Can an ANN of 2 neurons solve XOR?

Question

I know that an artificial neural network (ANN) of 3 neurons in 2 layers can solve XOR

Input1----Neuron1\
       \ /        \
       / \         +------->Neuron3
      /   \       /
Input2----Neuron2/

But to minify this ANN, can just 2 neurons (Neuron1 takes 2 inputs, Neuron2 take only 1 input) solve XOR?

Input1
      \
       \ Neuron1------->Neuron2
       / 
Input2/

The artificial neuron receives one or more inputs... https://en.wikipedia.org/wiki/Artificial_neuron

Bias input '1' is assumed to be always there in both diagrams.

Side notes:

Single neuron can solve xor but with additional input x1*x2 or x1+x2 https://www.quora.com/Why-cant-the-XOR-problem-be-solved-by-a-one-layer-perceptron/answer/Razvan-Popovici/log

The ANN form in second diagram may solve XOR with additional input like above to Neuron1 or Neuron2?

Answer 1

Of course it is possible. But before solving XOR problem with two neurons I want to discuss on linearly separability. A problem is linearly separable if only one hyperplane can make the decision boundary. (Hyperplane is just a plane drawn to differentiate the classes. For an N dimensional problem i.e, a problem having N features as inputs the hyperplane will be an N-1 dimensional plane.) So for a 2 input XOR problem the hyperplane will be an one dimensional plane that is a "line".

Now coming to the question, XOR is not linearly separable. Hence we cannot directly solve XOR problem with two neurons. Following images show no matter how many ways we draw a line in 2D space we cannot differentiate one side's output with the other. For example for the first one (0,1) and (1,0) both inputs makes XOR to give 1. But for the input (1,1) the output is 0 but we cannot make it separated and unfortunately they are falling in the same side.

So here we have two options to solve it:

1. Using hidden layer. But it will increase the number of neurons more than two.
2. Another option is to increase the dimensions.

Let's have an illustration how increasing dimensions can solve this problem keeping the the number of neurons 2.

For an analogy we can think XOR as a subtraction of AND from OR like below,

If you notice the upper figure, the first neuron will mimic logical AND after passing "v=(-1.5)+(x1*1)+(x2*1)" to some activation function and the output will be considered as 0 or 1 depending on v is negative or positive respectively (I am not getting into the details...hope you got the point). And the same way the next neuron will mimic logical OR.

So for the first three cases of the truth table the AND neuron will remain turned off. But for the last one (actually where OR is different from XOR) the AND neuron will be turned on providing a big negative value to the OR neuron which will overwhelm the total summation to negative as it is big enough to make the summation a negative number. So finally activation function of the second neuron will interpret it as 0.

By this way we can make XOR with 2 neurons.

Following two figures are also the solutions to your questions which I have collected: