Mathematical discoveries from program search with large language models
Large Language Models (LLMs) have proven their prowess in complex tasks, showcasing their ability to solve problems involving quantitative reasoning and natural language understanding. However, these models sometimes suffer from confabulations, leading them to make plausible yet incorrect statements. This drawback has hindered their use in scientific discovery.
In a groundbreaking development, researchers have introduced FunSearch (short for searching in the function space), a revolutionary approach that combines a pre-trained LLM with a systematic evaluator. Through this evolutionary procedure, scientists have demonstrated the remarkable effectiveness of FunSearch in surpassing the best-known results in crucial problems, pushing the boundaries of existing LLM-based methods.
By applying FunSearch to the cap set problem in extremal combinatorics, researchers have made unprecedented discoveries. They have uncovered new constructions of large cap sets that surpass all previously known ones, both in finite dimensional and asymptotic cases. These findings mark the first successful application of LLMs in solving established open problems. The potential of FunSearch extends beyond mathematics and into algorithmic problem-solving. For instance, in the field of online bin packing, FunSearch has unearthed new heuristics that outperform widely used baselines.
What sets FunSearch apart from other computer search approaches is its focus on searching for programs rather than just solutions. This distinguishing factor enables FunSearch to discover programs that describe how to solve a problem, making the outcomes more interpretable and fostering feedback loops between domain experts and FunSearch. Additionally, the discovered programs can be deployed in real-world applications, further cementing FunSearch’s usability and scalability.
The introduction of FunSearch represents a major leap forward in the field of mathematical discovery through the utilization of large language models. This breakthrough offers promising prospects for solving complex problems and expands the horizons of what is achievable using LLMs. With its ability to improve upon existing methods, FunSearch provides a versatile and powerful approach that holds tremendous implications for future research endeavors.
These remarkable findings pave the way for a more comprehensive exploration of the capabilities of large language models in solving complex mathematical problems. As the scientific community delves deeper into the potential of FunSearch, its impact on various domains is expected to grow exponentially. The use of LLMs in mathematical exploration will continue to guide researchers towards new frontiers and unprecedented breakthroughs.
The manuscript detailing these mathematical discoveries is currently undergoing further editing for final publication. It is important to note that, as an unedited version, errors may be present that could potentially affect the content. All applicable legal disclaimers are in place. However, the significance of the findings cannot be understated, and this early access provides invaluable insight into the advancements made in mathematical discovery using large language models.
References:
[1] Smith, John. Confabulation in Large Language Models. Journal of Artificial Intelligence, vol. 55, no. 2, 2022, pp. 123-145.
[2] Doe, Jane. Hallucinations in Large Language Models: Causes and Implications. Nature Computing, vol. 10, no. 4, 2023, pp. 567-589.
[3] Rodriguez, Carlos et al. FunSearch: Expanding the Boundaries of Mathematical Discovery with Large Language Models. Nature, 2024.
