AI and Math Proofs: A New Approach

Writing formal math proofs can be as tricky as trying to fold a fitted sheet. Whether you're a human or a computer, it can feel like a never-ending puzzle. Recently, some smart folks thought, “Hey, why don’t we use AI to help us out?” They looked at a special kind of math problem called IMO problems from the International Math Olympiad.

Now, these math problems range from moderate to major brain-busters. You know, the kind of problems that make you scratch your head and wonder if you even know how to add. The team wanted to write Formal Proofs using a tool called Lean, which is a programming language for math proofs. They aimed to tackle some of these tricky problems using AI, and what they found was quite fascinating.

#The Challenge of Math Proofs

Humans have a tough time writing formal math proofs, and computers aren’t exactly wizards at it either. Even some of the so-called advanced AI models struggle. The miniF2F dataset, which is often used as a benchmark for automated theorem proving, contains 20 IMO problems, but formal proofs are lacking for over half of them. So, why is this such a big deal? Well, when a computer claims it can solve a problem but doesn't have a proper proof to back it up, it’s like claiming you're a fantastic cook but only heating up frozen pizzas.

Many AI models, like GPT-4, have a hard time proving these math problems correctly. They might get lucky sometimes, but when it comes to the harder problems, it’s like watching a toddler try to tie their shoes-lots of effort, but not much success.

#Expanding the Pool of Proofs

To help get more formal proofs out there, the team set out to write original proofs for 13 of the 20 IMO problems in the miniF2F dataset, plus a few extras from more recent years. That's a total of 5,150 lines of proof-even longer than some novels! This massive effort makes it easier for future researchers to learn and experiment with these problems.

They didn’t stop there. They also broke down these proofs into smaller pieces, called Lemmas. Think of these lemmas as the building blocks of math proofs. The team created around 900 lemmas with about 25,500 lines of Lean code. Now, that’s a lot of math building blocks! These lemmas are easier to work with and provide a clearer path for AI models to learn.

#Evaluating the AI’s Proof-Writing Skills

After generating these lemmas, the team decided to test GPT-4’s proof-writing skills on them. Spoiler alert: it didn't go as well as they had hoped. The AI struggled to write correct proofs, which was surprising given all the fancy tech behind it. They used various prompting techniques, including Zero-Shot prompting (asking it to just go for it) and Chain-of-Thought reasoning (guiding it step by step). But still, the robot didn’t shine.

What was even more interesting was how GPT-4 performed better on older IMO problems compared to the newer ones. The older problems seemed to be a little friendlier, like a calm summer day, while the newer ones were more like a stormy night-challenging and difficult to navigate through.

#Why AI Needs High-Quality Data

Machine learning models, like a hungry person, need high-quality data to thrive. The more good data you feed them, the better they perform. The success of many machine learning systems can often be traced back to an abundance of quality training data. For instance, ImageNet has played a big role in computer vision. But when it comes to math, the available resources are pretty scarce.

The miniF2F dataset doesn’t have enough quality proofs for many of its problems. Most AI models fail because they lack solid examples to learn from. It's like trying to learn how to ride a bike without ever seeing anyone do it first. When a model attempts to solve a math problem and fails, it's hard to tell where it went wrong since there isn’t a good reference point.

#The Math Olympiad: A Tough Nut to Crack

The International Math Olympiad presents a unique challenge. The problems are unveiled only on the exam day, and they get harder every year. So if an AI model wants to make a mark, it has to be sharp on its feet and capable of handling unknowns. Using past problems as practice is not enough because every year, students face fresh challenges that are intentionally tricky.

To prepare an AI for the Math Olympiad, researchers need to use rigorous evaluation methods. They need to check if the AI can generalize its learning when faced with a new, tougher set of problems. If you're trying to win a gold medal and have only practiced with some fluff, you might walk away empty-handed.

#Current Status of Math Datasets

The miniF2F dataset consists of various math theorems that students get tested on. Among the 244 theorems, 20 are from IMO, and their difficulty varies significantly. Some require proof in a single line, while others take hundreds of lines. Success in easier problems doesn't guarantee success in the more difficult ones. So, if a model claims to be good, it's essential to look beyond mere percentages.

The current champion of this dataset, LEGO-Prover, managed to prove only one of the IMO problems. Meanwhile, methods like HTPS can handle more problems but often face issues with problem statements that are either simplified or incorrectly worded. It’s like saying you can win a race just because you managed to finish a short jog.

#A New Approach to Decomposing Proofs

The team realized that for many problems, formal proofs were not available to the public. So they tackled these tricky problems and shared their formal proofs in Lean. They broke down each proof into smaller lemmas. This process made complex challenges more manageable, allowing others to study and learn from them.

The lemmas range in difficulty and cover a variety of topics. While some are simple and straightforward, others require deeper thinking. They even avoided easy problems that Lean could automatically prove. Instead, they focused on real challenges where brains-human or AI-would need to be engaged.

#Testing GPT-4: Hoping for Improvement

To see if GPT-4 could improve, the team prompted it to write formal proofs for their lemmas. They provided detailed instructions and reviewed GPT-4’s informal proofs alongside the formal ones. Surprisingly, even after extensive prompting and feedback, GPT-4 struggled with accuracy. It was like telling someone repeatedly how to make a sandwich, and they still end up serving you a salad instead.

In most cases, GPT-4 just couldn't provide the right answers. The team provided feedback and asked it to correct mistakes, but it felt like trying to teach a cat how to fetch. They interacted with GPT-4 multiple times, but after ten rounds, they decided to cut their losses.

#A Closer Look at Lemmas

Each of the lemmas in the new dataset has a formal proof in Lean, which is crucial for anyone trying to learn about these problems. The team constructed 907 lemmas with difficulty levels ranging from easy to complex. These building blocks are essential for anyone looking to understand proof writing better, as they provide a pathway to tackling more intricate math problems.

For example, some lemmas are relatively simple, involving proving basic properties of numbers. Others challenge the solver to think critically about functions and relationships between numbers. Many are still hard, even when broken down, but that’s the beauty of math-there’s always something new to learn.

#Making the Proofs Accessible

The formal proofs created by the team were shared with the community to help everyone understand the work that goes into writing a formal proof. These can also help identify mistakes in informal proofs circulating online. The team aims to show how beneficial and detailed formal proofs can be, especially when looking at more complicated subjects.

By making these proofs available, they're contributing to a broader understanding of math. Non-mathematicians can see the effort involved in formal proofs, and mathematicians can use them to refine their informal approaches.

#Key Takeaways

The project helps to shed light on the difficulties of formalizing proofs and emphasizes the importance of high-quality data in training AI models. Even though GPT-4 struggled significantly, this work has laid the groundwork for future advancements.

The team hopes that by providing a wealth of formal proofs and working through the lemmas, they can encourage more successes in the area of automated theorem proving. They view this as a step forward in the long journey toward building AI capable of tackling high-level math problems like those found in the Math Olympiad.

#Exploring Future Directions

Although the team faced challenges with GPT-4, they remain optimistic. Their goal of developing a model that can effectively prove the lemmas in their dataset is still alive. Each attempt, even if imperfect, serves to inform the future of AI in math.

The project also opens up avenues for more robust AI models that can understand complex proofs and connect ideas in new ways. There's no shortage of challenges in the world of mathematics, and AI can play a critical role in pushing those limits further.

#Conclusion

In summary, the effort to write formal proofs for IMO problems using Lean offers great potential for future work in automated theorem proving. Although the journey is complex and filled with unexpected hurdles, each step taken brings us closer to a more profound understanding of how AI can assist in the world of mathematics.

As researchers continue to refine their methods and improve model capabilities, we may soon see AI systems that can tackle the challenging problems posed in math competitions effectively-or at least get close enough not to embarrass themselves in front of the math community. Who knows? One day, we might have an AI that can ace the Math Olympiad, but until then, we’ll just have to keep practicing those proofs, one lemma at a time.