Advertisement

Formal Languages over GF(2)

  • Ekaterina Bakinova
  • Artem Basharin
  • Igor Batmanov
  • Konstantin Lyubort
  • Alexander Okhotin
  • Elizaveta Sazhneva
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 10792)

Abstract

Variants of the union and concatenation operations on formal languages are investigated, in which Boolean logic in the definitions (that is, conjunction and disjunction) is replaced with the operations in the two-element field GF(2) (conjunction and exclusive OR). Union is thus replaced with symmetric difference, whereas concatenation gives rise to a new GF(2)-concatenation operation, which is notable for being invertible. All operations preserve regularity, and their state complexity is determined. Next, a new class of formal grammars based on GF(2)-operations is defined, and it is shown to have the same computational complexity as ordinary grammars with union and concatenation.

References

  1. 1.
    Albrecht, M.R., Bard, G.V., Hart, W.: Algorithm 898: efficient multiplication of dense matrices over GF(2). ACM Trans. Math. Softw. 37(1), 9:1–9:14 (2010)CrossRefzbMATHGoogle Scholar
  2. 2.
    Brent, R.P., Goldschlager, L.M.: A parallel algorithm for context-free parsing. Aust. Comp. Sci. Comm. 6(7), 1–10 (1984)Google Scholar
  3. 3.
    Damm, C.: Problems complete for \(\oplus \)L. Inf. Process. Lett. 36(5), 247–250 (1990)CrossRefzbMATHGoogle Scholar
  4. 4.
    Ginsburg, S., Rice, H.G.: Two families of languages related to ALGOL. J. ACM 9(3), 350–371 (1962)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Jeż, A.: Conjunctive grammars generate non-regular unary languages. Int. J. Found. Comput. Sci. 19(3), 597–615 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Leiss, E.L.: Unrestricted complementation in language equations over a one-letter alphabet. Theor. Comput. Sci. 132(2), 71–84 (1994)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Maslov, A.N.: Estimates of the number of states of finite automata. Sov. Math. Dokl. 11, 1373–1375 (1970)zbMATHGoogle Scholar
  8. 8.
    Okhotin, A.: Conjunctive and Boolean grammars: the true general case of the context-free grammars. Comput. Sci. Rev. 9, 27–59 (2013)CrossRefzbMATHGoogle Scholar
  9. 9.
    Okhotin, A.: Parsing by matrix multiplication generalized to Boolean grammars. Theor. Comput. Sci. 516, 101–120 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    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, Berlin (2009).  https://doi.org/10.1007/978-3-642-01492-5_7. Chap. 7CrossRefGoogle Scholar
  11. 11.
    Rossmanith, P., Rytter, W.: Observations on log(n) time parallel recognition of unambiguous cfl’s. Inf. Process. Lett. 44(5), 267–272 (1992)CrossRefzbMATHGoogle Scholar
  12. 12.
    Rytter, W.: On the recognition of context-free languages. In: Computation Theory - Proceedings of 5th Symposium, Zaborów, Poland, 3–8 December 1984, pp. 318–325 (1984)Google Scholar
  13. 13.
    Sudborough, I.H.: A note on tape-bounded complexity classes and linear context-free languages. J. ACM 22(4), 499–500 (1975)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Valiant, L.G.: General context-free recognition in less than cubic time. J. Comput. Syst. Sci. 10(2), 308–315 (1975)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Yu, S., Zhuang, Q., Salomaa, K.: The state complexities of some basic operations on regular languages. Theor. Comput. Sci. 125(2), 315–328 (1994)MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    van Zijl, L.: Magic numbers for symmetric difference NFAs. Int. J. Found. Comput. Sci. 16(5), 1027–1038 (2005)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer International Publishing AG, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Gymnasium №1KovrovRussia
  2. 2.School №179MoscowRussia
  3. 3.Moscow Institute of Physics and TechnologyDolgoprudnyRussia
  4. 4.St. Petersburg Academic UniversitySaint PetersburgRussia
  5. 5.St. Petersburg State UniversitySaint PetersburgRussia

Personalised recommendations