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Kernels of Sub-classes of Context-Free Languages

  • Martin KutribEmail author
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 12011)

Abstract

While the closure of a language family \(\mathscr {L}\) under certain language operations is the least family of languages which contains all members of \(\mathscr {L}\) and is closed under all of the operations, a kernel of \(\mathscr {L}\) is a greatest family of languages which is a subfamily of \(\mathscr {L}\) and is closed under all of the operations. Here we investigate properties of kernels of general language families and operations defined thereon as well as kernels of (deterministic) (linear) context-free languages with a focus on Boolean operations. While the closures of language families usually are unique, this uniqueness is not obvious for kernels. We consider properties of language families and operations that yield unique and non-unique, that is a set, of kernels. For the latter case, the question whether the union of all kernels coincides with the language family, or whether there are languages that do not belong to any kernel is addressed. Furthermore, the intersection of all kernels with respect to certain operations is studied in order to identify sets of languages that belong to all of these kernels.

Notes

Acknowledgment

The author would like to thank Henning Fernau for fruitful discussions at an early stage of the paper.

References

  1. 1.
    Bertsch, E., Nederhof, M.J.: Regular closure of deterministic languages. SIAM J. Comput. 29, 81–102 (1999)MathSciNetCrossRefGoogle Scholar
  2. 2.
    Fernau, H., Kutrib, M., Wendlandt, M.: Self-verifying pushdown automata. In: Non-Classical Models of Automata and Applications (NCMA 2017), vol. 329, pp. 103–117. Austrian Computer Society, Vienna (2017). books@ocg.atGoogle Scholar
  3. 3.
    Ginsburg, S.: The Mathematical Theory of Context-Free Languages. McGraw Hill, New York (1966)zbMATHGoogle Scholar
  4. 4.
    Ginsburg, S., Spanier, E.H.: Bounded ALGOL-like languages. Trans. Am. Math. Soc. 113, 333–368 (1964)MathSciNetzbMATHGoogle Scholar
  5. 5.
    Ginsburg, S., Spanier, E.H.: Finite-turn pushdown automata. SIAM J. Contr. 4, 429–453 (1966)MathSciNetCrossRefGoogle Scholar
  6. 6.
    Greibach, S.A.: The unsolvability of the recognition of linear context-free languages. J. ACM 13, 582–587 (1966)MathSciNetCrossRefGoogle Scholar
  7. 7.
    Harrison, M.A.: Introduction to Formal Language Theory. Addison-Wesley, Reading (1978)zbMATHGoogle Scholar
  8. 8.
    Ilie, L., Păun, G., Rozenberg, G., Salomaa, A.: On strongly context-free languages. Discrete Appl. Math. 103, 158–165 (2000)MathSciNetCrossRefGoogle Scholar
  9. 9.
    Jirásková, G.: State complexity of some operations on binary regular languages. Theoret. Comput. Sci. 330, 287–298 (2005)MathSciNetCrossRefGoogle Scholar
  10. 10.
    Kutrib, M., Malcher, A.: Finite turns and the regular closure of linear context-free languages. Discrete Appl. Math. 155, 2152–2164 (2007)MathSciNetCrossRefGoogle Scholar
  11. 11.
    Kutrib, M., Malcher, A., Wotschke, D.: The Boolean closure of linear context-free languages. Acta Inform. 45, 177–191 (2008)MathSciNetCrossRefGoogle Scholar
  12. 12.
    Okhotin, A.: Automaton representation of linear conjunctive languages. In: Ito, M., Toyama, M. (eds.) DLT 2002. LNCS, vol. 2450, pp. 393–404. Springer, Heidelberg (2003).  https://doi.org/10.1007/3-540-45005-X_35CrossRefGoogle Scholar
  13. 13.
    Parikh, R.J.: On context-free languages. J. ACM 13, 570–581 (1966)CrossRefGoogle Scholar
  14. 14.
    Wotschke, D.: Nondeterminism and Boolean operations in PDA’s. J. Comput. Syst. Sci. 16, 456–461 (1978)MathSciNetCrossRefGoogle Scholar
  15. 15.
    Wotschke, D.: The Boolean closures of the deterministic and nondeterministic context-free languages. In: Brauer, W. (ed.) GI 1973. LNCS, vol. 1, pp. 113–121. Springer, Heidelberg (1973).  https://doi.org/10.1007/3-540-06473-7_11CrossRefGoogle Scholar
  16. 16.
    Wotschke, D.: Degree-languages: a new concept of acceptance. J. Comput. Syst. Sci. 14(2), 187–209 (1977)MathSciNetCrossRefGoogle Scholar

Copyright information

© Springer Nature Switzerland AG 2020

Authors and Affiliations

  1. 1.Institut für InformatikUniversität GiessenGiessenGermany

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