computational complexity

, Volume 3, Issue 4, pp 307–318

BPP has subexponential time simulations unlessEXPTIME has publishable proofs

  • Lźszló Babai
  • Lance Fortnow
  • Noam Nisan
  • Avi Wigderson
Article

Abstract

We show thatBPP can be simulated in subexponential time for infinitely many input lengths unless exponential time
  • ℴ collapses to the second level of the polynomial-time hierarchy.

  • ℴ has polynomial-size circuits and

  • ℴ has publishable proofs (EXPTIME=MA).

We also show thatBPP is contained in subexponential time unless exponential time has publishable proofs for infinitely many input lengths. In addition, we showBPP can be simulated in subexponential time for infinitely many input lengths unless there exist unary languages inMA-P.

The proofs are based on the recent characterization of the power of multiprover interactive protocols and on random self-reducibility via low-degree polynomials. They exhibit an interplay between Boolean circuit simulation, interactive proofs and classical complexity classes. An important feature of this proof is that it does not relativize.

One of the ingredients of our proof is a lemma that states that ifEXPTIME has polynomial size circuits thenEXPTIME=MA. This extends previous work by Albert Meyer.

Key words

Complexity Classes Interactive Proof Systems 

Subject classifications

68Q15 

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References

  1. L. Adleman, Two theorems on random polynomial time, inProceedings of the 19th IEEE Symposium on Foundations of Computer Science, IEEE, New York, 1978, 75–83.Google Scholar
  2. L. Babai, Trading group theory for randomness, inProceedings of the 17th ACM Symposium on the Theory of Computing, ACM, New York, 1985, 421–429.Google Scholar
  3. L. Babai andL. Fortnow, Arithmetization: A new method in structural complexity theory,Computational Complexity,1:1 (1991), 41–66.Google Scholar
  4. L. Babai, L. Fortnow, andC. Lund, Non-deterministic exponential time has two-prover interactive protocols,Computational Complexity,1:1 (1991), 3–40.Google Scholar
  5. L. Babai andS. Moran, Arthur-Merlin games: a randomized proof system, and a hierarchy of complexity classes,Journal of Computer and System Sciences,36:2 (1988), 254–276.Google Scholar
  6. D. Beaver andJ. Feigenbaum, Hiding instances in multioracle queries, inProceedings of the 7th Symposium on Theoretical Aspects of Computer Science, volume 415 ofLecture Notes in Computer Science, Springer, Berlin, 1990, 37–48.Google Scholar
  7. M. Ben-Or, S. Goldwasser, J. Kilian, and A. Wigderson, Multiprover interactive proofs: How to remove intractability assumptions, inProceedings of the 20th ACM Symposium on the Theory of Computing, ACM, New York, 1988, 113–131.Google Scholar
  8. C. Bennet andJ. Gill, Relative to a random oracle,P A ≠ NPA ≠ co-NPA with probability one,SIAM Journal on Computing,10 (1981), 96–113.Google Scholar
  9. M. Blum and S. Kannan, Designing programs that check their work, inProceedings of the 21st ACM Symposium on the Theory of Computing, ACM, New York, 1989, 86–97.Google Scholar
  10. M. Blum, M. Luby, and R. Rubinfeld, Self-testing and self-correcting programs, with applications to numerical programs, inProceedings of the 22nd ACM Symposium on the Theory of Computing, ACM, New York, 1990, 73–83.Google Scholar
  11. M. Blum andS. Micali, How to generate cryptographically strong sequences of pseudo-random bits,SIAM Journal on Computing,13 (1984), 850–864.Google Scholar
  12. R. Boppana andR. Hirschfeld, Pseudorandom generators and complexity classes, inRandomness and Computation, volume 5 ofAdvances in Computing Research, S. Micali, ed., JAI Press, Greenwich, 1989, 1–26.Google Scholar
  13. O. Goldreich, H. Krawczyk, and M. Luby, On the existence of pseudorandom generators, inProceedings of the 29th IEEE Symposium on Foundations of Computer Science, IEEE, New York, 1988, 12–24.Google Scholar
  14. O. Goldreich and L. Levin, A hard-core predicate for all one-way functions, inProceedings of the 21st ACM Symposium on the Theory of Computing, ACM, New York, 1989, 25–32.Google Scholar
  15. S. Goldwasser, S. Micali, andC. Rackoff, The knowledge complexity of interactive proof-systems,SIAM Journal on Computing,18:1 (1989), 186–208.Google Scholar
  16. J. Håstad, Pseudo-random generators under uniform assumptions, inProceedings of the 22nd ACM Symposium on the Theory of Computing, ACM, New York, 1990, 395–404.Google Scholar
  17. J. Hartmanis, N. Immerman, andV. Sewelson, Sparse sets inNP-P: EXPTIME versusNEXPTIME, Information and Control,65 (1985), 158–181.Google Scholar
  18. H. Heller, On relativized exponential and probabilistic complexity classes,Information and Computation,71 (1986), 231–243.Google Scholar
  19. R. Impagliazzo, L. Levin, and M. Luby, Pseudo-random number generation from one-way functions, inProceedings of the 21st ACM Symposium on the Theory of Computing, ACM, New York, 1989, 12–24.Google Scholar
  20. R. Karp and R. Lipton, Some connections between nonuniform and uniform complexity classes, inProceedings of the 12th ACM Symposium on the Theory of Computing, ACM, New York, 1980, 302–309.Google Scholar
  21. L. Levin, One-way functions and pseudo-random generators,Combinatorica,7 (1987), 357–363.Google Scholar
  22. R. Lipton, New directions in testing, inDistributed Computing and Cryptography, volume 2 ofDIMACS Series in Discrete Mathematics and Theoretical Computer Science, J. Feigenbaum and M. Merritt, eds., American Mathematical Society, Providence, 1991, 191–202.Google Scholar
  23. C. Lund, L. Fortnow, H. Karloff, andN. Nisan, Algebraic methods for interactive proof systems,Journal of the ACM,39:4 (1992), 859–868.Google Scholar
  24. N. Nisan and A. Wigderson, Hardness vs. randomness, inProceedings of the 29th IEEE Symposium on Foundations of Computer Science, IEEE, New York, 1988, 2–11.Google Scholar
  25. A. Shamir, IP=PSPACE,Journal of the ACM,39:4 (1992), 869–877.Google Scholar
  26. A. Yao, Theory and applications of trapdoor functions, inProceedings of the 23rd IEEE Symposium on Foundations of Computer Science, IEEE, New York, 1982, 80–91.Google Scholar

Copyright information

© Birkhäuser Verlag 1993

Authors and Affiliations

  • Lźszló Babai
    • 1
  • Lance Fortnow
    • 1
  • Noam Nisan
    • 2
  • Avi Wigderson
    • 2
  1. 1.Department of Computer ScienceUniversity of ChicagoChicagoUSA
  2. 2.Department of Computer ScienceHebrew UniversityJerusalemIsrael

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