Turing_Variations

Turing_Variations - TuringsThesis Turings thesis (1930):...

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Turing’s Thesis  
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Turing’s thesis (1930): Any c o m puta tio n  c a rrie d o ut b y m e c hanic a l m e ans c a n b e  pe rfo rm e d b y a  Turing  Ma c hine
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Algorithm: An a lg o rithm  fo r a pro b le m  is  a   Turing  Mac hine  whic h s o lve s  the  pro b le m   The algorithm describes the steps of  the mechanical means This is easily translated to computation steps of a Turing machine
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When we say: There exists an algorithm We mean: There exists a Turing Machine that executes the algorithm
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Variations of the Turing Machine  
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Read-Write Head Control Unit a a c b a c b b a a Deterministic The Standard Model Infinite Tape (Left or Right)
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Variations of the Standard Model  Stay-Option   Semi-Infinite Tape  Off-Line  Multitape  Multidimensional  Nondeterministic Turing machines with: Different Turing Machine   Classes
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We  will pro ve : e ac h ne w c la s s  ha s  the  s a m e  po we r  with S ta nda rd Turing  Mac hine S a m e  Po we r o f two  m ac hine  c las s e s : b o th c la s s e s  ac c e pt the   s am e  s e t o f lang ua g e s (a c c e pt Turing -Re c o g niza b le  La ng uag e s )
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S a m e  Po we r o f two  c las s e s  m e a ns : fo r a ny m ac hine            o f firs t c la s s   1 M the re  is  a m ac hine            o f s e c o nd c la s s   2 M s uc h tha t:       ) ( ) ( 2 1 M L M L = a nd vic e -ve rs a
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A te c hniq ue  to  pro ve  s a m e  po we r. Simulation: S im ulate  the  m a c hine  o f o ne  c la s s with a  m a c hine  o f the  o the r c la s s Firs t C la s s   O rig inal Ma c hine 1 M 1 M 2 M      S e c o nd C la s s S im ula tio n Ma c hine s im ula te s 1 M
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C o nfig ura tio ns  in the  O rig ina l Mac hine ha ve  c o rre s po nding  c o nfig ura tio ns   in the  S im ula tio n Ma c hine n d d d 1 0 O rig ina l Ma c hine : S im ulatio n Mac hine : n d d d 1 0 1 M 2 M 1 M 2 M
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the  S im ula tio n Mac hine and the  O rig inal Mac hine ac c e pt the  s a m e  s tring s f d O rig ina l Ma c hine : S im ulatio n Mac hine : f d Ac c e pting  C o nfig uratio n ) ( ) ( 2 1 M L M L =
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Turing Machines with Stay-Option The head can stay in the same position a a c b a c b b a a Left, Right, Stay L,R,S: possible head moves
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Example: a a c b a c b b a a Time 1 b a c b a c b b a a Time 2 1 q 2 q 1 q 2 q S b a ,
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Stay-Option machines have the same power with  Standard Turing machines Theorem: Proof: 1.  S tay-O ptio n Ma c hine s    s im ula te  S ta nda rd Turing  m a c hine s   2. S tanda rd Turing  m ac hine s   s im ula te  S ta y-O ptio n m a c hine s
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 S tay-O ptio n Ma c hine s    s im ula te  S ta nda rd Turing  m a c hine s   Trivia l:
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Turing_Variations - TuringsThesis Turings thesis (1930):...

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