Java · Course navigation

Java: a two-layer network that computes XOR

Combine two hidden neurons so that a network solves the problem a single perceptron cannot.

On this page

Before you start

Java math functions and the perceptron lesson.

Combine two hidden neurons so that a network solves the problem a single perceptron cannot.
Predict the sample output and explain the state changes.
Complete the practice task and check the boundary cases.

The concept

A hidden layer lets a network build new features. One hidden neuron can learn OR and another can learn AND. The output neuron then fires when OR is on but AND is off, which is exactly XOR.

The program uses the sigmoid function, which maps any number smoothly into the range 0 to 1. Large weights make the neurons behave almost like on/off switches. Computing the outputs from given weights is called a forward pass. Training would find such weights automatically by backpropagation.

Code example

Java

import java.util.Locale;

class Main {
  static double sigmoid(double z) {
    return 1.0 / (1.0 + Math.exp(-z));
  }

  public static void main(String[] args) {
    int[][] inputs = {{0, 0}, {0, 1}, {1, 0}, {1, 1}};
    for (int[] in : inputs) {
      double or = sigmoid(20 * in[0] + 20 * in[1] - 10);
      double and = sigmoid(20 * in[0] + 20 * in[1] - 30);
      double out = sigmoid(20 * or - 20 * and - 10);
      System.out.printf(Locale.ROOT, "%d XOR %d = %.3f%n", in[0], in[1], out);
    }
  }
}

Output

0 XOR 0 = 0.000
0 XOR 1 = 1.000
1 XOR 0 = 1.000
1 XOR 1 = 0.000

Example details

Follow the code

Open the walkthrough to inspect the example one step at a time.

Show guided walkthrough

Guided walkthroughReady

1import java.util.Locale;

3class Main {

4 static double sigmoid(double z) {

5 return 1.0 / (1.0 + Math.exp(-z));

6 }

8 public static void main(String[] args) {

9 int[][] inputs = {{0, 0}, {0, 1}, {1, 0}, {1, 1}};

10 for (int[] in : inputs) {

11 double or = sigmoid(20 * in[0] + 20 * in[1] - 10);

12 double and = sigmoid(20 * in[0] + 20 * in[1] - 30);

13 double out = sigmoid(20 * or - 20 * and - 10);

14 System.out.printf(Locale.ROOT, "%d XOR %d = %.3f%n", in[0], in[1], out);

15 }

16 }

17}

Follow the example

Start to inspect the source, values, and output. These prepared steps explain the published sample; grouped loop steps describe multiple iterations.

View the complete dry-run table

Prepared trace of the published program. Some rows group repeated iterations.
What happens	State after step	Output so far
This hidden neuron is on when at least one input is 1.	`OR feature`	—
This one is on only when both inputs are 1.	`AND feature`	—
The output fires for OR but not AND.	`out ≈ OR and not AND`	—
The four outputs round to 0, 1, 1, 0.	`out ≈ OR and not AND`	0 XOR 0 = 0.000 0 XOR 1 = 1.000 1 XOR 0 = 1.000 1 XOR 1 = 0.000

Behind the scenes

How Java processes this code

The Java compiler produces bytecode. The JVM loads the class and invokes its main method.

Cost and common mistakes

O(1) per input for this fixed network; in general a forward pass costs about the number of weights.

Without a non-linear activation, stacking layers collapses into a single linear function and cannot compute XOR.
A sigmoid with very large inputs saturates, so its gradient becomes tiny and learning stalls.

Try it yourself

Why are two hidden neurons needed?

Reveal answer and reasoning

One neuron can draw only one line. Two hidden neurons give two features, OR and AND, and the output combines them with a third line.