On sparse oracles separating feasible complexity classes

  • Juris Hartmanis
  • Lane Hemachandra
Contributed Papers
Part of the Lecture Notes in Computer Science book series (LNCS, volume 210)


This note clarifies which oracles separate NP from P and which do not. In essence, we are changing our research paradigm from the study of which problems can be relativized in two conflicting ways to the study and characterization of the class of oracles achieving a specified relativization. Results of this type have the potential to yield deeper insights into the nature of relativization problems and focus our attention on new and interesting classes of languages.

A complete and transparent characterization of oracles that separate NP from P would resolve the long-standing P=?NP question. In this note, we settle a central case. We fully characterize the sparse oracles separating NP from P in worlds where P=NP. We display related results about coNP, E, NE, coNE, and PSPACE.


Computation Path Kolmogorov Complexity Advice Function Oracle Query Polynomial Hierarchy 
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]
    T. Baker, J. Gill, and R. Solovay, “Relativizations of the P=?NP Question,” SIAM Journal on Computing, 1975, pp. 431–442.Google Scholar
  2. [2]
    J. Balcázar, R. Book, T. Long, U. Schöning, and A. Selman, “Sparse Oracles and Uniform Complexity Classes,” FOCS 1984, pp. 308–313.Google Scholar
  3. [3]
    A. Chandra, D. Kozen, and L. Stockmeyer, “Alternation,” JACM, V. 26, #1, 1981.Google Scholar
  4. [4]
    M. Furst, J. Saxe, and M. Sipser, “Parity, Circuits, and the Polynomial-Time Hierarchy,” FOCS 1981, pp. 260–270.Google Scholar
  5. [5]
    W. Gasrarch, “Recursion Theoretic Techniques in Complexity Theory and Combinatorics,” Center for Research in Computing and Technology Report TR-09-85, Harvard University, May 1985.Google Scholar
  6. [6]
    J. Hartmanis, "Generalized Kolmogorov Complexity and the Structure of Feasible Computations,” Cornell Department of Computer Science Technical Report TR 83-573, September 1983.Google Scholar
  7. [7]
    J. Hartmanis, to appear in EATCS Bulletin.Google Scholar
  8. [8]
    J. Hartmanis and Y. Yesha, “Computation Times of NP Sets of Different Densities,” Theoretical Computer Science, V. 34, 1984, pp. 17–32.CrossRefGoogle Scholar
  9. [9]
    K-I. Ko, “On Self-reducibility and Weak P-Selectivity,” Journal of Computer and System Sciences, V. 26, 1983, pp. 209–221.CrossRefGoogle Scholar
  10. [10]
    R. Karp and R. Lipton, “Some Connections Between Nonuniform and Uniform Complexity Classes,” STOC 1980, pp. 302–309.Google Scholar
  11. [11]
    S. Mahaney “Sparse Complete Sets for NP: Solution of a Conjecture of Berman and Hartmanis,” FOCS 1980, pp. 54–60.Google Scholar
  12. [12]
    M. Sipser, “A Complexity Theoretic Approach to Randomness,” STOC 1983, pp. 330–335.Google Scholar
  13. [13]
    C. Wrathall, “Complete Sets and the Polynomial-time Hierarchy,” Theoretical Computer Science, V. 3, 1977, pp. 23–33.CrossRefGoogle Scholar
  14. [14]
    C. Yap, “Some Consequences of Non-uniform Conditions on Uniform Classes,” Theoretical Computer Science, V. 26, 1983, pp.287–300.CrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 1985

Authors and Affiliations

  • Juris Hartmanis
    • 1
  • Lane Hemachandra
    • 1
  1. 1.Department of Computer ScienceCornell UniversityUSA

Personalised recommendations