Regularity Problems for Weak Pushdown ω-Automata and Games

  • Christof Löding
  • Stefan Repke
Part of the Lecture Notes in Computer Science book series (LNCS, volume 7464)


We show that the regularity and equivalence problems are decidable for deterministic weak pushdown ω-automata, giving a partial answer to a question raised by Cohen and Gold in 1978. We prove the decidability by a reduction to the corresponding problems for deterministic pushdown automata on finite words. Furthermore, we consider the problem of deciding for pushdown games whether a winning strategy exists that can be implemented by a finite automaton. We show that this problem is already undecidable for games defined by one-counter automata or visibly pushdown automata with a safety condition.


Arena Rovan 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Alur, R., Madhusudan, P.: Visibly pushdown languages. In: STOC, pp. 202–211 (2004)Google Scholar
  2. 2.
    Baier, C., Katoen, J.-P.: Principles of Model Checking. MIT Press (2008)Google Scholar
  3. 3.
    Büchi, J.R., Landweber, L.H.: Solving sequential conditions by finite-state strategies. Transactions of the AMS 138, 295–311 (1969)Google Scholar
  4. 4.
    Cachat, T.: Symbolic Strategy Synthesis for Games on Pushdown Graphs. In: Widmayer, P., Triguero, F., Morales, R., Hennessy, M., Eidenbenz, S., Conejo, R. (eds.) ICALP 2002. LNCS, vol. 2380, pp. 704–715. Springer, Heidelberg (2002)CrossRefGoogle Scholar
  5. 5.
    Chandra, A.K., Kozen, D., Stockmeyer, L.J.: Alternation. J. ACM 28(1), 114–133 (1981)MathSciNetMATHCrossRefGoogle Scholar
  6. 6.
    Cohen, R.S., Gold, A.Y.: Omega-computations on deterministic pushdown machines. JCSS 16(3), 275–300 (1978)MathSciNetMATHGoogle Scholar
  7. 7.
    Esparza, J., Hansel, D., Rossmanith, P., Schwoon, S.: Efficient Algorithms for Model Checking Pushdown Systems. In: Emerson, E.A., Sistla, A.P. (eds.) CAV 2000. LNCS, vol. 1855, pp. 232–247. Springer, Heidelberg (2000)CrossRefGoogle Scholar
  8. 8.
    Hague, M., Ong, C.-H.L.: Winning Regions of Pushdown Parity Games: A Saturation Method. In: Bravetti, M., Zavattaro, G. (eds.) CONCUR 2009. LNCS, vol. 5710, pp. 384–398. Springer, Heidelberg (2009)CrossRefGoogle Scholar
  9. 9.
    Hopcroft, J.E., Ullman, J.D.: Introduction to Automata Theory, Languages, and Computation. Addison-Wesley (1979)Google Scholar
  10. 10.
    Löding, C.: Efficient minimization of deterministic weak omega-automata. Information Processing Letters 79(3), 105–109 (2001)MathSciNetMATHCrossRefGoogle Scholar
  11. 11.
    Perrin, D., Pin, J.-É.: Infinite words. Pure and Applied Mathematics, vol. 141. Elsevier (2004)Google Scholar
  12. 12.
    Segoufin, L., Sirangelo, C.: Constant-Memory Validation of Streaming XML Documents Against DTDs. In: Schwentick, T., Suciu, D. (eds.) ICDT 2007. LNCS, vol. 4353, pp. 299–313. Springer, Heidelberg (2006)CrossRefGoogle Scholar
  13. 13.
    Segoufin, L., Vianu, V.: Validating streaming XML documents. In: PODS, pp. 53–64 (2002)Google Scholar
  14. 14.
    Sénizergues, G.: L(A)=L(B)? decidability results from complete formal systems. Theor. Comput. Sci. 251(1-2), 1–166 (2001)MATHCrossRefGoogle Scholar
  15. 15.
    Staiger, L.: Finite-state ω-languages. JCSS 27(3), 434–448 (1983)MathSciNetMATHGoogle Scholar
  16. 16.
    Stearns, R.E.: A regularity test for pushdown machines. Information and Control 11(3), 323–340 (1967)MATHCrossRefGoogle Scholar
  17. 17.
    Straubing, H.: Finite Automata, Formal Logic, and Circuit Complexity. Birkhäuser, Basel (1994)MATHCrossRefGoogle Scholar
  18. 18.
    Valiant, L.G.: Regularity and related problems for deterministic pushdown automata. J. ACM 22(1), 1–10 (1975)MathSciNetMATHCrossRefGoogle Scholar
  19. 19.
    Walukiewicz, I.: Pushdown processes: Games and model checking. Information and Computation 164(2), 234–263 (2001)MathSciNetMATHCrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2012

Authors and Affiliations

  • Christof Löding
    • 1
  • Stefan Repke
    • 1
  1. 1.Lehrstuhl für Informatik 7RWTH AachenGermany

Personalised recommendations