Bridging AI and IMO Challenges: A Breakthrough in Formal Plane Geometry Systems

Through diligent effort and unwavering commitment, researchers embark on a multi-year journey to create a comprehensive formal planar geometry system to bridge the gap between challenging IMO-level problems and AI automated reasoning. This formal system allows modern AI models to deduce solutions for complex geometry problems in a human-readable, traceable, and verifiable manner. Their study introduces the Geometry Formalization Theory (GFT) to guide system development, resulting in FormalGeo, comprising geometric predicates and theorems. It also presents FGPS (Formal Geometry Problem Solver) in Python and the annotated FormalGeo7k dataset for AI integration. It discusses AI’s roles as a parser and solver, highlighting the system’s correctness and utility, with potential improvements through deep learning techniques.

In geometry problem-solving, various methods have been proposed, including Gelernter’s backward search, Nevins’ forward chaining, Wu’s algebraic approach, and Zhang’s point elimination method. Several formal systems and datasets have been created but often need more theoretical guidance and extensibility. AI-assisted systems like CL-based models, SCA, and GeoDRL aim to enhance success rates. Algebraic approaches and numerical parallel methods have also made significant contributions. Shared benchmarks and datasets have advanced research in AI-assisted geometric problem-solving.

Mathematics and computing share a mutually beneficial relationship, with computing both enabling mathematical work and providing a platform for formal mathematics. The advent of AI has expanded possibilities in computer-aided mathematical problem-solving. The Stanford 2021 AI100 report underscores the IMO grand challenge, seeking an AI system to generate machine-checkable proofs for formal problems and excel in the International Mathematical Olympiad, emphasizing the need for comprehensive mathematical formalization. While progress has been made in mechanizing mathematical problems, geometric problem formalization and mechanized solving face challenges, such as inconsistent knowledge representation and unreadable processes.

The research introduces a comprehensive plane geometry system, FormalGeo, comprising geometric predicates and theorems. It presents FGPS, a Python-based problem solver for geometry, offering interactive assistance and automated solving. FormalGeo7k, a dataset with formal language annotations for geometry problems, aids AI integration. The study aligns modern AI models with the system to enable deductive reasoning for challenging geometry problems. It proposes the GFT for system development, employing GDL and CDL for problem definitions. The backward depth-first search method shows low failure rates, with potential improvements through deep learning techniques.

FormalGeo is a comprehensive formal plane geometry system with 88 predicates and 196 theorems, enabling validation and solutions for challenging geometry problems. FGPS, a Python-based problem solver, offers interactive assistance and automated solving methods. The FormalGeo7k dataset, featuring 6,981 problems with formal annotations, facilitates AI integration. Modern AI models enhance the system, producing readable, traceable, and verifiable proofs. Experiments validate the GFT, and the FGPS’s backward depth-first search method achieves a low 2.42% failure rate, with the potential for further enhancement through deep learning techniques.

The approach introduces the GFT guiding geometric problem formalization and presents the FormalGeo system and FGPS solver. Experiments on the FormalGeo7k dataset validate GFT with a low 2.42% failure rate using backward depth-first search. Further improvements are proposed, including expanding predicates, annotating IMO-level datasets, and implementing deep learning techniques. Modern AI integration enables AI to offer readable, traceable, and verifiable geometry problem solutions. The availability of the FormalGeo7k dataset and FGPS source code promotes further research and development in automated geometric reasoning.

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