Size and Energy of Threshold Circuits Computing Mod Functions

• Authors: Kei Uchizawa;Takao Nishizeki;Eiji Takimoto
• Venue: MFCS '09 Proceedings of the 34th International Symposium on Mathematical Foundations of Computer Science 2009
• Year: 2009

Let C be a threshold logic circuit computing a Boolean function MOD$_m: \{ 0, 1\}^n \rightarrow \{0, 1\}$, where n 驴 1 and m 驴 2. Then C outputs "0" if the number of "1"s in an input x 驴 {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 MOD n + 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 驴 { 0, 1} n . In the paper, we prove that a very simple inequality n/(m 驴 1) ≤ s e holds for every circuit C computing MOD m . The inequality implies that there is a tradeoff between the size s and energy complexity e of threshold circuits computing MOD m , and yields a lower bound e = 驴((logn 驴 logm)/loglogn) 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 MOD m immediately follows. Our results on threshold circuits can be extended to a more general class of circuits, called unate circuits.