Administrivia

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Administrivia

• Any new students?
• Has everyone been to the course web site?
• First assignment ready pick it up on web
• Labs to begin Monday
• Location 007 Friend
• Details Pick up lab on web, read it, do it
• Laptops or desktops?
• Questions??

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Building a computer
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Why build a computer?

• Curiosity
• What really happens when I hit a key?
• Necessity
• Prerequisite to parts of the course
• Breadth
• Should understand whats changing the world

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Why NOT build a computer?

• Computers seem really complicated !
• Pentium III has over 28 MILLION components
• How can we hope to understand them ?

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Questions in building a house

• What are the basic components
• 2x4s, I-beams, plasterboard, ..
• Light fixtures, plumbing,
• What is glue for combining them
• Nails, screws, bolts, pipes,
• How do we model the process
• Architectural drawings, building codes,

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Questions in building a computer

• What are the basic components
• Logic gates
• What is glue for combining them
• Wires to build circuits
• How do we model the process
• Architectural drawings, design rules,
• Mathematical formulation

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Details

• Wires are made of silicon or chemicals
• Crossing wires make transistors
• Electrons either do or dont flow in wires
• Think of light switches turning current on or off
• Wire thickness currently about .135 micron
• 1 micron 10-6 meters
• Can fit 28 million transistors in less than 1x1
• Design must follow fabrication rules
• Non-crossing wires cant get too close
• Mathematical abstraction Boolean algebra

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What is Abstraction?

• Ignoring / Hiding details to capture essential
  common features
• Example for us Well ignore whether were
  talking about
• A Pentium II or a Pentium III
• A Mac or a PC
• Instead, well focus on what really makes a
  computer a computer.

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Abstraction (cont.)

• Real Life Example Brush your teeth
• Here, we ignore
• What kind of toothbrush
• What kind of toothpaste
• What youre wearing
• etc.
• These things arent important!
• Often called hand waving

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Abstraction (cont.)

• Abstraction allows us to understand things in a
  Modular fashion.
• If we had to spell out everything we did, our
  lives would seem really complicated.
• Same is true for computers.To understand them,
  we use abstraction.
• Working bottom up.

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Layers of Abstraction

• Build more and more powerful tools out of simpler
  ones.

Really Simple Stuff
Computers
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Layers of Abstraction (cont.)

• Each layer corresponds to a beautiful Big Idea
  of computer science.
• These ideas are the foundation for the digital
  revolution that everyone talks about.

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Layers of Abstraction (cont.)

• For Computers, What is theReally Simple Stuff?
• Answer 0s and 1s

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The secret lives of 0s and 1s
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A Simple Logic Puzzle

• Frank will go to the party if Ed goes
  AND Dan does NOT.
• Dan will go if Bob does NOT go OR if Carole goes.
• Ed will go to the party if Alice AND Bob go.
• Alice and Bob decide to go,but Carol stays home.
• Will Frank go to the party?

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A Simple Logic Puzzle

• Frank will go to the party if Ed goes
  AND Dan does NOT.
• Dan will go if Bob does NOT go OR if Carole goes.
• Ed will go to the party if Alice AND Bob go.
• Alice and Bob decide to go,but Carol stays home.
• Will Frank go to the party?
• Answer YES

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Using 0s and 1s

• What do 0s and 1s mean?
• For now, well take Natural meanings
• For example, if we have a variable Alice
  forwhether Alice goes to the party,
• If Alice goes, we write Alice 1
• If Alice doesnt, we write Alice 0

0 False
1 True
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Logic Gates

• Computers are circuits made of Logic Gates.
• Logic gates manipulate 0s and 1s (False and
  True) by letting electrons flow or not.
• Well look at three types of Logic Gates
• AND are all inputs true?
• OR is one input true?
• NOT flip the truth value

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AND Gate

• Zac will go to the party if Xena AND Yanni go.

X
Z
Y
X Y Z F F F F T F T F F T T
T
Truth Table
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AND Gate (cont.)

• Zac will go to the party if Xena AND Yanni go.

X
Z
Y
X Y Z 0 0 0 0 1 0 1 0 0 1 1
1
Truth Table
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AND Gate (cont.)

• AND Gate is a circuit that allows output current
  to flow if both inputs are flowing.

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AND Gate (cont.)
X
is shorthand for
W
Z
X
Y
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AND Gate (cont.)
W X Y Z 0 0 0 ? 0 0 1 ? 0 1 0 ? 0 1
1 ? 1 0 0 ? 1 0 1 ? 1 1 0 ? 1 1 1 ?
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AND Gate (cont.)
W X Y Z 0 0 0 0 0 0 1 0 0 1 0 0 0 1
1 0 1 0 0 0 1 0 1 0 1 1 0 0 1 1 1 1
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OR Gate

• Zac will go to the party if Xena OR Yanni go.

X
Z
Y
X Y Z F F F F T T T F T T T
T
Truth Table
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OR Gate (cont.)

• Zac will go to the party if Xena OR Yanni go.

X
Z
Y
X Y Z 0 0 0 0 1 1 1 0 1 1 1
1
Truth Table
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OR Gate (cont.)

• OR Gate is a circuit that allows output current
  to flow if either input is flowing.

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OR Gate (cont.)
W
Z
X
Y
is shorthand for
W
Z
X
Y
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OR Gate (cont.)
W
Z
W X Y Z 0 0 0 ? 0 0 1 ? 0 1 0 ? 0 1
1 ? 1 0 0 ? 1 0 1 ? 1 1 0 ? 1 1 1 ?
X
Y
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OR Gate (cont.)
W
Z
W X Y Z 0 0 0 0 0 0 1 1 0 1 0 1 0 1
1 1 1 0 0 1 1 0 1 1 1 1 0 1 1 1 1 1
X
Y
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NOT Gate

• Yanni will go to the party if Xena does NOT go.

Y
X
Shorthand
X
Y
X Y 0 1 1 0
Truth Table
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NOT Gate (cont.)

• NOT Gate is a circuit that reverse the sense of a
  flow.
• Logical complement.

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A Simple Logic Puzzle

• Frank will go to the party if Ed goes
  AND Dan does NOT.
• Dan will go if Bob does NOT go OR if Carole goes.
• Ed will go to the party if Alice AND Bob go.
• Alice and Bob decide to go,but Carol stays home.
• Will Frank go to the party?

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Logic Puzzle Circuit
Ed
Frank
Dan
Frank will go to the party if Ed goes
AND Dan does NOT.
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Logic Puzzle Circuit (cont.)
Ed
Frank
Dan
Frank will go to the party if Ed goes
AND Dan does NOT.
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Logic Puzzle Circuit (cont.)
Alice
Ed
Frank
Bob
Dan
Ed will go to the party if Alice AND Bob go.
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Logic Puzzle Circuit (cont.)
Alice
Ed
Frank
Bob
Dan
Carole
Dan will go if Bob does NOT go OR if Carole goes.
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Logic Puzzle Circuit (cont.)
1
Alice
Ed
Frank
1
Bob
Dan
0
Carole
Alice and Bob decide to go,but Carol stays home.
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Logic Puzzle Circuit (cont.)
1
1
Alice
Ed
Frank
1
1
Bob
0
Dan
0
0
Carole
Evaluation
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Logic Puzzle Circuit (cont.)
1
1
Alice
Ed
1
Frank
1
1
Bob
0
Dan
0
0
Carole
Evaluation
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Logic Puzzle Circuit (cont.)
1
1
Alice
Ed
1
Frank
1
1
Bob
0
Dan
0
0
0
Carole
Evaluation
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Logic Puzzle Circuit (cont.)
1
1
Alice
Ed
1
1
Frank
1
1
Bob
0
1
Dan
0
0
0
Carole
Evaluation
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Logic Puzzle Circuit (cont.)
1
1
Alice
Ed
1
1
1
Frank
1
1
Bob
0
1
Dan
0
0
0
Carole
Evaluation Complete! Answer Frank goes to the
party.
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Logic Puzzle Circuit (cont.)
1
Alice
Ed
Frank
1
Bob
Dan
1
Carole
What if Alice, Bob, and Carol all go to the
party?
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Logic Puzzle Circuit (cont.)
1
1
Alice
Ed
1
1
0
Frank
1
1
Bob
0
0
Dan
1
1
1
Carole
What if Alice, Bob, and Carol all go to the
party?
Answer Frank does NOT go to the party!
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Intermission

• Is it all clear?
• Should/Could we do another such problem?
• Light controllers
• Light fixture has 3 switches
• Light is on if an odd number of the switches are
  on

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Building Circuits

• Suppose someone gives us an arbitraryTruth
  Table.
• Can we build a circuit which satisfiesexactly
  that Truth Table?

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Our Logic Puzzle

• Frank will go to the party if Ed goes
  AND Dan does NOT.
• Dan will go if Bob does NOT go OR if Carole goes.
• Ed will go to the party if Alice AND Bob go.
• Suppose we made the truth table forthis puzzle.

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Logic Puzzle Circuit
Alice
Ed
Frank
Bob
Dan
Carole
The full circuit for the Logic Puzzle.
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Logic Puzzle Circuit (cont.)
Alice Bob Carole Frank 0 0 0 0 0 0 1 0 0
1 0 0 0 1 1 0 1 0 0 0 1 0 1 0 1 1 0 1 1
1 1 0
Note Frank goes only if AB1 and
C0
Truth Table for Logic Puzzle
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Recall AND Gate
A B C F 0 0 0 0 0 0 1 0 0 1 0 0 0 1
1 0 1 0 0 0 1 0 1 0 1 1 0 0 1 1 1 1
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Modified AND Gate
A
F
A B C F 0 0 0 0 0 0 1 0 0 1 0 0 0 1
1 0 1 0 0 0 1 0 1 0 1 1 0 1 1 1 1 0
B
C
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Modified AND Gate
A
F
A B C F 0 0 0 0 0 0 1 0 0 1 0 0 0 1
1 0 1 0 0 0 1 0 1 0 1 1 0 1 1 1 1 0
B
C
Note Frank goes only if AB1 and
C0. The modified AND also solves the
Logic Puzzle!
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Modified AND Gate
A
F
A B C F 0 0 0 0 0 0 1 0 0 1 0 0 0 1
1 0 1 0 0 0 1 0 1 0 1 1 0 1 1 1 1 0
B
C
?
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Sure
A
F
A B C F 0 0 0 0 0 0 1 1 0 1 0 0 0 1
1 0 1 0 0 0 1 0 1 0 1 1 0 0 1 1 1 0
B
C
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Similarly
A
F
A B C F 0 0 0 0 0 0 1 0 0 1 0 1 0 1
1 0 1 0 0 0 1 0 1 0 1 1 0 0 1 1 1 0
B
C
Similarly,we can makea circuit for any Truth
Tablewith only a single1 in its output.
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Given Any Truth Table
First, make circuitsfor each row in Truth Table
witha 1 in the output.
A B C F 0 0 0 0 0 0 1 1 0 1 0 1 0 1
1 0 1 0 0 0 1 0 1 0 1 1 0 1 1 1 1 0
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Any Truth Table (cont.)
A
A B C F 0 0 0 0 0 0 1 1 0 1 0 1 0 1
1 0 1 0 0 0 1 0 1 0 1 1 0 1 1 1 1 0
B
C
A
B
C
A
B
C
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Any Truth Table (cont.)
A
Finally, combine themwith an OR gate.
B
C
A
F
B
C
A
B
C
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Any Truth Table (cont.)
A
B
C
A
F
B
C

• The only way for F1 is if at least ONE of
  AND gates outputs 1.
• But each AND gate corresponds to a row
  in the Truth Table with a 1 in the output!

A
B
C
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Any Truth Table (cont.)
A B C F 0 0 0 0 0 0 1 1 0 1 0 1 0 1
1 0 1 0 0 0 1 0 1 0 1 1 0 1 1 1 1 0
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Another Example
First, make circuitsfor each row in Truth Table
witha 1 in the output.
A B C X 0 0 0 1 0 0 1 0 0 1 0 0 0 1
1 0 1 0 0 0 1 0 1 0 1 1 0 0 1 1 1 1
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Another Example (cont.)
A
A B C X 0 0 0 1 0 0 1 0 0 1 0 0 0 1
1 0 1 0 0 0 1 0 1 0 1 1 0 0 1 1 1 1
B
C
A
B
C
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Another Example (cont.)
A
Finally, combine themwith an OR gate.
B
C
X
A
B
C
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Another Example
A B C X 0 0 0 1 0 0 1 0 0 1 0 0 0 1
1 0 1 0 0 0 1 0 1 0 1 1 0 0 1 1 1 1
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Universality

• Note same idea works no matter how manyinput
  variables.
• So for ANY Truth Table we can write down,we can
  make a circuit for it using only 3 Logic Gates
  AND, OR, NOT
• This gives us a very powerful tool !
• Our first technical use of abstractionMake a
  circuit for that Truth Table.is something we
  can abstract and understand.

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Further Issues

• Some issues to think about on your own
• We know that AND,OR, and NOT are universal we
  can make a circuit for any Truth Table using just
  these gates ! What else is universal?
• Surprising answer There is a single gatecalled
  NAND which is universal all by itself !

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NAND Gate
X
Z
Y
X
Z
Y
X Y Z 0 0 1 0 1 1 1 0 1 1 1
0
Truth Table
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NOT from NAND Gate
X
Z
Fix Y at 1
Y
X
Z
X Y Z 0 0 1 0 1 1 1 0 1 1 1
0
1
X Y Z
0 1 1
1 1 0
Truth Table
Truth Table
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Further Issues (cont.)

• Some issues to think about on your own(if you
  want)
• We saw two circuits for the Logic Puzzlethat
  both worked but one (the modified AND) was much
  simpler than the other.
• Can you come up with methods for simplifying
  circuits?
• This is important for efficiency when you are
  dealing with very very large Truth Tables (which
  we will be in the future).

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Next Time Why are 0s and 1s allwe need?