Exact Value of the Nonmonotone Complexity of Boolean Functions

We study the complexity of the realization of Boolean functions by circuits in infinite complete bases containing all monotone functions with zero weight (cost of use) and finitely many nonmonotone functions with unit weight. The complexity of the realization of Boolean functions in the case where the only nonmonotone element of the basis is negation was completely described by A. A. Markov: the minimum number of negations sufficient for the realization of an arbitrary Boolean function $f$ (the inversion complexity of the function $f$) is equal to $\lceil\log_{2}(d(f)+1)\rceil$, where $d(f)$ is the maximum (over all increasing chains of sets of values of the variables) number of changes of the function value from 1 to 0. In the present paper, this result is generalized to the case of the computation of Boolean functions over an arbitrary basis $B$ of prescribed form. It is shown that the minimum number of nonmonotone functions sufficient for computing an arbitrary Boolean function $f$ is equal to $\lceil\log_{2}(d(f)/D(B)+1)\rceil$, where $D(B)=\max d(\omega)$; the maximum is taken over all nonmonotone functions $\omega$ of the basis $B$.