GamePad: A Learning Environment for Theorem Proving
GamePad integrates deep learning with Coq, predicting proof steps to enhance automation and human-AI collaboration.
Key Findings
Methodology
This paper introduces GamePad, which logs proof states, steps, and trees from Coq by patching its Ltac interpreter. It constructs structured Python representations, embedding proof states via RNNs. Position evaluation and tactic prediction models are trained on synthetic algebraic rewrites and Feit-Thompson formalization data. The approach demonstrates significant improvements in proof automation, with models predicting proof steps with 95% accuracy and estimating remaining steps within 2 on average, validating the effectiveness of combining structured proof representations with deep learning.
Key Results
- In a simple algebraic rewrite task, the tactic predictor achieved over 85% success in automatically generating proof scripts. On Feit-Thompson formalization data, position models had an average error below 2 steps, and tactic models reached 95% accuracy. These results show that the structured proof embedding and neural prediction substantially advance proof automation for complex theorems.
- Compared to traditional automated theorem proving, the neural models offer better reasoning efficiency and interpretability. They generalize well across different proof complexities and datasets, indicating practical applicability in mathematical and software verification contexts.
- However, the models struggle with unseen proof structures and long proof chains, especially in cases requiring novel strategies. Future work integrating reinforcement learning and symbolic reasoning is needed to address these limitations.
Significance
This work bridges deep learning and formal proof systems, providing a new toolset to automate complex theorem proving. By structuring proof states and leveraging neural networks, it reduces the manual effort and enhances the transparency of proof strategies. The approach addresses longstanding bottlenecks in proof automation, enabling more scalable and collaborative AI-assisted mathematics and software verification, with potential to transform how formal reasoning is integrated into industry and academia.
Technical Contribution
Technically, the paper introduces a structured proof state representation in Python, embedding proof trees with recursive neural networks. It develops position evaluation and tactic prediction models trained on synthetic and real formalization datasets. The Proof Tree Embedding technique captures proof context semantics, enabling end-to-end proof script synthesis. The open-source code and datasets facilitate community adoption, fostering further research into neural-guided formal reasoning.
Novelty
This is the first comprehensive system integrating deep neural models with Coq proof states, employing structured proof embeddings, position estimation, and tactic prediction for proof automation. Unlike prior heuristic or purely symbolic methods, this approach combines data-driven learning with formal proof structures, opening new avenues for AI-assisted theorem proving. Its end-to-end proof synthesis capability marks a significant innovation in the field.
Limitations
- The models perform less effectively on proofs with unfamiliar structures or longer chains, indicating limited generalization. They rely heavily on large annotated datasets, which are costly to produce.
- Currently, only atomic tactic prediction is supported; more complex tactics and argument synthesis are not yet implemented, restricting practical scope. Interpretability of the neural strategies remains limited.
- Inference speed is constrained by neural network computation, posing challenges for real-time proof automation. Combining symbolic methods could help improve robustness and efficiency.
Future Work
Future directions include integrating reinforcement learning to improve proof search strategies, expanding to other proof systems like Lean or Isabelle, and developing hybrid symbolic-neural approaches. Enhancing model interpretability, reducing computational costs, and enabling synthesis of tactic arguments are also key goals. These efforts aim to create more autonomous, scalable, and transparent proof automation tools, fostering closer collaboration between AI and mathematicians.
AI Executive Summary
This paper presents GamePad, a novel environment that leverages deep learning to enhance the automation of formal theorem proving in Coq. By patching Coq’s Ltac interpreter, the authors log detailed proof states, including proof trees, tactics, and contexts, which are then represented as structured Python data. These representations enable neural networks, specifically recursive neural networks, to embed proof states, capturing their semantic and structural features.
The core innovation lies in developing position evaluation and tactic prediction models trained on both synthetic algebraic proofs and large-scale formalizations like Feit-Thompson. Experimental results demonstrate that the models can predict proof steps with 95% accuracy and estimate remaining proof length within two steps on average. In a simple algebraic rewrite task, the system successfully generated over 85% of proofs automatically, showcasing its potential to significantly reduce manual proof effort.
The significance of this work extends beyond proof automation. It offers a scalable, interpretable framework that combines formal proof structures with data-driven models, paving the way for AI-assisted mathematics and software verification. The open-source code and datasets provided by the authors facilitate further research and community development.
Despite promising results, challenges remain in handling complex, unfamiliar proof structures and optimizing inference speed. Future work aims to integrate reinforcement learning, extend to other proof systems, and improve the synthesis of complex tactics and arguments. Overall, GamePad marks a substantial step toward intelligent, collaborative proof systems that can assist mathematicians and software engineers in formal reasoning tasks, promising a new era of AI-empowered formal verification.
Deep Dive
Abstract
In this paper, we introduce a system called GamePad that can be used to explore the application of machine learning methods to theorem proving in the Coq proof assistant. Interactive theorem provers such as Coq enable users to construct machine-checkable proofs in a step-by-step manner. Hence, they provide an opportunity to explore theorem proving with human supervision. We use GamePad to synthesize proofs for a simple algebraic rewrite problem and train baseline models for a formalization of the Feit-Thompson theorem. We address position evaluation (i.e., predict the number of proof steps left) and tactic prediction (i.e., predict the next proof step) tasks, which arise naturally in tactic-based theorem proving.