Size and Energy of Threshold Circuits Computing Mod Functions

Let C be a threshold logic circuit computing a Boolean function MODm : {0, 1}n → {0, 1}, where n ≥ 1 and m ≥ 2. Then C outputs “0” if the number of “1”s in an input x ∈ {0, 1}n to C is a multiple of m and, otherwise, C outputs “1.” The function MOD2 is the so-called PARITY function, and MODn+1 is the OR function. Let s be the size of the circuit C, that is, C consists of s threshold gates, and let e be the energy complexity of C, that is, at most e gates in C output “1” for any input x ∈ {0, 1}n. In the paper, we prove that a very simple inequality n/(m − 1) ≤ s holds for every circuit C computing MODm. The inequality implies that there is a tradeoff between the size s and energy complexity e of threshold circuits computing MODm, and yields a lower bound e = Ω((logn− logm)/ log log n) on e if s = O(polylog(n)). We actually obtain a general result on the so-called generalized mod function, from which the result on the ordinary mod function MODm immediately follows. Our results on threshold circuits can be extended to a more general class of circuits, called unate circuits.