Learning to Reason with Insight for Informal Theorem Proving
Proposed DeepInsightTheorem framework enhances informal theorem proving by identifying core techniques, significantly outperforming baselines.
Key Findings
Methodology
This study proposes a novel framework designed to cultivate insight in large language models (LLMs) for insightful reasoning in informal theorem proving. By constructing a hierarchical dataset named DeepInsightTheorem, which organizes informal proofs by explicitly extracting core techniques and proof sketches, the study designs a Progressive Multi-Stage SFT strategy that mimics the human learning process, guiding the model from basic proof writing to insightful thinking.
Key Results
- On challenging mathematical benchmarks, models using the DeepInsightTheorem framework significantly outperformed baseline models. For instance, performance on the FIMO dataset improved by 15.73%, and on the Putnam dataset by 37.01%.
- The experimental results demonstrate a significant enhancement in the model's ability to identify and apply core techniques, leading to superior performance in mathematical reasoning tasks.
- Ablation studies confirmed the effectiveness of the Progressive Multi-Stage Training Strategy, proving that this approach can effectively enhance the model's mathematical reasoning capabilities.
Significance
This research significantly enhances the performance of large language models in informal theorem proving by introducing an insight-driven reasoning paradigm. This approach is not only significant in academia, advancing the field of automated theorem proving, but also provides new solutions for industry applications requiring complex mathematical reasoning. By identifying and applying core techniques, the model can better understand and solve complex mathematical problems, overcoming the limitations of traditional methods in handling informal proofs.
Technical Contribution
Technical contributions include: 1) Proposing a new hierarchical dataset, DeepInsightTheorem, that organizes informal proofs by explicitly extracting core techniques; 2) Designing a Progressive Multi-Stage SFT strategy that mimics human learning processes to enhance the model's mathematical reasoning capabilities; 3) Experimentally validating the effectiveness of the insight-driven reasoning paradigm, significantly improving model performance on mathematical benchmarks.
Novelty
This study is the first to propose enhancing large language models' performance in informal theorem proving by identifying and applying core techniques. The innovation lies in incorporating human expert reasoning processes into model training, enabling the model to understand problems holistically and identify key techniques, significantly improving reasoning capabilities compared to previous research.
Limitations
- Despite the method's excellent performance on multiple benchmarks, the model may still struggle to identify core techniques when dealing with extremely complex mathematical problems.
- The training process requires substantial computational resources, which may not be feasible in resource-constrained environments.
- The method's effectiveness in specific domains has yet to be fully validated.
Future Work
Future research directions include: 1) Expanding the dataset's scale and diversity to cover more types of mathematical problems; 2) Optimizing the model's training process to reduce computational resource requirements; 3) Exploring the method's potential applications in other fields, such as complex problem-solving in physics and engineering.
AI Executive Summary
Automated theorem proving has long been a central goal in artificial intelligence, yet existing methods largely rely on formal proof systems, limiting their ability to handle informal theorem proving. Informal theorem proving aligns better with large language models' strengths in natural language processing, but a lack of insight, specifically the ability to recognize core techniques needed to solve complex problems, remains a primary bottleneck.
To address this issue, this study proposes a novel framework designed to cultivate insight in large language models, enabling them to perform insightful reasoning. Researchers constructed a hierarchical dataset named DeepInsightTheorem, which organizes informal proofs by explicitly extracting core techniques and proof sketches. To fully exploit this dataset, they designed a Progressive Multi-Stage SFT strategy that mimics the human learning process, guiding the model from basic proof writing to insightful thinking.
On challenging mathematical benchmarks, models using the DeepInsightTheorem framework significantly outperformed baseline models. These results demonstrate that by identifying and applying core techniques, the model can better understand and solve complex mathematical problems, overcoming the limitations of traditional methods in handling informal proofs.
This research is not only significant in academia, advancing the field of automated theorem proving, but also provides new solutions for industry applications requiring complex mathematical reasoning. By introducing an insight-driven reasoning paradigm, researchers have opened new avenues for applying large language models in informal theorem proving.
However, the method may still struggle to identify core techniques when dealing with extremely complex mathematical problems. Additionally, the training process requires substantial computational resources, which may not be feasible in resource-constrained environments. Future research directions include expanding the dataset's scale and diversity to cover more types of mathematical problems and optimizing the model's training process to reduce computational resource requirements.
Deep Analysis
Background
Automated theorem proving (ATP) has been a significant research direction in artificial intelligence. Traditional ATP methods often rely on formal proof systems like Lean, Coq, and Isabelle, which excel in handling formal proofs. However, with the development of large language models (LLMs), researchers have begun exploring the potential of applying LLMs to informal theorem proving. Informal theorem proving uses natural language and standard mathematical notation to generate proofs, aligning well with the strengths of modern LLMs. Yet, existing research has primarily focused on framework construction, with little attention paid to the proof generation mechanism and the key bottlenecks of LLM-based informal theorem proving.
Core Problem
The core problem in informal theorem proving is identifying the core techniques required to solve complex problems. Most automated theorem proving methods rely on formal proof systems, whereas informal theorem proving aligns better with large language models' strengths in natural language processing. However, a lack of insight, specifically the ability to recognize core techniques needed to solve complex problems, remains a primary bottleneck. Researchers argue that informal theorem proving requires first forming a big-picture view of the proof before eventually completing the full proof.
Innovation
The core innovations of this study include proposing a novel framework designed to cultivate insight in large language models, enabling them to perform insightful reasoning. Specific innovations include: 1) Constructing a hierarchical dataset named DeepInsightTheorem, which organizes informal proofs by explicitly extracting core techniques and proof sketches; 2) Designing a Progressive Multi-Stage SFT strategy that mimics the human learning process, guiding the model from basic proof writing to insightful thinking; 3) Introducing an insight-driven reasoning paradigm that significantly enhances large language models' performance in informal theorem proving.
Methodology
- �� Constructing the DeepInsightTheorem dataset: Organizing informal proofs by explicitly extracting core techniques and proof sketches.
- �� Designing a Progressive Multi-Stage SFT strategy: Mimicking the human learning process, guiding the model from basic proof writing to insightful thinking.
- �� Experimental validation: Testing the model's reasoning capabilities on challenging mathematical benchmarks.
- �� Ablation studies: Confirming the effectiveness of the Progressive Multi-Stage Training Strategy.
Experiments
The experimental design includes testing the model's reasoning capabilities on multiple mathematical benchmarks. The datasets used include FIMO, Putnam, and HMMT. Baseline models include Qwen2.5-7B and Llama3-8B. Experimental metrics include the model's ability to identify and apply core techniques and its performance in mathematical reasoning tasks. Ablation studies are conducted to confirm the effectiveness of the Progressive Multi-Stage Training Strategy.
Results
Experimental results show that models using the DeepInsightTheorem framework significantly outperformed baseline models in reasoning capabilities. For instance, performance on the FIMO dataset improved by 15.73%, and on the Putnam dataset by 37.01%. Ablation studies confirmed the effectiveness of the Progressive Multi-Stage Training Strategy, proving that this approach can effectively enhance the model's mathematical reasoning capabilities.
Applications
This method can be directly applied to real-world problems requiring complex mathematical reasoning, such as automated theorem proving, mathematics education, and scientific research. By identifying and applying core techniques, the model can better understand and solve complex mathematical problems, offering broad application prospects.
Limitations & Outlook
Despite the method's excellent performance on multiple benchmarks, the model may still struggle to identify core techniques when dealing with extremely complex mathematical problems. Additionally, the training process requires substantial computational resources, which may not be feasible in resource-constrained environments. Future research directions include expanding the dataset's scale and diversity to cover more types of mathematical problems and optimizing the model's training process to reduce computational resource requirements.
Plain Language Accessible to non-experts
Imagine you're in a kitchen trying to cook a complex dish, but you don't know where to start. Typically, you'd look at a recipe to understand the main steps and key techniques needed. Then, you'd follow the recipe step by step, ensuring each step is done correctly. This is similar to what we do in informal theorem proving. We need to identify the key techniques to solve the problem, just like identifying the key steps in cooking a dish. This way, we can better understand the problem and find a solution.
In this process, we use something called the DeepInsightTheorem dataset, which helps us identify these key techniques. This dataset is like a detailed recipe, telling us what to do at each step and why. By doing so, we can better understand and solve complex mathematical problems.
Additionally, we've designed a progressive multi-stage training strategy, which is like starting with simple dishes and gradually challenging more complex ones. This way, we can gradually improve our cooking skills, eventually being able to complete complex dishes independently.
In summary, this research is like cooking in a kitchen. By identifying key steps and techniques, we can better understand and solve complex problems.
ELI14 Explained like you're 14
Hey there! Have you ever wondered why some math problems look so complicated and hard to start? It's like playing a super tough game level where you need to find the key items and tricks to win.
In math, we also need to find the key techniques to solve problems, just like finding the secrets to beat a game level. This study helps us find these key techniques, making us better at solving math problems.
Researchers created something called the DeepInsightTheorem dataset, which is like a cheat sheet, telling us what the key techniques are for each math problem. With this cheat sheet, we can better understand the problem and find a solution.
They also designed a training method, like starting with easy levels and gradually challenging harder ones. This way, we can improve our math skills step by step and eventually solve those seemingly impossible problems. Isn't that cool?
Glossary
DeepInsightTheorem
A hierarchical dataset that organizes informal proofs by explicitly extracting core techniques and proof sketches.
Used to train large language models to identify and apply core techniques.
SFT (Supervised Fine-Tuning)
A training strategy that fine-tunes model performance by providing supervised signals.
Used to enhance model performance in mathematical reasoning tasks.
FIMO (Mathematical Competition Dataset)
A dataset used to test mathematical reasoning capabilities, containing challenging problems.
Used to validate model performance in mathematical reasoning tasks.
Putnam (Putnam Mathematical Competition)
A renowned college-level mathematics competition containing high-difficulty problems.
Used to test model reasoning capabilities on complex mathematical problems.
HMMT (Harvard-MIT Mathematics Tournament)
A high-level mathematics competition for high school students, containing various math problems.
Used to evaluate model performance across different mathematical domains.
Core Techniques
Key steps and methods required to solve complex mathematical problems.
Identifying and applying these techniques is crucial in informal theorem proving.
Insight-Driven Reasoning
Performing deep mathematical reasoning by identifying and applying core techniques.
Used to enhance large language models' performance in informal theorem proving.
Progressive Multi-Stage Training Strategy
A training method that mimics human learning processes, gradually enhancing model reasoning capabilities.
Guides the model from basic proof writing to insightful thinking.
Large Language Model (LLM)
An AI model capable of processing and generating natural language text.
Used for mathematical reasoning in informal theorem proving.
Informal Theorem Proving
Generating mathematical proofs using natural language and standard mathematical notation.
Aligns better with large language models' strengths compared to formal proof systems.
Open Questions Unanswered questions from this research
- 1 Current methods may struggle to identify core techniques when dealing with extremely complex mathematical problems. This is because such problems often involve multiple interrelated techniques, and existing datasets may not cover all possible technique combinations. Future research needs to expand the dataset's scale and diversity to cover more types of mathematical problems.
- 2 Although the Progressive Multi-Stage Training Strategy significantly enhances reasoning capabilities, its training process requires substantial computational resources. This limits its application in resource-constrained environments. Future research can explore more efficient training methods to reduce computational resource requirements.
- 3 Existing methods primarily focus on the mathematical domain, and their effectiveness in other domains has yet to be fully validated. For example, complex problem-solving in physics and engineering may require different techniques and methods. Future research can explore the method's potential applications in other fields.
- 4 While the DeepInsightTheorem dataset provides rich supervisory signals, in some cases, models may overly rely on these signals and overlook the overall structure of the problem. Future research can explore how to improve reasoning capabilities without relying on explicit signals.
- 5 Current evaluation methods primarily rely on human evaluation, which may lead to subjective results. Future research can explore more objective evaluation methods to improve the reliability of results.
Applications
Immediate Applications
Automated Theorem Proving
This method can be directly applied to automated theorem proving systems, helping to identify and apply core techniques to enhance reasoning capabilities.
Mathematics Education
By identifying and applying core techniques, this method can help students better understand and solve complex mathematical problems.
Scientific Research
In scientific research requiring complex mathematical reasoning, this method can help researchers better understand and solve problems.
Long-term Vision
Cross-Domain Applications
This method can be extended to other fields, such as complex problem-solving in physics and engineering, providing new solutions.
Intelligent Education Systems
In the future, this method can be used to develop intelligent education systems to help students learn individually and improve learning outcomes.
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. We propose $\mathtt{DeepInsightTheorem}$, a hierarchical dataset that structures informal proofs by explicitly extracting core techniques and proof sketches alongside the final proof. To fully exploit this dataset, we design a Progressive Multi-Stage SFT strategy that mimics the human learning process, guiding the model from basic proof writing to insightful thinking. Our experiments on challenging mathematical benchmarks demonstrate that this insight-aware generation strategy significantly outperforms baselines. These results demonstrate that teaching models to identify and apply core techniques can substantially improve their mathematical reasoning.
References (15)
PutnamBench: Evaluating Neural Theorem-Provers on the Putnam Mathematical Competition
G. Tsoukalas, Jasper Lee, J. Jennings et al.
MiniF2F: a cross-system benchmark for formal Olympiad-level mathematics
Kunhao Zheng, Jesse Michael Han, Stanislas Polu
ABEL: Sample Efficient Online Reinforcement Learning for Neural Theorem Proving
Fabian Gloeckle, Gabriel Synnaeve, Amaury Hayat
The Llama 3 Herd of Models
Abhimanyu Dubey, Abhinav Jauhri, Abhinav Pandey et al.
Hierarchical Memory for High-Efficiency Long-Term Reasoning in LLM Agents
Haoran Sun, Shaoning Zeng
Measuring Mathematical Problem Solving With the MATH Dataset
Dan Hendrycks, Collin Burns, Saurav Kadavath et al.
HyperTree Planning: Enhancing LLM Reasoning via Hierarchical Thinking
Runquan Gui, Zhihai Wang, Jie Wang et al.
Learning Formal Mathematics From Intrinsic Motivation
Gabriel Poesia, David Broman, Nick Haber et al.
Lean-STaR: Learning to Interleave Thinking and Proving
Haohan Lin, Zhiqing Sun, Yiming Yang et al.
miniCTX: Neural Theorem Proving with (Long-)Contexts
Jiewen Hu, Thomas (Hanwen) Zhu, S. Welleck
DeepSeek-R1: Incentivizing Reasoning Capability in LLMs via Reinforcement Learning
Adam Suma, Sam Dauncey
Formal Theorem Proving by Rewarding LLMs to Decompose Proofs Hierarchically
Kefan Dong, Arvind V. Mahankali, Tengyu Ma
Self-Discover: Large Language Models Self-Compose Reasoning Structures
Pei Zhou, J. Pujara, Xiang Ren et al.
NaturalProver: Grounded Mathematical Proof Generation with Language Models
S. Welleck, Jiacheng Liu, Ximing Lu et al.
ReasonFlux: Hierarchical LLM Reasoning via Scaling Thought Templates
Ling Yang, Zhaochen Yu, Bin Cui et al.