Boolean Functions & Shattering Demo

A perceptron with two inputs can compute 14 out of 16 Boolean functions — exactly those that are linearly separable. The two exceptions are XOR and XNOR: no single line can separate their positive from negative points. As input dimension grows, the fraction of computable functions collapses to zero. The VC dimension measures a model's capacity: a perceptron in ℝⁿ can shatter any n+1 points in general position.

All 16 Two-Input Boolean Functions

Click any cell to see details. Green border = linearly separable, red = not.

14 of 16 are linearly separable

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x₁	x₂	f(x)

VC Dimension & Shattering

Drag the 3 points below. A perceptron can shatter 3 points in general position (classify all 2³ = 8 labelings). If points become collinear, some labelings fail.

Labelings (8 total)

Click a labeling to test it. Blue = class 1, Red = class 0. Green highlight = separable, Red = not.

What is shattering?

A set of points is shattered by a model if the model can achieve every possible labeling. The VC dimension is the largest set size that can be shattered.

For a perceptron in ℝ²: VCdim = 3