Energy and depth of threshold circuits

In this paper we show that there is a close relationship between the energy complexity and the depth of threshold circuits computing any Boolean function although they have completely different physical meanings. Suppose that a Boolean function f can be computed by a threshold circuit C of energy complexity e and hence at most e threshold gates in C output “1” for any input to C. We prove that the function f can also be computed by a threshold circuit C′ of the depth 2e+1 and hence the parallel computation time of C′ is 2e+1. If the size of C is s, that is, there are s threshold gates in C, then the size s′ of C′ is s′=2es+1. Thus, if the size s of C is polynomial in the number n of input variables, then the size s′ of C′ is polynomial in n, too.