The AI that solved IMO Geometry Problems | Guest video by @Aleph0

Key Concepts:

• Alpha Geometry: An AI model by Google DeepMind that solves geometry problems.
• International Mathematical Olympiad (IMO): A high school-level competitive math contest.
• Deductive Database (DD): A hardcoded list of geometric rules used for theorem proving.
• Algebraic Reasoning (AR): The ability to solve systems of linear equations.
• Auxiliary Constructions: Extra lines or shapes added to a diagram to solve geometry problems.
• Language Model: An AI model trained to produce auxiliary constructions.
• Synthetic Data Generation: Creating training data by randomly generating geometric diagrams and deducing theorems.

1. Introduction to Alpha Geometry and its Context

• In January 2024, Google DeepMind released Alpha Geometry, an AI model capable of solving geometry problems from the International Mathematical Olympiad (IMO).
• The IMO is a prestigious high school math competition where over 100 countries send six representatives each year.
• Alpha Geometry was tested on 30 IMO geometry problems and solved 25, surpassing the performance of a silver medalist.
• The video highlights that a non-AI technique was able to solve 18 out of 30 problems before AI was even introduced.

2. The Power of Non-AI Logic: Deductive Database (DD)

• The video emphasizes the surprising effectiveness of a 25-year-old non-AI technique based on logic and equation solving.
• The approach involves using a database of geometric rules to deduce new theorems.
• Example: Two simple geometric facts (opposite angles of intersecting lines are equal, angles inside a "Z" formed by parallel lines are equal) can prove a non-trivial theorem.
• In October 2000, researchers Tu, Gao, and Zhang created a database of 75 geometry rules.
• The researchers used a niche language specifically designed for geometry to represent the problems.
  • Example: Representing a triangle with two equal sides having two equal angles in the specialized language:
    • ABC points (three points A, B, C)
    • AB equals AC (sides AB and AC have the same length)
    • Goal, angle ABC, equals angle BCA (prove angles ABC and BCA are equal)
• This method is called Deductive Database (DD), which starts with a database of facts and deduces new theorems.
• DD alone solved 7 out of 30 IMO geometry problems.

3. Enhancing DD with Algebraic Reasoning (AR)

• DD's limitation is its inability to solve equations.
• Thelz's theorem (angle in a semicircle is a right angle) is used as an example to demonstrate the need for equation solving.
• The proof involves drawing a third line to the center, identifying isosceles triangles, and solving linear equations.
• DeepMind researchers added Algebraic Reasoning (AR), which solves systems of linear equations using linear algebra.
• DD and AR are used alternatingly: DD runs until it stops, then AR runs until it stops, and the process repeats. This is called DD + AR.
• DD + AR solved 14 out of 30 IMO geometry problems.
• Adding human-coded heuristics further improved the performance to 18 out of 30 problems, nearly a bronze medal at the IMO.

4. The Role of AI: Overcoming the Limitation of Auxiliary Constructions

• The model's weakness is its struggle with auxiliary constructions (adding extra lines or shapes to the diagram).
• Example: Proving the sum of angles in a triangle is 180 degrees requires drawing parallel lines at the top and bottom of the triangle.
• Auxiliary constructions are crucial for solving many hard geometry problems, but they create an infinite search space for machines.
• AI is used to address this limitation by building a language model specifically for producing auxiliary constructions.
• The language model takes the problem statement and proof steps as input (in the geometry coding language) and outputs an auxiliary construction.

5. Alpha Geometry's Architecture and Workflow

• Alpha Geometry combines the language model (for auxiliary constructions) with DD + AR (for logical deduction).
• Workflow:
  1. Input the problem statement into the language model, which returns an auxiliary construction.
  2. Give the new diagram to DD + AR, which outputs a series of steps.
  3. Feed the output back into the language model for another auxiliary construction.
  4. Repeat until the problem is solved or time runs out.
• The language model acts as the "creative brain," while DD + AR acts as the "logical brain."

6. Training Data Generation: Synthetic Proof Examples

• The lack of sufficient IMO geometry problems for training data is addressed by generating synthetic data.
• Process:
  1. Randomly plot points and lines on the plane.
  2. Use DD + AR to deduce theorems.
  3. Erase portions of the diagram to create a problem where auxiliary constructions are needed to solve it.
• This process generated hundreds of millions of synthetic proof examples, including 9 million requiring at least one auxiliary construction.
• The most complex synthetic proof had 247 steps with two auxiliary constructions.

7. Performance and Significance of Alpha Geometry

• Performance:
  • DD: 7/30 problems solved
  • DD + AR: 14/30 problems solved
  • DD + AR + human heuristics: 18/30 problems solved
  • DD + AR + language model (fine-tuned): 25/30 problems solved
• Significance:
  • Alpha Geometry demonstrates machines' ability to think and reason like humans, combining creativity and logic.
  • This approach is applicable to problem-solving in various domains, including science, medicine, and engineering.

8. Updates and Future Directions

• Since Alpha Geometry's announcement in January 2024, there have been significant updates in AI's ability to solve math contest problems.
• Multiple groups have achieved gold-level performance on all IMO problems, not just geometry.
• Some models, like those from Google DeepMind and OpenAI, are using natural language without domain-specific languages.

9. Call for Participation

• The creator is working on a video about the role of AI in math research and is collecting interviews and stories.
• A Google form is provided for individuals to share their experiences with using AI tools in math research or as students.

10. Conclusion

• Alpha Geometry represents a significant step towards AI systems that can combine creative and logical reasoning to solve complex problems.
• The video emphasizes the importance of both AI and non-AI techniques in achieving this milestone.
• The future of AI in problem-solving across various domains is promising.