TURING MACHINE STRUCTURE AND OPERATIONAL FUNCTIONALITY

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TURING MACHINE STRUCTURE AND OPERATIONAL FUNCTIONALITY Powered By Docstoc
					Structure and Operational Functionality of

   The Turing Machine

              11 — 13 — 2008



             ¨         ¨
            Soren Wellhofer
                                    History
                    Structure and Definition
                                   Samples
                                   Varieties
                              Computability
                                 References


Table of Contents

  1 History


  2 Structure and Definition


  3 Samples


  4 Varieties


  5 Computability


  6 References

                             o         o
                            S¨ren Wellh¨fer    Structure and Operational Functionality of The Turing Machine
                                  History
                  Structure and Definition
                                 Samples
                                 Varieties
                            Computability
                               References


Time frame — The 20th century



  What were mathematicians working on?

    • Rediscover Theory of Numbers
    • Reduction of all math to fundamental logic




                           o         o
                          S¨ren Wellh¨fer    Structure and Operational Functionality of The Turing Machine
                                  History
                  Structure and Definition
                                 Samples
                                 Varieties
                            Computability
                               References


Time frame — The 20th century



  What were mathematicians working on?

    • Rediscover Theory of Numbers
    • Reduction of all math to fundamental logic
    • Arithmetics/computations by means of automatic formal
      system




                           o         o
                          S¨ren Wellh¨fer    Structure and Operational Functionality of The Turing Machine
                                  History
                  Structure and Definition
                                 Samples
                                 Varieties
                            Computability
                               References


Time frame — The 20th century



  What were mathematicians working on?

    • Rediscover Theory of Numbers
    • Reduction of all math to fundamental logic
    • Arithmetics/computations by means of automatic formal
      system




                           o         o
                          S¨ren Wellh¨fer    Structure and Operational Functionality of The Turing Machine
                                  History
                  Structure and Definition
                                 Samples
                                 Varieties
                            Computability
                               References


Time frame — The 20th century



  What were mathematicians working on?

    • Rediscover Theory of Numbers
    • Reduction of all math to fundamental logic
    • Arithmetics/computations by means of automatic formal
      system




                           o         o
                          S¨ren Wellh¨fer    Structure and Operational Functionality of The Turing Machine
                                  History
                  Structure and Definition
                                 Samples
                                 Varieties
                            Computability
                               References


What does it mean to be computable?


  Turing’s achievements

    • Proof of the possibility of a symbol-processing machine
    • Simple operations according to rules




                           o         o
                          S¨ren Wellh¨fer    Structure and Operational Functionality of The Turing Machine
                                  History
                  Structure and Definition
                                 Samples
                                 Varieties
                            Computability
                               References


What does it mean to be computable?


  Turing’s achievements

    • Proof of the possibility of a symbol-processing machine
    • Simple operations according to rules
    • Instruction tables for machine’s moves = formal system




                           o         o
                          S¨ren Wellh¨fer    Structure and Operational Functionality of The Turing Machine
                                   History
                   Structure and Definition
                                  Samples
                                  Varieties
                             Computability
                                References


What does it mean to be computable?


  Turing’s achievements

    • Proof of the possibility of a symbol-processing machine
    • Simple operations according to rules
    • Instruction tables for machine’s moves = formal system
    • All computations logically feasible!




                            o         o
                           S¨ren Wellh¨fer    Structure and Operational Functionality of The Turing Machine
                                   History
                   Structure and Definition
                                  Samples
                                  Varieties
                             Computability
                                References


What does it mean to be computable?


  Turing’s achievements

    • Proof of the possibility of a symbol-processing machine
    • Simple operations according to rules
    • Instruction tables for machine’s moves = formal system
    • All computations logically feasible!
   → wrote first programmes




                            o         o
                           S¨ren Wellh¨fer    Structure and Operational Functionality of The Turing Machine
                                   History
                   Structure and Definition
                                  Samples
                                  Varieties
                             Computability
                                References


What does it mean to be computable?


  Turing’s achievements

    • Proof of the possibility of a symbol-processing machine
    • Simple operations according to rules
    • Instruction tables for machine’s moves = formal system
    • All computations logically feasible!
   → wrote first programmes




                            o         o
                           S¨ren Wellh¨fer    Structure and Operational Functionality of The Turing Machine
                                 History
                 Structure and Definition
                                            General Structure
                                Samples
                                            Formal definition
                                Varieties
                                            Action table
                           Computability
                              References




Elements of the Turing Machine

  • Read/Write Head
  • Infinetly long tape
  • Divided into cells
  • Cells containing symbols




                          o         o
                         S¨ren Wellh¨fer    Structure and Operational Functionality of The Turing Machine
                                 History
                 Structure and Definition
                                            General Structure
                                Samples
                                            Formal definition
                                Varieties
                                            Action table
                           Computability
                              References




Elements of the Turing Machine

  • Read/Write Head
  • Infinetly long tape
  • Divided into cells
  • Cells containing symbols
  • Action table = the program




                          o         o
                         S¨ren Wellh¨fer    Structure and Operational Functionality of The Turing Machine
                                 History
                 Structure and Definition
                                            General Structure
                                Samples
                                            Formal definition
                                Varieties
                                            Action table
                           Computability
                              References




Elements of the Turing Machine

  • Read/Write Head
  • Infinetly long tape
  • Divided into cells
  • Cells containing symbols
  • Action table = the program




                          o         o
                         S¨ren Wellh¨fer    Structure and Operational Functionality of The Turing Machine
                                History
                Structure and Definition
                                           General Structure
                               Samples
                                           Formal definition
                               Varieties
                                           Action table
                          Computability
                             References


The Turing Machine




    0
    1   0
        1   1
            0    0              0           0                  0               1
                                                                               0     1
                                                                                     0      0
                                                                                            1



                         Infinetly long ...
                         o         o
                        S¨ren Wellh¨fer    Structure and Operational Functionality of The Turing Machine
                                History
                Structure and Definition
                                           General Structure
                               Samples
                                           Formal definition
                               Varieties
                                           Action table
                          Computability
                             References


The Turing Machine


        Tape




    0
    1   0
        1   1
            0    0              0           0                  0               1
                                                                               0     1
                                                                                     0      0
                                                                                            1



                         Infinetly long ...
                         o         o
                        S¨ren Wellh¨fer    Structure and Operational Functionality of The Turing Machine
                                History
                Structure and Definition
                                           General Structure
                               Samples
                                           Formal definition
                               Varieties
                                           Action table
                          Computability
                             References


The Turing Machine


        Tape




    0
    1   0
        1   1
            0    0              0           0                  0               1
                                                                               0     1
                                                                                     0      0
                                                                                            1



                         Infinetly long ...
                         o         o
                        S¨ren Wellh¨fer    Structure and Operational Functionality of The Turing Machine
                                History
                Structure and Definition
                                           General Structure
                               Samples
                                           Formal definition
                               Varieties
                                           Action table
                          Computability
                             References


The Turing Machine


        Tape




    0
    1   0
        1   1
            0    0              0           0                  0               1
                                                                               0     1
                                                                                     0      0
                                                                                            1



                         Infinetly long ...
                         o         o
                        S¨ren Wellh¨fer    Structure and Operational Functionality of The Turing Machine
                                History
                Structure and Definition
                                           General Structure
                               Samples
                                           Formal definition
                               Varieties
                                           Action table
                          Computability
                             References


The Turing Machine


        Tape




    0
    1   0
        1   1
            0    0              0           0                  0               1
                                                                               0     1
                                                                                     0      0
                                                                                            1



                         Infinetly long ...
                         o         o
                        S¨ren Wellh¨fer    Structure and Operational Functionality of The Turing Machine
                                History
                Structure and Definition
                                           General Structure
                               Samples
                                           Formal definition
                               Varieties
                                           Action table
                          Computability
                             References


The Turing Machine

                                                Read/Write Head

        Tape




    0
    1   1
        0   0
            1    0              0           0                  0               1
                                                                               0     0
                                                                                     1      1
                                                                                            0



                         Infinetly long ...
                         o         o
                        S¨ren Wellh¨fer    Structure and Operational Functionality of The Turing Machine
                                History
                Structure and Definition
                                           General Structure
                               Samples
                                           Formal definition
                               Varieties
                                           Action table
                          Computability
                             References


The Turing Machine

                                                Read/Write Head

        Tape




    0
    1   1
        0   0
            1    0              0           0                  0               1
                                                                               0     0
                                                                                     1      1
                                                                                            0



                         Infinetly long ...
                         o         o
                        S¨ren Wellh¨fer    Structure and Operational Functionality of The Turing Machine
                                History
                Structure and Definition
                                           General Structure
                               Samples
                                           Formal definition
                               Varieties
                                           Action table
                          Computability
                             References


The Turing Machine

                                                Read/Write Head

        Tape




                               A


    0
    1   1
        0   0
            1    0              0           0                  0               1
                                                                               0     0
                                                                                     1      1
                                                                                            0



                         Infinetly long ...
                         o         o
                        S¨ren Wellh¨fer    Structure and Operational Functionality of The Turing Machine
                                History
                Structure and Definition
                                           General Structure
                               Samples
                                           Formal definition
                               Varieties
                                           Action table
                          Computability
                             References


The Turing Machine

                                                Read/Write Head

        Tape
                                                                           Machine’s State


                               A


    0
    1   1
        0   0
            1    0              0           0                  0               1
                                                                               0     0
                                                                                     1      1
                                                                                            0



                         Infinetly long ...
                         o         o
                        S¨ren Wellh¨fer    Structure and Operational Functionality of The Turing Machine
                                History
                Structure and Definition
                                           General Structure
                               Samples
                                           Formal definition
                               Varieties
                                           Action table
                          Computability
                             References


The Turing Machine

                                                Read/Write Head

        Tape
                                                                           Machine’s State


                               A


    0
    1   1
        0   0
            1    0              0           0                  0               1
                                                                               0     0
                                                                                     1      1
                                                                                            0



                         Infinetly long ...
                         o         o
                        S¨ren Wellh¨fer    Structure and Operational Functionality of The Turing Machine
                                 History
                 Structure and Definition
                                            General Structure
                                Samples
                                            Formal definition
                                Varieties
                                            Action table
                           Computability
                              References


The Turing Machine

                                                 Read/Write Head

  Now action according
  to state table                                                            Machine’s State


                                A


     1
     0   0
         1   0
             1    0              0           0                  0               0
                                                                                1     1
                                                                                      0      1
                                                                                             0




                          o         o
                         S¨ren Wellh¨fer    Structure and Operational Functionality of The Turing Machine
                                 History
                 Structure and Definition
                                            General Structure
                                Samples
                                            Formal definition
                                Varieties
                                            Action table
                           Computability
                              References


The Turing Machine
  State table might say:                         Read/Write Head
  When in state A reading 0:
  write 1, move right, change
  state to B.
                                                                            Machine’s State


                                A


     1
     0   0
         1   0
             1    0              0           0                  0               0
                                                                                1     1
                                                                                      0      1
                                                                                             0




                          o         o
                         S¨ren Wellh¨fer    Structure and Operational Functionality of The Turing Machine
                                 History
                 Structure and Definition
                                            General Structure
                                Samples
                                            Formal definition
                                Varieties
                                            Action table
                           Computability
                              References


The Turing Machine
  State table might say:                         Read/Write Head
  When in state A reading 0:
  write 1, move right, change
  state to B.
                                                                            Machine’s State


                                A


     1
     0   0
         1   0
             1    0              1           0                  0               0
                                                                                1     1
                                                                                      0      1
                                                                                             0




                          o         o
                         S¨ren Wellh¨fer    Structure and Operational Functionality of The Turing Machine
                                      History
                      Structure and Definition
                                                 General Structure
                                     Samples
                                                 Formal definition
                                     Varieties
                                                 Action table
                                Computability
                                   References


The Turing Machine
  State table might say:                              Read/Write Head
  When in state A reading 0:
  write 1, move right, change
  state to B.
                                                                                 Machine’s State



         Move right                  A


     1
     0    0
          1    0
               1       0              1           0                  0               0
                                                                                     1     1
                                                                                           0      1
                                                                                                  0




                               o         o
                              S¨ren Wellh¨fer    Structure and Operational Functionality of The Turing Machine
                                 History
                 Structure and Definition
                                            General Structure
                                Samples
                                            Formal definition
                                Varieties
                                            Action table
                           Computability
                              References


The Turing Machine
  State table might say:
  When in state A reading 0:
  write 1, move right, change
  state to B.



                 Move right                   A


     1
     0   0
         1   0
             1    0              1           0                  0               0
                                                                                1     1
                                                                                      0      1
                                                                                             0




                          o         o
                         S¨ren Wellh¨fer    Structure and Operational Functionality of The Turing Machine
                                   History
                   Structure and Definition
                                              General Structure
                                  Samples
                                              Formal definition
                                  Varieties
                                              Action table
                             Computability
                                References


The Turing Machine
  State table might say:
  When in state A reading 0:
  write 1, move right, change
  state to B.


             New state B
                                                B


     1
     0   0
         1    0
              1     0              1           0                  0               0
                                                                                  1     1
                                                                                        0      1
                                                                                               0




                            o         o
                           S¨ren Wellh¨fer    Structure and Operational Functionality of The Turing Machine
                                  History
                  Structure and Definition
                                             General Structure
                                 Samples
                                             Formal definition
                                 Varieties
                                             Action table
                            Computability
                               References


The Turing Machine


  Now read again, etc ...


  ... until final configuration.


                                               B


     0
     1   1
         0   0
             1     0              1           0                  0               1
                                                                                 0     0
                                                                                       1      1
                                                                                              0




                           o         o
                          S¨ren Wellh¨fer    Structure and Operational Functionality of The Turing Machine
                               History
               Structure and Definition
                                          General Structure
                              Samples
                                          Formal definition
                              Varieties
                                          Action table
                         Computability
                            References


Formal definition — 7-tupel

               M = S, Γ, b, Σ, δ, s0 , F

       S                                      Finite set of states




                        o         o
                       S¨ren Wellh¨fer    Structure and Operational Functionality of The Turing Machine
                               History
               Structure and Definition
                                          General Structure
                              Samples
                                          Formal definition
                              Varieties
                                          Action table
                         Computability
                            References


Formal definition — 7-tupel

               M = S, Γ, b, Σ, δ, s0 , F

       S                                      Finite set of states
       Γ                                      Finite set of symbols




                        o         o
                       S¨ren Wellh¨fer    Structure and Operational Functionality of The Turing Machine
                               History
               Structure and Definition
                                          General Structure
                              Samples
                                          Formal definition
                              Varieties
                                          Action table
                         Computability
                            References


Formal definition — 7-tupel

               M = S, Γ, b, Σ, δ, s0 , F

       S                                      Finite set of states
       Γ                                      Finite set of symbols
       b∈Γ                                    Blank symbol




                        o         o
                       S¨ren Wellh¨fer    Structure and Operational Functionality of The Turing Machine
                               History
               Structure and Definition
                                          General Structure
                              Samples
                                          Formal definition
                              Varieties
                                          Action table
                         Computability
                            References


Formal definition — 7-tupel

                M = S, Γ, b, Σ, δ, s0 , F

       S                                      Finite set of states
       Γ                                      Finite set of symbols
       b∈Γ                                    Blank symbol
       Σ ⊆ Γ \ {b}                            Input symbols




                        o         o
                       S¨ren Wellh¨fer    Structure and Operational Functionality of The Turing Machine
                                History
                Structure and Definition
                                           General Structure
                               Samples
                                           Formal definition
                               Varieties
                                           Action table
                          Computability
                             References


Formal definition — 7-tupel

                M = S, Γ, b, Σ, δ, s0 , F

       S                                       Finite set of states
       Γ                                       Finite set of symbols
       b∈Γ                                     Blank symbol
       Σ ⊆ Γ \ {b}                             Input symbols
       δ : S × Γ × {L, R}                      Finite set of states




                         o         o
                        S¨ren Wellh¨fer    Structure and Operational Functionality of The Turing Machine
                                History
                Structure and Definition
                                           General Structure
                               Samples
                                           Formal definition
                               Varieties
                                           Action table
                          Computability
                             References


Formal definition — 7-tupel

                M = S, Γ, b, Σ, δ, s0 , F

       S                                       Finite set of states
       Γ                                       Finite set of symbols
       b∈Γ                                     Blank symbol
       Σ ⊆ Γ \ {b}                             Input symbols
       δ : S × Γ × {L, R}                      Finite set of states
       s0 ∈ S                                  Initial state




                         o         o
                        S¨ren Wellh¨fer    Structure and Operational Functionality of The Turing Machine
                                History
                Structure and Definition
                                           General Structure
                               Samples
                                           Formal definition
                               Varieties
                                           Action table
                          Computability
                             References


Formal definition — 7-tupel

                M = S, Γ, b, Σ, δ, s0 , F

       S                                       Finite set of states
       Γ                                       Finite set of symbols
       b∈Γ                                     Blank symbol
       Σ ⊆ Γ \ {b}                             Input symbols
       δ : S × Γ × {L, R}                      Finite set of states
       s0 ∈ S                                  Initial state
       F ⊆S                                    Accepting states



                         o         o
                        S¨ren Wellh¨fer    Structure and Operational Functionality of The Turing Machine
                                History
                Structure and Definition
                                           General Structure
                               Samples
                                           Formal definition
                               Varieties
                                           Action table
                          Computability
                             References


Formal definition — 7-tupel

                M = S, Γ, b, Σ, δ, s0 , F

       S                                       Finite set of states
       Γ                                       Finite set of symbols
       b∈Γ                                     Blank symbol
       Σ ⊆ Γ \ {b}                             Input symbols
       δ : S × Γ × {L, R}                      Finite set of states
       s0 ∈ S                                  Initial state
       F ⊆S                                    Accepting states



                         o         o
                        S¨ren Wellh¨fer    Structure and Operational Functionality of The Turing Machine
                                History
                Structure and Definition
                                           General Structure
                               Samples
                                           Formal definition
                               Varieties
                                           Action table
                          Computability
                             References


Formal definition — 7-tupel

                M = S, Γ, b, Σ, δ, s0 , F

       S                                       Finite set of states
       Γ                                       Finite set of symbols
       b∈Γ                                     Blank symbol
       Σ ⊆ Γ \ {b}                             Input symbols
       δ : S × Γ × {L, R}                      Finite set of states
       s0 ∈ S                                  Initial state
       F ⊆S                                    Accepting states



                         o         o
                        S¨ren Wellh¨fer    Structure and Operational Functionality of The Turing Machine
                                History
                Structure and Definition
                                           General Structure
                               Samples
                                           Formal definition
                               Varieties
                                           Action table
                          Computability
                             References


Action table/Transition function δ — Quintupel


                         si aj −→ si1 aj1 dk

            When in state si reading symbol aj :




                         o         o
                        S¨ren Wellh¨fer    Structure and Operational Functionality of The Turing Machine
                                History
                Structure and Definition
                                           General Structure
                               Samples
                                           Formal definition
                               Varieties
                                           Action table
                          Computability
                             References


Action table/Transition function δ — Quintupel


                         si aj −→ si1 aj1 dk

            When in state si reading symbol aj :

              • write symbol aj1




                         o         o
                        S¨ren Wellh¨fer    Structure and Operational Functionality of The Turing Machine
                                History
                Structure and Definition
                                           General Structure
                               Samples
                                           Formal definition
                               Varieties
                                           Action table
                          Computability
                             References


Action table/Transition function δ — Quintupel


                         si aj −→ si1 aj1 dk

            When in state si reading symbol aj :

              • write symbol aj1
              • move into dk , k ∈ {L, R}




                         o         o
                        S¨ren Wellh¨fer    Structure and Operational Functionality of The Turing Machine
                                History
                Structure and Definition
                                           General Structure
                               Samples
                                           Formal definition
                               Varieties
                                           Action table
                          Computability
                             References


Action table/Transition function δ — Quintupel


                         si aj −→ si1 aj1 dk

            When in state si reading symbol aj :

              • write symbol aj1
              • move into dk , k ∈ {L, R}
              • change state to si1




                         o         o
                        S¨ren Wellh¨fer    Structure and Operational Functionality of The Turing Machine
                                History
                Structure and Definition
                                           General Structure
                               Samples
                                           Formal definition
                               Varieties
                                           Action table
                          Computability
                             References


Action table/Transition function δ — Quintupel


                         si aj −→ si1 aj1 dk

            When in state si reading symbol aj :

              • write symbol aj1
              • move into dk , k ∈ {L, R}
              • change state to si1




                         o         o
                        S¨ren Wellh¨fer    Structure and Operational Functionality of The Turing Machine
                                History
                Structure and Definition
                                           General Structure
                               Samples
                                           Formal definition
                               Varieties
                                           Action table
                          Computability
                             References


Action table/Transition function δ — Quintupel


                         si aj −→ si1 aj1 dk

            When in state si reading symbol aj :

              • write symbol aj1
              • move into dk , k ∈ {L, R}
              • change state to si1




                         o         o
                        S¨ren Wellh¨fer    Structure and Operational Functionality of The Turing Machine
                                History
                Structure and Definition
                                           Addition
                               Samples
                                           Class activity
                               Varieties
                                           Complement
                          Computability
                             References


Unary Numbers




        n−→u                                   n=m
        1−→X                                   u... number of Xs
        5−→XXXXX                               n... natural number




                         o         o
                        S¨ren Wellh¨fer    Structure and Operational Functionality of The Turing Machine
                              History
              Structure and Definition
                                         Addition
                             Samples
                                         Class activity
                             Varieties
                                         Complement
                        Computability
                           References


Unary Number Addition — Self-performed




               Simulating a Turing Machine:

             Now it’s your turn!



                       o         o
                      S¨ren Wellh¨fer    Structure and Operational Functionality of The Turing Machine
                              History
              Structure and Definition
                                         Addition
                             Samples
                                         Class activity
                             Varieties
                                         Complement
                        Computability
                           References


Unary Number Addition Machine


              S ={0, 1, HALT}              Γ={B, X, +}
              F ={HALT}                    Σ={X, +}




                       o         o
                      S¨ren Wellh¨fer    Structure and Operational Functionality of The Turing Machine
                              History
              Structure and Definition
                                         Addition
                             Samples
                                         Class activity
                             Varieties
                                         Complement
                        Computability
                           References


Unary Number Addition Machine


              S ={0, 1, HALT}              Γ={B, X, +}
              F ={HALT}                    Σ={X, +}
              s0 =0                        b=B




                       o         o
                      S¨ren Wellh¨fer    Structure and Operational Functionality of The Turing Machine
                                    History
                    Structure and Definition
                                               Addition
                                   Samples
                                               Class activity
                                   Varieties
                                               Complement
                              Computability
                                 References


Unary Number Addition Machine


                    S ={0, 1, HALT}              Γ={B, X, +}
                    F ={HALT}                    Σ={X, +}
                    s0 =0                        b=B

   Action table δ            si = 0                                        si = 1




                             o         o
                            S¨ren Wellh¨fer    Structure and Operational Functionality of The Turing Machine
                                       History
                       Structure and Definition
                                                  Addition
                                      Samples
                                                  Class activity
                                      Varieties
                                                  Complement
                                 Computability
                                    References


Unary Number Addition Machine


                       S ={0, 1, HALT}              Γ={B, X, +}
                       F ={HALT}                    Σ={X, +}
                       s0 =0                        b=B

   Action table δ             si = 0                                  si = 1
    aj = X          si1 = 0; aj1 = X; dK = R                si1 = HALT; aj1 = B; dK = R




                                o         o
                               S¨ren Wellh¨fer    Structure and Operational Functionality of The Turing Machine
                                       History
                       Structure and Definition
                                                  Addition
                                      Samples
                                                  Class activity
                                      Varieties
                                                  Complement
                                 Computability
                                    References


Unary Number Addition Machine


                       S ={0, 1, HALT}              Γ={B, X, +}
                       F ={HALT}                    Σ={X, +}
                       s0 =0                        b=B

   Action table δ             si = 0                                  si = 1
    aj = X          si1 = 0; aj1 = X; dK = R                si1 = HALT; aj1 = B; dK = R
    aj = +          si1 = 0; aj1 = X; dK = R                             —




                                o         o
                               S¨ren Wellh¨fer    Structure and Operational Functionality of The Turing Machine
                                       History
                       Structure and Definition
                                                  Addition
                                      Samples
                                                  Class activity
                                      Varieties
                                                  Complement
                                 Computability
                                    References


Unary Number Addition Machine


                       S ={0, 1, HALT}              Γ={B, X, +}
                       F ={HALT}                    Σ={X, +}
                       s0 =0                        b=B

   Action table δ             si = 0                                  si = 1
    aj = X          si1 = 0; aj1 = X; dK = R                si1 = HALT; aj1 = B; dK = R
    aj = +          si1 = 0; aj1 = X; dK = R                             —
    aj = B          si1 = 1; aj1 = B; dK = L                             —




                                o         o
                               S¨ren Wellh¨fer    Structure and Operational Functionality of The Turing Machine
                                       History
                       Structure and Definition
                                                  Addition
                                      Samples
                                                  Class activity
                                      Varieties
                                                  Complement
                                 Computability
                                    References


Unary Number Addition Machine


                       S ={0, 1, HALT}              Γ={B, X, +}
                       F ={HALT}                    Σ={X, +}
                       s0 =0                        b=B

   Action table δ             si = 0                                  si = 1
    aj = X          si1 = 0; aj1 = X; dK = R                si1 = HALT; aj1 = B; dK = R
    aj = +          si1 = 0; aj1 = X; dK = R                             —
    aj = B          si1 = 1; aj1 = B; dK = L                             —




                                o         o
                               S¨ren Wellh¨fer    Structure and Operational Functionality of The Turing Machine
                             History
             Structure and Definition
                                        Addition
                            Samples
                                        Class activity
                            Varieties
                                        Complement
                       Computability
                          References


A Complement Machine



               S ={0, HALT}              Γ={B, 0, 1}
               F ={HALT}                 Σ={0, 1}




                      o         o
                     S¨ren Wellh¨fer    Structure and Operational Functionality of The Turing Machine
                             History
             Structure and Definition
                                        Addition
                            Samples
                                        Class activity
                            Varieties
                                        Complement
                       Computability
                          References


A Complement Machine



               S ={0, HALT}              Γ={B, 0, 1}
               F ={HALT}                 Σ={0, 1}
               s0 =0                     b=B




                      o         o
                     S¨ren Wellh¨fer    Structure and Operational Functionality of The Turing Machine
                                 History
                 Structure and Definition
                                             Addition
                                Samples
                                             Class activity
                                Varieties
                                             Complement
                           Computability
                              References


A Complement Machine



                   S ={0, HALT}               Γ={B, 0, 1}
                   F ={HALT}                  Σ={0, 1}
                   s0 =0                      b=B
          Action table δ                       si = 0




                            o         o
                           S¨ren Wellh¨fer   Structure and Operational Functionality of The Turing Machine
                                 History
                 Structure and Definition
                                             Addition
                                Samples
                                             Class activity
                                Varieties
                                             Complement
                           Computability
                              References


A Complement Machine



                   S ={0, HALT}             Γ={B, 0, 1}
                   F ={HALT}                Σ={0, 1}
                   s0 =0                    b=B
          Action table δ                     si = 0
           aj = 0                  si1 = 0; aj1 = 1; dK = R




                            o         o
                           S¨ren Wellh¨fer   Structure and Operational Functionality of The Turing Machine
                                 History
                 Structure and Definition
                                             Addition
                                Samples
                                             Class activity
                                Varieties
                                             Complement
                           Computability
                              References


A Complement Machine



                   S ={0, HALT}             Γ={B, 0, 1}
                   F ={HALT}                Σ={0, 1}
                   s0 =0                    b=B
          Action table δ                     si = 0
           aj = 0                  si1 = 0; aj1 = 1; dK = R
           aj = 1                  si1 = 0; aj1 = 0; dK = R




                            o         o
                           S¨ren Wellh¨fer   Structure and Operational Functionality of The Turing Machine
                                 History
                 Structure and Definition
                                             Addition
                                Samples
                                             Class activity
                                Varieties
                                             Complement
                           Computability
                              References


A Complement Machine



                   S ={0, HALT}            Γ={B, 0, 1}
                   F ={HALT}               Σ={0, 1}
                   s0 =0                   b=B
          Action table δ                    si = 0
           aj = 0                 si1 = 0; aj1 = 1; dK = R
           aj = 1                 si1 = 0; aj1 = 0; dK = R
           aj = B              si1 = HALT; aj1 = B; dK = R




                            o         o
                           S¨ren Wellh¨fer   Structure and Operational Functionality of The Turing Machine
                                 History
                 Structure and Definition
                                             Addition
                                Samples
                                             Class activity
                                Varieties
                                             Complement
                           Computability
                              References


A Complement Machine



                   S ={0, HALT}            Γ={B, 0, 1}
                   F ={HALT}               Σ={0, 1}
                   s0 =0                   b=B
          Action table δ                    si = 0
           aj = 0                 si1 = 0; aj1 = 1; dK = R
           aj = 1                 si1 = 0; aj1 = 0; dK = R
           aj = B              si1 = HALT; aj1 = B; dK = R




                            o         o
                           S¨ren Wellh¨fer   Structure and Operational Functionality of The Turing Machine
                                  History
                                             Variations
                  Structure and Definition
                                             4-tupel
                                 Samples
                                             State diagrams
                                 Varieties
                                             Instantaneous description
                            Computability
                                             Universal Turing Machine
                               References


Variations of the TMs ...



  ... provably equivalent

    • Two-way infinite tapes
    • Arbitrary movement of the head
    • Arbitrary numbers of read-write heads
    • Arbitrary finite alphabet




                           o         o
                          S¨ren Wellh¨fer    Structure and Operational Functionality of The Turing Machine
                                  History
                                             Variations
                  Structure and Definition
                                             4-tupel
                                 Samples
                                             State diagrams
                                 Varieties
                                             Instantaneous description
                            Computability
                                             Universal Turing Machine
                               References


Variations of the TMs ...



  ... provably equivalent

    • Two-way infinite tapes
    • Arbitrary movement of the head
    • Arbitrary numbers of read-write heads
    • Arbitrary finite alphabet




                           o         o
                          S¨ren Wellh¨fer    Structure and Operational Functionality of The Turing Machine
                                 History
                                            Variations
                 Structure and Definition
                                            4-tupel
                                Samples
                                            State diagrams
                                Varieties
                                            Instantaneous description
                           Computability
                                            Universal Turing Machine
                              References


4-tupel representation


                              asi −→ si1 dk

             When in state si reading symbol a:




                          o         o
                         S¨ren Wellh¨fer    Structure and Operational Functionality of The Turing Machine
                                 History
                                            Variations
                 Structure and Definition
                                            4-tupel
                                Samples
                                            State diagrams
                                Varieties
                                            Instantaneous description
                           Computability
                                            Universal Turing Machine
                              References


4-tupel representation


                              asi −→ si1 dk

             When in state si reading symbol a:

               • change state to si1




                          o         o
                         S¨ren Wellh¨fer    Structure and Operational Functionality of The Turing Machine
                                 History
                                            Variations
                 Structure and Definition
                                            4-tupel
                                Samples
                                            State diagrams
                                Varieties
                                            Instantaneous description
                           Computability
                                            Universal Turing Machine
                              References


4-tupel representation


                              asi −→ si1 dk

             When in state si reading symbol a:

               • change state to si1
               • Take action d: move right/left or
                  write




                          o         o
                         S¨ren Wellh¨fer    Structure and Operational Functionality of The Turing Machine
                                 History
                                            Variations
                 Structure and Definition
                                            4-tupel
                                Samples
                                            State diagrams
                                Varieties
                                            Instantaneous description
                           Computability
                                            Universal Turing Machine
                              References


4-tupel representation


                              asi −→ si1 dk

             When in state si reading symbol a:

               • change state to si1
               • Take action d: move right/left or
                  write




                          o         o
                         S¨ren Wellh¨fer    Structure and Operational Functionality of The Turing Machine
                      History
                                 Variations
      Structure and Definition
                                 4-tupel
                     Samples
                                 State diagrams
                     Varieties
                                 Instantaneous description
                Computability
                                 Universal Turing Machine
                   References




State diagram: successor of a unary number



               o         o
              S¨ren Wellh¨fer    Structure and Operational Functionality of The Turing Machine
                History
                           Variations
Structure and Definition
                           4-tupel
               Samples
                           State diagrams
               Varieties
                           Instantaneous description
          Computability
                           Universal Turing Machine
             References




         o         o
        S¨ren Wellh¨fer    Structure and Operational Functionality of The Turing Machine
                                 History
                                            Variations
                 Structure and Definition
                                            4-tupel
                                Samples
                                            State diagrams
                                Varieties
                                            Instantaneous description
                           Computability
                                            Universal Turing Machine
                              References


Instantaneous description




           A Turing Machine in an instantaneous state.




                          o         o
                         S¨ren Wellh¨fer    Structure and Operational Functionality of The Turing Machine
                                  History
                                             Variations
                  Structure and Definition
                                             4-tupel
                                 Samples
                                             State diagrams
                                 Varieties
                                             Instantaneous description
                            Computability
                                             Universal Turing Machine
                               References


Universal Turing Machine


  Features
    • Emulates δ of other Turing Machines
    • Von Neumann architecture




                           o         o
                          S¨ren Wellh¨fer    Structure and Operational Functionality of The Turing Machine
                                  History
                                             Variations
                  Structure and Definition
                                             4-tupel
                                 Samples
                                             State diagrams
                                 Varieties
                                             Instantaneous description
                            Computability
                                             Universal Turing Machine
                               References


Universal Turing Machine


  Features
    • Emulates δ of other Turing Machines
    • Von Neumann architecture
    • Turing-completeness




                           o         o
                          S¨ren Wellh¨fer    Structure and Operational Functionality of The Turing Machine
                                  History
                                             Variations
                  Structure and Definition
                                             4-tupel
                                 Samples
                                             State diagrams
                                 Varieties
                                             Instantaneous description
                            Computability
                                             Universal Turing Machine
                               References


Universal Turing Machine


  Features
    • Emulates δ of other Turing Machines
    • Von Neumann architecture
    • Turing-completeness


       ... 010110101111 00 101011011 00 110110110111 00
                       11010111011110 ...




                           o         o
                          S¨ren Wellh¨fer    Structure and Operational Functionality of The Turing Machine
                                  History
                                             Variations
                  Structure and Definition
                                             4-tupel
                                 Samples
                                             State diagrams
                                 Varieties
                                             Instantaneous description
                            Computability
                                             Universal Turing Machine
                               References


Universal Turing Machine


  Features
    • Emulates δ of other Turing Machines
    • Von Neumann architecture
    • Turing-completeness


       ... 010110101111 00 101011011 00 110110110111 00
                       11010111011110 ...




                           o         o
                          S¨ren Wellh¨fer    Structure and Operational Functionality of The Turing Machine
                                  History
                  Structure and Definition
                                 Samples
                                 Varieties
                            Computability
                               References


Computability


   Computable
     • Any number if
       TM-representable
              √
       (π, e,   )
     • Numerical functions
       (+, −, × , ÷)




                           o         o
                          S¨ren Wellh¨fer    Structure and Operational Functionality of The Turing Machine
                                  History
                  Structure and Definition
                                 Samples
                                 Varieties
                            Computability
                               References


Computability


   Computable                                   Incomputable
     • Any number if                                • Entscheidungsproblem: Will
       TM-representable                                 any algorithm A with
              √
       (π, e,   )                                       arbitrary input I halt?
     • Numerical functions                              h(A, I ) is incomputable
       (+, −, × , ÷)




                           o         o
                          S¨ren Wellh¨fer    Structure and Operational Functionality of The Turing Machine
                                  History
                  Structure and Definition
                                 Samples
                                 Varieties
                            Computability
                               References


Computability


   Computable                                   Incomputable
     • Any number if                                • Entscheidungsproblem: Will
       TM-representable                                 any algorithm A with
              √
       (π, e,   )                                       arbitrary input I halt?
     • Numerical functions                              h(A, I ) is incomputable
       (+, −, × , ÷)                                • “Busy beaver” function
                                                        Σ(n)




                           o         o
                          S¨ren Wellh¨fer    Structure and Operational Functionality of The Turing Machine
                                  History
                  Structure and Definition
                                 Samples
                                 Varieties
                            Computability
                               References


Computability


   Computable                                   Incomputable
     • Any number if                                • Entscheidungsproblem: Will
       TM-representable                                 any algorithm A with
              √
       (π, e,   )                                       arbitrary input I halt?
     • Numerical functions                              h(A, I ) is incomputable
       (+, −, × , ÷)                                • “Busy beaver” function
                                                        Σ(n)




                           o         o
                          S¨ren Wellh¨fer    Structure and Operational Functionality of The Turing Machine
                                   History
                   Structure and Definition
                                  Samples
                                  Varieties
                             Computability
                                References


Church?Turing thesis

     “Every effectively calculable function is a computable
                           function.”



    • Effectively calculable: produced intuitively
    • Computable function: computable by a Turing Machine




                            o         o
                           S¨ren Wellh¨fer    Structure and Operational Functionality of The Turing Machine
                                   History
                   Structure and Definition
                                  Samples
                                  Varieties
                             Computability
                                References


Church?Turing thesis

     “Every effectively calculable function is a computable
                           function.”



    • Effectively calculable: produced intuitively
    • Computable function: computable by a Turing Machine




    Solvable by humans = Solvable by machines (algorithm)

                            o         o
                           S¨ren Wellh¨fer    Structure and Operational Functionality of The Turing Machine
                                   History
                   Structure and Definition
                                  Samples
                                  Varieties
                             Computability
                                References


Church?Turing thesis

     “Every effectively calculable function is a computable
                           function.”



    • Effectively calculable: produced intuitively
    • Computable function: computable by a Turing Machine




    Solvable by humans = Solvable by machines (algorithm)

                            o         o
                           S¨ren Wellh¨fer    Structure and Operational Functionality of The Turing Machine
                             History
             Structure and Definition
                            Samples
                            Varieties
                       Computability
                          References


The Universal Turing Machine: A Half-Century Survey
R. Herken
New York: Oxford University Press, 1988.
[http://plato.stanford.edu/entries/turing-machine/]
Turing Machines
Stanford Encyclopedia of Philosophy, 04.11.1995


[http://www.intelligentedu.com/turing machines examples.html]
Turing Machines: Examples
Jaime Soffer, 2005.
[http://en.wikipedia.org/wiki] Turing Machine, Busy Beaver,
Computability, Turing-completeness, Entscheidungsproblem,
11-03-2008

                      o         o
                     S¨ren Wellh¨fer    Structure and Operational Functionality of The Turing Machine

				
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