The perceptron learning rule is used to train a simple (single layer) perceptron. In order to train a perceptron, you need to have a set of linearly separable test cases. Linearly separable means that the required outputs must be separable by a single straight line. A set of inputs defining an XOR function are not linerally seperable, and thus the XOR function cannot be reproduced by a simple perceptron.

The perceptron learning rule basically boils down to adding or subtracting the input vector from the weight vector and adding or subtracting from the bias depending on an error signal.

As example, say you have four test cases and say that the weight vector and the bias are both zero:

• p1 = (0 0), t1 = 0
• p2 = (1 0), t2 = 0
• p2 = (0 1), t2 = 0
• p2 = (1 1), t2 = 1

These cases define a boolean AND. Starting with the last case, since it is the only one that will make a difference at the start, we have:

a = hardlim((1 1)(0 0)T + 0) = 0

We now have an error in the output: e = t - a = 1 - 0. We can correct this error:

Wnew = Wold + p1 (e) = (0 0) + (1 1) = (1 1)

bnew = bold + e = 1

We should now iterate through the cases again, to make sure they all still work:

a = hardlim((0 0)(1 1)T + 1) = 1, e = t - a = -1

Wnew = Wold + p1 (e) = (1 1) + (0 0) = (1 1)

bnew = bold + e = 0

... until e = 0 for all cases.

After iterating over the cases until they all produce the correct output (i.e. e = 0), the network will have learned to classify all of the inputs correctly.

The exact final configuration of the network will depend on the order in which the test cases are evaluated. There are many possible configurations that give the correct answers.

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