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 S, and the positive sensitivity ps(f) of the Boolean function f implemented by this circuit. The lower estimate E(Sigma) >= left perpendicularps(f)-1mright perpendicular 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 ps(f)) lower estimate for the switching complexity.