Skip to main content
SpringerLink
Log in

Temporal Logic Verification of Stochastic Systems Using Barrier Certificates

  • Conference paper
  • First Online:
Book cover Automated Technology for Verification and Analysis (ATVA 2018)

Part of the book series: Lecture Notes in Computer Science ((LNPSE,volume 11138))

Abstract

This paper presents a methodology for temporal logic verification of discrete-time stochastic systems. Our goal is to find a lower bound on the probability that a complex temporal property is satisfied by finite traces of the system. Desired temporal properties of the system are expressed using a fragment of linear temporal logic, called safe LTL over finite traces. We propose to use barrier certificates for computations of such lower bounds, which is computationally much more efficient than the existing discretization-based approaches. The new approach is discretization-free and does not suffer from the curse of dimensionality caused by discretizing state sets. The proposed approach relies on decomposing the negation of the specification into a union of sequential reachabilities and then using barrier certificates to compute upper bounds for these reachability probabilities. We demonstrate the effectiveness of the proposed approach on case studies with linear and polynomial dynamics.

This work was supported in part by the German Research Foundation (DFG) through the grant ZA 873/1-1 and the TUM International Graduate School of Science and Engineering (IGSSE).

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

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 39.99
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 54.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

References

  1. Abate, A., Katoen, J.P., Mereacre, A.: Quantitative automata model checking of autonomous stochastic hybrid systems. In: Proceedings of the 14th International Conference on Hybrid systems: Computation and Control, pp. 83–92. ACM (2011)

    Google Scholar 

  2. Ayala, A.I.M., Andersson, S.B., Belta, C.: Probabilistic control from time-bounded temporal logic specifications in dynamic environments. In: 2012 IEEE International Conference on Robotics and Automation, pp. 4705–4710 (2012)

    Google Scholar 

  3. Baier, C., Katoen, J.P., Larsen, K.G.: Principles of Model Checking. MIT press, Cambridge (2008)

    Google Scholar 

  4. De Giacomo, G., Vardi, M.Y.: Linear temporal logic and linear dynamic logic on finite traces. In: International Joint Conference on Artificial Intelligence, vol. 13, pp. 854–860 (2013)

    Google Scholar 

  5. De Giacomo, G., Vardi, M.Y.: Synthesis for LTL and LDL on finite traces. In: International Joint Conference on Artificial Intelligence, vol. 15, pp. 1558–1564 (2015)

    Google Scholar 

  6. Dimitrova, R., Majumdar, R.: Deductive control synthesis for alternating-time logics. In: 2014 International Conference on Embedded Software (EMSOFT), pp. 1–10 (2014)

    Google Scholar 

  7. Duret-Lutz, A., Lewkowicz, A., Fauchille, A., Michaud, T., Renault, É., Xu, L.: Spot 2.0 — a framework for LTL and \(\omega \)-automata manipulation. In: Artho, C., Legay, A., Peled, D. (eds.) ATVA 2016. LNCS, vol. 9938, pp. 122–129. Springer, Cham (2016). https://doi.org/10.1007/978-3-319-46520-3_8

    Chapter  Google Scholar 

  8. Henriksen, J.G., Jensen, J., Jørgensen, M., Klarlund, N., Paige, R., Rauhe, T., Sandholm, A.: Mona: monadic second-order logic in practice. In: Brinksma, E., Cleaveland, W.R., Larsen, K.G., Margaria, T., Steffen, B. (eds.) TACAS 1995. LNCS, vol. 1019, pp. 89–110. Springer, Heidelberg (1995). https://doi.org/10.1007/3-540-60630-0_5

    Chapter  Google Scholar 

  9. Huang, C., Chen, X., Lin, W., Yang, Z., Li, X.: Probabilistic safety verification of stochastic hybrid systems using barrier certificates. ACM Trans. Embed. Comput. Syst. 16(5s), 186 (2017)

    Article  Google Scholar 

  10. Jagtap, P., Zamani, M.: QUEST: a tool for state-space quantization-free synthesis of symbolic controllers. In: Bertrand, N., Bortolussi, L. (eds.) QEST 2017. LNCS, vol. 10503, pp. 309–313. Springer, Cham (2017). https://doi.org/10.1007/978-3-319-66335-7_21

    Chapter  Google Scholar 

  11. Klein, J., Baier, C.: Experiments with deterministic \(\omega \)-automata for formulas of linear temporal logic. Theor. Comput. Sci. 363(2), 182–195 (2006)

    Google Scholar 

  12. Kupferman, O., Vardi, M.: Model checking of safety properties. In: International Conference on Computer Aided Verification, pp. 172–183. Springer, Berlin (1999)

    Chapter  Google Scholar 

  13. Kushner, H.J.: On the stability of stochastic dynamical systems. Proc. Natl. Acad. Sci. 53(1), 8–12 (1965)

    Article  MathSciNet  Google Scholar 

  14. Lahijanian, M., Andersson, S.B., Belta, C.: Formal verification and synthesis for discrete-time stochastic systems. IEEE Trans. Autom. Control 60(8), 2031–2045 (2015)

    Article  MathSciNet  Google Scholar 

  15. Lavaei, A., Soudjani, S., Zamani, M.: From dissipativity theory to compositional construction of finite Markov decision processes. In: Hybrid Systems: Computation and Control (HSCC), pp. 21–30. ACM, New York (2018)

    Google Scholar 

  16. Maity, D., Baras, J.S.: Motion planning in dynamic environments with bounded time temporal logic specifications. In: 2015 23rd Mediterranean Conference on Control and Automation, pp. 940–946 (2015)

    Google Scholar 

  17. Parrilo, P.A.: Semidefinite programming relaxations for semialgebraic problems. Math. Program. 96(2), 293–320 (2003)

    Article  MathSciNet  Google Scholar 

  18. Postoyan, R., Nesic, D.: Time-triggered control of nonlinear discrete-time systems. In: 2016 IEEE 55th Conference on Decision and Control, pp. 6814–6819 (2016)

    Google Scholar 

  19. Prajna, S., Jadbabaie, A., Pappas, G.J.: A framework for worst-case and stochastic safety verification using barrier certificates. IEEE Trans. Autom. Control 52(8), 1415–1428 (2007)

    Article  MathSciNet  Google Scholar 

  20. Prajna, S., Papachristodoulou, A., Parrilo, P.A.: Introducing SOSTOOLS: a general purpose sum of squares programming solver. In: Proceedings of the 41st IEEE Conference on Decision and Control, vol. 1, pp. 741–746 (2002). http://www.cds.caltech.edu/sostools/

  21. Russell, S.J., Norvig, P.: Artificial Intelligence: A Modern Approach, 2nd edn. Pearson Education, London (2003)

    MATH  Google Scholar 

  22. Saha, I., Ramaithitima, R., Kumar, V., Pappas, G.J., Seshia, S.A.: Automated composition of motion primitives for multi-robot systems from safe LTL specifications. In: 2014 IEEE/RSJ International Conference on Intelligent Robots and Systems, pp. 1525–1532 (2014)

    Google Scholar 

  23. Esmaeil Zadeh Soudjani, S., Abate, A.: Precise approximations of the probability distribution of a Markov process in time: an application to probabilistic invariance. In: Ábrahám, E., Havelund, K. (eds.) TACAS 2014. LNCS, vol. 8413, pp. 547–561. Springer, Heidelberg (2014). https://doi.org/10.1007/978-3-642-54862-8_45

    Chapter  MATH  Google Scholar 

  24. Soudjani, S., Abate, A., Majumdar, R.: Dynamic Bayesian networks as formal abstractions of structured stochastic processes. In: 26th International Conference on Concurrency Theory, pp. 1–14. Dagstuhl Publishing, Madrid (2015)

    Google Scholar 

  25. Soudjani, S., Abate, A.: Adaptive and sequential gridding procedures for the abstraction and verification of stochastic processes. SIAM J. Appl. Dyn. Syst. 12(2), 921–956 (2013)

    Article  MathSciNet  Google Scholar 

  26. Soudjani, S., Gevaerts, C., Abate, A.: FAUST\(^2\): formal abstractions of uncountable-state stochastic processes. In: Baier, C., Tinelli, C. (eds.) Tools and Algorithms for the Construction and Analysis of Systems, pp. 272–286. Springer, Berlin (2015)

    Google Scholar 

  27. Steinhardt, J., Tedrake, R.: Finite-time regional verification of stochastic non-linear systems. Int. J. Robot. Res. 31(7), 901–923 (2012)

    Article  Google Scholar 

  28. Sturm, J.F.: Using SeDuMi 1.02, a MATLAB toolbox for optimization over symmetric cones. Optim. Methods Softw. 11(1–4), 625–653 (1999). http://sedumi.ie.lehigh.edu/

    Article  MathSciNet  Google Scholar 

  29. Tabuada, P.: Verification and Control of Hybrid Systems: A Symbolic Approach. Springer Science & Business Media, Berlin (2009)

    Chapter  Google Scholar 

  30. Tkachev, I., Abate, A.: Formula-free finite abstractions for linear temporal verification of stochastic hybrid systems. In: Proceedings of the 16th International Conference on Hybrid Systems: Computation and Control, pp. 283–292. ACM (2013)

    Google Scholar 

  31. Vincent, T.L., Yu, J.: Control of a chaotic system. Dyn. Control 1(1), 35–52 (1991)

    Article  MathSciNet  Google Scholar 

  32. Wisniewski, R., Bujorianu, M.L.: Stochastic safety analysis of stochastic hybrid systems. In: 2017 IEEE 56th Annual Conference on Decision and Control, pp. 2390–2395 (2017)

    Google Scholar 

  33. Wongpiromsarn, T., Topcu, U., Lamperski, A.: Automata theory meets barrier certificates: temporal logic verification of nonlinear systems. IEEE Trans. Autom. Control 61(11), 3344–3355 (2016)

    Article  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Technical University of Munich, Munich, Germany

    Pushpak Jagtap & Majid Zamani

  2. Newcastle University, Newcastle upon Tyne, United Kingdom

    Sadegh Soudjani

Authors
  1. Pushpak Jagtap
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Sadegh Soudjani
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Majid Zamani
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Sadegh Soudjani .

Editor information

Editors and Affiliations

  1. Microsoft Research, Redmond, WA, USA

    Shuvendu K. Lahiri

  2. University of Southern California, Los Angeles, CA, USA

    Chao Wang

Rights and permissions

Reprints and permissions

Copyright information

© 2018 Springer Nature Switzerland AG

About this paper

Check for updates. Verify currency and authenticity via CrossMark

Cite this paper

Jagtap, P., Soudjani, S., Zamani, M. (2018). Temporal Logic Verification of Stochastic Systems Using Barrier Certificates. In: Lahiri, S., Wang, C. (eds) Automated Technology for Verification and Analysis. ATVA 2018. Lecture Notes in Computer Science(), vol 11138. Springer, Cham. https://doi.org/10.1007/978-3-030-01090-4_11

Download citation

  • DOI: https://doi.org/10.1007/978-3-030-01090-4_11

  • Published:

  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-030-01089-8

  • Online ISBN: 978-3-030-01090-4

  • eBook Packages: Computer ScienceComputer Science (R0)

Publish with us

Policies and ethics

Search

Navigation