Mathematical systems theory

, Volume 28, Issue 2, pp 117–140 | Cite as

Adaptive logspace reducibility and parallel time

  • C. Àlvarez
  • J. L. Balcázar
  • B. Jenner


We discuss two notions of functional oracle for logarithmic space-bounded machines, which differ in whether there is only one oracle tape for both the query and the answer or a separate tape for the answer, which can still be read while the next query is already being constructed. The first notion turns out to be basically nonadaptive, behaving like access to an oracle set. The second notion, on the other hand, is adaptive. By imposing appropriate bounds on the number of functional oracle queries made in this computation model, we obtain new characterizations of the NC and AC hierarchies; thus the number of oracle queries can be considered as a measure of parallel time. Using this characterization of parallel classes, we solve open questions of Wilson.


Computation Model Computational Mathematic Parallel Classis Parallel Time Oracle Query 
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.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. [1]
    C. Àlvarez, B. Jenner, A very hard log-space counting class,Theoretical Computer Science 107 (1993), 3–30.Google Scholar
  2. [2]
    J. L. Balcázar, J. Díaz, J. Gabarró,Structural Complexity, Vol. I, Springer-Verlag, Berlin, 1988.Google Scholar
  3. [3]
    J. L. Balcázar, J. Díaz, J. Gabarró,Structural Complexity, Vol. II, Springer-Verlag, Berlin, 1990.Google Scholar
  4. [4]
    D. A. Mix Barrington, N. Immerman, H. Straubing, On uniformity within NC1,Journal of Computer and System Sciences 41(3) (1990), 274–306.Google Scholar
  5. [5]
    A. Borodin, On relating time and space to size and depth,SIAM Journal of Computing 6(4) (1977), 733–744.Google Scholar
  6. [6]
    A. Borodin, S. A. Cook, P. Dymond, W. L. Ruzzo, M. Tompa, Two applications of complementation via inductive counting,SIAM Journal of Computing 18(3) (1989), 559–578.Google Scholar
  7. [7]
    S. Buss, The Boolean formula value problem is in ALOGTIME,Proc. 19th Annual ACM Symposium on Theory of Computing (1987), pp. 123–131.Google Scholar
  8. [8]
    S. R. Buss, L. Hay, On truth-table reducibility to SAT and the difference hierarchy over NP,Proc. 3rd Structure in Complexity Theory Conference (1988), pp. 224–233.Google Scholar
  9. [9]
    J. Castro, C. Seara, Characterizations of some complexity classes between Θ2p and Δ2p. Report LSI-90-27, Universitat Politècnica de Catalunya, Barcelona.Google Scholar
  10. [10]
    J. Castro, C. Seara, The Θ-operator and the low hierarchy. Report LSI-92-16-R, Universitat Politècnica de Catalunya, Barcelona.Google Scholar
  11. [11]
    S. A. Cook, A taxonomy of problems with fast parallel algorithms,Information and Control 64 (1985), 2–22.Google Scholar
  12. [12]
    L. A. Hemachandra, The strong exponential hierarchy collapses,Proc. 19th Annual ACM Symposium on Theory of Computing (1987), pp. 110–122.Google Scholar
  13. [13]
    J.-W. Hong, On simularity and duality of computation (I),Information and Control 62 (1984), 109–128.Google Scholar
  14. [14]
    J.-W. Hong,Computation: Computability, Similarity and Duality, Pitman, London, 1986.Google Scholar
  15. [15]
    B. Jenner, B. Kirsig, Alternierung und Logarithmischer Platz, Dissertation (in German), Universität Hamburg, 1989.Google Scholar
  16. [16]
    J. Kadin, PNP[log n] and sparse Turing-complete sets for NP,Proc. 2nd Structure in Complexity Theory Conference (1987), pp. 33–40.Google Scholar
  17. [17]
    M. W. Krentel, The complexity of optimization problems,Proc. 18th Annual ACM Symposium on Theory of Computing (1986), pp. 69–76.Google Scholar
  18. [18]
    R. Ladner, N. Lynch, Relativization of questions about log space computability,Mathematical Systems Theory 10 (1976), 19–32.Google Scholar
  19. [19]
    R. Ladner, N. Lynch, A. Selman, Comparison of polynomial-time reducibilities,Theoretical Computer Science 1 (1975), 103–123.Google Scholar
  20. [20]
    W. Ruzzo, On uniform circuit complexity,Journal of Computer and System Sciences 22 (1981), 365–383.Google Scholar
  21. [21]
    K. Wagner, Bounded query classes,SIAM Journal on Computing 19(5) (1990), 833–846.Google Scholar
  22. [22]
    C. B. Wilson, Relativized NC,Mathematical Systems Theory 20 (1987), 13–29.Google Scholar
  23. [23]
    C. B. Wilson, Decomposing NC and AC,SIAM Journal on Computing 19(2) (1990), 384–396. (Preliminary version at 4th Structure in Complexity Theory Conference, 1989.)Google Scholar

Copyright information

© Springer-Verlag 1995

Authors and Affiliations

  • C. Àlvarez
    • 1
  • J. L. Balcázar
    • 1
  • B. Jenner
    • 2
  1. 1.Department of Software (Llenguatges i Sistemes Informàtics)Universitat Politècnica de CatalunyaBarcelonaSpain
  2. 2.Fakultät für InformatikTechnische Universität MünchenMünchenGermany

Personalised recommendations