Model Checking Freeze LTL over One-Counter Automata

  • Stéphane Demri
  • Ranko Lazić
  • Arnaud Sangnier
Part of the Lecture Notes in Computer Science book series (LNCS, volume 4962)


We study complexity issues related to the model-checking problem for LTL with registers (a.k.a. freeze LTL) over one-counter automata. We consider several classes of one-counter automata (mainly deterministic vs. nondeterministic) and several syntactic fragments (restriction on the number of registers and on the use of propositional variables for control locations). The logic has the ability to store a counter value and to test it later against the current counter value. By introducing a non-trivial abstraction on counter values, we show that model checking LTL with registers over deterministic one-counter automata is PSpace-complete with infinite accepting runs. By constrast, we prove that model checking LTL with registers over nondeterministic one-counter automata is \(\Sigma_1^1\)-complete [resp. \(\Sigma_1^0\)-complete] in the infinitary [resp. finitary] case even if only one register is used and with no propositional variable. This makes a difference with the facts that several verification problems for one-counter automata are known to be decidable with relatively low complexity, and that finitary satisfiability for LTL with a unique register is decidable. Our results pave the way for model-checking LTL with registers over other classes of operational models, such as reversal-bounded counter machines and deterministic pushdown systems.


Model Check Temporal Logic Propositional Variable Hybrid Logic Data Word 
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., Dill, D.: A theory of timed automata. TCS 126, 183–235 (1994)zbMATHCrossRefMathSciNetGoogle Scholar
  2. 2.
    Alur, R., Henzinger, T.: A really temporal logic. In: FOCS 1989, pp. 164–169. IEEE, Los Alamitos (1989)Google Scholar
  3. 3.
    Bernholtz, O., Vardi, M., Wolper, P.: An automata-theoretic approach to branching-time model checking. In: Dill, D.L. (ed.) CAV 1994. LNCS, vol. 818, pp. 142–155. Springer, Heidelberg (1994)Google Scholar
  4. 4.
    Bojańczyk, M., David, C., Muscholl, A., Schwentick, T., Segoufin, L.: Two-variable logic on data trees and XML reasoning. In: PODS 2006, pp. 10–19 (2006)Google Scholar
  5. 5.
    Bojańczyk, M., Muscholl, A., Schwentick, T., Segoufin, L., David, C.: Two-variable logic on words with data. In: LICS 2006, pp. 7–16. IEEE, Los Alamitos (2006)Google Scholar
  6. 6.
    Bouajjani, A., Jurski, Y., Sighireanu, M.: A generic framework for reasoning about dynamic networks of infinite-state processes. In: Grumberg, O., Huth, M. (eds.) TACAS 2007. LNCS, vol. 4424, pp. 690–705. Springer, Heidelberg (2007)CrossRefGoogle Scholar
  7. 7.
    Bouyer, P., Petit, A., Thérien, D.: An algebraic approach to data languages and timed languages. I & C 182(2), 137–162 (2003)zbMATHGoogle Scholar
  8. 8.
    Dang, Z., Ibarra, O., Pietro, P.S.: Liveness verification of reversal-bounded multicounter machines with a free counter. In: Hariharan, R., Mukund, M., Vinay, V. (eds.) FSTTCS 2001. LNCS, vol. 2245, pp. 132–143. Springer, Heidelberg (2001)CrossRefGoogle Scholar
  9. 9.
    Demri, S., Lazić, R.: LTL with the freeze quantifier and register automata. In: LICS 2006, pp. 17–26. IEEE, Los Alamitos (2006)Google Scholar
  10. 10.
    Demri, S., Lazić, R., Nowak, D.: On the freeze quantifier in constraint LTL: Decidability and complexity. I & C 205(1), 2–24 (2007)zbMATHGoogle Scholar
  11. 11.
    Demri, S., Lazić, R., Sangnier, A.: Model checking freeze LTL over one-counter automata. Research report, Laboratoire Spécification et Vérification, ENS Cachan (2008)Google Scholar
  12. 12.
    Franceschet, M., de Rijke, M.: Model checking hybrid logics (with an application to semistructured data). Journal of Applied Logic 4(3), 279–304 (2006)zbMATHCrossRefMathSciNetGoogle Scholar
  13. 13.
    Goranko, V.: Hierarchies of modal and temporal logics with references pointers. Journal of Logic, Language, and Information 5, 1–24 (1996)zbMATHCrossRefMathSciNetGoogle Scholar
  14. 14.
    Jančar, P., Kučera, A., Moller, F., Sawa, Z.: DP lower bounds for equivalence-checking and model-checking of one-counter automata. I & C 188(1), 1–19 (2004)Google Scholar
  15. 15.
    Jurdziński, M., Lazić, R.: Alternation-free modal mu-calculus for data trees. In: LICS 2007, pp. 131–140 (2007)Google Scholar
  16. 16.
    Kupferman, O., Vardi, M.: Memoryful Branching-Time Logic. In: LICS 2006, pp. 265–274. IEEE, Los Alamitos (2006)Google Scholar
  17. 17.
    Kupferman, O., Vardi, M.Y., Wolper, P.: An automata-theoretic approach to branching-time model checking. JACM 47(2), 312–360 (2000)CrossRefMathSciNetGoogle Scholar
  18. 18.
    Laroussinie, F., Markey, N., Schnoebelen, P.: Temporal logic with forgettable past. In: LICS 2002, pp. 383–392. IEEE, Los Alamitos (2002)Google Scholar
  19. 19.
    Lazić, R.: Safely freezing LTL. In: Arun-Kumar, S., Garg, N. (eds.) FSTTCS 2006. LNCS, vol. 4337, pp. 381–392. Springer, Heidelberg (2006)CrossRefGoogle Scholar
  20. 20.
    Markey, N., Schnoebelen, P.: Model checking a path. In: Amadio, R.M., Lugiez, D. (eds.) CONCUR 2003. LNCS, vol. 2761, pp. 251–261. Springer, Heidelberg (2003)Google Scholar
  21. 21.
    Neven, F., Schwentick, T., Vianu, V.: Finite state machines for strings over infinite alphabets. TOCL 5(3), 403–435 (2004)CrossRefMathSciNetGoogle Scholar
  22. 22.
    Ouaknine, J., Worrell, J.: On Metric Temporal Logic and faulty Turing machines. In: Aceto, L., Ingólfsdóttir, A. (eds.) FOSSACS 2006. LNCS, vol. 3921, pp. 217–230. Springer, Heidelberg (2006)CrossRefGoogle Scholar
  23. 23.
    Ouaknine, J., Worrell, J.: On the decidability and complexity of metric temporal logic over finite words. Logical Methods in Computer Science 3(1:8), 1–27 (2007)MathSciNetGoogle Scholar
  24. 24.
    Schwentick, T., Weber, V.: Bounded-variable fragments of hybrid logics. In: Thomas, W., Weil, P. (eds.) STACS 2007. LNCS, vol. 4393, pp. 561–572. Springer, Heidelberg (2007)CrossRefGoogle Scholar
  25. 25.
    Segoufin, L.: Automata and logics for words and trees over an infinite alphabet. In: Ésik, Z. (ed.) CSL 2006. LNCS, vol. 4207, pp. 41–57. Springer, Heidelberg (2006)CrossRefGoogle Scholar
  26. 26.
    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
  27. 27.
    ten Cate, B., Franceschet, M.: On the complexity of hybrid logics with binders. In: Ong, L. (ed.) CSL 2005. LNCS, vol. 3634, pp. 339–354. Springer, Heidelberg (2005)CrossRefGoogle Scholar
  28. 28.
    Vardi, M.: Alternating automata and program verification. In: van Leeuwen, J. (ed.) Computer Science Today. LNCS, vol. 1000, pp. 471–485. Springer, Heidelberg (1995)CrossRefGoogle Scholar
  29. 29.
    Vardi, M.: Alternating automata: unifying truth and validity checking for temporal logics. In: McCune, W. (ed.) CADE 1997. LNCS, vol. 1249, pp. 191–206. Springer, Heidelberg (1997)Google Scholar
  30. 30.
    Vardi, M., Wolper, P.: Automata-theoretic techniques for modal logics of programs. JCSS 32, 183–221 (1986)zbMATHMathSciNetGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2008

Authors and Affiliations

  • Stéphane Demri
    • 1
  • Ranko Lazić
    • 3
  • Arnaud Sangnier
    • 1
    • 2
  1. 1.LSV, ENS Cachan, CNRS, INRIA 
  2. 2.EDF R&D 
  3. 3.Department of Computer ScienceUniversity of WarwickUK

Personalised recommendations