How to run a 2-layer perceptron to solve XOR

Question

XOR is not solvable by using a single perceptron with standard scalar product and unit step function.

This article suggests using 3 perceptron to make a network: http://toritris.weebly.com/perceptron-5-xor-how--why-neurons-work-together.html

I'm trying to run the 3-perceptron network this way but it doesn't produce correct results for XOR:

//pseudocode
class perceptron {

  constructor(training_data) {
    this.training_data = training_data   
  }

  train() {
    iterate multiple times over training data
    to train weights
  }

  unit_step(value) {
    if (value<0) return 0
    else return 1
  }

  compute(input) {
    weights = this.train()
    sum     = scalar_product(input,weights)
    return unit_step(sum)
  }
}

The above perceptron can solve NOT, AND, OR bit operations correctly. This is how I use 3 perceptrons to solve XOR:

AND_perceptron = perceptron([
  {Input:[0,0],Output:0},
  {Input:[0,1],Output:0},
  {Input:[1,0],Output:0},
  {Input:[1,1],Output:1}
])

OR_perceptron = perceptron([
  {Input:[0,0],Output:0},
  {Input:[0,1],Output:1},
  {Input:[1,0],Output:1},
  {Input:[1,1],Output:1}
])

XOR_perceptron = perceptron([
  {Input:[0,0],Output:0},
  {Input:[0,1],Output:1},
  {Input:[1,0],Output:1},
  {Input:[1,1],Output:0}
])

test_x1 = 0
test_x2 = 1 

//first layer of perceptrons
and_result   = AND_perceptron.compute(test_x1,test_x2)
or_result    = OR_perceptron.compute(test_x1,test_x2)

//second layer
final_result = XOR_perceptron.compute(and_result,or_result)

The final_result above is not consistent, sometimes 0, sometimes 1. It seems I run the 2 layers wrongly. How to run these 3 perceptrons in 2 layers the correct way?

Answer 1

If you want to build a neural network with logical connectives (and, or, not), you have to consider the following equivalences regarding xor:

A xor B ≡ (A ∨ B) ∧ ¬(A ∧ B) ≡ (A ∨ B) ∧ (¬A ∨ ¬B) ≡ (A ∧ ¬B) ∨ (¬A ∧ B)

So you would need at least three and- or or-perceptrons and one negation if you want to use your perceptrons if I understand them correctly. In the article they use three perceprons with special weights for the xor. These are not the same as and- and or-perceptrons.