Difference between revisions of "Boolean function"
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| − | In [[mathematics]], a '''finitary boolean function''' is a [[function (mathematics)|function]] of the form | + | In [[mathematics]], a '''finitary boolean function''' is a [[function (mathematics)|function]] of the form <math>f : \mathbb{B}^k \to \mathbb{B},</math> where <math>\mathbb{B} = \{ 0, 1 \}</math> is a [[boolean domain]] and where <math>k\!</math> is a nonnegative integer. In the case where <math>k = 0,\!</math> the function is simply a constant element of <math>\mathbb{B}.</math> |
More generally, a function of the form ''f'' : ''X'' → '''B''', where ''X'' is an arbitrary set, is a ''[[boolean-valued function]]''. If ''X'' = '''M''' = {1, 2, 3, …}, then ''f'' is a ''binary sequence'', that is, an infinite [[sequence]] of 0's and 1's. If ''X'' = [''k''] = {1, 2, 3, …, ''k''}, then ''f'' is ''binary sequence'' of length ''k''. | More generally, a function of the form ''f'' : ''X'' → '''B''', where ''X'' is an arbitrary set, is a ''[[boolean-valued function]]''. If ''X'' = '''M''' = {1, 2, 3, …}, then ''f'' is a ''binary sequence'', that is, an infinite [[sequence]] of 0's and 1's. If ''X'' = [''k''] = {1, 2, 3, …, ''k''}, then ''f'' is ''binary sequence'' of length ''k''. | ||
Revision as of 00:26, 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 \(\mathbb{B}.\)
More generally, a function of the form f : X → B, where X is an arbitrary set, is a boolean-valued function. If X = M = {1, 2, 3, …}, then f is a binary sequence, that is, an infinite sequence of 0's and 1's. If X = [k] = {1, 2, 3, …, k}, then f is binary sequence of length k.
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.
Algebraic normal form
A boolean function can be written uniquely as a sum (XOR) of products (AND). This is known as the algebraic normal form (ANF).
| \(f(x_1, x_2, \ldots , x_n) = \!\) | \(a_0 + \!\) |
| \(a_1x_1 + a_2x_2 + \ldots + a_nx_n + \!\) | |
| \(a_{1,2}x_1x_2 + a_{n-1,n}x_{n-1}x_n + \!\) | |
| \(\ldots + \!\) | |
| \(a_{1,2,\ldots,n}x_1x_2\ldots x_n \!\) |
The values of the sequence \(a_0, a_1, \ldots, a_{1, 2, \ldots, n}\) can therefore also uniquely represent a boolean function. The algebraic degree of a boolean function is defined as the highest number of \(x_i\) that appear in a product term. Thus \(f(x_1, x_2, x_3) = x_1 + x_3\) has degree 1 (linear), whereas \(f(x_1, x_2, x_3) = x_1 + x_1 x_2 x_3\) has degree 3 (cubic).
See also
Template:Col-breakTemplate:Col-breakTemplate:Col-breakTemplate:Col-endExternal links
- Boolean Planet — boolean functions in cryptography.
Document history
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.
