Advertisement

On adaptive dlogtime and polylogtime reductions

  • Carme àlvarez
  • Birgit Jenner
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 775)

Abstract

We investigate properties of the relativized NC and AC hierarchies in their DLOGTIME-, respectively, ALOGTIME-uniform setting and show that these hierarchies can be characterized in terms of adaptive reducibility in deterministic (poly)logarithmic time, i.e. in time O(log n)i for i≥0. Using this characterization, we substantially generalize various previous results concerning the structure of the NC and AC hierarchies.

Keywords

Boolean Circuit Input Gate Circuit Family Oracle Function Gate Number 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    M. Ajtai, σ11 formulae on finite structures, Ann. Pure Appl. Logic, 24 (1983), 1–48.Google Scholar
  2. 2.
    C. àlvarez, Polylogtime and Logspace Adaptive Reductions, Tech. Report LSI-93-42-R, Dept. LSI, Universitat Politècnica de Catalunya, 1993.Google Scholar
  3. 3.
    C. àlvarez, J.L. Balcázar and B. Jenner, Adaptive logspace reducibility and parallel time, to appear in Math. Systems Theory. (preliminary version in Proc. 8th STACS, LNCS 480 (Springer, Berlin, 1991), 422–433.)Google Scholar
  4. 4.
    C. àlvarez and B. Jenner, A very hard log-space counting class, Theoret. Computer Science 107 (1993), 3–30.Google Scholar
  5. 5.
    C. àlvarez and B. Jenner, On adaptive dlogtime and polylogtime Reductions, Tech. Report LSI-93-41-R, Dept. LSI, Universitat Politècnica de Catalunya, 1993.Google Scholar
  6. 6.
    J.L. Balcázar, Adaptive logspace and depth-bounded reducibilities, Proc. 6th Structure in Complexity Theory Conference (1991), 240–254.Google Scholar
  7. 7.
    J.L. Balcázar, A. Lozano, J. Torán, The Complexity of Algorithmic Problems on Succinct Instances, R. Baeza-Yates, U. Manber (eds.), Computer Science, Plenum Press, New York, (1992), 351–377.Google Scholar
  8. 8.
    D.A. Mix Barrington, N. Immerman, H. Straubing, On uniformity within NC1, J. of Comp. and System Sci. 41, 3 (1990), 274–306.Google Scholar
  9. 9.
    A. Borodin, S.A. Cook, P.W. Dymond, W.L. Ruzzo, M. Tompa, Two applications of inductive counting for complementation problems, SIAM J. of Comput. 18, 3 (1989), 559–578.Google Scholar
  10. 10.
    S.R. Buss, The formula value problem is in ALOGTIME, Proc. 19th ACM STOC Symp., 1987, 123–131.Google Scholar
  11. 11.
    S. Buss, S. Cook, A. Gupta, V Ramachandran, An optimal parallel algorithm for formula evaluation, typescript, Univ. of Toronto, 1989.Google Scholar
  12. 12.
    J. Castro, C. Seara, Characterizations of some complexity classes between θ2p and δ2p, Proc. 9th STACS, LNCS 577 (Springer, Berlin, 1992), 305–319.Google Scholar
  13. 13.
    A.K. Chandra, D. Kozen, L.J. Stockmeyer, Alternation, J. of the ACM 28 (1981), 114–133.Google Scholar
  14. 14.
    A.K. Chandra, L.J. Stockmeyer, U. Vishkin, Constant depth reducibility, SIAM J. of Comput. 13 (1984), 423–439.Google Scholar
  15. 15.
    S.A. Cook, A taxonomy of problems, with fast parallel algorithms, Information and Control 64 (1985), 2–22.Google Scholar
  16. 16.
    M. Furst, J.B. Saxe, M. Sipser, Parity, circuits, and the polynomial-time hierarchy, Math. Systems Theory 17 (1984), 13–27.Google Scholar
  17. 17.
    B. Jenner, J. Torán, Parallel queries to NP, Proc. 8th Structure in Complexity Theory Conference (1993), 280–291. (to appear in Theoret. Computer Science)Google Scholar
  18. 18.
    R. Ladner, The circuit value problem is log space complete for P, SIGACT News 7 (1975), 18–20.Google Scholar
  19. 19.
    W.L. Ruzzo, On uniform circuit complexity, J. of Comput. and System Sci. 22 (1981), 365–383.Google Scholar
  20. 20.
    M. Sipser, Borel sets and circuit complexity, Proc. 15th ACM STOC Symp. (1983), 61–69.Google Scholar
  21. 21.
    L. Stockmeyer, The polynomial-time hierarchy, Theoret. Computer Science 3 (1977), pp 1–22.Google Scholar
  22. 22.
    I.H. Sudborough, On the tape complexity of deterministic context-free languages, J. of the ACM 25 (1978), 405–414.Google Scholar
  23. 23.
    K.W. Wagner, Bounded Query Classes, SIAM J. of Comput. 19, 5 (1990), 833–846.Google Scholar
  24. 24.
    C.B. Wilson, Relativized NC, Math. Systems Theory 20 (1987), 13–29.Google Scholar
  25. 25.
    C.B. Wilson, Decomposing NC and AC, SIAM J. of Comput. 19, 2 (1990), pp. 384–396.Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 1994

Authors and Affiliations

  • Carme àlvarez
    • 1
  • Birgit Jenner
    • 2
  1. 1.Dept. L.S.I.Universitat Politècnica de CatalunyaBarcelonaSpain
  2. 2.Fakultät für InformatikTechnische Universität MünchenMünchenGermany

Personalised recommendations