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A Semantic Search Engine for Mathlib4 (2403.13310v2)

Published 20 Mar 2024 in cs.IR, cs.LG, and cs.LO

Abstract: The interactive theorem prover Lean enables the verification of formal mathematical proofs and is backed by an expanding community. Central to this ecosystem is its mathematical library, mathlib4, which lays the groundwork for the formalization of an expanding range of mathematical theories. However, searching for theorems in mathlib4 can be challenging. To successfully search in mathlib4, users often need to be familiar with its naming conventions or documentation strings. Therefore, creating a semantic search engine that can be used easily by individuals with varying familiarity with mathlib4 is very important. In this paper, we present a semantic search engine (https://leansearch.net/) for mathlib4 that accepts informal queries and finds the relevant theorems. We also establish a benchmark for assessing the performance of various search engines for mathlib4.

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Summary

  • The paper introduces a novel semantic search engine for mathlib4 that transforms formal theorem statements into informal queries to simplify retrieval.
  • It employs a vector embedding model and query augmentation to align informal queries with formal theorem representations, surpassing baseline methods.
  • Experimental benchmarks demonstrate significant improvements in theorem retrieval accuracy, enhancing access to formal mathematical proofs for diverse users.

Semantic Search Engine for Mathlib4: Enhancing Theorem Retrieval with Informal Queries

Introduction to Theorem Retrieval Challenges

Lean, a state-of-the-art interactive theorem prover, supports the verification of formal mathematical proofs, facilitated by its extensive mathematical library, mathlib4. A significant challenge within this ecosystem lies in theorem retrieval due to the formal language structure of mathlib4, which often requires users to be familiar with specific naming conventions or documentation strings. This paper addresses these challenges by introducing a semantic search engine for mathlib4 designed to accept informal queries and enhance theorem retrieval efficiency. The proposed engine significantly benefits users, especially beginners, by simplifying the process of locating relevant theorems.

Related Work

Existing text retrieval methods, including both classical approaches like BM25 and modern deep neural networks, primarily focus on capturing semantic similarities between documents and queries. Despite their effectiveness in general text retrieval tasks, these methods fall short in mathematical information retrieval (MIR) due to the structured nature of mathematical formulas. In the domain of MIR, both classical structure search methods and recent dense retrievers have shown promise. However, these approaches have not been fully explored or adapted for searching within formal mathematical libraries, such as mathlib4.

Methodology

Our method involves three main steps: informalizing the formal statements in mathlib4, constructing a semantic search engine tailored for mathlib4 based on these informal-formal theorem pairs, and augmenting user queries to improve retrieval relevance. To transform the formal statements of theorems into informal language comprehensible to diverse users, we employ a LLM. The resulting informal-formal theorem pairs are then embedded into a vector space model, facilitating semantic search capabilities. Furthermore, query augmentation enriches user input with formal expressions and additional context, leveraging LLMs to enhance search precision.

Mathlib4 Semantic Search Benchmark

To evaluate the effectiveness of various search engines for mathlib4, we establish a comprehensive benchmark. This benchmark comprises a curated set of queries, relevance judgments for theorem-query pairs, and a collection of performance metrics tailored to the MIR context. Our benchmark spans a broad range of mathematical topics, offering a robust foundation for the systematic assessment of search engines dedicated to mathlib4.

Experimental Results

Comparative analyses of multiple embedding models reveal that the proposed search engine substantially outperforms existing methods, including Moogle and other baseline embedding models. The augmentation of queries and the incorporation of informal-formal theorem pairs into the search corpus are pivotal to achieving significant improvements in theorem retrieval. These enhancements are evident across various performance metrics, underscoring the utility of our approach in practical theorem search tasks within mathlib4.

Future Directions

The development of the semantic search engine for mathlib4 marks a critical step toward improving access to formal mathematical libraries. Looking ahead, efforts will focus on refining the informalization process of formal statements, expanding the Mathlib4 semantic search benchmark, and exploring fine-tuning of text embedding models for enriched theorem retrieval. By addressing these aspects, we aim to further elevate the efficiency and accuracy of theorem search in mathlib4, thereby supporting a wider range of users in their engagement with formal mathematical proofs.

In conclusion, the semantic search engine presented in this paper not only alleviates the challenges faced by users in navigating mathlib4 but also sets a foundation for future advancements in the retrieval of mathematical theorems from formal libraries. Through continuous development and refinement, this engine has the potential to significantly contribute to the accessibility and usability of formal mathematical resources.

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