Computing All MOD-Functions Simultaneously

Abstract

The fundamental symmetric functions are $\textrm{EX}_k^n$ (equal to 1 if the sum of n input bits is exactly k), $\textrm{TH}_k^n$ (the sum is at least k), and $\textrm{MOD}_{m,r}^n$ (the sum is congruent to r modulo m). It is well known that all these functions and in fact any symmetric Boolean function have linear circuit size.

Simple counting shows that the circuit complexity of computing n symmetric functions is Ω(n ^{2 − o(1)}) for almost all tuples of n symmetric functions. It is well-known that all EX and TH functions (i.e., for all 0 ≤ k ≤ n) of n input bits can be computed by linear size circuits.

In this short note, we investigate the circuit complexity of computing all MOD functions (for all 1 ≤ m ≤ n). We prove an O(n) upper bound for computing the set of functions

$$\{\ensuremath{\textrm{MOD}}_{1,r}^n, \ensuremath{\textrm{MOD}}_{2,r}^n, \dots, \ensuremath{\textrm{MOD}}_{n,r}^n\}$$

and an O(nloglogn) upper bound for

$$\{\ensuremath{\textrm{MOD}}_{1,r_1}^n, \ensuremath{\textrm{MOD}}_{2,r_2}^n, \dots, \ensuremath{\textrm{MOD}}_{n,r_n}^n\},$$

where r ₁, r ₂, …, r _n are arbitrary integers.

Research is partially supported by Russian Foundation for Basic Research (11-01-00760-a and 11-01-12135-ofi-m-2011), RAS Program for Fundamental Research, and JSPS KAKENHI 23700020. Part of this research was done while the second author was visiting the University of Kyoto.