AlphaProof and AlphaGeometry2 Wins Silver In International Math Olympiad

Artificial intelligence has significantly advanced in various fields, and its potential to address complex mathematical problems is increasingly evident. As of July 25th 2024, AlphaProof and AlphaGeometry2 achieved remarkable success by solving problems at the International Mathematical Olympiad (IMO) at a silver-medal level.

Deepmind has a rather well written account of this with clear inner workings of the models in their article here. I have no intention to replicate this.

In this blog I delve into the mathematical capabilities of these AI systems and explore the intricate math concepts they mastered.

Understanding the Mathematical Challenges

The IMO presents six exceptionally challenging problems in algebra, combinatorics, geometry, and number theory. Each problem demands deep understanding and innovative problem-solving skills, often requiring solutions involving multiple steps and intricate reasoning. To appreciate the achievements of AlphaProof and AlphaGeometry2, we need to explore the types of problems they solved.

Algebra

Algebra problems in the IMO often require a blend of polynomial equations, inequalities, and functional equations. Solving these problems involves:

• Polynomial Manipulations: Expanding, factoring, and simplifying polynomials. For example, dealing with polynomials​ and finding roots or specific values requires understanding their properties and transformations.

• Inequalities: Using techniques like the Arithmetic Mean-Geometric Mean (AM-GM) inequality, Cauchy-Schwarz inequality, and others. For example, proving that for non-negative numbers a,b,c the inequality holds involves understanding and applying these principles.

• Functional Equations: Finding functions that satisfy given conditions, often involving substitution and transformation techniques. An example problem could be finding all functions:

AlphaProof demonstrated its algebraic capabilities by solving two challenging algebra problems, including one of the hardest in the competition, involving non-obvious constructions and intricate proof steps.

Number Theory

Number theory problems often involve properties of integers, prime numbers, modular arithmetic, and Diophantine equations. Key techniques include:

• Modular Arithmetic: Working with remainders and congruences to simplify problems. For example, solving congruences as shown below involves understanding residue classes and their properties.

• Prime Factorization: Breaking down numbers into prime components to find solutions. For example, proving that the number of divisors of the formulation below requires knowledge of prime factorization.

• Diophantine Equations: Solving equations that require integer solutions, often using methods like the Euclidean algorithm or properties of quadratic forms. For instance, finding integer solutions to equations below (Pell’s equation) involves deep number-theoretic insights.

AlphaProof’s number theory solution showcased deep reasoning and proof strategies, highlighting its ability to handle complex integer properties and relationships.

Geometry

Geometry problems require a deep understanding of shapes, angles, distances, and their properties. Key techniques include:

• Synthetic Geometry: Using classical geometric constructions and theorems. For instance, proving properties of triangles, circles, and other geometric figures using Euclidean geometry principles.
• Coordinate Geometry: Applying algebraic methods to geometric problems by placing them on a coordinate plane. For example, finding the equation of a circle passing through given points involves using distance formulas and algebraic manipulation.
• Trigonometry: Utilizing relationships between angles and sides in triangles and other shapes. For example, solving problems involving the sine and cosine rules or proving identities like sin⁡(A+B)=sin⁡A.cos⁡B+cos⁡A.sin⁡B.

AlphaGeometry2 excelled in solving a geometry problem by employing advanced geometric constructions and proof techniques.

Reinforcement Learning and Formal Mathematics

AlphaProof utilizes reinforcement learning, where the system learns to make decisions by receiving rewards or penalties based on its actions. This is coupled with formal mathematics, where proofs are written in a formal language (Lean in this case) that allows for rigorous verification of correctness.

Formal Language (Lean): A mathematical language that ensures proofs are logically sound. AlphaProof translates natural language problems into Lean, generating a formal proof. For instance, converting the statement “For all integers n≥2n, the sum of the reciprocals of the primes less than or equal to n is less than 2” into a formal Lean statement involves expressing it as:

AlphaZero Algorithm: Originally designed for mastering games like chess and Go, this algorithm trains AlphaProof to search for proof steps efficiently, improving with each iteration. AlphaZero, used in AlphaProof, starts with random proof steps and iteratively improves by playing millions of self-play games, each reinforcing successful strategies.

AlphaProof’s ability to prove or disprove mathematical statements involves:

• Problem Translation: Converting informal problems into formal statements.
• Proof Search: Exploring potential proof steps and validating them.
• Reinforcement Learning Loop: Iteratively refining its proof strategies based on successful and unsuccessful attempts.

Neuro-Symbolic Hybrid Systems

AlphaGeometry2 combines neural networks with symbolic reasoning to tackle geometric problems. This hybrid approach leverages the strengths of both:

• Neural Networks: Good at pattern recognition and handling large amounts of data. For example, a neural network trained on thousands of geometric diagrams learns to recognize and categorize shapes, angles, and their relationships.
• Symbolic Reasoning: Effective for precise mathematical manipulations and logical deductions. For example, using symbolic reasoning, AlphaGeometry2 can deduce that two triangles are similar by matching their angles and proportional sides, then applying this knowledge to prove a broader geometric property.

Future Directions

The achievement of AlphaProof and AlphaGeometr 2 at the IMO is not just a milestone in AI research but also a glimpse into the future of mathematical problem-solving. These systems demonstrate that AI can:

• Assist Mathematicians: By providing insights, generating hypotheses, and verifying proofs, AI can probably become a valuable tool for mathematicians. For instance, an AI system could suggest potential steps in a long proof, highlight possible errors, or propose alternative methods to approach a problem.
• Advance Education: AI-driven tools can help students learn complex mathematical concepts by providing step-by-step solutions and explanations. An AI tutor could guide students through difficult topics like calculus or linear algebra, offering personalized feedback and additional practice problems.
• Tackle Unsolved Problems: With continued development, AI systems could address some of the most challenging open problems in mathematics. AI could contribute to solving long-standing conjectures like the Riemann Hypothesis or the P vs. NP problem by exploring vast mathematical landscapes and proposing novel approaches.