XOR Problem
The XOR problem (exclusive OR problem) is a fundamental example in machine learning and artificial intelligence that highlights the limitations of linear models. The logical XOR operation returns true only when the two inputs are different (0-1 or 1-0). Graphically, the positive and negative points of the XOR problem cannot be separated by a single straight line, making it a non-linear problem.
In the 1960s, this problem was widely studied because it demonstrated that simple perceptrons (linear neuron models) were unable to solve it. This led to a temporary questioning of neural approaches and contributed to what is sometimes referred to as the "AI winter." However, the emergence of multi-layer perceptrons (MLPs) in the 1980s, coupled with algorithms such as backpropagation, showed that more complex architectures could effectively solve the XOR problem.
Today, the XOR problem is used as a pedagogical example to illustrate the necessity of non-linearity in learning models. Activation functions such as sigmoid, ReLU, or tanh allow neural networks to learn non-linear decision boundaries and overcome the limitations of simple perceptrons. In this sense, the XOR problem is often presented as a key step in understanding why deep neural networks are so powerful.
The XOR problem is one of the most famous didactic examples in artificial intelligence, highlighting the limitations of linear models. In the logical XOR operation, the output is true only when the inputs differ (0-1 or 1-0). Geometrically, this creates a configuration where positive and negative points cannot be separated by a single straight line, making the problem intrinsically non-linear.
In the 1960s, XOR became emblematic of the shortcomings of simple perceptrons, which can only handle linearly separable tasks. Marvin Minsky and Seymour Papert’s book Perceptrons (1969) formalized this limitation, contributing to skepticism about neural networks and to what later became known as the AI winter.
The breakthrough came in the 1980s, when multi-layer perceptrons (MLPs) and the backpropagation algorithm demonstrated that introducing non-linear activation functions allowed networks to solve XOR efficiently. This not only revived interest in neural networks but also paved the way for modern deep learning.
Today, XOR is often used as a teaching example. It illustrates why neural networks need hidden layers and non-linearities like sigmoid, tanh, or ReLU to model complex decision boundaries. Despite its simplicity, XOR remains a symbolic milestone: it shows that the power of neural networks lies in their ability to learn hierarchical, non-linear representations.
References:
- Wikipedia – XOR Problem: https://en.wikipedia.org/wiki/XOR_gate#Perceptrons
- Minsky & Papert, Perceptrons (1969)
- Towards Data Science – The XOR Problem and Neural Networks: https://towardsdatascience.com/the-xor-problem-in-neural-networks-46dfef3f46f1
