Boolean Functions with External Parameters

In this paper, we consider Boolean functions $f(\tilde x,\tilde z)$ depending on two groups of variables, $\tilde x$ and $\tilde z$. The first group contains standard Boolean variables, which are subjected to the operations of renaming, identification, and substitution of formulas. No operations are allowed on the variable $\tilde z$ of the second group. The variables $\tilde z$ represent the influence of the environment on the function $f$. Let $\gamma$ be a finite system of such functions. Assume that is $g(\tilde x)$ a standard Boolean function. We study the conditions for realizability of the function $g(\tilde x)$ by a circuit of functional elements in the basis $\gamma$.