Ontological Layer: Physical Boolean Evaluation

Logic Gates

The Silicon Bricks of the Digital Mind

Mathematics Made Physical

Boolean Algebra is purely symbolic until it is given a physical body. Logic gates are the electronic circuits that answer mathematical questions with voltage. An AND gate physically executes boolean multiplication; an OR gate executes boolean addition. By wiring these atomic decision-makers together, we transition from abstract algebraic expressions into functional digital hardware capable of computation.

BOOLEAN EQUATION → LOGIC SYMBOL → PHYSICAL GATE → VOLTAGE IN → TRUTH TABLE EVALUATION → UNIVERSAL REPLACEMENT → VOLTAGE OUT

The 15 Gate Realms

Ontological Mapping: From Symbols to Silicon

GROUP 01

Gate Fundamentals

Logic gate concept
Inputs and outputs
Binary logic levels
Boolean function ↔ physical gate

GROUP 02

NOT Gate (Inverter)

Input → complement
Inverter bubble symbol
Boolean expression (A')
Active-high / active-low

GROUP 03

AND Gate

Boolean multiplication (A·B)
Requires ALL inputs High
Series switch equivalent
Multi-input AND expansion

GROUP 04

OR Gate

Boolean addition (A+B)
Requires ANY input High
Parallel switch equivalent
Multi-input OR expansion

GROUP 05

NAND Gate

NOT + AND (A·B)'
Outputs Low only if ALL High
Universal-gate property
Hardware cost efficiency

GROUP 06

NOR Gate

NOT + OR (A+B)'
Outputs High only if ALL Low
Universal-gate property
CMOS pull-up/pull-down logic

GROUP 07

XOR Gate

Exclusive-OR (A ⊕ B)
Inequality detector (A ≠ B)
Odd-parity interpretation
Foundation of Binary Adders

GROUP 08

XNOR Gate

Exclusive-NOR (A ⊙ B)
Equality detector (A = B)
Even-parity interpretation
Comparator applications

GROUP 09

Truth Tables

2ⁿ combinations matrix
Evaluating intermediate nodes
Output-column mapping
Circuit ↔ table conversion

GROUP 10

Logic Symbols

IEEE Standard shapes
Inversion bubbles (state flip)
Active-low signals (bars)
Visual symbol interpretation

GROUP 11

Gate Equivalence

De Morgan visual transformations
Bubbled-OR ↔ NAND
Bubbled-AND ↔ NOR
Simplifying circuit schematics

GROUP 12

Universal Logic

NAND-only realization
NOR-only realization
Arbitrary Boolean functions
IC manufacturing efficiency

GROUP 13-14

Practical Hardware

Propagation delay (t_pd)
Logic voltage thresholds
Fan-in & Fan-out limits
Power dissipation (CMOS/TTL)

GROUP 14 (CONT)

Fan-Out Limits

Current sourcing/sinking
Driving multiple gate inputs
Capacitive load slowing speed
Buffer insertion

FINAL PROTOCOL

15 — The Gate Implementation Lab

F = A ⊕ B = A'B + AB'

F = ( (A(AB)')' · (B(AB)')' )'

GATES = 4x NAND2

[1] WRITE EXPRESSION [2] DRAW TRUTH TABLE [3] DE MORGAN CONVERSION [4] NAND/NOR MAPPING [5] TRACE PATH DELAY

A circuit diagram is a physical algorithm. By mastering universal gate conversion and predicting propagation delays, you ensure the mathematical truth of Realm 04 survives the electrical realities of Realm 05. This prepares the logic for the complex combinatorial circuits of Realm 07.

SIMULATION RUNNING