Turing's thesis states that any computation that can be performed by a mechanical device can also be performed by a Turing machine. The document then discusses several variations of the standard Turing machine model, including machines with additional capabilities like staying in the same position or having multiple tapes. It proves that all of these machine variations have the same computational power as the standard Turing machine by demonstrating how to simulate one type of machine with another.

1. Fall 2006 Costas Busch - RPI 1
Turing’s Thesis

2. Fall 2006 Costas Busch - RPI 2
Turing’s thesis (1930):
Any computation carried out
by mechanical means
can be performed by a Turing Machine

3. Fall 2006 Costas Busch - RPI 3
Algorithm:
An algorithm for a problem is a
Turing Machine which solves the problem
The algorithm describes the steps of
the mechanical means
This is easily translated to computation steps
of a Turing machine

4. Fall 2006 Costas Busch - RPI 4
When we say: There exists an algorithm
We mean: There exists a Turing Machine
that executes the algorithm

5. Fall 2006 Costas Busch - RPI 5
Variations
of the
Turing Machine

6. Fall 2006 Costas Busch - RPI 6
Read-Write Head
Control Unit
◊◊ a a c ◊◊ ◊b a cb b a a
Deterministic
The Standard Model
Infinite Tape
(Left or Right)

7. Fall 2006 Costas Busch - RPI 7
Variations of the Standard Model
• Stay-Option
• Semi-Infinite Tape
• Off-Line
• Multitape
• Multidimensional
• Nondeterministic
Turing machines with:
Different Turing Machine Classes

8. Fall 2006 Costas Busch - RPI 8
We will prove:
each new class has the same power
with Standard Turing Machine
Same Power of two machine classes:
both classes accept the
same set of languages
(accept Turing-Recognizable Languages)

9. Fall 2006 Costas Busch - RPI 9
Same Power of two classes means:
for any machine of first class1M
there is a machine of second class2M
such that: )()( 21 MLML =
and vice-versa

10. Fall 2006 Costas Busch - RPI 10
A technique to prove same power.Simulation:
Simulate the machine of one class
with a machine of the other class
First Class
Original Machine
1M 1M
2M
Second Class
Simulation Machine
simulates 1M

11. Fall 2006 Costas Busch - RPI 11
Configurations in the Original Machine
have corresponding configurations
in the Simulation Machine
nddd  10Original Machine:
Simulation Machine: nddd ′′′
∗∗∗
 10
1M
2M
1M
2M

12. Fall 2006 Costas Busch - RPI 12
the Simulation Machine
and the Original Machine
accept the same strings
fdOriginal Machine:
Simulation Machine: fd′
Accepting Configuration
)()( 21 MLML =

13. Fall 2006 Costas Busch - RPI 13
Turing Machines with Stay-Option
The head can stay in the same position
◊◊ a a c ◊◊ ◊b a cb b a a
Left, Right, Stay
L,R,S: possible head moves

14. Fall 2006 Costas Busch - RPI 14
Example:
◊◊ a a c ◊◊ ◊b a cb b a a
Time 1
◊◊ b a c ◊◊ ◊b a cb b a a
Time 2
1q 2q
1q
2q
Sba ,→

15. Fall 2006 Costas Busch - RPI 15
Stay-Option machines
have the same power with
Standard Turing machines
Theorem:
Proof: 1. Stay-Option Machines
simulate Standard Turing machines
2. Standard Turing machines
simulate Stay-Option machines

16. Fall 2006 Costas Busch - RPI 16
1. Stay-Option Machines
simulate Standard Turing machines
Trivial: any standard Turing machine
is also a Stay-Option machine

17. Fall 2006 Costas Busch - RPI 17
2. Standard Turing machines
simulate Stay-Option machines
We need to simulate the stay head option
with two head moves, one left and one right

18. Fall 2006 Costas Busch - RPI 18
1q 2q
Sba ,→
1q
Lba ,→
2q
Rxx ,→
Stay-Option Machine
Simulation in Standard Machine
For every possible tape symbol x

19. Fall 2006 Costas Busch - RPI 19
1q 2q
Lba ,→
1q 2q
Lba ,→
Stay-Option Machine
Simulation in Standard Machine
Similar for Right moves
For other transitions nothing changes

20. Fall 2006 Costas Busch - RPI 20
example of simulation
◊ ◊a a b a
1q
Stay-Option Machine:
1
◊ ◊b a b a
2q
2
1q 2qSba ,→
Simulation in Standard Machine:
◊ ◊a a b a
1q
1
◊ ◊b a b a
2q
2
◊ ◊b a b a
3q
3
END OF PROOF

21. Fall 2006 Costas Busch - RPI 21
Multiple Track Tape
◊
◊
◊
◊
◊
◊
b
d
a
b
b
a
a
c
track 1
track 2
One symbol ),( ba
One head
A useful trick to perform more
complicated simulations
One Tape

22. Fall 2006 Costas Busch - RPI 22
◊
◊
◊
◊
◊
◊
b
d
a
b
b
a
a
c
track 1
track 2
1q 2q
Ldcab ),,(),( →
1q
◊
◊
◊
◊
◊
◊
b
d
a
b
c
d
a
c
track 1
track 2
2q

23. Fall 2006 Costas Busch - RPI 23
Semi-Infinite Tape
.........a b a c ◊ ◊
• When the head moves left from the border,
it returns to the same position
The head extends infinitely only to the right
• Initial position is the leftmost cell

24. Fall 2006 Costas Busch - RPI 24
Semi-Infinite machines
have the same power with
Standard Turing machines
Theorem:
Proof:
2. Semi-Infinite Machines
simulate Standard Turing machines
1. Standard Turing machines
simulate Semi-Infinite machines

25. Fall 2006 Costas Busch - RPI 25
1. Standard Turing machines simulate
Semi-Infinite machines:
a. insert special symbol
at left of input string
#
a b a c ◊ ◊#
b. Add a self-loop
to every state
(except states with no
outgoing transitions)
R,##→
Standard Turing Machine
◊◊

26. Fall 2006 Costas Busch - RPI 26
2. Semi-Infinite tape machines simulate
Standard Turing machines:
Standard machine
.........
Semi-Infinite tape machine
..................
Squeeze infinity of both directions
in one direction

27. Fall 2006 Costas Busch - RPI 27
Standard machine
.........
Semi-Infinite tape machine with two tracks
..................
reference point
#
#
Right part
Left part
◊ ◊ ◊a b c d e
ac b
d e ◊ ◊
◊
◊
◊

28. Fall 2006 Costas Busch - RPI 28
1q
2q
R
q2
L
q1
L
q2 R
q1
Left part Right part
Standard machine
Semi-Infinite tape machine

29. Fall 2006 Costas Busch - RPI 29
1q 2q
Rga ,→
Standard machine
L
q1
L
q2
Lgxax ),,(),( →
R
q1
R
q2
Rxgxa ),,(),( →
Semi-Infinite tape machine
Left part
Right part
For all tape symbols x

30. Fall 2006 Costas Busch - RPI 30
Standard machine
.................. ◊ ◊ ◊a b c d e
1q
.........
Semi-Infinite tape machine
#
#
Right part
Left part ac b
d e ◊ ◊
◊
◊
◊
L
q1
Time 1

31. Fall 2006 Costas Busch - RPI 31
Time 2
◊ ◊ ◊g b c d e
2q
#
#
Right part
Left part gc b
d e ◊ ◊
◊
◊
◊
L
q2
Standard machine
..................
.........
Semi-Infinite tape machine

32. Fall 2006 Costas Busch - RPI 32
L
q1
R
q1
R),#,(#)#,(# →
Semi-Infinite tape machine
Left part
At the border:
R
q1
L
q1
R),#,(#)#,(# →
Right part

33. Fall 2006 Costas Busch - RPI 33
.........
Semi-Infinite tape machine
#
#
Right part
Left part gc b
d e ◊ ◊
◊
◊
◊
L
q1
.........#
#
Right part
Left part gc b
d e ◊ ◊
◊
◊
◊
R
q1
Time 1
Time 2
END OF PROOF

34. Fall 2006 Costas Busch - RPI 34
The Off-Line Machine
Control Unit
Input File
Tape
read-only (once)
a b c
d eg ◊ ◊◊◊
read-write
Input string
Appears on
input file only
(state machine)
Input string

35. Fall 2006 Costas Busch - RPI 35
Off-Line machines
have the same power with
Standard Turing machines
Theorem:
Proof: 1. Off-Line machines
simulate Standard Turing machines
2. Standard Turing machines
simulate Off-Line machines

36. Fall 2006 Costas Busch - RPI 36
1. Off-line machines simulate
Standard Turing Machines
Off-line machine:
1. Copy input file to tape
2. Continue computation as in
Standard Turing machine

37. Fall 2006 Costas Busch - RPI 37
1. Copy input file to tape
Input File
a b c ◊◊ ◊
Tape
a b c◊ ◊ ◊
Standard machine
Off-line machine
a b c

38. Fall 2006 Costas Busch - RPI 38
2. Do computations as in Turing machine
Input File
a b c ◊◊ ◊
Tape
a b c◊ ◊ ◊
a b c
1q
1q
Standard machine
Off-line machine

39. Fall 2006 Costas Busch - RPI 39
2. Standard Turing machines simulate
Off-Line machines:
Use a Standard machine with
a four-track tape to keep track of
the Off-line input file and tape contents

40. Fall 2006 Costas Busch - RPI 40
Input File
a b c ◊◊ ◊
Tape
Off-line Machine
e f gd
Standard Machine -- Four track tape
a b c d
e f g
0 0 0
0 0
1
1
Input File
head position
Tape
head position
#
#

41. Fall 2006 Costas Busch - RPI 41
a b c d
e f g
0 0 0
0 0
1
1
Input File
head position
Tape
head position
#
#
Repeat for each state transition:
1. Return to reference point
2. Find current input file symbol
3. Find current tape symbol
4. Make transition
Reference point (uses special symbol # )
#
#
END OF PROOF

42. Fall 2006 Costas Busch - RPI 42
Multi-tape Turing Machines
◊◊ a b c ◊◊ e f g
Control unit
Tape 1 Tape 2
Input string
Input string appears on Tape 1
(state machine)

43. Fall 2006 Costas Busch - RPI 43
◊◊ a b c ◊◊ e f g
1q 1q
◊◊ a g c ◊◊ e d g
2q 2q
Time 1
Time 2
RLdgfb ,),,(),( →
1q 2q
Tape 1 Tape 2
Tape 1 Tape 2

44. Fall 2006 Costas Busch - RPI 44
Multi-tape machines
have the same power with
Standard Turing machines
Theorem:
Proof: 1. Multi-tape machines
simulate Standard Turing machines
2. Standard Turing machines
simulate Multi-tape machines

45. Fall 2006 Costas Busch - RPI 45
1. Multi-tape machines simulate
Standard Turing Machines:
Trivial: Use just one tape

46. Fall 2006 Costas Busch - RPI 46
2. Standard Turing machines simulate
Multi-tape machines:
• Uses a multi-track tape to simulate
the multiple tapes
• A tape of the Multi-tape machine
corresponds to a pair of tracks
Standard machine:

47. Fall 2006 Costas Busch - RPI 47
◊◊ a b c h◊ e f g ◊
Multi-tape Machine
Tape 1 Tape 2
Standard machine with four track tape
a b c
e f g
0 0
0 0
1
1
Tape 1
head position
Tape 2
head position
h
0

48. Fall 2006 Costas Busch - RPI 48
Repeat for each state transition:
1. Return to reference point
2. Find current symbol in Tape 1
3. Find current symbol in Tape 2
4. Make transition
a b c
e f g
0 0
0 0
1
1
Tape 1
head position
Tape 2
head position
h
0
#
#
#
#
Reference point
END OF PROOF

49. Fall 2006 Costas Busch - RPI 49
( steps)
}{ nn
baL =
Standard Turing machine:
Go back and forth times)( 2
nO
2-tape machine:
1. Copy to tape 2n
b
2. Compare on tape 1
and tape 2
)(nO
n
b
n
a
to match the a’s with the b’s
( steps))(nO
)( 2
nO time
)(nO time
Same power doesn’t imply same speed:

50. Fall 2006 Costas Busch - RPI 50
Multidimensional Turing Machines
x
y
a
b
c
2-dimensional tape
HEAD
Position: +2, -1
MOVES: L,R,U,D
U: up D: down
◊
◊
◊
◊

51. Fall 2006 Costas Busch - RPI 51
Multidimensional machines
have the same power with
Standard Turing machines
Theorem:
Proof: 1. Multidimensional machines
simulate Standard Turing machines
2. Standard Turing machines
simulate Multi-Dimensional machines

52. Fall 2006 Costas Busch - RPI 52
1. Multidimensional machines simulate
Standard Turing machines
Trivial: Use one dimension

53. Fall 2006 Costas Busch - RPI 53
2. Standard Turing machines simulate
Multidimensional machines
Standard machine:
• Use a two track tape
• Store symbols in track 1
• Store coordinates in track 2

54. Fall 2006 Costas Busch - RPI 54
x
y
a
b
c◊
◊
◊
◊
a
1
b
#
symbols
coordinates
2-dimensional machine
Standard Machine
1 # 2 # 1−
c
# − 1
1q
1q

55. Fall 2006 Costas Busch - RPI 55
Repeat for each transition followed
in the 2-dimensional machine:
1. Update current symbol
2. Compute coordinates of next position
3. Go to new position
Standard machine:
END OF PROOF

56. Fall 2006 Costas Busch - RPI 56
Nondeterministic Turing Machines
Lba ,→
Rca ,→
1q
2q
3q
Allows Non Deterministic Choices
Choice 1
Choice 2

57. Fall 2006 Costas Busch - RPI 57
a b c◊ ◊
1q
Lba ,→
Rca ,→
1q
2q
3q
Time 0
Time 1
b b c◊ ◊
2q
c b c◊ ◊
3q
Choice 1
Choice 2

58. Fall 2006 Costas Busch - RPI 58
Input string is accepted if
there is a computation:
w
yqxwq f
∗
0
Initial configuration Final Configuration
Any accept state
There is a computation:

59. Fall 2006 Costas Busch - RPI 59
Nondeterministic machines
have the same power with
Standard Turing machines
Theorem:
Proof: 1. Nondeterministic machines
simulate Standard Turing machines
2. Standard Turing machines
simulate Nondeterministic machines

60. Fall 2006 Costas Busch - RPI 60
1. Nondeterministic Machines simulate
Standard (deterministic) Turing Machines
Trivial: every deterministic machine
is also nondeterministic

61. Fall 2006 Costas Busch - RPI 61
2. Standard (deterministic) Turing machines
simulate Nondeterministic machines:
• Stores all possible computations
of the non-deterministic machine
on the 2-dimensional tape
Deterministic machine:
• Uses a 2-dimensional tape
(which is equivalent to 1-dimensional tape)

62. Fall 2006 Costas Busch - RPI 62
All possible computation paths
Initial state
Step 1
Step 2
Step i
Step i+1
acceptreject
infinite
path

63. Fall 2006 Costas Busch - RPI 63
The Deterministic Turing machine
simulates all possible computation paths:
•in a breadth-first search fashion
•simultaneously
•step-by-step

64. Fall 2006 Costas Busch - RPI 64
a b c◊ ◊
1q
Lba ,→
Rca ,→
1q
2q
3q
Time 0
NonDeterministic machine
Deterministic machine
a b c
1q
# # # # ##
#
#
# # #
#
#
# #
current
configuration

65. Fall 2006 Costas Busch - RPI 65
Lba ,→
Rca ,→
1q
2q
3q
b b c
2q
# # # # ##
#
# #
#
# #
Computation 1
b b c◊ ◊
2q
Choice 1
c b c◊ ◊
3q
Choice 2
c b c
3q ## Computation 2
NonDeterministic machine
Deterministic machine
Time 1

66. Fall 2006 Costas Busch - RPI 66
Repeat
For each configuration in current step
of non-deterministic machine,
if there are two or more choices:
1. Replicate configuration
2. Change the state in the replicas
END OF PROOF
Deterministic Turing machine
Until either the input string is accepted
or rejected in all configurations

67. Fall 2006 Costas Busch - RPI 67
The simulation takes in the worst case
exponential time compared to the shortest
accepting path length of the
nondeterministic machine
Remark: