In an era marked by the rapid development of large language models like ChatGPT, scientists have made a significant breakthrough in the field of mathematical education. A novel recursion-based proof method, known as POETRY (PrOving thEorem RecursivelY), holds immense potential for revolutionizing the way mathematical theories are proven and understood.
The Limitations of Existing Proof Methods
Despite the impressive capabilities of large language models, their mathematical reasoning abilities still fall short. Previous research in this domain has primarily relied on data-driven approaches, which face challenges in the realm of natural language mathematics. Mathematical data often lacks automatic verifiability, necessitating manual verification by human experts, a process that can be time-consuming and resource-intensive.
The Rise of Formally-Based Systems
To address these challenges, researchers have turned to formally-based systems like Metamath, Lean, and Isabelle. These systems enable the automatic verification of mathematical theorems by translating them into a formal language. This has paved the way for the development of neural-symbolic methods that combine the strengths of formal systems and large language models, such as OpenAI’s GPT-f, Thor, and Leandojo.
The GPT-f Approach and its Limitations
GPT-f and similar methods leverage the collaboration between formal systems and large language models to prove theorems. However, the search for more complex proofs can be challenging due to the infinite nature of the search space, which can surpass even the vast search space of围棋 (Go) games.
The POETRY Method: A New Approach
To address this issue, researchers at Sun Yat-sen University have proposed the POETRY method. Inspired by the way humans prove mathematical theorems, this method allows for the recursive filling of proofs. By utilizing the Sorry keyword in formal systems, researchers can skip the proof of intermediate lemmas, enabling the large language model to focus on proving the current goal.
This approach effectively breaks down a long proof into multiple layers, with each layer requiring a shorter sub-proof. It also allows for the deferral of intermediate lemma proofs, significantly reducing the complexity of the search process.
Experimental Results and Implications
Experimental results demonstrate that the POETRY method can significantly improve proof length. In one study, the method was able to find proofs that were up to 26 steps longer than those found by the GPT-f method on the PISA dataset.
Potential Applications and Future Directions
The POETRY method holds immense potential for various applications, including enhancing the reliability of AI-based mathematical education products and preventing knowledge deception. Furthermore, this approach could lead to significant advancements in various scientific disciplines.
Researchers are now exploring the extension of the POETRY method to other formal systems, such as Lean and Hol. As this method gains traction, it is poised to revolutionize the field of mathematical education and contribute to the broader advancement of artificial intelligence and mathematical research.
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