Depth of functions of k-valued logic in finite bases

Abstract

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 DB(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 αB such that DB(n) ∼ αBn for n → ∞.