On resource-bounded measure and pseudorandomness
In this paper we extend a key result of Nisan and Wigderson  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 2 αn , 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  regarding the derandomization of the classes BP·σ k P and BP·θ k P under plausible measure theoretic assumptions. As a consequence, if θ 2 P does not have p-measure 0, then AM ∩ coAM is low for θ 2 P . Thus, in this case, the graph isomorphism problem is low for θ 2 P . 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.
Unable to display preview. Download preview PDF.
- 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
- 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
- 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
- 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
- 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
- 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