Theoretical Computer Science
Martingale families and dimension in P
Theoretical Computer Science
The isomorphism conjecture for constant depth reductions
Journal of Computer and System Sciences
Martingale families and dimension in p
CiE'06 Proceedings of the Second conference on Computability in Europe: logical Approaches to Computational Barriers
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Resource-bounded measure theory is a study of complexity classes via an adaptation of the probabilistic method. The central hypothesis in this theory is the assertion that NP does not have measure zero in Exponential Time. This is a quantitative strengthening of NP/spl ne/P. We show that the analog in P of this hypothesis fails dramatically. In fact, we show that NTIME[n/sup 1/11/] has measure zero in P. These follow as consequences of our main theorem that the collection of languages accepted by constant-depth nearly exponential-size circuits has measure zero at polynomial time. In contrast, we show that the class AC/sup 0//sub 4/[/spl oplus/] of languages accepted by depth-4 polynomial-size circuits with AND, OR, NOT, and PARITY gates does not have measure zero at polynomial time. Our proof is based on techniques from circuit complexity theory and pseudorandom generators.