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T URING M ACHINE ALAK ROY. Assistant Professor Dept. of CSE NIT Agartala CSE-501 F ORMAL L ANGUAGE & A UTOMATA T HEORY Aug-Dec,2010

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

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

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4 A Turing Machine Tape Read-Write head Control Unit

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

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

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

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

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

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

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11 States & Transitions Read Write Move Left Move Right

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12 Example: Time 1 current state

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Time Time 2

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Time Time 2 Example:

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Time Time 2 Example:

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

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

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

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19 Example: No possible transition HALT!!!

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

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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

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

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23 Time 0

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24 Time 1

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25 Time 2

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26 Time 3

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27 Time 4 Halt & Accept

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28 Rejection Example Time 0

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29 Time 1 No possible Transition Halt & Reject

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30 Infinite Loop Example A Turing machine for language

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31 Time 0

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32 Time 1

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33 Time 2

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34 Time 2 Time 3 Time 4 Time 5 Infinite loop

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

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F ORMAL D EFINITIONS FOR T URING M ACHINES 36

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37 Transition Function

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38 Transition Function

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

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40 Configuration Instantaneous description:

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41 Time 4Time 5 A Move:

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42 Time 4Time 5 Time 6Time 7

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43 Equivalent notation:

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44 Initial configuration: Input string

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45 The Accepted Language For any Turing Machine Initial stateFinal state

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

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C OMPUTING F UNCTIONS WITH T URING M ACHINES 47

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48 A function Domain: Result Region: has:

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

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

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

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

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

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

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55 Start Finish final state initial state

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

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57 Turing machine for function

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58 Execution Example: Time 0 Final Result (2)

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59 Time 0

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60 Time 1

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61 Time 2

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62 Time 3

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63 Time 4

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64 Time 5

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65 Time 6

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66 Time 7

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67 Time 8

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68 Time 9

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69 Time 10

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70 Time 11

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71 HALT & accept Time 12

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

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73 Start Finish final state initial state

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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

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75 Turing Machine for

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76 Example Start Finish

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77 Block Diagram Turing Machine inputoutput

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T URING S T HESIS 78

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

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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

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

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

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

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

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An Any TM Simulator Input: Output: result of running m on w Universal Turing Machine m w Output Tape for running TM- m starting on tape w

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Universal Turing Machine (UTM) functional notation of UTM: U(mw) = 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. 86

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Halting Problem 87

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T HANK YOU 88

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