The Order Type of Scattered Context-Free Orderings of Rank One Is Computable

  • Kitti Gelle
  • Szabolcs IvánEmail author
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 12011)


A linear ordering is called context-free if it is the lexicographic ordering of some context-free language and is called scattered if it has no dense subordering. Each scattered ordering has an associated ordinal, called its rank. It is known that the isomorphism problem of context-free orderings is undecidable in general. In this paper we show that it is decidable whether a context-free ordering is scattered with rank at most one, and if so, its order type is effectively computable.



Ministry of Human Capacities, Hungary grant 20391-3/2018/FEKUSTRAT is acknowledged. Szabolcs Iván was supported by the János Bolyai Scholarship of the Hungarian Academy of Sciences. Kitti Gelle was supported by the ÚNKP-19-3-SZTE-86 New National Excellence Program of the Ministry of Human Capacities.


  1. 1.
    Bloom, S.L., Ésik, Z.: Algebraic ordinals. Fundam. Inform. 99(4), 383–407 (2010)MathSciNetCrossRefGoogle Scholar
  2. 2.
    Bloom, S.L., Ésik, Z.: The equational theory of regular words. Inf. Comput. 197(1), 55–89 (2005)MathSciNetCrossRefGoogle Scholar
  3. 3.
    Ésik, Z.: Scattered context-free linear orderings. In: Mauri, G., Leporati, A. (eds.) DLT 2011. LNCS, vol. 6795, pp. 216–227. Springer, Heidelberg (2011). Scholar
  4. 4.
    Ésik, Z.: An undecidable property of context-free linear orders. Inf. Process. Lett. 111(3), 107–109 (2011)CrossRefGoogle Scholar
  5. 5.
    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). Scholar
  6. 6.
    Gelle, K., Iván, S.: On the order type of scattered context-free orderings. In: The Tenth International Symposium on Games, Automata, Logics, and Formal Verification, September 2–3, 2019, pp. 169–182 (2019)CrossRefGoogle Scholar
  7. 7.
    Gelle, K., Iván, S.: The ordinal generated by an ordinal grammar is computable. Theoret. Comput. Sci. 793, 1–13 (2019)MathSciNetCrossRefGoogle Scholar
  8. 8.
    Heilbrunner, S.: An algorithm for the solution of fixed-point equations for infinite words. RAIRO - Theoret. Inf. Appl. Informatique Théorique et Applications 14(2), 131–141 (1980)MathSciNetCrossRefGoogle Scholar
  9. 9.
    Hopcroft, J.E., Ullman, J.D.: Introduction to Automata Theory, Languages, and Computation. Addison-Wesley Publishing Company, Reading (1979)zbMATHGoogle Scholar
  10. 10.
    Lohrey, M., Mathissen, C.: Isomorphism of regular trees and words. Inf. Comput. 224, 71–105 (2013)MathSciNetCrossRefGoogle Scholar
  11. 11.
    Rosenstein, J.: Linear Orderings. Pure and Applied Mathematics. Elsevier Science, Amsterdam (1982)zbMATHGoogle Scholar
  12. 12.
    Stark, J.A.: Ordinal arithmetic (2015).
  13. 13.
    Thomas, W.: On frontiers of regular trees. ITA 20(4), 371–381 (1986)MathSciNetzbMATHGoogle Scholar

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Authors and Affiliations

  1. 1.Department of Computer ScienceUniversity of SzegedSzegedHungary

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