What is TongGeometry
TongGeometry is a tree search-based geometric model jointly launched by the Beijing Academy of Artificial Intelligence and the Institute for Artificial Intelligence at Peking University, specifically used for proposing and solving Olympiad-level geometry problems. TongGeometry, based on efficient search algorithms and large-scale parallel computing, has established the most extensive library of geometric theorems to date, discovering 6.7 billion geometric theorems that require auxiliary constructions, of which 4.1 billion have geometric symmetry. TongGeometry can generate high-quality competition problems and can autonomously complete proofs. TongGeometry surpasses gold medalist performance for the first time in solving geometry problems from the International Mathematical Olympiad (IMO) and can run efficiently on consumer-grade computers.
Main features of TongGeometry
- Geometry problem generation: Automatically proposes high-quality Olympiad geometry problems, covering a variety of types from basic to complex, and supports the generation of symmetry and auxiliary constructions.
- Theorem discovery and proof: Based on tree search and neuro-symbolic reasoning, autonomously discovers and proves geometric theorems, including complex auxiliary constructions.
- Problem evaluation and screening: Screens out problems suitable for competitions based on scoring criteria, evaluating them according to difficulty and innovation.
- Educational resource generation: Provides abundant geometry problems and proof examples for education and research, promoting the popularization of geometry teaching.
Technical principles of TongGeometry
- Tree search and guided problem generation: Uses a tree search algorithm to start from basic geometric elements and gradually construct complex geometry problems. Based on a combination of backward tracking (starting from the goal) and forward reasoning (gradually constructing auxiliary constructions), it generates geometry problems that require auxiliary constructions.
- Neuro-symbolic reasoning: Combines neural networks and symbolic reasoning; the policy model generates auxiliary constructions, and the value model estimates solution steps.
- Large-scale parallel computing: Uses large-scale parallel computing resources (such as thousands of CPU cores) to explore the space of geometry problems within a limited time and generate billions of problems. Based on efficient search strategies and caching mechanisms, it accelerates the problem generation and screening process.
- Auxiliary construction and theorem proving: Automatically identifies and generates auxiliary constructions (such as auxiliary lines, circles, etc.), filling in the key steps in geometric proofs. Uses the Deductive Database method for theorem proving, combined with the full-angle method to generate human-readable proofs.
Project address of TongGeometry
- arXiv technical paper：https://arxiv.org/pdf/2412.10673
Application scenarios of TongGeometry
- Mathematics competitions: Generates high-quality geometry competition problems, assisting competition problem setting and contestant training.
- Mathematics education: Provides teaching resources and supports personalized learning and online education platforms.
- Artificial intelligence research: Serves as an experimental platform for geometry reasoning research, promoting algorithm development and model optimization.
- Mathematics research: Assists in discovering and verifying geometric theorems and building problem libraries.
- Educational assessment: Supports competition problem setting and student ability assessment.