On the structure, complexity, and depth of the circuits over the basis $\{ \&, \vee\} $ realizing step Boolean functions

The step Boolean function is a function of the algebra of logic of $n$ Boolean variables, $n =1,2,\ldots$, reducing to $1$ on all of the sets of an $n$-dimensional unit cube, the ordinal numbers of which are not lower than the given set. In this paper, the problem of synthesis of circuits over the basis $\{ \&, \vee\} $ realizing step Boolean functions was considered. The optimized structure of the given circuits was studied with regard to complexity and depth. Step functions often appear in theoretical and applied tasks as an auxiliary subfunctions. For instance, an $n$-bit adder contains such a subfunction.