A generalisation of Parikh's theorem in formal language theory

  • J. van Leeuwen
Monday Morning
Part of the Lecture Notes in Computer Science book series (LNCS, volume 14)


We show that when a family of languages F has a few appropriate closure-properties, all languages algebraic over F are still equivalent to languages in F when occurrences of symbols are permuted. At the same time, the methods used imply a new and simple algebraic proof of Parikh's original theorem, directly transforming an arbitrary context-free grammar into a letter-equivalent regular grammar. Further applications are discussed.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. [1]
    Aho, A. V., and J. D. Ullman, The theory of parsing, translation and compiling (vol.I), Prentice-Hall, Englewood Cliffs, 1973.Google Scholar
  2. [2]
    Ginsburg, S., The mathematical theory of context-free languages, McGraw Hill, New York, 1966.Google Scholar
  3. [3]
    Ginsburg, S., and H. G. Rice, Two families of languages related to ALGOL, JACM 9 (1962) 350–371.CrossRefGoogle Scholar
  4. [4]
    Ginsburg, S., and E. H. Spanier, AFLs with the semilinear property, J. Comp. Syst. Sci. 5 (1971) 365–396.Google Scholar
  5. [5]
    Greibach, S., A generalization of Parikh's semilinear theorem, Discr. Math. 2 (1972) 347–355.CrossRefGoogle Scholar
  6. [6]
    Parikh, R. J., On context-free languages, JACM 13 (1966) 570–581.CrossRefGoogle Scholar
  7. [7]
    Pilling, D. L., Commutative regular equations and Parikh's theorem, J. London Math. Soc. (II) 6 (1973) 663–666.Google Scholar
  8. [8]
    Salomaa, A., Formal languages, Acad. Press, New York, 1973.Google Scholar
  9. [9]
    Van Leeuwen, J., Relative grammaticality, (to appear).Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 1974

Authors and Affiliations

  • J. van Leeuwen
    • 1
    • 2
  1. 1.Department of Computer ScienceState University of New York at BuffaloAmherst
  2. 2.Department of MathematicsState University of UtrechtUtrechtNetherlands

Personalised recommendations