Constant depth circuits and the Lutz hypothesis

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≠P. We show that the analog in P of this hypothesis fails dramatically. In fact, we show that NTIME[n^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 AC⁰₄[⊕] 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