Language Equations with Symmetric Difference

  • Alexander Okhotin
Part of the Lecture Notes in Computer Science book series (LNCS, volume 3967)


Systems of language equations used by Ginsburg and Rice (“Two families of languages related to ALGOL”, JACM, 1962) to represent context-free grammars are modified to use the symmetric difference operation instead of union. Contrary to a natural expectation that these two types of equations should have incomparable expressive power, it is shown that equations with symmetric difference can express every recursive set by their unique solutions, every recursively enumerable set by their least solutions and every co-recursively-enumerable set by their greatest solutions. The solution existence problem is Π1-complete, the existence of a unique, a least or a greatest solution is Π2-complete, while the existence of finitely many solutions is Σ3-complete.


Boolean Operation Turing Machine Expressive Power Great Solution Input Alphabet 
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  1. 1.
    Aiken, A., Kozen, D., Vardi, M.Y., Wimmers, E.L.: The complexity of set constraints. In: Meinke, K., Börger, E., Gurevich, Y. (eds.) CSL 1993. LNCS, vol. 832, pp. 1–17. Springer, Heidelberg (1994)CrossRefGoogle Scholar
  2. 2.
    Autebert, J., Berstel, J., Boasson, L.: Context-free languages and pushdown automata. In: Rozenberg, Salomaa (eds.) Handbook of Formal Languages, vol. 1, pp. 111–174. Springer, Berlin (1997)CrossRefGoogle Scholar
  3. 3.
    Baader, F., Küsters, R.: Unification in a description logic with transitive closure of roles. In: Nieuwenhuis, R., Voronkov, A. (eds.) LPAR 2001. LNCS, vol. 2250, pp. 217–232. Springer, Heidelberg (2001)CrossRefGoogle Scholar
  4. 4.
    Baker, B.S., Book, R.V.: Reversal-bounded multipushdown machines. Journal of Computer and System Sciences 8, 315–332 (1974)MathSciNetCrossRefMATHGoogle Scholar
  5. 5.
    Bala, S.: Regular language matching and other decidable cases of the satisfiability problem for constraints between regular open terms. In: Diekert, V., Habib, M. (eds.) STACS 2004. LNCS, vol. 2996, pp. 596–607. Springer, Heidelberg (2004)CrossRefGoogle Scholar
  6. 6.
    Charatonik, W.: Set constraints in some equational theories. Information and Computation 142, 40–75 (1998)MathSciNetCrossRefMATHGoogle Scholar
  7. 7.
    Conway, J.H.: Regular Algebra and Finite Machines. Chapman and Hall, Boca Raton (1971)MATHGoogle Scholar
  8. 8.
    Ginsburg, S., Rice, H.G.: Two families of languages related to ALGOL. Journal of the ACM 9, 350–371 (1962)MathSciNetCrossRefMATHGoogle Scholar
  9. 9.
    Hartmanis, J.: Context-free languages and Turing machine computations. In: Proceedings of Symposia in Applied Mathematics, vol. 19, pp. 42–51. AMS (1967)Google Scholar
  10. 10.
    Kunc, M.: The power of commuting with finite sets of words. In: Diekert, V., Durand, B. (eds.) STACS 2005. LNCS, vol. 3404, pp. 569–580. Springer, Heidelberg (2005)CrossRefGoogle Scholar
  11. 11.
    Meyer, A.R., Rabinovich, A.M.: Valid identity problem for shuffle regular expressions. Journal of Automata, Languages and Combinatorics 7(1), 109–125 (2002)MathSciNetMATHGoogle Scholar
  12. 12.
    Okhotin, A.: Conjunctive grammars and systems of language equations. Programming and Computer Software 28(5), 243–249 (2002)MathSciNetCrossRefMATHGoogle Scholar
  13. 13.
    Okhotin, A.: Decision problems for language equations with Boolean operations. In: Baeten, J.C.M., Lenstra, J.K., Parrow, J., Woeginger, G.J. (eds.) ICALP 2003. LNCS, vol. 2719, pp. 239–251. Springer, Heidelberg (2003); Full journal version submittedGoogle Scholar
  14. 14.
    Okhotin, A.: Sistemy yazykovykh uravnenii i zamknutye klassy funktsii algebry logiki (Systems of language equations and closed classes of logic algebra functions) in Russian. In: Proceedings of the Fifth International conference Discrete models in the theory of control systems, pp. 56–64 (2003)Google Scholar
  15. 15.
    Okhotin, A.: Boolean grammars. Information and Computation 194(1), 19–48 (2004)MathSciNetCrossRefMATHGoogle Scholar
  16. 16.
    Okhotin, A.: The dual of concatenation. Theoretical Computer Science 345(2–3), 425–447 (2005)MathSciNetCrossRefMATHGoogle Scholar
  17. 17.
    Okhotin, A.: Unresolved systems of language equations: expressive power and decision problems. Theoretical Computer Science 349(3), 283–308 (2005)MathSciNetCrossRefMATHGoogle Scholar
  18. 18.
    Okhotin, A.: Strict language inequalities and their decision problems. In: Jedrzejowicz, J., Szepietowski, A. (eds.) MFCS 2005. LNCS, vol. 3618, pp. 708–719. Springer, Heidelberg (2005)CrossRefGoogle Scholar
  19. 19.
    Okhotin, A., Yakimova, O.: On language equations with complementation, TUCS Technical Report No 735, Turku, Finland (December 2005)Google Scholar
  20. 20.
    Post, E.L.: The two-valued iterative systems of mathematical logic (1941)Google Scholar
  21. 21.
    Yablonski, S.V., Gavrilov, G.P., Kudryavtsev, V.B.: Funktsii algebry logiki i klassy Posta. Functions of logic algebra and the classes of Post (1966) (in Russian)Google Scholar
  22. 22.
    van Zijl, L.: On binary ⊕-NFAs and succinct descriptions of regular languages. Theoretical Computer Science 313(1), 159–172 (2004)MathSciNetCrossRefMATHGoogle Scholar

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© Springer-Verlag Berlin Heidelberg 2006

Authors and Affiliations

  • Alexander Okhotin
    • 1
  1. 1.Department of MathematicsUniversity of TurkuTurkuFinland

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