1. ALAK ROY. Assistant Professor Dept. of CSE NIT Agartala
Turing Machine
CSE-501
Formal Language & Automata Theory
Aug-Dec,2010
ALAK ROY.
Assistant Professor
Dept. of CSE
NIT Agartala

2. The Language Hierarchy? ?
Context-Free Languages
Regular Languages

3. Languages accepted by
Turing Machines
Context-Free Languages
Regular Languages

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

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

6. ......
......
Read-Write head
The head at each time step:
1. Reads a symbol
2. Writes a symbol
3. Moves Left or Right

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

8. Time 1
......
......
Time 2
......
......
1. Reads
2. Writes
3. Moves Right

9. The Input String Input string Blank symbol ...... ...... head
Head starts at the leftmost position
of the input string

10. Input string
Blank symbol
......
......
head
Remark: the input string is never empty

11. States & Transitions
Write
Read
Move Left
Move Right

12. Example:
Time 1
......
......
current state

13. Time 1
......
......
Time 2
......
......

14. Example:
Time 1
......
......
Time 2
......
......

15. Example:
Time 1
......
......
Time 2
......
......

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

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

18. Halting The machine halts if there are
no possible transitions to follow

19. Example:
......
......
No possible transition
HALT!!!

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

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

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

23. Time 0

24. Time 1

25. Time 2

26. Time 3

27. Time 4
Halt & Accept

28. Rejection Example
Time 0

29. Time 1
No possible Transition
Halt & Reject

30. Infinite Loop Example
A Turing machine
for language

31. Time 0

32. Time 1

33. Time 2

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

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

36. Formal Definitions for Turing Machines

37. Transition Function

38. Transition Function

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

40. Configuration
Instantaneous description:

41. Time 4
Time 5
A Move:

42. Time 4
Time 5
Time 6
Time 7

43. Equivalent notation:

44. Initial configuration:
Input string

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

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

47. Computing Functions with Turing Machines

48. A function
has:
Result Region:
Domain:

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

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

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

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

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

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

55. Start
initial state
Finish
final state

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

57. Turing machine for function

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

59. Time 0

60. Time 1

61. Time 2

62. Time 3

63. Time 4

64. Time 5

65. Time 6

66. Time 7

67. Time 8

68. Time 9

69. Time 10

70. Time 11

71. Time 12
HALT & accept

72. Another Example – SELF STUDY
The function
is computable
is integer
Turing Machine:
Input string:
unary
Output string:
unary

73. Start
initial state
Finish
final state

74. 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

75. Turing Machine for

76. Example
Start
Finish

77. Block Diagram
Turing
Machine
input
output

78. Turing’s Thesis

79. Turing’s thesis: Any computation carried out by mechanical means
can be performed by a Turing Machine
(1930)

80. Computer Science Law:
A computation is mechanical
if and only if
it can be performed by a Turing Machine
There is no known model of computation
more powerful than Turing Machines

81. Definition of Algorithm:
An algorithm for function
is a
Turing Machine which computes

82. Algorithms are Turing Machines
When we say:
There exists an algorithm
We mean:
There exists a Turing Machine
that executes the algorithm

83. Construct a Turing Machine:
Construct a TM to accept the set L of all strings over {0,1} ending with 010.

84. Can one TM simulate every TM?
Can a TM simulate a TM?
Yes, obviously.
Can one TM simulate every TM?

85. Universal Turing Machine An Any TM Simulator
Input: < Description of some TM m, w >
Output: result of running m on w m
Universal
Turing
Machine
Output Tape
for running
TM- m starting on tape w w

86. Universal Turing Machine (UTM)
functional notation of UTM:
U(“m”“w”) = “m(w)”
takes 2 arguments:
m : a description of a Turing Machine TM
w : description of an input string w
have 2 properties:
UTM halts on input “m” “w” if & only if TM halts on input “w”.

87. Halting Problem

88. Thank you