Lec7 Print Turing Machines Stanford

8/10/2019 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 ## #

..

.



FINITESTATE

CONTROL

0 01

FINITESTATE

CONTROL 0 01 ## #

..

.




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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

Lec7 Print Turing Machines Stanford

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