Theory-Level Autoformalization: From Isolated Statements to Unified Formal Knowledge Bases

2026-07-14

AI Summary

Existing approaches to automatic formalization are largely confined to isolated statements and struggle to capture the intricate dependency structures among axioms, definitions, and lemmas within mathematical theories. This work introduces a novel paradigm—*theory-level automatic formalization*—which systematically advocates shifting from statement-level to integrated theory-level formalization. By constructing formalized mathematical libraries, modeling dependency graphs, and establishing mappings from natural to formal languages, the approach enables machine-verifiable translation of entire mathematical theories along with their internal structures. The paper delineates core challenges inherent to this direction, proposes three viable research pathways, and releases a comprehensive survey resource to catalyze further progress in the field.

Abstract

Autoformalization translates informal natural language into formal, machine-verifiable languages. While most work focuses on individual statements, real formalization efforts are inherently theory-level: they require an entire web of axioms, definitions, and lemmas before target theorems can even be stated. In this position paper, we argue for theory-level autoformalization: formalizing complete theories, including all their inter-dependencies, as structured libraries. We examine the significance of this shift, address alternative views, identify open challenges, and propose three promising paths forward. Our survey of autoformalization is available at https://github.com/marcusm117/Awesome-Autoformalization.

Problem

• autoformalization
• theory-level formalization
• formal knowledge bases
• inter-dependencies
• structured libraries

Innovation

• theory-level autoformalization
• formal knowledge bases
• structured libraries
• inter-dependencies
• machine-verifiable languages