Mathematical systems theory

, Volume 3, Issue 2, pp 119–124 | Cite as

On the equivalence and containment problems for context-free languages

  • J. E. Hopcroft


LetG andG0 be context-free grammars. Necessary and sufficient conditions onG0 are obtained for the decidability ofL(G0)\( \subseteq \)L((G) It is also shown that it is undecidable for whichG0,L(G)\( \subseteq \) is decidable. Furthermore, given thatL(G)\( \subseteq \) is decidable for a fixedG0, there is no effective procedure to determine the algorithm which decidesL(G)\( \subseteq \) IfL(G0) is a regular set,L(G) = L(G0) is decidable if and only ifL(G0) is bounded. However, there exist non-regular, unboundedL(G0) for whichL(G) = L(G0) is decidable.


Computational Mathematic Effective Procedure Containment Problem 
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  1. [1]
    S. Greibach, A note on undecidable properties of formal languages. SDC Document TM-738/038/00, August 1967.Google Scholar
  2. [2]
    S. Ginsburg andE. H. Spanier, BoundedAlgol-like languages,Trans. Amer. Math. Soc. 113 (1964), 333–368.Google Scholar

Copyright information

© Swets & Zeitlinger B.V. 1969

Authors and Affiliations

  • J. E. Hopcroft
    • 1
  1. 1.Cornell UniversityUSA

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