Lec7 Print Turing Machines Stanford
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Transcript of Lec7 Print Turing Machines Stanford
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CS154
Turing Machines
Turing Machine
FINITESTATE
CONTROL
INFINITE TAPE
I N P U T
q0q1
A
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0 0, R
read write move
, R
qaccept
qreject
0 0, R
, R
This Turing machine decides the language {0}
= {0}
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0 0, R
read write move
, R
qaccept
0 0, R
, R
0 0, R
, L
This Turing machine recognizes the language {0}
= {0}
Turing Machines versus DFAs
TM can both write to an read from the ta!e
The hea can move left and right
The in!ut oesn"t have to be rea entirel#$
Acce!t an %e&ect ta'e immeiate e((ect
an the com!utation can continue (urther
)even$ (orever* a(ter all in!ut has been rea
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Deciing the language + = { ,-, . , {0$/} }
STATE
0 1 1 # 0 1 10 #1 0 1X XX
1. If theres no # on the tape (or more than one #), reject.
2. While there is a bit to the left of #,
Replace the first bit ith !, an" chec if the first bit bto the ri$ht of the # is i"entical. (If not, reject.)
Replace that bit b ith an ! too.
%. If theres a bit to the ri$ht of #, then reject else accept
De(inition1 A Turing Machine is a 23tu!le
T = )4$ $ 5$ $ 60$ 6acce!t$ 6re&ect*$ ,here1
4 is a (inite set o( states
5 is the ta!e al!habet$ ,here 5 an 5
60 4 is the start state
is the in!ut al!habet$ ,here
1 4 5 7 4 5 {+$ %}
6acce!t 4 is the acce!t state
6re&ect 4 is the re&ect state$ an 6re&ect 6acce!t
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Turing Machine 8on(igurations
//0/06200//0 {45}*corres!ons to the configuration1
q7
1 0 0 0 0 01 1 1 1
De(ining Acce!tance an %e&ection (or TMs
+et 8/ an 89 be con(igurations o( a TM M
De(inition: 8/ yields 89 i( M is in con(iguration 89a(ter running M in con(iguration 8/ (or one ste!
;u!!ose )6/$ b* = )69$ c$ +*
Then aa6/bb #iels a69acb
;u!!ose )6/$ a* = )69$ c$ %*
Then cab6/a #iels cabc69
+et , an M be a Turing machineM accepts , i( there are con(igs 80$ 8/$ :::$ 8'$ s:t:
80 = 60, 8i #iels 8i
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A TM M recognizes a language +
i( M acce!ts eactl# those strings in +
A TM M decides a language + i( M acce!ts all
strings in + an re&ects all strings not in +
A language + is calle recogni>able or
recursivel# enumerable )r:e:*
i( some TM recogni>es +
A language + is calle eciable or recursivei( some TM ecies +
w L ?
accept reject
TM
yes no
w *
L is decidable
(recursive)
w L ?
accept reject or loop
TM
yes no
w *
L is recognizable
(recursively enumerable)
Theorem1 + is eciable
i(( both + an + are recogni>able
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%ecall1 ?iven + *$ e(ine + 1= @ +
o,B An# ieasB
M/ al,a#s acce!ts $ ,hen is in +
M9 al,a#s acce!ts $ ,hen isn"t in +
Theorem1 + is eciablei(( both + an + are recogni>able
?iven1
a TM M/ that recogni>es + an
a TM M9 that recogni>es +$
,ant to buil a ne, machine M that decides +
Theorem1 + is eciable
i(( both + an + are recogni>able
;imulate M/ )* on one ta!e$ M9 )* on another:Eactl# one o( the t,o ,ill acce!t
I( M/ acce!ts then acce!t
I( M9 acce!ts then re&ect
%ecall1 ?iven + *$ e(ine + 1= @ +
?iven1
a TM M/ that recogni>es + an
a TM M9 that recogni>es +$
,ant to buil a ne, machine M that decides +
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{ 0 . n C 0 }9n
1. Seep from left to ri$ht, cross o&t e'er other 0
2. If in step 1, the tape ha" onl one 0, accept
%. If in step 1, the tape ha" an o"" n&mber of 0s,
reject
4. o'e the hea" bac to the first inp&t smbol.
5. *o to step 1.
Turing Machine P;ED8DE1
Why does this work?
Iea1 Ever# time ,e return to stage /$
the number o( 0"s on the ta!e has been halve:
;te! /0 , R
, R
qacceptqreject
0 x, R
x x, R, R
x x, R
0 0, L
x x, L
x x, R
, L, R
0 x, R0 0, R
, Rx x, R
q0 q1
q2
q3
q4
{ 0 . n C 0 }9n
;te! 9
;te!
;te! G
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Multita!e Turing Machines
1 4 5' 7 4 5' {+$%}'
FINITESTATE
CONTROLk
FINITESTATE
CONTROL
0 01
FINITESTATE
CONTROL 0 01 ## #
..
.
Theorem1 Ever# Multita!e Turing Machine can be
trans(orme into a single ta!e Turing Machine
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FINITESTATE
CONTROL
0 01
FINITESTATE
CONTROL 0 01 ## #
..
.
Theorem1 Ever# Multita!e Turing Machine can be
trans(orme into a single ta!e Turing Machine
FINITESTATE
CONTROL
0 01
FINITESTATE
CONTROL 0 01 ## #
..
.
Theorem1 Ever# Multita!e Turing Machine can be
trans(orme into a single ta!e Turing Machine
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Theorem1 Ever# Multita!e Turing Machine can be
trans(orme into a single ta!e Turing Machine
FINITESTATE
CONTROL
0 01
FINITESTATE
CONTROL 0 01 # # #.
. .
Theorem1 Ever# noneterministic Turing machine N
can be trans(orme into a single ta!e Turing
Machine M that recognizes the language +)N*:
Noneterministic Turing Machines
Proo( Iea )more etails in ;i!ser*Pic' a natural orering on all strings in {4 5 -}
M),*1 For all strings D {4 5 -} in the orering$
8hec' i( D = 80-LLLL -8' ,here 80$ H$8' is some
acce!ting com!utation histor# (or N on ,:
I( so$ accept.
ave multi!le transitions (or a state$ s#mbol !air
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e can encoe Turing Machines as bit strings
0n/0m/0'/0s/0t/0r/0u/ H
n states
m tape symbols(first k are input
symbols)
start
state
acceptstate
rejectstate
blanksymbol
) )!$ i*$ )6$ &$ +* * = 0!/0i/06/0&/0
) )!$ i*$ )6$ &$ %* * = 0!/0i/06/0&/00
;imilarl#$ ,e can encoe DFAs an NFAs asbit strings$ an , as bit strings
ADFA = { )J$ ,* . J encoes a DFA over some $an J acce!ts , }
ANFA = { )J$ ,* . J encoes an NFA$ J acce!ts , }
ATM = { )M$ ,* . M encoes a TM$ M acce!ts , }
For e(ine b)* to be its binar# encoing
For $ # $ e(ine thepair of x and yas
)$ #* 1= 0.b)*./ b)* b)#*
Then ,e e(ine the (ollo,ing languages over {0$/}1
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ATM = { )M$ ,* . M encoes a TM over some $
, encoes a string over an M acce!ts ,}
ATM = { )M$ ,* . M oes not acce!t , }
Technical Note1
e"ll use an ecoing o( !airs$ TMs$ an strings so that
everybinar# string ecoes to some !air )M$ ,*
I( {0$/} oesn"t ecoe to )M$,* in the usual ,a#$
then ,e e(ine that ecoes to the !air )D$ K*
,here D is a Lumm# TM that acce!ts nothing:
Then$ ,e can e(ine the com!lement o( ATM ver# sim!l#1
niversal Turing Machines
Theorem1 There is a Turing machine
,hich ta'es as in!ut1
+ the coe o( an arbitrar# TM M
+ an an in!ut string ,
such that acce!ts )M$ ,*M acce!ts ,:
This is afundamental!ro!ert# o( TMs1
There is a Turing Machine that
can run arbitrar# Turing Machine coe
Note that DFAsONFAs o nothave this !ro!ert#:
That is$ ADFA an ANFA are not regular:
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The 8hurch3Turing Thesis
Ever#one"s
Intuitive Notion
o( Algorithms
= Turing Machines
This is not a theorem
it is a falsifiable scientific hypothesis.
An it has been thoroughl# teste