Depth of Functions of k-Valued Logic in Finite Bases

Realization of functions of the k-valued logic by circuits is considered over an arbitrary finite complete basis B. Asymptotic behavior of the Shannon function D-B (n) of the circuit depth over B is examined. The value DB (n) is the minimal depth sufficient to realize every function of the k-valued logic of n variables by a circuit over B. It is shown that for each natural k >= 2 and for any finite complete basis B there exists a positive constant alpha(B) such that D-B (n) similar to alpha(n)(B) for n -> infinity .