On Languages Reducible to Algorithmically RandomLanguages
SIAM Journal on Computing
An observation on probability versus randomness with applications to complexity classes
Mathematical Systems Theory
BPP has subexponential time simulations unless EXPTIME has publishable proofs
Computational Complexity
STOC '00 Proceedings of the thirty-second annual ACM symposium on Theory of computing
Kolmogorov entropy in the context of computability theory
Theoretical Computer Science
Randomness vs time: derandomization under a uniform assumption
Journal of Computer and System Sciences
SIAM Journal on Computing
Pseudorandomness and Average-Case Complexity Via Uniform Reductions
Computational Complexity
Limitations of Hardness vs. Randomness under Uniform Reductions
APPROX '08 / RANDOM '08 Proceedings of the 11th international workshop, APPROX 2008, and 12th international workshop, RANDOM 2008 on Approximation, Randomization and Combinatorial Optimization: Algorithms and Techniques
An Introduction to Kolmogorov Complexity and Its Applications
An Introduction to Kolmogorov Complexity and Its Applications
Derandomizing from Random Strings
CCC '10 Proceedings of the 2010 IEEE 25th Annual Conference on Computational Complexity
Lower bounds for reducibility to the Kolmogorov random strings
CiE'10 Proceedings of the Programs, proofs, process and 6th international conference on Computability in Europe
The pervasive reach of resource-bounded Kolmogorov complexity in computational complexity theory
Journal of Computer and System Sciences
Curiouser and curiouser: the link between incompressibility and complexity
CiE'12 Proceedings of the 8th Turing Centenary conference on Computability in Europe: how the world computes
Randomness, computation and mathematics
CiE'12 Proceedings of the 8th Turing Centenary conference on Computability in Europe: how the world computes
Limits on the computational power of random strings
Information and Computation
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How powerful is the set of random strings? What can one say about a set A that is efficiently reducible to R, the set of Kolmogorov-random strings? We present the first upper bound on the class of computable sets in PR and NPR. The two mostwidely-studied notions ofKolmogorov complexity are the "plain" complexity C(x) and "prefix" complexity K(x); this gives two ways to define the set "R": RC and RK. (Of course, each different choice of universal Turing machine U in the definition of C and K yields another variant RCU or RKU.) Previous work on the power of "R" (for any of these variants [1,2,9]) has shown - BPP ⊆ {A : A≤ttpR}. - PSPACE ⊆ PR. - NEXP ⊆ NPR. Since these inclusions hold irrespective of low-level details of how "R" is defined, we have e.g.: NEXP ⊆ Δ10∩∩U NPRKU. (δ10 is the class of computable sets.) Our main contribution is to present the first upper bounds on the complexity of sets that are efficiently reducible to RKU. We show: - BPP ⊆ Δ10∩∩U{A : A≤ttpRKU} ⊆ PSPACE. - NEXP ⊆ Δ10 ∩∩U NPRKU ⊆ EXPSPACE. Hence, in particular, PSPACE is sandwiched between the class of sets Turingand truth-table-reducible to R. As a side-product, we obtain new insight into the limits of techniques for derandomization from uniform hardness assumptions.