Constant Depth Circuits and the Lutz Hypothesis

The central hypothesis in the theory of resource-bounded measure 6] is the assertion that NP does not have measure 0 in Exponential Time. This is a quantitative strengthening of the assertion that NP does not equal P. We show that the analog in P of this hypothesis fails dramatically. In fact, we show that nondeterministic time n to the power (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 AND-OR-NOT circuits has measure zero at polynomial time. In contrast, we show that the class 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.