The game of Set has always fascinated mathematicians with its intriguing problems. Now, a breakthrough in artificial intelligence (AI) using large language models (LLMs) is showing that AI can assist mathematicians in generating new solutions. This AI system, named FunSearch, has made significant progress in solving Set-inspired problems in combinatorics, a field of mathematics that focuses on counting the possible arrangements of sets with a finite number of objects. Published in Nature on December 14, FunSearch has the potential to be applied to various mathematical and computer science questions.
According to Pushmeet Kohli, the head of AI for Science team at Google DeepMind, FunSearch’s accomplishment is unprecedented: “This is the first time anyone has shown that an LLM-based system can go beyond what was known by mathematicians and computer scientists. It’s not just novel, it’s more effective than anything else that exists today.” Previously, AI researchers used LLMs to solve math problems with known solutions. FunSearch, on the other hand, is a mathematical chatbot that formulates requests for a specially trained LLM, asking it to write short computer programs to generate solutions for specific math problems. The system quickly evaluates these solutions and provides feedback to the LLM, striving to improve with each iteration.
FunSearch’s approach is to use the LLM as a “creativity engine,” as described by Bernardino Romera-Paredes, a computer scientist at DeepMind. Although not every program generated by the LLM is useful, some are groundbreaking while others are incorrect and may not even run. By discarding the incorrect programs and testing the output of the correct ones, FunSearch refines its solutions. The team put FunSearch to the test using the “cap set problem,” which originated from the game Set. Each Set card displays one, two, or three symbols that are identical in color, shape, and shading, with three possible options for each feature. This gives rise to a total of 81 possibilities. Mathematicians have already determined that if at least 21 cards are turned over, a player is guaranteed to find a set. However, for more complex versions of the game with five or more properties, there are still unanswered questions.
Specifically, mathematicians have yet to discover the minimum number of cards that must be revealed to guarantee a solution when there are “n” properties, where “n” can be any whole number. FunSearch tackles this problem by framing it in the context of discrete geometry, equivalent to finding certain arrangements of three points in an n-dimensional space. The FunSearch team managed to improve the lower bound for n=8 by generating sets of cards that satisfy all the requirements of the game. Although they haven’t proven that further improvements are impossible, they have achieved a construction that surpasses previous knowledge.
One notable aspect of FunSearch is that people can examine the successful programs created by the LLM and learn from them, as highlighted by co-author Jordan Ellenberg, a mathematician at the University of Wisconsin–Madison. This distinguishes FunSearch from other AI applications, where the AI is often considered a black box. Ellenberg sees this technology as a way to establish novel modes of collaboration between humans and machines. Rather than replacing human mathematicians, FunSearch acts as a force multiplier, enhancing their abilities.
The success of FunSearch represents a significant advancement in the capabilities of AI in the realm of mathematics and computer science. By combining the power of large language models with human ingenuity, mathematicians not only have a new tool to generate solutions but also a means to learn from these solutions and further expand their understanding of complex mathematical problems. As AI continues to evolve, it is clear that it can serve as a valuable ally for creativity, exploration, and pushing the boundaries of what we know.
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