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On Distinguishing NC\(^1\) and NL

  • Andreas Krebs
  • Klaus-Jörn LangeEmail author
  • Michael Ludwig
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 9168)

Abstract

We obtain results within the area of dense completeness, which describes a close relation between families of formal languages and complexity classes. Previously we were able show that this relation exists between counter languages and \(\mathbf {NL}\) but not between the regular languages and \(\mathbf {NC^1}\).

We narrow the gap between the regular languages and the counter languages by considering visibly counter languages. It turns out that they are not densely complete for \(\mathbf {NC^1}\). At the same time we found a restricted counter automaton model which is densely complete for \(\mathbf {NL}\).

Besides counter automata we show more positive examples in terms of L-systems.

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References

  1. AM04.
    Alur, R., Madhusudan, P.: Visibly pushdown languages. In: Babai, L. (ed.) Proceedings of the 36th Annual ACM Symposium on Theory of Computing, June 13–16, pp. 202–211. ACM, Chicago (2004)Google Scholar
  2. BCST92.
    Barrington, D.A.M., Compton, K.J., Straubing, H., Thérien, D.: Regular Languages in NC\({^1}\). J. Comput. Syst. Sci. 44(3), 478–499 (1992)Google Scholar
  3. BLS06.
    Bárány, V., Löding, C., Serre, O.: Regularity problems for visibly pushdown languages. In: Durand, B., Thomas, W. (eds.) STACS 2006. LNCS, vol. 3884, pp. 420–431. Springer, Heidelberg (2006) CrossRefGoogle Scholar
  4. Dym88.
    Dymond, P.W.: Input-Driven Languages are in log n Depth. Inf. Process. Lett. 26(5), 247–250 (1988)MathSciNetCrossRefGoogle Scholar
  5. FSS84.
    Furst, M.L., Saxe, J.B., Sipser, M.: Parity, Circuits, and the Polynomial-Time Hierarchy. Mathematical Systems Theory 17(1), 13–27 (1984)MathSciNetCrossRefGoogle Scholar
  6. Hås86.
    Håstad, J.: Almost optimal lower bounds for small depth circuits. In: Hartmanis, J. (ed.) Proceedings of the 18th Annual ACM Symposium on Theory of Computing, May 28–30, pp. 6–20. ACM, Berkeley (1986)Google Scholar
  7. KL12.
    Krebs, A., Lange, K.-J.: Dense completeness. In: Yen, H.-C., Ibarra, O.H. (eds.) DLT 2012. LNCS, vol. 7410, pp. 178–189. Springer, Heidelberg (2012) CrossRefGoogle Scholar
  8. KLL15.
    Krebs, A., Lange, K.-L., Ludwig, M.: Visibly counter languages and constant depth circuits. In: Mayr, E.W., Ollinger, N. (eds) 32nd International Symposium on Theoretical Aspects of Computer Science, STACS 2015, March 4–7. LIPIcs, vol. 30, pp. 594–607. Schloss Dagstuhl - Leibniz-Zentrum fuer Informatik, Garching (2015)Google Scholar
  9. Meh80.
    Mehlhorn, K.: Pebbling moutain ranges and its application of DCFL-Recognition. In: de Bakker, J.W., van Leeuwen, J. (eds) Proceedings of the Automata, Languages and Programming, 7th Colloquium, Noordweijkerhout, July 14–18, The Netherland. LNCS, vol. 85, pp. 422–435. Springer, Heidelberg (1980)Google Scholar
  10. RS80.
    Rozenberg, G., Salomaa, A.: Mathematical Theory of L Systems. Academic Press Inc., Orlando (1980) zbMATHGoogle Scholar
  11. Smo87.
    Smolensky, R.: Algebraic methods in the theory of lower bounds for boolean circuit complexity. In: Aho, A.V. (ed.) Proceedings of the 19th Annual ACM Symposium on Theory of Computing, pp. 77–82. ACM, New York (1987)Google Scholar
  12. vL75.
    van Leeuwen, J.: The Membership Question for ET0L-Languages is Polynomially Complete. Inf. Process. Lett. 3(5), 138–143 (1975)CrossRefGoogle Scholar
  13. vL76.
    van Leeuwen, J.: Variations of a new machine model. In: 17th Annual Symposium on Foundations of Computer Science, October 25–27, pp. 228–235. IEEE Computer Society, Texas (1976)Google Scholar
  14. Vol90.
    Vollmer, H.: The gap-language-technique revisited. In: Börger, E., Böuning, H.K., Richter, M.M., Schönfeld, W. (eds) CSL 1990. LNCS, vol. 533, pp. 389–399. Springer, Heidelberg (1990)Google Scholar
  15. Vol99.
    Vollmer, H.: Introduction to circuit complexity - a uniform approach. Texts in theoretical computer science. Springer (1999)Google Scholar

Copyright information

© Springer International Publishing Switzerland 2015

Authors and Affiliations

  • Andreas Krebs
    • 1
  • Klaus-Jörn Lange
    • 1
    Email author
  • Michael Ludwig
    • 1
  1. 1.WSI - University of TübingenTübingenGermany

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