Vector Barrier Certificates and Comparison Systems

  • Andrew SogokonEmail author
  • Khalil GhorbalEmail author
  • Yong Kiam Tan
  • André Platzer
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 10951)


Vector Lyapunov functions are a multi-dimensional extension of the more familiar (scalar) Lyapunov functions, commonly used to prove stability properties in systems of non-linear ordinary differential equations (ODEs). This paper explores an analogous vector extension for so-called barrier certificates used in safety verification. As with vector Lyapunov functions, the approach hinges on constructing appropriate comparison systems, i.e., related differential equation systems from which properties of the original system may be inferred. The paper presents an accessible development of the approach, demonstrates that most previous notions of barrier certificate are special cases of comparison systems, and discusses the potential applications of vector barrier certificates in safety verification and invariant synthesis.


Ordinary differential equations Safety verification Vector barrier certificates Comparison systems 



The authors would like to thank the FM 2018 reviewers for their feedback, constructive criticisms and suggestions, and extend special thanks to Dr. Stefan Mitsch and Brandon Bohrer at Carnegie Mellon University for their detailed comments and scrutiny.


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© Springer International Publishing AG, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Computer Science DepartmentCarnegie Mellon UniversityPittsburghUSA
  2. 2.InriaRennesFrance

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