Scalable Verification of AI-Controlled CPS: Abstraction Techniques Revealed

Table of Contents

1. Introduction to Formal Verification of Cyber-Physical Systems (CPS)
2. Model-Based Design for CPS
3. State of the Art in CPS Verification
4. Exploring Abstraction Techniques for CPS Analysis
   • Post Computation of Dynamical Systems
   • Output Range Analysis of Neural Networks
   • Interval Neural Networks
5. Formal Verification of Cyber-Physical Systems
   • Mathematical Models and Specifications
   • Verification Algorithms
6. Safety Analysis of AI-Controlled CPS
   • Challenges in Classical Control Design
   • AI-Based Controllers for CPS
7. Abstraction-Based Analysis for Neural Networks
   • Interval Neural Network Abstraction
   • Encoding Interval Neural Networks as M ILP Constraints
8. Experimental Evaluation of the Abstraction-Based Analysis Algorithm
   • Effect of Abstraction Granularity on Verification Time and Precision
   • Impact of Partitioning Strategies on Output Range Precision
9. Conclusion and Future Directions

Introduction

In this article, we will discuss the abstraction techniques used for analyzing cyber-physical systems (CPS) with AI-based controllers. Formal verification plays a crucial role in ensuring the reliability of CPS, especially in highly safety-critical environments. We will explore various abstraction techniques and their applications in CPS analysis, focusing on the post computation of dynamical systems and output range analysis of neural networks. Additionally, we will delve into the challenges posed by AI-based controllers in CPS and discuss how abstraction-based analysis can address these challenges. Lastly, we will Present the experimental evaluation of an abstraction-based analysis algorithm and highlight the trade-off between verification time and precision. Let's dive in!

🔍 Key Points:

• Formal verification is essential for building and deploying reliable CPS.
• Abstraction techniques play a crucial role in CPS analysis.
• AI-based controllers pose challenges in classical control design.
• Abstraction-based analysis can address challenges in AI-controlled CPS.
• Experimental evaluation sheds light on verification time and precision.

Model-Based Design for CPS

Before diving into the details of abstraction techniques and analysis, let's first understand the concept of model-based design for CPS. Model-based design provides a promising methodology for designing and analyzing CPS. It involves modeling the physical system, often using differential equations, and designing the controller based on this model. The design and analysis of the closed-loop system can be performed using modern platforms that automate the generation of code from the high-level controller design. While simulations are commonly used for analysis, they may not provide the level of guarantee required for safety-critical systems. Formal verification techniques offer a more rigorous approach to ensure the correctness of CPS. Let's explore formal verification in the context of cyber-physical systems.

Formal Verification of Cyber-Physical Systems

Formal verification involves the use of mathematical models and specifications to analyze the correctness of a system. In the context of cyber-physical systems, formal verification plays a crucial role in certifying the reliability of software-controlled physical systems. The process consists of a mathematical model of the system to be analyzed and a mathematical specification that captures the Notion of correctness. Verification algorithms take the model and specification as input and output either a proof of correctness or a counterexample, which represents a software bug. Formal verification provides guarantees for the correctness of a system, making it essential in safety-critical industries such as aerospace. Let's delve deeper into the components of formal verification in the context of cyber-physical systems.

Mathematical Models and Specifications

In formal verification, the first step is to construct a mathematical model of the system under analysis. The model captures the behavior of the system and its interactions with the environment. In the case of cyber-physical systems, the model incorporates the physical dynamics of the system along with the software-controlled components. The model is typically represented using mathematical notations, such as differential equations, Automata, or hybrid systems.

Once the model is constructed, a mathematical specification is defined to capture the correctness requirements of the system. The specification defines the expected behavior of the system and serves as a benchmark for verification. It specifies properties such as safety, liveness, or temporal logic constraints. Formal verification algorithms analyze the model and specification to determine if the system satisfies the specified requirements. This process involves checking the model against the specification using techniques such as theorem proving, model checking, or abstract interpretation. Let's explore the verification algorithms used in formal verification.

Verification Algorithms

Verification algorithms take the model and specification as input and perform analysis to determine if the model satisfies the specification. These algorithms can provide a proof of correctness, along with the states of the system that satisfy the specification, or they can provide a counterexample, which represents a violation of the specification.

In the context of cyber-physical systems, verification algorithms need to tackle the complexity of the combined software and physical dynamics. Simulation-based approaches, where the system is repeatedly executed from different initial states and checked against the specification, are commonly used. However, simulations alone may not provide the level of guarantee required for safety-critical systems. Formal verification techniques offer a more rigorous approach to analyzing cyber-physical systems, ensuring the correctness of the system behavior.

Recent advancements in formal verification have led to the development of various tools and approaches specific to the analysis of hybrid systems, symbolically representing linear and nonlinear dynamics, simulation-based techniques, theorem provers, and constraint-solving-based methods. These tools cater to different classes of dynamics and offer different approaches to verification. Let's now explore the challenges posed by AI-controlled CPS and their implications for formal verification.

🤔 Did You Know?

The aerospace industry recommends using formal analysis to certify commercial aircraft software due to its importance in ensuring the safety and reliability of the systems.

🔍 Key Points:

• Formal verification involves mathematical models and specifications.
• Verification algorithms provide proofs of correctness or counterexamples.
• Simulation-based approaches are commonly used for analysis.
• Formal verification techniques offer a more rigorous approach for safety-critical systems.
• Recent advancements have led to the development of specialized tools for different classes of dynamics.

Safety Analysis of AI-Controlled CPS

The emergence of AI-based controllers has revolutionized the field of cyber-physical systems. Traditional control design methodologies are being replaced by learning-based components, such as artificial neural networks, to handle the complexities and uncertainties of modern CPS. However, this shift introduces new challenges for safety analysis. In this section, we will explore the challenges posed by AI-controlled CPS and discuss how abstraction-based analysis can address these challenges.

Challenges in Classical Control Design

Classic control design methodologies rely on the formulation of a control algorithm based on a model of the physical system. The control algorithm is typically a state-feedback controller that maps the current state of the system to an appropriate control input. However, in highly dynamic and uncertain environments, classical control design falls short. The complexities and uncertainties of such systems make it challenging to develop accurate mathematical models and design controllers that can handle these uncertainties effectively. This is where AI-based controllers come into play.

AI-Based Controllers for CPS

AI-based controllers, particularly those utilizing artificial neural networks, offer a more flexible and adaptive approach to control design. These controllers can learn from data and adapt their behavior based on the observed environment. For example, in autonomous driving, an AI-based controller can receive image inputs of the environment and compute the distance to other vehicles and obstacles, allowing it to make informed control decisions.

While AI-based controllers provide enhanced capabilities, their safety analysis poses new challenges. Formal verification techniques that were effective for classical control design need to be extended to handle the complexities introduced by AI-based controllers. Abstraction-based analysis offers a promising approach to address these challenges.

Abstraction-Based Analysis for Neural Networks

Abstraction-based analysis provides a powerful technique for analyzing complex systems, including neural networks used in AI-controlled CPS. The goal of abstraction is to construct a smaller system that over-approximates the behavior of the original system, enabling more efficient analysis. In the case of neural networks, this involves abstracting the network into an interval neural network and encoding it as a set of constraints amenable to analysis using mathematical techniques such as M ILP (Mixed Integer Linear Programming).

Interval Neural Network Abstraction

The abstraction process involves partitioning the nodes of each layer in the neural network and merging them to form abstract nodes in the interval neural network. The weights and biases of the edges in the neural network are replaced by intervals, capturing the range of possible values. The abstraction process ensures that the input-output relation of the original neural network is preserved, albeit with some over-approximation.

To construct the interval neural network, the interval hull of the weight and bias intervals is taken, and the abstraction partitions are scaled according to the number of nodes being merged. This scaling ensures that the over-approximation is done correctly and prevents information loss during the merging process.

Encoding Interval Neural Networks as M ILP Constraints

Once the interval neural network is constructed, it can be encoded as a set of constraints using M ILP techniques. Encoding involves replacing the interval weights and biases with their lower and upper bounds, respectively, in appropriate inequalities. This encoding ensures that the constraints capture the same input-output relation as the interval neural network.

The encoded M ILP constraints can then be solved using solvers such as Gurobi. The solution yields the range of output values, providing insights into the behavior and correctness of the system. By analyzing the output range, potential safety violations can be detected, and appropriate measures can be taken to ensure the reliability of the AI-controlled CPS.

Experimental Evaluation of the Abstraction-Based Analysis Algorithm

To evaluate the effectiveness of the abstraction-based analysis algorithm, we conducted experiments using a benchmark neural network known as Cekic, which consists of six Hidden layers with fifty neurons in each layer. The experiments focused on analyzing the impact of different abstraction granularity on verification time and output range precision.

The results showed that the abstraction time and encoding time increased with the number of abstract nodes. However, the absolute time taken remained relatively small, less than a Second in all cases. This demonstrates that abstraction and encoding are not the bottlenecks in the analysis process.

Furthermore, the experiments revealed that the precision of the output range improved as the number of abstract nodes increased. This indicates that the interval neural network, constructed through abstraction, becomes more precise and closely approximates the behavior of the original neural network. However, the precise output range varied based on the specific partitioning strategy employed.

These findings highlight the trade-off between verification time and precision in abstraction-based analysis. Different partitioning strategies and heuristics can be explored to improve the precision of the output range and further optimize the analysis process. Future research will focus on developing efficient partitioning algorithms and extending the analysis to handle more complex activation functions and closed-loop CPS scenarios.

🔍 Key Points:

• AI-based controllers revolutionize CPS.
• Classical control design faces challenges in dynamic and uncertain environments.
• Abstraction-based analysis addresses challenges in AI-controlled CPS.
• Abstraction produces an interval neural network over-approximating the original system.
• Encoding interval neural networks as M ILP constraints enables safety analysis.
• Experimental evaluation reveals trade-off between verification time and precision.
• Partitioning strategies impact output range precision.

Conclusion and Future Directions

Formal verification of AI-controlled cyber-physical systems is essential for ensuring their reliability and safety in highly safety-critical environments. Abstraction-based analysis has proven to be a powerful technique for analyzing complex systems, such as neural networks used in CPS. By abstracting the neural network into an interval neural network and encoding it as M ILP constraints, we can efficiently analyze the system and obtain insights into its behavior.

Our experimental evaluation highlights the trade-off between verification time and precision in abstraction-based analysis. The choice of abstraction granularity and partitioning strategy significantly impacts the output range precision. Future research directions include exploring heuristics for partitioning algorithms, extending the analysis to handle more complex activation functions, and analyzing the whole CPS with neural networks in closed-loop scenarios.

The increasingly complex and safety-critical nature of AI-controlled cyber-physical systems necessitates the development of robust formal verification techniques. Through ongoing research and collaboration between academia and industry, we can continue to advance the field of formal verification and ensure the reliability of AI-controlled CPS.

🔍 Key Points:

• Formal verification is crucial for reliability and safety in AI-controlled CPS.
• Abstraction-based analysis is a powerful technique for analyzing neural networks.
• Future directions include developing heuristics, handling complex activation functions, and analyzing closed-loop scenarios.
• Collaboration and research are essential for advancing the field of formal verification in CPS.

Resources:

• European Symposium on Programming (EUROCRYPT), 2019.