Relativized circuit complexity
Journal of Computer and System Sciences
Complexity measures for public-key cryptosystems
SIAM Journal on Computing - Special issue on cryptography
A pseudorandom oracle characterization of BPP
SIAM Journal on Computing
Kolmogorov complexity and Hausdorff dimension
Information and Computation
Journal of Computer and System Sciences
P = BPP if E requires exponential circuits: derandomizing the XOR lemma
STOC '97 Proceedings of the twenty-ninth annual ACM symposium on Theory of computing
FOCS '02 Proceedings of the 43rd Symposium on Foundations of Computer Science
When Worlds Collide: Derandomization, Lower Bounds, and Kolmogorov Complexity
FST TCS '01 Proceedings of the 21st Conference on Foundations of Software Technology and Theoretical Computer Science
Comparing Notions of Full Derandomization
CCC '01 Proceedings of the 16th Annual Conference on Computational Complexity
Dimension in Complexity Classes
SIAM Journal on Computing
Effective fractal dimension: foundations and applications
Effective fractal dimension: foundations and applications
Hausdorff dimension and oracle constructions
Theoretical Computer Science
Dimension, entropy rates, and compression
Journal of Computer and System Sciences
Measure on small complexity classes, with applications for BPP
SFCS '94 Proceedings of the 35th Annual Symposium on Foundations of Computer Science
One-sided versus two-sided error in probabilistic computation
STACS'99 Proceedings of the 16th annual conference on Theoretical aspects of computer science
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We use derandomization to show that sequences of positive pspace-dimension – in fact, even positive Δ$^{\rm p}_{\rm k}$-dimension for suitable k – have, for many purposes, the full power of random oracles. For example, we show that, if S is any binary sequence whose Δ$^{\rm p}_{\rm 3}$-dimension is positive, then BPP ⊆ PS and, moreover, every BPP promise problem is PS-separable. We prove analogous results at higher levels of the polynomial-time hierarchy. The dimension-almost-class of a complexity class $\mathcal{C}$, denoted by dimalmost-$\mathcal{C}$, is the class consisting of all problems A such that $A \in \mathcal{C}^S$ for all but a Hausdorff dimension 0 set of oracles S. Our results yield several characterizations of complexity classes, such as BPP = dimalmost-P and AM = dimalmost-NP, that refine previously known results on almost-classes. They also yield results, such as Promise-BPP = almost-P-Sep = dimalmost-P-Sep, in which even the almost-class appears to be a new characterization.