  The minimum number of NOT gates in a Boolean circuit computing a Boolean
function is called the inversion complexity of the function. In 1957, A. A.
Markov determined the inversion complexity of every Boolean function and proved
that $\lceil\log_{2}(d(f)+1)\rceil$ NOT gates are necessary and sufficient to
compute any Boolean function $f$ (where $d(f)$ is maximum number of value
changes from 1 to 0 over all increasing chains of tuples of variables values).
In this paper we consider Boolean circuits over an arbitrary basis that
consists of all monotone functions (with zero weight) and finite nonempty set
of non-monotone functions (with unit weight). It is shown that the minimal
sufficient for a realization of the Boolean function $f$ number of non-monotone
gates is equal to $\lceil\log_{2}(d(f)+1)\rceil - O(1)$. Similar extends of
another classical result of A. A. Markov for the inversion complexity of system
of Boolean functions has been obtained.
