TLDR;
This video explores Google DeepMind's Alpha Geometry, an AI model capable of solving complex geometry problems from the International Mathematical Olympiad (IMO). It highlights the surprising effectiveness of a 25-year-old non-AI technique that uses logic and equation solving, achieving a bronze medal level performance before AI was even introduced. The video then explains how Alpha Geometry integrates this logical model with an AI component to achieve even better results, focusing on the AI's ability to make auxiliary constructions. Finally, it touches on the broader implications of AI in problem-solving across various domains.
- Alpha Geometry, an AI model by Google DeepMind, can solve IMO geometry problems.
- A non-AI technique using logic and equation solving already performs at a bronze medal level.
- Alpha Geometry combines this technique with AI to make auxiliary constructions, significantly improving its problem-solving capabilities.
What's surprising [0:00]
In January 2024, Google DeepMind introduced Alpha Geometry, an AI model designed to tackle 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 geometry problems and successfully solved 25, surpassing the performance of a silver medalist. The most surprising aspect is that a 25-year-old non-AI technique, relying solely on logic and equation solving, could already solve 18 out of the 25 problems, which is equivalent to a bronze medal at the IMO. The AI component was crucial for solving the remaining problems that the logical model couldn't handle. The DeepMind team integrated the logical model with AI to achieve the impressive result of solving 25 out of 30 problems.
Solve without AI [1:33]
The challenge is to create a bot that can solve IMO geometry problems without using AI. The approach involves leveraging key facts and theorems in geometry. For instance, the video mentions that when two lines intersect, the opposite angles are equal, and when parallel lines are intersected by a transversal, the alternate interior angles are equal. Using just these two facts, one can prove non-trivial theorems. In October 2000, researchers Tu, Gao, and Zhang developed a database of 75 geometry rules. By applying these rules in different combinations, they could solve complex geometry problems. These rules range from simple to complex. To input these diagrams into a computer, the authors used a specialized language designed for geometry, which involves defining points, lines, and goals. This method is called deductive database (DD), where new theorems are deduced from a database of key facts. The DD method solved 7 out of 30 IMO geometry problems. While this isn't a high number, it's impressive that a brute-force approach could solve any IMO problems at all, given their difficulty.
Where AI comes in [7:10]
The model's fundamental weakness lies in its inability to make auxiliary constructions, which are extra lines or shapes added to the diagram to help solve the problem. For example, to prove that the sum of angles in a triangle is 180 degrees, one can draw two parallel lines at the top and bottom of the triangle. These auxiliary constructions are essential for solving many hard geometry problems. The DeepMind team built a language model specifically to generate these auxiliary constructions. The input to the language model is the problem statement and the steps in the proof produced so far, using the geometry coding language. The output is an auxiliary construction, such as an extra point or figure on the diagram. Alpha Geometry works by first inputting the problem statement into the language model, which returns an auxiliary construction. This new diagram is then given to DD plus AR, which outputs a series of steps until it terminates. This output is fed back into the language model to produce another auxiliary construction, and the process repeats until the problem is solved or time runs out. The language model acts as the creative brain, while DD plus AR serves as the logical brain.
Grant's comments [12:48]
This video is part of a guest video series and features Aditya Chakravarti from the Aleph Not channel, which explains higher-level math topics in a concise manner. Since Alpha Geometry was announced in January 2024, there have been significant updates in the field of AI solving math contest problems. Multiple groups have achieved gold-level performance on all problems, not just geometry, using natural language processing without domain-specific languages. The video pairs well with an upcoming video about the role of AI in math research. The creator is collecting interviews and stories from mathematicians who have incorporated AI tools in their research or studies, whether the AI was useful or not.