The Caucal Hierarchy of Infinite Graphs in Terms of Logic and Higher-Order Pushdown Automata

  • Arnaud Carayol
  • Stefan Wöhrle
Part of the Lecture Notes in Computer Science book series (LNCS, volume 2914)

Abstract

In this paper we give two equivalent characterizations of the Caucal hierarchy, a hierarchy of infinite graphs with a decidable monadic second-order (MSO) theory. It is obtained by iterating the graph transformations of unfolding and inverse rational mapping. The first characterization sticks to this hierarchical approach, replacing the language-theoretic operation of a rational mapping by an MSO-transduction and the unfolding by the treegraph operation. The second characterization is non-iterative. We show that the family of graphs of the Caucal hierarchy coincides with the family of graphs obtained as the ε-closure of configuration graphs of higher-order pushdown automata.

While the different characterizations of the graph family show their robustness and thus also their importance, the characterization in terms of higher-order pushdown automata additionally yields that the graph hierarchy is indeed strict.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Blumensath, A.: Prefix-recognisable graphs and monadic second-order logic. Technical Report AIB-2001-06, RWTH Aachen (2001)Google Scholar
  2. 2.
    Cachat, T.: Higher order pushdown automata, the Caucal hierarchy of graphs and parity games. In: Baeten, J.C.M., Lenstra, J.K., Parrow, J., Woeginger, G.J. (eds.) ICALP 2003. LNCS, vol. 2719, pp. 556–569. Springer, Heidelberg (2003)CrossRefGoogle Scholar
  3. 3.
    Carayol, A., Colcombet, T.: On equivalent representations of infinite structures. In: Baeten, J.C.M., Lenstra, J.K., Parrow, J., Woeginger, G.J. (eds.) ICALP 2003. LNCS, vol. 2719, pp. 599–610. Springer, Heidelberg (2003)CrossRefGoogle Scholar
  4. 4.
    Carayol, A., Wöhrle, S.: The Caucal hierarchy of infinite graphs in terms of logic and higher-order pushdown automata. Technical report, RWTH Aachen (2003)Google Scholar
  5. 5.
    Carton, O., Thomas, W.: The monadic theory of morphic infinite words and generalizations. In: Nielsen, M., Rovan, B. (eds.) MFCS 2000. LNCS, vol. 1893, pp. 275–284. Springer, Heidelberg (2000)CrossRefGoogle Scholar
  6. 6.
    Caucal, D.: On infinite transition graphs having a decidable monadic theory. In: Meyer auf der Heide, F., Monien, B. (eds.) ICALP 1996. LNCS, vol. 1099, pp. 194–205. Springer, Heidelberg (1996)Google Scholar
  7. 7.
    Caucal, D.: On infinite terms having a decidable monadic theory. In: Diks, K., Rytter, W. (eds.) MFCS 2002. LNCS, vol. 2420, pp. 165–176. Springer, Heidelberg (2002)CrossRefGoogle Scholar
  8. 8.
    Courcelle, B.: Monadic second-order definable graph transductions: A survey. Theoretical Computer Science 126, 53–75 (1994)MATHCrossRefMathSciNetGoogle Scholar
  9. 9.
    Courcelle, B., Walukiewicz, I.: Monadic second-order logic, graph coverings and unfoldings of transition systems. Annals of Pure and Applied Logic 92, 35–62 (1998)MATHCrossRefMathSciNetGoogle Scholar
  10. 10.
    Damm, W.: An algebraic extension of the Chomsky-hierarchy. In: Proceedings of the 8th International Symposium on Mathematical Foundations of Computer Science. LNCS, vol. 74, pp. 266–276. Springer, Heidelberg (1979)Google Scholar
  11. 11.
    Damm, W.: The IO and OI hierarchies. Theoretical Computer Science 20, 95–208 (1982)MATHCrossRefMathSciNetGoogle Scholar
  12. 12.
    Damm, W., Goerdt, A.: An automata-theoretical characterization of the OI hierarchy. Information and Control 71, 1–32 (1986)MATHCrossRefMathSciNetGoogle Scholar
  13. 13.
    Ebbinghaus, H.D., Flum, J.: Finite Model Theory. Springer, Heidelberg (1995)MATHGoogle Scholar
  14. 14.
    Engelfriet, J.: Iterated stack automata and complexity classes. Information and Computation 95, 21–75 (1991)CrossRefMathSciNetGoogle Scholar
  15. 15.
    Knapik, T., Niwiński, D., Urzyczyn, P.: Higher-order pusdown trees are easy. In: Nielsen, M., Engberg, U. (eds.) FOSSACS 2002. T. Knapik, D. Niwiński, and P. Urzyczyn, vol. 2303, pp. 205–222. Springer, Heidelberg (2002)CrossRefGoogle Scholar
  16. 16.
    Stirling, C.: Decidability of bisimulation equivalence for pushdown processes (submitted)Google Scholar
  17. 17.
    Thomas, W.: Constructing infinite graphs with a decidable MSO-theory. In: Rovan, B., Vojtáš, P. (eds.) MFCS 2003. LNCS, vol. 2747, pp. 113–124. Springer, Heidelberg (2003)CrossRefGoogle Scholar
  18. 18.
    Walukiewicz, I.: Monadic second-order logic on tree-like structures. Theoretical Computer Science 275, 311–346 (2002)MATHCrossRefMathSciNetGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2003

Authors and Affiliations

  • Arnaud Carayol
    • 1
  • Stefan Wöhrle
    • 2
  1. 1.IRISA RennesFrance
  2. 2.Lehrstuhl für Informatik 7, RWTH AachenGermany

Personalised recommendations