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A Boolean function with multiple outputs, : {,} {,} with > is a vectorial or vector-valued Boolean function (an S-box in symmetric cryptography). [ 6 ] There are 2 2 k {\displaystyle 2^{2^{k}}} different Boolean functions with k {\displaystyle k} arguments; equal to the number of different truth tables with 2 k {\displaystyle 2^{k}} entries.
There are 2 n+1 symmetric n-ary Boolean functions. Instead of the truth table , traditionally used to represent Boolean functions, one may use a more compact representation for an n -variable symmetric Boolean function: the ( n + 1)-vector, whose i -th entry ( i = 0, ..., n ) is the value of the function on an input vector with i ones.
The Boolean function is said to be linearly separable provided these two sets of points are linearly separable. The number of distinct Boolean functions is 2 2 n {\displaystyle 2^{2^{n}}} where n is the number of variables passed into the function.
In two-valued logic, there are sixteen possible truth functions, also called Boolean functions, of two inputs P and Q. Any of these functions corresponds to a truth table of a certain logical connective in classical logic, including several degenerate cases such as a function not depending on one or both of its arguments. Truth and falsehood ...
In the binary case, there are four possible inputs, viz. (T, T), (T, F), (F, T), and (F, F), thus yielding sixteen possible binary truth functions – in general, there are n-ary truth functions for each natural number n. The sixteen possible binary truth functions are listed in the table below.
Perhaps there were features derivable from the original features that were important for identifying the ugly duckling. The set of booleans in the vector can be extended with new features computed as boolean functions of the original features. The only canonical way to do this is to extend it with all possible Boolean functions.
Similarly, a Boolean function of degree at most depends on at most coordinates, making it a junta (a function depending on a constant number of coordinates), where is an absolute constant equal to at least 1.5, and at most 4.41, as shown by Wellens. [5]
State space of a Boolean Network with N=4 nodes and K=1 links per node. Nodes can be either switched on (red) or off (blue). Thin (black) arrows symbolise the inputs of the Boolean function which is a simple "copy"-function for each node. The thick (grey) arrows show what a synchronous update does.