CS251 - Computer Organization and Design - Spring 2008
Lecture 3 - Digital Logic
Practical Details
- Office hours
- Textbook chapter on CD: Appendix B
- sections B.1, B.2, B.3, B.7, B.8, B.9, B.10
Digital Logic
The Magic of Digital
The operational amplifier
The Magic of Logic
Whitehead and Russell
Many subsequent alternate formulations
- Emil Post: production systems (condition => action)
- Alonzo Church: lambda calculus
- Kleene closures come in here
- Alan Turing
Combinational Logic
Boolean Logic
Laws of thought, discovered/created/formalized by George Boole
- AND (\dot, usually omitted)
- NOT (\turnstyle, overbar more common)
- OR (+)
AND is the easiest to understand: OR, NOT harder
NOT and either AND or OR is enough to produce any Boolean expression
Rules of Boolean Algebra
TRUE = 1; FALSE = 0.
- Excluded middle: NOT NOT X = X
- Identity: X + 0 = X; X1 = X
- Zero/one: X + 1 = 1; X0 = 0
- Idempotence: X+X = X; XX = X
- Inverse: X + NOT X = 1; (NOT X)X = 0
- Commutativity: X + Y = Y + X; XY = YX
- Associativity: (X + Y) + Z = X + (Y + Z); (XY)Z = X(YZ)
- Distribution: X(Y + Z) = XY + XZ; X + YZ = (X + Y)(X + Z)
- DeMorgan: NOT ( X + Y) = (NOT X)(NOT Y); NOT (XY) = (NOT X) + (NOT
Y)
Connection to Abstract Algebra
XOR can stand in for NOT
- x XOR y = (x AND NOT y) OR (NOT x AND y)
Rules of logic are the same as the algebra of Z_2
- {0, 1} with the usual laws of multiplication and addition, modulo
2.
- The only unusual things are
- Use XOR instead of OR: 1+1=0.
- NOT is minus: 0-1=1; 1-1=0.
Truth Tables
Normal Forms
Disjunctive
Conjunctive
Pictures of Gates
Transistors
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