Proposing and solving olympiad geometry with guided tree search (2412.10673v1)

Abstract: Mathematics olympiads are prestigious competitions, with problem proposing and solving highly honored. Building artificial intelligence that proposes and solves olympiads presents an unresolved challenge in automated theorem discovery and proving, especially in geometry for its combination of numerical and spatial elements. We introduce TongGeometry, a Euclidean geometry system supporting tree-search-based guided problem proposing and solving. The efficient geometry system establishes the most extensive repository of geometry theorems to date: within the same computational budget as the existing state-of-the-art, TongGeometry discovers 6.7 billion geometry theorems requiring auxiliary constructions, including 4.1 billion exhibiting geometric symmetry. Among them, 10 theorems were proposed to regional mathematical olympiads with 3 of TongGeometry's proposals selected in real competitions, earning spots in a national team qualifying exam or a top civil olympiad in China and the US. Guided by fine-tuned LLMs, TongGeometry solved all International Mathematical Olympiad geometry in IMO-AG-30, outperforming gold medalists for the first time. It also surpasses the existing state-of-the-art across a broader spectrum of olympiad-level problems. The full capabilities of the system can be utilized on a consumer-grade machine, making the model more accessible and fostering widespread democratization of its use. By analogy, unlike existing systems that merely solve problems like students, TongGeometry acts like a geometry coach, discovering, presenting, and proving theorems.

• The paper presents TongGeometry as the first AI system that both proposes and solves complex olympiad geometry problems using guided tree search.
• It discovers 6.7 billion geometric theorems, including 4.1 billion with symmetry, by combining numeric and spatial reasoning.
• TongGeometry outperforms IMO gold medalists on benchmark datasets, achieving superior results with consumer-grade computational resources.

Overview of "Proposing and Solving Olympiad Geometry with Guided Tree Search"

The paper presents TongGeometry, an innovative system developed for the automated proposing and solving of geometry problems at the level of mathematical olympiads. This system addresses the dual challenges associated with these prestigious competitions: problem discovery and theorem proving. TongGeometry stands out for its unique capability to function as both a problem proposer and solver, leveraging a guided tree-search methodology that incorporates both numeric and spatial reasoning—a particularly challenging combination in the domain of geometry.

Key Contributions

TongGeometry significantly enhances the landscape of automated theorem proving and geometry problem-solving through several notable contributions:

1. Massive Discovery of Theorems: The system successfully discovers an unprecedented 6.7 billion geometry theorems, with 4.1 billion exhibiting geometric symmetry. This achievement expands the repository of known theorems significantly beyond existing systems like AlphaGeometry.
2. Real-World Impact: Among these theorems, ten were proposed for inclusion in mathematical olympiads, and three were selected for actual competitions, illustrating the practical utility and acceptance of TongGeometry's outputs in traditional competition settings.
3. Superior Problem Solving: TongGeometry outperforms gold medalists on the International Mathematical Olympiad (IMO) geometry problems within the IMO-AG-30 dataset, making it the first AI system to surpass such a level of human expertise in this domain.
4. Efficiency and Accessibility: Notably, the system's remarkable performance is achievable using consumer-grade computational resources, democratizing access to advanced geometric problem-solving capabilities.

Methodological Insights

TongGeometry employs a neuro-symbolic approach, utilizing guided tree-search algorithms fortified by LLMs to effectively navigate the theorem discovery and problem-solving space. It combines backward tracing for problem construction and forward chaining for theorem proving. The system also incorporates a deduction-driven deductive database enhanced by actor-critic style neural models for efficient auxiliary construction—a pivotal element in conjecturing and verification processes.

Results and Evaluation

TongGeometry's dominance is evidenced on standard benchmarks. It achieves 30/30 solves on the IMO-AG-30 dataset, clearly surpassing the capabilities of both its predecessor systems and human competitors, including average contestants and award recipients at the IMO. When evaluated on the newly curated MO-TG-225 dataset, the system achieves a high solve rate, further demonstrating its robust performance across diverse problem sets.

Implications and Future Directions

The implications of TongGeometry extend beyond immediate benchmarking results. By functioning as a mathematical "coach," capable of formulating and solving problems rather than merely executing pre-defined problem-solving routines, it contributes significantly to the understanding of automated theorem proving in synthetic environments.

Future advancements could include further refining its problem assessment and selection algorithms, potentially incorporating even richer datasets for deep learning models, and expanding its capabilities to tackle more complex and novel problem configurations. Additionally, the success of TongGeometry might inspire analogous developments in other mathematical domains, thus broadening its impact within both educational and research contexts.

In conclusion, TongGeometry represents a significant step forward in computational geometry, offering a robust framework for both theorem discovery and olympiad-style problem solving. The system's integration of symbolic and numeric reasoning, coupled with its real-world applicability, underscores its potential as a powerful tool for advancing mathematical research and education.