Explaining Interactive Theorem Proving Tactic Prediction Using Inductive Logic Programming

Concepts de base

This paper proposes a novel approach to improve tactic prediction in Interactive Theorem Provers (ITPs) by leveraging Inductive Logic Programming (ILP) to learn explainable rules for tactic selection, enhancing both the accuracy and interpretability of existing methods like k-NN.

Résumé

Bibliographic Information:

Zhang, L., Cerna, D.M., & Kaliszyk, C. (2024). Learning Rules Explaining Interactive Theorem Proving Tactic Prediction. arXiv preprint arXiv:2411.01188v1.

Research Objective:

This paper aims to address the challenges of tactic prediction in ITPs, particularly focusing on improving the accuracy and interpretability of existing machine learning-based approaches.

Methodology:

The authors propose using Inductive Logic Programming (ILP) to learn rules explaining tactic prediction. They represent the problem as an ILP task and enrich the feature space by encoding computationally expensive properties as background knowledge predicates. This enriched feature space is then used to learn rules explaining when a tactic is applicable to a given proof state. These learned rules are then used to filter the output of an existing k-NN tactic selection approach.

Key Findings:

• The proposed approach of combining ILP and k-NN improves the accuracy of tactic suggestions in Tactician, a prominent tactic prediction system for Coq.
• Feature predicates, which dynamically calculate features based on the representation of the Abstract Syntax Tree (AST) of the proof state, are shown to learn more precise rules compared to representation predicates.
• The use of anonymous predicates, which abstract away specific identifiers, further enhances the generalization ability of the learned rules.

Main Conclusions:

This research demonstrates the potential of ILP as a valuable tool for improving tactic suggestion methods in ITPs. The learned rules provide explainable predictions, enhancing user understanding and trust in the suggested tactics.

Significance:

This work represents the first application of ILP to interactive theorem proving, opening up new avenues for research in this domain. The proposed approach has the potential to significantly improve the usability and effectiveness of ITPs, making formal verification more accessible to a wider range of users.

Limitations and Future Research:

• The current approach does not generalize tactics with different arguments and struggles to describe inherently complex tactics like induction.
• Future work could explore the use of stronger ILP systems, capture relations between tactic arguments and their referred objects, and investigate the application of ILP to other ITP tasks.

Stats

Coq's standard library, consisting of 41 theories and 151,678 proof states, was used as the benchmark dataset.
The theory "Structures" was chosen for training due to its balanced distribution of tactics.
Testing was performed on theories that do not depend on "Structures," including "rtauto," "FSets," "Wellfounded," "funind," "btauto," "nsatz," and "MSets."
The validation dataset comprised five randomly chosen theories: "PArith," "Relations," "Bool," "Logic," and "Lists."
A timeout of ten minutes was set for the ILP learning process.

Citations

"This is the first time an investigation has considered ILP as a tool for improving tactic suggestion methods for ITPs."
"Our hypothesis is that features of proof state defined through logic programs can be used to learn rules which can be used to filter the output of a k-NN model to improve accuracy."
"In addition to improved performance, our approach produces rules to explain the predictions."

Idées clés tirées de

Learning Rules Explaining Interactive Theorem Proving Tactic Prediction

by Liao Zhang, ... à arxiv.org 11-05-2024

https://arxiv.org/pdf/2411.01188.pdf

Questions plus approfondies

How can the proposed ILP-based approach be integrated with other machine learning techniques beyond k-NN to further enhance tactic prediction in ITPs?

This question explores the exciting intersection of Inductive Logic Programming (ILP) with other machine learning techniques for improving tactic prediction in Interactive Theorem Provers (ITPs). Here's a breakdown of potential integration strategies:
1. Ensemble Methods:

Combining ILP with Probabilistic Models: ILP's strength lies in learning interpretable rules, while probabilistic models like Bayesian Networks or Hidden Markov Models excel at capturing uncertainty and dependencies in sequential data (like proof steps). An ensemble could leverage ILP rules as high-level constraints or features within a probabilistic framework. This would provide both interpretability and a principled way to handle uncertainty in tactic selection.
Boosting with ILP Rules: Boosting algorithms like AdaBoost could be adapted to incorporate ILP-learned rules as weak learners. The boosting process would iteratively weight training examples and rules to focus on challenging proof states, potentially leading to a more robust and accurate tactic prediction model.
2. Deep Learning Integration:

ILP for Feature Engineering: Deep learning models often benefit from carefully engineered features. ILP could be used to automatically discover complex, non-linear relationships between proof state elements, which can then be fed as features into a Convolutional Neural Network (CNN) or a Graph Neural Network (GNN) for tactic prediction. This leverages ILP's symbolic reasoning capabilities to enhance the representation power of deep learning.
Knowledge Distillation: Train a deep learning model (student) to mimic the behavior of an ILP-based tactic predictor (teacher). This can transfer the knowledge captured by ILP rules into a more scalable deep learning architecture, potentially improving prediction speed without sacrificing too much accuracy or interpretability.
3. Reinforcement Learning:

ILP as Reward Shaping: In a reinforcement learning setting for theorem proving, ILP rules could be used to provide intermediate rewards for taking actions (applying tactics) that align with the learned rules. This reward shaping can guide the RL agent towards promising proof paths more efficiently.
ILP for Policy Initialization: An ILP-learned policy can serve as a good starting point for an RL agent, reducing the exploration space and accelerating the learning process.
Challenges and Considerations:

Data Efficiency: ILP typically requires more labeled data than some other techniques. Strategies for efficient data augmentation or active learning would be crucial.
Scalability: Integrating ILP with complex models like deep networks requires careful optimization to ensure reasonable training and prediction times.
Interpretability Trade-offs: While ILP enhances interpretability, striking a balance between explainability and the performance gains from other techniques is essential.

Could the limitations of the current approach in handling complex tactics like induction be addressed by incorporating domain-specific knowledge or heuristics into the ILP learning process?

Yes, incorporating domain-specific knowledge and heuristics into the ILP learning process holds significant promise for addressing the limitations of handling complex tactics like induction. Here's how:
1. Background Knowledge Enhancement:

Induction Schemes: Provide ILP with background knowledge about common induction schemes used in the specific theorem proving domain. This could include structural induction on data types, well-founded induction, or domain-specific induction principles.
Tactical Lemmas: Include a library of previously proven lemmas that are frequently used in inductive proofs. ILP can then learn rules that recognize situations where these lemmas are applicable.
Type Information: Leverage the rich type information available in ITPs to guide the ILP system. Rules can be learned that take into account the types of variables and expressions, making the induction hypothesis generation more targeted.
2. Heuristic Guidance:

Ripple-Down Rules: Integrate ILP with a ripple-down rules system. This allows for the incremental refinement of learned rules by human experts, capturing subtle patterns and exceptions that are difficult for ILP to learn from data alone.
Constraint-Based ILP: Use a constraint logic programming (CLP) system as the underlying engine for ILP. This enables the incorporation of domain-specific constraints and heuristics directly into the learning process, guiding the search for relevant rules.
3. Example Selection and Weighting:

Prioritize Inductive Proofs: During training, focus on examples involving inductive proofs. This allows ILP to specialize in learning rules relevant to induction.
Weight Complex Examples: Assign higher weights to complex inductive proofs, ensuring that the learned rules are robust to challenging cases.
Benefits of Domain Knowledge:

Improved Accuracy: By incorporating domain-specific knowledge, ILP can learn more accurate rules for complex tactics, leading to better tactic prediction.
Enhanced Interpretability: Rules learned with domain knowledge are more likely to align with human intuition, making them easier for users to understand and trust.
Targeted Learning: Domain knowledge helps focus the ILP learning process on relevant aspects of the problem, potentially reducing training time and improving data efficiency.
Challenges:

Knowledge Acquisition: Acquiring and formalizing domain-specific knowledge can be time-consuming and require expertise.
Overfitting: Care must be taken to avoid overfitting to the specific knowledge provided, ensuring that the learned rules generalize well to unseen proofs.

What are the potential implications of using explainable AI techniques like ILP in formal verification for ensuring the reliability and trustworthiness of critical software systems?

Using explainable AI (XAI) techniques like ILP in formal verification has profound implications for enhancing the reliability and trustworthiness of critical software systems. Here's an exploration of the potential benefits:
1. Increased Trust and Confidence:

Transparent Decision-Making: ILP provides human-readable rules that explain why a particular verification step (e.g., tactic application) is recommended. This transparency fosters trust in the verification process, especially for critical systems where errors can have severe consequences.
Auditable Verification: The rule-based nature of ILP makes the verification process more auditable. Experts can inspect the learned rules to ensure they are sound, complete, and align with domain-specific requirements.
2. Error Detection and Correction:

Identifying False Positives/Negatives: Explainable models help identify cases where the verification system might be overly optimistic (false positives) or pessimistic (false negatives). By understanding the reasoning behind these errors, developers can refine the verification process and improve its accuracy.
Debugging Assistance: When errors are detected, ILP rules can provide insights into the underlying causes. This aids developers in pinpointing the source of the problem and devising appropriate fixes.
3. Knowledge Discovery and Formalization:

Extracting Verification Expertise: ILP can be used to extract knowledge from existing formally verified systems or from the expertise of human verification engineers. This knowledge can then be reused and shared, facilitating the development of more reliable software.
Formalizing Best Practices: ILP-learned rules can contribute to the formalization of best practices and design patterns for developing critical systems. This promotes consistency and reduces the likelihood of errors.
4. Regulatory Compliance and Certification:

Explainability for Certification: In safety-critical domains like avionics or medical devices, certification authorities often require evidence of a system's reliability. Explainable AI techniques like ILP can provide the necessary transparency and justification for certification.
Compliance with Ethical Standards: As AI systems play an increasingly important role in critical applications, there is growing demand for ethical and responsible AI development. Explainable AI aligns with these principles by providing insights into the decision-making process.
Challenges and Considerations:

Complexity vs. Explainability: Finding the right balance between the complexity of the verification system and the interpretability of the explanations is crucial. Overly complex rules might be difficult for humans to understand.
Scalability to Large Systems: Applying ILP to the verification of extremely large and complex software systems remains a challenge. Techniques for modularization and abstraction are needed.
Human-in-the-Loop Validation: While ILP enhances explainability, human experts are still essential for validating the learned rules and ensuring their correctness.
In conclusion, integrating explainable AI techniques like ILP into formal verification holds immense potential for building more reliable and trustworthy critical software systems. By providing transparency, facilitating error detection, and enabling knowledge discovery, XAI can contribute significantly to the development of safer and more dependable software for critical applications.