Leveraging Trial-and-Error Data to Enhance Large Language Model Performance in Intuitionistic Propositional Logic Theorem Proving

Core Concepts

Incorporating trial-and-error information during training and inference can significantly improve the performance of large language models in solving intuitionistic propositional logic theorems compared to models trained only on successful proof paths.

Abstract

The paper presents a new dataset, PropL, for intuitionistic propositional logic theorems formalized in the Lean theorem prover. The dataset includes complete proof search information, capturing both successful and failed proof attempts.
The authors fine-tune a large language model, TRIALMASTER, on the PropL dataset, which contains trial-and-error data. They compare TRIALMASTER's performance to a conventional depth-first search (DFS) system trained only on successful proof paths.
The key findings are:

TRIALMASTER outperforms the DFS system on out-of-distribution theorems, achieving a higher proof search success rate with a lower search cost.
The trial-and-error information enables TRIALMASTER to effectively learn backtracking capabilities, allowing it to navigate the proof search space more efficiently.
An ablation study confirms the importance of training with trial-and-error data, as a model trained without it performs significantly worse.
The authors also find that training on shorter proof paths with trial-and-error information leads to better performance than training on longer proof paths.
The paper demonstrates the benefits of incorporating trial-and-error data in training and inference for large language models tackling theorem-proving tasks in the domain of intuitionistic propositional logic.

Stats

The dataset PropL contains 200,000 theorems of intuitionistic propositional logic, with 109,887 theorems used for training and 1,000 each for in-distribution and out-of-distribution testing.

Quotes

"Intuitively, a tactic that leads to a failed search path would indicate that similar tactics should receive less attention during the following trials."
"We demonstrate the benefit of training models that additionally learn from failed search paths."

Key Insights Distilled From

Learn from Failure

by Chenyang An,... at arxiv.org 04-12-2024

https://arxiv.org/pdf/2404.07382.pdf

Deeper Inquiries

How can the trial-and-error learning approach be extended to other mathematical domains beyond intuitionistic propositional logic?

In order to extend the trial-and-error learning approach to other mathematical domains, several key steps can be taken:

Dataset Creation: Just like in the context of intuitionistic propositional logic, creating comprehensive datasets with trial-and-error information for other mathematical domains is crucial. This involves generating a wide range of theorems and proofs, including both successful and failed attempts.

Model Training: Utilizing large language models (LLMs) for training on these datasets is essential. The models should be fine-tuned on proofs with trial-and-error information to learn from both successful and unsuccessful proof attempts.

Inference Process: During the inference stage, the model should be prompted with the entire proof tree, including failed attempts and backtracking information. This allows the model to make informed decisions based on past experiences.

Evaluation and Iteration: Continuous evaluation of the model's performance on a diverse set of mathematical theorems is necessary. Iterative improvements can be made based on the insights gained from the model's behavior.

By following these steps and adapting the trial-and-error learning approach to different mathematical domains, it is possible to enhance the model's ability to reason and prove theorems effectively across a wide range of mathematical concepts.

What are the potential limitations or drawbacks of relying too heavily on trial-and-error data during training, and how can they be mitigated?

While leveraging trial-and-error data during training can be beneficial, there are potential limitations and drawbacks to consider:

Overfitting to Failed Attempts: Relying too heavily on trial-and-error data may cause the model to overfit to unsuccessful proof paths, leading to suboptimal performance on unseen data. This can be mitigated by balancing the training data with a mix of successful and failed attempts.

Increased Training Complexity: Training on a large amount of trial-and-error data can increase the complexity of the model and training process, potentially leading to longer training times and higher computational costs. This can be addressed by optimizing the training pipeline and dataset curation process.

Limited Generalization: Models trained extensively on trial-and-error data may struggle to generalize to new and diverse mathematical theorems outside the training set. Regularization techniques and diverse dataset creation can help improve generalization.

Biased Learning: Depending too much on trial-and-error data may introduce biases into the model, affecting its decision-making process. Regular monitoring and bias correction strategies can help mitigate this issue.

To mitigate these limitations, it is essential to strike a balance between learning from failed attempts and successful proofs, optimize the training process, ensure dataset diversity, and regularly evaluate the model's performance on a variety of tasks.

Could the insights from this work on leveraging failed proof attempts be applied to improve human-AI collaborative theorem proving systems?

The insights gained from leveraging failed proof attempts in AI theorem proving systems can indeed be applied to enhance human-AI collaborative theorem proving systems in the following ways:

Error Analysis and Feedback: By analyzing failed proof attempts, AI systems can provide valuable feedback to human collaborators, highlighting potential pitfalls and guiding them towards more effective proof strategies.

Suggesting Alternative Approaches: AI systems can leverage insights from failed attempts to suggest alternative proof strategies or tactics to human collaborators, enhancing their problem-solving capabilities.

Real-Time Assistance: During collaborative theorem proving sessions, AI systems can dynamically adapt their suggestions based on the ongoing proof attempts, incorporating feedback from both successful and failed paths.

Educational Tool: The AI system can serve as an educational tool for human collaborators, showcasing common errors, backtracking strategies, and successful proof paths to enhance their understanding of theorem proving techniques.

By integrating the learnings from failed proof attempts into human-AI collaborative theorem proving systems, the overall efficiency, accuracy, and learning experience of the collaboration can be significantly improved.