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Theory of Computing Systems

, Volume 63, Issue 3, pp 386–393 | Cite as

Nondeterminisic Sublinear Time Has Measure 0 in P

  • John M. HitchcockEmail author
  • Adewale Sekoni
Article
  • 22 Downloads

Abstract

The measure hypothesis is a quantitative strengthening of the \(\mathrm {P} \neq \text {NP}\) conjecture which asserts that \(\text {NP}\) is a nonnegligible subset of \(\text {EXP}\). Cai et al. (1997) showed that the analogue of this hypothesis in \(\mathrm {P}\) is false. In particular, they showed that \(\text {NTIME}[n^{1/11}]\) has measure 0 in \(\mathrm {P}\). We improve on their result to show that the class of all languages decidable in nondeterministic sublinear time has measure 0 in \(\mathrm {P}\). Our result is based on DNF width and holds for all four major notions of measure on \(\mathrm {P}\).

Keywords

Nondeterministic time DNF width Resource-bounded measure 

Notes

References

  1. 1.
    Allender, E., Rubinstein, R.: P-printable sets. SIAM J. Comput. 17, 1193–1202 (1988)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Allender, E., Strauss, M.: Measure on small complexity classes with applications for BPP. In: Proceedings of the 35th Symposium on Foundations of Computer Science, pp. 807–818. IEEE Computer Society (1994)Google Scholar
  3. 3.
    Allender, E., Strauss, M.: Measure on P Robustness of the notion. In: International Symposium on Mathematical Foundations of Computer Science, pp. 129–138. Springer (1995)Google Scholar
  4. 4.
    Ambos-Spies, K., Mayordomo, E.: Resource-bounded measure and randomness. In: Sorbi, A. (ed.) Complexity, Logic and Recursion Theory, Lecture Notes in Pure and Applied Mathematics, pp 1–47. Marcel Dekker, New York (1997)Google Scholar
  5. 5.
    Cai, J., Sivakumar, D., Strauss, M.: Constant-depth circuits and the Lutz hypothesis. In: Proceedings of the 38th Symposium on Foundations of Computer Science, pp. 595–604. IEEE Computer Society (1997)Google Scholar
  6. 6.
    Crama, Y., Hammer, P.L.: Boolean Functions - Theory, Algorithms, and Applications, volume 142 of Encyclopedia of Mathematics and Its Applications. Cambridge University Press (2011)Google Scholar
  7. 7.
    Lutz, J.H.: Almost everywhere high nonuniform complexity. J. Comput. Syst. Sci. 44(2), 220–258 (1992)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Lutz, J.H.: The quantitative structure of exponential time. In: Hemaspaandra, L.A., Selman, A.L. (eds.) Complexity Theory Retrospective II, pp 225–254. Springer (1997)Google Scholar
  9. 9.
    Lutz, J.H.: Dimension in complexity classes. SIAM J. Comput. 32(5), 1236–1259 (2003)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Lutz, J.H., Mayordomo, E.: Cook versus Karp-Levin: Separating completeness notions if NP is not small. Theor. Comput. Sci. 164(1–2), 141–163 (1996)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Lutz, J.H., Mayordomo, E.: Twelve problems in resource-bounded measure. Bull. European Assoc. Theor. Comput. Sci. 68, 64–80 (1999). Also in Current Trends in Theoretical Computer Science: Entering the 21st Century, pages 83–101 World Scientific Publishing, 2001MathSciNetzbMATHGoogle Scholar
  12. 12.
    Moser, P.: Martingale families and dimension in P. Theor. Comput. Sci. 400 (1-3), 46–61 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    O’Donnell, R.: Analysis of Boolean Functions. Cambridge University Press (2014)Google Scholar
  14. 14.
    Strauss, M.: Measure on P: Strength of the notion. Inf. Comput. 136(1), 1–23 (1997)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Department of Computer ScienceUniversity of WyomingLaramieUSA

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