Training a First-Order Theorem Prover from Synthetic Data

A major challenge in applying machine learning to automated theorem proving is the scarcity of training data, which is a key ingredient in training successful deep learning models. To tackle this problem, we propose an approach that relies on training purely with synthetically generated theorems, without any human data aside from axioms. We use these theorems to train a neurally-guided saturationbased prover. Our neural prover outperforms the state-of-the-art E-prover on this synthetic data in both time and search steps, and shows significant transfer to the unseen human-written theorems from the TPTP library, where it solves 72% of first-order problems without equality. Most work applying machine learning to theorem proving takes the following approach: 1) pick a dataset of formalized mathematics, such as Mizar or Metamath, or the standard library of a major proof assistant such as HOL-Light or Coq; 2) split the dataset into train and test; 3) use imitation learning or reinforcement learning on the training set to learn a policy; and finally 4) evaluate the policy on the test set (Loos et al. (2017), Bansal et al. (2019), Yang & Deng (2019), Han et al. (2021), Polu & Sutskever (2020)). Such methods are fundamentally limited by the size of the training set, particularly when relying on deep neural networks (Kaplan et al., 2020). Unfortunately, unlike in computer vision and natural language processing, theorem proving datasets are comparatively tiny.