HARPA Summary

The source paper is "LeanDojo: Theorem Proving with Retrieval-Augmented Language Models" (247 citations, 2023, ID: 87875a07976c26f82705de1fc70041169e5d652b). This idea builds on a progression of related work [96a9546fdb2f147ced3a03d4c98b6c900ecd1b8c, dd18782960f9ee4c66b79e1518b342ad3f8d19e7, 98b607e7cb84e1a5c87c8a49562ae35435e6722d, cddb552f6c3464a54a02b0b64b2d1af56c086606, b16c7d45183b9d595ab64301be019741b1528860, f8b5ee53c3410f20049e7def47bd52403fa388e3, 0b419e86492123e7c9634855ff4edf2d2de4562a, ee71447d68f3d8666c974b8199e330e19aebf263, 4ba57555bef02f988f2ed3bab2f102733dc55221, 4ed896f45e143b31dca465b2153fc7fc93ca3fdf].

The analysis reveals a progression in enhancing the capabilities of LLMs for theorem proving and mathematical reasoning. The source paper introduced LeanDojo, which provided an open-source platform for theorem proving with retrieval-augmented LLMs. Subsequent papers built on this by addressing limitations such as fixed theorem libraries, integrating in-context learning, and enhancing interaction with proof environments. They also explored process-oriented models and reinforcement learning to improve reasoning accuracy. A research idea that advances this field could focus on combining these advancements to develop a comprehensive framework that integrates modular skill libraries, in-context learning, and process-oriented reinforcement learning for theorem proving.

Hypothesis

Integrating Dynamic Skill Addition with Subgoal-based Demonstration Learning will improve the accuracy and efficiency of LLMs in theorem proving tasks compared to using either approach independently.

Research Gap

Existing research has not extensively explored the combination of Dynamic Skill Addition and Subgoal-based Demonstration Learning to enhance the efficiency and accuracy of theorem proving in LLMs. This gap is crucial as it could lead to more efficient use of computational resources and improved theorem proving capabilities.

Hypothesis Elements

Independent variable: Integration of Dynamic Skill Addition with Subgoal-based Demonstration Learning

Dependent variable: Accuracy (success rate in proving theorems) and efficiency (time taken to complete proofs) of LLMs in theorem proving tasks

Comparison groups: Three systems: 1) Dynamic Skill Addition (DSA) only, 2) Subgoal-based Demonstration Learning (SDL) only, 3) Integrated system combining DSA and SDL

Baseline/control: Systems using either Dynamic Skill Addition or Subgoal-based Demonstration Learning independently

Assumptions: Dynamic Skill Addition enhances adaptability by enriching the skill library; Subgoal-based Demonstration Learning improves efficiency by breaking down tasks into manageable subgoals

Measurement method: Accuracy measured by success rate on the miniF2F dataset; efficiency measured by time taken to complete theorem proving tasks

Overview

The research aims to test the hypothesis that integrating Dynamic Skill Addition with Subgoal-based Demonstration Learning can significantly enhance the accuracy and efficiency of LLMs in theorem proving tasks. Dynamic Skill Addition allows the system to evolve its skill library by adding new, validated skills during the proving process, thereby expanding its capabilities over time. Subgoal-based Demonstration Learning, on the other hand, breaks down complex theorem proving tasks into smaller, manageable subgoals, facilitating more efficient learning and problem-solving. By combining these two approaches, the system is expected to leverage the strengths of both: the adaptability and growth of the skill library from Dynamic Skill Addition, and the structured, efficient problem-solving strategy from Subgoal-based Demonstration Learning. This combination addresses the gap in existing research by providing a novel approach that could lead to more efficient use of computational resources and improved theorem proving capabilities. The expected outcome is a system that not only proves theorems more accurately but also does so in a time-efficient manner, making it a valuable tool for formal mathematics and automated reasoning.

Background

Dynamic Skill Addition: Dynamic Skill Addition involves the continuous evolution of the skill library by adding new skills during the theorem proving process. This is achieved by prompting an LLM to generate new skills, which are then validated and stored in the library. This approach allows the system to tackle increasingly complex problems by enriching the library with reusable skills. The expected role of Dynamic Skill Addition is to enhance the system's adaptability and problem-solving capabilities by providing a growing repository of skills that can be leveraged during the proving process.

Subgoal-based Demonstration Learning: Subgoal-based Demonstration Learning restructures complex theorem proving tasks into smaller, manageable subgoals. This approach improves learning efficiency by allowing the system to focus on individual components of a problem, thereby facilitating more effective problem-solving. The expected role of this variable is to enhance the system's efficiency by providing a structured framework for tackling complex proofs, reducing the computational overhead and time required to complete theorem proving tasks.

Implementation

The hypothesis will be implemented by integrating Dynamic Skill Addition and Subgoal-based Demonstration Learning into a unified framework for theorem proving. The Dynamic Skill Addition module will prompt the LLM to generate new skills during the proving process, which will be validated and added to the skill library. This module will be responsible for expanding the system's capabilities by continuously enriching the library with new, reusable skills. The Subgoal-based Demonstration Learning module will decompose complex theorem proving tasks into smaller subgoals, allowing the system to tackle each subgoal individually. This module will provide a structured framework for proof generation, enhancing the system's efficiency by reducing the computational overhead associated with complex proofs. The integration of these modules will occur at the logic level, where the outputs of the Dynamic Skill Addition module will be used by the Subgoal-based Demonstration Learning module to inform the selection and execution of subgoals. The data flow will involve the LLM generating new skills, which are validated and stored in the library, and then used to guide the subgoal-based proof generation process. The expected outcome is a system that can prove theorems more accurately and efficiently by leveraging the strengths of both Dynamic Skill Addition and Subgoal-based Demonstration Learning.

Operationalization Information

Please implement an experiment to test the hypothesis that integrating Dynamic Skill Addition with Subgoal-based Demonstration Learning will improve the accuracy and efficiency of LLMs in theorem proving tasks compared to using either approach independently.

Experiment Overview

This experiment will compare three systems for theorem proving using LLMs:
1. Baseline 1: Dynamic Skill Addition (DSA) only
2. Baseline 2: Subgoal-based Demonstration Learning (SDL) only
3. Experimental: Integrated system combining DSA and SDL

The experiment will evaluate these systems on the miniF2F dataset, measuring both accuracy (success rate in proving theorems) and efficiency (time taken to complete proofs).

Implementation Requirements

Pilot Mode Configuration

Implement a global variable PILOT_MODE with three possible settings:
- MINI_PILOT: Use only 5 theorems from the miniF2F training set for quick debugging
- PILOT: Use 50 theorems from the training set and 20 from the validation set
- FULL_EXPERIMENT: Use the complete miniF2F dataset (training, validation, and test sets)

Start by running the MINI_PILOT, then if successful, run the PILOT. Do not run the FULL_EXPERIMENT automatically - this will be manually triggered after human verification of the pilot results.

Data Preparation

Download and prepare the miniF2F dataset
Create appropriate splits for the different pilot modes
Implement functions to load and preprocess theorems for the LLM

Core Modules

1. Dynamic Skill Addition (DSA) Module

Implement a skill library data structure to store reusable skills/tactics
Create a mechanism for the LLM to generate new skills during theorem proving
Implement validation logic to verify generated skills before adding them to the library
Design a retrieval mechanism to access relevant skills from the library during proving

2. Subgoal-based Demonstration Learning (SDL) Module

Implement logic to decompose theorem proving tasks into smaller subgoals
Create a mechanism for the LLM to identify and generate appropriate subgoals
Implement tracking of subgoal progress and dependencies
Design a framework for learning from demonstrations at the subgoal level

3. Integrated System

Implement the integration of DSA and SDL modules
Design the data flow where skills from DSA inform subgoal selection and execution in SDL
Ensure that new skills can be generated based on subgoal requirements
Implement mechanisms for subgoals to leverage the skill library effectively

LLM Integration

Use an appropriate LLM (e.g., GPT-4) for theorem proving
Implement appropriate prompting strategies for each system variant
Ensure consistent LLM usage across all three systems for fair comparison

Evaluation Framework

Implement metrics for accuracy (success rate on proving theorems)
Implement metrics for efficiency (time taken to complete proofs)
Create a logging system to record detailed information about each proof attempt
Implement statistical analysis to compare the three systems (including bootstrap resampling for significance testing)

Experiment Execution

For each pilot mode:

Run all three systems (DSA only, SDL only, Integrated) on the same set of theorems
Record success rates, time taken, and other relevant metrics
Generate detailed logs of the proving process for each system
Perform statistical analysis to compare the performance of the three systems

Reporting

Generate a comprehensive report with the following sections:
Summary of results (success rates, average time per theorem)
Comparative analysis of the three systems
Statistical significance of differences
Detailed case studies of specific theorems where the integrated system performed notably better or worse
Analysis of the skill library growth and utilization
Analysis of subgoal generation and completion

Technical Details

DSA Module Specifics

Skills should be stored as structured data with:
Skill name/identifier
Skill description/implementation
Applicable contexts/conditions
Success metrics/history
The validation process should verify that generated skills are:
Syntactically correct
Logically sound
Novel compared to existing skills
Useful for the current or future theorem proving tasks

SDL Module Specifics

Subgoals should be represented with:
Subgoal description/statement
Dependencies on other subgoals
Status (pending, in-progress, completed, failed)
Proof steps/tactics used
The subgoal decomposition should be dynamic and adaptive based on the theorem complexity

Integration Specifics

The integrated system should:
Use subgoal decomposition to guide the theorem proving process
Leverage the skill library to address specific subgoals
Generate new skills when existing skills are insufficient for a subgoal
Update the skill library based on successful subgoal completions

Please implement this experiment with clean, modular code and comprehensive documentation. Ensure that all results are reproducible and that the experiment can be easily extended for future research.

Paper ID

Motivation