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ALAK ROY. Assistant Professor Dept. of CSE NIT Agartala

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Presentation on theme: "ALAK ROY. Assistant Professor Dept. of CSE NIT Agartala"— Presentation transcript:

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


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