Relations between energy complexity measures of Boolean networks and positive sensitivity of Boolean functions

We study relationships between lower estimates for the energy complexity $E(\Sigma)$, the switching complexity $S(\Sigma)$ of a normalized Boolean network $\Sigma$, and the positive sensitivity $\operatorname{ps}(f)$ of the Boolean function $f$ implemented by this circuit. The lower estimate $E(\Sigma)\geqslant \lfloor\frac{\operatorname{ps}(f)-1}{m}\rfloor$ is proved for a sufficiently wide class of bases consisting of monotone Boolean functions of at most $m$ variables, the negation gate, and the Boolean constants $0$ and $1$. For the switching complexity of circuits, we construct a counterexample which shows that, for the standard basis of elements of the disjunction, conjunction, and negation, there do not exist a linear (with respect to $\operatorname{ps}(f)$) lower estimate for the switching complexity.