Written by: Jonathan Handjojo

In a groundbreaking achievement, researchers at the Google DeepMind team have unveiled AlphaGeometry, an artificial intelligence (AI) system that rivals human Olympiad gold medalists in solving complex geometry problems. The development signifies a significant leap forward in the capabilities of AI systems, demonstrating their prowess in logical reasoning and problem-solving.

The Challenge of Olympiad Geometry Problems 

The International Mathematical Olympiad (IMO) has long been a platform for showcasing the mathematical talents of high-school students. The geometry problems presented in the Olympiad are known for their complexity, requiring a deep understanding of spatial relationships, shapes, and distances. These challenges have traditionally posed difficulties for AI systems due to their intricate nature and the need for advanced reasoning skills. 

Figure 1

How AlphaGeometry Solves a Problem

Source: Google DeepMind

AlphaGeometry’s Approach 

AlphaGeometry adopts a neuro-symbolic approach, combining a neural language model with a symbolic deduction engine. In Figure 1, given the problem diagram and its theorem premises (left), AlphaGeometry (middle) first uses its symbolic engine to deduce new statements about the diagram until the solution is found or new statements are exhausted. If no solution is found, AlphaGeometry’s language model adds one potentially useful construct (blue), opening new paths of deduction for the symbolic engine. This loop continues until a solution is found (right). This synergistic combination allows the AI system to provide fast and intuitive ideas through the language model, while the deduction engine ensures deliberate and rational decision-making. This dual-system setup mirrors the concept of “thinking, fast and slow,” where one system excels at identifying patterns, and the other relies on formal logic to arrive at conclusions.

Impressive Results 

In a benchmarking test featuring 30 Olympiad geometry problems (IMO-AG-30), AlphaGeometry exhibited remarkable performance. The AI system solved 25 problems within the standard Olympiad time limit, surpassing the capabilities of previous state-of-the-art systems. For comparison, the average human gold medalist typically solves 25.9 problems, highlighting AlphaGeometry’s achievement in approaching human-level proficiency. 

Figure 2

IMO Performance

Source: Google DeepMind

One of the key challenges in training AI systems for geometry problems is the lack of reasoning skills and training data. AlphaGeometry addresses this by introducing a method to generate a vast pool of synthetic training data. The system generated one billion random diagrams of geometric objects, deriving relationships between points and lines in each diagram. This extensive dataset, filtered to include 100 million unique examples (Figure 3), enabled AlphaGeometry to train without relying on human demonstrations, overcoming the data bottleneck. 

Figure 3

Example of Training Data

Source: Google DeepMind

DeepMind’s newest version, AlphaGeometry 2, has been trained on 10x more synthetic data than its predecessor and incorporates the language model based on Gemini. This helped the new model solve much more challenging geometry problems including questions containing movements of objects (i.e. transformations), and equations of angles, ratios, or distances (i.e. proportionality theorems, distance formula, and trig relations). Combined with its new symbolic engine, it is now 100 times faster than its predecessor and capable of solving problem 4 in IMO 2024 within an astonishing 19 seconds. Combined with this speed, AlphaGeometry 2 was also was able to solve 83% of all historical IMO geometry problems from the past 25 years.

Conclusion

AlphaGeometry’s success in solving Olympiad-level geometry problems signifies a major milestone in developing AI systems with advanced reasoning capabilities. As researchers continue to push the boundaries of AI in mathematics, the implications for broader applications in science, technology, and artificial intelligence are promising. While AlphaGeometry currently focuses on geometry problems in the IMO, its broader impact lies in advancing reasoning for next-generation AI systems. The open-sourcing of AlphaGeometry’s code and model reflects a commitment to fostering collaboration and exploration in mathematics, science, and AI along with the expansion of human knowledge.

References and Sources

Source 1 Trinh, T., & Luong, T. (2024, January 17). AlphaGeometry: An Olympiad-level AI system for geometry. Google DeepMind. https://deepmind.google/discover/blog/alphageometry-an-olympiad-level-ai-system-for-geometry/

Source 2 AlphaProof and AlphaGeometry teams AI achieves silver-medal standard solving International Mathematical Olympiad problems. (2024, May 14). Google DeepMind. https://deepmind.google/discover/blog/ai-solves-imo-problems-at-silver-medal-level/