Logic Values, Combinational and Sequential, Combinational Logic | CS 35101, Study notes of Computer Architecture and Organization

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Computer Architecture

Appendix B

Fall 2005

Department of Computer Science

Kent State University

Logic Values

• Logic '0'
  • (^) False, off, deasserted, low
• Logic '1'
  • True, on, asserted, high

Combinational Logic

• Built from gates
• Can be described using a truth table or a

Boolean equation

• It is possible to translate between all three

representations

AND Gate 1 0 0 1 1 1 0 1 0 0 0 0 A B F F = A ⋅ B

NOT Gate

1 0 0 1 A F

F =

A

Inversion Bubbles

• (^) Inverters are so commonly used that designers have developed a shorthand notation
• Instead of using explicit inverters, you can attach bubbles to the inputs or outputs of other gates

B

A A

B

NAND Gate

• The NAND gate is the combination of an

NOT gate with an AND gate.

The Bubble in front of the gate is an inverter.

Combinational logic

• How would your describe the output of this

combinational logic circuit?

NAND and NOR gates

• The NAND and NOR gates are very

popular as they can be connected in more

ways that the simple AND and OR gates.

Exercise 1 – a bit challenging

Complete the Truth Table for the NAND and NOR Gates

1 1 1 0 0 1 0 0 Input Output 1 1 1 0 0 1 0 0 Input Output NAND NOR Hint: Think of the AND and OR truth tables. The outputs for the NAND and NOR are inverted.

Universal Gates

• Any combinational function can be built

from AND, OR and NOT gates

• However, there are universal gates that

alone can implement any function

• NAND and NOR are two such gates
• NAND and NOR are AND and OR gates

with inverted outputs

Combining Logic Gates

Add the boolean expressions at the output for each AND gate

A B Y C A•B B•C

Combining Logic: Truth Table

Given : Y = (A•B)+(B•C)

1 1 1 1 1 1 0 1 1 0 1 0 1 0 0 0 0 1 1 1 0 1 0 0 0 0 1 0 0 0 0 0 A B C Y

Laws of Boolean Algebra , Babbage (B.3,pB6)

· Identity law: A + 0 = A and A • 1 = A · Zero and One laws: A + 1 = 1 and A • 0 = 0 · Inverse Laws: A + A = 1 and A • A = 0 · Commutative laws: A + B = B + A and A • B = B • A (switch inputs) · Associative laws: A + (B + C) = (A + B) + C and A • (B • C) = (A • B) • C (reorder inputs) · Distributive laws: A • (B + C) = (A • B) + (A • C) and A + (B • C) = (A + B) • (A + C)

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· De Morgan’s laws: (A + B) = (A) • (B) and

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(A • B) = (A) + (B) (invert inputs)