Unit 1: The Model of the Computer - david-abel.github.io · ‣ State Machines! ... ‣ Michael’s domino OR gate: ... Identical logical formulas, dramatically different speed of

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February 5th, 2016

Dave Abel

Unit 1: The Model of the Computer

Today’s Takeaway

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‣ Earlier this week: computers are doing logic with gates!

‣ Today: Memory, Logic, and input/output = Computer!

Outline for Today

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‣ Review of Gates

- AND, OR, NOT, Composite Gates

- Domino Gates, XOR, Adding with Gates

‣ State Machines!

‣ Model of a Computer

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Gates: NOT

1 0N

ot

0 1

P NOT(P)

1 0

0 1

Not

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Gates: NOT

Not

1 0

Q: What is this, physically?

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Gates: NOT

Not

1 0

Q: What is this, physically?

A: High voltage electrical pulses!

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Gates: NOT

1 0

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Gates: NOT

P NOT(P)

1 0

0 1

1 0

0 1

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Gates: ANDP Q AND( P Q)

1 1 1 1 1

1 0 0 1 0

0 1 0 0 1

0 0 0 0 0

AND

1

1AND(P,Q) = 1

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Gates: ORP Q OR( P Q)

1 1 1 1 1

1 0 1 1 0

0 1 1 0 1

0 0 0 0 0

OR

OR(P,Q) = 10

1

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Gates

OR

P

Q

OR(P,Q)

AND

P

Q

AND(P,Q)

Not

P NOT(P)

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Gates: Composition

OR(P,NOT(Q))

A: OR

POR(P,NOT(Q))

Not

Q NOT(Q)

P Q OR( P NOT( Q)

T T T T F T

T F T T T F

F T F F F T

F F T F T F

Domino Gates

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P

Q

OR(P,Q)

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Domino Gates

Could It Work?

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‣ Michael’s domino OR gate: 24 dominoes

‣ The first pentium processor had 3.3 Mill transistors, or roughly 800k gates.

‣ So we need around 20 Mill dominoes

‣ World record for domino topple: 4.5 Mill

‣ Pentium: computes 60 Mill times a second

‣ Dominoes? Takes awhile to set up…

Two Different Notions of Speed

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1. Electricity vs. Dominos falling over

2. Do we have the shortest computation for that problem?

Problem: Speed!

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P NOT(NOT(P))=

Problem: Speed!

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P NOT(NOT(P))

# of Gates: 0 2

Problem: Speed!

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P NOT(NOT(NOT(NOT(P))))

# of Gates: 0 4

Problem: Speed!

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P NOT(NOT(NOT(NOT(NOT(P)))))

# of Gates: 0 6

Problem: Speed!

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P NOT(NOT(NOT(NOT(NOT(P)))))…….

# of Gates: 0 6024

Problem: Speed!

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# of Gates: 0 6024

Think about that many domino gates…

Problem: Speed!

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# of Gates: 0 6024

Identical logical formulas, dramatically different speed of computation!

Think about that many domino gates…

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One Last Gate… XORP Q XOR( P Q)

1 1 0 1 1

1 0 1 1 0

0 1 1 0 1

0 0 0 0 0

XOR

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1 1 0 1 1

1 0 1 1 0

0 1 1 0 1

0 0 0 0 0

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1 1 0 1 1

1 0 1 1 0

0 1 1 0 1

0 0 0 0 0

XOR

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Arithmetic and LogicWe know how to do addition, subtraction, and

multiplication in binary…

But we’ve claimed that computers used AND, OR, NOT gates at their core to do everything…

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Arithmetic and LogicWe know how to do addition, subtraction, and

multiplication in binary…

But we’ve claimed that computers used AND, OR, NOT gates at their core to do everything…

Q: Can we represent addition with logical gates?

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Arithmetic and Gates

0+ 00 0

1+ 00 1

0+ 10 1

1+ 11 0

1


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0+ 00 0

1+ 00 1

0+ 10 1

1+ 11 0

1

XOR

Hint: you’ll need these:

AND


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XOR

AND

topNumber

bottomNumbertwo’s digit

one’s digit

A:

0+ 00 0

1+ 00 1

0+ 10 1

1+ 11 0

1

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0+ 00 0

1+ 00 1

0+ 10 1

1+ 11 0

1

XOR

AND

0

0two’s digit

one’s digit

A:

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0+ 00 0

1+ 00 1

0+ 10 1

1+ 11 0

1

XOR

AND

0

0two’s digit

one’s digit

A:

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0+ 00 0

1+ 00 1

0+ 10 1

1+ 11 0

1

XOR

AND

0

0two’s digit

0

A:

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0+ 00 0

1+ 00 1

0+ 10 1

1+ 11 0

1

XOR

AND

0

0two’s digit

0

A:

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0+ 00 0

1+ 00 1

0+ 10 1

1+ 11 0

1

XOR

AND

0

00

0

A:

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0+ 00 0

1+ 00 1

0+ 10 1

1+ 11 0

1

XOR

AND

1

1two’s digit

one’s digit

A:

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0+ 00 0

1+ 00 1

0+ 10 1

1+ 11 0

1

XOR

AND

1

1two’s digit

one’s digit

A:

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0+ 00 0

1+ 00 1

0+ 10 1

1+ 11 0

1

XOR

AND

1

1two’s digit

0

A:

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0+ 00 0

1+ 00 1

0+ 10 1

1+ 11 0

1

XOR

AND

1

1two’s digit

0

A:

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0+ 00 0

1+ 00 1

0+ 10 1

1+ 11 0

1

XOR

AND

1

11

0

A:

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0+ 00 0

1+ 00 1

0+ 10 1

1+ 11 0

1

XOR

AND

1

11

0

A:

Binary Adder

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Truth Table to Formula

Idea: with a certain set of logical functions, we can represent all

possible logical formulas!

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Truth Table to Gate

Idea: with a certain set of logical functions gates, we can represent all

possible logical formulas!

Abstraction:

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OR

P

Q

OR(P,Q)

AND

P

Q

AND(P,Q)

Not

P NOT(P)

all possible logical

formulas!

Abstraction:

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Low level programs

Abstraction:

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Low level programs

High level programs

Abstraction:

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Low level programs

High level programs

Your ideas!=

State Machines

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State Machines: Example

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Suppose we want a blinking christmas light:


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O = was the light on a second ago?N = is the light on now?


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oldLight = was the light on a second ago?


newLight = is the light on now?


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newLight = is the light on now?Q: How can we write the value of newLight in

terms of oldLight?


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newLight = is the light on now?

A: newLight = NOT(oldLight)


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Not

newLightoldLight

…1 second delay


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Not

newLightoldLight

…1 second delay


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Not

newLightoldLight

…1 second delay


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Not

newLightoldLight

…1 second delay


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Not

newLightoldLight

…1 second delay


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Not

newLightoldLight

…1 second delay


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ONSTATE

OFFSTATE

A More Complicated Example: Sliding Doors

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isPersonLeft isPersonRight


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isPersonLeft isPersonRight


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isPersonRightisPersonLeft


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OR doorOpen

DOORCLOSED

DOOROPEN


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DOORCLOSED

DOOROPEN


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DOORCLOSED

DOOROPEN


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OR(left,right)

DOORCLOSED

DOOROPEN


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OR(left,right)

NOT(OR(left,right))

DOORCLOSED

DOOROPEN


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OR(left,right)

NOT(OR(left,right))

left = 0right = 0Memory:

readread

DOORCLOSED

DOOROPEN


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Q: How else can we write this with logic?

NOT(OR(left,right))

OR(left,right)

DOORCLOSED

DOOROPEN


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Q: How else can we write this with logic?A: AND(NOT(left),NOT(right))

NOT(OR(left,right))

OR(left,right)

Side Note: DeMorgan’s Law

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NOT(OR(left,right))

AND(NOT(left),NOT(right))

=


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P Q AND( NOT(P) NOT(Q))

1 1 0 0 01 0 0 0 10 1 0 1 00 0 1 1 1

P Q NOT( OR(P,Q))

1 1 0 11 0 0 10 1 0 10 0 1 0


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P Q AND( NOT(P) NOT(Q))

1 1 0 0 01 0 0 0 10 1 0 1 00 0 1 1 1

P Q NOT( OR(P,Q))

1 1 0 11 0 0 10 1 0 10 0 1 0

DOORCLOSED

DOOROPEN

State Machines: Sliding Doors

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DOORCLOSED

DOOROPEN

State Machines: Sliding Doors

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Sensors! Input! Reading from Memory!

The Computer

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‣ Internal states (memory)

‣ Complex logic relating bits/information

‣ Mechanism for setting bits (input)

‣ Mechanism for displaying bits (output)

The Computer

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The Computer

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The Computer

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The Computer

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The Computer

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https://xkcd.com/505/

https://xkcd.com/505/

The Church-Turing Thesis

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Alonzo Church Alan Turing


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“There is only one notion of computation”


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(and anything with these properties is doing it)

‣ Internal states

‣ Complex logic relating bits




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Things that can be computed, period.


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Things a domino computer could compute

before the sun goes supernova


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Things a regular computer can compute before the

sun goes supernova


Things a domino computer could compute

before the sun goes supernova


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Things a regular computer can compute before the sun

goes supernova


Things a domino computer could compute before the

sun goes supernova

Q: What, if anything, is out here?


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goes supernova



sun goes supernova

???



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goes supernova



sun goes supernova

???

A: Theory unit!



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‣ Some other questions:

- Is the human brain carrying out computation?

- Is the comic right? Or are there things in the world that feel like they aren’t computation?

- Is “randomness” computable? How?

- Is there one “fastest” model of a computer? (e.g. X is to the transistor as the transistor is to the domino)

The Computer

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‣ Internal states (memory)




Q: What are these in our computers?

The Computer

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‣ Internal states





The Computer

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‣ Internal states





Memory

The Computer

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‣ Internal states





Memory

CPU

The Computer

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‣ Internal states





Memory

CPU

Mouse

Keyboard

The Computer

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‣ Internal states





Memory

CPU

Mouse

Keyboard

Monitor

Next Time

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‣ Internal states




Memory

CPU

Mouse

Keyboard

Monitor

Programming lets us define the

“complex logic”!

Unit 1: The Model of the Computer - david-abel.github.io · ‣ State Machines! ... ‣ Michael’s domino OR gate: ... Identical logical formulas, dramatically different speed of

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