RobotnixAcademy

Week 02 · lesson

Investigation: Find the Boundary

system flow

From Represented Inputs to a Tested Layered Output

  1. Inputs

    Declare the represented values x1 and x2.

  2. Parameters

    Hold weights and biases fixed during inference.

  3. Weighted Score

    Multiply each input by its weight, add the results, then add bias.

  4. Activation

    Apply the declared step rule to produce a binary unit output.

  5. Hidden Features

    Pass h1 and h2 into the output unit as a changed representation.

  6. Test and Boundary

    Compare all four outputs with XOR targets and preserve a counterexample after one parameter changes.

Read this concept flow as plain text
  1. Inputs. Declare the represented values x1 and x2. SUPPLIED DOMAIN: FOUR BINARY CASES.
  2. Parameters. Hold weights and biases fixed during inference. INSPECT: WEIGHTS + BIAS.
  3. Weighted Score. Multiply each input by its weight, add the results, then add bias. CALCULATION: Σ wx + b.
  4. Activation. Apply the declared step rule to produce a binary unit output. RULE: SCORE ≥ 0.
  5. Hidden Features. Pass h1 and h2 into the output unit as a changed representation. LAYER: OR-LIKE + AND-LIKE.
  6. Test and Boundary. Compare all four outputs with XOR targets and preserve a counterexample after one parameter changes. EVIDENCE: 4/4 + CHANGED CASE.

A perceptron combines represented inputs into one score and compares that score with a threshold. The arithmetic is simple enough to inspect:

score = (x1 × w1) + (x2 × w2) + bias
output = 1 when score ≥ 0; otherwise output = 0

That calculation creates a linear decision boundary. With two inputs, the boundary is a straight line. Changing a weight rotates or shifts the line; changing the bias shifts it without changing the input values.

Completed example: an AND-like rule

Use w1 = 1, w2 = 1, and bias = -1.5.

x1x2CalculationScoreOutput
00(0×1) + (0×1) - 1.5-1.50
01(0×1) + (1×1) - 1.5-0.50
10(1×1) + (0×1) - 1.5-0.50
11(1×1) + (1×1) - 1.50.51

The output is 1 only when both inputs are 1. The table is stronger evidence than checking one successful case because it tests the complete two-input binary domain.

Training and inference are different events

During training, an algorithm adjusts parameters using examples and an objective. During inference, the fitted parameters are held fixed while a new represented input moves through the calculation. A model can infer without learning anything new from that individual case.

Do not say “the neuron decided to increase its weight.” The training procedure changed a number according to a declared update rule. The perceptron has no goal, feeling, or self-explanation.

Why XOR exposes the limit

XOR expects output 1 when exactly one input is 1:

x1x2XOR target
000
011
101
110

No single straight boundary separates the two positive cases from the two negative cases. Repeatedly changing the same perceptron's weights cannot repair the representation limit. A layered network can create intermediate features and combine more than one boundary.

Investigation

  1. Recalculate the four AND-like cases above.
  2. Change only the bias from -1.5 to -0.5 and recalculate.
  3. Name every output that changed.
  4. Attempt to choose one set of two weights and one bias for XOR.
  5. When one case fails, preserve it rather than hiding it.

Success means your arithmetic is reproducible and your conclusion distinguishes a parameter defect from a model-capacity limit.

Vocabulary lab

Flip the idea, not just the card

Explain the term before you reveal the back. Then compare your explanation with the definition, example, and warning.

1 / 5
Read all terms without animation
perceptron
A binary classifier that combines weighted inputs, adds a bias, and applies a threshold. Example: The AND-like perceptron outputs one only for input 1,1. Do not confuse it with: A perceptron is a mathematical model, not a biological neuron or a complete deep network.
weight
A parameter multiplying one represented input before the values are combined. Example: A weight of one preserves a binary input's contribution to the score. Do not confuse it with: A weight is fitted or selected; it is not evidence that the feature causes the outcome.
bias
A parameter added to the weighted sum before the activation rule is applied. Example: A bias of negative 1.5 makes the AND-like threshold harder to cross. Do not confuse it with: This mathematical bias is not the same meaning as social or dataset bias.
linear decision boundary
A straight dividing line produced by a linear score in a two-feature space. Example: One side receives output zero and the other receives output one. Do not confuse it with: A single linear boundary cannot separate the XOR pattern.
inference
Using fixed fitted parameters to calculate an output for a represented input. Example: The four table rows are four inference cases. Do not confuse it with: Inference does not automatically update the parameters.