2013 IEEE 54th Annual Symposium on Foundations of Computer Science (1991)

San Juan, Puerto Rico

Oct. 1, 1991 to Oct. 4, 1991

ISBN: 0-8186-2445-0

pp: 602-611

B. Schmeltz , Tech. Hochschule Darmstadt, Germany

R. Reischuk , Tech. Hochschule Darmstadt, Germany

ABSTRACT

Boolean circuits in which gates independently make errors with probability (at most) epsilon are considered. It is shown that the critical number crit(f) of a function f yields lower bound Omega (crit(f) log crit (f)) for the noisy circuit size. The lower bound is proved for an even stronger computational model, static Boolean decision trees with erroneous answers. A decision tree is static if the questions it asks do not depend on previous answers. The depth of such a tree provides a lower bound on the number of gates that depend directly on some input and hence on the size of a noisy circuit. Furthermore, it is shown that an Omega (n log n) lower bound holds for almost all Boolean n-input functions with respect to the depth of noisy dynamic decision trees. This bound is the best possible and implies that almost all n-input Boolean functions have noisy decision tree complexity Theta (n log n) in the static as well as in the dynamic case.

INDEX TERMS

noisy decision tree complexity, tree depth, noisy circuits, lower bound, Boolean circuits, critical number, computational model, static Boolean decision trees, erroneous answers, questions, answers, noisy dynamic decision trees, Boolean functions

CITATION

B. Schmeltz,
R. Reischuk,
"Reliable computation with noisy circuits and decision trees-a general n log n lower bound",

*2013 IEEE 54th Annual Symposium on Foundations of Computer Science*, vol. 00, no. , pp. 602-611, 1991, doi:10.1109/SFCS.1991.185425