Consider learning the logical ``or'' function.

We'll use a single perceptron with three inputs (x_1, x_2, x_3) and a
single output ($l$). Note that the input x_3 is an artificial input
(i.e. the bias node) will remain fixed at -1.

Our examples are:


   example      x_1    x_2    x_3     l
   ====================================
   1             0      0     -1      0
   2             0      1     -1      1
   3             1      0     -1      1
   4             1      1     -1      1


We'll set the learning rate to 0.5 and initialize the weights to 0's: 
<w_1,w_2,w_3>=<0,0,0> 

Epoch One
=========

Example 1 <0,0,-1>: 

  dot product of example and weights is <0,0,-1> x <0,0,0> = 0
  since 0 >= threshold of 0, output = 1
  desired output = 0
  error = -1

  so update weights:
  <0,0,0> + <.5 (learning rate) x 0 (x_1) x -1 (error),
             .5 x 0 x -1,
             .5 x -1 x -1> = <0,0,0> + <0,0,.5> = <0,0,.5>

  <w_1,w_2,w_3>=<0,0,.5>


Example 2 <0,1,-1>: 

  dot product of example and weights is <0,1,-1> x <0,0,.5> = -.5
  since -.5 < threshold of 0, output = 0
  desired output = 1
  error = 1

  so update weights:
  <0,0,.5> + <.5 x 0 x 1, .5 x 1 x 1, .5 x -1 x 1> = 
  <0,0,.5> + <0,.5,-.5> = <0,.5,0>

  <w_1,w_2,w_3>=<0,.5,0>

Example 3 <1,0,-1>: 

  dot product of example and weights is <1,0,-1> x <0,.5,0> = 0
  since 0 >= threshold of 0, output = 1
  desired output = 1
  error = 0 
  example is correct, so no weight updates.
  <w_1,w_2,w_3>=<0,.5,0>

Example 4 <1,1,-1>: 

  dot product of example and weights is <1,1,-1> x <0,.5,0> = .5
  since .5 >= threshold of 0, output = 1
  desired output = 1
  error = 0 
  example is correct, so no weight updates.
  <w_1,w_2,w_3>=<0,.5,0>


Epoch Two
=========

Example 1 <0,0,-1>: 

  dot product of example and weights is <0,0,-1> x <0,.5,0> = 0
  since 0 >= threshold of 0, output = 1
  desired output = 0
  error = -1

  so update weights:
  <0,.5,0> + <.5 x 0 x -1, .5 x 0 x -1, .5 x -1 x -1> = 
  <0,.5,0> + <0,0,.5> = <0,.5,.5>

  <w_1,w_2,w_3>=<0,.5,.5>

Example 2 <0,1,-1>: 

  dot product of example and weights is <0,1,-1> x <0,.5,.5> = 0
  since 0 >= threshold of 0, output = 1
  desired output = 1
  error = 0
  example is correct, so no weight updates.
  <w_1,w_2,w_3>=<0,.5,.5>

Example 3 <1,0,-1>: 

  dot product of example and weights is <1,0,-1> x <0,.5,.5> = -.5
  since -.5 < threshold of 0, output = 0
  desired output = 1
  error = 1

  so update weights:
  <0,.5,.5> + <.5 x 1 x 1, .5 x 0 x 1, .5 x -1 x 1> = 
  <0,.5,.5> + <.5,0,-.5> = <.5,.5,0>

  <w_1,w_2,w_3>=<.5,.5,0>

Example 4 <1,1,-1>: 

  dot product of example and weights is <1,1,-1> x <.5,.5,0> = 1
  since 1 >= threshold of 0, output = 1
  desired output = 1
  error = 0 
  example is correct, so no weight updates.
  <w_1,w_2,w_3>=<.5,.5,0>


Epoch Three
===========

Example 1 <0,0,-1>: 

  dot product of example and weights is <0,0,-1> x <.5,.5,0> = 0
  since 0 >= threshold of 0, output = 1
  desired output = 0
  error = -1

  so update weights:
  <.5,.5,0> + <.5 x 0 x -1, .5 x 0 x -1, .5 x -1 x -1> = 
  <.5,.5,0> + <0,0,.5> = <.5,.5,.5>

  <w_1,w_2,w_3>=<.5,.5,.5>

Example 2 <0,1,-1>: 

  dot product of example and weights is <0,1,-1> x <.5,.5,.5> = 0
  since 0 >= threshold of 0, output = 1
  desired output = 1
  error = 0
  example is correct, so no weight updates.
  <w_1,w_2,w_3>=<.5,.5,.5>

Example 3 <1,0,-1>: 

  dot product of example and weights is <1,0,-1> x <.5,.5,.5> = 0
  since 0 >= threshold of 0, output = 1
  desired output = 1
  error = 0
  example is correct, so no weight updates.
  <w_1,w_2,w_3>=<.5,.5,.5>

Example 4 <1,1,-1>: 

  dot product of example and weights is <1,1,-1> x <.5,.5,.5> = .5
  since .5 >= threshold of 0, output = 1
  desired output = 1
  error = 0 
  example is correct, so no weight updates.
  <w_1,w_2,w_3>=<.5,.5,.5>

Epoch Four
==========

Every example is correct.

Example 1 <0,0,-1>: 

  dot product of example and weights is <0,0,-1> x <.5,.5,.5> = -.5
  since -.5 < threshold of 0, output = 0
  desired output = 0
  example is correct, so no weight updates.
  <w_1,w_2,w_3>=<.5,.5,.5>
  
Example 2 <0,1,-1>: 

  dot product of example and weights is <0,1,-1> x <.5,.5,.5> = 0
  since 0 >= threshold of 0, output = 1
  desired output = 1
  example is correct, so no weight updates.
  <w_1,w_2,w_3>=<.5,.5,.5>

Example 3 <1,0,-1>: 

  dot product of example and weights is <1,0,-1> x <.5,.5,.5> = 0
  since 0 >= threshold of 0, output = 1
  desired output = 1
  example is correct, so no weight updates.
  <w_1,w_2,w_3>=<.5,.5,.5>

Example 4 <1,1,-1>: 

  dot product of example and weights is <1,1,-1> x <.5,.5,.5> = .5
  since .5 >= threshold of 0, output = 1
  desired output = 1
  example is correct, so no weight updates.
  <w_1,w_2,w_3>=<.5,.5,.5>