Decidability of MSO Theories of Tree Structures

  • Angelo Montanari
  • Gabriele Puppis
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 3328)


In this paper we provide an automaton-based solution to the decision problem for a large set of monadic second-order theories of deterministic tree structures. We achieve it in two steps: first, we reduce the considered problem to the problem of determining, for any Rabin tree automaton, whether it accepts a given tree; then, we exploit a suitable notion of tree equivalence to reduce (a number of instances of ) the latter problem to the decidable case of regular trees. We prove that such a reduction works for a large class of trees, that we call residually regular trees. We conclude the paper with a short discussion of related work.


Tree Structure Graph Structure Regular Tree Tree Automaton Complete Binary 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.


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  1. 1.
    Barthelmann, K.: On equational simple graphs. Technical Report 9, Universität Mainz, Institut für Informatik (1997)Google Scholar
  2. 2.
    Büchi, J.R.: On a decision method in restricted second order arithmetic. In: Proceedings of the International Congress on Logic, Methodology and Philosophy of Science, pp. 1–11. Stanford University Press, Stanford (1960)Google Scholar
  3. 3.
    Blumensath, A.: Prefix-recognizable graphs and monadic second-order logic. Technical Report AIB-06-2001, RWTH Aachen (2001)Google Scholar
  4. 4.
    Blumensath, A., Gradel, E.: Automatic structures. Logic in Computer Science, 51–62 (2000)Google Scholar
  5. 5.
    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
  6. 6.
    Carayol, A., Wöhrle, S.: The caucal hierarchy of infinite graphs in terms of logic and higher-order pushdown automata. In: Pandya, P.K., Radhakrishnan, J. (eds.) FSTTCS 2003. LNCS, vol. 2914, pp. 112–123. Springer, Heidelberg (2003)CrossRefGoogle Scholar
  7. 7.
    Carton, O., Thomas, W.: The monadic theory of morphic infinite words and generalizations. Information and Computation 176, 51–65 (2002)zbMATHCrossRefMathSciNetGoogle Scholar
  8. 8.
    Caucal, D.: On the regular structure of prefix rewriting. Theoretical Computer Science 106, 61–86 (1992)CrossRefMathSciNetGoogle Scholar
  9. 9.
    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
  10. 10.
    Caucal, D.: On infinite transition graphs having a decidable monadic theory. Theoretical Computer Science 290, 79–115 (2003)zbMATHCrossRefMathSciNetGoogle Scholar
  11. 11.
    Courcelle, B.: The monadic second-order logic of graphs II: Infinite graphs of bounded tree width. Mathematical Systems Theory 21, 187–221 (1989)zbMATHCrossRefMathSciNetGoogle Scholar
  12. 12.
    Courcelle, B.: Graph rewriting: An algebraic and logic approach. In: Van Leeuwen, J. (ed.) Handbook of Theoretical Computer Science, pp. 193–242. Elsevier, Amsterdam (1990)Google Scholar
  13. 13.
    Courcelle, B.: Monadic second-order graph transductions: a survey. Theoretical Computer Science 126, 53–75 (1994)zbMATHCrossRefMathSciNetGoogle Scholar
  14. 14.
    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)zbMATHCrossRefMathSciNetGoogle Scholar
  15. 15.
    Elgot, C.C., Rabin, M.O.: Decidability and undecidability of extensions of second (first) order theory of (generalized) successor. Journal of Symbolic Logic 31(2), 169–181 (1966)zbMATHCrossRefGoogle Scholar
  16. 16.
    Montanari, A., Peron, A., Policriti, A.: Extending Kamp’s theorem to model time granularity. Journal of Logic and Computation 12(4), 641–678 (2002)zbMATHCrossRefMathSciNetGoogle Scholar
  17. 17.
    Montanari, A., Puppis, G.: Decidability of MSO theories of tree structures. Research Report 01, Dipartimento di Matematica e Informatica, Universita di Udine, Italy (2004)Google Scholar
  18. 18.
    Montanari, A., Puppis, G.: Decidability of the theory of the totally unbounded ω-layered structure. In: Proceedings of the 11th International Symposium on Temporal Representation and Reasoning TIME, pp. 156–160 (2004)Google Scholar
  19. 19.
    Muller, D., Schupp, P.: The theory of ends, pushdown automata, and second-order logics. Theoretical Computer Science 37, 51–75 (1985)zbMATHCrossRefMathSciNetGoogle Scholar
  20. 20.
    Rabin, M.O.: Decidability of second-order theories and automata on infinite trees. Transactions of the American Mathematical Society 141, 1–35 (1969)zbMATHCrossRefMathSciNetGoogle Scholar
  21. 21.
    Thomas, W.: Languages, automata, and logic. In: Rozemberg, G., Salomaa, A. (eds.) Handbook of Formal Languages, vol. 3, pp. 389–455. Springer, Heidelberg (1997)Google Scholar
  22. 22.
    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
  23. 23.
    Walukiewicz, I.: Monadic second-order logic on tree-like structures. Theoretical Computer Science 275, 311–346 (2002)zbMATHCrossRefMathSciNetGoogle Scholar
  24. 24.
    Zhang, G.: Automata, boolean matrices, and ultimate periodicity. Information and Computation 152(1), 138–154 (1999)zbMATHCrossRefMathSciNetGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2004

Authors and Affiliations

  • Angelo Montanari
    • 1
  • Gabriele Puppis
    • 1
  1. 1.Dipartimento di Matematica e InformaticaUniversitè di UdineUdineItaly

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