Higher-Order Pushdown Trees Are Easy

  • Teodor Knapik
  • Damian Niwiński
  • Paweł Urzyczyn
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 2303)

Abstract

We show that the monadicsec ond-order theory of an infinite tree recognized by a higher-order pushdown automaton of any level is decidable. We also show that trees recognized by pushdown automata of level n coincide with trees generated by safe higher-order grammars of level n. Our decidability result extends the result of Courcelle on algebraic(pushdo wn of level 1) trees and our own result on trees of level 2.

References

  1. 1.
    D. Caucal. On infinite transition graphs having a decidable monadic second-order theory. In F. Meyer auf der Heide and B. Monien, editors, 23th International Colloquium on Automata Languages and Programming, LNCS 1099, pages 194–205, 1996. A long version will appear in TCS.Google Scholar
  2. 2.
    B. Courcelle. The monadic second-order theory of graphs IX: Machines and their behaviours. Theoretical Comput. Sci., 151:125–162, 1995.MATHCrossRefMathSciNetGoogle Scholar
  3. 3.
    B. Courcelle and T. Knapik. The evaluation of first-order substitution is monadic second-order compatible. Theoretical Comput. Sci., 2002. To appear.Google Scholar
  4. 4.
    W. Damm. The IO-and OI-hierarchies. Theoretical Comput. Sci., 20(2):95–208, 1982.MATHCrossRefMathSciNetGoogle Scholar
  5. 5.
    W. Damm and A. Goerdt. An automata-theoreticc haracterization of the OIhierarchy. Information and Control, 71:1–32, 1986.MATHCrossRefMathSciNetGoogle Scholar
  6. 6.
    J. Engelfriet. Iterated push-down automata and complexity classes. In Proc. 15th STOC, pages 365–373, 1983.Google Scholar
  7. 7.
    H. Hungar. Model checking and higher-order recursion. In L. Pacholski, M. Kutyłowski and T. Wierzbicki, editors, Mathematical Foundations of Computer Science 1999, LNCS 1672, pages 149–159, 1999.CrossRefGoogle Scholar
  8. 8.
    B. Jacobs and J. Rutten. A tutorial on (co)algebras and (co)induction. Bulletin of EATCS, 1997(62):222–259.Google Scholar
  9. 9.
    A.J. Kfoury, J. Tiuryn and P. Urzyczyn. On the expressive power of finitely typed and universally polymorphic recursive procedures. Theoretical Comput. Sci., 93:1–41, 1992.MATHCrossRefMathSciNetGoogle Scholar
  10. 10.
    A. Kfoury and P. Urzyczyn. Finitely typed functional programs, part II: comparisons to imperative languages. Report, Boston University, 1988.Google Scholar
  11. 11.
    T. Knapik, D. Niwiński, and P. Urzyczyn. Deciding monadic theories of hyperalgebraictrees. In Typed Lambda Calculi and Applications, 5th International Conference, LNCS 2044, pages 253–267. Springer-Verlag, 2001.CrossRefGoogle Scholar
  12. 12.
    W. Kowalczyk, D. Niwiński, and J. Tiuryn. A generalization of of Cook’s auxiliarypushdown-automata theorem. Fundamenta Informaticae, 12:497–506, 1989.MATHMathSciNetGoogle Scholar
  13. 13.
    O. Kupferman and M. Vardi. An automata-theoreticapproach to reasoning about infinite-state systems. In Computer Aided Verification, Proc. 12th Int. Conference, Lecture Notes in Computer Science. Springer-Verlag, 2000.Google Scholar
  14. 14.
    D. Muller and P. Schupp. The theory of ends, pushdown automata, and secondorder logic. Theoretical Comput. Sci., 37:51–75, 1985.MATHCrossRefMathSciNetGoogle Scholar
  15. 15.
    M. O. Rabin. Decidability of second-order theories and automata on infinite trees. Trans. Amer. Soc, 141:1–35, 1969.MATHCrossRefMathSciNetGoogle Scholar
  16. 16.
    W. Thomas. Languages, automata, and logic. In G. Rozenberg and A. Salomaa, editors, Handbook of Formal Languages, volume 3, pages 389–455. Springer-Verlag, 1997.Google Scholar
  17. 17.
    J. Tiuryn. Higher-order arrays and stacks in programming: An application of complexity theory to logics of programs. In Proc. 12th MFCS, LNCS 233, pages 177–198. Springer-Verlag, 1986.Google Scholar
  18. 18.
    I. Walukiewicz. Pushdown processes: Games and model checking. Information and Computation, 164(2):234–263, 2001.MATHCrossRefMathSciNetGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2002

Authors and Affiliations

  • Teodor Knapik
    • 1
  • Damian Niwiński
    • 2
  • Paweł Urzyczyn
    • 2
  1. 1.Université de la RéunionSaint Denis Messageries Cedex 9, Réunion
  2. 2.Institute of InformaticsWarsaw UniversityWarszawaPoland

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