Depth of Boolean functions over an arbitrary infinite basis

Abstract

Realization of Boolean functions by circuits is considered over an arbitrary infinite complete basis. The depth of a circuit is defined as the greatest number of functional elements constituting a directed path from an input of the circuit to its output. The Shannon function of the depth is defined for a positive integer n as the minimal depth DB(n) of the circuits sufficient to realize every Boolean function on n variables over a basis B. It is shown that, for each infinite basis B, either there exists a constant β ⩾ 1 such that DB(n) = β for all sufficiently large n or there exist an integer constant γ ⩾ 2 and a constant δ such that logγn ⩽ DB(n) ⩽ logγn + δ for all n.