Exclusive Or (XOR) refers to a situation whereby a decision is based
on one, and only one, of two conditions being satisfied. For
instance, if I dislike crowds I may decide to go to the beach if it
is sunny, or if it is a bank holiday, but not both.
This is a Boolian logic function, and if each condition is assigned
a value of 1 if it is met, and 0 if it is not met, then a table can
be drawn up representing this, with values or 0 and 1 indicating
the decision:
1 0
1 0 1
0 1 0
This is a difficult concept for the human mind to grasp; we usually
function using and/or conditions. For a network, it constitutes the
basis of the XOR problem. How can a network be constructed so that
it will arrive at the the correct output when the input data is based
on XOR reasoning, and has nothing in common?
A network consists of a set of units that are each connected to all
the units of the next layer, but can only communicate with
each other by means of very simple signals.
The basic 2 layer perceptron is not capable of such processing, and
this was Minsky's critique. What is required to accomplish XOR
processing is a network such that
if the input is a pair of binary digits (which can be 0 or 1),
and the output is another binary pair, for the output value to be 1
of one of the inputs is 1, but 0 if neither or both is 1.
The answer to how this kind of decision can be made lies in the
network having one or more hidden layers between the input and output
layers. The units of the hidden layer are isolated from the networks
environment, and the connections pass from the input layer through
the hidden layer to the output layer. Each unit at a level is
connected to all units of the next higher layer.
The units can only transmit simple numerical values - the input
receives 1 or 0 and sends an output value of 1 or 0 along each of its
connections with other units. Each connection has a weight which is
either positive, negative or 0, and each unit has a bias. The
incoming value is multiplied by the weight on each of
its connections, and the sum of the products is added to the bias
that is associated with each unit. The resulting value is then
assigned an activation value of 0 or 1, according to the threshold of the
unit, and if the unit is thus activated it continues to propogate its
value to the output layer via another weighted connection.
Another advantage of this system is that changing the weights
allows a network to learn from past experience,
and thus improve its performance through the process of
backpropogation.
tion.
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