How deeply does Math and Architectures of Deep Learning cover logical XOR?

Explains truth assignments, geometric nonseparability, and an MLP decomposition into complemented conjunctions, stopping short of learned parameter values.

Logical XOR

As explained in Math and Architectures of Deep Learning

A Boolean operation that outputs true when exactly one of its two binary inputs is true.

Logical XOR, or exclusive OR, returns 1 when exactly one of two binary inputs is 1. It returns 0 when both inputs agree: both are 0 or both are 1. For inputs x0 and x1, XOR can be written as ¬x0x1 + x0¬x1. section 7.2.1 section 7.5.2

The two-input truth table is:

0, 0 → 0: neither input is true. section 7.4.1
0, 1 → 1: only the second input is true. section 7.5.2
1, 0 → 1: only the first input is true. section 7.5.2
1, 1 → 0: both inputs are true, so the exclusive condition fails. section 7.2.1

Rock-music friendship recommendation

Let each input indicate whether one person likes rock music. XOR identifies disagreement: one person likes rock and the other does not. Applying NOT to XOR instead recommends friendship when both like rock or both dislike it: friendship = ¬(rockA ⊕ rockB). section 7.2.1

XOR matters for neural networks because its positive points, (0,1) and (1,0), cannot be separated from its negative points, (0,0) and (1,1), by one straight line. A single perceptron therefore cannot model XOR. Multiple perceptrons can form the required nonlinear partition. section 7.4.1 section 7.4

Figure 7.13 MLP for the logical function corresponding to table 7.1

(a) Perceptron for x̄₀x₁

(b) Perceptron for x₀x̄₁

Inspect the MLP construction: implement x̄0x1 and x0x̄1 with separate perceptrons, then combine the two terms with OR. section 7.5.2

How Math and Architectures of Deep Learning covers it

Cover of Math and Architectures of Deep Learning

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Also explained in

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The Well-Grounded Rubyist, Fourth EditionBitwise XOR is a binary operation that produces 1 where exactly one corresponding input bit is 1 and produces 0 where both corresponding input bits are equal.
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Classic Computer Science Problems in Swift: Essential techniques for practicing programmersThe exclusive-or operation that returns true for exactly one true operand and false when both operands have the same truth value.

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Related concepts

boolean operationkind of · A Boolean operation combines or transforms truth-valued inputs according to logical rules to produce a truth-valued result.
CNOT gateimplemented by · A CNOT gate is a two-qubit controlled-NOT gate that flips a target qubit when its control qubit is one.
conditional logicequivalent to · Conditional logic is a programming technique that selects execution paths, actions, or result values according to evaluated conditions.
perceptroncontrasts with · A perceptron is a single-layer computational model and linear classifier that combines weighted inputs with a bias and applies a threshold or activation rule.
one-time padused by · A one-time pad is a symmetric encryption technique that combines plaintext with a truly random, equally long secret key that is never reused.

This explainer was generated by Manning's knowledge graph from the text of Math and Architectures of Deep Learning. The section links point to the parts of the book it draws on.