Almost everywhere high nonuniform complexity
Journal of Computer and System Sciences
Measure, Stochasticity, and the Density of Hard Languages
SIAM Journal on Computing
Theoretical Computer Science
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SIAM Journal on Computing
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Information and Computation
Genericity and measure for exponential time
MFCS '94 Selected papers from the 19th international symposium on Mathematical foundations of computer science
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Equivalence of Measures of Complexity Classes
SIAM Journal on Computing
A Generalization of Resource-Bounded Measure, With an Application (Extended Abstract)
STACS '98 Proceedings of the 15th Annual Symposium on Theoretical Aspects of Computer Science
Resource-Bounded Randomness and Compressibility with Respect to Nonuniform Measures
RANDOM '97 Proceedings of the International Workshop on Randomization and Approximation Techniques in Computer Science
COCO '98 Proceedings of the Thirteenth Annual IEEE Conference on Computational Complexity
COCO '98 Proceedings of the Thirteenth Annual IEEE Conference on Computational Complexity
Pseudorandom generators, measure theory, and natural proofs
FOCS '95 Proceedings of the 36th Annual Symposium on Foundations of Computer Science
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SFCS '94 Proceedings of the 35th Annual Symposium on Foundations of Computer Science
Process complexity and effective random tests
Journal of Computer and System Sciences
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The resource-bounded measures of certain classes of languages are shown to be invariant under certain changes in the underlying probability measure. Specifically, for any real number δ 0, any polynomial-time computable sequence β = (β0, β1, ...) of biases βi Ɛ [δ, 1 - δ], and any class C of languages that is closed upwards or downwards under positive, polynomial-time truth-table reductions with linear bounds on number and length of queries, it is shown that the following two conditions are equivalent. (1) C has p-measure 0 relative to the probability measure given by β. (2) C has p-measure 0 relative to the uniform probability measure. The analogous equivalences are established for measure in E and measure in E2. (Breutzmann and Lutz [5] established this invariance for classes C that are closed downwards under slightly more powerful reductions, but nothing was known about invariance for classes that are closed upwards.) The proof introduces two new techniques, namely, the contraction of a martingale for one probability measure to a martingale for an induced probability measure, and a new, improved positive bias reduction of one bias sequence to another. Consequences for the BPP versus E problem and small span theorems are derived.