Advertisement

Languages of Nested Trees

  • Rajeev Alur
  • Swarat Chaudhuri
  • P. Madhusudan
Part of the Lecture Notes in Computer Science book series (LNCS, volume 4144)

Abstract

We study languages of nested trees—structures obtained by augmenting trees with sets of nested jump-edges. These graphs can naturally model branching behaviors of pushdown programs, so that the problem of branching-time software model checking may be phrased as a membership question for such languages. We define finite-state automata accepting such languages—these automata can pass states along jump-edges as well as tree edges. We find that the model-checking problem for these automata on pushdown systems is EXPTIME-complete, and that their alternating versions are expressively equivalent to NT-μ, a recently proposed temporal logic for nested trees that can express a variety of branching-time, “context-free” requirements. We also show that monadic second order logic (MSO) cannot exploit the structure: MSO on nested trees is too strong in the sense that it has an undecidable model checking problem, and seems too weak to capture NT-μ.

Keywords

Model Check Winning Strategy Tree Automaton Tree Edge Nest Tree 
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.

References

  1. 1.
    Alur, R., Chaudhuri, S., Madhusudan, P.: A fixpoint calculus for local and global program flows. In: Proc. of POPL 2006, pp. 153–165 (2006)Google Scholar
  2. 2.
    Alur, R., Chaudhuri, S., Madhusudan, P.: Languages of nested trees. University of Pennsylvania Technical Report MS-CIS-06-10 (2006)Google Scholar
  3. 3.
    Alur, R., Madhusudan, P.: Visibly pushdown languages. In: Proc. of STOC 2004, pp. 202–211 (2004)Google Scholar
  4. 4.
    Alur, R., Madhusudan, P.: Adding nesting structure to words. In: H. Ibarra, O., Dang, Z. (eds.) DLT 2006. LNCS, vol. 4036, pp. 1–13. Springer, Heidelberg (2006)CrossRefGoogle Scholar
  5. 5.
    Ball, T., Rajamani, S.: The SLAM project: debugging system software via static analysis. In: Proc. of POPL 2002, pp. 1–3 (2002)Google Scholar
  6. 6.
    Burkart, O., Steffen, B.: Model checking the full modal mu-calculus for infinite sequential processes. Theoretical Computer Science 221, 251–270 (1999)MATHCrossRefMathSciNetGoogle Scholar
  7. 7.
    Caucal, D.: On infinite transition graphs having a decidable monadic theory. Theor. Comput. Sci. 290(1), 79–115 (2003)MATHCrossRefMathSciNetGoogle Scholar
  8. 8.
    Comon, H., Dauchet, M., Gilleron, R., Lugiez, D., Tison, S., Tommasi, M.: Tree automata techniques and applications. Draft (2003)Google Scholar
  9. 9.
    Emerson, E.A., Jutla, C.S.: Tree automata, mu-calculus, and determinacy. In: Proc. of FOCS 1991, pp. 368–377 (1991)Google Scholar
  10. 10.
    Grädel, E., Thomas, W., Wilke, T. (eds.): Automata, Logics, and Infinite Games. LNCS, vol. 2500. Springer, Heidelberg (2002)MATHGoogle Scholar
  11. 11.
    Henzinger, T.A., Jhala, R., Majumdar, R., Necula, G.C., Sutre, G., Weimer, W.: Temporal-safety proofs for systems code. In: Brinksma, E., Larsen, K.G. (eds.) CAV 2002. LNCS, vol. 2404, pp. 526–538. Springer, Heidelberg (2002)CrossRefGoogle Scholar
  12. 12.
    Hopcroft, J.E., Ullman, J.D.: Introduction to automata theory, languages, and computation. Addison-Wesley, Reading (1979)MATHGoogle Scholar
  13. 13.
    Janin, D., Walukiewicz, I.: On the expressive completeness of the propositional mu-calculus with respect to monadic second order logic. In: Sassone, V., Montanari, U. (eds.) CONCUR 1996. LNCS, vol. 1119, pp. 263–277. Springer, Heidelberg (1996)Google Scholar
  14. 14.
    Kozen, D.: Results on the propositional mu-calculus. Theoretical Computer Science 27, 333–354 (1983)MATHCrossRefMathSciNetGoogle Scholar
  15. 15.
    Kupferman, O., Piterman, N., Vardi, M.Y.: Pushdown specifications. In: Baaz, M., Voronkov, A. (eds.) LPAR 2002. LNCS (LNAI), vol. 2514, pp. 262–277. Springer, Heidelberg (2002)CrossRefGoogle Scholar
  16. 16.
    Löding, C.: Private communicationGoogle Scholar
  17. 17.
    Löding, 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)CrossRefGoogle Scholar
  18. 18.
    McMillan, K.L.: Symbolic model checking: an approach to the state explosion problem. Kluwer Academic Publishers, Dordrecht (1993)Google Scholar
  19. 19.
    Muller, D.E., Schupp, P.E.: Alternating automata on infinite trees. Theor. Comput. Sci. 54(2-3), 267–276 (1987)MATHCrossRefMathSciNetGoogle Scholar
  20. 20.
    Muller, D.E., Schupp, P.E.: The theory of ends, pushdown automata, and second-order logic. Theoretical Computer Science 37, 51–75 (1985)MATHCrossRefMathSciNetGoogle Scholar
  21. 21.
    Rabin, M.O.: Decidability of second order theories and automata on infinite trees. Transactions of the AMS 141, 1–35 (1969)MATHMathSciNetGoogle Scholar
  22. 22.
    Reps, T., Horwitz, S., Sagiv, S.: Precise interprocedural dataflow analysis via graph reachability. In: Proc. of POPL 1995, pp. 49–61 (1995)Google Scholar
  23. 23.
    Walukiewicz, I.: Pushdown processes: Games and model-checking. Information and Computation 164(2), 234–263 (2001)MATHCrossRefMathSciNetGoogle Scholar
  24. 24.
    Walukiewicz, I.: Monadic second-order logic on tree-like structures. Theor. Comput. Sci. 275(1-2), 311–346 (2002)MATHCrossRefMathSciNetGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2006

Authors and Affiliations

  • Rajeev Alur
    • 1
  • Swarat Chaudhuri
    • 1
  • P. Madhusudan
    • 2
  1. 1.University of PennsylvaniaUSA
  2. 2.University of Illinois at Urbana-ChampaignUSA

Personalised recommendations