Modeling Computers Finite State Machines Turing Machines

1. Class 14:
Modeling
Computers
Monday’s
class will
meet in Rice
Hall Bagel
Shop Area
cs1120 Fall 2011
David Evans
23 September 2011

2. Plan
Modeling Computers
Turing’s Model
Programming Turing Machines
2

3. What makes a
good model?
Copernicus
F = GM1M2 / R2
Ptolomy Newton
3

4. How should we model a Computer?
Turing invented the
Colossus (1944)
model we’ll use today
in 1936. What
“computer” was he
modeling?
Cray-1 (1976)
Apollo Guidance
Computer (1969)
Palm Pre (2009)
Apple II (1977) Flickr: louisvolant
Honeywell Kitchen Computer (1969) 4

5. “Computers” before WWII
5

6. Mechanical Computing
6

7. Modeling Computers
Input
Without it, we can’t describe a problem
Output
Without it, we can’t get an answer
Processing
Need some way of getting from the input to the
output
Memory: Need to keep track of what we are doing
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8. Modeling Input
Punch Cards
Altair BASIC Paper Tape, 1976
Engelbart’s mouse and keypad
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9. Turing’s Model
“Computing is normally done by
writing certain symbols on
paper. We may suppose this
paper is divided into squares like
a child’s arithmetic book.”
Alan Turing, On computable
numbers, with an application to the
Entscheidungsproblem, 1936
9

10. Modeling Pencil and Paper
... ...
# C S S A 7 2 3
How long should the tape be?
Infinitely long! We are modeling a computer, not
building one. Our model should not have silly practical
limitations (like a real computer does).
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11. Modeling Output
Blinking lights are
cool, but hard to
model
Use the tape: output
is what is written
on the tape at the
end
Connection Machine CM-5, 1993
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12. Modeling Processing (Brains)
Look at the
current state of
the computation
Follow simple
rules about what
to do next
12

13. Modeling Processing
Evaluation Rules
Given an input on our tape, how do we evaluate to
produce the output
What do we need:
Read what is on the tape at the current square
Move the tape one square in either direction
Write into the current square
0 0 1 1 0 0 1 0 0 0
Is that enough to model a computer?
13

14. Modeling Processing
Read, write and move is not enough
We also need to keep track of what we are
doing:
How do we know whether to read, write or move at
each step?
How do we know when we’re done?
What do we need for this?
14

15. Finite State Machines
0 1
0
Start 1 2
1
#
HALT
15

16. Hmmm…maybe we don’t need those
infinite tapes after all?
not a ( not a
paren paren
Start 1 2
)
)
# What if the
next input symbol
ERROR HALT is ( in state 2?
16

17. How many states do we need?
not a ( not a
paren paren
not a
Start 1 2 paren
(
)
)
)
3 not a
paren
#
(
ERROR HALT 4
)
(
17

18. Finite State Machine
There are lots of things we can’t compute
with only a finite number of states
Solutions:
“Infinite” State Machine
Hard to define, draw, and reason about
Add an infinite tape to the Finite State
Machine
18

19. Modeling Processing (Brains)
Follow simple rules
Remember what you are doing
“For the present I shall only
say that the justification lies
in the fact that the human
memory is necessarily
limited.” Alan Turing
19

20. FSM + Infinite Tape
Start:
FSM in Start State
Input on Infinite Tape
Pointer to start of input
Step:
Read one input symbol from tape
Write symbol on tape, and move L or R one square
Follow transition rule from current state
Finish:
Transition to halt state
20

21. Turing’s Model: Turing Machine
... # 1 0 1 1 0 1 1 1 0 1 1 0 1 1 1 # ...
Infinite Tape: Finite set of symbols, one in each square
Can read/write one square each step
Start Input: # Controller:
Input: 1 Limited (finite)
Write: #
Write: 0 number of states
Move:
Move:
1
Follow rules based
2 Input: 0
on current state
and read symbol
Write: #
Input: 1 Input: 0 Move: Halt Write one square
Write: 1 Write: 0 each step, move
Move: Move:
3
left or right or
halt, change state
21

22. Take on faith for now: we will argue this more convincingly near the end of the course!
Church-Turing Thesis
All mechanical computers are
equally powerful (except for
practical limits like memory
size, time, display, energy, etc.) Alonzo Church, 1903-1995
Some Turing machine can simulate
any mechanical computer
Any computer that can simulate a
Turing machine, can simulate
any mechanical computer
Alan Turing, 1912-1954 22

23. Turing Machine Summary
Model Input, Output, and Scratch Memory
Infinite tape, divided into discrete squares
Each square can contain a single symbol from
a finite alphabet
Model Processing
Finite State Machine
Keep track of a finite state (in your head)
Follow transition rules:
next state = f(current state, input symbol)
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24. Monday’s class will
Example meet in Rice Hall
Bagel Shop Area
24

25. Charge
PS4 Posted Today, due Monday 3 October
Exam 1: out October 7, due October 12
Covers:
Problem Sets 1-4 including PS Comments
Course Book Chapters 1-6
Classes 1-18
Monday’s class in Rice Hall
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