Alternating Automata and a Temporal Fixpoint Calculus for Visibly Pushdown Languages

  • Laura Bozzelli
Part of the Lecture Notes in Computer Science book series (LNCS, volume 4703)

Abstract

We investigate various classes of alternating automata for visibly pushdown languages (VPL) over infinite words. First, we show that alternating visibly pushdown automata (AVPA) are exactly as expressive as their nondeterministic counterpart (NVPA) but basic decision problems for AVPA are 2Exptime-complete. Due to this high complexity, we introduce a new class of alternating automata called alternating jump automata (AJA). AJA extend classical alternating finite-state automata over infinite words by also allowing non-local moves. A non-local forward move leads a copy of the automaton from a call input position to the matching-return position. We also allow local and non-local backward moves. We show that one-way AJA and two-way AJA have the same expressiveness and capture exactly the class of VPL. Moreover, boolean operations for AJA are easy and basic decision problems such as emptiness, universality, and pushdown model-checking for parity two-way AJA are Exptime-complete. Finally, we consider a linear-time fixpoint calculus which subsumes the full linear-time μ-calculus (with both forward and backward modalities) and the logic CaRet and captures exactly the class of VPL. We show that formulas of this logic can be linearly translated into parity two-way AJA, and vice versa. As a consequence satisfiability and pushdown model checking for this logic are Exptime-complete.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Alur, R., Chaudhuri, S., Madhusudan, P.: A fixpoint calculus for local and global program flows. In: Proc. 33rd POPL, pp. 153–165. ACM Press, New York (2006)Google Scholar
  2. 2.
    Alur, R., Chaudhuri, S., Madhusudan, P.: Languages of nested trees. In: Ball, T., Jones, R.B. (eds.) CAV 2006. LNCS, vol. 4144, pp. 329–342. Springer, Heidelberg (2006)CrossRefGoogle Scholar
  3. 3.
    Alur, R., Etessami, K., Madhusudan, P.: A Temporal Logic of Nested Calls and Returns. In: Jensen, K., Podelski, A. (eds.) TACAS 2004. LNCS, vol. 2988, pp. 467–481. Springer, Heidelberg (2004)Google Scholar
  4. 4.
    Alur, R., Madhusudan, P.: Visibly pushdown languages. In: Proc. 36th STOC, pp. 202–211. ACM Press, New York (2004)Google Scholar
  5. 5.
    Alur, R., Madhusudan, P.: Adding nesting structure to words. In: Developments in Language Theory, pp. 1–13 (2006)Google Scholar
  6. 6.
    Ball, T., Rajamani, S.: Bebop: a symbolic model checker for boolean programs. In: Havelund, K., Penix, J., Visser, W. (eds.) SPIN Model Checking and Software Verification. LNCS, vol. 1885, pp. 113–130. Springer, Heidelberg (2000)CrossRefGoogle Scholar
  7. 7.
    Bouajjani, A., Esparza, J., Maler, O.: Reachability Analysis of Pushdown Automata: Application to Model-Checking. In: Mazurkiewicz, A., Winkowski, J. (eds.) CONCUR 1997. LNCS, vol. 1243, pp. 135–150. Springer, Heidelberg (1997)Google Scholar
  8. 8.
    Chatterjee, K., Ma, D., Majumdar, R., Zhao, T., Henzinger, T.A., Palsberg, J.: Stack size analysis for interrupt-driven programs. In: Cousot, R. (ed.) SAS 2003. LNCS, vol. 2694, pp. 109–126. Springer, Heidelberg (2003)CrossRefGoogle Scholar
  9. 9.
    Chen, H., Wagner, D.: Mops: an infrastructure for examining security properties of software. In: Proc. 9th CCS, pp. 235–244. ACM Press, New York (2002)Google Scholar
  10. 10.
    Esparza, J., Kucera, A., Schwoon, S.: Model checking LTL with regular valuations for pushdown systems. Information and Computation 186(2), 355–376 (2003)MATHCrossRefMathSciNetGoogle Scholar
  11. 11.
    Loding, C., Madhusudan, P., Serre, O.: Visibly pushdown games. In: Lodaya, K., Mahajan, M. (eds.) FSTTCS 2004. LNCS, vol. 3328, pp. 408–420. Springer, Heidelberg (2004)Google Scholar
  12. 12.
    Muller, D.E., Schupp, P.E.: Alternating Automata on Infinite Trees. Theoretical Computer Science 54, 267–276 (1987)MATHCrossRefMathSciNetGoogle Scholar
  13. 13.
    Muller, D.E., Schupp, P.E.: Simulating alternating tree automata by nondeterministic automata: new results and new proofs of the Theorems of Rabin, McNaughton and Safra. Theoretical Computer Science 141(1-2), 69–107 (1995)MATHCrossRefMathSciNetGoogle Scholar
  14. 14.
    Vardi, M.Y.: A temporal fixpoint calculus. In: Proc. 15th Annual POPL, pp. 250–259. ACM Press, New York (1988)Google Scholar
  15. 15.
    Vardi, M.Y.: Reasoning about the past with two-way automata. In: Larsen, K.G., Skyum, S., Winskel, G. (eds.) ICALP 1998. LNCS, vol. 1443, pp. 628–641. Springer, Heidelberg (1998)CrossRefGoogle Scholar
  16. 16.
    Walukiewicz, I.: Pushdown processes: Games and Model Checking. In: Alur, R., Henzinger, T.A. (eds.) CAV 1996. LNCS, vol. 1102, pp. 62–74. Springer, Heidelberg (1996)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2007

Authors and Affiliations

  • Laura Bozzelli
    • 1
  1. 1.Università di Napoli Federico II , Via Cintia, 80126 - NapoliItaly

Personalised recommendations