Olympiad geometry solved by AI

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Google DeepMind has developed an AI system called AlphaGeometry that can solve complex geometry problems, including those from the International Mathematical Olympiad (IMO)234. This represents a significant step towards machines with more human-like reasoning skills2. AlphaGeometry was tested on a set of 30 geometry problems from the IMO, and it was able to solve 25 of them34. This performance is nearly at the level of a human gold medalist in the IMO3. The system combines a neural language model that generates intuitive ideas and a symbolic deduction engine that verifies them using formal logic and rules5. When presented with a geometry problem, AlphaGeometry first attempts to generate a proof using its symbolic engine. If it cannot do so using the symbolic engine alone, the language model adds a new point or line to the diagram, opening up additional possibilities for the symbolic engine to continue searching for a solution2. The development of AlphaGeometry involved creating a custom language embedded with several dozen basic rules of geometry. The team then wrote a program to automatically generate 100 million 'proofs', which were essentially random sequences of simple but logically unassailable steps. AlphaGeometry was trained on these machine-generated proofs, allowing it to solve problems by guessing one step after the other4. The system's performance surpasses the previous state-of-the-art approach for geometry problems, and it represents an advancement in AI reasoning in mathematics6. Despite its success, the researchers acknowledge the limitations and challenges of their work, such as the need for more human-readable proofs and the scalability to more complex problems5.
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