Why can't Perceptron implement XOR gate?

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In summary, the conversation discussed the perceptron training rule and its limitations in implementing XOR. It was mentioned that the perceptron has a tri-state output and that the condition for the output state y = -1 is incorrect. It was also acknowledged that XOR cannot be implemented with a single binary perceptron due to the conditions for the output states. The conversation also touched upon the issue of weights remaining unchanged between epochs and the need to randomize the order of training data between each epoch to achieve convergence.
  • #1
shivajikobardan
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Homework Statement
perceptron xor gate
Relevant Equations
xor gate truth table
1645020070348.png

1645020094661.png


As you can see this is perceptron training rule. I want to learn how it can't implement xor. After 1 epoch/iteration, I get w1=0, w2=0,b=0. does this keeps on repeating? Anyone can implement this algorithm to code in python? Would be nice to see what happens in long run.
 
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  • #2
  • That is not a binary perceptron, it has a tri-state output in {-1, 0, 1}.
  • The condition for the output state y = -1 is incorrect; can you see why?
  • It is true that you cannot implement XOR with a single binary perceptron: can you think why?
    how can you get ## b + \sum w_i x_i ## to be < ## \theta ## when x = (0, 0) and x = (1, 1)?
  • If the weights are unchanged at the end of an epoch and you run exactly the same training data in the next epoch then obviously the weights will be the same at the end of that epoch as well. We do something between epochs to prevent this, can you remember or work out what this is?
 
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  • #3
pbuk said:
  • That is not a binary perceptron, it has a tri-state output in {-1, 0, 1}.
yeah -1,0,1 are values
pbuk said:
  • The condition for the output state y = -1 is incorrect; can you see why?
why?
pbuk said:
  • It is true that you cannot implement XOR with a single binary perceptron: can you think why?
    how can you get ## b + \sum w_i x_i ## to be < ## \theta ## when x = (0, 0) and x = (1, 1)?

pbuk said:
  • If the weights are unchanged at the end of an epoch and you run exactly the same training data in the next epoch then obviously the weights will be the same at the end of that epoch as well. We do something between epochs to prevent this, can you remember or work out what this is?
oh yeah that's what i thought. idk what we do between epochs? do we randomize? but i don't think i need to go to that depth tho.
 
  • #4
shivajikobardan said:
yeah -1,0,1 are values
And the first line of the question is "this algorithm is suitable for bipolar input vectors with a bipolar target". Do you see the problem?

shivajikobardan said:
why?
Because values ## - \theta \le y_{in} \lt \theta ## satisfy the condtions for both ## y = 0 ## and ## y = -1 ##.

shivajikobardan said:
idk what we do between epochs? do we randomize? but i don't think i need to go to that depth tho.
What do you mean "go to that depth"? You need to keep training until you achieve convergence, in this case when ## y = t ## for all training data. And yes, between each training epoch you randomize the order of the training data.

Of course if you are trying to create an XOR with a single binary perceptron you will never achieve convergence for the reason hinted at above.
 
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1. Why is the Perceptron unable to implement the XOR gate?

The Perceptron is a type of artificial neural network that uses a single layer of neurons to perform binary classification. It works by assigning weights to the input features and calculating a weighted sum, which is then passed through an activation function to produce a binary output. However, the XOR gate is a non-linear function that cannot be represented by a single layer of neurons, making it impossible for the Perceptron to implement.

2. Can the Perceptron be modified to implement the XOR gate?

No, the Perceptron cannot be modified to implement the XOR gate. This is because the Perceptron's single layer of neurons is limited in its ability to handle non-linearly separable data. To implement the XOR gate, a multi-layer neural network with hidden layers and non-linear activation functions is needed.

3. Are there any other limitations of the Perceptron?

Yes, the Perceptron has several limitations. Apart from being unable to implement non-linear functions like the XOR gate, it also struggles with data that is not linearly separable. Additionally, it is prone to overfitting and cannot handle complex datasets with multiple classes.

4. What is the role of activation functions in the Perceptron?

The activation function in the Perceptron is responsible for transforming the weighted sum of inputs into a binary output. It introduces non-linearity to the model and allows the Perceptron to learn and classify non-linearly separable data. However, since the XOR gate is a non-linear function, it cannot be represented by a single activation function.

5. Is the Perceptron still used in modern machine learning applications?

Yes, the Perceptron is still used in some modern machine learning applications, but it has been largely replaced by more advanced neural network architectures. However, the concept of the Perceptron is still relevant and serves as the building block for more complex neural networks, such as multi-layer perceptrons and deep learning models.

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