The Role of Entropy in Guiding a Connection Prover

TL;DR

This paper investigates how entropy regularization in neural network-guided theorem proving enhances proof search efficiency, demonstrating that controlling confidence levels in inference selection significantly improves performance.

Contribution

It introduces entropy regularization for neural network guidance in theorem proving, showing improved proof search effectiveness over traditional methods.

Findings

Entropy regularization improves proof success rates.

Overconfidence in neural guidance hampers search efficiency.

Proper entropy control enhances theorem prover performance.

Abstract

In this work we study how to learn good algorithms for selecting reasoning steps in theorem proving. We explore this in the connection tableau calculus implemented by leanCoP where the partial tableau provides a clean and compact notion of a state to which a limited number of inferences can be applied. We start by incorporating a state-of-the-art learning algorithm -- a graph neural network (GNN) -- into the plCoP theorem prover. Then we use it to observe the system's behaviour in a reinforcement learning setting, i.e., when learning inference guidance from successful Monte-Carlo tree searches on many problems. Despite its better pattern matching capability, the GNN initially performs worse than a simpler previously used learning algorithm. We observe that the simpler algorithm is less confident, i.e., its recommendations have higher entropy. This leads us to explore how the entropy of the inference selection implemented via the neural network influences the proof search. This is related to research in human decision-making under uncertainty, and in particular the probability matching theory. Our main result shows that a proper entropy regularisation, i.e., training the GNN not to be overconfident, greatly improves plCoP's performance on a large mathematical corpus.