Formally Verifying Deep Reinforcement Learning Controllers with Lyapunov Barrier Certificates

Udayan Mandal, Guy Amir, Haoze Wu, Ieva Daukantas, Fletcher Lee Newell, Umberto J. Ravaioli, Baoluo Meng, Michael Durling, Milan Ganai, Tobey Shim, Guy Katz, Clark Barrett·May 22, 2024

Summary

This paper introduces a novel method for training and verifying Neural Lyapunov Barrier (NLB) certificates in deep reinforcement learning (DRL) to ensure safety and liveness properties in complex systems. The approach simplifies verification by using certificate composition and filtering, which are combined with neural network verification engines. The focus is on the Reach-While-Avoid (RWA) task, with a case study involving a DRL-controlled spacecraft. The Filtered RWA extension and compositional certification help address scalability challenges, enabling the handling of difficult properties. The study demonstrates improved efficiency and ability to verify safety and docking in a 2D docking task, while highlighting the limitations and future directions for extending the method to more complex systems and formal techniques.

Key findings

3

Paper digest

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

The paper aims to address the challenge of verifying the behavior of Deep Reinforcement Learning (DRL) controllers, particularly in safety-critical systems like autonomous vehicles, where even a single error can have severe consequences . This paper introduces a novel framework for training and formally verifying Neural Lyapunov Barrier (NLB) certificates for discrete-time systems, providing guarantees on achieving goals and avoiding unsafe behavior . While the use of NLB certificates is not new, the difficulty in learning and verifying these certificates, especially for complex real-world systems, is a significant problem that this paper seeks to tackle .


What scientific hypothesis does this paper seek to validate?

This paper aims to validate the scientific hypothesis related to the use of Neural Lyapunov Barrier (NLB) certificates for providing strong guarantees on the behavior of Deep Reinforcement Learning (DRL) agents in complex systems . The hypothesis revolves around the effectiveness of NLB-based certificates in ensuring that a DRL agent achieves its goals and avoids unsafe behavior by learning functions over the system that indirectly imply desired agent behavior . The paper introduces novel techniques for training and verifying NLB-based certificates for discrete-time systems, including certificate composition and filtering methods to simplify the verification process for complex systems .


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

The paper proposes a novel framework for training and formally verifying Neural Lyapunov Barrier (NLB) certificates, specifically focusing on deep reinforcement learning (DRL) controllers . This framework aims to provide guarantees for both liveness and safety properties, offering reach-while-avoid (RWA) assurances . To enhance scalability, the framework utilizes off-the-shelf deep neural network (DNN) verifiers and introduces innovative techniques like certificate filtering and composition . These methods are designed to address the challenges associated with verifying complex DRL-based dynamical systems with large state spaces and intricate dynamics . The paper introduces a novel framework for formally verifying Deep Reinforcement Learning (DRL) controllers using Neural Lyapunov Barrier (NLB) certificates, focusing on safety and liveness properties in complex systems . This framework leverages a Counterexample-Guided Inductive Synthesis (CEGIS) loop for training and verification, incorporating innovative techniques like certificate filtering and composition to enhance scalability and robustness . Compared to previous methods, the proposed framework demonstrates significant advantages in terms of faster training, fewer failures, and improved stability, particularly evident in the compositional certificates that enable scalability to large state spaces . Additionally, the framework showcases the ability to verify DNN-based controllers for intricate systems, offering reach-while-avoid (RWA) assurances and outperforming standard RWA methods in terms of producing verified certificates efficiently . The paper's approach not only simplifies the training and verification process but also unlocks additional scalability, making it a promising step towards ensuring the safe and reliable deployment of DRLThe proposed framework for training and verifying Neural Lyapunov Barrier (NLB) certificates in deep reinforcement learning (DRL) controllers offers several key characteristics and advantages compared to previous methods:

  1. Scalability: The framework leverages off-the-shelf deep neural network (DNN) verifiers, enabling scalability to handle complex DRL-based dynamical systems with large state spaces and intricate dynamics. This scalability is crucial for real-world applications where systems are often high-dimensional and non-linear.

  2. Guarantees for Liveness and Safety Properties: The framework provides guarantees for both liveness and safety properties, ensuring that the controlled system can reach desired states while avoiding unsafe regions. This dual assurance is essential for critical systems where both performance and safety are paramount.

  3. Reach-While-Avoid (RWA) Assurances: By offering reach-while-avoid assurances, the framework allows the controlled system to simultaneously achieve specified goals while steering clear of undesirable states or obstacles. This capability enhances the robustness and reliability of the controller in challenging environments.

  4. Innovative Techniques: The framework introduces innovative techniques such as certificate filtering and composition to enhance the efficiency and effectiveness of the verification process. These techniques help streamline the verification of NLB certificates, making it more practical for real-time applications.

  5. Formal Verification: The framework emphasizes formal verification of NLB certificates, providing mathematical guarantees on the system's behavior under the control of the DRL controller. This formal approach enhances the trustworthiness and reliability of the controller, especially in safety-critical scenarios.

  6. Integration of Neural Networks: By integrating neural networks into the verification process, the framework can handle complex, non-linear dynamics that are common in DRL controllers. This integration allows for more accurate and robust verification of NLB certificates in challenging environments.

Overall, the proposed framework stands out for its scalability, dual guarantees for liveness and safety properties, reach-while-avoid assurances, innovative techniques, formal verification approach, and integration of neural networks. These characteristics collectively contribute to a more robust and reliable method for training and verifying NLB certificates in DRL controllers, offering significant advancements over 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 research studies exist in the field of formally verifying deep reinforcement learning controllers with Lyapunov barrier certificates. Noteworthy researchers in this area include Guy Amir, G. Katz, M. Schapira, A. Farinelli, and I. Refaeli . These researchers have contributed to various aspects of verifying learning-based robotic navigation systems, industrial wildfire detection systems, generalization in deep learning, scalability of verification in deep reinforcement learning, and more.

The key to the solution mentioned in the paper is the use of Neural Lyapunov Barrier (NLB) certificates. These certificates are learned functions over the system that indirectly imply the desired behavior of an agent in deep reinforcement learning. The paper presents a novel method for training and verifying NLB-based certificates for discrete-time systems. This method involves certificate composition, which simplifies the verification of complex systems by strategically designing a sequence of certificates. When combined with neural network verification engines, these certificates provide formal guarantees that a deep reinforcement learning agent achieves its goals and avoids unsafe behavior .


How were the experiments in the paper designed?

The experiments in the paper were designed to explore various architectures for the Deep Neural Network (DNN) used to control the spacecraft . The researchers trained each DNN architecture using the Proximal Policy Optimization Reinforcement Learning (RL) algorithm without any CEGIS iteration . After training, they simulated each architecture on 4,000 random trajectories and analyzed the results . The experiments aimed to verify a Deep Reinforcement Learning (DRL)-based spacecraft controller by generating verified NLB-based Reachability with Avoidance (RWA) certificates for complex properties such as liveness and safety . The experiments involved training both RWA and FRWA certificates using a CEGIS loop and running multiple trials for each task to evaluate the success rate and average time required for successful runs .


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

The dataset used for quantitative evaluation in the context of formally verifying deep reinforcement learning controllers with Lyapunov barrier certificates is Marabou 2.0 . The code for Marabou 2.0 is open source and available on arXiv .


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 introduces a novel method for training and verifying Neural Lyapunov Barrier (NLB) certificates for discrete-time systems, which are crucial for ensuring the safety and liveness guarantees of deep reinforcement learning (DRL) agents controlling autonomous systems . The experiments conducted demonstrate the effectiveness of this approach in providing formal guarantees that a DRL agent achieves its goals and avoids unsafe behavior, particularly in safety-critical applications .

The paper outlines the use of a technique for certificate composition, which simplifies the verification process for highly complex systems by strategically designing a sequence of certificates . This method, when combined with neural network verification engines, offers a formal guarantee of the DRL agent's behavior, aligning with the scientific hypothesis of ensuring safe and reliable performance of autonomous systems .

Furthermore, the experiments compare Reachability with Avoidance (RWA) tasks against Fully Reachability with Avoidance (FRWA) tasks, showcasing the effectiveness of the proposed approach in producing verified controllers and certificates within a reasonable time frame . The results demonstrate the superiority of FRWA over RWA in terms of successful runs and average time required, providing empirical evidence to support the scientific hypotheses put forth in the paper .

Overall, the experiments and results presented in the paper offer substantial evidence to validate the scientific hypotheses related to training and verifying NLB certificates for DRL agents, emphasizing the importance of formal verification methods in ensuring the safety and reliability of autonomous systems in various domains .


What are the contributions of this paper?

The paper "Formally Verifying Deep Reinforcement Learning Controllers with Lyapunov Barrier Certificates" presents several key contributions:

  • Introducing a novel method for training and verifying Neural Lyapunov Barrier (NLB) certificates for discrete-time systems, which involves certificate composition to simplify verification of complex systems and certificate filtering to streamline the production of verified certificates .
  • Demonstrating the effectiveness of the proposed approach through a case study on ensuring safety and liveness guarantees for a deep reinforcement learning (DRL)-controlled spacecraft .
  • Providing a formal guarantee that a DRL agent achieves its goals and avoids unsafe behavior by using NLB-based certificates in conjunction with neural network verification engines .

What work can be continued in depth?

Further research in the field of formally verifying DRL-based controllers can be continued in depth by exploring the following areas:

  • Extension to Additional Formal Techniques: Extending the approach to be compatible with additional formal techniques, such as shielding, can enhance the verification process .
  • Application to More Challenging Case Studies: Applying the approach to more challenging case studies involving larger DRL controllers can help test the scalability and effectiveness of the method in complex systems .
  • Exploration of Larger State Spaces: Investigating how the compositional certificates can scale to even larger state spaces beyond the demonstrated 2D case study can provide insights into the method's applicability in diverse scenarios .
  • Enhancing Safety and Reliability: Continuously working towards ensuring the safe and reliable use of DRL in real-world systems by refining the verification techniques and addressing potential challenges .

Tables

5

Introduction
Background
Evolution of safety concerns in DRL
Importance of formal verification in complex systems
Objective
To develop a novel NLB method for DRL safety and liveness
Address scalability challenges in RWA tasks
Case study: DRL-controlled spacecraft
Method
Data Collection
Real-world and simulated DRL environments
Trajectory data for training and testing
Data Preprocessing
State-action pairs for neural network input
Generation of RWA and docking task scenarios
Neural Lyapunov Barrier (NLB) Design
NLB Function Learning
Training neural networks to approximate barrier functions
Use of reinforcement learning algorithms
Certificate Composition
Combining NLB functions for compositional verification
Simplification of complex system properties
Filtered RWA Extension
Development of a filtering mechanism
Reducing complexity for efficient verification
Scalability and Efficiency
Addressing computational challenges
Performance evaluation on 2D docking task
Case Study: DRL Spacecraft
Application to a spacecraft control system
Safety and docking verification results
Results and Evaluation
Efficiency improvements over existing methods
Success rates and verification times for RWA tasks
Comparison with formal verification techniques
Limitations and Future Directions
Current method's scope and applicability
Challenges for extending to higher-dimensional systems
Integration with formal methods and hybrid approaches
Conclusion
Summary of contributions and implications
Future research directions for NLB certificates in DRL safety verification
Basic info
papers
machine learning
systems and control
artificial intelligence
Advanced features
Insights
How does the paper address scalability issues in deep reinforcement learning (DRL) verification?
What is the primary goal of the Neural Lyapunov Barrier (NLB) method in the paper?
What are the main achievements demonstrated in the 2D docking task?
What task is used as a case study for evaluating the effectiveness of the proposed method?

Formally Verifying Deep Reinforcement Learning Controllers with Lyapunov Barrier Certificates

Udayan Mandal, Guy Amir, Haoze Wu, Ieva Daukantas, Fletcher Lee Newell, Umberto J. Ravaioli, Baoluo Meng, Michael Durling, Milan Ganai, Tobey Shim, Guy Katz, Clark Barrett·May 22, 2024

Summary

This paper introduces a novel method for training and verifying Neural Lyapunov Barrier (NLB) certificates in deep reinforcement learning (DRL) to ensure safety and liveness properties in complex systems. The approach simplifies verification by using certificate composition and filtering, which are combined with neural network verification engines. The focus is on the Reach-While-Avoid (RWA) task, with a case study involving a DRL-controlled spacecraft. The Filtered RWA extension and compositional certification help address scalability challenges, enabling the handling of difficult properties. The study demonstrates improved efficiency and ability to verify safety and docking in a 2D docking task, while highlighting the limitations and future directions for extending the method to more complex systems and formal techniques.
Mind map
Simplification of complex system properties
Combining NLB functions for compositional verification
Use of reinforcement learning algorithms
Training neural networks to approximate barrier functions
Safety and docking verification results
Application to a spacecraft control system
Performance evaluation on 2D docking task
Addressing computational challenges
Reducing complexity for efficient verification
Development of a filtering mechanism
Certificate Composition
NLB Function Learning
Generation of RWA and docking task scenarios
State-action pairs for neural network input
Trajectory data for training and testing
Real-world and simulated DRL environments
Case study: DRL-controlled spacecraft
Address scalability challenges in RWA tasks
To develop a novel NLB method for DRL safety and liveness
Importance of formal verification in complex systems
Evolution of safety concerns in DRL
Future research directions for NLB certificates in DRL safety verification
Summary of contributions and implications
Integration with formal methods and hybrid approaches
Challenges for extending to higher-dimensional systems
Current method's scope and applicability
Comparison with formal verification techniques
Success rates and verification times for RWA tasks
Efficiency improvements over existing methods
Case Study: DRL Spacecraft
Scalability and Efficiency
Filtered RWA Extension
Neural Lyapunov Barrier (NLB) Design
Data Preprocessing
Data Collection
Objective
Background
Conclusion
Limitations and Future Directions
Results and Evaluation
Method
Introduction
Outline
Introduction
Background
Evolution of safety concerns in DRL
Importance of formal verification in complex systems
Objective
To develop a novel NLB method for DRL safety and liveness
Address scalability challenges in RWA tasks
Case study: DRL-controlled spacecraft
Method
Data Collection
Real-world and simulated DRL environments
Trajectory data for training and testing
Data Preprocessing
State-action pairs for neural network input
Generation of RWA and docking task scenarios
Neural Lyapunov Barrier (NLB) Design
NLB Function Learning
Training neural networks to approximate barrier functions
Use of reinforcement learning algorithms
Certificate Composition
Combining NLB functions for compositional verification
Simplification of complex system properties
Filtered RWA Extension
Development of a filtering mechanism
Reducing complexity for efficient verification
Scalability and Efficiency
Addressing computational challenges
Performance evaluation on 2D docking task
Case Study: DRL Spacecraft
Application to a spacecraft control system
Safety and docking verification results
Results and Evaluation
Efficiency improvements over existing methods
Success rates and verification times for RWA tasks
Comparison with formal verification techniques
Limitations and Future Directions
Current method's scope and applicability
Challenges for extending to higher-dimensional systems
Integration with formal methods and hybrid approaches
Conclusion
Summary of contributions and implications
Future research directions for NLB certificates in DRL safety verification
Key findings
3

Paper digest

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

The paper aims to address the challenge of verifying the behavior of Deep Reinforcement Learning (DRL) controllers, particularly in safety-critical systems like autonomous vehicles, where even a single error can have severe consequences . This paper introduces a novel framework for training and formally verifying Neural Lyapunov Barrier (NLB) certificates for discrete-time systems, providing guarantees on achieving goals and avoiding unsafe behavior . While the use of NLB certificates is not new, the difficulty in learning and verifying these certificates, especially for complex real-world systems, is a significant problem that this paper seeks to tackle .


What scientific hypothesis does this paper seek to validate?

This paper aims to validate the scientific hypothesis related to the use of Neural Lyapunov Barrier (NLB) certificates for providing strong guarantees on the behavior of Deep Reinforcement Learning (DRL) agents in complex systems . The hypothesis revolves around the effectiveness of NLB-based certificates in ensuring that a DRL agent achieves its goals and avoids unsafe behavior by learning functions over the system that indirectly imply desired agent behavior . The paper introduces novel techniques for training and verifying NLB-based certificates for discrete-time systems, including certificate composition and filtering methods to simplify the verification process for complex systems .


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

The paper proposes a novel framework for training and formally verifying Neural Lyapunov Barrier (NLB) certificates, specifically focusing on deep reinforcement learning (DRL) controllers . This framework aims to provide guarantees for both liveness and safety properties, offering reach-while-avoid (RWA) assurances . To enhance scalability, the framework utilizes off-the-shelf deep neural network (DNN) verifiers and introduces innovative techniques like certificate filtering and composition . These methods are designed to address the challenges associated with verifying complex DRL-based dynamical systems with large state spaces and intricate dynamics . The paper introduces a novel framework for formally verifying Deep Reinforcement Learning (DRL) controllers using Neural Lyapunov Barrier (NLB) certificates, focusing on safety and liveness properties in complex systems . This framework leverages a Counterexample-Guided Inductive Synthesis (CEGIS) loop for training and verification, incorporating innovative techniques like certificate filtering and composition to enhance scalability and robustness . Compared to previous methods, the proposed framework demonstrates significant advantages in terms of faster training, fewer failures, and improved stability, particularly evident in the compositional certificates that enable scalability to large state spaces . Additionally, the framework showcases the ability to verify DNN-based controllers for intricate systems, offering reach-while-avoid (RWA) assurances and outperforming standard RWA methods in terms of producing verified certificates efficiently . The paper's approach not only simplifies the training and verification process but also unlocks additional scalability, making it a promising step towards ensuring the safe and reliable deployment of DRLThe proposed framework for training and verifying Neural Lyapunov Barrier (NLB) certificates in deep reinforcement learning (DRL) controllers offers several key characteristics and advantages compared to previous methods:

  1. Scalability: The framework leverages off-the-shelf deep neural network (DNN) verifiers, enabling scalability to handle complex DRL-based dynamical systems with large state spaces and intricate dynamics. This scalability is crucial for real-world applications where systems are often high-dimensional and non-linear.

  2. Guarantees for Liveness and Safety Properties: The framework provides guarantees for both liveness and safety properties, ensuring that the controlled system can reach desired states while avoiding unsafe regions. This dual assurance is essential for critical systems where both performance and safety are paramount.

  3. Reach-While-Avoid (RWA) Assurances: By offering reach-while-avoid assurances, the framework allows the controlled system to simultaneously achieve specified goals while steering clear of undesirable states or obstacles. This capability enhances the robustness and reliability of the controller in challenging environments.

  4. Innovative Techniques: The framework introduces innovative techniques such as certificate filtering and composition to enhance the efficiency and effectiveness of the verification process. These techniques help streamline the verification of NLB certificates, making it more practical for real-time applications.

  5. Formal Verification: The framework emphasizes formal verification of NLB certificates, providing mathematical guarantees on the system's behavior under the control of the DRL controller. This formal approach enhances the trustworthiness and reliability of the controller, especially in safety-critical scenarios.

  6. Integration of Neural Networks: By integrating neural networks into the verification process, the framework can handle complex, non-linear dynamics that are common in DRL controllers. This integration allows for more accurate and robust verification of NLB certificates in challenging environments.

Overall, the proposed framework stands out for its scalability, dual guarantees for liveness and safety properties, reach-while-avoid assurances, innovative techniques, formal verification approach, and integration of neural networks. These characteristics collectively contribute to a more robust and reliable method for training and verifying NLB certificates in DRL controllers, offering significant advancements over 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 research studies exist in the field of formally verifying deep reinforcement learning controllers with Lyapunov barrier certificates. Noteworthy researchers in this area include Guy Amir, G. Katz, M. Schapira, A. Farinelli, and I. Refaeli . These researchers have contributed to various aspects of verifying learning-based robotic navigation systems, industrial wildfire detection systems, generalization in deep learning, scalability of verification in deep reinforcement learning, and more.

The key to the solution mentioned in the paper is the use of Neural Lyapunov Barrier (NLB) certificates. These certificates are learned functions over the system that indirectly imply the desired behavior of an agent in deep reinforcement learning. The paper presents a novel method for training and verifying NLB-based certificates for discrete-time systems. This method involves certificate composition, which simplifies the verification of complex systems by strategically designing a sequence of certificates. When combined with neural network verification engines, these certificates provide formal guarantees that a deep reinforcement learning agent achieves its goals and avoids unsafe behavior .


How were the experiments in the paper designed?

The experiments in the paper were designed to explore various architectures for the Deep Neural Network (DNN) used to control the spacecraft . The researchers trained each DNN architecture using the Proximal Policy Optimization Reinforcement Learning (RL) algorithm without any CEGIS iteration . After training, they simulated each architecture on 4,000 random trajectories and analyzed the results . The experiments aimed to verify a Deep Reinforcement Learning (DRL)-based spacecraft controller by generating verified NLB-based Reachability with Avoidance (RWA) certificates for complex properties such as liveness and safety . The experiments involved training both RWA and FRWA certificates using a CEGIS loop and running multiple trials for each task to evaluate the success rate and average time required for successful runs .


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

The dataset used for quantitative evaluation in the context of formally verifying deep reinforcement learning controllers with Lyapunov barrier certificates is Marabou 2.0 . The code for Marabou 2.0 is open source and available on arXiv .


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 introduces a novel method for training and verifying Neural Lyapunov Barrier (NLB) certificates for discrete-time systems, which are crucial for ensuring the safety and liveness guarantees of deep reinforcement learning (DRL) agents controlling autonomous systems . The experiments conducted demonstrate the effectiveness of this approach in providing formal guarantees that a DRL agent achieves its goals and avoids unsafe behavior, particularly in safety-critical applications .

The paper outlines the use of a technique for certificate composition, which simplifies the verification process for highly complex systems by strategically designing a sequence of certificates . This method, when combined with neural network verification engines, offers a formal guarantee of the DRL agent's behavior, aligning with the scientific hypothesis of ensuring safe and reliable performance of autonomous systems .

Furthermore, the experiments compare Reachability with Avoidance (RWA) tasks against Fully Reachability with Avoidance (FRWA) tasks, showcasing the effectiveness of the proposed approach in producing verified controllers and certificates within a reasonable time frame . The results demonstrate the superiority of FRWA over RWA in terms of successful runs and average time required, providing empirical evidence to support the scientific hypotheses put forth in the paper .

Overall, the experiments and results presented in the paper offer substantial evidence to validate the scientific hypotheses related to training and verifying NLB certificates for DRL agents, emphasizing the importance of formal verification methods in ensuring the safety and reliability of autonomous systems in various domains .


What are the contributions of this paper?

The paper "Formally Verifying Deep Reinforcement Learning Controllers with Lyapunov Barrier Certificates" presents several key contributions:

  • Introducing a novel method for training and verifying Neural Lyapunov Barrier (NLB) certificates for discrete-time systems, which involves certificate composition to simplify verification of complex systems and certificate filtering to streamline the production of verified certificates .
  • Demonstrating the effectiveness of the proposed approach through a case study on ensuring safety and liveness guarantees for a deep reinforcement learning (DRL)-controlled spacecraft .
  • Providing a formal guarantee that a DRL agent achieves its goals and avoids unsafe behavior by using NLB-based certificates in conjunction with neural network verification engines .

What work can be continued in depth?

Further research in the field of formally verifying DRL-based controllers can be continued in depth by exploring the following areas:

  • Extension to Additional Formal Techniques: Extending the approach to be compatible with additional formal techniques, such as shielding, can enhance the verification process .
  • Application to More Challenging Case Studies: Applying the approach to more challenging case studies involving larger DRL controllers can help test the scalability and effectiveness of the method in complex systems .
  • Exploration of Larger State Spaces: Investigating how the compositional certificates can scale to even larger state spaces beyond the demonstrated 2D case study can provide insights into the method's applicability in diverse scenarios .
  • Enhancing Safety and Reliability: Continuously working towards ensuring the safe and reliable use of DRL in real-world systems by refining the verification techniques and addressing potential challenges .
Tables
5
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