Making Written Theorems Explorable by Grounding Them in Formal Representations
Summary: arXiv:2604.02598v1 Announce Type: cross
Abstract: LLM-generated explanations can make technical content more accessible, but there is a ceiling on what they can support interactively. Because LLM outputs are static text, they cannot be executed or stepped through. We argue that grounding explanations in a formalized representation enables interactive affordances beyond what static text supports. We instantiate this idea for mathematical proof comprehension with explorable theorems, a system that uses LLMs to translate a theorem and its written proof into Lean, a programming language for machine-checked proofs, and links the written proof with the Lean code. Readers can work through the proof at a step-level granularity, test custom examples or counterexamples, and trace the logical dependencies bridging each step. Each worked-out step is produced by executing the Lean proof on that example and extracting its intermediate state. A user study ($n = 16$) shows potential advantages of this approach: in a proof-reading task, participants who had access to the provided explorability features gave better, more correct, and more detailed answers to comprehension questions, demonstrating a stronger overall understanding of the underlying mathematics.
Introduction
The increasing complexity of mathematical content has led to a growing need for tools that enhance comprehension and accessibility. Traditional written proofs, while rigorous, often pose challenges for readers seeking to understand the underlying principles. Recent advancements in large language models (LLMs) have shown promise in generating explanations that simplify complex ideas. However, these models have limitations when it comes to interactivity.
Challenges of Static Text
Static text, such as that generated by LLMs, can provide explanations and clarify concepts but lacks the ability to engage users actively. This makes it difficult for readers to explore mathematical ideas deeply or test their understanding through interaction. As a result, there is a pressing need to develop systems that enhance the learning experience by making mathematical proofs more explorable.
Grounding in Formal Representations
To address this challenge, researchers have proposed grounding written explanations in formal representations. This approach integrates LLM-generated text with formal methods, allowing for a richer, more interactive learning environment. One implementation of this idea is the system known as “explorable theorems.”
Explorable Theorems System
The explorable theorems system utilizes LLMs to convert a theorem and its proof into Lean, a programming language designed for machine-checked proofs. This dual representation provides readers with the ability to:
- Work through the proof step-by-step.
- Test custom examples or counterexamples against the theorem.
- Trace logical dependencies between each step of the proof.
By executing the Lean proof for each step, the system extracts intermediate states, allowing users to visualize the progression of the proof and better grasp its logical structure.
User Study Insights
A user study conducted with 16 participants revealed significant advantages of the explorable theorems approach. Participants who had access to interactive features demonstrated:
- Improved accuracy in answering comprehension questions.
- More detailed responses that reflected a deeper understanding of the mathematical concepts.
- A stronger overall grasp of the proof structure and logic.
These findings suggest that integrating interactive elements into the study of mathematics can enhance learning outcomes and foster a more profound engagement with the material.
Conclusion
The development of explorable theorems marks a significant step forward in making mathematical proofs more accessible and engaging. By grounding written explanations in formal representations, educators and learners can benefit from a more interactive and effective learning experience. This innovative approach paves the way for future advancements in the accessibility of complex technical content.