Logic and Compositional Verification of Hybrid Systems

(Invited Tutorial)
  • André Platzer
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 6806)


Hybrid systems are models for complex physical systems and have become a widely used concept for understanding their behavior. Many applications are safety-critical, including car, railway, and air traffic control, robotics, physical-chemical process control, and biomedical devices. Hybrid systems analysis studies how we can build computerised controllers for physical systems which are guaranteed to meet their design goals. The continuous dynamics of hybrid systems can be modeled by differential equations, the discrete dynamics by a combination of discrete state-transitions and conditional execution. The discrete and continuous dynamics interact to form hybrid systems, which makes them quite challenging for verification.

In this tutorial, we survey state-of-the-art verification techniques for hybrid systems. In particular, we focus on a coherent logical approach for systematic hybrid systems analysis. We survey theory, practice, and applications, and show how hybrid systems can be verified in the hybrid systems verification tool KeYmaera. KeYmaera has been used successfully to verify safety, reactivity, controllability, and liveness properties, including collision freedom in air traffic, car, and railway control systems. It has also been used to verify properties of electrical circuits.


Model Check Hybrid System Dynamic Logic Hybrid Automaton Continuous Dynamic 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


  1. 1.
    Alur, R., Courcoubetis, C., Halbwachs, N., Henzinger, T.A., Ho, P.H., Nicollin, X., Olivero, A., Sifakis, J., Yovine, S.: The algorithmic analysis of hybrid systems. Theor. Comput. Sci. 138(1), 3–34 (1995)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Alur, R., Courcoubetis, C., Henzinger, T.A., Ho, P.H.: Hybrid automata: An algorithmic approach to the specification and verification of hybrid systems. In: Grossman, et al. (eds.) [18], pp. 209–229Google Scholar
  3. 3.
    Alur, R., Sontag, E.D., Henzinger, T.A. (eds.): HS 1995. LNCS, vol. 1066. Springer, Heidelberg (1996)Google Scholar
  4. 4.
    Beckert, B., Hähnle, R., Schmitt, P.H. (eds.): Verification of Object-Oriented Software. LNCS (LNAI), vol. 4334. Springer, Heidelberg (2007)Google Scholar
  5. 5.
    van Beek, D.A., Man, K.L., Reniers, M.A., Rooda, J.E., Schiffelers, R.R.H.: Syntax and consistent equation semantics of hybrid Chi. J. Log. Algebr. Program. 68(1-2), 129–210 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    van Beek, D.A., Reniers, M.A., Schiffelers, R.R.H., Rooda, J.E.: Concrete syntax and semantics of the compositional interchange format for hybrid systems. In: 17th IFAC World Congress (2008)Google Scholar
  7. 7.
    Bergstra, J.A., Middelburg, C.A.: Process algebra for hybrid systems. Theor. Comput. Sci. 335(2-3), 215–280 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Branicky, M.S.: General hybrid dynamical systems: Modeling, analysis, and control. In: Alur, et al. (eds.) [3], pp. 186–200Google Scholar
  9. 9.
    Branicky, M.S.: Studies in Hybrid Systems: Modeling, Analysis, and Control. Ph.D. thesis, Dept. Elec. Eng. and Computer Sci. Massachusetts Inst. Technol. Cambridge, MA (1995)Google Scholar
  10. 10.
    Branicky, M.S., Borkar, V.S., Mitter, S.K.: A unified framework for hybrid control: Model and optimal control theory. IEEE T. Automat. Contr. 43(1), 31–45 (1998)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Cassez, F., Larsen, K.G.: The impressive power of stopwatches. In: CONCUR, pp. 138–152 (2000)Google Scholar
  12. 12.
    Chaochen, Z., Ji, W., Ravn, A.P.: A formal description of hybrid systems. In: Alur, et al. (eds.) [3], pp. 511–530Google Scholar
  13. 13.
    Chutinan, A., Krogh, B.H.: Computational techniques for hybrid system verification. IEEE T. Automat. Contr. 48(1), 64–75 (2003)MathSciNetCrossRefGoogle Scholar
  14. 14.
    Clarke, E.M., Fehnker, A., Han, Z., Krogh, B.H., Ouaknine, J., Stursberg, O., Theobald, M.: Abstraction and counterexample-guided refinement in model checking of hybrid systems. Int. J. Found. Comput. Sci. 14(4), 583–604 (2003)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Cuijpers, P.J.L., Reniers, M.A.: Hybrid process algebra. J. Log. Algebr. Program. 62(2), 191–245 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Davoren, J.M., Nerode, A.: Logics for hybrid systems, vol. 88(7), pp. 985–1010. IEEE, Los Alamitos (2000)Google Scholar
  17. 17.
    Frehse, G.: PHAVer: algorithmic verification of hybrid systems past HyTech. STTT 10(3), 263–279 (2008)CrossRefzbMATHGoogle Scholar
  18. 18.
    Grossman, R.L., Ravn, A.P., Rischel, H., Nerode, A. (eds.): Hybrid Systems 1991 and HS 1992. LNCS, vol. 736. Springer, Heidelberg (1993)Google Scholar
  19. 19.
    Harel, D., Kozen, D., Tiuryn, J.: Dynamic logic. MIT Press, Cambridge (2000)zbMATHGoogle Scholar
  20. 20.
    Henzinger, T.A.: The theory of hybrid automata. In: LICS, pp. 278–292. IEEE Computer Society, Los Alamitos (1996)Google Scholar
  21. 21.
    Jifeng, H.: From CSP to hybrid systems. In: Roscoe, A.W. (ed.) A classical mind: essays in honour of C. A. R. Hoare, pp. 171–189. Prentice Hall, Hertfordshire (1994)Google Scholar
  22. 22.
    Kesten, Y., Manna, Z., Pnueli, A.: Verification of clocked and hybrid systems. Acta Inf. 36(11), 837–912 (2000)MathSciNetCrossRefzbMATHGoogle Scholar
  23. 23.
    Kozen, D.: Kleene algebra with tests. ACM Trans. Program. Lang. Syst. 19(3), 427–443 (1997)CrossRefzbMATHGoogle Scholar
  24. 24.
    Manna, Z., Sipma, H.: Deductive verification of hybrid systems using STeP. In: Henzinger, T.A., Sastry, S.S. (eds.) HSCC 1998. LNCS, vol. 1386, pp. 305–318. Springer, Heidelberg (1998)CrossRefGoogle Scholar
  25. 25.
    Mitchell, I., Bayen, A.M., Tomlin, C.: A time-dependent Hamilton-Jacobi formulation of reachable sets for continuous dynamic games. IEEE T. Automat. Contr. 50(7), 947–957 (2005)MathSciNetCrossRefGoogle Scholar
  26. 26.
    Mysore, V., Piazza, C., Mishra, B.: Algorithmic algebraic model checking II: Decidability of semi-algebraic model checking and its applications to systems biology. In: Peled, D.A., Tsay, Y.-K. (eds.) ATVA 2005. LNCS, vol. 3707, pp. 217–233. Springer, Heidelberg (2005)CrossRefGoogle Scholar
  27. 27.
    Nicollin, X., Olivero, A., Sifakis, J., Yovine, S.: An approach to the description and analysis of hybrid systems. In: Grossman, et al. (eds.) [18], pp. 149–178Google Scholar
  28. 28.
    Perko, L.: Differential equations and dynamical systems. Springer, New York (1991)CrossRefzbMATHGoogle Scholar
  29. 29.
    Platzer, A.: Differential dynamic logic for verifying parametric hybrid systems. In: Olivetti, N. (ed.) TABLEAUX 2007. LNCS (LNAI), vol. 4548, pp. 216–232. Springer, Heidelberg (2007)CrossRefGoogle Scholar
  30. 30.
    Platzer, A.: Differential dynamic logic for hybrid systems. J. Autom. Reas. 41(2), 143–189 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  31. 31.
    Platzer, A.: Differential-algebraic dynamic logic for differential-algebraic programs. J. Log. Comput. 20(1), 309–352 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  32. 32.
    Platzer, A.: Logical Analysis of Hybrid Systems: Proving Theorems for Complex Dynamics. Springer, Heidelberg (2010)CrossRefzbMATHGoogle Scholar
  33. 33.
    Platzer, A.: Quantified differential dynamic logic for distributed hybrid systems. In: Dawar, A., Veith, H. (eds.) CSL 2010. LNCS, vol. 6247, pp. 469–483. Springer, Heidelberg (2010)CrossRefGoogle Scholar
  34. 34.
    Platzer, A.: Quantified differential invariants. In: Frazzoli, E., Grosu, R. (eds.) HSCC, pp. 63–72. ACM Press, New York (2011)Google Scholar
  35. 35.
    Platzer, A.: Stochastic differential dynamic logic for stochastic hybrid programs. In: Bjørner, N., Sofronie-Stokkermans, V. (eds.) CADE. LNCS. Springer, Heidelberg (2011)Google Scholar
  36. 36.
    Platzer, A., Clarke, E.M.: The image computation problem in hybrid systems model checking. In: Bemporad, A., Bicchi, A., Buttazzo, G. (eds.) HSCC 2007. LNCS, vol. 4416, pp. 473–486. Springer, Heidelberg (2007)CrossRefGoogle Scholar
  37. 37.
    Platzer, A., Clarke, E.M.: Computing differential invariants of hybrid systems as fixedpoints. In: Gupta, A., Malik, S. (eds.) CAV 2008. LNCS, vol. 5123, pp. 176–189. Springer, Heidelberg (2008)CrossRefGoogle Scholar
  38. 38.
    Platzer, A., Clarke, E.M.: Formal verification of curved flight collision avoidance maneuvers: A case study. In: Cavalcanti, A., Dams, D.R. (eds.) FM 2009. LNCS, vol. 5850, pp. 547–562. Springer, Heidelberg (2009)CrossRefGoogle Scholar
  39. 39.
    Platzer, A., Quesel, J.-D.: KeYmaera: A hybrid theorem prover for hybrid systems (System description). In: Armando, A., Baumgartner, P., Dowek, G. (eds.) IJCAR 2008. LNCS (LNAI), vol. 5195, pp. 171–178. Springer, Heidelberg (2008)CrossRefGoogle Scholar
  40. 40.
    Platzer, A., Quesel, J.-D.: European Train Control System: A Case Study in Formal Verification. In: Breitman, K., Cavalcanti, A. (eds.) ICFEM 2009. LNCS, vol. 5885, pp. 246–265. Springer, Heidelberg (2009)CrossRefGoogle Scholar
  41. 41.
    Ratschan, S., She, Z.: Safety verification of hybrid systems by constraint propagation-based abstraction refinement. Trans. on Embedded Computing Sys. 6(1), 8 (2007)CrossRefzbMATHGoogle Scholar
  42. 42.
    Rönkkö, M., Ravn, A.P., Sere, K.: Hybrid action systems. Theor. Comput. Sci. 290(1), 937–973 (2003)MathSciNetCrossRefzbMATHGoogle Scholar
  43. 43.
    Tavernini, L.: Differential automata and their discrete simulators. Non-Linear Anal. 11(6), 665–683 (1987)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2011

Authors and Affiliations

  • André Platzer
    • 1
  1. 1.Computer Science DepartmentCarnegie Mellon UniversityPittsburghUSA

Personalised recommendations