Boolean Analysis & Neural Nets

Why are some logic functions easy for AI to learn, while others are nearly impossible? The answer lies in the Fourier Spectrum.

1. The Notation Shift

Standard logic uses {0, 1}. But math and physics prefer {-1, 1}. This simple translation turns logical rules into algebraic equations.

Computer Science
0 / 1
Bit
Fourier Analysis
-1 / +1
Sign (Spin)

The Parity "Trick"

Logic Rule Count 1s. Is count Odd?
Algebraic Rule \( f(x) = x_1 \cdot x_2 \cdot x_3 ... \)
"Multiplying the inputs allows us to detect 'sameness' or 'difference' purely with math."

2. Interactive Spectrum Analyzer

Select a function (n=3) to see its Fourier "DNA".

Truth Table ({-1, 1} World)

x1 x2 x3 Output

Fourier Spectrum (Coefficients Squared) Total Mass: 1.0

Neural Net Difficulty
Easy

High linear mass. A single perceptron can learn this easily.

Linearity (Degree-1 Mass)
0% (Pure Noise) 75%

Low Frequency (Easy)

Functions like AND, OR, and MAJORITY have large coefficients on single variables (Degree 1). Neural nets find these gradients immediately.

High Frequency (Hard)

PARITY has all its mass on the highest interaction term. It looks like random noise to a perceptron. It requires depth (layers) to solve.

The Takeaway

Fourier Analysis gives us an "X-Ray" view of a boolean function's complexity before we even try to train a network on it.