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1

Turing

Machin

es

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2

TuringMachine

ATuring

machineisadevicethatma

nipulates

symbolsonastripoftapeaccording

toatableof

rules.De

spiteitssimplicity,aTuring

machinecanb

e

adapted

tosimulatethelogicofanycomputer

algorithm

,andispartic

ularlyusefulinexplainingth

e

functions

ofaCPUinsideacomputer

.

The"Tur

ing"machinewasdescribed

byAlanTuring

[in1936]whocalledit

an"a(utomatic)-machine".

TheTuringmachineis

notintendeda

sapractical

computin

gtechnology,

butratheras

ahypothetical

devicere

presentingac

omputingmachine.

Turing

machineshelpcomputerscientistsun

derstandthe

limitsofmechanicalcomputation.

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Turingmachine

ATur

ingmachin

eisthesimplestform

ofac

omputer.

Thiswast

hefirst

comp

uterinvented(onpaperonly).

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4

PrinciplesofaTuringm

achine

ATuringmachinec

onsistsof:

Atapewhichisdividedintocells,onenexttotheother.Eachc

ellc

ontainsasymbolfrom

somefinitealp

habet.Thealphabetcon

tainsaspecialblanksym

bol(herewrittenas'B')and

oneormoreothersymbols.

TheTuring

machineisalwayssuppliedwithasmuchtape

asit

needsforitscomputation.

Cellsthathavenotbeenwrittentobeforeareassumedtobe

filledwiththeblanksymbol.Insomem

odelsthetapehasaleftendmarkedwithaspe

cial

symbol;thetapeextendsorisindefinitelyextensibletotheright.

Aheadthatcanreadandwritesymbolsonthetapeandmove

thetapeleftandrighto

ne

(andonlyone)

cella

tatime.

Insomemodelstheheadmoves

andthetapeisstationary.

Afinitetable(occasionallycalledanac

tiontableortransitionfunction)ofinstructions

(usuallyquintu

ples[5-tuples]:qiajqi1aj1dk,

butsometimes4-tuples)that,giventhe

state(qi)them

achineiscurrentlyinan

dthesymbol(aj)itisreadingonthetape(symb

ol

currentlyunderthehead)tellsthemac

hinetodothefollowing

insequence(forthe5-tuple

models):

Eithereraseor

writeasymbol(instead

ofaj,writeaj1),andth

en

Movethehead

(whichisdescribedbydkandcanhavevalues:

'L'foronestepleftor'R'for

onesteprightor'N'forstayinginthesameplace),andthen

Assumethesameoranewstateasprescribed(gotostateqi1

).

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5

FunctionOfT

uringMachine

1.

ATurin

gmachinecanbothwriteonthetapeand

readfromit.

2.

Theread-writehead

canmovebothtotheleftan

d

totheright.

3.

Thetapeisinfinite.

4.

Thespecialstatesforrejectingand

acceptingtak

e

immediateeffect.

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A

ThinkingMachine

e.gS

ucces

sor

Program

SampleRules:

Ifread1,

write0,goright,rep

eat.

Ifread0,

write1,H

ALT!

Ifread

,write1,H

ALT!

Letsseehowtheya

recarried

outona

pieceofpaperth

atcontainsthe

revers

ebinaryre

presentationof47:

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A

ThinkingMachine

e.gS

ucces

sor

Program

Ifread1,

write0,goright,rep

eat.

Ifread0,

write1,H

ALT!

Ifread

,write1,H

ALT!

1

0

1

1

1

1

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A

ThinkingMachine

EGS

uccess

or

Pro

gram

Ifread1,

write0,goright,rep

eat.

Ifread0,

write1,H

ALT!

Ifread

,write1,H

ALT!

1

0

1

1

1

0

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A

ThinkingMachine

e.gS

ucces

sor

Program

Ifread1,

write0,goright,rep

eat.

Ifread0,

write1,H

ALT!

Ifread

,write1,H

ALT!

1

0

1

1

0

0

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A

ThinkingMachine

e.gS

ucces

sor

Program

Ifread1,

write0,goright,rep

eat.

Ifread0,

write1,H

ALT!

Ifread

,write1,H

ALT!

1

0

1

0

0

0

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A

ThinkingMachine

e.gS

ucces

sor

Program

Ifread1,

write0,goright,rep

eat.

Ifread0,

write1,H

ALT!

Ifread

,write1,H

ALT!

1

1

0

0

0

0

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A

ThinkingMachine

e.gS

ucces

sor

Program

Sothesuccessorso

utputon1

11101was

00001

1whichis

thereversebinary

repres

entationof48.

Similarly,

thesucces

sorof127

shouldbe

128:

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A

ThinkingMachine

e.gS

ucces

sor

Program

Ifread1,

write0,goright,rep

eat.

Ifread0,

write1,H

ALT!

Ifread

,write1,H

ALT!

1

1

1

1

1

1

1

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A

ThinkingMachine

e.gS

ucces

sor

Program

Ifread1,

write0,goright,rep

eat.

Ifread0,

write1,H

ALT!

Ifread

,write1,H

ALT!

1

1

1

1

1

1

0

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A

ThinkingMachine

e.gS

ucces

sor

Program

Ifread1,

write0,goright,rep

eat.

Ifread0,

write1,H

ALT!

Ifread

,write1,H

ALT!

1

1

1

1

1

0

0

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A

ThinkingMachine

e.gS

ucces

sor

Program

Ifread1,

write0,goright,rep

eat.

Ifread0,

write1,H

ALT!

Ifread

,write1,H

ALT!

1

1

1

1

0

0

0

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A

ThinkingMachine

e.gS

ucces

sor

Program

Ifread1,

write0,goright,rep

eat.

Ifread0,

write1,H

ALT!

Ifread

,write1,H

ALT!

1

1

1

0

0

0

0

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A

ThinkingMachine

e.gS

ucces

sor

Program

Ifread1,

write0,goright,rep

eat.

Ifread0,

write1,H

ALT!

Ifread

,write1,H

ALT!

1

0

0

0

0

0

0

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A

ThinkingMachine

e.gS

ucces

sor

Program

Ifread1,

write0,goright,rep

eat.

Ifread0,

write1,H

ALT!

Ifread

,write1,H

ALT!

0

0

0

0

0

0

0

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A

ThinkingMachine

e.gS

ucces

sor

Program

Ifread1,

write0,goright,rep

eat.

Ifread0,

write1,H

ALT!

Ifread

,write1,H

ALT! 1

0

0

0

0

0

0

0