Depth of Boolean functions over an arbitrary infinite basis
O. M. Kasim-Zade · Journal of Applied and Industrial Mathematics · 2008
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 D B (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 D B (n) = β for all sufficiently large n or there exist an integer constant γ ⩾ 2 and a constant δ such that log γ n ⩽ D B (n) ⩽ log γ n + δ for all n.