In a groundbreaking achievement, AI systems developed by Google DeepMind have attained a silver medal-level score in the 2024 International Mathematical Olympiad (IMO), a prestigious global competition for young mathematicians. The AI models, named AlphaProof and AlphaGeometry 2, successfully solved four out of six complex math problems, scoring 28 out of 42 points. This places them among the top 58 out of 609 contestants, demonstrating a remarkable advancement in mathematical reasoning and AI capabilities.
AlphaProof is a new reinforcement-learning-based system designed for formal mathematical reasoning. It combines a fine-tuned version of the Gemini language model with the AlphaZero reinforcement learning algorithm, which has previously excelled in mastering games like chess, shogi, and Go. AlphaProof translates natural language problem statements into formal mathematical language, creating a vast library of formal problems. It then uses a solver network to search for proofs or disproofs in the Lean formal language, progressively training itself to solve more complex issues through continuous learning.
AlphaGeometry 2, an enhanced version of the earlier AlphaGeometry system, is a neurosymbolic hybrid model based on the Gemini language model. It has been trained extensively on synthetic data, enabling it to tackle more challenging geometry problems. AlphaGeometry 2 employs a symbolic engine significantly faster than its predecessor and utilizes a knowledge-sharing mechanism for advanced problem-solving.
During the IMO 2024, the combined efforts of AlphaProof and AlphaGeometry 2 resulted in solving two algebra problems, one number theory problem, and one geometry problem. Notably, AlphaProof solved the hardest problem in the competition, which only five human contestants could solve. However, the two combinatorics problems still needed to be solved.
AlphaProof’s formal approach to reasoning allowed it to generate and verify solution candidates, reinforcing its language model with each proven solution. This iterative learning process enabled the system to tackle increasingly difficult problems, leading to its success in the competition. On the other hand, AlphaGeometry 2’s rapid problem-solving capability was highlighted when it solved a geometry problem just 19 seconds after its formalization.
This achievement marks a significant milestone in applying AI to complex problem-solving and mathematical reasoning. The success of AlphaProof and AlphaGeometry 2 demonstrates the potential of combining LLMs with powerful search mechanisms, such as reinforcement learning, to solve intricate mathematical problems. The ability of AI systems to perform at a level comparable to some of the world’s best young mathematicians suggests a promising future where AI can assist in exploring new hypotheses, solving long-standing problems, and streamlining the proof process in mathematics.
The research and development teams behind AlphaProof and AlphaGeometry 2 continue to refine their models and explore new approaches to enhance AI’s mathematical reasoning capabilities further. As these systems become more advanced, they can revolutionize how mathematicians and scientists approach problem-solving and discovery. The success of AlphaProof and AlphaGeometry 2 at the IMO 2024 is a testament to the rapid advancements in AI and its growing role in complex domains such as mathematics. This achievement paves the way for future innovations and collaborations between AI and human experts, driving progress in science and technology.
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