Mathematical systems theory

, Volume 28, Issue 3, pp 199–213 | Cite as

Pumping lemmas for the control language hierarchy

  • M. A. Palis
  • S. M. Shende


We investigate a progression of grammatically defined language families, thecontrol language hierarchy. This hierarchy has been studied recently from the perspective of providing a linguistic framework for natural language syntax. We exhibit a progression of pumping lemmas, one for each family in the hierarchy, thereby showing that the hierarchy is strictly separable.


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  1. [1]
    A. V. Aho. Indexed grammars—an extension to context free grammars.J. Assoc. Comput. Mech., 15:647–671, 1968.Google Scholar
  2. [2]
    S. Ginsburg and E. H. Spanier, Control sets on grammars.Math. Systems Theory, 2:159–177, 1968.Google Scholar
  3. [3]
    M. A. Harrison.Introduction to Formal Language Theory. Addison-Wesley, Reading, MA, 1978.Google Scholar
  4. [4]
    G. T. Herman and G. Rozenberg.Developmental Systems and Languages. North-Holland, Amsterdam, 1975.Google Scholar
  5. [5]
    O. H. Ibarra. Simple matrix languages.Inform, and Control, 17:359–394, 1970.Google Scholar
  6. [6]
    A. K. Joshi, L. S. Levy, and M. Takahashi. Tree adjunct grammars.J. Comput. System Sci., 10(1), 1975.Google Scholar
  7. [7]
    T. Kasai. An hierarchy between context-free and context-sensitive languages.J. Comput. System Sci., 4:492–508, 1970.Google Scholar
  8. [8]
    N. A. Khabbaz. A geometric hierarchy of languages.J. Comput. System Sci., 8:142–157, 1974.Google Scholar
  9. [9]
    M. A. Palis and S. Shende. Upper bounds on recognition of a hierarchy of non-context-free languages.Theoret. Comput. Sci., 98(2):289–319, May 1992.Google Scholar
  10. [10]
    D. J. Rozenkrantz. Programmed grammars and classes of formal languages.J. Assoc. Comput. Mech., 16:107–131, 1969.Google Scholar
  11. [11]
    A. Salomaa. Matrix grammars with a leftmost restriction.Inform, and Control, 20:143–149, 1970.Google Scholar
  12. [12]
    A. Salomaa.Formal Languages. Academic Press, New York, 1973.Google Scholar
  13. [13]
    J. W. Thatcher. Tree automata: an informal survey. In A. V. Aho, editor,Currents in the Theory of Computing, pp. 143–172. Prentice-Hall, Englewood Cliffs, NJ, 1973.Google Scholar
  14. [14]
    K. Vijay-Shanker. A Study of Tree Adjoining Grammars. Ph.D. thesis, University of Pennsylvania, Philadelphia, PA, 1987.Google Scholar
  15. [15]
    K. Vijay-Shanker and D. J. Weir. The equivalence of four extensions of context-free grammars.Math. Systems Theory, 27:511–546, 1994.Google Scholar
  16. [16]
    D. J. Weir. Characterizing Mildly Context-Sensitive Grammar Formalisms. Ph.D. Thesis, University of Pennsylvania, Philadelphia, PA, September 1988.Google Scholar
  17. [17]
    D. J. Weir. A geometric hierarchy beyond context-free languages.Theoret. Comput. Sci., 104:235–261, 1992.Google Scholar

Copyright information

© Springer-Verlag New York Inc. 1995

Authors and Affiliations

  • M. A. Palis
    • 1
  • S. M. Shende
    • 2
  1. 1.Department of Electrical and Computer EngineeringNew Jersey Institute of TechnologyNewarkUSA
  2. 2.Department of Computer Science and EngineeringUniversity of NebraskaLincolnUSA

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