Minimum-Time Reachability in Timed Games

  • Thomas Brihaye
  • Thomas A. Henzinger
  • Vinayak S. Prabhu
  • Jean-François Raskin
Part of the Lecture Notes in Computer Science book series (LNCS, volume 4596)


We consider the minimum-time reachability problem in concurrent two-player timed automaton game structures. We show how to compute the minimum time needed by a player to reach a target location against all possible choices of the opponent. We do not put any syntactic restriction on the game structure, nor do we require any player to guarantee time divergence. We only require players to use receptive strategies which do not block time. The minimal time is computed in part using a fixpoint expression, which we show can be evaluated on equivalence classes of a non-trivial extension of the clock-region equivalence relation for timed automata.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Adler, B., de Alfaro, L., Faella, M.: Average reward timed games. In: Pettersson, P., Yi, W. (eds.) FORMATS 2005. LNCS, vol. 3829, pp. 65–80. Springer, Heidelberg (2005)CrossRefGoogle Scholar
  2. 2.
    Alur, R., Bernadsky, M., Madhusudan, P.: Optimal reachability for weighted timed games. In: Díaz, J., Karhumäki, J., Lepistö, A., Sannella, D. (eds.) ICALP 2004. LNCS, vol. 3142, pp. 122–133. Springer, Heidelberg (2004)Google Scholar
  3. 3.
    Alur, R., Dill, D.: A theory of timed automata. Theor. Comput. Sci. 126(2), 183–235 (1994)MATHCrossRefMathSciNetGoogle Scholar
  4. 4.
    Alur, R., Henzinger, T.: Modularity for timed and hybrid systems. In: Mazurkiewicz, A., Winkowski, J. (eds.) CONCUR 1997. LNCS, vol. 1243, pp. 74–88. Springer, Heidelberg (1997)Google Scholar
  5. 5.
    Asarin, E., Maler, O.: As soon as possible: Time optimal control for timed automata. In: Vaandrager, F.W., van Schuppen, J.H. (eds.) HSCC 1999. LNCS, vol. 1569, pp. 19–30. Springer, Heidelberg (1999)CrossRefGoogle Scholar
  6. 6.
    Bouyer, P., Cassez, F., Fleury, E., Larsen, K.G.: Optimal strategies in priced timed game automata. In: Lodaya, K., Mahajan, M. (eds.) FSTTCS 2004. LNCS, vol. 3328, pp. 148–160. Springer, Heidelberg (2004)Google Scholar
  7. 7.
    Bouyer, P., D’Souza, D., Madhusudan, P., Petit, A.: Timed control with partial observability. In: Hunt Jr., W.A., Somenzi, F. (eds.) CAV 2003. LNCS, vol. 2725, pp. 180–192. Springer, Heidelberg (2003)Google Scholar
  8. 8.
    Brihaye, T., Bruyère, V., Raskin, J.: On optimal timed strategies. In: Pettersson, P., Yi, W. (eds.) FORMATS 2005. LNCS, vol. 3829, pp. 49–64. Springer, Heidelberg (2005)CrossRefGoogle Scholar
  9. 9.
    Cassez, F., David, A., Fleury, E., Larsen, K., Lime, D.: Efficient on-the-fly algorithms for the analysis of timed games. In: Abadi, M., de Alfaro, L. (eds.) CONCUR 2005. LNCS, vol. 3653, pp. 66–80. Springer, Heidelberg (2005)CrossRefGoogle Scholar
  10. 10.
    Cassez, F., Henzinger, T., Raskin, J.-F.: A comparison of control problems for timed and hybrid systems. In: Tomlin, C.J., Greenstreet, M.R. (eds.) HSCC 2002. LNCS, vol. 2289, pp. 134–148. Springer, Heidelberg (2002)CrossRefGoogle Scholar
  11. 11.
    Courcoubetis, C., Yannakakis, M.: Minimum and maximum delay problems in real-time systems. Formal Methods in System Design 1(4), 385–415 (1992)MATHCrossRefGoogle Scholar
  12. 12.
    de Alfaro, L., Faella, M., Henzinger, T., Majumdar, R., Stoelinga, M.: The element of surprise in timed games. In: Amadio, R.M., Lugiez, D. (eds.) CONCUR 2003. LNCS, vol. 2761, pp. 144–158. Springer, Heidelberg (2003)Google Scholar
  13. 13.
    de Alfaro, L., Henzinger, T., Majumdar, R.: From verification to control: Dynamic programs for omega-regular objectives. In: LICS 2001, pp. 279–290. IEEE Computer Society Press, Los Alamitos (2001)Google Scholar
  14. 14.
    D’Souza, D., Madhusudan, P.: Timed control synthesis for external specifications. In: Alt, H., Ferreira, A. (eds.) STACS 2002. LNCS, vol. 2285, pp. 571–582. Springer, Heidelberg (2002)CrossRefGoogle Scholar
  15. 15.
    Faella, M., Torre, S.L., Murano, A.: Dense real-time games. In: LICS 2002, pp. 167–176. IEEE Computer Society Press, Los Alamitos (2002)Google Scholar
  16. 16.
    Henzinger, T., Kopke, P.: Discrete-time control for rectangular hybrid automata. Theoretical Computer Science 221, 369–392 (1999)MATHCrossRefMathSciNetGoogle Scholar
  17. 17.
    Henzinger, T., Prabhu, V.: Timed alternating-time temporal logic. In: Asarin, E., Bouyer, P. (eds.) FORMATS 2006. LNCS, vol. 4202, pp. 1–17. Springer, Heidelberg (2006)CrossRefGoogle Scholar
  18. 18.
    Jurdziński, M., Trivedi, A.: Reachability-time games on timed automata. In: ICALP 2007. LNCS, Springer, Heidelberg (2007)Google Scholar
  19. 19.
    Maler, O., Pnueli, A., Sifakis, J.: On the synthesis of discrete controllers for timed systems (an extended abstract). In: Mayr, E.W., Puech, C. (eds.) STACS 1995. LNCS, vol. 900, pp. 229–242. Springer, Heidelberg (1995)Google Scholar
  20. 20.
    Pnueli, A., Asarin, E., Maler, O., Sifakis, J.: Controller synthesis for timed automata. In: Proc. System Structure and Control, Elsevier, Amsterdam (1998)Google Scholar
  21. 21.
    Segala, R., Gawlick, R., Søgaard-Andersen, J., Lynch, N.: Liveness in timed and untimed systems. Inf. Comput. 141(2), 119–171 (1998)MATHCrossRefGoogle Scholar
  22. 22.
    Wong-Toi, H., Hoffmann, G.: The control of dense real-time discrete event systems. In: Proc. of 30th Conf. Decision and Control, pp. 1527–1528 (1991)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2007

Authors and Affiliations

  • Thomas Brihaye
    • 1
  • Thomas A. Henzinger
    • 2
  • Vinayak S. Prabhu
    • 3
  • Jean-François Raskin
    • 4
  1. 1.LSV-CNRS & ENS de Cachan 
  2. 2.Department of Computer and Communication Sciences, EPFL 
  3. 3.Department of Electrical Engineering & Computer Sciences, UC Berkeley 
  4. 4.Département d’Informatique, Université Libre de Bruxelles 

Personalised recommendations