1. Lecture 14-16 Ref. handout page 57 -65
Turing Machines
Lecture 14-16
Ref. handout page

2. Alan Turing
Great mathematician
The father of
the modern computer

3. Bletchley Park is near Milton Keynes Half hour from London
Alan Turing’s Statue
at Bletchley Park
Bletchley Park
is near Milton Keynes 
Half hour from London

4. Historical Notes
For a long time it was believed that any mathematical problem could, at least in principle, be proved from the basic axioms
1931, Kurt Gödel proposed the theorem of undecidability – there exist theorems which can neither be proved or disproved
Later, Alonso Church and Alan Turing also found other problems which had no algorithmic solution

5. Alan Turing was interested in
whether there was a way to define which problems were/were not decidable (computable)?
can we create a machine to simulate the human brain so that those computable problems can be solved automatically?
, Turing was working on a paper, “computable numbers”. The Turing machine
in this paper turned out to be the simplest prototype of all computers!

6. Turing’s Idea
thinks
reads
a+b*x/y
makes notes
State of mind changes

7. The Turing Machines control unit ...... ...... head
Current
state
control
unit
Read/Write
head
......
......
(Infinitely long) tape, one symbol per square

8. A Turing Machine Conceptualization- - -
reset -
start -

9. Actions of a Turing Machine
depending on current state
and
current tape symbol
Change
state
Write a new symbol
onto the current tape square
Move left/right one square

10. State Transitions A B
X,Y / L
read write move

11. State Transitions – comparing with FA and PDAB
X,Y / L
read write move for TM
input value on stack/ op of stack for PDA

12. A State Transition Table
read
write
move
new 1 a b L 2 R 3 c 2 1 3

13. The Infinite Tape a b a c a Initial position of read/write
head (the left most) a b a c a
Initial data (finite)
Blank tape (infinite)

14. A Turing Machine Example
Test for a palindrome
□ □ □ □
aabbabbaaa
□ □ □ □ □ □
If ‘a’ then replace with space
go to right hand end
check for ‘a’
if not found halt -> error
else go to left hand end
do the
same
for ‘b’

15. Cont. Transition Table for checking ‘a’
state
read
write
move
New state
start a □
findA b
deleteA

16. Answer (cont. TM for detecting palindromes)
a,a/R
b,b/R
findA
delA
□,□/L
a,□/L
a,□/R
a,a/L
b,b/L
b,b/L
□,□/L
start
□,□/L
yes no
return
□,□/L
a,a/L
b,□/R
b,□/L
findB
□,□/L
delB
a,a/R
b,b/R
□,□/R

17. Memo for In-class test 14 [ /5]
questions
my answers
correct answers
comments 1 2 3 4 5

18. Turing Machine Tricks Write as many y’ as there are x’s before
□ □ □ □ x x x x □ □ □ □
after
□ □ □ □ x x x x □ y y y y □ □ □ □

19. Answer

20. Unary Numbers 0 x x x x x + x x x = x x x x x x x x ? + ? = ? x x x
......
x x x x + x x x = x x x x x x
? ? = ?

21. The Limitations of TMs control unit ...... ...... Control Unit
does one thing only
control
unit
‘hardware’
specific
......
......

22. The 2-Tape Turing Machine
control
unit
data tape
program tape
......
......
program tape stores the description
of a 1-tape TM’s transition table

23. The 2-tape TM – The Universal TM
control
unit
Data tape
Program tape
......
......

24. 2-Tape Transitions tape 1 tape 2 state read write move new
Includes ‘no move’

25. extras, e.g. the current state
The Emulator
control
unit
data tape
program tape
......
......
extras, e.g. the current state

26. An Example of UTM (Universal Turing Machine)
Adding one to a binary number
e.g.
before □ □ □ □ □ □ □ □
after □ □ □ □ □ □ □ □
before □ □ □ □ □ □ □ □
after □ □ □ □ □ □ □ □
before □ □ □ □ □ □ □ □
after □ □ □ □ □ □ □

27. An Example of UTM Cont.
How to go about writing a TM for adding one (a single-tape first)?
Idea (algorithm):
Move the head to the right most position
If the right most is 0, replace it by 1,stop
move to left one space
If the current value is ‘0’ or a blank replace
it by 1, stop
Go to step 3

28. An Example of UTM Cont. start 1 R check □ L _ stop carry start check
state
read
write
mov
new
start 1 R
check □ L _
stop
carry
□,□ /L
start
check
1,0,L
1,0,L
0,1,_
carry
stop
0,1,_
□,1, _
A TM which adds one in binary

29. Cont. change to 3-tape TM Program tape S 1 1 R S S 0 0 R S S □ □ L C
Data tape
Extra tape S

30. What’s so Special? The lever – many lifting jobs The wheel –
lots of uses
UTM –
anything you can
‘program’
UTM

31. Memo for In-class test 15 [ /5]
questions
my answers
correct answers
comments 1 2 3 4 5

32. A Real Computer Control unit: processor Data tape: user memory
Program tape:
program memory
Extra tape:
system memory
I/O facilities added

33. TMs and ‘Real’ Computer - 2
Write a TM which simulates a PC TM
Anything a real computer can do,a TM can do

34. TMs and ‘Real’ Computer - 1
Write a Java program which simulates a TM TM
Anything a TM can do, a real computer can do

35. The Church-Turing Thesis
Anything which can be computed can be computed by a Turing Machine
corollary
Anything which can’t be done by a TM
can’t be done by any computer

36. So ...... If there is anything a TM can’t do it probably can’t be done
But may be a TM can do anything
we can imagine (?)
Is there anything which
TMs can’t do???

37. Decision Problems
It is harder to say what we will never be able to do rather than what we can do (computable).
Consider only simpler problems – decision problems where the answer is ‘yea’ or ‘no’.
Consider only TMs with two halt states ‘yes’ and ‘no’.
A problem is decidable if we can have a TM for it which eventually enters either the ‘yes’ or ‘no’ state depending on its input.

38. The Halting Problem ...... In searching of an undecidable problem.
The best-known such problem is called the halting problem.
Given an arbitrary program with an arbitrary input, can we make a Turing Machine to test whether the program stop or loop forever?

39. The Barber Paradox (by Bertrand Russell )
Once upon a time, a village barber put a notice outside his shop
I shave all and only those
men in the village who do
not shave themselves.
Q. Does the barber shave
himself?

40. The Barber Paradox U = { every man in the village }
S = { men shave themselves }
U \ S = { men don’t shave themselves}
Where does the barber belong to, S or U \ S ?

41. The Barber Paradox – trouble in either ways
Assume: the barber does shave himself
Any man in this village who shaves himself is not shaved by barber.
Therefore, the barber does not shave himself.
Assume: the barber doesn’t shave himself
Any man in this village is shaved by the barber if and only if he is not shaved by himself. Therefore, the barber does shaves himself.

42. Why is the Barber Paradox important?
A paradox with importance to mathematical logic and set theory.
It was constructed to demonstrate the self-contradictory nature of the elementary set theory
It underlies the proof of Alan Turing’s proof of the undesirability of the halting problem.

43. Program Testing – (another way to look at the halting problem)
A Java program is stored in a file
The title is called ‘abc.java’
I have tested ‘abc’ with different data
It always works
It’s never got into an infinite loop
Can I be sure it never will for any input?

44. Automated Program Testing
A file containing
a program
A file of test data
Prog Test
MyTest
infinite loop /halt

45. The Java Prog Test Class
public class Prog Test {
Public boolean halt(String prog, String data)
if prog halts when given data as input
return true;
else
return false; }

46. Using Prog Test import Prog Text public class MyTest {
static public void main(String[] args)
{ String s = args[0];
if(ProgTest.halt(s,s))
// loop forever
while(true) continue; }

47. Running MyTest java MyTest MyTest.java MyTest (program To test)
with itself
as data
command to
run java

48. Suppose It Loops static public void main(String[] args)
{ String s = args[0]; s = MyTest.java
if(ProgTest.halt(s,s))
// loop forever
while(true) continue; }
ProgTest.halt(“MyTest.java, “MyTest.java”)

49. The Dilemma If it halts it loops forever ...
If it loops forever it halts ...

50. The Solution The result is nonsense – MyTest must halt or loop
We got into this mess by assuming that
ProgTest.halts could be written
So ProgTest.halts doesn’t (can’t) exist.

51. Conclusion There are some things computer can’t do
ProgTest.halts can’t be written
No program exists which can tell whether any given program halts for any given data
There are some things computer can’t do

52. Memo for In-class test 16 [ /5]
questions
my answers
correct answers
comments 1 2 3 4 5