P-Selectivity, Immunity, and the Power of One Bit

  • Lane A. Hemaspaandra
  • Leen Torenvliet
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 3831)

Abstract

We prove that P-sel, the class of all P-selective sets, is EXP-immune, but is not EXP/1-immune. That is, we prove that some infinite P-selective set has no infinite EXP-time subset, but we also prove that every infinite P-selective set has some infinite subset in EXP/1. Informally put, the immunity of P-sel is so fragile that it is pierced by a single bit of information.

The above claims follow from broader results that we obtain about the immunity of the P-selective sets. In particular, we prove that for every recursive function f, P-sel is DTIME(f)-immune. Yet we also prove that P-sel is not \({\it \Pi}^{p}_{2}\)/1-immune.

Keywords

Turing Machine SIAM Journal Recursive Function Random Oracle Selector Function 
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.
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References

  1. 1.
    Berman, L.: On the Structure of Complete Sets. In: Proceedings of the 17th IEEE Symposium on Foundations of Computer Science, October 1976, pp. 76–80. IEEE Computer Society, Los Alamitos (1976)Google Scholar
  2. 2.
    Cai, J.: Probability one Separation of the Boolean Hierarchy. In: Brandenburg, F.J., Wirsing, M., Vidal-Naquet, G. (eds.) STACS 1987. LNCS, vol. 247, pp. 148–158. Springer, Heidelberg (1987)CrossRefGoogle Scholar
  3. 3.
    Cai, J., Gundermann, T., Hartmanis, J., Hemachandra, L., Sewelson, V., Wagner, K., Wechsung, G.: The Boolean Hierarchy I: Structural Properties. SIAM Journal on Computing 17(6), 1232–1252 (1988)CrossRefMathSciNetMATHGoogle Scholar
  4. 4.
    Hemaspaandra, L., Hempel, H., Nickelsen, A.: Algebraic Properties for Selector Functions. SIAM Journal on Computing 33(6), 1309–1337 (2004)CrossRefMathSciNetMATHGoogle Scholar
  5. 5.
    Hemaspaandra, E., Hemaspaandra, L., Watanabe, O.: The Complexity of Kings. Technical Report TR-870, Department of Computer Science, University of Rochester, Rochester, NY (June 2005)Google Scholar
  6. 6.
    Hemaspaandra, L., Naik, A., Ogihara, M., Selman, A.: Computing Solutions Uniquely Collapses the Polynomial Hierarchy. SIAM Journal on Computing 25(4), 697–708 (1996)CrossRefMathSciNetMATHGoogle Scholar
  7. 7.
    Hemaspaandra, L., Ogihara, M.: The Complexity Theory Companion. Springer, Heidelberg (2002)MATHGoogle Scholar
  8. 8.
    Hemaspaandra, L., Ogihara, M., Zaki, M., Zimand, M.: The Complexity of Finding Top-Toda-Equivalence-Class Members MINOR PANIC: hem-ogi-zak-zim: Missing year/issue/voume, and remove. Theory of Computing Systems (to appear); Farach-Colton, M. (ed.) LATIN 2004. LNCS, vol. 2976, pp. 90–99. Springer, Heidelberg (2004)Google Scholar
  9. 9.
    Hemaspaandra, L., Torenvliet, L.: Optimal Advice. Theoretical Computer Science 154(2), 367–377 (1996)CrossRefMathSciNetMATHGoogle Scholar
  10. 10.
    Hemaspaandra, L., Torenvliet, L.: Theory of Semi-Feasible Algorithms. Springer, Heidelberg (2003)MATHGoogle Scholar
  11. 11.
    Karp, R., Lipton, R.: Some Connections between Nonuniform and Uniform Complexity Classes. In: Proceedings of the 12th ACM Symposium on Theory of Computing, pp. 302–309. ACM Press, New York (1980); An extended version has also appeared as: Turing Machines that Take Advice, L’Enseignement Mathématique. 2nd series, vol. 28, pp. 191–209 (1982)Google Scholar
  12. 12.
    Ko, K.: On Self-Reducibility and Weak P-Selectivity. Journal of Computer and System Sciences 26(2), 209–221 (1983)CrossRefMathSciNetMATHGoogle Scholar
  13. 13.
    Landau, H.: On Dominance Relations and the Structure of Animal Societies, III: The Condition for Score Structure. Bulletin of Mathematical Biophysics 15(2), 143–148 (1953)CrossRefGoogle Scholar
  14. 14.
    Nickelsen, A., Tantau, T.: On Reachability in Graphs with Bounded Independence Number. In: Ibarra, O.H., Zhang, L. (eds.) COCOON 2002. LNCS, vol. 2387, pp. 554–563. Springer, Heidelberg (2002)CrossRefGoogle Scholar
  15. 15.
    Nickelsen, A., Tantau, T.: Partial information classes. SIGACT News 34(1), 32–46 (2003)CrossRefGoogle Scholar
  16. 16.
    Rogers Jr., H.: The Theory of Recursive Functions and Effective Computability. McGraw-Hill, New York (1967)Google Scholar
  17. 17.
    Schöning, U., Book, R.: Immunity, Relativization, and Nondeterminism. SIAM Journal on Computing 13, 329–337 (1984)CrossRefMathSciNetMATHGoogle Scholar
  18. 18.
    Selman, A.: P-Selective Sets, Tally Languages, and the Behavior of Polynomial Time Reducibilities on NP. Mathematical Systems Theory 13(1), 55–65 (1979)CrossRefMathSciNetMATHGoogle Scholar
  19. 19.
    Selman, A.: Some Observations on NP Real Numbers and P-Selective Sets. Journal of Computer and System Sciences 23(3), 326–332 (1981)CrossRefMathSciNetMATHGoogle Scholar
  20. 20.
    Selman, A.: Analogues of Semirecursive Sets and Effective Reducibilities to the Study of NP Complexity. Information and Control 52(1), 36–51 (1982)CrossRefMathSciNetMATHGoogle Scholar
  21. 21.
    Selman, A.: Reductions on NP and P-Selective Sets. Theoretical Computer Science 19(3), 287–304 (1982)CrossRefMathSciNetMATHGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2006

Authors and Affiliations

  • Lane A. Hemaspaandra
    • 1
  • Leen Torenvliet
    • 2
  1. 1.Department of Computer ScienceUniversity of RochesterRochesterUSA
  2. 2.ILLCUniversity of AmsterdamAmsterdamThe Netherlands

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