The Hardest Language for Conjunctive Grammars

  • Alexander OkhotinEmail author
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 9691)


A famous theorem by Greibach (“The hardest context-free language”, SIAM J. Comp., 1973) states that there exists such a context-free language \(L_0\) that every context-free language over any alphabet is reducible to \(L_0\) by a homomorphic reduction—in other words, is representable as an inverse homomorphic image \(h^{-1}(L_0)\), for a suitable homomorphism h. This paper establishes similar characterizations for conjunctive grammars, that is, for grammars extended with a conjunction operator.


Normal Form Parse Tree Membership Problem Nonterminal Symbol Language Equation 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


  1. 1.
    Aizikowitz, T., Kaminski, M.: LR(0) conjunctive grammars and deterministic synchronized alternating pushdown automata. In: Kulikov, A., Vereshchagin, N. (eds.) CSR 2011. LNCS, vol. 6651, pp. 345–358. Springer, Heidelberg (2011)CrossRefGoogle Scholar
  2. 2.
    Autebert, J.-M.: Non-principalité du cylindre des langages à compteur. Math. Syst. Theory 11(1), 157–167 (1977)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Barash, M., Okhotin, A.: An extension of context-free grammars with one-sided context specifications. Inf. Comput. 237, 268–293 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Barash, M., Okhotin, A.: Two-sided context specifications in formal grammars. Theoret. Comput. Sci. 591, 134–153 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Boasson, L., Nivat, M.: Le cylindre des langages linéaires. Math. Syst. Theory 11, 147–155 (1977)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Čulík II, K., Maurer, H.A.: On simple representations of language families. RAIRO Informatique Théorique et Appl. 13(3), 241–250 (1979)MathSciNetzbMATHGoogle Scholar
  7. 7.
    Greibach, S.A.: A new normal-form theorem for context-free phrase structure grammars. J. ACM 12, 42–52 (1965)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Greibach, S.A.: The hardest context-free language. SIAM J. Comput. 2(4), 304–310 (1973)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Greibach, S.A.: Jump PDA’s and hierarchies of deterministic context-free languages. SIAM J. Comput. 3(2), 111–127 (1974)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Kountouriotis, V., Nomikos, C., Rondogiannis, P.: Well-founded semantics for Boolean grammars. Inf. Comput. 207(9), 945–967 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Lehtinen, T., Okhotin, A.: Boolean grammars and GSM mappings. Int. J. Found. Comput. Sci. 21(5), 799–815 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Okhotin, A.: Conjunctive grammars. J. Automata Lang. Comb. 6(4), 519–535 (2001)MathSciNetzbMATHGoogle Scholar
  13. 13.
    Okhotin, A.: Conjunctive grammars and systems of language equations. Program. Comput. Sci. 28(5), 243–249 (2002)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Okhotin, A.: Boolean grammars. Inf. Comput. 194(1), 19–48 (2004)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Okhotin, A.: The dual of concatenation. Theoret. Comput. Sci. 345(2–3), 425–447 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Okhotin, A.: Conjunctive and Boolean grammars: the true general case of the context-free grammars. Comput. Sci. Rev. 9, 27–59 (2013)CrossRefzbMATHGoogle Scholar
  17. 17.
    Okhotin, A.: Parsing by matrix multiplication generalized to Boolean grammars. Theoret. Comput. Sci. 516, 101–120 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Okhotin, A., Reitwießner, C.: Conjunctive grammars with restricted disjunction. Theoret. Comput. Sci. 411(26–28), 2559–2571 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Seki, H., Matsumura, T., Fujii, M., Kasami, T.: On multiple context-free grammars. Theoret. Comput. Sci. 88(2), 191–229 (1991)MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Vijay-Shanker, K., Weir, D.J., Joshi, A.K.: Characterizing structural descriptions produced by various grammatical formalisms. In: 25th Annual Meeting of the Association for Computational Linguistics (ACL 1987), pp. 104–111 (1987)Google Scholar

Copyright information

© Springer International Publishing Switzerland 2016

Authors and Affiliations

  1. 1.Department of Mathematics and StatisticsUniversity of TurkuTurkuFinland

Personalised recommendations