On the Average Sensitivity of Testing Square-Free Numbers

Abstract

We study combinatorial complexity characteristics of a Boolean function related to a natural number theoretic problem. In particular we obtain a linear lower bound on the average sensitivity of the Boolean function deciding whether a given integer is square-free. This result allows us to derive a quadratic lower bound for the formula size complexity of testing square-free numbers and a linear lower bound on the average decision tree depth. We also obtain lower bounds on the degrees of exact and approximative polynomial representations of this function.