Skip to main content
SpringerLink
Log in

On resource-bounded measure and pseudorandomness

  • Conference paper
  • First Online:
Foundations of Software Technology and Theoretical Computer Science (FSTTCS 1997)

Part of the book series: Lecture Notes in Computer Science ((LNCS,volume 1346))

Abstract

In this paper we extend a key result of Nisan and Wigderson [17] to the nondeterministic setting: for all α > 0 we show that if there is a language in E = DTIME(2O(n)) that is hard to approximate by nondeterministic circuits of size 2αn, then there is a pseudorandom generator that can be used to derandomize BP·NP (in symbols, BP·NP = NP). By applying this extension we are able to answer some open questions in [14] regarding the derandomization of the classes BP·σ Pk and BP·θ P k under plausible measure theoretic assumptions. As a consequence, if θ P2 does not have p-measure 0, then AM ∩ coAM is low for θ P2 . Thus, in this case, the graph isomorphism problem is low for θ P2 . By using the Nisan-Wigderson design of a pseudorandom generator we unconditionally show the inclusion MA ⊑ ZPPNPNP and that MA∩ coMA is low for ZPPNP.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution

Institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. E. Allender and M. Strauss. Measure on small complexity classes with applications for BPP. In Proc. 35th IEEE Symposium on the Foundations of Computer Science, 807–818. IEEE Computer Society Press, 1994.

    Google Scholar 

  2. A. Andreev, A. Clementi, and J. Rolim. Hitting sets derandomize BPP. In Proc. 23rd International Colloquium on Automata, Languages, and Programming, Lecture Notes in Computer Science #1099, 357–368. Springer-Verlag, 1996.

    Google Scholar 

  3. A. Andreev, A. Clementi, and J. Rolim. Worst-case hardness suffices for derandomization: a new method for hardness-randomness trade-offs. In Proc. 24th International Colloquium on Automata, Languages, and Programming, Lecture Notes in Computer Science #1256. Springer-Verlag, 1997.

    Google Scholar 

  4. L. Babai. Trading group theory for randomness. In Proc. 17th ACM Symposium on Theory of Computing, 421–429. ACM Press, 1985.

    Google Scholar 

  5. J. Balcázar, J. Díaz, and J. Gabarró. Structural Complexity II. Springer-Verlag, 1990.

    Google Scholar 

  6. J. Balcázar, J. Díaz, and J. Gabarró. Structural Complexity I. Springer-Verlag, second edition, 1995.

    Google Scholar 

  7. M. Bellare and S. Goldwasser. The complexity of decision versus search. SIAM Journal on Computing, 23:97–119, 1994.

    Article  MATH  MathSciNet  Google Scholar 

  8. L. Fortnow and N. Reingold. PP is closed under truth-table reductions. Information and Computation, 124(1):1–6, 1996.

    Article  MATH  MathSciNet  Google Scholar 

  9. R. Impagliazzo and A. Wigderson. P=BPP unless E has sub-exponential circuits: derandomizing the XOR lemma. In Proc. 29rd ACM Symposium on Theory of Computing. ACM Press, 1997.

    Google Scholar 

  10. J. Kadin. PNP[log n] and sparse Turing-complete sets for NP. Journal of Computer and System Sciences, 39:282–298, 1989.

    Article  MATH  MathSciNet  Google Scholar 

  11. R. M. Karp and R. J. Lipton. Some connections between nonuniform and uniform complexity classes. In Proc. 12th ACM Symposium on Theory of Computing, 302–309. ACM Press, 1980.

    Google Scholar 

  12. J. H. Lutz. Almost every where high nonuniform complexity. Journal of Computer and System Sciences, 44:220–258, 1992.

    Article  MATH  MathSciNet  Google Scholar 

  13. J. H. Lutz. A pseudorandom oracle characterization of BPP. SIAM Journal on Computing, 22:1075–1086, 1993.

    Article  MATH  MathSciNet  Google Scholar 

  14. J. H. Lutz. Observations on measure and lowness for δ P2 . Theory of Computing Systems, 30:429–442, 1997.

    Article  MATH  MathSciNet  Google Scholar 

  15. J. H. Lutz and E. Mayordomo. Cook versus Karp-Levin: separating reducibilities if NP is not small. Theoretical Computer Science, 164:141–163, 1996.

    Article  MATH  MathSciNet  Google Scholar 

  16. J. H. Lutz and W. J. Schmidt. Circuit size relative to pseudorandom oracles. Theoretical Computer Science, 107:95–120, 1993.

    Article  MATH  MathSciNet  Google Scholar 

  17. N. Nisan and A. Wigderson. Hardness vs randomness. Journal of Computer and System Sciences, 49:149–167, 1994.

    Article  MATH  MathSciNet  Google Scholar 

  18. C. Papadimitriou. Computational Complexity, Addison-Wesley, 1994.

    Google Scholar 

  19. S. Rudich. Super-bits, demi-bits, and NQP-natural proofs. In Proc. 1st Intern. Symp. on Randomization and Approximation Techniques in Computer Science (Random'97), Lecture Notes in Computer Science #1269. Springer-Verlag, 1997.

    Google Scholar 

  20. M. Santha. Relativized Arthur-Merlin versus Merlin-Arthur games. Information and Computation, 80(1):44–49, 1989.

    Article  MATH  MathSciNet  Google Scholar 

  21. U. Schöning. Probabilistic complexity classes and lowness. Journal of Computer and System Sciences, 39:84–100, 1989.

    Article  MATH  MathSciNet  Google Scholar 

  22. A. Shamir. On the generation of cryptographically strong pseudo-random sequences. In Proc. 8th International Colloquium on Automata, Languages, and Programming, Lecture Notes in Computer Science #62, 544–550. Springer-Verlag, 1981.

    Google Scholar 

  23. S. Skyum and L.G. Valiant. A complexity theory based on boolean algebra. Journal of the ACM, 32:484–502, 1985.

    Article  MATH  MathSciNet  Google Scholar 

  24. S. Toda. PP is as hard as the polynomial-time hierarchy. SIAM Journal on Computing, 20:865–877, 1991.

    Article  MATH  MathSciNet  Google Scholar 

  25. A. C. Yao. Theory and applications of trapdoor functions. In Proc. 23rd IEEE Symposium on the Foundations of Computer Science, 80–91. IEEE Computer Society Press, 1982.

    Google Scholar 

  26. S. Zachos and M. Fürer. Probabilistic quantifiers vs. distrustful adversaries. In Proc. 7th Conference on Foundations of Software Technology and Theoretical Computer Science, Lecture Notes in Computer Science #287, 443–455. Springer-Verlag, 1987.

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Institute of Mathematical Sciences, 600113, Chennai, India

    V. Arvind

  2. Theoretische Informatik, UniversitÄt Ulm, D-89069, Ulm, Germany

    J. Köbler

Authors
  1. V. Arvind
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. J. Köbler
    View author publications

    You can also search for this author in PubMed Google Scholar

Editor information

S. Ramesh G Sivakumar

Rights and permissions

Reprints and permissions

Copyright information

© 1997 Springer-Verlag Berlin Heidelberg

About this paper

Cite this paper

Arvind, V., Köbler, J. (1997). On resource-bounded measure and pseudorandomness. In: Ramesh, S., Sivakumar, G. (eds) Foundations of Software Technology and Theoretical Computer Science. FSTTCS 1997. Lecture Notes in Computer Science, vol 1346. Springer, Berlin, Heidelberg. https://doi.org/10.1007/BFb0058034

Download citation

  • DOI: https://doi.org/10.1007/BFb0058034

  • Published:

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-63876-6

  • Online ISBN: 978-3-540-69659-9

  • eBook Packages: Springer Book Archive

Publish with us

Policies and ethics

Search

Navigation