Difference between revisions of "Iterated majority"
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\textrm{maj}(f(x^{(1)}),f(x^{(2)}),f(x^{(3)}) & \textrm{otherwise}, \end{cases}</math> | \textrm{maj}(f(x^{(1)}),f(x^{(2)}),f(x^{(3)}) & \textrm{otherwise}, \end{cases}</math> | ||
| − | where <math>x^{(1)} = (x_1,x_2\ldots x_{n/3})</math>, <math>x^{(2)} = (x_{n/3+1},\ldots x_{2n/3})</math> and <math>x^{(3)} = (x_{2n/3+1},\ldots x_{n})</math> | + | where <math>x^{(1)} = (x_1,x_2\ldots x_{n/3})</math>, <math>x^{(2)} = (x_{n/3+1},\ldots x_{2n/3})</math> and <math>x^{(3)} = (x_{2n/3+1},\ldots x_{n})</math>. |
== Properties == | == Properties == | ||
* TODO | * TODO | ||
| + | * TODO: add influence. | ||
== References == | == References == | ||
| − | [[Category:monotone function]] [[Category:balanced function]] | + | [[Category:symmetric function]] [[Category:monotone function]] [[Category:balanced function]] |
Revision as of 13:39, 6 June 2019
Definition
Let [math]n = 3^k[/math]. The iterated majority function [math]f:\{-1,1\}^n \to \{-1,1\}[/math] is a recursively-defined variation on the majority function:
[math] f(x) = \begin{cases} \textrm{maj}(x), & \textrm{if}~ n = 3 \\ \textrm{maj}(f(x^{(1)}),f(x^{(2)}),f(x^{(3)}) & \textrm{otherwise}, \end{cases}[/math]
where [math]x^{(1)} = (x_1,x_2\ldots x_{n/3})[/math], [math]x^{(2)} = (x_{n/3+1},\ldots x_{2n/3})[/math] and [math]x^{(3)} = (x_{2n/3+1},\ldots x_{n})[/math].
Properties
- TODO
- TODO: add influence.
