A boolean operation is a function whose inputs and output take binary values: 0 (“no” or “don’t fire”) and 1 (“yes” or “fire”). Logical operations combine decisions produced by other machines, making them useful as compact components in larger systems. section 7.2.1
Common operations express different combinations of binary conditions:
- OR outputs 1 when at least one condition is active, such as braking when a self-driving car detects a person, vehicle, or bend. section 7.2.1
- AND outputs 1 when all required conditions are active, such as a cat fleeing only when an object is both hard and sharp. section 7.2.1
- NOT reverses a binary decision, such as sounding an alarm when a face detector fails to recognize an authorized person. section 7.2.1
Every boolean operation can be represented by a truth table. For each row whose output is 1, form an AND of the relevant inputs or their complements; then combine those terms with OR. This construction expresses arbitrary logical functions with a multilayer perceptron (MLP). section 7.5.2
Constructing XOR with an MLP
XOR is y = x̄₀x₁ + x₀x̄₁. The two rows producing y = 1 become separate perceptrons: one computes x̄₀x₁ and the other computes x₀x̄₁. A final perceptron performs OR over their outputs. A single perceptron cannot model XOR because its decision boundary cannot separate the required points, whereas the two-layer MLP can. section 7.5.2 section 7.4
