DeepMath - Deep Sequence Models for Premise Selection (1606.04442v2)

Abstract: We study the effectiveness of neural sequence models for premise selection in automated theorem proving, one of the main bottlenecks in the formalization of mathematics. We propose a two stage approach for this task that yields good results for the premise selection task on the Mizar corpus while avoiding the hand-engineered features of existing state-of-the-art models. To our knowledge, this is the first time deep learning has been applied to theorem proving on a large scale.

• The paper shows that deep neural networks can effectively support automated premise selection in theorem proving by eliminating the need for hand-engineered features.
• It empirically compares network architectures, revealing that convolutional models with semantic embeddings outperform other approaches for selecting relevant premises.
• The research implies that enhanced neural premise selection substantially boosts ATP efficiency, paving the way for more reliable and scalable automated reasoning systems.

Summary of "DeepMath - Deep Sequence Models for Premise Selection"

The paper "DeepMath - Deep Sequence Models for Premise Selection" investigates the application of deep neural networks for premise selection in automated theorem proving (ATP). The authors address two significant bottlenecks in formalized mathematics: the lack of semantic parsing tools for informal mathematics (autoformalization) and the limited capacity of ATP systems to deduce formalized proofs on a large scale. This research primarily focuses on enhancing the effectiveness of automated reasoning by proposing a novel approach to premise selection, utilizing neural sequence models that avoid the complexities of hand-engineered features typical in conventional state-of-the-art models.

Key Contributions

1. Neural Network Application in Theorem Proving: This paper marks the first large-scale implementation of deep learning for automated theorem proving. The findings demonstrate that neural network models can effectively support large-scale logical reasoning, circumventing the need for manually crafted features.
2. Comparison of Network Architectures: The paper provides an empirical evaluation of various neural network architectures, such as convolutional networks (CNNs), recurrent networks (RNNs), and hybrid models. The results indicated that CNNs, particularly with semantic-aware definition embeddings for function symbols, yield superior performance for premise selection tasks.
3. Improved Premise Selection Methodology: The research introduces a method that employs definition-based embeddings to enhance the generalization capabilities of formulas containing infrequently occurring symbols. When combined with previous methods that employ hand-engineered features, these neural network models produce even stronger results.
4. Practical Evaluation and Experimentation: Using the Mizar Mathematical Library (MML), a large corpus of formalized mathematical proofs, the paper shows that neural-network-based premise selection can be effectively utilized alongside ATP systems, significantly boosting their proof-search capabilities.

Implications and Future Directions

The findings have several practical implications. Enhancing premise selection in ATP systems can substantially improve the efficiency and usability of theorem proving tools, contributing to more rapid and cost-effective verification of complex software and hardware systems. The research also impacts the field of automated reasoning in large theories, suggesting that deep learning can play a pivotal role in bridging the gap between human-assisted and fully automated proof generation.

Theoretically, the success of deep learning in this context may inspire further exploration into more sophisticated neural architectures and learning strategies tailored to the unique task of theorem proving. Future developments might include integrating more advanced hierarchical or recursive models that better capture the logical structures inherent in mathematical proofs. Additionally, the introduction of more refined optimization techniques, such as curriculum learning, could further enhance the training and performance of neural networks in theorem proving applications.

In conclusion, the paper provides evidence that neural networks, when properly adapted and applied, offer a promising new tool for improving automated reasoning systems. As the field of artificial intelligence continues to evolve, the potential for even deeper integration of machine learning with formalized mathematical reasoning seems increasingly plausible, inviting further investigation and advancement.