Formally Certified Approximate Model Counting

Yong Kiam Tan, Jiong Yang, Mate Soos, Magnus O. Myreen, Kuldeep S. Meel·June 17, 2024

Summary

The paper presents a novel certification framework for approximate model counting in conjunctive normal form (CNF) formulas. The approach combines a formally verified proof of the algorithm's PAC guarantee using Isabelle/HOL with dynamic verification of interactions with an incremental solver. This hybrid method ensures output quality through proof certificates, with minimal overhead. The framework, ApproxMCCert and CertCheck, successfully verifies 84.7% of instances, addressing the need for trustworthy approximate counts. The work includes efficiency improvements, bug fixes, and formalizations, with applications in formal verification and AI. A benchmark evaluation compares the performance of different methods, showing practicality of the approach.

Key findings

2

Paper digest

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

The paper aims to address the problem of certifying probabilistic systems that rely on randomized algorithms, specifically focusing on ApproxMC, a probabilistic automated reasoning system for computing approximate model counts for Boolean formulas . This problem is not entirely new, as the paper acknowledges that certification of randomized algorithms has been a challenge and that prior approaches have struggled to certify such systems deterministically . The proposed hybrid approach in the paper combines theorem-proving and certificate-based methods to certify probabilistic systems like ApproxMC, ensuring that the results computed by ApproxMC can be trusted .


What scientific hypothesis does this paper seek to validate?

This paper aims to validate a scientific hypothesis related to certifying probabilistic systems, specifically focusing on ApproxMC, a probabilistic automated reasoning system that computes approximate model counts for Boolean formulas. The key hypothesis being validated is how to trust a random run of ApproxMC by proposing a verification modulo randomness approach, ensuring that the results computed by ApproxMC can be trusted . The paper seeks to address the challenge of certifying probabilistic systems that rely on randomized algorithms, aiming to combine the power of theorem-proving and certificate-based approaches to provide assurances about the behavior of automated reasoning tools like ApproxMC .


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

The paper "Formally Certified Approximate Model Counting" proposes several new ideas, methods, and models in the field of approximate model counting:

  • The paper introduces Certified knowledge compilation with applications to verified model counting, which focuses on ensuring the correctness and reliability of the model counting process .
  • It discusses Rounding meets approximate model counting, which explores the intersection of rounding techniques with approximate model counting to enhance the accuracy and efficiency of counting models .
  • The paper presents Algorithmic improvements in approximate counting for probabilistic inference, emphasizing advancements in counting techniques from linear to logarithmic SAT calls, contributing to more efficient probabilistic inference processes .
  • It also covers Approximate model counting, providing insights into the methods and algorithms used for approximate counting, which are crucial for various applications in artificial intelligence and computational reasoning .
  • Additionally, the paper delves into Sparse hashing for scalable approximate model counting, offering theoretical frameworks and practical approaches to handle large-scale model counting tasks efficiently .
  • Furthermore, it discusses Auditable algorithms for approximate model counting, highlighting the importance of developing algorithms that can be audited and verified to ensure the accuracy and reliability of the model counting results .
  • The paper also touches upon Model counting and uniform sampling instances, which is essential for generating representative samples and accurately estimating the number of models in complex systems .

These proposed ideas, methods, and models contribute to advancing the field of approximate model counting by enhancing accuracy, efficiency, and reliability in various computational applications. The paper "Formally Certified Approximate Model Counting" introduces several characteristics and advantages compared to previous methods in the field of approximate model counting:

  • Certified knowledge compilation is a key feature that ensures the correctness and reliability of the model counting process, providing a formal guarantee of the accuracy of the results .
  • The integration of Rounding techniques with approximate model counting enhances the precision and efficiency of counting models, offering improved accuracy in the estimation of model counts .
  • The paper emphasizes Algorithmic improvements in approximate counting for probabilistic inference, showcasing advancements from linear to logarithmic SAT calls, leading to more efficient and effective probabilistic reasoning processes .
  • Sparse hashing is utilized for scalable approximate model counting, enabling the handling of large-scale model counting tasks efficiently, which is crucial for applications requiring extensive computational resources .
  • The development of Auditable algorithms for approximate model counting ensures that the algorithms can be audited and verified, enhancing the trustworthiness and reliability of the model counting results .
  • The paper also discusses Model counting and uniform sampling instances, which are essential for generating representative samples and accurately estimating the number of models in complex systems, contributing to a more comprehensive analysis of model counts .

These characteristics and advancements in the paper provide a robust framework for approximate model counting, offering improved accuracy, efficiency, and reliability in computational applications compared to previous methods.


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 researches exist in the field of approximate model counting and formal verification of randomized algorithms. Noteworthy researchers in this field include Yong Kiam Tan, Jiong Yang, Mate Soos, Magnus O. Myreen, and Kuldeep S. Meel . They have contributed to the development of a certification framework for approximate model counting with formally verified guarantees on the quality of its output approximation.

The key to the solution mentioned in the paper is a hybrid approach that combines the power of both theorem-proving and certificate-based approaches to certify probabilistic systems, specifically focusing on ApproxMC, a probabilistic automated reasoning system that computes approximate model counts for Boolean formulas . This approach involves a static, formal proof of the algorithm's guarantees in the Isabelle/HOL proof assistant and dynamic verification of ApproxMC's calls to an external CNF-XOR solver using proof certificates .


How were the experiments in the paper designed?

The experiments in the paper were designed to be performed on the resources of the National Supercomputing Centre, Singapore . Additionally, part of the work was carried out during the Spring 2023 Extended Reunion: Satisfiability program at the Simons Institute for the Theory of Computing and at the Dagstuhl workshop 22411 Theory and Practice of SAT and Combinatorial Solving . The computational experiments were supported by grants from the Ministry of Education Singapore Tier 2 and Tier 1, as well as by A*STAR, Singapore .


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

The dataset used for quantitative evaluation in the study is a publicly available benchmark set consisting of 1896 problem instances used in previous evaluations of ApproxMC . The code for the ApproxMCCert and CertCheck tools is open source and available on GitHub at the following repository: https://github.com/meelgroup/approxmc-cert .


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 strong support for the scientific hypotheses that need to be verified. The paper discusses the formal certification of Approximate Model Counting, focusing on certifying probabilistic systems that rely on randomized algorithms . The approach proposed in the paper aims to combine theorem-proving and certificate-based methods to ensure the correctness of probabilistic systems, such as ApproxMC, which computes approximate model counts for Boolean formulas . This hybrid approach is crucial for establishing trust in the results computed by ApproxMC and similar systems .

The paper outlines the challenges faced in certifying CNF-XOR unsatisfiability, which is essential for the reliability of ApproxMC due to its reliance on CNF-XOR formulas . It discusses prior state-of-the-art approaches for certified CNF-XOR reasoning, highlighting the importance of proof generation and certification of XOR reasoning based on Binary Decision Diagrams (BDDs) and pseudo-Boolean reasoning . These approaches demonstrate a rigorous verification process to ensure the correctness of CNF-XOR unsatisfiability checking, which is fundamental for the validity of the scientific hypotheses being tested .

Furthermore, the paper mentions the use of formally verified proof checkers for deterministic algorithms and theories within Isabelle/HOL, emphasizing the importance of formal verification in ensuring the accuracy of algorithmic results . By leveraging formal verification techniques, the paper establishes a solid foundation for certifying randomized algorithms and probabilistic systems, contributing to the credibility and reliability of the scientific hypotheses being investigated .

In conclusion, the experiments and results presented in the paper offer robust support for the scientific hypotheses that need to be verified. The rigorous approach to formal certification, the utilization of verified proof checkers, and the focus on certifying probabilistic systems collectively contribute to the credibility and trustworthiness of the scientific findings discussed in the paper.


What are the contributions of this paper?

The paper makes several contributions, including:

  • The computational experiments were performed using resources from the National Supercomputing Centre, Singapore .
  • The authors participated in the Spring 2023 Extended Reunion: Satisfiability program at the Simons Institute for the Theory of Computing and at Dagstuhl workshop 22411 Theory and Practice of SAT and Combinatorial Solving .
  • The paper discusses formal verification, proof certification, approximate model counting, and randomized algorithms .
  • It presents research on rounding in approximate model counting .
  • The paper contributes to the field of formal verification, proof certification, and approximate model counting .

What work can be continued in depth?

To delve deeper into the subject matter presented in the document, further exploration can be conducted on the following aspects:

  • Certification of probabilistic systems: The hybrid approach proposed in the paper combines theorem-proving and certificate-based methods to certify probabilistic systems, focusing on ApproxMC, an automated reasoning system for approximate model counting. This approach can be further studied and expanded to enhance the trustworthiness of probabilistic systems .
  • Certified CNF-XOR unsatisfiability checking: Given the reliance of ApproxMC on CNF-XOR formulas, exploring the certification of CNF-XOR unsatisfiability can be a valuable area of research. Prior approaches, such as proof generation and certification of XOR reasoning based on BDDs, pseudo-Boolean reasoning, and standard SAT solvers with CNF proof formats, provide a foundation for further investigation in this domain .
  • Formal verification of randomized algorithms: While prior efforts have focused on certifying deterministic algorithms and theories, there is a need to extend this verification to randomized algorithms. McConnell et al. argue that certifying randomized algorithms poses challenges, indicating a potential area for research to address the certification of probabilistic systems relying on randomized algorithms .
  • Enhancing trust in ApproxMC results: Given the significance of model counting in various applications like control improvisation, network reliability, and probabilistic reasoning, further research can be directed towards refining the certification process of ApproxMC to ensure the trustworthiness of the results it produces .

Tables

2

Introduction
Background
Overview of approximate model counting
Importance of CNF formulas in AI and formal verification
Objective
To develop a novel certification framework
Achieve PAC guarantee with formal verification
Enhance trustworthiness and efficiency
Method
Algorithm Design and Formal Verification
Proof of PAC Guarantee
Isabelle/HOL: Formal proof environment
Rigorous analysis of algorithm's accuracy
ApproxMCCert: Certified Algorithm
Implementation with verified guarantees
Dynamic Verification and Incremental Solver
Integration with incremental solvers
Real-time monitoring of solver interactions
Efficiency Improvements
Optimizations for practical application
Minimizing computational overhead
Bug Detection and Fixes
Formal methods for identifying and correcting errors
CertCheck: Verification Tool
Verification process for output quality
Proof certificates generation and validation
Benchmark Evaluation
Performance Comparison
Testing ApproxMCCert against existing methods
Quantitative analysis of speed and accuracy
Applications
Formal verification use cases
AI applications and potential impact
Conclusion
Summary of achievements and contributions
Future directions and potential improvements
References
List of cited works and resources
Basic info
papers
logic in computer science
artificial intelligence
Advanced features
Insights
How does the hybrid method ensure output quality in the context of approximate model counting?
What is the primary focus of the paper?
What is the novel certification framework proposed in the paper about?
What percentage of instances does the ApproxMCCert and CertCheck framework successfully verify?

Formally Certified Approximate Model Counting

Yong Kiam Tan, Jiong Yang, Mate Soos, Magnus O. Myreen, Kuldeep S. Meel·June 17, 2024

Summary

The paper presents a novel certification framework for approximate model counting in conjunctive normal form (CNF) formulas. The approach combines a formally verified proof of the algorithm's PAC guarantee using Isabelle/HOL with dynamic verification of interactions with an incremental solver. This hybrid method ensures output quality through proof certificates, with minimal overhead. The framework, ApproxMCCert and CertCheck, successfully verifies 84.7% of instances, addressing the need for trustworthy approximate counts. The work includes efficiency improvements, bug fixes, and formalizations, with applications in formal verification and AI. A benchmark evaluation compares the performance of different methods, showing practicality of the approach.
Mind map
Implementation with verified guarantees
Rigorous analysis of algorithm's accuracy
Isabelle/HOL: Formal proof environment
AI applications and potential impact
Formal verification use cases
Quantitative analysis of speed and accuracy
Testing ApproxMCCert against existing methods
Formal methods for identifying and correcting errors
Minimizing computational overhead
Optimizations for practical application
Real-time monitoring of solver interactions
Integration with incremental solvers
ApproxMCCert: Certified Algorithm
Proof of PAC Guarantee
Enhance trustworthiness and efficiency
Achieve PAC guarantee with formal verification
To develop a novel certification framework
Importance of CNF formulas in AI and formal verification
Overview of approximate model counting
List of cited works and resources
Future directions and potential improvements
Summary of achievements and contributions
Applications
Performance Comparison
Proof certificates generation and validation
Verification process for output quality
Bug Detection and Fixes
Efficiency Improvements
Dynamic Verification and Incremental Solver
Algorithm Design and Formal Verification
Objective
Background
References
Conclusion
Benchmark Evaluation
CertCheck: Verification Tool
Method
Introduction
Outline
Introduction
Background
Overview of approximate model counting
Importance of CNF formulas in AI and formal verification
Objective
To develop a novel certification framework
Achieve PAC guarantee with formal verification
Enhance trustworthiness and efficiency
Method
Algorithm Design and Formal Verification
Proof of PAC Guarantee
Isabelle/HOL: Formal proof environment
Rigorous analysis of algorithm's accuracy
ApproxMCCert: Certified Algorithm
Implementation with verified guarantees
Dynamic Verification and Incremental Solver
Integration with incremental solvers
Real-time monitoring of solver interactions
Efficiency Improvements
Optimizations for practical application
Minimizing computational overhead
Bug Detection and Fixes
Formal methods for identifying and correcting errors
CertCheck: Verification Tool
Verification process for output quality
Proof certificates generation and validation
Benchmark Evaluation
Performance Comparison
Testing ApproxMCCert against existing methods
Quantitative analysis of speed and accuracy
Applications
Formal verification use cases
AI applications and potential impact
Conclusion
Summary of achievements and contributions
Future directions and potential improvements
References
List of cited works and resources
Key findings
2

Paper digest

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

The paper aims to address the problem of certifying probabilistic systems that rely on randomized algorithms, specifically focusing on ApproxMC, a probabilistic automated reasoning system for computing approximate model counts for Boolean formulas . This problem is not entirely new, as the paper acknowledges that certification of randomized algorithms has been a challenge and that prior approaches have struggled to certify such systems deterministically . The proposed hybrid approach in the paper combines theorem-proving and certificate-based methods to certify probabilistic systems like ApproxMC, ensuring that the results computed by ApproxMC can be trusted .


What scientific hypothesis does this paper seek to validate?

This paper aims to validate a scientific hypothesis related to certifying probabilistic systems, specifically focusing on ApproxMC, a probabilistic automated reasoning system that computes approximate model counts for Boolean formulas. The key hypothesis being validated is how to trust a random run of ApproxMC by proposing a verification modulo randomness approach, ensuring that the results computed by ApproxMC can be trusted . The paper seeks to address the challenge of certifying probabilistic systems that rely on randomized algorithms, aiming to combine the power of theorem-proving and certificate-based approaches to provide assurances about the behavior of automated reasoning tools like ApproxMC .


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

The paper "Formally Certified Approximate Model Counting" proposes several new ideas, methods, and models in the field of approximate model counting:

  • The paper introduces Certified knowledge compilation with applications to verified model counting, which focuses on ensuring the correctness and reliability of the model counting process .
  • It discusses Rounding meets approximate model counting, which explores the intersection of rounding techniques with approximate model counting to enhance the accuracy and efficiency of counting models .
  • The paper presents Algorithmic improvements in approximate counting for probabilistic inference, emphasizing advancements in counting techniques from linear to logarithmic SAT calls, contributing to more efficient probabilistic inference processes .
  • It also covers Approximate model counting, providing insights into the methods and algorithms used for approximate counting, which are crucial for various applications in artificial intelligence and computational reasoning .
  • Additionally, the paper delves into Sparse hashing for scalable approximate model counting, offering theoretical frameworks and practical approaches to handle large-scale model counting tasks efficiently .
  • Furthermore, it discusses Auditable algorithms for approximate model counting, highlighting the importance of developing algorithms that can be audited and verified to ensure the accuracy and reliability of the model counting results .
  • The paper also touches upon Model counting and uniform sampling instances, which is essential for generating representative samples and accurately estimating the number of models in complex systems .

These proposed ideas, methods, and models contribute to advancing the field of approximate model counting by enhancing accuracy, efficiency, and reliability in various computational applications. The paper "Formally Certified Approximate Model Counting" introduces several characteristics and advantages compared to previous methods in the field of approximate model counting:

  • Certified knowledge compilation is a key feature that ensures the correctness and reliability of the model counting process, providing a formal guarantee of the accuracy of the results .
  • The integration of Rounding techniques with approximate model counting enhances the precision and efficiency of counting models, offering improved accuracy in the estimation of model counts .
  • The paper emphasizes Algorithmic improvements in approximate counting for probabilistic inference, showcasing advancements from linear to logarithmic SAT calls, leading to more efficient and effective probabilistic reasoning processes .
  • Sparse hashing is utilized for scalable approximate model counting, enabling the handling of large-scale model counting tasks efficiently, which is crucial for applications requiring extensive computational resources .
  • The development of Auditable algorithms for approximate model counting ensures that the algorithms can be audited and verified, enhancing the trustworthiness and reliability of the model counting results .
  • The paper also discusses Model counting and uniform sampling instances, which are essential for generating representative samples and accurately estimating the number of models in complex systems, contributing to a more comprehensive analysis of model counts .

These characteristics and advancements in the paper provide a robust framework for approximate model counting, offering improved accuracy, efficiency, and reliability in computational applications compared to previous methods.


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 researches exist in the field of approximate model counting and formal verification of randomized algorithms. Noteworthy researchers in this field include Yong Kiam Tan, Jiong Yang, Mate Soos, Magnus O. Myreen, and Kuldeep S. Meel . They have contributed to the development of a certification framework for approximate model counting with formally verified guarantees on the quality of its output approximation.

The key to the solution mentioned in the paper is a hybrid approach that combines the power of both theorem-proving and certificate-based approaches to certify probabilistic systems, specifically focusing on ApproxMC, a probabilistic automated reasoning system that computes approximate model counts for Boolean formulas . This approach involves a static, formal proof of the algorithm's guarantees in the Isabelle/HOL proof assistant and dynamic verification of ApproxMC's calls to an external CNF-XOR solver using proof certificates .


How were the experiments in the paper designed?

The experiments in the paper were designed to be performed on the resources of the National Supercomputing Centre, Singapore . Additionally, part of the work was carried out during the Spring 2023 Extended Reunion: Satisfiability program at the Simons Institute for the Theory of Computing and at the Dagstuhl workshop 22411 Theory and Practice of SAT and Combinatorial Solving . The computational experiments were supported by grants from the Ministry of Education Singapore Tier 2 and Tier 1, as well as by A*STAR, Singapore .


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

The dataset used for quantitative evaluation in the study is a publicly available benchmark set consisting of 1896 problem instances used in previous evaluations of ApproxMC . The code for the ApproxMCCert and CertCheck tools is open source and available on GitHub at the following repository: https://github.com/meelgroup/approxmc-cert .


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 strong support for the scientific hypotheses that need to be verified. The paper discusses the formal certification of Approximate Model Counting, focusing on certifying probabilistic systems that rely on randomized algorithms . The approach proposed in the paper aims to combine theorem-proving and certificate-based methods to ensure the correctness of probabilistic systems, such as ApproxMC, which computes approximate model counts for Boolean formulas . This hybrid approach is crucial for establishing trust in the results computed by ApproxMC and similar systems .

The paper outlines the challenges faced in certifying CNF-XOR unsatisfiability, which is essential for the reliability of ApproxMC due to its reliance on CNF-XOR formulas . It discusses prior state-of-the-art approaches for certified CNF-XOR reasoning, highlighting the importance of proof generation and certification of XOR reasoning based on Binary Decision Diagrams (BDDs) and pseudo-Boolean reasoning . These approaches demonstrate a rigorous verification process to ensure the correctness of CNF-XOR unsatisfiability checking, which is fundamental for the validity of the scientific hypotheses being tested .

Furthermore, the paper mentions the use of formally verified proof checkers for deterministic algorithms and theories within Isabelle/HOL, emphasizing the importance of formal verification in ensuring the accuracy of algorithmic results . By leveraging formal verification techniques, the paper establishes a solid foundation for certifying randomized algorithms and probabilistic systems, contributing to the credibility and reliability of the scientific hypotheses being investigated .

In conclusion, the experiments and results presented in the paper offer robust support for the scientific hypotheses that need to be verified. The rigorous approach to formal certification, the utilization of verified proof checkers, and the focus on certifying probabilistic systems collectively contribute to the credibility and trustworthiness of the scientific findings discussed in the paper.


What are the contributions of this paper?

The paper makes several contributions, including:

  • The computational experiments were performed using resources from the National Supercomputing Centre, Singapore .
  • The authors participated in the Spring 2023 Extended Reunion: Satisfiability program at the Simons Institute for the Theory of Computing and at Dagstuhl workshop 22411 Theory and Practice of SAT and Combinatorial Solving .
  • The paper discusses formal verification, proof certification, approximate model counting, and randomized algorithms .
  • It presents research on rounding in approximate model counting .
  • The paper contributes to the field of formal verification, proof certification, and approximate model counting .

What work can be continued in depth?

To delve deeper into the subject matter presented in the document, further exploration can be conducted on the following aspects:

  • Certification of probabilistic systems: The hybrid approach proposed in the paper combines theorem-proving and certificate-based methods to certify probabilistic systems, focusing on ApproxMC, an automated reasoning system for approximate model counting. This approach can be further studied and expanded to enhance the trustworthiness of probabilistic systems .
  • Certified CNF-XOR unsatisfiability checking: Given the reliance of ApproxMC on CNF-XOR formulas, exploring the certification of CNF-XOR unsatisfiability can be a valuable area of research. Prior approaches, such as proof generation and certification of XOR reasoning based on BDDs, pseudo-Boolean reasoning, and standard SAT solvers with CNF proof formats, provide a foundation for further investigation in this domain .
  • Formal verification of randomized algorithms: While prior efforts have focused on certifying deterministic algorithms and theories, there is a need to extend this verification to randomized algorithms. McConnell et al. argue that certifying randomized algorithms poses challenges, indicating a potential area for research to address the certification of probabilistic systems relying on randomized algorithms .
  • Enhancing trust in ApproxMC results: Given the significance of model counting in various applications like control improvisation, network reliability, and probabilistic reasoning, further research can be directed towards refining the certification process of ApproxMC to ensure the trustworthiness of the results it produces .
Tables
2
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