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

Abstract

Abstract We study relationships between lower estimates for the energy complexity E(Σ), the switching complexity S(Σ) of a normalized Boolean network Σ, and the positive sensitivity ps(f) of the Boolean function f implemented by this circuit. The lower estimate E ( Σ ) ⩾ ⌊ ps ( f ) − 1 m ⌋ $\begin{array}{} \displaystyle E({\it\Sigma})\geqslant \lfloor\frac{\text{ps}(f)-1}{m}\rfloor \end{array}$ 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.