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 Formal Definitions for Turing Machines

37. 37 Transition Function

38. 38 Transition Function

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

40. 40 Configuration Instantaneous description:

41. 41 Time 4Time 5 A Move:

42. 42 Time 4Time 5 Time 6Time 7

43. 43 Equivalent notation:

44. 44 Initial configuration: Input string

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

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

47. 47 Computing Functions with Turing Machines

48. 48 A function Domain: Result Region: has:

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

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

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

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

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

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

55. 55 Start Finish final state initial state

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

57. 57 Turing machine for function

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

59. 59 Time 0

60. 60 Time 1

61. 61 Time 2

62. 62 Time 3

63. 63 Time 4

64. 64 Time 5

65. 65 Time 6

66. 66 Time 7

67. 67 Time 8

68. 68 Time 9

69. 69 Time 10

70. 70 Time 11

71. 71 HALT & accept Time 12

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

73. 73 Start Finish final state initial state

74. 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. 75 Turing Machine for

76. 76 Example Start Finish

77. 77 Another Example The function is computable if

78. 78 Turing Machine for Input: Output: or if

79. 79 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

80. 80 Combining Turing Machines

81. 81 Block Diagram Turing Machine inputoutput

82. 82 Example: if Comparer Adder Eraser

83. 83 Turing’s Thesis

84. 84 Turing’s thesis: Any computation carried out by mechanical means can be performed by a Turing Machine (1930)

85. 85 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

86. 86 Definition of Algorithm: An algorithm for function is a Turing Machine which computes

87. 87 When we say: There exists an algorithm Algorithms are Turing Machines We mean: There exists a Turing Machine that executes the algorithm