Advertisement

Hardness Hypotheses, Derandomization, and Circuit Complexity

  • John M. Hitchcock
  • A. Pavan
Part of the Lecture Notes in Computer Science book series (LNCS, volume 3328)

Abstract

We consider three complexity-theoretic hypotheses that have been studied in different contexts and shown to have many plausible consequences.

The Measure Hypothesis: NP does not have p-measure 0.

The pseudo-NP Hypothesis: there is an NP Language L such that any DTIME \(2^{{n^\epsilon}}\) Language L’ can be distinguished from L by an NP refuter.

The NP-Machine Hypothesis: there is an NP machine accepting 0* for which no \(2^{{n^\epsilon}}\)-time machine can find infinitely many accepting computations.

We show that the NP-machine hypothesis is implied by each of the first two. Previously, no relationships were known among these three hypotheses. Moreover, we unify previous work by showing that several derandomization and circuit-size lower bounds that are known to follow from the first two hypotheses also follow from the NP-machine hypothesis. We also consider UP versions of the above hypotheses as well as related immunity and scaled dimension hypotheses.

Keywords

Boolean Function Circuit Complexity Measure Hypothesis Oracle Gate Oracle Circuit 
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.
    Allender, E.: When the worlds collide: derandomization, lower bounds and kolmogorov complexity. In: Foundations of Software Technology and Theoretical Computer Science, pp. 1–15. Springer, Heidelberg (2001)Google Scholar
  2. 2.
    Allender, E., Strauss, M.: Measure on small complexity classes with applications for BPP. In: Proceedings of the 35th Annual Symposium on Foundations of Computer Science, pp. 807–818. IEEE Computer Society, Los Alamitos (1994)CrossRefGoogle Scholar
  3. 3.
    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
  4. 4.
    Balcázar, J., Schöning, U.: Bi-immune sets for complexity classes. Mathematical Systems Theory 18(1), 1–18 (1985)zbMATHCrossRefMathSciNetGoogle Scholar
  5. 5.
    Bshouty, N., Cleve, R., Kannan, S., Gavalda, R., Tamon, C.: Oracles and queries that are sufficient for exact learning. Journal of Computer and System Sciences 52, 421–433 (1996)zbMATHCrossRefMathSciNetGoogle Scholar
  6. 6.
    Fenner, S., Fortnow, L., Naik, A., Rogers, J.: Inverting onto functions. Information and Computation 186(1), 90–103 (2003)zbMATHCrossRefMathSciNetGoogle Scholar
  7. 7.
    Glaßer, C., Pavan, A., Selman, A.L., Sengupta, S.: Properties of NP-complete sets. In: Proceedings of the 19th IEEE Conference on Computational Complexity, pp. 184–197. IEEE Computer Society, Los Alamitos (2004)CrossRefGoogle Scholar
  8. 8.
    Hemaspaandra, L.A., Rothe, J., Wechsung, G.: Easy sets and hard certificate schemes. Acta Informatica 34(11), 859–879 (1997)zbMATHCrossRefMathSciNetGoogle Scholar
  9. 9.
    Hitchcock, J.M.: Small spans in scaled dimension. SIAM Journal on Computing.(to appear)Google Scholar
  10. 10.
    Hitchcock, J.M.: MAX3SAT is exponentially hard to approximate if NP has positive dimension. Theoretical Computer Science 289(1), 861–869 (2002)zbMATHCrossRefMathSciNetGoogle Scholar
  11. 11.
    Hitchcock, J.M., Lutz, J.H., Mayordomo, E.: Scaled dimension and nonuniform complexity. Journal of Computer and System Sciences 69(2), 97–122 (2004)zbMATHCrossRefMathSciNetGoogle Scholar
  12. 12.
    Impagliazzo, R., Kabanets, V., Wigderson, A.: In search of an easy witness: Exponential time vs. probabilistic polynomial time. Journal of Computer and System Sciences 65, 672–694 (2002)zbMATHCrossRefMathSciNetGoogle Scholar
  13. 13.
    Impagliazzo, R., Moser, P.: A zero-one law for RP. In: Proceedings of the 18th IEEE Conference on Computational Complexity, pp. 43–47. IEEE Computer Society, Los Alamitos (2003)Google Scholar
  14. 14.
    Impagliazzo, R., Wigderson, A.: P = BPP if E requires exponential circuits: Derandomizing the XOR lemma. In: Proceedings of the 29th ACM Symposium on Theory of Computing, pp. 220–229 (1997)Google Scholar
  15. 15.
    Kabanets, V.: Easiness assumptions and hardness tests: Trading time for zero error. Journal of Computer and System Sciences 63, 236–252 (2001)zbMATHCrossRefMathSciNetGoogle Scholar
  16. 16.
    Kabanets, V.: Derandomization: A brief overview. Bulletin of the EATCS 76, 88–103 (2002)zbMATHMathSciNetGoogle Scholar
  17. 17.
    Kannan, R.: Circuit-size lower bounds and non-reducibility to sparse sets. Information and Control 55, 40–56 (1982)zbMATHCrossRefMathSciNetGoogle Scholar
  18. 18.
    Klivans, A., van Melkebeek, D.: Graph nonisomorphism has subexponential size proofs unless the polynomial-time hierarchy collapses. SIAM Journal on Computing 31, 1501–1526 (2002)zbMATHCrossRefMathSciNetGoogle Scholar
  19. 19.
    Köbler, J., Watanabe, O.: New collapse consequences of NP having small circuits. SIAM Journal on Computing 28(1), 311–324 (1998)zbMATHCrossRefGoogle Scholar
  20. 20.
    Lu, C.-J.: Derandomizing Arthur-Merlin games under uniform assumptions. Computational Complexity 10(3), 247–259 (2001)CrossRefMathSciNetGoogle Scholar
  21. 21.
    Lutz, J.H.: Almost everywhere high nonuniform complexity. Journal of Computer and System Sciences 44(2), 220–258 (1992)zbMATHCrossRefMathSciNetGoogle Scholar
  22. 22.
    Lutz, J.H.: Observations on measure and lowness for \(\Delta^{\mathrm{P}}_2\). Theory of Computing Systems 30(4), 429–442 (1997)zbMATHMathSciNetGoogle Scholar
  23. 23.
    Lutz, J.H.: The quantitative structure of exponential time. In: Hemaspaandra, L.A., Selman, A.L. (eds.) Complexity Theory Retrospective II, pp. 225–260. Springer, New York (1997)Google Scholar
  24. 24.
    Lutz, J.H., Mayordomo, E.: Measure, stochasticity, and the density of hard languages. SIAM Journal on Computing 23(4), 762–779 (1994)zbMATHCrossRefMathSciNetGoogle Scholar
  25. 25.
    Lutz, J.H., Mayordomo, E.: Cook versus Karp-Levin: Separating completeness notions if NP is not small. Theoretical Computer Science 164, 141–163 (1996)zbMATHCrossRefMathSciNetGoogle Scholar
  26. 26.
    Miltersen, P.B.: Derandomizing complexity classes. In: Handbook of Randomized Computing, vol. II, pp. 843–934. Kluwer, Dordrecht (2001)Google Scholar
  27. 27.
    Pavan, A., Selman, A.L.: Separation of NP-completeness notions. SIAM Journal on Computing 31(3), 906–918 (2002)zbMATHCrossRefMathSciNetGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2004

Authors and Affiliations

  • John M. Hitchcock
    • 1
  • A. Pavan
    • 2
  1. 1.Department of Computer ScienceUniversity of Wyoming 
  2. 2.Department of Computer ScienceIowa State University 

Personalised recommendations