Parity
Definition
TODO: extend the page to include parity of some specific [math]k[/math] bits out of the entire [math]n[/math].
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. [1]
- A constant-depth Boolean circuit must have exponential size in order to represent the parity function. [2]
- Parity over [math]n[/math] bits is odd when [math]n[/math] is odd, and even when [math]n[/math] is even.
References
- ↑ Ingo Wegener, Randall J. Pruim, Complexity Theory, 2005, Template:Isbn, p. 260
- ↑ Johan Håstad, Computational limitations of small depth circuits, 1987.
