Parity

Definition

The [math]n[/math]-variable parity function is the Boolean function [math]f:\{-1,1\}^n\to\{-1,1\}[/math] defined by [math]f(x)=x_1 \cdot x_2 \cdot \ldots \cdot x_n[/math]. In other words, [math]f(x)=1[/math] if and only if the number of ones in the vector [math]x\in\{-1,1\}^n[/math] is odd.

Properties

• Parity only depends on the number of ones and is therefore a symmetric Boolean function.
• Over [math]\{-1,1\}[/math], the representation of the parity function is exactly the monomial [math]x_1 x_2 \ldots x_n[/math]. Thus all of its Fourier mass rests on the highest level.
• The n-variable parity function and its negation are the only Boolean functions for which all disjunctive normal forms have the maximal number of 2 ^n − 1 monomials of length n and all conjunctive normal forms have the maximal number of 2 ^n − 1 clauses of length n.
• A constant-depth Boolean circuit must have exponential size in order to represent the parity function.