XOR problem

The XOR problem highlights the inability of a single-layer perceptron to learn non-linearly separable functions. A single-layer perceptron can only learn linear decision boundaries. Since the XOR function requires a non-linear boundary to separate the inputs that result in true and false outputs, a single-layer perceptron cannot solve it. This limitation was a significant factor in the early stagnation of neural network research. The solution to the XOR problem is achieved by using multi-layer perceptrons (MLPs) with hidden layers. These hidden layers introduce non-linear transformations of the input data, allowing the network to learn complex, non-linear decision boundaries and accurately predict the XOR function's output. The demonstration that MLPs could solve the XOR problem was a key step in the resurgence of neural network research and the development of more sophisticated architectures.