Difference between revisions of "Boolean function"

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A '''boolean mask operation''' on boolean-valued functions combines values point-wise, for example, by [[exclusive disjunction|XOR]], or other [[boolean operator]]s.
 
A '''boolean mask operation''' on boolean-valued functions combines values point-wise, for example, by [[exclusive disjunction|XOR]], or other [[boolean operator]]s.
  
==Algebraic normal form==
 
  
A boolean function can be written uniquely as a sum ([[exclusive disjunction|XOR]]) of products ([[logical conjunction|AND]]). This is known as the [[algebraic normal form]] (ANF).
 
 
{| cellpadding="4"
 
|-
 
|<math>f(x_1, x_2, \ldots , x_n) = \!</math>
 
|<math>a_0 + \!</math>
 
|-
 
|
 
|<math>a_1x_1 + a_2x_2 + \ldots + a_nx_n + \!</math>
 
|-
 
|
 
|<math>a_{1,2}x_1x_2 + a_{n-1,n}x_{n-1}x_n + \!</math>
 
|-
 
|
 
|<math>\ldots + \!</math>
 
|-
 
|
 
|<math>a_{1,2,\ldots,n}x_1x_2\ldots x_n \!</math>
 
|}
 
 
The values of the sequence <math>a_0, a_1, \ldots, a_{1, 2, \ldots, n}</math> can therefore also uniquely represent a boolean function. The algebraic degree of a boolean function is defined as the highest number of <math>x_i</math> that appear in a product term. Thus <math>f(x_1, x_2, x_3) = x_1 + x_3</math> has degree 1 (linear), whereas <math>f(x_1, x_2, x_3) = x_1 + x_1 x_2 x_3</math> has degree 3 (cubic).
 
  
 
==See also==
 
==See also==

Revision as of 13:10, 22 October 2008

In mathematics, a finitary boolean function is a function of the form \(f : \mathbb{B}^k \to \mathbb{B},\) where \(\mathbb{B} = \{ 0, 1 \}\) is a boolean domain and where \(k\!\) is a nonnegative integer. In the case where \(k = 0,\!\) the function is simply a constant element of \(\mathbb{B}.\)

There are \(2^{2^k}\) such functions. These play a basic role in questions of complexity theory as well as the design of circuits and chips for digital computers. The properties of boolean functions play a critical role in cryptography, particularly in the design of symmetric key algorithms (see S-box).

A boolean mask operation on boolean-valued functions combines values point-wise, for example, by XOR, or other boolean operators.


See also

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Portions of the above article were adapted from the following sources under the GNU Free Documentation License, under other applicable licenses, or by permission of the copyright holders.