Difference between revisions of "Boolean function"

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In [[mathematics]], a '''finitary boolean function''' is a [[function (mathematics)|function]] of the form ''f''&nbsp;:&nbsp;'''B'''<sup>''k''</sup>&nbsp;&rarr;&nbsp;'''B''', where '''B'''&nbsp;=&nbsp;{0,&nbsp;1} is a ''[[boolean domain]]'' and where ''k'' is a nonnegative integer.  In the case where ''k''&nbsp;=&nbsp;0, the "function" is simply a constant element of '''B'''.
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<font size="3">&#9758;</font> This page belongs to resource collections on [[Logic Live|Logic]] and [[Inquiry Live|Inquiry]].
  
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More generally, a function of the form ''f''&nbsp;:&nbsp;''X''&nbsp;&rarr;&nbsp;'''B''', where ''X'' is an arbitrary set, is a ''[[boolean-valued function]]''.  If ''X'' = '''M''' = {1,&nbsp;2,&nbsp;3,&nbsp;&hellip;}, then ''f'' is a ''binary sequence'', that is, an infinite [[sequence]] of 0's and 1's.  If ''X'' = [''k''] = {1,&nbsp;2,&nbsp;3,&nbsp;&hellip;,&nbsp;''k''}, then ''f'' is ''binary sequence'' of length ''k''.
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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>
  
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There are <math>2^{2^k}</math> such functions.  These play a basic role in questions of [[complexity theory]] as well as the design of circuits and chips for [[digital computer]]s. The properties of boolean functions play a critical role in [[cryptography]], particularly in the design of [[symmetric key algorithm]]s (see [[S-box]]).
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There are <math>2^{2^k}</math> such functions.  These play a basic role in questions of [[complexity theory]] as well as the design of circuits and chips for digital computers.
  
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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.
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==Syllabus==
  
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==Algebraic normal form==
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===Focal nodes===
  
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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).
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* [[Inquiry Live]]
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* [[Logic Live]]
  
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{| cellpadding="4"
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===Peer nodes===
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|-
 
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|<math>f(x_1, x_2, \ldots , x_n) = \!</math>
 
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|<math>a_0 + \!</math>
 
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|-
 
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|<math>a_1x_1 + a_2x_2 + \ldots + a_nx_n + \!</math>
 
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|-
 
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|<math>a_{1,2}x_1x_2 + a_{n-1,n}x_{n-1}x_n + \!</math>
 
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|-
 
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|<math>\ldots + \!</math>
 
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|-
 
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|<math>a_{1,2,\ldots,n}x_1x_2\ldots x_n \!</math>
 
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|}
 
  
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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).
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* [http://intersci.ss.uci.edu/wiki/index.php/Boolean_function Boolean Function @ InterSciWiki]
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* [http://mywikibiz.com/Boolean_function Boolean Function @ MyWikiBiz]
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* [http://ref.subwiki.org/wiki/Boolean_function Boolean Function @ Subject Wikis]
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* [http://en.wikiversity.org/wiki/Boolean_function Boolean Function @ Wikiversity]
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* [http://beta.wikiversity.org/wiki/Boolean_function Boolean Function @ Wikiversity Beta]
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===Logical operators===
  
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==See also==
 
 
{{col-begin}}
 
{{col-begin}}
 
{{col-break}}
 
{{col-break}}
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* [[Algebra of sets]]
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* [[Exclusive disjunction]]
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* [[Boolean algebra]]
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* [[Logical conjunction]]
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* [[List of Boolean algebra topics|Boolean algebra topics]]
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* [[Logical disjunction]]
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* [[Logical equality]]
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{{col-break}}
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* [[Logical implication]]
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* [[Logical NAND]]
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* [[Logical NNOR]]
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* [[Logical negation|Negation]]
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{{col-end}}
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===Related topics===
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{{col-begin}}
 
{{col-break}}
 
{{col-break}}
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* [[Ampheck]]
 
* [[Boolean domain]]
 
* [[Boolean domain]]
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* [[Boolean logic]]
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* [[Boolean function]]
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* [[Boolean value]]
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* [[Boolean-valued function]]
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* [[Differential logic]]
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{{col-break}}
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* [[Logical graph]]
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* [[Minimal negation operator]]
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* [[Multigrade operator]]
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* [[Parametric operator]]
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* [[Peirce's law]]
 
{{col-break}}
 
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* [[Boolean-valued function]]
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* [[Propositional calculus]]
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* [[Logical connective]]
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* [[Sole sufficient operator]]
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* [[Truth function]]
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* [[Truth table]]
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* [[Universe of discourse]]
 
* [[Zeroth order logic]]
 
* [[Zeroth order logic]]
 
{{col-end}}
 
{{col-end}}
  
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==External links==
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===Relational concepts===
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* [[Continuous predicate]]
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* [[Hypostatic abstraction]]
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* [[Logic of relatives]]
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* [[Logical matrix]]
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* [[Relation (mathematics)|Relation]]
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* [[Relation composition]]
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* [[Relation construction]]
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* [[Relation reduction]]
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* [[Relation theory]]
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* [[Relative term]]
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* [[Sign relation]]
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* [[Triadic relation]]
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{{col-end}}
  
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* [http://www.isi.qut.edu.au/people/fuller/ Boolean Planet] — boolean functions in cryptography.
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===Information, Inquiry===
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{{col-break}}
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* [[Inquiry]]
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* [[Dynamics of inquiry]]
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* [[Semeiotic]]
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* [[Logic of information]]
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* [[Descriptive science]]
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* [[Normative science]]
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* [[Pragmatic maxim]]
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* [[Truth theory]]
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{{col-end}}
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===Related articles===
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* [http://intersci.ss.uci.edu/wiki/index.php/Cactus_Language Cactus Language]
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* [http://intersci.ss.uci.edu/wiki/index.php/Futures_Of_Logical_Graphs Futures Of Logical Graphs]
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* [http://intersci.ss.uci.edu/wiki/index.php/Propositional_Equation_Reasoning_Systems Propositional Equation Reasoning Systems]
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{{col-break}}
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* [http://intersci.ss.uci.edu/wiki/index.php/Differential_Logic_:_Introduction Differential Logic : Introduction]
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* [http://intersci.ss.uci.edu/wiki/index.php/Differential_Propositional_Calculus Differential Propositional Calculus]
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* [http://intersci.ss.uci.edu/wiki/index.php/Differential_Logic_and_Dynamic_Systems_2.0 Differential Logic and Dynamic Systems]
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{{col-break}}
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* [http://intersci.ss.uci.edu/wiki/index.php/Prospects_for_Inquiry_Driven_Systems Prospects for Inquiry Driven Systems]
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* [http://intersci.ss.uci.edu/wiki/index.php/Introduction_to_Inquiry_Driven_Systems Introduction to Inquiry Driven Systems]
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* [http://intersci.ss.uci.edu/wiki/index.php/Inquiry_Driven_Systems Inquiry Driven Systems : Inquiry Into Inquiry]
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{{col-end}}
  
 
==Document history==
 
==Document history==
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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.
 
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.
  
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* [http://www.getwiki.net/-Boolean_function Boolean function], [http://www.getwiki.net/ GetWiki].
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* [http://intersci.ss.uci.edu/wiki/index.php/Boolean_function Boolean Function], [http://intersci.ss.uci.edu/ InterSciWiki]
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* [http://mywikibiz.com/Boolean_function Boolean Function], [http://mywikibiz.com/ MyWikiBiz]
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* [http://wikinfo.org/index.php/Boolean_function Boolean function], [http://wikinfo.org/index.php/Main_Page Wikinfo].
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* [http://planetmath.org/BooleanValuedFunction Boolean Function], [http://planetmath.org/ PlanetMath]
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* [http://wikinfo.org/w/index.php/Boolean_function Boolean Function], [http://wikinfo.org/w/ Wikinfo]
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* [http://en.wikipedia.org/wiki/Boolean_function Boolean function], [http://en.wikipedia.org/ Wikipedia].
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* [http://en.wikiversity.org/wiki/Boolean_function Boolean Function], [http://en.wikiversity.org/ Wikiversity]
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* [http://beta.wikiversity.org/wiki/Boolean_function Boolean Function], [http://beta.wikiversity.org/ Wikiversity Beta]
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* [http://en.wikipedia.org/w/index.php?title=Boolean_function&oldid=60886833 Boolean Function], [http://en.wikipedia.org/ Wikipedia]
  
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[[Category:Boolean algebra]]
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[[Category:Inquiry]]
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[[Category:Cryptography]]
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[[Category:Open Educational Resource]]
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[[Category:Peer Educational Resource]]
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[[Category:Combinatorics]]
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[[Category:Computer Science]]
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[[Category:Discrete Mathematics]]
 
[[Category:Logic]]
 
[[Category:Logic]]
 
[[Category:Mathematics]]
 
[[Category:Mathematics]]

Latest revision as of 20:24, 5 November 2015

☞ This page belongs to resource collections on Logic and Inquiry.

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.

Syllabus

Focal nodes

Peer nodes

Logical operators

Template:Col-breakTemplate:Col-breakTemplate:Col-end

Related topics

Template:Col-breakTemplate:Col-breakTemplate:Col-breakTemplate:Col-end

Relational concepts

Template:Col-breakTemplate:Col-breakTemplate:Col-breakTemplate:Col-end

Information, Inquiry

Template:Col-breakTemplate:Col-breakTemplate:Col-breakTemplate:Col-breakTemplate:Col-end

Related articles

Template:Col-breakTemplate:Col-breakTemplate:Col-breakTemplate:Col-end

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.

Retrieved from "https://mywikibiz.com/index.php?title=Boolean_function&oldid=466830"