3D-Prover: Diversity Driven Theorem Proving With Determinantal Point Processes

A key challenge in automated formal reasoning is the intractable search space, which grows exponentially with the depth of the proof. This branching is caused by the large number of candidate proof tactics which can be applied to a given goal. Nonetheless, many of these tactics are semantically similar or lead to an execution error, wasting valuable resources in both cases. We address the problem of effectively pruning this search, using only synthetic data generated from previous proof attempts. We first demonstrate that it is possible to generate semantically aware tactic representations which capture the effect on the proving environment, likelihood of success, and execution time. We then propose a novel filtering mechanism which leverages these representations to select semantically diverse and high quality tactics, using Determinantal Point Processes. Our approach, 3D- Prover, is designed to be general, and to augment any underlying tactic generator. We demonstrate the effectiveness of 3D-Prover on the miniF2F and LeanDojo benchmarks by augmenting popular open source proving LLMs. We show that our approach leads to an increase in the overall proof rate, as well as a significant improvement in the tactic success rate, execution time and diversity. We make our code available at https://github.com/sean-lamont/3D-Prover.