Learning to Reason with Insight for Informal Theorem Proving
Summary: arXiv:2604.16278v1 Announce Type: new
Abstract
Although most of the automated theorem-proving approaches depend on formal proof systems, informal theorem proving can align better with large language models’ (LLMs) strength in natural language processing. In this work, we identify a primary bottleneck in informal theorem proving as a lack of insight, namely the difficulty of recognizing the core techniques required to solve complex problems.
To address this, we propose a novel framework designed to cultivate this essential reasoning skill and enable LLMs to perform insightful reasoning.
Introduction
The field of automated theorem proving has traditionally relied on formal proof systems, which can be rigid and challenging for many applications. In contrast, informal theorem proving leverages the strengths of large language models, enabling more intuitive and flexible reasoning processes. However, a significant challenge remains: the need for models to develop insight into the core techniques necessary for effective problem-solving.
Identifying the Bottleneck
One of the primary obstacles in informal theorem proving is the lack of insight among existing models. This limitation often hinders their ability to discern and apply the fundamental strategies required to tackle intricate mathematical problems. By recognizing this issue, researchers can better formulate approaches that enhance the reasoning capabilities of LLMs.
Introducing the DeepInsightTheorem Framework
To overcome the identified bottlenecks, we introduce a groundbreaking framework known as DeepInsightTheorem. This framework is centered around the development of a hierarchical dataset that meticulously structures informal proofs. The dataset explicitly extracts core techniques and proof sketches, in addition to the final proofs themselves.
Progressive Multi-Stage SFT Strategy
To leverage the potential of the DeepInsightTheorem dataset, we have designed a Progressive Multi-Stage SFT (Supervised Fine-Tuning) strategy. This innovative approach mirrors the human learning process, guiding models from basic proof writing to more sophisticated and insightful reasoning.
- Stage 1: Basic proof writing skills are developed.
- Stage 2: The focus shifts to identifying core techniques.
- Stage 3: Models are trained to generate insights and apply them effectively.
Experimental Results
In our experiments, we evaluated the performance of our insight-aware generation strategy against several challenging mathematical benchmarks. The results were promising, demonstrating that the insight-enhanced approach significantly outperforms existing baselines.
Conclusion
Our findings underscore the importance of teaching models to identify and apply core techniques. By enhancing their ability to reason insightfully, we can substantially improve their mathematical reasoning capabilities. This work not only paves the way for more advanced theorem proving but also opens new avenues for research in the intersection of artificial intelligence and mathematics.