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In [[Boolean logic]], the '''majority function''' (also called the '''median operator''')  is a [[function (mathematics)|function]] from ''n'' inputs to one output. The value of the operation is false when ''n''/2 or more arguments are false, and true otherwise.
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Alternatively, representing true values as 1 and false values as 0, we may use the formula


:<math>\operatorname{Majority} \left ( p_1,\dots,p_n \right ) =  \left \lfloor \frac{1}{2} +  \frac{\left(\sum_{i=1}^n  p_i\right) - 1/2}{n} \right \rfloor. </math>
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The "&minus;1/2" in the formula serves to break ties in favor of zeros when ''n'' is even. If the term "&minus;1/2" is omitted, the formula can be used for a function that breaks ties in favor of ones.
  <li>[http://www.sebalo.info/spip/spip.php?article13 http://www.sebalo.info/spip/spip.php?article13]</li>
 
 
== Boolean circuits ==
  <li>[http://www.syygax.com/forum.php?mod=viewthread&tid=764780 http://www.syygax.com/forum.php?mod=viewthread&tid=764780]</li>
A ''majority gate'' is a [[logical gate]] used in [[circuit complexity]] and other applications of [[Boolean circuits]]. A majority gate returns true if and only if more than 50% of its inputs are true.
 
 
  <li>[http://www.usep.edu.ph/Campuses/Obrero/Colleges/CEd/index.php/forum/2-welcome-mat/39653-vibram-sydney-4#39653 http://www.usep.edu.ph/Campuses/Obrero/Colleges/CEd/index.php/forum/2-welcome-mat/39653-vibram-sydney-4#39653]</li>
For instance, in a [[Adder (electronics)|full adder]], the carry output is found by applying a majority function to the three inputs, although frequently this part of the adder is broken down into several simpler logical gates.
 
 
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A major result in [[circuit complexity]] asserts that the majority function cannot be computed by [[AC0|AC0 circuits]] of subexponential size.
 
 
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== Monotone formulae for majority ==
 
 
</ul>
For ''n'' = 1 the median operator is just the unary identity operation ''x''. For ''n'' = 3 the ternary median operator can be expressed using conjunction and disjunction as ''xy'' + ''yz'' + ''zx''. Remarkably this expression denotes the same operation independently of whether the symbol + is interpreted as [[disjunction|inclusive or]] or [[exclusive or]].
 
For an arbitrary ''n'' there exists a monotone formula for majority of size O(''n''<sup>5.3</sup>) {{harv|Valiant|1984}}. This is proved using [[probabilistic method]]. Thus, this formula is non-constructive. However, one can obtain an explicit formula for majority of polynomial size using a [[sorting network]] of [[Miklós Ajtai|Ajtai]], [[János Komlós (mathematician)|Komlós]], and [[Endre Szemerédi|Szemerédi]].
 
== Properties ==
 
Among the properties of the ternary median operator &lt; x,y,z &gt; are:
#  &lt; x,y,y &gt; = y
#  &lt; x,y,z &gt; = &lt; z,x,y &gt;
#  &lt; x,y,z &gt; = &lt; x,z,y &gt;
#   &lt; &lt; x,w,y &gt;, w,z &gt; = &lt; x,w, &lt; y,w,z &gt; &gt;
 
An abstract system satisfying these as axioms is a [[median algebra]].
 
==References==
*{{Cite journal
| first = L. | last = Valiant | authorlink = Leslie Valiant
| title = Short monotone formulae for the majority function
| journal = Journal of Algorithms
| volume = 5
| issue = 3 | year = 1984 | pages = 363–366
| doi = 10.1016/0196-6774(84)90016-6
| ref = harv
| postscript = <!--None-->}}.
* {{cite book | last=Knuth | first=Donald E. | authorlink=Donald Knuth | title=Introduction to combinatorial algorithms and Boolean functions | series=[[The Art of Computer Programming]] | volume=4a | year=2008 | isbn=0-321-53496-4 | pages=64–74 | publisher=Addison-Wesley | location=Upper Saddle River, NJ }}
 
==See also==
{{Commonscat-inline|Majority functions}}
* [[Boolean algebra (structure)]]
* [[Boolean algebras canonically defined]]
* [[Majority problem (cellular automaton)]]
 
[[Category:Logic gates]]
[[Category:Circuit complexity]]
[[Category:Boolean algebra]]

Latest revision as of 04:30, 27 December 2014

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As in many extra parts of society, the harder you work the more the reward. Adrasan, two valleys south of Cirali, is a strip of pensions and modest hotels on a gracefully curving beach bookended by pineclad spurs running down in the rocky heights above.

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Napster's service is compatible with more than 75 music devices, including those by Creative, iRiver and Samsung. He is a member of the British Association of Dermatologists. Accept, and embrace, who you are. Even if recovered, illegally removed fossils have diminished scientific value, says Xu Xing of the Institute of Vertebrate Paleontology and Paleoanthropology (IVPP) in Beijing: "If it's not collected right, a fossil loses its contextthe layer it was found in and its relationship to other fossils.".

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