On the depth of $k$-valued logic functions over arbitrary bases

The behavior of the Shannon function of the depth of $k$-valued logic functions realized by circuits over an arbitrary complete basis is examined. For all $k$, $k \ge 3$, for an arbitrary basis of $k$-valued logic functions, the existence of the asymptotic behavior of the Shannon function of the depth is established. The asymptotic behavior is linear for finite bases and it is constant or logarithmic for infinite bases. Thus, the complete picture of asymptotic behavior of the Shannon function of the depth is obtained for all $k$, $k \ge 2$.