We study recursive timed automata that extend timed automata with recursion. Timed automata, as introduced by Alur and Dill, are finite automata accompanied by a finite set of real-valued variables called clocks. Recursive timed automata are finite collections of timed automata extended with special states that correspond to (potentially recursive) invocations of other timed automata from their collection. During an invocation of a timed automaton, our model permits passing the values of clocks using both pass-by-value and pass-by-reference mechanisms. We study the natural reachability and termination (reachability with empty invocation stack) problems for recursive timed automata. We show that these problems are decidable (in many cases with the same complexity as the reachability problem on timed automata) for recursive timed automata satisfying the following condition: during each invocation either all clocks are passed by reference or none is passed by reference. Furthermore, we show that for recursive timed automata that violate this condition reachability/termination problems are undecidable for automata with as few as three clocks. We also establish similar results for two-player game extension of our model against reachability/termination objective.


Label Transition System Local Clock Reachability Problem Global Clock Exit Node 
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  1. 1.
    Alur, R., Benedikt, M., Etessami, K., Godefroid, P., Reps, T., Yannakakis, M.: Analysis of recursive state machines. ACM Transactions on Programming Languages and Systems 27, 786–818 (2005)CrossRefGoogle Scholar
  2. 2.
    Alur, R., Dill, D.: A theory of timed automata. Theor. Comput. Sci. 126 (1994)Google Scholar
  3. 3.
    Alur, R., Yannakakis, M.: Model checking of hierarchical state machines. In: ACM SIGSOFT 1998, pp. 175–188 (1998)Google Scholar
  4. 4.
    Ball, T., Rajamani, S.: The slam toolkit. In: Berry, G., Comon, H., Finkel, A. (eds.) CAV 2001. LNCS, vol. 2102, pp. 260–264. Springer, Heidelberg (2001)CrossRefGoogle Scholar
  5. 5.
    Bouajjani, A., Echahed, R., Robbana, R.: On the automatic verification of systems with continuous variables and unbounded discrete data structures. In: Antsaklis, P.J., Kohn, W., Nerode, A., Sastry, S.S. (eds.) HS 1994. LNCS, vol. 999, pp. 64–85. Springer, Heidelberg (1995)CrossRefGoogle Scholar
  6. 6.
    Bouajjani, A., Echahed, R., Robbana, R.: Verification of context-free timed systems using linear hybrid observers. In: Dill, D.L. (ed.) CAV 1994. LNCS, vol. 818, pp. 118–131. Springer, Heidelberg (1994)CrossRefGoogle Scholar
  7. 7.
    Bouchy, F., Finkel, A., Sangnier, A.: Reachability in timed counter systems. Electronic Notes in Theoretical Computer Science 239, 167–178 (2009); Joint Proceedings of the 8th, 9th, and 10th International Workshops on Verification of Infinite-State Systems (INFINITY 2006, 2007, 2008) Google Scholar
  8. 8.
    Buttazzo, G.C.: Hard Real-time Computing Systems: Predictable Scheduling Algorithms and Applications. Springer, Santa Clara (2004)zbMATHGoogle Scholar
  9. 9.
    Cassez, F., Jessen, J.J., Larsen, K.G., Raskin, J.-F., Reynier, P.-A.: Automatic synthesis of robust and optimal controllers — an industrial case study. In: Majumdar, R., Tabuada, P. (eds.) HSCC 2009. LNCS, vol. 5469, pp. 90–104. Springer, Heidelberg (2009)CrossRefGoogle Scholar
  10. 10.
    Dang, Z.: Binary reachability analysis of pushdown timed automata with dense clocks. In: Berry, G., Comon, H., Finkel, A. (eds.) CAV 2001. LNCS, vol. 2102, pp. 506–517. Springer, Heidelberg (2001)CrossRefGoogle Scholar
  11. 11.
    Dang, Z.: Pushdown timed automata: a binary reachability characterization and safety verification. Theor. Comput. Sci. 302(1-3), 93–121 (2003)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Demri, S., Gascon, R.: The Effects of Bounding Syntactic Resources on Presburger LTL. J. Logic Computation 19(6), 1541–1575 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Emmi, M., Majumdar, R.: Decision problems for the verification of real-time software. In: Hybrid Systems: Computation and Control, pp. 200–211 (2006)Google Scholar
  14. 14.
    Etessami, K., Yannakakis, M.: Recursive markov decision processes and recursive stochastic games. In: Caires, L., Italiano, G.F., Monteiro, L., Palamidessi, C., Yung, M. (eds.) ICALP 2005. LNCS, vol. 3580, pp. 891–903. Springer, Heidelberg (2005)CrossRefGoogle Scholar
  15. 15.
    Etessami, K.: Analysis of recursive game graphs using data flow equations. In: Steffen, B., Levi, G. (eds.) VMCAI 2004. LNCS, vol. 2937, pp. 282–296. Springer, Heidelberg (2004)CrossRefGoogle Scholar
  16. 16.
    Etessami, K., Wojtczak, D., Yannakakis, M.: Quasi-Birth-Death processes, Tree-like QBDs, Probabilistic 1-Counter Automata, and Pushdown Systems. Performance Evaluation 67(9), 837–857 (2010); Special Issue of QEST 2008 (2008)CrossRefGoogle Scholar
  17. 17.
    Graf, S., Saidi, H.: Construction of abstract state graphs with PVS. In: Grumberg, O. (ed.) CAV 1997. LNCS, vol. 1254, pp. 72–83. Springer, Heidelberg (1997)CrossRefGoogle Scholar
  18. 18.
    Jancar, P., Sawa, Z.: A note on emptiness for alternating finite automata with a one-letter alphabet. Inf. Process. Lett. 104(5), 164–167 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Minsky, M.L.: Computation: finite and infinite machines. Prentice-Hall, Inc., Englewood Cliffs (1967)zbMATHGoogle Scholar
  20. 20.
    Serre, O.: Parity games played on transition graphs of one-counter processes. In: Aceto, L., Ingólfsdóttir, A. (eds.) FOSSACS 2006. LNCS, vol. 3921, pp. 337–351. Springer, Heidelberg (2006)CrossRefGoogle Scholar
  21. 21.
    Trivedi, A., Wojtczak, D.: Recursive timed automata. Oxford University Computing Laboratory technical report, RR-10-09 (2010)Google Scholar
  22. 22.
    Walukiewicz, I.: Pushdown processes: Games and model checking, pp. 62–74 (1996)Google Scholar

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© Springer-Verlag Berlin Heidelberg 2010

Authors and Affiliations

  • Ashutosh Trivedi
    • 1
  • Dominik Wojtczak
    • 1
  1. 1.Computing LaboratoryOxford UniversityUK

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