based on self-supervised learning within the proof assistant HOL4. A close interaction between the machine learning modules and the HOL4 library is achieved by the choice of tree neural networks (TNNs) as machine learning models and the internal use of HOL4 terms to represent tree structures of TNNs. Recursive improvement is possible when a task is expressed as a search problem. In this case, a Monte Carlo Tree Search (MCTS) algorithm guided by a TNN can be used to explore the search space and produce better examples for training the next TNN. As an illustration, term synthesis tasks on combinators and Diophantine equations are specified and learned. We achieve a success rate of 65% on combinator synthesis problems outperforming state-of-the-art ATPs run with their best general set of strategies. We set a precedent for statistically guided synthesis of Diophantine equations by solving 78.5% of the generated test problems.