Build a Bot to Solve Geometry Problems Without AI

How to Build a Bot to Solve Geometry Problems Without AI

This article guides you through constructing a computational approach to solving geometry problems, inspired by techniques that predated advanced AI. We will explore how a system based on logical rules and algebraic manipulation can tackle complex geometric theorems, laying the groundwork for understanding the power of computational reasoning.

Overview of What You Will Learn

You will learn about the foundational principles of representing geometric knowledge computationally, how to implement a deductive system using a database of geometric rules, and how to integrate algebraic reasoning to solve problems that require equation manipulation. By the end of this guide, you will understand the core components of a non-AI system capable of proving geometric theorems and appreciate its strengths and limitations.

Prerequisites

• Basic understanding of geometric principles (e.g., angles, lines, parallel lines, triangles).
• Familiarity with logical deduction and algebraic equations.

Step 1: Understanding Geometric Rules and Deductive Databases

The first step in building a computational geometry solver is to represent geometric knowledge in a format that a computer can understand. This involves identifying fundamental geometric facts and rules.

1. Identify Key Geometric Facts: Start with basic axioms and theorems. For example:
   • When two lines intersect, vertically opposite angles are equal.
   • If two parallel lines are intersected by a transversal, the alternate interior angles are equal (forming a ‘Z’ shape).
2. Formulate a Database of Rules: Researchers like Tu, Gao, and Zhang proposed creating a comprehensive database of geometric rules. This database can contain dozens or even hundreds of rules, ranging from simple definitions to more complex theorems.
3. Represent Problems and Solutions: Geometric problems and their solutions need to be encoded. This can be done using specialized languages or formats that describe points, lines, angles, and their relationships. For instance, a theorem stating that an isosceles triangle has equal base angles could be represented as:
   • Input: Points A, B, C. Side AB = Side AC.
   • Goal: Angle ABC = Angle BCA.
4. Implement the Deductive Database (DD): This module systematically applies the rules from the database to the given problem. It attempts to derive new facts from the initial conditions and previously proven facts. This process is akin to a logical proof, where each step is justified by a rule from the database. The goal is to reach the desired conclusion (e.g., proving two angles are equal).

Expert Note: The effectiveness of the DD system heavily depends on the completeness and accuracy of its rule database. A larger, more comprehensive set of rules allows the system to tackle a wider range of problems.

Step 2: Incorporating Algebraic Reasoning (AR)

A significant limitation of purely logical deduction is its inability to solve equations, which are often crucial for proving geometric theorems. The integration of an algebraic reasoning module addresses this gap.

1. Identify the Need for Equations: Many geometry problems require calculating angle measures or lengths, which can only be done by setting up and solving algebraic equations. A classic example is Thales’s Theorem (the angle inscribed in a semicircle is a right angle), which can be elegantly proven using algebraic manipulation of angle sums and properties of isosceles triangles formed by radii.

2. Implement Algebraic Reasoning (AR): This module takes the geometric information derived by the DD module (e.g., relationships between angles and lengths) and formulates it into a system of linear equations. Standard linear algebra techniques can then be used to solve these equations.
3. Combine DD and AR: The power comes from alternating between the two modules. First, the DD module applies its logical rules. If it reaches a point where equations are needed, the AR module is invoked. The results from AR can then provide new geometric facts (e.g., a specific angle is 90 degrees) that the DD module can use to continue its deductions. This iterative process, often referred to as ‘DD plus AR’, significantly expands the problem-solving capabilities.

Tip: The AR module is particularly effective when dealing with problems involving numerical values for angles or lengths, or when proving relationships that depend on precise measurements.

Step 3: Adding Human-Coded Heuristics

While DD plus AR is powerful, certain types of problems remain challenging. Incorporating human expertise in the form of heuristics can further boost performance.

1. Understand Heuristics: Heuristics are practical methods or rules of thumb, often derived from human experience, that guide the problem-solving process. In the context of geometry, these could be strategies like ‘if you see a circle, consider drawing a radius’ or ‘look for isosceles triangles’.
2. Implement Human-Coded Heuristics: These are specific rules or procedures, carefully crafted by human experts, that are added to the system. They can help the DD plus AR process to make more effective choices, prune unproductive paths, or recognize common problem-solving patterns more quickly.
3. Observe Performance Improvement: By adding these heuristics, the system’s ability to solve problems increases. For instance, a system that solves 14 out of 30 problems with DD plus AR might jump to 18 out of 30 with the addition of well-designed heuristics.

Expert Note: Human-coded heuristics represent a bridge between pure computational logic and human intuition. They encode distilled wisdom from years of human problem-solving experience.

Step 4: Addressing the Challenge of Auxiliary Constructions

Many difficult geometry problems require the addition of new lines or shapes (auxiliary constructions) to the original diagram to make the proof possible. This is a major hurdle for purely deductive systems.

1. Recognize the Need for Auxiliary Lines: Consider the proof that the sum of angles in a triangle is 180 degrees. A common method involves drawing two parallel lines through one vertex, parallel to the opposite side. This step is not explicitly given in the problem but is crucial for the proof.
2. The Infinite Search Space Problem: For a computer, deciding *what* auxiliary construction to add at *any* given step is incredibly difficult. There are infinitely many possibilities, and most are irrelevant. This combinatorial explosion makes it hard for non-AI systems to proceed.
3. AI’s Role in Creativity: This is where AI, specifically language models, can play a significant role. An AI component can be trained to suggest plausible auxiliary constructions based on the current state of the problem and the proof steps taken so far. It acts as the ‘creative’ part of the system, proposing novel ideas.

Warning: Without a mechanism to generate auxiliary constructions, even a powerful DD plus AR system will fail on many challenging IMO geometry problems.

Step 5: The Alpha Geometry Approach – Integrating Logic and Creativity

The Alpha Geometry model developed by DeepMind offers a sophisticated solution by combining a powerful logical engine with an AI-driven creative component.

1. The Language Model for Constructions: A language model is specifically trained to generate auxiliary constructions. It takes the problem statement and the current proof steps as input and outputs a suggestion for an additional line, point, or shape.
2. The Iterative Process: The overall system works in a loop:
   • Input the problem into the language model to get an auxiliary construction.
   • Feed the original problem *plus* the new construction into the DD plus AR engine.
   • The DD plus AR engine performs deductions until it can no longer proceed.
   • Feed the new state of the problem (original plus construction plus deductions) back into the language model for another construction.
   • Repeat until the problem is solved or a time limit is reached.
3. Synergy of Modules: In this architecture, the language model serves as the ‘creative brain,’ proposing ingenious auxiliary constructions, while the DD plus AR engine acts as the ‘logical brain,’ rigorously deducing consequences from these proposals and established facts.

Analogy: Think of it like a human mathematician: the creative part brainstorms a clever idea (auxiliary construction), and the logical part rigorously checks and expands upon that idea (DD plus AR).

Step 6: Generating Training Data for the AI

A key challenge in training AI models for complex tasks like theorem proving is the scarcity of large, high-quality datasets. The Alpha Geometry team addressed this by generating their own synthetic data.

1. Forward Generation: Start by randomly placing points and drawing lines on a plane.
2. Deduce Everything: Use the DD plus AR system to derive all possible consequences and theorems from this random configuration. This establishes a rich set of known facts and relationships.
3. Backward Engineering: Select a subset of the derived facts (a theorem) and then ‘erase’ some of the intermediate steps or constructions that were necessary to reach it.
4. Formulate the Problem: The remaining diagram and the target theorem now form a synthetic geometry problem. To solve it, the AI would need to rediscover the ‘erased’ auxiliary constructions and follow the logical path.
5. Scale Up: This process was used to generate millions of synthetic proof examples, including a significant number that required auxiliary constructions. This massive dataset allowed the language model to be trained effectively.

Tip: This synthetic data generation technique is a powerful method for creating training datasets for AI systems in domains where real-world data is scarce or difficult to annotate.

Conclusion

By combining a robust deductive database with algebraic reasoning, and crucially, by using an AI language model to propose creative auxiliary constructions, the Alpha Geometry system achieved remarkable success. This approach highlights how integrating logical rigor with creative exploration can lead to powerful problem-solving capabilities, offering a glimpse into the future of AI in complex reasoning domains beyond geometry.

Source: The AI that solved IMO Geometry Problems | Guest video by @Aleph0 (YouTube)