A Context-Free Linear Ordering with an Undecidable First-Order Theory

  • Arnaud Carayol
  • Zoltán Ésik
Part of the Lecture Notes in Computer Science book series (LNCS, volume 7604)


The words of a context-free language, ordered by the lexicographic ordering, form a context-free linear ordering. It is well-known that the linear orderings associated with deterministic context-free languages have a decidable monadic second-order theory. In stark contrast, we give an example of a context-free language whose lexicographic ordering has an undecidable first-order theory.


Disjoint Union Linear Ordering Regular Language Order Type Countable Word 
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.


  1. 1.
    Bloom, S.L., Choffrut, C.: Long words: the theory of concatenation and omega-power. Theoretical Computer Science 259, 533–548 (2001)MathSciNetzbMATHCrossRefGoogle Scholar
  2. 2.
    Bloom, S.L., Ésik, Z.: Deciding whether the frontier of a regular tree is scattered. Fundamenta Informaticae 55, 1–21 (2003)MathSciNetzbMATHGoogle Scholar
  3. 3.
    Bloom, S.L., Ésik, Z.: The equational theory of regular words. Information and Computation 197, 55–89 (2005)MathSciNetzbMATHCrossRefGoogle Scholar
  4. 4.
    Bloom, S.L., Ésik, Z.: Regular and Algebraic Words and Ordinals. In: Mossakowski, T., Montanari, U., Haveraaen, M. (eds.) CALCO 2007. LNCS, vol. 4624, pp. 1–15. Springer, Heidelberg (2007)CrossRefGoogle Scholar
  5. 5.
    Bloom, S.L., Ésik, Z.: Algebraic ordinals. Fundamenta Informaticae 99, 383–407 (2010)MathSciNetzbMATHGoogle Scholar
  6. 6.
    Bloom, S.L., Ésik, Z.: Algebraic linear orderings. Int. J. Foundations of Computer Science 22, 491–515 (2011)zbMATHCrossRefGoogle Scholar
  7. 7.
    Braud, L., Carayol, A.: Linear Orders in the Pushdown Hierarchy. In: Abramsky, S., Gavoille, C., Kirchner, C., Meyer auf der Heide, F., Spirakis, P.G. (eds.) ICALP 2010. LNCS, vol. 6199, pp. 88–99. Springer, Heidelberg (2010)CrossRefGoogle Scholar
  8. 8.
    Carayol, A., Ésik, Z.: The FC-rank of a context-free language (to appear)Google Scholar
  9. 9.
    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
  10. 10.
    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
  11. 11.
    Caucal, D.: On infinite transition graphs having a decidable monadic theory. Theoretical Computer Science 290, 79–115 (2003)MathSciNetzbMATHCrossRefGoogle Scholar
  12. 12.
    Courcelle, B.: Frontiers of infinite trees. Theoretical Informatics and Applications 12, 319–337 (1978)MathSciNetzbMATHGoogle Scholar
  13. 13.
    Courcelle, B.: Fundamental properties of infinite trees. Theoretical Computer Science 25, 95–169 (1983)MathSciNetzbMATHCrossRefGoogle Scholar
  14. 14.
    Delhommé, C.: Automaticité des ordinaux et des graphes homogènes. C. R. Acad. Sci. Paris, Ser. I 339, 5–10 (2004)zbMATHCrossRefGoogle Scholar
  15. 15.
    Ésik, Z.: Representing small ordinals by finite automata. In: 12th Workshop Descriptional Complexity of Formal Systems, Saskatoon, Canada. EPTCS, vol. 31, pp. 78–87 (2010)Google Scholar
  16. 16.
    Ésik, Z.: An undecidable property of context-free linear orders. Information Processing Letters 111, 107–109 (2010)CrossRefGoogle Scholar
  17. 17.
    Ésik, Z.: Scattered Context-Free Linear Orderings. In: Mauri, G., Leporati, A. (eds.) DLT 2011. LNCS, vol. 6795, pp. 216–227. Springer, Heidelberg (2011)CrossRefGoogle Scholar
  18. 18.
    Ésik, Z., Iván, S.: Hausdorff Rank of Scattered Context-Free Linear Orders. In: Fernández-Baca, D. (ed.) LATIN 2012. LNCS, vol. 7256, pp. 291–302. Springer, Heidelberg (2012)CrossRefGoogle Scholar
  19. 19.
    Hague, M., Murawski, A., Ong, C.-H.L., Serre, O.: Collapsible Pushdown Automata and Recursion Schemes. In: LICS 2008, pp. 452–461. IEEE (2008)Google Scholar
  20. 20.
    Heilbrunner, S.: An algorithm for the solution of fixed-point equations for infinite words. Theoretical Informatics and Applications 14, 131–141 (1980)MathSciNetzbMATHGoogle Scholar
  21. 21.
    Hopcroft, J.E., Ullman, J.D.: Introduction to Automata Theory, Languages and Computation. Addison-Wesley (1979)Google Scholar
  22. 22.
    Khoussainov, B., Rubin, S., Stephan, F.: Automatic linear orders and trees. ACM Trans. Comput. Log. 6, 625–700 (2005)MathSciNetCrossRefGoogle Scholar
  23. 23.
    Lohrey, M.: Private communication (2012)Google Scholar
  24. 24.
    Lohrey, M., Mathissen, C.: Isomorphism of Regular Trees and Words. In: Aceto, L., Henzinger, M., Sgall, J. (eds.) ICALP 2011, Part II. LNCS, vol. 6756, pp. 210–221. Springer, Heidelberg (2011)CrossRefGoogle Scholar
  25. 25.
    Luke Ong, C.-H.: On model-checking trees generated by higher-order recursion schemes. In: LICS 2006, pp. 81–90. IEEE Press (2006)Google Scholar
  26. 26.
    Parys, P.: Higher-order stacks can not replace nondeterminism. Note published on the authors webpage (3 p.) (February 2010)Google Scholar
  27. 27.
    Rosenstein, J.G.: Linear Orderings. Pure and Applied Mathematics, vol. 98. Academic Press (1982)Google Scholar
  28. 28.
    Thomas, W.: On frontiers of regular trees. Theoretical Informatics and Applications 20, 371–381 (1986)zbMATHGoogle Scholar

Copyright information

© IFIP International Federation for Information Processing 2012

Authors and Affiliations

  • Arnaud Carayol
    • 1
  • Zoltán Ésik
    • 2
  1. 1.Laboratoire d’Informatique Gaspard-MongeUniversité Paris-EstFrance
  2. 2.Institute of InformaticsUniversity of SzegedHungary

Personalised recommendations