Majority function

In Boolean logic, the majority function (also called the median operator) is the Boolean function that evaluates to false when half or more arguments are false and true otherwise, i.e. the value of the function equals the value of the majority of the inputs. Representing true values as 1 and false values as 0, we may use the (real-valued) formula:

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

The "−1/2" in the formula serves to break ties in favor of zeros when the number of arguments 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.

Most applications deliberately force an odd number of inputs so they don't have to deal with the question of what happens when exactly half the inputs are 0 and exactly half the inputs are 1. The few systems that calculate the majority function on an even number of inputs are often biased towards "0"—they produce "0" when exactly half the inputs are 0 -- for example, a 4-input majority gate has a 0 output only when two or more 0's appear at its inputs. In a few systems, the tie can be broken randomly.