Skip to main content
SpringerLink
Log in
Book cover

International Computer Science Symposium in Russia

CSR 2006: Computer Science – Theory and Applications pp 292–303Cite as

Language Equations with Symmetric Difference

  • Conference paper

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 3967))

Abstract

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.

Supported by the Academy of Finland under grant 206039.

This is a preview of subscription content, log in via an institution.

Chapter
USD   29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD   84.99
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD   109.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Learn about institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  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)

    Chapter  Google Scholar 

  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)

    Chapter  Google Scholar 

  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)

    Chapter  Google Scholar 

  4. Baker, B.S., Book, R.V.: Reversal-bounded multipushdown machines. Journal of Computer and System Sciences 8, 315–332 (1974)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Chapter  Google Scholar 

  6. Charatonik, W.: Set constraints in some equational theories. Information and Computation 142, 40–75 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  7. Conway, J.H.: Regular Algebra and Finite Machines. Chapman and Hall, Boca Raton (1971)

    MATH  Google Scholar 

  8. Ginsburg, S., Rice, H.G.: Two families of languages related to ALGOL. Journal of the ACM 9, 350–371 (1962)

    Article  MathSciNet  MATH  Google Scholar 

  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. 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)

    Chapter  Google Scholar 

  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)

    MathSciNet  MATH  Google Scholar 

  12. Okhotin, A.: Conjunctive grammars and systems of language equations. Programming and Computer Software 28(5), 243–249 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  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 submitted

    Google Scholar 

  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. Okhotin, A.: Boolean grammars. Information and Computation 194(1), 19–48 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  16. Okhotin, A.: The dual of concatenation. Theoretical Computer Science 345(2–3), 425–447 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  17. Okhotin, A.: Unresolved systems of language equations: expressive power and decision problems. Theoretical Computer Science 349(3), 283–308 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Chapter  Google Scholar 

  19. Okhotin, A., Yakimova, O.: On language equations with complementation, TUCS Technical Report No 735, Turku, Finland (December 2005)

    Google Scholar 

  20. Post, E.L.: The two-valued iterative systems of mathematical logic (1941)

    Google Scholar 

  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. van Zijl, L.: On binary ⊕-NFAs and succinct descriptions of regular languages. Theoretical Computer Science 313(1), 159–172 (2004)

    Article  MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics, University of Turku, FIN-20014, Turku, Finland

    Alexander Okhotin

Authors
  1. Alexander Okhotin
    View author publications

    You can also search for this author in PubMed Google Scholar

Editor information

Editors and Affiliations

  1. IRMAR, Université de Rennes, Campus de Beaulieu, 35042, Rennes Cedex, France

    Dima Grigoriev

  2. Intel Corporation, JF1-13, 2111 NE 25th Avenue, 97124, Hillsboro, OR, USA

    John Harrison

  3. Steklov Institute of Mathematics at St. Petersburg, 27 Fontanka, St., 191023, Petersburg, Russia

    Edward A. Hirsch

Rights and permissions

Reprints and permissions

Copyright information

© 2006 Springer-Verlag Berlin Heidelberg

About this paper

Cite this paper

Okhotin, A. (2006). Language Equations with Symmetric Difference. In: Grigoriev, D., Harrison, J., Hirsch, E.A. (eds) Computer Science – Theory and Applications. CSR 2006. Lecture Notes in Computer Science, vol 3967. Springer, Berlin, Heidelberg. https://doi.org/10.1007/11753728_30

Download citation

  • DOI: https://doi.org/10.1007/11753728_30

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-34166-6

  • Online ISBN: 978-3-540-34168-0

  • eBook Packages: Computer ScienceComputer Science (R0)

Publish with us

Policies and ethics

Search

Navigation