Skip to main content
SpringerLink
Log in

Quantifier-free encoding of invariants for hybrid systems

  • Published:
Formal Methods in System Design Aims and scope Submit manuscript

Abstract

Hybrid systems are a clean modeling framework for embedded systems, which feature integrated discrete and continuous dynamics. A well-known source of complexity comes from the time invariants, which represent an implicit quantification of a constraint over all time points of a continuous transition.

Emerging techniques based on Satisfiability Modulo Theory (SMT) have been found promising for the verification and validation of hybrid systems because they combine discrete reasoning with solvers for first-order theories. However, these techniques are efficient for quantifier-free theories and the current approaches have so far either ignored time invariants or have been limited to hybrid systems with linear constraints.

In this paper, we propose a new method that encodes a class of hybrid systems into transition systems with quantifier-free formulas. The method does not rely on expensive quantifier elimination procedures. Rather, it exploits the sequential nature of the transition system to split the continuous evolution enforcing the invariants on the discrete time points. This way, we can encode all hybrid systems whose invariants can be expressed in terms of polynomial constraints. This pushes the application of SMT-based techniques beyond the standard linear case.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2

Similar content being viewed by others

Notes

  1. This conversion is not currently automated.

  2. This definition is sufficient to represent input BU where \(B \in\mathbb{R}^{n \times m}\) and \(U : \mathbb{R}^{m} \rightarrow \mathbb{R}^{n}\)

  3. Note that u l cannot be an eigenvalue of the system. This condition is necessary to get a solution where δ appears only as exponent of e, thus enabling the removal of the exponential function via substitution.

  4. http://nusmv.fbk.eu/.

  5. http://isat.gforge.avacs.org/.

  6. http://www.usna.edu/cs/~qepcad/B/QEPCAD.html.

  7. http://redlog.dolzmann.de/.

  8. http://research.microsoft.com/en-us/um/redmond/projects/z3/.

  9. http://smtrat.sourceforge.net/.

  10. http://cs.nyu.edu/acsys/cvc3/.

  11. http://cl-informatik.uibk.ac.at/software/minismt/.

  12. http://homepages.inf.ed.ac.uk/s0793114/rahd/.

  13. http://code.google.com/p/hydlogic/.

  14. http://www.cs.cmu.edu/~sicung/dReal/.

  15. http://isat.gforge.avacs.org/.

References

  1. Ábrahám E, Becker B, Klaedtke F, Steffen M (2005) Optimizing bounded model checking for linear hybrid systems. In: VMCAI, pp 396–412

    Google Scholar 

  2. Alur R (2011) Formal verification of hybrid systems. In: EMSOFT, pp 273–278

    Google Scholar 

  3. Alur R, Courcoubetis C, Henzinger TA, Ho P-H (1992) Hybrid automata: an algorithmic approach to the specification and verification of hybrid systems. In: Hybrid systems, pp 209–229

    Google Scholar 

  4. Asarin E, Dang T, Maler O, Bournez O (2000) Approximate reachability analysis of piecewise-linear dynamical systems. In: HSCC, pp 20–31

    Google Scholar 

  5. Audemard G, Bozzano M, Cimatti A, Sebastiani R (2005) Verifying industrial hybrid systems with MathSAT. Electron Notes Theor Comput Sci 119(2):17–32

    Article  Google Scholar 

  6. Barrett CW, Sebastiani R, Seshia SA, Tinelli C (2009) Satisfiability modulo theories. In: Handbook of satisfiability, pp 825–885

    Google Scholar 

  7. Biere A, Cimatti A, Clarke EM, Zhu Y (1999) Symbolic model checking without BDDs. In: TACAS, pp 193–207

    Google Scholar 

  8. Bozzano M, Cimatti A, Katoen J-P, Nguyen VY, Noll T, Roveri M (2011) Safety, dependability and performance analysis of extended AADL models. Comput J 54(5):754–775

    Article  Google Scholar 

  9. Bozzano M, Cimatti A, Katoen J-P, Nguyen VY, Noll T, Roveri M, Wimmer R (2010) A model checker for AADL. In: CAV, pp 562–565

    Google Scholar 

  10. Bu L, Cimatti A, Li X, Mover S, Tonetta S (2010) Model checking of hybrid systems using shallow synchronization. In: FORTE

    Google Scholar 

  11. Bu L, Zhao J, Li X (2010) Path-oriented reachability verification of a class of nonlinear hybrid automata using convex programming. In: VMCAI, pp 78–94

    Google Scholar 

  12. Casagrande A, Casey K, Falchi R, Piazza C, Ruperti B, Vizzotto G, Mishra B (2007) Translating time-course gene expression profiles into semi-algebraic hybrid automata via dimensionality reduction. In: AB, pp 51–65

    Google Scholar 

  13. Cimatti A, Mover S, Tonetta S (2011) Efficient scenario verification for hybrid automata. In: CAV, pp 317–332

    Google Scholar 

  14. Cimatti A, Mover S, Tonetta S (2011) HyDI: a language for symbolic hybrid systems with discrete interaction. In: EUROMICRO-SEAA

    Google Scholar 

  15. Cimatti A, Mover S, Tonetta S (2011) Proving and explaining the unfeasibility of message sequence charts for hybrid systems. In: FMCAD

    Google Scholar 

  16. Cimatti A, Mover S, Tonetta S (2012) A quantifier-free SMT encoding of non-linear hybrid automata. In: FMCAD, pp 187–195

    Google Scholar 

  17. Cimatti A, Roveri M, Tonetta S (2009) Requirements validation for hybrid systems. In: CAV, pp 188–203

    Google Scholar 

  18. de Alfaro L, Manna Z (1995) Verification in continuous time by discrete reasoning. In: AMAST, pp 292–306

    Google Scholar 

  19. Dolzmann A, Sturm T, Weispfenning V (1998) Real quantifier elimination in practice. In: Algorithmic algebra and number theory. Springer, Berlin, pp 221–247

    Google Scholar 

  20. Eggers A, Fränzle M, Herde C (2008) SAT modulo ODE: a direct SAT approach to hybrid systems. In: ATVA, pp 171–185

    Google Scholar 

  21. Eggers A, Ramdani N, Nedialkov N, Fränzle M (2011) Improving SAT modulo ODE for hybrid systems analysis by combining different enclosure methods. In: SEFM, pp 172–187

    Google Scholar 

  22. Fränzle M (2001) What will be eventually true of polynomial hybrid automata? In: TACS, pp 340–359

    Google Scholar 

  23. Frehse G, Le Guernic C, Donzé A, Cotton S, Ray R, Lebeltel O, Ripado R, Girard A, Dang T, Maler O (2011) SpaceEx: scalable verification of hybrid systems. In: CAV, pp 379–395

    Google Scholar 

  24. Graf S, Saïdi H (1997) Construction of abstract state graphs with PVS. In: CAV, pp 72–83

    Google Scholar 

  25. Henzinger TA, Ho P-H, Wong-Toi H (1998) Algorithmic analysis of nonlinear hybrid systems. IEEE Trans Autom Control 43(4):540–554

    Article  MathSciNet  MATH  Google Scholar 

  26. Herde C, Eggers A, Fränzle M, Teige T (2008) Analysis of hybrid systems using HySAT. In: ICONS, pp 196–201

    Google Scholar 

  27. Ishii D, Ueda K, Hosobe H (2011) An interval-based SAT modulo ODE solver for model checking nonlinear hybrid systems. Int J Softw Tools Technol Transfer 13(5):449–461

    Article  Google Scholar 

  28. Jha S, Brady BA, Seshia SA (2007) Symbolic reachability analysis of lazy linear hybrid automata. In: FORMATS, pp 241–256

    Google Scholar 

  29. King T, Barrett C (2011) Exploring and categorizing error spaces using BMC and SMT. In: SMT

    Google Scholar 

  30. Lafferriere G, Pappas GJ, Yovine S (2001) Symbolic reachability computation for families of linear vector fields. J Symb Comput 32(3):231–253

    Article  MathSciNet  MATH  Google Scholar 

  31. McMillan KL (2003) Interpolation and SAT-based model checking. In: CAV, pp 1–13

    Google Scholar 

  32. Mover S, Cimatti A, Tiwari A, Tonetta S (2013) Time-aware relational abstractions for hybrid systems. In: EMSOFT, pp 1–10

    Google Scholar 

  33. Plaku E, Kavraki LE, Vardi MY (2009) Hybrid systems: from verification to falsification by combining motion planning and discrete search. Form Methods Syst Des 34(2):157–182

    Article  MATH  Google Scholar 

  34. Platzer A, Clarke EM (2007) The image computation problem in hybrid systems model checking. In: HSCC, pp 473–486

    Google Scholar 

  35. Platzer A, Clarke EM (2009) Formal verification of curved flight collision avoidance maneuvers: a case study. In: FM, pp 547–562

    Google Scholar 

  36. Rabinovich AM (1998) On the decidability of continuous time specification formalisms. J Log Comput 8(5):669–678

    Article  MathSciNet  MATH  Google Scholar 

  37. Sankaranarayanan S, Tiwari A (2011) Relational abstractions for continuous and hybrid systems. In: CAV, pp 686–702

    Google Scholar 

  38. Sheeran M, Singh S, Stålmarck G (2000) Checking safety properties using induction and a SAT-solver. In: FMCAD, pp 108–125

    Google Scholar 

  39. Tiwari A (2008) Abstractions for hybrid systems. Form Methods Syst Des 32(1):57–83

    Article  MATH  Google Scholar 

  40. Tonetta S (2009) Abstract model checking without computing the abstraction. In: FM, pp 89–105

    Google Scholar 

  41. Yushtein Y, Bozzano M, Cimatti A, Katoen J-P, Nguyen VY, Noll Th, Olive X, Roveri M (2011) System-software co-engineering: dependability and safety perspective. In: SMC-IT. IEEE Comput. Sci., Los Alamitos, pp 18–25

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Fondazione Bruno Kessler, Povo (TN), Trento, Italy

    Alessandro Cimatti, Sergio Mover & Stefano Tonetta

Authors
  1. Alessandro Cimatti
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Sergio Mover
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Stefano Tonetta
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Sergio Mover.

Additional information

This paper presents in a coherent and expanded form material that appears in the conference venue [16].

Rights and permissions

Reprints and permissions

About this article

Cite this article

Cimatti, A., Mover, S. & Tonetta, S. Quantifier-free encoding of invariants for hybrid systems. Form Methods Syst Des 45, 165–188 (2014). https://doi.org/10.1007/s10703-013-0202-8

Download citation

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10703-013-0202-8

Keywords

Search

Navigation