Proceedings 38th Annual Symposium on Foundations of Computer Science (1997)
Miami Beach, FL
Oct. 19, 1997 to Oct. 22, 1997
Jin-Yi Cai , State Univ. of New York, Buffalo, NY, USA
D. Sivakumar , State Univ. of New York, Buffalo, NY, USA
M. Strauss , State Univ. of New York, Buffalo, NY, USA
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.
computational complexity; Lutz hypothesis; complexity classes; NP; Exponential Time; constant-depth; nearly exponential-size circuits; circuit complexity theory; pseudorandom generators; constant depth circuits
J. Cai, M. Strauss and D. Sivakumar, "Constant depth circuits and the Lutz hypothesis," Proceedings 38th Annual Symposium on Foundations of Computer Science(FOCS), Miami Beach, FL, 1997, pp. 595.