- The paper introduces a transformer-based model that learns from extensive HOL4 proof libraries and achieves up to 93.7% accuracy in top tactic recommendations.
- It employs multiple transformer models, with RoBERTa emerging as the most effective after rigorous hyperparameter tuning and grid search.
- The study paves the way for enhanced automated theorem proving by improving tactic prediction, with future work aimed at full proof generation and broader application.
Proof Recommendation System for the HOL4 Theorem Prover
This paper presents the development and evaluation of a proof recommendation system tailored for the HOL4 theorem prover. Utilizing a transformer-based model architecture, the system is designed to recognize patterns inherent in theorem proving within HOL4 environments and to recommend probable subsequent tactics in proof development. This is achieved by analyzing extensive proof libraries to predict the next logical move from a sequence of previously applied tactics.
The construction of this system involved several critical steps. Initially, a comprehensive dataset of HOL4 proofs was assembled. The proofs were abstracted to focus solely on the tactics used in proving theorems and lemmas. The authors utilized data from five distinct HOL4 theories developed by the Hardware Verification Group (HVG) at Concordia University, complemented by an existing dataset based on real arithmetic theory. These datasets were unified into a single amalgamated dataset for training and testing purposes.
For the model development phase, the researchers employed several transformer-based LLMs, namely BERT, RoBERTa, and T5. These models were examined to identify which provided the most effective performance in predicting proof steps. The main performance metric used was the n-correctness rate—a measure reflecting how often the actual necessary next step is included within the top n recommendations. Through fine-tuning and hyperparameter optimization via grid search, RoBERTa emerged as the most effective model, especially for n = 7 in top tactic predictions at k=1 (predicting one future tactic step).
Results indicate that RoBERTa achieved varying degrees of success across different datasets. It demonstrated a high accuracy of 97.8% for Dataset 4 and a notable 89.8% accuracy when analyzing the combined Dataset 7. However, the performance diminished when attempting to predict two future tactics, highlighting the model's limitations stemming from increasing combinatorial complexity. Such variance in prediction accuracy is attributed to the inherent structural differences among the datasets, with some exhibiting uniform characteristics due to their single-author origin, while others offered more diverse theorem representations sourced from broader mathematical libraries.
When compared against previous AI-assisted theorem prover tools, the HOL4 proof recommendation system exhibits superior performance in tactic prediction. The paper notes that previous research efforts reported lower accuracy rates, such as 70% accuracy for top-three recommendations. In contrast, this system attains up to 93.7% accuracy for the top ten recommendations. This advancement underscores the potential of integrating advanced LLMs into theorem proving systems, providing a more robust framework than traditional approaches.
Future directions for this research include expanding the model's applicability to a broader range of HOL4 theories and refining the user interface for better integration. There is also an aspiration to enable the automatic generation of complete proofs. This will likely involve utilizing advanced tree search algorithms to tackle the exponentially increasing complexity of proof sequences, a challenge inherent in real-world applications of automated theorem proving.
These efforts are poised to significantly enhance the performance and accuracy of theorem proving in automated environments, presenting profound implications for both the development of proof systems and the theoretical exploration within the domain of automated reasoning and formal methods.