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)


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.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  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