BPP has subexponential time simulations unlessEXPTIME has publishable proofs
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 wordsComplexity Classes Interactive Proof Systems
- 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
- 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
- L. Babai andL. Fortnow, Arithmetization: A new method in structural complexity theory,Computational Complexity,1:1 (1991), 41–66.Google Scholar
- L. Babai, L. Fortnow, andC. Lund, Non-deterministic exponential time has two-prover interactive protocols,Computational Complexity,1:1 (1991), 3–40.Google Scholar
- 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
- 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
- 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
- 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
- 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
- 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
- M. Blum andS. Micali, How to generate cryptographically strong sequences of pseudo-random bits,SIAM Journal on Computing,13 (1984), 850–864.Google Scholar
- 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
- 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
- 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
- S. Goldwasser, S. Micali, andC. Rackoff, The knowledge complexity of interactive proof-systems,SIAM Journal on Computing,18:1 (1989), 186–208.Google Scholar
- 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
- J. Hartmanis, N. Immerman, andV. Sewelson, Sparse sets inNP-P: EXPTIME versusNEXPTIME, Information and Control,65 (1985), 158–181.Google Scholar
- H. Heller, On relativized exponential and probabilistic complexity classes,Information and Computation,71 (1986), 231–243.Google Scholar
- 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
- 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
- L. Levin, One-way functions and pseudo-random generators,Combinatorica,7 (1987), 357–363.Google Scholar
- 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
- 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
- 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
- A. Shamir, IP=PSPACE,Journal of the ACM,39:4 (1992), 869–877.Google Scholar
- 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