Difference between revisions of "Category:Balanced function"
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This also has a probabilistic view: if <math>X</math> is a uniformly random vector in <math>\{-1,1\}^n</math>, then <math>\mathbb{E}f(X) = 0</math>. | This also has a probabilistic view: if <math>X</math> is a uniformly random vector in <math>\{-1,1\}^n</math>, then <math>\mathbb{E}f(X) = 0</math>. | ||
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| + | Many times, a sequence of Boolean functions <math>(f_n)_{n\in \mathbb{N}}</math> is said to be balanced, or "asymptotically balanced", if <math>\lim_{n\to \infty} \mathbb{E}f(x) = 0</math>. | ||
A function which is not balanced is [[:Category:Biased function|biased]]. | A function which is not balanced is [[:Category:Biased function|biased]]. | ||
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== Properties == | == Properties == | ||
Revision as of 14:08, 5 September 2018
Definition
A Boolean function [math]f:\{-1,1\}^n \to \{-1,1\}[/math] is balanced if it obtains the value 1 on exactly half of its inputs. This can be written as:
[math]\sum_{x\in\{-1,1\}^n} f(x) = 0[/math].
This also has a probabilistic view: if [math]X[/math] is a uniformly random vector in [math]\{-1,1\}^n[/math], then [math]\mathbb{E}f(X) = 0[/math].
Many times, a sequence of Boolean functions [math](f_n)_{n\in \mathbb{N}}[/math] is said to be balanced, or "asymptotically balanced", if [math]\lim_{n\to \infty} \mathbb{E}f(x) = 0[/math].
A function which is not balanced is biased.
Properties
- TODO
Examples of balanced functions
References
Pages in category "Balanced function"
The following 17 pages are in this category, out of 17 total.
