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> They "cheat" by feeding how all elements compares the their neighbours at each step. Like, if you give the algorithm that input for free what's the point.

How is that "cheating"? The algorithm is still learning how to do a comparison-sort, which has direct applications (assuming it performs well).

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So do they feed such information only when training the system?

You can generate that input only when needed. So whenever the sorting algorithm asks whether A[i] > A[i+1], you can perform the comparison there and then. So the answer to your question is yes.

The input is always needed, the neural net is an iterative algorithm that views the "local information" gathered from the array on every time step. It's asking continuously whether e.g. `A[i] > A[i+1]`.

No, the neural net needs to receive a feature vector before each instruction in order to sort the array. So whenever you want to sort things, you need to create that vector over and over.

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