Some Non-semi-decidability Problems for Linear and Deterministic Context-Free Languages

  • Henning Bordihn
  • Markus Holzer
  • Martin Kutrib
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 3317)


We investigate the operation problem for linear and deterministic context-free languages: Fix an operation on formal languages. Given linear (deterministic, respectively) context-free languages, is the application of this operation to the given languages still a linear (deterministic, respectively) context-free language? Besides the classical operations, for which the linear and deterministic context-free languages are not closed, we also consider the recently introduced root and power operation. We show non-semi-decidability for all of the aforementioned operations, if the underlying alphabet contains at least two letters. The non-semi-decidability and thus the undecidability for the power operation solves an open problem stated in [4].


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  1. 1.
    Bader, C., Moura, A.: A generalization of Ogden’s lemma. Journal of the ACM 29, 404–407 (1982)zbMATHCrossRefMathSciNetGoogle Scholar
  2. 2.
    Baker, B.S., Book, R.V.: Reversal-bounded multipushdown machines. Journal of Computer and System Sciences 8, 315–332 (1974)zbMATHMathSciNetCrossRefGoogle Scholar
  3. 3.
    Bar-Hillel, Y., Perles, M., Shamir, E.: On formal properties of simple phrase structure grammars. Zeitschrift für Phonetik, Sprachwissenschaft und Kommunikationsforschung 14, 143–177 (1961)zbMATHMathSciNetGoogle Scholar
  4. 4.
    Bordihn, H.: Context-freeness of the power of context-free languages is undecidable. Theoretical Computer Science (2003) (to appear)Google Scholar
  5. 5.
    Cachat, T.: The power of one-letter rational languages. In: Kuich, W., Rozenberg, G., Salomaa, A. (eds.) DLT 2001. LNCS, vol. 2295, pp. 145–154. Springer, Heidelberg (2002)CrossRefGoogle Scholar
  6. 6.
    Calbrix, H., Nivat, M.: Prefix and period languages and rational ω-languages. In: Developments in Language Theory II. At the Crossroads of Mathematics, Computer Science and Biology, pp. 341–349. World Scientific, Singapore (1996)Google Scholar
  7. 7.
    Ginsburg, S., Rose, G.F.: Some recursively unsolvable problem in ALGOL-like languages. Journal of the ACM 10, 29–47 (1963)zbMATHCrossRefMathSciNetGoogle Scholar
  8. 8.
    Ginsburg, S., Spanier, E.H.: Bounded ALGOL-like languages. Transactions of the American Matematical Society 113, 333–368 (1964)zbMATHCrossRefMathSciNetGoogle Scholar
  9. 9.
    Greibach, S.A.: The undecidability of ambiguity problem for minimal linear grammars. Information and Control 6, 117–125 (1963)CrossRefMathSciNetGoogle Scholar
  10. 10.
    Greibach, S.A.: The unsolvability of the recognition of linear context-free languages. Journal of the ACM 13, 582–587 (1966)zbMATHCrossRefMathSciNetGoogle Scholar
  11. 11.
    Gross, M.: Inherent ambiguity of minimal linear grammars. Information and Control 7(3), 366–368 (1963)CrossRefGoogle Scholar
  12. 12.
    Hartmanis, J.: Context-free languages and Turing machine computations. In: Proceedings of Symposia in Applied Mathematics, vol. 19. American Mathematical Society, Providence (1967)Google Scholar
  13. 13.
    Hopcroft, J.E., Ullman, J.D.: Introduction to Automata Theory, Languages and Computation. Addison-Wesley, Reading (1979)zbMATHGoogle Scholar
  14. 14.
    Horváth, S., Leupold, P., Lischke, G.: Roots and powers of regular languages. In: Ito, M., Toyama, M. (eds.) DLT 2002. LNCS, vol. 2450, pp. 220–230. Springer, Heidelberg (2003)CrossRefGoogle Scholar
  15. 15.
    McNaughton, R.: Parenthesis grammars. Journal of the ACM 14, 490–500 (1967)zbMATHCrossRefMathSciNetGoogle Scholar
  16. 16.
    Salomaa, A.: Formal Languages. ACM Monograph Series. Academic Press, London (1973)zbMATHGoogle Scholar
  17. 17.
    Sénizergues, G.: L(A)=L(B)? Decidability results from complete formal systems. Theoretical Computer Science 251, 1–166 (2003)CrossRefGoogle Scholar
  18. 18.
    Wood, D.: Theory of Computation. John Wiley & Sons, Chichester (1987)zbMATHGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2005

Authors and Affiliations

  • Henning Bordihn
    • 1
  • Markus Holzer
    • 2
  • Martin Kutrib
    • 3
  1. 1.Institut für InformatikUniversität PotsdamPotsdamGermany
  2. 2.Institut für InformatikTechnische Universität MünchenGarching bei MünchenGermany
  3. 3.Institut für InformatikUniversität GiessenGiessenGermany

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