miniCodeProps: a Minimal Benchmark for Proving Code Properties

Evan Lohn, Sean Welleck·June 16, 2024

Summary

The paper introduces miniCodeProps, a benchmark for assessing large language models' ability to generate proofs for code properties in the Lean proof assistant. It consists of 177 simple, self-contained program specifications, focusing on tasks like list and binary tree manipulation, with varying levels of difficulty. Current LLM-based provers struggle, managing to prove only about 25% of the cases. miniCodeProps aims to facilitate research on automated theorem proving for formally verified code by providing a controlled environment that differentiates it from other mathematical theorem proving benchmarks. The benchmark is derived from Haskell programs, with a mix of function definitions, results, and termination lemmas, and is publicly available for researchers to improve and advance the field. The study evaluates the performance of several language models, revealing limitations in transferring neural theorem proving to code verification tasks.

Key findings

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Paper digest

What problem does the paper attempt to solve? Is this a new problem?

The paper aims to address the challenge of evaluating the capabilities of large language models (LLMs) for program verification, code generation, and theorem proving . This is not a new problem, as previous work has explored using machine learning techniques based on LLMs for tasks like automatically generating code, proving mathematical theorems, and generating proofs of properties from verification projects . The paper focuses on assessing the effectiveness of LLMs in these areas, highlighting the need for better automation to facilitate code verification and theorem proving processes .


What scientific hypothesis does this paper seek to validate?

This paper aims to validate the scientific hypothesis related to the effectiveness of machine learning techniques based on large language models (LLMs) in automating various aspects of code verification, such as generating code, proving mathematical theorems in interactive theorem provers (ITPs), and generating proofs of properties from verification projects . The paper discusses the challenges in evaluating the capabilities of LLMs for program verification, especially in complex projects with dependencies like the CompCert compiler or Archive of Formal Proofs, highlighting the need to isolate and understand the core capabilities and weaknesses of these models .


What new ideas, methods, or models does the paper propose? What are the characteristics and advantages compared to previous methods?

The paper introduces several innovative ideas, methods, and models in the field of automated theorem proving and code verification:

  • Neural Theorem Proving with Growing Libraries: The LEGO-prover is a neural theorem proving approach that utilizes growing libraries to enhance its capabilities .
  • In-Context Learning Agent for Formal Theorem-Proving: The paper presents an in-context learning agent designed for formal theorem-proving tasks, contributing to advancements in this area .
  • Llemma: An Open Language Model for Mathematics: The Llemma model is introduced as an open language model specifically tailored for mathematical tasks, showcasing the potential of language models in theorem proving .
  • Machine Learning Techniques with Large Language Models (LLMs): The paper explores the use of large language models in automatically generating code, proving mathematical theorems, and generating proofs of properties in verification projects .
  • LeanDojo: Theorem Proving with Retrieval-Augmented Language Models: The LeanDojo model is proposed for theorem proving, incorporating retrieval-augmented language models to enhance the theorem proving process .
  • Formal Mathematics Statement Curriculum Learning: The concept of formal mathematics statement curriculum learning is introduced, highlighting a novel approach to learning and applying mathematical statements .
  • Hypertree Proof Search for Neural Theorem Proving: The paper discusses the Hypertree proof search method, which aims to improve neural theorem proving by enhancing the search process .
  • Draft, Sketch, and Prove Approach: The Draft, Sketch, and Prove methodology is presented as a strategy to guide formal theorem provers using informal proofs, showcasing a unique approach to theorem proving .
  • Generative Language Modeling for Automated Theorem Proving: The paper explores the use of generative language modeling techniques to automate theorem proving tasks, demonstrating the potential of such models in this domain .

These innovative ideas, methods, and models collectively contribute to advancing the field of automated theorem proving and code verification, showcasing the potential of machine learning and large language models in enhancing the efficiency and effectiveness of theorem proving processes. The paper introduces novel approaches in automated theorem proving and code verification, offering distinct characteristics and advantages compared to previous methods:

  • Full Proof Generation via Few-Shot Prompting: The paper employs a few-shot prompting approach for full proof generation, where the model generates potential proofs verified by a proof checker like the Lean 4 kernel . This method allows for the generation of proofs based on examples of code, properties, and proofs not seen in the benchmark, showcasing adaptability and potential for improvement .
  • Next-Step Tactic Generation: The experimental setup includes next-step tactic generation using frameworks like LLMStep, providing suggestions for the next tactic to use in the proof based on the proof state and file context . This approach enhances the efficiency of theorem proving by guiding the model in selecting appropriate tactics for proof construction.
  • Neural Theorem Proving with Growing Libraries: The LEGO-prover introduces neural theorem proving with growing libraries, enabling the model to expand its knowledge base and improve theorem proving capabilities . This characteristic enhances the model's ability to handle a wider range of mathematical problems and code verification tasks.
  • In-Context Learning Agent for Formal Theorem-Proving: The in-context learning agent presented in the paper focuses on formal theorem-proving tasks, contributing to advancements in this domain . This approach allows the model to learn and adapt within the context of theorem proving, leading to more effective and accurate results.
  • Generative Language Modeling for Automated Theorem Proving: The paper explores the use of generative language modeling techniques for automated theorem proving, showcasing the potential of such models in enhancing the efficiency and effectiveness of theorem proving processes . This approach leverages the power of large language models to automate theorem proving tasks and generate proofs of properties in verification projects.
  • LeanDojo: Theorem Proving with Retrieval-Augmented Language Models: The LeanDojo model integrates retrieval-augmented language models for theorem proving, enhancing the theorem proving process by incorporating retrieval mechanisms . This characteristic enables the model to access relevant information and improve its theorem proving capabilities.
  • Formal Mathematics Statement Curriculum Learning: The concept of formal mathematics statement curriculum learning is introduced, offering a unique approach to learning and applying mathematical statements . This method enhances the model's understanding of mathematical concepts and improves its performance in theorem proving tasks.

These characteristics and advantages highlight the innovative nature of the proposed methods in automated theorem proving and code verification, showcasing advancements in leveraging machine learning techniques and large language models to enhance the efficiency, accuracy, and adaptability of theorem proving processes.


Do any related researches exist? Who are the noteworthy researchers on this topic in this field?What is the key to the solution mentioned in the paper?

Several related research works exist in the field of theorem proving and code properties. Noteworthy researchers in this area include Makarius Wenzel, Lawrence C Paulson, Tobias Nipkow, The Coq Development Team, Leonardo de Moura, Soonho Kong, Jeremy Avigad, Floris Van Doorn, Jakob von Raumer, Chuyue Sun, Ying Sheng, Oded Padon, Clark Barrett, Haiming Wang, Huajian Xin, Chuanyang Zheng, and many others . These researchers have contributed to various aspects of theorem proving, formal verification, and code properties.

The key to the solution mentioned in the paper "miniCodeProps: a Minimal Benchmark for Proving Code Properties" involves the development of future ITP (Interactive Theorem Proving) code verification agents . The paper discusses the potential of machine learning techniques based on large language models (LLMs) in automatically generating code, proving mathematical theorems in ITPs, and generating proofs of properties from verification projects . The focus is on improving automation in interactive theorem proving to make it easier to verify code and enhance the capabilities of LLMs for program verification.


How were the experiments in the paper designed?

The experiments in the paper were designed with two main setups: Next-Step Tactic Generation and Full Proof Generation . In the Next-Step Tactic Generation setup, the experiments utilized the LLMStep framework, modified to provide cumulative log probabilities along with next step suggestions from each backend language model . The Pythia2.8B experiments were conducted with proof state only as input, while other models received proof state + dependencies as input . These experiments were run on a single A100 GPU with 30 CPU cores running Ubuntu 20.04 .

For the Full Proof Generation setup, a few-shot prompting approach was employed, where the prompt contained examples of code, properties, and proofs not seen in the benchmark . The evaluation mode used proof state + dependencies, although in practice, only the dependencies and the theorem statement itself were used . The experiments were performed using the OpenAI chat completions API with GPT4-turbo . The results of the experiments were displayed in Table 3, showing the performance of the baseline approaches in proving program specifications .


What is the dataset used for quantitative evaluation? Is the code open source?

The dataset used for quantitative evaluation in the context of proving code properties is derived from Tons of Inductive Problems (TIP) . The programs and specifications in this dataset are written in Haskell and are translated into Lean for the benchmark miniCodeProps. The code used in the benchmark is open source as it is sourced from Tons of Inductive Problems, which is available on GitHub .


Do the experiments and results in the paper provide good support for the scientific hypotheses that need to be verified? Please analyze.

The experiments and results presented in the paper provide valuable insights into the scientific hypotheses that need to be verified in the field of automated code verification using machine learning techniques based on large language models (LLMs) . The experiments focus on two main approaches: full proof generation via few-shot prompting a language model and next-step tactic generation . The results indicate that while the baseline approaches struggled with proving complex function properties, there were some successes in proving termination lemmas, suggesting the potential for improvement in neural theorem proving approaches for code verification in the future .

The paper discusses the challenges and opportunities in automating formal code verification, highlighting the historical development of automation for formal verification of code and the shift towards machine learning-based approaches in interactive theorem proving . Previous methods focused on Coq, such as ProverBot9000, ASTactic, TacTok, Diva, and Passport, while recent work explores prompting strategies in Coq and proof generation and repair in Isabelle's Archive of Formal Proofs . The absence of machine learning-based proof automation targeting program verification in Lean is noted, despite advancements in automated mathematical proving .

Overall, the experiments and results in the paper contribute to the understanding of the challenges and potential advancements in automated code verification using machine learning techniques, providing a foundation for further research and development in this field . The insights gained from the experiments help in evaluating the capabilities of large language models for program verification and highlight the need for continued exploration and innovation in this area .


What are the contributions of this paper?

The paper makes several contributions, including:

  • Introducing a minimal benchmark, miniCodeProps, for proving code properties .
  • Exploring formal mathematics statement curriculum learning .
  • Presenting Hypertree proof search for neural theorem proving .
  • Proposing LeanDojo, a theorem proving approach with retrieval-augmented language models .
  • Guiding formal theorem provers with informal proofs through Draft, sketch, and prove methodology .
  • Evaluating large language models trained on code .
  • Surveying deep learning for theorem proving .
  • Developing Baldur for whole-proof generation and repair using large language models .
  • Investigating learning to prove theorems via interacting with proof assistants .
  • Providing a survey of deep learning for mathematical reasoning .
  • Introducing minif2f, a cross-system benchmark for formal olympiad-level mathematics .

What work can be continued in depth?

Continuing the work in the field of automated code verification and interactive theorem proving offers several avenues for further exploration and development . Some potential areas for in-depth research include:

  • Exploring prompting strategies: Designing prompting strategies tailored to mathematical problems, such as conditioning generation on informal proofs or examples from Mathlib, could enhance the automation of formal code verification .
  • Machine learning-based proof automation: Further investigation into machine learning-based proof automation, particularly targeting program verification in Lean, could be a valuable area of study .
  • Enhancing automation for formal verification: Building on the rich history of automation for formal code verification, there is room for improvement and innovation in developing more efficient and effective automation techniques .
  • Investigating transferability of neural theorem proving: Research focusing on the transferability of neural theorem proving approaches from mathematics to interactive theorem proving for code verification could lead to advancements in this field .
  • Benchmark development: Continuous refinement and expansion of benchmarks like miniCodeProps can provide valuable insights into the capabilities and limitations of different verification methods, guiding future research directions .
  • Studying automation in different theorem proving systems: Exploring automation techniques in various theorem proving systems like Coq, Dafny, and Rust can help in understanding the strengths and weaknesses of different approaches and improving overall automation capabilities .
Basic info
papers
software engineering
machine learning
artificial intelligence
Advanced features
Insights
How does miniCodeProps contribute to the research on automated theorem proving for formally verified code?
How many program specifications are included in the benchmark, and what are they mainly about?
What is miniCodeProps, and what does it evaluate in large language models?
What percentage of cases can current LLM-based provers manage to prove in the miniCodeProps benchmark?

miniCodeProps: a Minimal Benchmark for Proving Code Properties

Evan Lohn, Sean Welleck·June 16, 2024

Summary

The paper introduces miniCodeProps, a benchmark for assessing large language models' ability to generate proofs for code properties in the Lean proof assistant. It consists of 177 simple, self-contained program specifications, focusing on tasks like list and binary tree manipulation, with varying levels of difficulty. Current LLM-based provers struggle, managing to prove only about 25% of the cases. miniCodeProps aims to facilitate research on automated theorem proving for formally verified code by providing a controlled environment that differentiates it from other mathematical theorem proving benchmarks. The benchmark is derived from Haskell programs, with a mix of function definitions, results, and termination lemmas, and is publicly available for researchers to improve and advance the field. The study evaluates the performance of several language models, revealing limitations in transferring neural theorem proving to code verification tasks.
Mind map
Performance Metrics
Program Specifications
Future Directions
Limitations and Challenges
Model Evaluation
Data Preprocessing
Data Collection
Objective
Background
Conclusion
Discussion
Method
Introduction
Key findings
2

Paper digest

What problem does the paper attempt to solve? Is this a new problem?

The paper aims to address the challenge of evaluating the capabilities of large language models (LLMs) for program verification, code generation, and theorem proving . This is not a new problem, as previous work has explored using machine learning techniques based on LLMs for tasks like automatically generating code, proving mathematical theorems, and generating proofs of properties from verification projects . The paper focuses on assessing the effectiveness of LLMs in these areas, highlighting the need for better automation to facilitate code verification and theorem proving processes .


What scientific hypothesis does this paper seek to validate?

This paper aims to validate the scientific hypothesis related to the effectiveness of machine learning techniques based on large language models (LLMs) in automating various aspects of code verification, such as generating code, proving mathematical theorems in interactive theorem provers (ITPs), and generating proofs of properties from verification projects . The paper discusses the challenges in evaluating the capabilities of LLMs for program verification, especially in complex projects with dependencies like the CompCert compiler or Archive of Formal Proofs, highlighting the need to isolate and understand the core capabilities and weaknesses of these models .


What new ideas, methods, or models does the paper propose? What are the characteristics and advantages compared to previous methods?

The paper introduces several innovative ideas, methods, and models in the field of automated theorem proving and code verification:

  • Neural Theorem Proving with Growing Libraries: The LEGO-prover is a neural theorem proving approach that utilizes growing libraries to enhance its capabilities .
  • In-Context Learning Agent for Formal Theorem-Proving: The paper presents an in-context learning agent designed for formal theorem-proving tasks, contributing to advancements in this area .
  • Llemma: An Open Language Model for Mathematics: The Llemma model is introduced as an open language model specifically tailored for mathematical tasks, showcasing the potential of language models in theorem proving .
  • Machine Learning Techniques with Large Language Models (LLMs): The paper explores the use of large language models in automatically generating code, proving mathematical theorems, and generating proofs of properties in verification projects .
  • LeanDojo: Theorem Proving with Retrieval-Augmented Language Models: The LeanDojo model is proposed for theorem proving, incorporating retrieval-augmented language models to enhance the theorem proving process .
  • Formal Mathematics Statement Curriculum Learning: The concept of formal mathematics statement curriculum learning is introduced, highlighting a novel approach to learning and applying mathematical statements .
  • Hypertree Proof Search for Neural Theorem Proving: The paper discusses the Hypertree proof search method, which aims to improve neural theorem proving by enhancing the search process .
  • Draft, Sketch, and Prove Approach: The Draft, Sketch, and Prove methodology is presented as a strategy to guide formal theorem provers using informal proofs, showcasing a unique approach to theorem proving .
  • Generative Language Modeling for Automated Theorem Proving: The paper explores the use of generative language modeling techniques to automate theorem proving tasks, demonstrating the potential of such models in this domain .

These innovative ideas, methods, and models collectively contribute to advancing the field of automated theorem proving and code verification, showcasing the potential of machine learning and large language models in enhancing the efficiency and effectiveness of theorem proving processes. The paper introduces novel approaches in automated theorem proving and code verification, offering distinct characteristics and advantages compared to previous methods:

  • Full Proof Generation via Few-Shot Prompting: The paper employs a few-shot prompting approach for full proof generation, where the model generates potential proofs verified by a proof checker like the Lean 4 kernel . This method allows for the generation of proofs based on examples of code, properties, and proofs not seen in the benchmark, showcasing adaptability and potential for improvement .
  • Next-Step Tactic Generation: The experimental setup includes next-step tactic generation using frameworks like LLMStep, providing suggestions for the next tactic to use in the proof based on the proof state and file context . This approach enhances the efficiency of theorem proving by guiding the model in selecting appropriate tactics for proof construction.
  • Neural Theorem Proving with Growing Libraries: The LEGO-prover introduces neural theorem proving with growing libraries, enabling the model to expand its knowledge base and improve theorem proving capabilities . This characteristic enhances the model's ability to handle a wider range of mathematical problems and code verification tasks.
  • In-Context Learning Agent for Formal Theorem-Proving: The in-context learning agent presented in the paper focuses on formal theorem-proving tasks, contributing to advancements in this domain . This approach allows the model to learn and adapt within the context of theorem proving, leading to more effective and accurate results.
  • Generative Language Modeling for Automated Theorem Proving: The paper explores the use of generative language modeling techniques for automated theorem proving, showcasing the potential of such models in enhancing the efficiency and effectiveness of theorem proving processes . This approach leverages the power of large language models to automate theorem proving tasks and generate proofs of properties in verification projects.
  • LeanDojo: Theorem Proving with Retrieval-Augmented Language Models: The LeanDojo model integrates retrieval-augmented language models for theorem proving, enhancing the theorem proving process by incorporating retrieval mechanisms . This characteristic enables the model to access relevant information and improve its theorem proving capabilities.
  • Formal Mathematics Statement Curriculum Learning: The concept of formal mathematics statement curriculum learning is introduced, offering a unique approach to learning and applying mathematical statements . This method enhances the model's understanding of mathematical concepts and improves its performance in theorem proving tasks.

These characteristics and advantages highlight the innovative nature of the proposed methods in automated theorem proving and code verification, showcasing advancements in leveraging machine learning techniques and large language models to enhance the efficiency, accuracy, and adaptability of theorem proving processes.


Do any related researches exist? Who are the noteworthy researchers on this topic in this field?What is the key to the solution mentioned in the paper?

Several related research works exist in the field of theorem proving and code properties. Noteworthy researchers in this area include Makarius Wenzel, Lawrence C Paulson, Tobias Nipkow, The Coq Development Team, Leonardo de Moura, Soonho Kong, Jeremy Avigad, Floris Van Doorn, Jakob von Raumer, Chuyue Sun, Ying Sheng, Oded Padon, Clark Barrett, Haiming Wang, Huajian Xin, Chuanyang Zheng, and many others . These researchers have contributed to various aspects of theorem proving, formal verification, and code properties.

The key to the solution mentioned in the paper "miniCodeProps: a Minimal Benchmark for Proving Code Properties" involves the development of future ITP (Interactive Theorem Proving) code verification agents . The paper discusses the potential of machine learning techniques based on large language models (LLMs) in automatically generating code, proving mathematical theorems in ITPs, and generating proofs of properties from verification projects . The focus is on improving automation in interactive theorem proving to make it easier to verify code and enhance the capabilities of LLMs for program verification.


How were the experiments in the paper designed?

The experiments in the paper were designed with two main setups: Next-Step Tactic Generation and Full Proof Generation . In the Next-Step Tactic Generation setup, the experiments utilized the LLMStep framework, modified to provide cumulative log probabilities along with next step suggestions from each backend language model . The Pythia2.8B experiments were conducted with proof state only as input, while other models received proof state + dependencies as input . These experiments were run on a single A100 GPU with 30 CPU cores running Ubuntu 20.04 .

For the Full Proof Generation setup, a few-shot prompting approach was employed, where the prompt contained examples of code, properties, and proofs not seen in the benchmark . The evaluation mode used proof state + dependencies, although in practice, only the dependencies and the theorem statement itself were used . The experiments were performed using the OpenAI chat completions API with GPT4-turbo . The results of the experiments were displayed in Table 3, showing the performance of the baseline approaches in proving program specifications .


What is the dataset used for quantitative evaluation? Is the code open source?

The dataset used for quantitative evaluation in the context of proving code properties is derived from Tons of Inductive Problems (TIP) . The programs and specifications in this dataset are written in Haskell and are translated into Lean for the benchmark miniCodeProps. The code used in the benchmark is open source as it is sourced from Tons of Inductive Problems, which is available on GitHub .


Do the experiments and results in the paper provide good support for the scientific hypotheses that need to be verified? Please analyze.

The experiments and results presented in the paper provide valuable insights into the scientific hypotheses that need to be verified in the field of automated code verification using machine learning techniques based on large language models (LLMs) . The experiments focus on two main approaches: full proof generation via few-shot prompting a language model and next-step tactic generation . The results indicate that while the baseline approaches struggled with proving complex function properties, there were some successes in proving termination lemmas, suggesting the potential for improvement in neural theorem proving approaches for code verification in the future .

The paper discusses the challenges and opportunities in automating formal code verification, highlighting the historical development of automation for formal verification of code and the shift towards machine learning-based approaches in interactive theorem proving . Previous methods focused on Coq, such as ProverBot9000, ASTactic, TacTok, Diva, and Passport, while recent work explores prompting strategies in Coq and proof generation and repair in Isabelle's Archive of Formal Proofs . The absence of machine learning-based proof automation targeting program verification in Lean is noted, despite advancements in automated mathematical proving .

Overall, the experiments and results in the paper contribute to the understanding of the challenges and potential advancements in automated code verification using machine learning techniques, providing a foundation for further research and development in this field . The insights gained from the experiments help in evaluating the capabilities of large language models for program verification and highlight the need for continued exploration and innovation in this area .


What are the contributions of this paper?

The paper makes several contributions, including:

  • Introducing a minimal benchmark, miniCodeProps, for proving code properties .
  • Exploring formal mathematics statement curriculum learning .
  • Presenting Hypertree proof search for neural theorem proving .
  • Proposing LeanDojo, a theorem proving approach with retrieval-augmented language models .
  • Guiding formal theorem provers with informal proofs through Draft, sketch, and prove methodology .
  • Evaluating large language models trained on code .
  • Surveying deep learning for theorem proving .
  • Developing Baldur for whole-proof generation and repair using large language models .
  • Investigating learning to prove theorems via interacting with proof assistants .
  • Providing a survey of deep learning for mathematical reasoning .
  • Introducing minif2f, a cross-system benchmark for formal olympiad-level mathematics .

What work can be continued in depth?

Continuing the work in the field of automated code verification and interactive theorem proving offers several avenues for further exploration and development . Some potential areas for in-depth research include:

  • Exploring prompting strategies: Designing prompting strategies tailored to mathematical problems, such as conditioning generation on informal proofs or examples from Mathlib, could enhance the automation of formal code verification .
  • Machine learning-based proof automation: Further investigation into machine learning-based proof automation, particularly targeting program verification in Lean, could be a valuable area of study .
  • Enhancing automation for formal verification: Building on the rich history of automation for formal code verification, there is room for improvement and innovation in developing more efficient and effective automation techniques .
  • Investigating transferability of neural theorem proving: Research focusing on the transferability of neural theorem proving approaches from mathematics to interactive theorem proving for code verification could lead to advancements in this field .
  • Benchmark development: Continuous refinement and expansion of benchmarks like miniCodeProps can provide valuable insights into the capabilities and limitations of different verification methods, guiding future research directions .
  • Studying automation in different theorem proving systems: Exploring automation techniques in various theorem proving systems like Coq, Dafny, and Rust can help in understanding the strengths and weaknesses of different approaches and improving overall automation capabilities .
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