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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
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| :<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 "−1/2" in the formula serves to break ties in favor of zeros when ''n'' is even. If the term "−1/2" is omitted, the formula can be used for a function that breaks ties in favor of ones.
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| == Boolean circuits ==
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| 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.
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| 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 ==
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| 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]].
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| 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]].
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| == Properties ==
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| Among the properties of the ternary median operator < x,y,z > are:
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| # < x,y,y > = y
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| # < x,y,z > = < z,x,y >
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| # < x,y,z > = < x,z,y >
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| # < < x,w,y >, w,z > = < x,w, < y,w,z > >
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| An abstract system satisfying these as axioms is a [[median algebra]].
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| ==References==
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| *{{Cite journal
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| | first = L. | last = Valiant | authorlink = Leslie Valiant
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| | title = Short monotone formulae for the majority function
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| | journal = Journal of Algorithms
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| | volume = 5
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| | issue = 3 | year = 1984 | pages = 363–366
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| | doi = 10.1016/0196-6774(84)90016-6
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| | ref = harv
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| | postscript = <!--None-->}}.
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| * {{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 }}
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| ==See also==
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| {{Commonscat-inline|Majority functions}}
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| * [[Boolean algebra (structure)]]
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| * [[Boolean algebras canonically defined]]
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| * [[Majority problem (cellular automaton)]]
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| [[Category:Logic gates]]
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| [[Category:Circuit complexity]]
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| [[Category:Boolean algebra]]
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