1. 1 Turing Machines

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

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

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

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

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

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

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

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

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

11. 11 States & Transitions Read Write Move Left Move Right

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

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

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

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

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

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

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

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

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

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

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

23. 23 Time 0

24. 24 Time 1

25. 25 Time 2

26. 26 Time 3

27. 27 Time 4 Halt & Accept

28. 28 Rejection Example Time 0

29. 29 Time 1 No possible Transition Halt & Reject

30. 30 Infinite Loop Example A Turing machine for language

31. 31 Time 0

32. 32 Time 1

33. 33 Time 2

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

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

36. 36 Another Turing Machine Example Turing machine for the language

37. 37 Time 0

38. 38 Time 1

39. 39 Time 2

40. 40 Time 3

41. 41 Time 4

42. 42 Time 5

43. 43 Time 6

44. 44 Time 7

45. 45 Time 8

46. 46 Time 9

47. 47 Time 10

48. 48 Time 11

49. 49 Time 12

50. 50 Halt & Accept Time 13

51. 51 If we modify the machine for the language we can easily construct a machine for the language Observation:

52. 52 Formal Definitions for Turing Machines

53. 53 Transition Function

54. 54 Transition Function

55. 55 Turing Machine: States Input alphabet Tape alphabet Transition function Initial state blank Final states

56. 56 Configuration Instantaneous description:

57. 57 Time 4Time 5 A Move:

58. 58 Time 4Time 5 Time 6Time 7

59. 59 Equivalent notation:

60. 60 Initial configuration: Input string

61. 61 The Accepted Language For any Turing Machine Initial stateFinal state

62. 62 Standard Turing Machine Deterministic Infinite tape in both directions Tape is the input/output file The machine we described is the standard:

63. 63 Computing Functions with Turing Machines

64. 64 A function Domain: Result Region: has:

65. 65 A function may have many parameters: Example: Addition function

66. 66 Integer Domain Unary: Binary: Decimal: 11111 101 5 We prefer unary representation: easier to manipulate with Turing machines

67. 67 Definition: A function is computable if there is a Turing Machine such that: Initial configurationFinal configuration Domain final stateinitial state For all

68. 68 Initial Configuration Final Configuration A function is computable if there is a Turing Machine such that: In other words: Domain For all

69. 69 Example The function is computable Turing Machine: Input string: unary Output string: unary are integers

70. 70 Start initial state The 0 is the delimiter that separates the two numbers

71. 71 Start Finish final state initial state

72. 72 Finish final state The 0 helps when we use the result for other operations

73. 73 Turing machine for function

74. 74 Execution Example: Time 0 Final Result (2)

75. 75 Time 0

76. 76 Time 1

77. 77 Time 2

78. 78 Time 3

79. 79 Time 4

80. 80 Time 5

81. 81 Time 6

82. 82 Time 7

83. 83 Time 8

84. 84 Time 9

85. 85 Time 10

86. 86 Time 11

87. 87 HALT & accept Time 12

88. 88 Another Example The function is computable Turing Machine: Input string: unary Output string:unary is integer

89. 89 Start Finish final state initial state

90. 90 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

91. 91 Turing Machine for

92. 92 Example Start Finish

93. 93 Another Example The function is computable if

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

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

96. 96 Combining Turing Machines

97. 97 Block Diagram Turing Machine inputoutput

98. 98 Example: if Comparer Adder Eraser