1. Introduction to Turing Machines1 1 1 1 1 1 1 1 1 1 1 1 1

2. Alan Turing -- 1912 – 1954 Mathematician, a computer scientis
Influantial
Development of computer sciences
provided an influential formalisation
of the concept of the algorithm
and computation with the
Turing Machine
Turing Test contribute to the debate of AI
Can machines think?
One of the 100 more imp. People in 20th century
21st on BBC nation wide poll of the 100 Greatest britons

3. The Turing Machine A TM consists of an infinite length tape, on which1 1 1 1 1 1 1 1 1 1 1 1 1 1
A TM consists of an infinite length tape, on which
input is provided as a finite sequence of symbols.
A head reads the input tape. The TM starts at start
state s0. On reading an input symbol it optionally
replaces it with another symbol, changes its
internal state and moves one cell to the right or left.
In the above diagram, TM states are depicted by different colours. So, when the head moves, its colour also changes indicating that it has changed its internal state.

4. The Turing Machine A TM is defined as:
TM = <S, T, s0, d, H> where,
S is a set of TM states
T is a set of tape symbols
s is the start state
H  S is a set of halting states
d : S x T S x T x {L,R}
is the transition function
A TM begins computation at state s0. At each computational step, it reads the present tape symbol, changes its internal state, optionally writes another symbol onto tape, and moves one cell to the left or right in the tape.
The TM halts computation (and accepts the input string) when it reaches one of the halt states in H.

5. A Turing Machine
Tape
......
......
Read-Write head
Control Unit

6. The Tape No boundaries -- infinite length ...... ......
Read-Write head
The head moves Left or Right

7. ......
......
Read-Write head
The head at each time step:
1. Reads a symbol
2. Writes a symbol
3. Moves Left or Right

8. Example:
Time 0
......
......
Time 1
...... d
1. Reads
2. Writes d
3. Moves Left

9. Time 1
......
...... b s
Time 2
......
...... f s b
1. Reads
2. Writes f
3. Moves Right

10. The Input String Input string Blank symbol ...... ...... # # # # #
head
Head starts at the leftmost position of the input string.
The input string is never empty

11. States & Transitions
Write
Read
Move Left
Move Right

12. R d a , ® Example: Time 1 ...... ...... a current state Time 2 ......

13. Example:
Time 1
......
...... a
Time 2
......
...... b

14. L g , ® # Example: Time 1 ...... ...... Time 2 ...... ...... # # # # #

15. Determinism Not Allowed Turing Machines are deterministic Allowed
No lambda transitions allowed

16. Partial Transition Function
Example: #
......
Allowed:
No transition
for input symbol

17. Halting The machine halts if there are
no possible transitions to follow
......
...... # # # # #
No possible transition
HALT!!!

18. Final States Not Allowed Allowed
Final states have no outgoing transitions
In a final state the machine halts

19. Acceptance If machine halts Accept Input in a final state
in a non-final state or
If machine enters
an infinite loop
Reject Input

20. Turing Machine Example
A Turing machine that accepts the language:

21. Time 0

22. Time 1

23. Time 2

24. Time 3

25. Time 4
Halt & Accept

26. Rejection Example
Time 0

27. Time 1
No possible Transition
Halt & Reject

28. Infinite Loop Example
A Turing machine
for language

29. Time 0

30. Time 1

31. Time 2

32. Time 2
Time 3
Infinite loop
Time 4
Time 5

33. Because of the infinite loop:
The final state cannot be reached
The machine never halts
The input is not accepted

34. Another Turing Machine Example
Turing machine for the language

35. Time 0

36. Time 1

37. Time 2

38. Time 3

39. Time 4

40. Time 5

41. Time 6

42. Time 7

43. Time 8

44. Time 9

45. Time 10

46. Time 11

47. Time 12

48. Time 13
Halt & Accept

49. Observation:
If we modify the
machine for the language
we can easily construct
a machine for the language

50. Formal Definitions for Turing Machines

51. Transition Function

52. Transition Function

53. Turing Machine:
Input
alphabet
Tape
alphabet
States
Transition
function
Final
states
Initial
state
blank

54. Configuration
Instantaneous description:

55. Time 4
Time 5
A Move:

56. Time 4
Time 5
Time 6
Time 7

57. Equivalent notation:

58. Initial configuration:
Input string

59. The Accepted Language
For any Turing Machine
Initial state
Final state

60. Standard Turing Machine
The machine we described is the standard:
Deterministic
Infinite tape in both directions
Tape is the input/output file

61. Computing Functions with Turing Machines

62. A function
has:
Result Region:
Domain:

63. A function may have many parameters:
Example:
Addition function

64. Integer Domain
Decimal: 5
Binary:
101
We prefer unary representation:
easier to manipulate with Turing machines
Unary:
11111

65. Definition:
A function is computable if
there is a Turing Machine such that:
Initial configuration
Final configuration
initial state
final state
For all
Domain

66. In other words:
A function is computable if
there is a Turing Machine such that:
Initial
Configuration
Final
Configuration
For all
Domain

67. Example The function is computable are integers Turing Machine:
Input string:
unary
Output string:
unary

68. Start
initial state
The 0 is the delimiter that
separates the two numbers

69. Start
initial state
Finish
final state

70. The 0 helps when we use
the result for other operations
Finish
final state

71. Turing machine for function

72. Execution Example:
Time 0
(2)
(2)
Final Result

73. Time 0

74. Time 1

75. Time 2

76. Time 3

77. Time 4

78. Time 5

79. Time 6

80. Time 7

81. Time 8

82. Time 9

83. Time 10

84. Time 11

85. Time 12
HALT & accept

86. Another Example The function is computable is integer Turing Machine:
Input string:
unary
Output string:
unary

87. Start
initial state
Finish
final state

88. Turing Machine Pseudocode for
Replace every 1 with $
Repeat:
Find rightmost $, replace it with 1
Go to right end, insert 1
Until no more $ remain

89. Turing Machine for

90. Example
Start
Finish

91. Simple TM Examples Turing Machine U+1:
Given a string of 1s on a tape (followed by
an infinite number of 0s), add one more 1 at the
end of the string.
# ……. 
# ……….
A tape is denoted by # …. Where # is the start of the tape.

92. Simple TM Examples TM: U+1 d(s0, 1) |-- (s0, 1, R)
d(s0, 0) |-- (h, 1, STOP)
#s ….. 
#1s ….. 
#11s ….. 
#111s ….. 
#1111s000000….. 
#11111h0000….. STOP
TM computations are illustrated in the form #s … which shows the TM to be in state s, and at the beginning of the tape.

93. Another Example if
The function if
is computable

94. Turing Machine for if if
Input:
Output: or

95. Turing Machine Pseudocode:
Repeat
Match a 1 from with a 1 from
Until all of or is matched
If a 1 from is not matched
erase tape, write 1
else
erase tape, write 0

96. Combining Turing Machines

97. Block Diagram
Turing
Machine
input
output

98. Example: if if
Adder
Comparer
Eraser

99. Turing’s Thesis Any mathematical problem solving that
can be described by a mechanical procedure
(algorithm) can be modeled by a
Turing machine.
All computers today perform only mechanical
problem solving. They are no more expressive
than a Turing machine.
A philosophical question: Do digital computers today perform only mechanical computations? What about our thinking? Do we think mechanically? Can we reduce our thought processes to an algorithm?

100. Turing’s Thesis
Turing’s thesis is not a “theorem” there is no “proof” for the thesis.
The theorem may be refuted by showing at least one task that is performed by a digital computer which cannot be performed by a Turing machine.
Many contentions have been made to this end. However, till date there have not been any conclusive evidence to refute Turing’s thesis.
Although Turing’s a-machine were the most popular, there are many more machines proposed by Turing. Basically they are meant to be more powerful than a-machines. These include the c-machines (choice machines), o-machines (oracle machines) and u-machines (unorganized machines). However, there have been no conclusive proofs about whether they are more expressive than a-machines.

101. Conclusions
TMs are at a level that is much below the assembly language of any typical microprocessor.
So in the practical world, TMs are more useful in what they cannot do rather than in what they can.

102. Lab2
Write some simpleTuring machine programs QUESTIONS?