On relativized polynomial and exponential computations
SIAM Journal on Computing
Separating the polynomial-time hierarchy by oracles
Proc. 26th annual symposium on Foundations of computer science
On relativized exponential and probabilistic complexity classes
Information and Control
Computational limitations of small-depth circuits
Computational limitations of small-depth circuits
Algebraic methods for interactive proof systems
Journal of the ACM (JACM)
Journal of the ACM (JACM)
The random oracle hypothesis is false
Journal of Computer and System Sciences
On collapsing the polynomial-time hierarchy
Information Processing Letters
Relativized worlds with an infinite hierarchy
Information Processing Letters
Note: fractal dimension and logarithmic loss unpredictability
Theoretical Computer Science
Dimension in Complexity Classes
SIAM Journal on Computing
Scaled dimension and nonuniform complexity
Journal of Computer and System Sciences
Dimension characterizations of complexity classes
MFCS'06 Proceedings of the 31st international conference on Mathematical Foundations of Computer Science
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Bennett and Gill [Relative to a random oracle A, PA ≠ NPA ≠ co-NPA with probability 1, SIAM J. Comput. 10 (1981) 96-113] proved that PA ≠ NPA relative to a random oracle A, or in other words, that the set O[P=NP] = {A | PA = NPA} has Lebesgue measure 0. In contrast, we show that O[P=NP] has Hausdorff dimension 1.This follows from a much more general theorem: if there is a relativizable and paddable oracle construction for a complexity-theoretic statement Φ, then the set of oracles relative to which Φ holds has Hausdorff dimension 1.We give several other applications including proofs that the polynomial-time hierarchy is infinite relative to a Hausdorff dimension 1 set of oracles and that PA ≠ NPA ∩ coNPA relative to a Hausdorff dimension 1 set of oracles.