computational complexity

, Volume 21, Issue 1, pp 3–61 | Cite as

Pseudorandom Generators, Typically-Correct Derandomization, and Circuit Lower Bounds

  • Jeff Kinne
  • Dieter van Melkebeek
  • Ronen Shaltiel
Article

Abstract

The area of derandomization attempts to provide efficient deterministic simulations of randomized algorithms in various algorithmic settings. Goldreich and Wigderson introduced a notion of “typically-correct” deterministic simulations, which are allowed to err on few inputs. In this paper, we further the study of typically-correct derandomization in two ways.

First, we develop a generic approach for constructing typically-correct derandomizations based on seed-extending pseudorandom generators, which are pseudorandom generators that reveal their seed. We use our approach to obtain both conditional and unconditional typically-correct derandomization results in various algorithmic settings. We show that our technique strictly generalizes an earlier approach by Shaltiel based on randomness extractors and simplifies the proofs of some known results. We also demonstrate that our approach is applicable in algorithmic settings where earlier work did not apply. For example, we present a typically-correct polynomial-time simulation for every language in BPP based on a hardness assumption that is (seemingly) weaker than the ones used in earlier work.

Second, we investigate whether typically-correct derandomization of BPP implies circuit lower bounds. Extending the work of Kabanets and Impagliazzo for the zero-error case, we establish a positive answer for error rates in the range considered by Goldreich and Wigderson. In doing so, we provide a simpler proof of the zero-error result. Our proof scales better than the original one and does not rely on the result by Impagliazzo, Kabanets, and Wigderson that NEXP having polynomialsize circuits implies that NEXP coincides with EXP.

Keywords

Typically-correct derandomization pseudorandom generators circuit lower bounds randomized algorithms 

Subject classification

68Q10 68Q15 68Q17 68Q25 03D15 

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References

  1. Scott Aaronson Dieter van Melkebeek (2010). A note on circuit lower bounds from derandomization. Electronic Colloquium on Computational Complexity (ECCC), 17(105).Google Scholar
  2. Aaronson Scott, Wigderson Avi (2009) Algebrization: A new barrier in complexity theory. ACM Transactions on Computation Theory, 1(1): 1–54CrossRefGoogle Scholar
  3. László Babai, Lance Fortnow, Leonid A. Levin & Mario Szegedy (1991). Checking computations in polylogarithmic time. In Proceedings of the ACM Symposium on Theory of Computing (STOC), pages 21–31.Google Scholar
  4. László Babai, Lance Fortnow, Noam Nisan & Avi Wigderson (1993). BPP has subexponential time simulations unless EXPTIME has publishable proofs. Computational Complexity, 3, 307–318.Google Scholar
  5. Babai László, Fortnow Lance, Nisan Noam, Wigderson Avi (1993) BPP has subexponential time simulations unless EXPTIME has publishable proofs. Computational Complexity 3: 307–318MathSciNetMATHCrossRefGoogle Scholar
  6. Aviad Cohen & Avi Wigderson (1989). Dispersers, deterministic amplification, & weak random sources (extended abstract). In Proceedings of the IEEE Symposium on Foundations of Computer Science (FOCS), pages 14–19.Google Scholar
  7. Oded Goldreich, Noam Nisan & Avi Wigderson (1995). On Yao’s XOR-lemma. Electronic Colloquium on Computational Complexity (ECCC), 2(50).Google Scholar
  8. Oded Goldreich & Avi Wigderson (2000). On pseudorandomness with respect to deterministic observers. In Carleton Scientific, editor, International Colloquium on Automata, Languages and Programming (ICALP), pages 77–84.Google Scholar
  9. Oded Goldreich & Avi Wigderson (2002). Derandomization that is rarely wrong from short advice that is typically good. In Proceedings of the International Workshop on Randomization and Computation (RANDOM), pages 209–223.Google Scholar
  10. Gutfreund Dan, Shaltiel Ronen, Ta-Shma Amnon (2003) Uniform hardness versus randomness tradeoffs for Arthur-Merlin games. Computational Complexity 12(3-4): 85–130MathSciNetMATHCrossRefGoogle Scholar
  11. Håstad Johan (1987) Computational limitations of small-depth circuits. MIT Press, Cambridge, MA, USA.Google Scholar
  12. Impagliazzo Russell, Kabanets Valentine, Wigderson Avi (2002) In search of an easy witness: exponential time vs probabilistic polynomial time. Journal of Computer and System Sciences 65(4): 672–694MathSciNetMATHCrossRefGoogle Scholar
  13. Russell Impagliazzo (1995). Hard-core distributions for somewhat hard problems. In Proceedings of the IEEE Symposium on Foundations of Computer Science (FOCS), pages 538–545.Google Scholar
  14. Russell Impagliazzo & Avi Wigderson (1997). P = BPP if E requires exponential circuits: Derandomizing the XOR lemma. In Proceedings of the ACM Symposium on Theory of Computing (STOC), pages 220–229.Google Scholar
  15. Impagliazzo Russell, Wigderson Avi (2001) Randomness vs time: Derandomization under a uniform assumption. Journal of Computer and System Sciences 63(4): 672–688MathSciNetMATHCrossRefGoogle Scholar
  16. Russell Impagliazzo & David Zuckerman (1989). How to recycle random bits. In Proceedings of the IEEE Symposium on Foundations of Computer Science (FOCS), pages 248–253.Google Scholar
  17. Kabanets Valentine (2001) Easiness assumptions and hardness tests: Trading time for zero error. Journal of Computer and System Sciences 63(2): 236–252MathSciNetMATHCrossRefGoogle Scholar
  18. Kabanets Valentine, Impagliazzo Russell (2004) Derandomizing polynomial identity tests means proving circuit lower bounds. Computational Complexity 13(1/2): 1–46MathSciNetMATHCrossRefGoogle Scholar
  19. Kannan Ravi (1982) Circuit-size lower bounds and nonreducibility to sparse sets. Information and Control 55(1): 40–56MathSciNetMATHCrossRefGoogle Scholar
  20. Klivans Adam R., van Melkebeek Dieter (2002) Graph nonisomorphism has subexponential size proofs unless the polynomial-time hierarchy collapses. SIAM Journal on Computing 31(5): 1501–1526MathSciNetMATHCrossRefGoogle Scholar
  21. van Melkebeek Dieter, Santhanam Rahul (2005) Holographic proofs and derandmization. SIAM Journal on Computing 35(1): 59–90MathSciNetMATHCrossRefGoogle Scholar
  22. Peter Bro Miltersen (2001). Derandomizing complexity classes. In Handbook of Randomized Computing, pages 843–941. Kluwer Academic Publishers.Google Scholar
  23. Miltersen Peter Bro, Vinodchandran N. V. (2005) Derandomizing Arthur-Merlin games using hitting sets. Computational Complexity 14(3): 256–279MathSciNetMATHCrossRefGoogle Scholar
  24. Newman Ilan (1991) Private vs common random bits in communication complexity. Information Processing Letters 39(2): 67–71MathSciNetMATHCrossRefGoogle Scholar
  25. Nisan Noam (1991) Pseudorandom bits for constant depth circuits. Combinatorica 11(1): 63–70MathSciNetMATHCrossRefGoogle Scholar
  26. Nisan Noam (1993) On read-once vs multiple access to randomness in logspace. Theoretical Computer Science 107(1): 135–144MathSciNetMATHCrossRefGoogle Scholar
  27. Nisan Noam, Wigderson Avi (1994) Hardness vs randomness. Journal of Computer and System Sciences 49(2): 149–167MathSciNetMATHCrossRefGoogle Scholar
  28. Omer Reingold (2008). Undirected connectivity in log-space. Journal of the ACM, 55(4).Google Scholar
  29. Ronen Shaltiel (2009). Weak derandomization of weak algorithms: explicit versions of Yao’s lemma. In Proceedings of the IEEE Conference on Computational Complexity.Google Scholar
  30. Shaltiel Ronen, Umans Christopher (2005) Simple extractors for all min-entropies and a new pseudorandom generator. Journal of the ACM 52(2): 172–216MathSciNetCrossRefGoogle Scholar
  31. Shaltiel Ronen, Umans Christopher (2006) Pseudorandomness for approximate counting and sampling. Computational Complexity 15(4): 298–341MathSciNetMATHCrossRefGoogle Scholar
  32. Ronen Shaltiel & Christopher Umans (2007). Low-end uniform hardness vs. randomness tradeoffs for AM. In Proceedings of the ACM Symposium on Theory of Computing (STOC), pages 430–439.Google Scholar
  33. Shamir Adi (1992) IP = PSPACE. Journal of the ACM 39(4): 869–877MathSciNetMATHCrossRefGoogle Scholar
  34. Saks Michael E., Zhou Shiyu (1999) BPHSPACE(S) \({\subseteq}\) DSPACE(S3/2). Journal of Computer and System Sciences 58(2): 376–403MathSciNetMATHCrossRefGoogle Scholar
  35. Toda Seinosuke (1991) PP is as hard as the polynomial-time hierarchy. SIAM Journal on Computing 20(5): 865–877MathSciNetMATHCrossRefGoogle Scholar
  36. Trevisan Luca, Vadhan Salil P. (2007) Pseudorandomness and average-case complexity via uniform reductions. Computational Complexity 16(4): 331–364MathSciNetMATHCrossRefGoogle Scholar
  37. Viola Emanuele (2005) The complexity of constructing pseudorandom generators from hard functions. Computational Complexity 13(3-4): 147–188MathSciNetCrossRefGoogle Scholar
  38. Viola Emanuele (2006) Pseudorandom bits for constant-depth circuits with few arbitrary symmetric gates. SIAM Journal on Computing 36(5): 1387–1403MathSciNetCrossRefGoogle Scholar
  39. Wilson Christopher B. (1985) Relativized circuit complexity. Journal of Computer and System Sciences 31(2): 169–181MathSciNetMATHCrossRefGoogle Scholar
  40. Zanko Viktoria (1991) #P-completeness via many-one reductions. International Journal of Foundations of Computer Science 2(1): 77–82MathSciNetMATHCrossRefGoogle Scholar
  41. Marius Zimand (2006). Exposure-resilient extractors. In Proceedings of the IEEE Conference on Computational Complexity, pages 61–72.Google Scholar
  42. Zimand Marius (2008) Exposure-resilient extractors and the derandomization of probabilistic sublinear time. Computational Complexity, 17(2): 220–253MathSciNetMATHCrossRefGoogle Scholar

Copyright information

© Springer Basel AG 2011

Authors and Affiliations

  • Jeff Kinne
    • 1
  • Dieter van Melkebeek
    • 2
  • Ronen Shaltiel
    • 3
  1. 1.Department of Math and Computer ScienceIndiana State UniversityTerre HauteUSA
  2. 2.Department of Computer SciencesUniversity of Wisconsin-MadisonMadisonUSA
  3. 3.Department of Computer ScienceUniversity of HaifaHaifaIsrael

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