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A barrier certificate, defined over the states of a dynamical system, is a real-valued function whose zero level set characterizes an in- ductively verifiable state invariant separating reachable states from unsafe ones. When combined with powerful decision procedures— such as sum-of-squares programming (SOS) or satisfiability-modulo- theory solvers (SMT)—barrier certificates enable an automated de- ductive verification approach to safety. The barrier certificate ap- proach has been extended to refute LTL and l -regular specifications by separating consecutive transitions of corresponding l -automata in the hope of denying all accepting runs. Unsurprisingly, such tactics are bound to be conservative as refutation of recurrence properties requires reasoning about the well-foundedness of the transitive closure of the transition relation. This paper introduces the notion of closure certificates as a natural extension of barrier certificates from state invariants to transition invariants. We aug- ment these definitions with SOS and SMT based characterization for automating the search of closure certificates and demonstrate their effectiveness over some case studies.
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Vector barrier certificates and comparison systems
Vector Lyapunov functions are a multi-dimensional extension of the more familiar (scalar) Lyapunov functions, commonly used to prove stability properties in systems described by 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.
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- Award ID(s):
- 1739629
- PAR ID:
- 10064721
- Date Published:
- Journal Name:
- FM 2018: Formal Methods - 22nd International Symposium, Oxford, UK, July 15-17, 2018, Proceedings
- Volume:
- 10951
- Page Range / eLocation ID:
- 418-437
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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- Free Publicly Accessible Full Text
-
Accepted Manuscript1.0
- Conference Paper:
- https://doi.org/10.1007/978-3-319-95582-7_25
