1. Theory of computing, part 4

2. 1Introduction 2Theoretical background Biochemistry/molecular biology 3Theoretical background computer science 4History of the field 5Splicing systems 6P systems 7Hairpins 8Detection techniques 9Micro technology introduction 10Microchips and fluidics 11Self assembly 12Regulatory networks 13Molecular motors 14DNA nanowires 15Protein computers 16DNA computing - summery 17Presentation of essay and discussion Course outline

3. Turing machines

4. Regular Languages Context-Free Languages The language hierarchy

5. Regular Languages Context-Free Languages Languages accepted by Turing Machines The language hierarchy

6. ...... Tape Read-Write head Control Unit A Turing machine

7. ...... Read-Write head No boundaries -- infinite length The head moves Left or Right The tape

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

9. ...... Time 0...... Time 1 1. Reads 2. Writes 3. Moves Left Example

10. ...... Time 1...... Time 2 Example 1. Reads 2. Writes 3. Moves Left

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

12. Read Write Move Left Move Right States and transitions

13. ...... Time 1 current state Example

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

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

16. ...... Time 1...... Time 2 Example

17. Allowed Not Allowed Turing Machines are deterministic Determinism

18. ...... No transition for input symbol Allowed: Example: partial transition function

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

20. ...... No possible transitionHALT Example

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

22. 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 Acceptance

23. A Turing machine that accepts language a * Turing machine example

24. Time 0 Turing machine example

25. Time 1 Turing machine example

26. Time 2 Turing machine example

27. Time 3 Turing machine example

28. Time 4 Halt & Accept Turing machine example

29. Time 0 Rejection example

30. Time 1 No possible Transition Halt & Reject Rejection example

31. Another Turing machine for language a * and is this one correct??? Infinite loop example

32. Time 0 Infinite loop example

33. Time 1 Infinite loop example

34. Time 2 Infinite loop example

35. Time 2 Time 3 Time 4 Time 5... Infinite Loop Infinite loop example

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

37. Turing machine for the language Another Turing machine example

38. Time 0 Another Turing machine example

39. Time 1 Another Turing machine example

40. Time 2 Another Turing machine example

41. Time 3 Another Turing machine example

42. Time 4 Another Turing machine example

43. Time 5 Another Turing machine example

44. Time 6 Another Turing machine example

45. Time 7 Another Turing machine example

46. Time 8 Another Turing machine example

47. Time 9 Another Turing machine example

48. Time 10 Another Turing machine example

49. Time 11 Another Turing machine example

50. Time 12 Another Turing machine example

51. Halt & Accept Time 13 Another Turing machine example

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

53. Formal definitions

54. Transition function

56. States Input alphabetTape alphabet Transition function Initial stateblank Final states Turing machine

57. Instantaneous description: Configuration

58. Time 4Time 5 A Move: Configuration

59. Time 4Time 5 Time 6Time 7 Configuration

60. Equivalent notation: Configuration

61. Input string Initial configuration

62. For any Turing Machine Initial state Final state The accepted language

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

64. Computing functions

65. A function Domain:Result Region: has: Functions

66. A function may have many parameters Example: Addition function Functions

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

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

69. Initial Configuration Final Configuration Functions definition A function is computable if there is a Turing Machine such that: DomainFor all

70. The functionis computable Turing Machine: Input string:unary Output string:unary are integers Example

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

72. Start Finish final state initial state Example

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

74. Turing machine for function Turing machine example

75. Execution Example: Time 0 Final Result (2) Turing machine example

76. Time 0 Turing machine example

77. Time 1 Turing machine example

78. Time 2 Turing machine example

79. Time 3 Turing machine example

80. Time 4 Turing machine example

81. Time 5 Turing machine example

82. Time 6 Turing machine example

83. Time 7 Turing machine example

84. Time 8 Turing machine example

85. Time 9 Turing machine example

86. Time 10 Turing machine example

87. Time 11 Turing machine example

88. HALT & accept Time 12 Turing machine example

89. The functionis computable Turing Machine: Input string:unary Output string:unary is integer Another example

90. Start Finish final state initial state

91. 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 Pseudocode

92. Turing Machine for Example

93. Start Finish Example

94. The function is computable if Another example

95. Turing Machine for Input: Output: or if Another example

96. 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 Pseudocode

97. Combining Turing machines

98. Block Diagram Turing Machineinputoutput Combining Turing machines

99. if Comparer Adder Eraser Example

100. Turing’s thesis

101. Do Turing machines have the same power with a digital computer? Intuitive answer: Yes There is no formal answer!!! Question: Turing’s thesis

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

103. 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 Computer science law

104. An algorithm for function is a Turing Machine which computes Definition of an algorithm

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