Advertisement

On the Expressive Power of GF(2)-Grammars

  • Vladislav MakarovEmail author
  • Alexander Okhotin
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 11376)

Abstract

GF(2)-grammars, recently introduced by Bakinova et al. (“Formal languages over GF(2)”, LATA 2018), are a variant of ordinary context-free grammars, in which the disjunction is replaced by exclusive OR, whereas the classical concatenation is replaced by a new operation called GF(2)-concatenation: \(K \odot L\) is the set of all strings with an odd number of partitions into a concatenation of a string in K and a string in L. This paper establishes several results on the family of languages defined by these grammars. Over the unary alphabet, GF(2)-grammars define exactly the 2-automatic sets. No language of the form \(\{a^n b^{f(n)} \,\mid \, n \geqslant 1\}\), with uniformly superlinear f, can be described by any GF(2)-grammar. The family is not closed under union, intersection, classical concatenation and Kleene star, non-erasing homomorphisms. On the other hand, this family is closed under injective nondeterministic finite transductions, and contains a hardest language under reductions by homomorphisms.

References

  1. 1.
    Allouche, J.-P., Shallit, J.: Automatic Sequences: Theory, Applications, Generalizations. Cambridge University Press, Cambridge (2003)CrossRefGoogle Scholar
  2. 2.
    Bakinova, E., Basharin, A., Batmanov, I., Lyubort, K., Okhotin, A., Sazhneva, E.: Formal languages over GF(2). In: Klein, S.T., Martín-Vide, C., Shapira, D. (eds.) LATA 2018. LNCS, vol. 10792, pp. 68–79. Springer, Cham (2018).  https://doi.org/10.1007/978-3-319-77313-1_5CrossRefGoogle Scholar
  3. 3.
    Barash, M., Okhotin, A.: An extension of context-free grammars with one-sided context specifications. Inf. Comput. 237, 268–293 (2014).  https://doi.org/10.1016/j.ic.2014.03.003MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Buchholz, T., Kutrib, M.: On time computability of functions in one-way cellular automata. Acta Informatica 35(4), 329–352 (1998).  https://doi.org/10.1007/s002360050123MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Christol, G.: Ensembles presque periodiques \(k\)-reconnaissables. Theor. Comput. Sci. 9, 141–145 (1979).  https://doi.org/10.1016/0304-3975(79)90011-2MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Eilenberg, S.: Automata, Languages and Machines, vol. 1. Academic Press, Cambridge (1974)zbMATHGoogle Scholar
  7. 7.
    Forejt, V., Jančar, P., Kiefer, S., Worrell, J.: Language equivalence of probabilistic pushdown automata. Inf. Comput. 237, 1–11 (2014).  https://doi.org/10.1016/j.ic.2014.04.003MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Ginsburg, S., Rice, H.G.: Two families of languages related to ALGOL. J. ACM 9, 350–371 (1962).  https://doi.org/10.1145/321127.321132MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Ginsburg, S., Spanier, E.H.: Quotients of context-free languages. J. ACM 10(4), 487–492 (1963).  https://doi.org/10.1145/321186.321191MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Greibach, S.A.: The hardest context-free language. SIAM J. Comput. 2(4), 304–310 (1973).  https://doi.org/10.1137/0202025MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Ibarra, O.H., Kim, S.M.: Characterizations and computational complexity of systolic trellis automata. Theor. Comput. Sci. 29, 123–153 (1984).  https://doi.org/10.1016/0304-3975(84)90015-XMathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Jeż, A.: Conjunctive grammars can generate non-regular unary languages. Int. J. Found. Comput. Sci. 19(3), 597–615 (2008).  https://doi.org/10.1142/S012905410800584XCrossRefzbMATHGoogle Scholar
  13. 13.
    Knuth, D.E.: Context-free multilanguages. In: Theoretical Studies in Computer Science, pp. 1–13. Academic Press, Cambridge (1992)Google Scholar
  14. 14.
    Okhotin, A.: On the equivalence of linear conjunctive grammars to trellis automata. RAIRO Informatique Théorique et Applications 38(1), 69–88 (2004).  https://doi.org/10.1051/ita:2004004MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Okhotin, A.: Conjunctive and Boolean grammars: the true general case of the context-free grammars. Comput. Sci. Rev. 9, 27–59 (2013).  https://doi.org/10.1016/j.cosrev.2013.06.001CrossRefzbMATHGoogle Scholar
  16. 16.
    Okhotin, A.: Input-driven languages are linear conjunctive. Theor. Comput. Sci. 618, 52–71 (2016).  https://doi.org/10.1016/j.tcs.2016.01.007MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Okhotin, A.: Underlying principles and recurring ideas of formal grammars. In: Klein, S.T., Martín-Vide, C., Shapira, D. (eds.) LATA 2018. LNCS, vol. 10792, pp. 36–59. Springer, Cham (2018).  https://doi.org/10.1007/978-3-319-77313-1_3CrossRefzbMATHGoogle Scholar
  18. 18.
    Petre, I., Salomaa, A.: Algebraic systems and pushdown automata. In: Droste, M., Kuich, W., Vogler, H. (eds.) Handbook of Weighted Automata, pp. 257–289. Springer, Heidelberg (2009).  https://doi.org/10.1007/978-3-642-01492-5_7CrossRefGoogle Scholar
  19. 19.
    Seki, H., Matsumura, T., Fujii, M., Kasami, T.: On multiple context-free grammars. Theor. Comput. Sci. 88(2), 191–229 (1991).  https://doi.org/10.1016/0304-3975(91)90374-BMathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Terrier, V.: On real-time one-way cellular array. Theor. Comput. Sci. 141(1–2), 331–335 (1995).  https://doi.org/10.1016/0304-3975(94)00212-2MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer Nature Switzerland AG 2019

Authors and Affiliations

  1. 1.St. Petersburg State UniversitySaint PetersburgRussia

Personalised recommendations