Category and measure in complexity classes
SIAM Journal on Computing
Almost everywhere high nonuniform complexity
Journal of Computer and System Sciences
Cook versus Karp-Levin: separating completeness notions if NP is not small
Theoretical Computer Science
Measure on P: strength of the notion
Information and Computation
The quantitative structure of exponential time
Complexity theory retrospective II
Graph nonisomorphism has subexponential size proofs unless the polynomial-time hierarchy collapses
STOC '99 Proceedings of the thirty-first annual ACM symposium on Theory of computing
The zero-one law holds for BPP
Theoretical Computer Science
Dimension in Complexity Classes
COCO '00 Proceedings of the 15th Annual IEEE Conference on Computational Complexity
Constant depth circuits and the Lutz hypothesis
FOCS '97 Proceedings of the 38th Annual Symposium on Foundations of Computer Science
The dimensions of individual strings and sequences
Information and Computation
Theoretical Computer Science
Theoretical Computer Science
Journal of Computer and System Sciences - Special issue on COLT 2002
Measure on small complexity classes, with applications for BPP
SFCS '94 Proceedings of the 35th Annual Symposium on Foundations of Computer Science
Theoretical Computer Science
Baire categories on small complexity classes and meager--comeager laws
Information and Computation
Resource-bounded measure on probabilistic classes
Information Processing Letters
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We introduce a new measure notion on small complexity classes (called F-measure), based on martingale families, that gets rid of some drawbacks of previous measure notions: it can be used to define dimension because martingale families can make money on all strings, and it yields random sequences with an equal frequency of 0's and 1's. As applications to F-measure, we answer a question raised in [1] by improving their result to: or almost every language A decidable in subexponential time, PA =BPPA. We show that almost all languages in PSPACE do not have small non-uniform complexity. We compare F-measure to previous notions and prove that martingale families are strictly stronger than Γ-measure [1] , we also discuss the limitations of martingale families concerning finite unions. We observe that all classes closed under polynomial many-one reductions have measure zero in EXP iff they have measure zero in SUBEXP. We use martingale families to introduce a natural generalization of Lutz resource-bounded dimension [13] on P , which meets the intuition behind Lutz's notion. We show that P-dimension lies between finite-state dimension and dimension on E. We prove an analogue to the Theorem of Eggleston in P , i.e. the class of languages whose characteristic sequence contains 1's with frequency α, has dimension the Shannon entropy of α in P .