On Finite-Index Indexed Grammars and Their Restrictions

  • Flavio D’AlessandroEmail author
  • Oscar H. Ibarra
  • Ian McQuillan
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 10168)


The family, \({\mathsf{{\mathcal {L}}(IND_{LIN})}}\), of languages generated by linear indexed grammars has been studied in the literature. It is known that the Parikh image of every language in \({\mathsf{{\mathcal {L}}(IND_{LIN})}}\) is semi-linear. However, there are bounded semi-linear languages that are not in \({\mathsf{{\mathcal {L}}(IND_{LIN})}}\). Here, we look at larger families of (restricted) indexed languages and study their properties, their relationships, and their decidability properties.


Indexed languages Finite-index Full trios Semi-linearity Bounded languages ET0L languages 


  1. 1.
    Aho, A.V.: Indexed grammars—an extension of context-free grammars. J. ACM 15, 647–671 (1968)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Aho, A.V.: Nested stack automata. J. ACM 16, 383–406 (1969)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Berstel, J.: Transductions and Context-Free Languages. B.B. Teubner, Stuttgart (1979)CrossRefzbMATHGoogle Scholar
  4. 4.
    Dassow, J., Pǎun, G.: Regulated Rewriting in Formal Language Theory. EATCS Monographs on Theoretical Computer Science, vol. 18. Springer, Berlin (1989)CrossRefzbMATHGoogle Scholar
  5. 5.
    Duske, J., Parchmann, R.: Linear indexed languages. Theoret. Comput. Sci. 32(1–2), 47–60 (1984)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Gazdar, G.: Applicability of indexed grammars to natural languages. In: Reyle, U., Rohrer, C. (eds.) Natural Language Parsing and Linguistic Theories, vol. 35, pp. 69–94. Springer, Dordrecht (1988)CrossRefGoogle Scholar
  7. 7.
    Ginsburg, S.: The Mathematical Theory of Context-Free Languages. McGraw-Hill, Inc., New York (1966)zbMATHGoogle Scholar
  8. 8.
    Harrison, M.A.: Introduction to Formal Language Theory. Addison-Wesley Series in Computer Science. Addison-Wesley Pub. Co., Reading (1978)zbMATHGoogle Scholar
  9. 9.
    Hopcroft, J.E., Ullman, J.D.: Introduction to Automata Theory, Languages, and Computation. Addison-Wesley, Reading (1979)zbMATHGoogle Scholar
  10. 10.
    Ibarra, O.H.: Reversal-bounded multicounter machines and their decision problems. J. ACM 25(1), 116–133 (1978)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Ibarra, O.H., McQuillan, I.: On bounded semilinear languages, counter machines, and finite-index ET0L. In: Han, Y.-S., Salomaa, K. (eds.) CIAA 2016. LNCS, vol. 9705, pp. 138–149. Springer, Heidelberg (2016). doi: 10.1007/978-3-319-40946-7_12 CrossRefGoogle Scholar
  12. 12.
    Ibarra, O.H., McQuillan, I.: On families of full trios containing counter machine languages. In: Brlek, S., Reutenauer, C. (eds.) DLT 2016. LNCS, vol. 9840, pp. 216–228. Springer, Heidelberg (2016). doi: 10.1007/978-3-662-53132-7_18 CrossRefGoogle Scholar
  13. 13.
    Rozenberg, G., Salomaa, A.: The Mathematical Theory of L Systems. Academic Press, Inc., New York (1980)zbMATHGoogle Scholar
  14. 14.
    Rozenberg, G., Vermeir, D.: On ET0L systems of finite index. Inf. Control 38, 103–133 (1978)CrossRefzbMATHGoogle Scholar
  15. 15.
    Rozenberg, G., Vermeir, D.: On the effect of the finite index restriction on several families of grammars. Inf. Control 39, 284–302 (1978)MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Rozoy, B.: The Dyck language \(D_1^{\prime *}\) is not generated by any matrix grammar of finite index. Inf. Comput. 74(1), 64–89 (1987)MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Vijay-Shanker, K., Weir, D.J.: The equivalence of four extensions of context-free grammars. Math. Syst. Theor. 27(6), 511–546 (1994)MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Zetzsche, G.: An approach to computing downward closures. In: Halldórsson, M.M., Iwama, K., Kobayashi, N., Speckmann, B. (eds.) ICALP 2015. LNCS, vol. 9135, pp. 440–451. Springer, Heidelberg (2015). doi: 10.1007/978-3-662-47666-6_35 CrossRefGoogle Scholar

Copyright information

© Springer International Publishing AG 2017

Authors and Affiliations

  • Flavio D’Alessandro
    • 1
    • 2
    Email author
  • Oscar H. Ibarra
    • 3
  • Ian McQuillan
    • 4
  1. 1.Department of MathematicsSapienza University of RomeRomeItaly
  2. 2.Department of MathematicsBoğaziçi UniversityBebekTurkey
  3. 3.Department of Computer ScienceUniversity of CaliforniaSanta BarbaraUSA
  4. 4.Department of Computer ScienceUniversity of SaskatchewanSaskatoonCanada

Personalised recommendations