Boolean functions · gates · combinational logic · state

Digital Logic & Circuits

Digital logic turns binary relationships into physical circuits. Boolean functions define desired behavior, gates implement those functions, combinational networks compose them into useful operations, and sequential circuits add memory so hardware can evolve through time.

Logic composition

Hardware complexity grows by composing small exact behaviors.

A gate is simple enough to describe with a tiny truth table. Useful hardware emerges by wiring many such functions together and, when memory is needed, feeding state forward through time.

Boolean function

Specify a relationship between binary inputs and outputs using algebra, a truth table, or another exact logical description.

Gate network

Implement Boolean operations with composable gates such as NOT, AND, OR, XOR, NAND, and NOR.

Combinational circuit

Combine gates so outputs depend only on the inputs present now: adders, multiplexers, encoders, decoders, comparators, and more.

Sequential circuit

Add stored state and feedback so future behavior depends on both current input and remembered history.

Gate truth tables · reference

A truth table defines the function independently of the circuit used to implement it.

01¬A

NOT

invert one bit

0 → 11 → 0

02A ∧ B

AND

1 only when both inputs are 1

00 → 001 → 010 → 011 → 1

03A ∨ B

OR

1 when at least one input is 1

00 → 001 → 110 → 111 → 1

04A ⊕ B

XOR

1 when the inputs differ

00 → 001 → 110 → 111 → 0

Combinational instrument · one-bit addition

FULL_ADDER

1-BIT ALU SLICE

SUM = A ⊕ B ⊕ C

COUT = AB + C(A ⊕ B)

SUM (S)

C-OUT

0 + 0 + 0 = 00 (Binary)

0 (Decimal)

A full adder combines XOR, AND, and OR relationships to add two bits plus an incoming carry. Chaining full adders creates wider binary addition.

Sequential logic · state across time

Memory changes the equation from output = f(input) to next state = f(input, current state).

Combinational logic answers a question about the present inputs. Sequential logic lets the circuit's past affect what happens next.

Latch / flip-flop

Stores one bit of state using a circuit whose output can persist after the initiating input changes.

Register

Groups multiple state elements so a word of binary data can be held and updated together.

Counter

Uses stored state plus combinational next-state logic to move through a defined sequence of binary states.

Finite-state controller

Combines a state register with next-state and output logic to coordinate multi-step hardware behavior.

Hardware ArchitectureSee how logic, registers, memory, and control become an executing processor.Formal LogicStudy symbolic validity and inference independently of the physical circuits that implement Boolean operations.AutomataCompare physical state machines with abstract finite-state computation and language recognition.