On Local Characterization of Global Timed Bisimulation for Abstract Continuous-Time Systems

  • Ievgen Ivanov
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 9608)


We consider two notions of timed bisimulation on states of continuous-time dynamical systems: global and local timed bisimulation. By analogy with the notion of a bisimulation relation on states of a labeled transition system which requires the existence of matching transitions starting from states in such a relation, local timed bisimulation requires the existence of sufficiently short (locally defined) matching trajectories. Global timed bisimulation requires the existence of arbitrarily long matching trajectories. For continuous-time systems the notion of a global bisimulation is stronger than the notion of a local bisimulation and its definition has a non-local character. In this paper we give a local characterization of global timed bisimulation. More specifically, we consider a large class of abstract dynamical systems called Nondeterministic Complete Markovian Systems (NCMS) which covers various concrete continuous and discrete-continuous (hybrid) dynamical models and introduce the notion of an \(f^+\)-timed bisimulation, where \(f^+\) is a so called extensibility measure. This notion has a local character. We prove that it is equivalent to global timed bisimulation on states of a NCMS. In this way we give a local characterization of the notion of a global timed bisimulation.


Bisimulation Cyber-physical system Dynamical system Continuous time Local characterization 


  1. 1.
    Milner, R.: Communication and Concurrency. Prentice-Hall Inc., Upper Saddle River (1989)zbMATHGoogle Scholar
  2. 2.
    Park, D.: Concurrency and automata on infinite sequences. In: Deussen, P. (ed.) Theoretical Computer Science. LNCS, vol. 104, pp. 167–183. Springer, Heidelberg (1981)CrossRefGoogle Scholar
  3. 3.
    Sangiorgi, D.: Introduction to Bisimulation and Coinduction. Cambridge University Press, New York (2011)CrossRefzbMATHGoogle Scholar
  4. 4.
    Sangiorgi, D.: On the origins of bisimulation and coinduction. ACM Trans. Program. Lang. Syst. 31(4), 15:1–15:41 (2009)CrossRefzbMATHGoogle Scholar
  5. 5.
    Sangiorgi, D.: On the bisimulation proof method. Mathematical. Struct. Comput. Sci. 8(5), 447–479 (1998)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Shi, J., Wan, J., Yan, H., Suo, H.: A survey of cyber-physical systems. In: 2011 International Conference on Wireless Communications and Signal Processing (WCSP), pp. 1–6. IEEE (2011)Google Scholar
  7. 7.
    Baheti, R., Gill, H.: Cyber-physical systems. Impact Control Technol. 12, 161–166 (2011)Google Scholar
  8. 8.
    Lee, E., Seshia, S.: Introduction to Embedded Systems: A Cyber-physical Systems Approach., Berkeley (2013)Google Scholar
  9. 9.
    Sifakis, J.: Rigorous design of cyber-physical systems. In: 2012 International Conference on Embedded Computer Systems (SAMOS), p. 319. IEEE (2012)Google Scholar
  10. 10.
    Bouissou, O., Chapoutot, A.: An operational semantics for Simulink’s simulation engine. In: Proceedings of 13th ACM SIGPLAN/SIGBED International Conference on Languages, Compilers, Tools and Theory for Embedded Systems, pp. 129–138. ACM (2012)Google Scholar
  11. 11.
    Simulink - Simulation and Model-Based Design.
  12. 12.
  13. 13.
    Campbell, S., Chancelier, J.P., Nikoukhah, R.: Modeling and Simulation in Scilab/Scicos with ScicosLab 4.4. Springer, New York (2005)zbMATHGoogle Scholar
  14. 14.
    Feiler, P., Gluch, D., Hudak, J.: The architecture analysis and design language (AADL): an introduction. Technical report CMU/SEI-2006-TN-011, Carnegie-Mellon University (2006)Google Scholar
  15. 15.
    Alur, R., Henzinger, T.A., Lafferriere, G., Pappas, G.J.: Discrete abstractions of hybrid systems. Proc. IEEE 88(7), 971–984 (2000)CrossRefGoogle Scholar
  16. 16.
    Pappas, G.: Bisimilar linear systems. Automatica 39(12), 2035–2047 (2003)MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    van der Schaft, A.: Equivalence of dynamical systems by bisimulation. IEEE Trans. Autom. Control 49(12), 2160–2172 (2004)MathSciNetCrossRefGoogle Scholar
  18. 18.
    van der Schaft, A.: Equivalence of hybrid dynamical systems. In: Proceedings of 16th International Symposium on Mathematical Theory of Networks and Systems, Leuven, Belgium, 5–9 July 2004Google Scholar
  19. 19.
    Julius, A., van der Schaft, A.: Bisimulation as congruence in the behavioral setting. In: Proceedings of 44th IEEE Conference on Decision and Control and the European Control Conference 2005, Seville, Spain, 12–15 December 2005Google Scholar
  20. 20.
    Pola, G., van der Schaft, A., Di Benedetto, M.: Equivalence of switching linear systems by bisimulation. Int. J. Control 79(1), 74–92 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    Schmuck, A., Raisch, J.: Simulation and bisimulation over multiple time scales in a behavioral setting (2014). CoRR abs/1402.3484Google Scholar
  22. 22.
    Cuijpers, P.J.L., Reniers, M.A.: Lost in translation: hybrid-time flows vs. real-time transitions. In: Egerstedt, M., Mishra, B. (eds.) HSCC 2008. LNCS, vol. 4981, pp. 116–129. Springer, Heidelberg (2008)CrossRefGoogle Scholar
  23. 23.
    Polderman, J., Willems, J.: Introduction to Mathematical Systems Theory: A Behavioral Approach. Springer, Berlin (1997)zbMATHGoogle Scholar
  24. 24.
    Haghverdi, E., Tabuada, P., Pappas, G.: Unifying bisimulation relations for discrete and continuous systems. In: Proceedings of International Symposium MTNS2002, South (2002)Google Scholar
  25. 25.
    Haghverdi, E., Tabuada, P., Pappas, G.: Bisimulation relations for dynamical, control, and hybrid systems. Theoret. Comput. Sci. 342(2–3), 229–261 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  26. 26.
    Davoren, J.M., Tabuada, P.: On simulations and bisimulations of general flow systems. In: Bemporad, A., Bicchi, A., Buttazzo, G. (eds.) HSCC 2007. LNCS, vol. 4416, pp. 145–158. Springer, Heidelberg (2007)CrossRefGoogle Scholar
  27. 27.
    Tripakis, S., Yovine, S.: Analysis of timed systems using time-abstracting bisimulations. Form. Methods Syst. Des. 18(1), 25–68 (2001)CrossRefzbMATHGoogle Scholar
  28. 28.
    Ivanov, I.: A criterion for existence of global-in-time trajectories of non-deterministic Markovian systems. Commun. Comput. Inf. Sci. (CCIS) 347, 111–130 (2012)CrossRefGoogle Scholar
  29. 29.
    Ivanov, I.: On representations of abstract systems with partial inputs and outputs. In: Gopal, T.V., Agrawal, M., Li, A., Cooper, S.B. (eds.) TAMC 2014. LNCS, vol. 8402, pp. 104–123. Springer, Heidelberg (2014)CrossRefGoogle Scholar
  30. 30.
    Ivanov, I., Nikitchenko, M., Abraham, U.: On a decidable formal theory for abstract continuous-time dynamical systems. Commun. Comput. Inf. Sci. (CCIS) 469, 78–99 (2014)CrossRefGoogle Scholar
  31. 31.
    Ivanov, I.: An abstract block formalism for engineering systems. In: CEUR Workshop Proceedings, ICTERI, vol. 1000, pp. 448–463. (2013)Google Scholar
  32. 32.
    Ivanov, I.: On existence of total input-output pairs of abstract time systems. Commun. Comput. Inf. Sci. (CCIS) 412, 308–331 (2013)CrossRefGoogle Scholar
  33. 33.
    Hájek, O.: Theory of processes, I. Czechoslovak Math. J. 17, 159–199 (1967)MathSciNetzbMATHGoogle Scholar
  34. 34.
    Hájek, O.: Theory of processes, II. Czechoslovak Math. J. 17(3), 372–398 (1967)MathSciNetzbMATHGoogle Scholar
  35. 35.
    Alur, R., Courcoubetis, C., Dill, D.: Model checking in dense real-time. Inf. Comput. 104, 2–34 (1993)MathSciNetCrossRefzbMATHGoogle Scholar
  36. 36.
    Liberzon, D.: Switching in Systems and Control (Systems and Control: Foundations and Applications). Birkhauser Boston Inc., Boston (2003)CrossRefzbMATHGoogle Scholar
  37. 37.
    Alur, R., Courcoubetis, C., Halbwachs, N., Henzinger, T., Ho, P.H., Nicollin, X., Olivero, A., Sifakis, J., Yovine, S.: The algorithmic analysis of hybrid systems. Theoret. Comput. Sci. 138(1), 3–34 (1995)MathSciNetCrossRefzbMATHGoogle Scholar
  38. 38.
    Goebel, R., Sanfelice, R.G., Teel, A.: Hybrid dynamical systems. IEEE Control Syst. 29(2), 28–93 (2009)MathSciNetCrossRefGoogle Scholar

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© IFIP International Federation for Information Processing 2016

Authors and Affiliations

  1. 1.Taras Shevchenko National University of KyivKyivUkraine

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