Automated Synthesis of Stochastic Cyber-Physical Systems: A Robust Approach

Project Information

Auto-CyPheR

Grant agreement ID: 101089047

DOI

10.3030/101089047

EC signature date 27 July 2023

Start date 1 November 2023

End date 31 October 2028

Funded under

• European Research Council (ERC)

Total cost

€ 1 993 756,00

EU contribution

€ 1 993 756,00

Coordinated by

MAX-PLANCK-GESELLSCHAFT ZUR FORDERUNG DER WISSENSCHAFTEN EV
Germany

Periodic Reporting for period 1 - Auto-CyPheR (Automated Synthesis of Stochastic Cyber-Physical Systems: A Robust Approach)

Reporting period: 2023-11-01 to 2026-04-30

Summary of the context and overall objectives of the project

Cyber-physical systems (CPS) are complex systems with tight interactions between cyber elements and physical components. The cyber elements are control algorithms implemented by computer-based software. Developing the embedded control software for CPS is currently ad hoc and error-prone, which has created costly undesired behaviours, particularly in safety-critical applications. Examples of such undesired behaviours include frequency deviation in power networks causing outages or blackouts (e.g. in Jan 2021 in EU, affecting 200k households), crash of airplanes due to software bugs (Boeing 737 Max, costs 15.9 billion euros) or autonomous cars having software bugs (Toyota recalled 65,000 cars in 2015). Nowadays, most of the costs of CPS design is spent on ensuring that the system meets all the requirements especially when it is working in uncertain conditions. In order to design reliable CPS and to reduce the costs of such a design, I propose a novel robust synthesis approach that computes automatically the control software from high-level requirements. This novel approach creates a paradigm shift in CPS design as it computes control software in a push-button manner and eliminates time-consuming, costly post-validation steps. The approach tackles the CPS complexity by developing new abstraction techniques that are compositional and robust to model uncertainties, which will be integrated in a unified framework for automating the design of the control software. This project is high-risk because it requires merging and re-thinking different design methodologies from multiple disciplines including control theory, computer science, and probability theory. It is high-gain since it will transform the design principles of CPS to enable designing reliable and safe CPS working in uncertain conditions.

Work performed from the beginning of the project to the end of the period covered by the report and main results achieved so far

During the reporting period, the project has focused on developing theoretical, algorithmic, and data-driven methods for the synthesis and verification of stochastic cyber-physical systems under uncertainty, in line with the objectives of the action. The work performed has advanced abstraction-based control design, robust synthesis under epistemic and aleatoric uncertainty, learning-enabled verification, and scalable methods for stochastic systems, while also establishing software tools to support these developments.

A central activity has been the development of **abstraction-based methods for stochastic systems**, enabling the construction of finite models and controllers with formal probabilistic guarantees. Progress has been achieved on learning abstractions from data, robust abstraction under uncertainty, and multi-objective synthesis for temporal logic specifications. These activities address core scientific goals of the project and provide a foundation for correct-by-design control of complex stochastic systems.

The project has also advanced **learning-enabled verification and certification methods**, including approaches for learning safety and performance certificates from data while maintaining formal guarantees. These developments strengthen the integration between machine learning and formal synthesis and enable verification in settings where accurate system models are not available.

Another key strand of work concerns **robust simulation relations and compositional abstraction-based synthesis**, which support scalable analysis and controller design for large and interconnected stochastic systems. These methods enable reasoning about uncertain systems through structured abstractions and provide the basis for compositional design and verification workflows.

In parallel with theoretical developments, the project has implemented **software tools** that operationalise the proposed methodologies. The LUCID tool supports learning-enabled certification and uncertainty-aware analysis of stochastic dynamical systems, while the SySCoRe toolset supports robust abstraction and formal synthesis for continuous-state stochastic systems. These tools facilitate experimentation, reproducibility, and methodological transfer.

The main scientific achievements to date include:

* robust abstraction and simulation techniques under epistemic uncertainty;
* advances in temporal logic synthesis and certificate-based reasoning;
* the development of data-driven abstraction methods with formal guarantees;
* learning-based construction of certificates for verification and synthesis;
* scalable methods and tool-supported implementations for stochastic systems.

Overall, the work performed during this reporting period has established a coherent methodological foundation for abstraction-based, learning-enabled, and robust synthesis of stochastic cyber-physical systems, positioning the project for further integration of theoretical advances, tool development, and application-driven validation in the next phase.

Progress beyond the state of the art and expected potential impact (including the socio-economic impact and the wider societal implications of the project so far)

The project has generated several results that advance the state of the art in the synthesis and verification of stochastic cyber-physical systems and open new directions at the interface of control theory, formal methods, and machine learning.

A central advance is the development of **data-driven abstractions and certification methods with formal guarantees**, enabling the synthesis and verification of stochastic systems even when precise models are unavailable. This bridges a long-standing gap between learning-based modelling and formal correctness guarantees, and introduces a new paradigm in which data-driven and formal approaches can be combined in a principled manner.

Further advances include **robust abstraction and simulation techniques under epistemic and aleatoric uncertainty**, new optimisation-based methods for satisfying temporal logic specifications, and certificate-based approaches that reduce conservatism in verification and synthesis. Together, these results extend abstraction-based control design beyond model-based settings and enable its application to data-rich and uncertain environments.

The project has also produced **tool-supported implementations** (e.g. LUCID and SySCoRe), demonstrating the feasibility of translating theoretical advances into usable methodologies for analysis and controller synthesis of stochastic systems. In addition, the development of discrete abstraction techniques based on Gaussian process models and the associated patent application highlight the novelty and translational potential of the research.

These results have the potential to influence multiple domains, including safety-critical autonomous systems, energy systems, and AI-enabled cyber-physical infrastructures, by enabling reliable decision-making under uncertainty and limited model knowledge.

To ensure further uptake and long-term impact, several elements are important:

* continued research on scalability, compositionality, and high-dimensional systems;
* further development, validation, and dissemination of software tools;
* demonstration of methods in representative application domains;
* support for intellectual property management and knowledge transfer where appropriate;
* engagement with international research communities to promote adoption of abstraction-based and learning-enabled formal synthesis methods.

Overall, the project’s results provide a foundation for a new generation of methods that integrate learning, abstraction, and formal guarantees, with significant potential for both scientific and applied impact as the project progresses.