Denotational Semantics of Hybrid Automata

  • Abbas Edalat
  • Dirk Pattinson
Part of the Lecture Notes in Computer Science book series (LNCS, volume 3921)


We introduce a denotational semantics for non-linear hybrid automata, and relate it to the operational semantics given in terms of hybrid trajectories. The semantics is defined as least fixpoint of an operator on the continuous domain of functions of time that take values in the lattice of compact subsets of n-dimensional Euclidean space. The semantic function assigns to every point in time the set of states the automaton can visit at that time, starting from one of its initial states. Our main results are the correctness and computational adequacy of the denotational semantics with respect to the operational semantics and the fact that the denotational semantics is computable.


Model Checker Operational Semantic Iterate Function System Reachable State Label Transition System 
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  1. 1.
    Abramsky, S., Jung, A.: Domain Theory. In: Abramsky, S., Gabbay, D., Maibaum, T.S.E. (eds.) Handbook of Logic in Computer Science, vol. 3, Clarendon Press (1994)Google Scholar
  2. 2.
    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. Comp. Sci 138(1), 3–34 (1995)MathSciNetCrossRefMATHGoogle Scholar
  3. 3.
    Alur, R., Henzinger, T., Ho, P.-H.: Automatic symbolic verification of embedded systems. IEEE Transactions on Software Engineering 22(3), 181–201 (1996)CrossRefGoogle Scholar
  4. 4.
    Aubin, J.-P.: Viability Theory. Birkhäuser (1991)Google Scholar
  5. 5.
    Edalat, A.: Dynamical systems, measures and fractals via domain theory. Information and Computation 120(1), 32–48 (1995)MathSciNetCrossRefMATHGoogle Scholar
  6. 6.
    Edalat, A.: Power domains and iterated function systems. Information and Computation 124, 182–197 (1996)MathSciNetCrossRefMATHGoogle Scholar
  7. 7.
    Edalat, A., Krznarić, M., Lieutier, A.: Domain-theoretic solution of differential equations (scalar fields). In: Proceedings of MFPS XIX. Elect. Notes in Theoret. Comput. Sci, vol. 83 (2004)Google Scholar
  8. 8.
    Edalat, A., Lieutier, A.: Domain theory and differential calculus (functions of one variable. Math. Struct. Comp. Sci. 14 (2004)Google Scholar
  9. 9.
    Edalat, A., Pattinson, D.: A domain theoretic account of picard’s theorem. In: Díaz, J., Karhumäki, J., Lepistö, A., Sannella, D. (eds.) ICALP 2004. LNCS, vol. 3142, pp. 494–505. Springer, Heidelberg (2004)CrossRefGoogle Scholar
  10. 10.
    Edalat, A., Pattinson, D.: Domain theoretic solutions of initial value problems for unbounded vector fields. In: Escardó, M. (ed.) Proc. MFPS XXI. Electr. Notes in Theoret. Comp. Sci ( to appear, 2005)Google Scholar
  11. 11.
    Gierz, G., Hofmann, K.H., Keimel, K., Lawson, J.D., Mislove, M., Scott, D.: Continuous Lattices and Domains. Cambridge University Press, Cambridge (2003)CrossRefMATHGoogle Scholar
  12. 12.
    Henzinger, T.: The theory of hybrid automata. In: Inan, M., Kurshan, R. (eds.) CADE 1984. NATO ASI Series F: Computer and Systems Sciences, vol. 170, pp. 265–292. Springer, Heidelberg (2000)CrossRefGoogle Scholar
  13. 13.
    Henzinger, T., Ho, P.-H., Wong-Toi, H.: HYTECH: A model checker for hybrid systems. International Journal on Software Tools for Technology Transfer 1(1–2), 110–122 (1997)CrossRefMATHGoogle Scholar
  14. 14.
    Henzinger, T., Ho, P.-H., Wong-Toi, H.: Algorithmic analysis of nonlinear hybrid systems. IEEE Transactions on Automatic Control 43, 540–554 (1998)MathSciNetCrossRefMATHGoogle Scholar
  15. 15.
    Henzinger, T., Horowitz, B., Majumdar, R., Wong-Toi, H.: Beyond HYTECH: Hybrid systems analysis using interval numerical methods. In: Lynch, N.A., Krogh, B.H. (eds.) HSCC 2000. LNCS, vol. 1790, pp. 130–144. Springer, Heidelberg (2000)CrossRefGoogle Scholar
  16. 16.
    Hutchinson, J.E.: Fractals and self-similarity. Indiana University Mathematics Journal 30, 713–747 (1981)MathSciNetCrossRefMATHGoogle Scholar
  17. 17.
    Lygeros, J., Godbole, D., Sastry, S.: Verified hybrid controllers for automated vehicles. IEEE Transactions on Automatic Control 43(4), 522–539 (1998)MathSciNetCrossRefMATHGoogle Scholar
  18. 18.
    Müller, O., Stauner, T.: Modelling and verification using linear hybrid automata – a case study. Mathematical and Computer Modelling of Dynamical Systems 6(1), 71–89 (2000)CrossRefMATHGoogle Scholar
  19. 19.
    Simic, S., Johansson, K., Sastry, S., Lygeros, J.: Towards a geometric theory of hybrid systems. In: Lynch, N.A., Krogh, B.H. (eds.) HSCC 2000. LNCS, vol. 1790, pp. 421–436. Springer, Heidelberg (2000)CrossRefGoogle Scholar
  20. 20.
    Tomlin, C., Pappas, G., Sastry, S.: Conflict resolution for air traffic management: A study in muti-agent hybrid systems. IEEE Transactions on Automatic Control 43(4), 509–521 (1998)MathSciNetCrossRefMATHGoogle Scholar
  21. 21.
    Weihrauch, K.: Computable Analysis. Springer, Heidelberg (2000)CrossRefMATHGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2006

Authors and Affiliations

  • Abbas Edalat
    • 1
  • Dirk Pattinson
    • 2
  1. 1.Department of ComputingImperial College LondonUK
  2. 2.Department of Computer ScienceUniversity of LeicesterUK

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