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In Boolean logic, the majority function (also called the median operator) is a 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. Alternatively, representing true values as 1 and false values as 0, we may use the formula

$\operatorname {Majority} \left(p_{1},\dots ,p_{n}\right)=\left\lfloor {\frac {1}{2}}+{\frac {\left(\sum _{i=1}^{n}p_{i}\right)-1/2}{n}}\right\rfloor .$

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.

Boolean circuits

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.

For instance, in a 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.

A major result in circuit complexity asserts that the majority function cannot be computed by AC0 circuits of subexponential size.

Monotone formulae for majority

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 inclusive or or exclusive or.

For an arbitrary n there exists a monotone formula for majority of size O(n^5.3) Template:Harv. 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 Ajtai, Komlós, and Szemerédi.

Properties

Among the properties of the ternary median operator < x,y,z > are:

1. < x,y,y > = y
2. < x,y,z > = < z,x,y >
3. < x,y,z > = < x,z,y >
4. < < x,w,y >, w,z > = < x,w, < y,w,z > >

An abstract system satisfying these as axioms is a median algebra.

References

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See also

• Boolean algebra (structure)
• Boolean algebras canonically defined
• Majority problem (cellular automaton)