Chapter7

TuringMachines

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AGeneralModelofComputation

• Bothfiniteautomataandpushdownautomataaremodelsofcomputation– Eachreceivesaninputstringandexecutesanalgorithmtoobtainananswer,followingasetofrulesspecifictothemachinetype

• Itiseasytofindexamplesoflanguagesthatcannotbeacceptedbecauseofthemachine’slimitations:– AnFAcannotaccept{xcxr |x {a,b}*}– APDAcannotacceptAnBnCn ={anbncn |n 0}orL ={xcx |x {a,b}*}

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AGeneralModelofComputation(cont’d.)

• APDA‐likemachinewithtwo stackscanacceptAnBnCn

• AnFAwithaqueue insteadofastackcanacceptL• Inbothcases,itmightseemthatamachineisbeingspecificallydevelopedtohandleonelanguage,butitturnsoutthatboththesedeviceshavesubstantiallymorecomputingpowerthaneitheranFAoraPDA

• Eitheroneisareasonablecandidateforamodelofgeneral‐purposecomputation

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AGeneralModelofComputation(cont’d.)

• TheabstractmodelwewillstudyinsteadistheTuringmachine– Itisnotobtainedbyaddingdatastructuresontoafiniteautomaton

– Rather,itpredatestheFAandPDAmodels(AlanTuring’scontributionsdatefromthe1930’s)

• ATuringmachineisnotjust thenextstepbeyondapushdownautomaton– AccordingtotheChurch‐Turingthesis,itisageneralmodelofcomputation,potentiallyabletoexecuteanyalgorithm

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AGeneralModelofComputation(cont’d.)

• Turing’sobjectivewastodemonstratetheinherentlimitationsofalgorithmicmethods.Thisiswhyhewantedhisdevicetobeabletoexecuteanyalgorithmthatahumancomputercould

• Toformulatehiscomputationalmodel,heconsideredahumanbeingworkingwithapencilandpaper

• Asaresult,hepostulatedthatthestepsacomputertakesshouldincludethese:– Examineanindividualsymbolonthepaper– Eraseasymbolorreplaceitbyanother– Transferattentionfromonesymboltoanearbyone

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AGeneralModelofComputation(cont’d.)

• Forsimplicity,Turingspecifiedalineartapewhichhasaleftendandispotentiallyinfinitetotheright– Thetapeismarkedoffintosquareseachofwhichholdsonesymbol

– Wewillsometimesassignconsecutivenumberstothesquares,butthat’snotpartofthemodel

• Wevisualizethereadingandwritingasbeingdonebyatapehead,whichatanytimeiscenteredonasinglesquare

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AGeneralModelofComputation(cont’d.)

• InourversionofaTuringmachine,asinglemoveisdeterminedbythecurrentstate(correspondingtothe“stateofmind”ofthehumancomputer)andthecurrenttapesymbolandhasthreeparts– Changingfromthecurrentstatetoanotherstate– Replacingthesymbolinthesquarebyanother– Leavingthetapeheadwhereitis,movingitonesquaretotheleft,ormovingitonesquaretotheright

• Theinputstringisassumedtobeonthetapeinitially

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AGeneralModelofComputation(cont’d.)

• Thetapeprovidesthememoryneededduringcomputationandservesastheoutputdevice

• OnecrucialdifferencebetweenaTuringmachineandanFAorPDAisthataTuringmachineisnotrestrictedtoasinglepassthroughtheinput

• WewillfocusontwoprimaryobjectivesofaTuringmachine– Acceptingalanguage– Computingafunction

• Thefirstissimilartowhatwe’vedonesofar

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AGeneralModelofComputation(cont’d.)

• ATuringmachinewillhavetwohalt states,onedenotingacceptanceandtheotherrejection.(Morethantwoareunnecessary;unlikeanFA,thecompleteinputstringisonthetapeinitially,andaseparateanswerforeachprefixisnotrequired)

• UnlikeFAsandPDAs(oratleastPDAswithoutΛ‐transitions),Turingmachinesmayneverstop– Thiswillturnouttobeimportant

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AGeneralModelofComputation(cont’d.)

• Definition7.1:ATuringMachine(TM)isa5‐tupleT =(Q,,,q0,),where:– Q isafinitesetofstates

• Thetwohaltstatesha andhr arenotelementsofQ– Theinputalphabet andthetapealphabet arebothfinitesets,with • Theblanksymbol isnotanelementof

– q0,theinitialstate,isanelementofQ– Thetransitionfunctionis

:Q ( ∪ {}) (Q ∪ {ha,hr}) ( ∪ {}) {R,L,S}

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AGeneralModelofComputation(cont’d.)

• Weinterpret(p,X)=(q,Y,D)tomean:whenT isinstatep andthesymbolinthecurrentsquareisX,theTMreplacesX byY inthatsquare,changestostateq,andmovesthetapeheadonesquaretotheright,ormovesonesquaretotheleft,ordoesn’tmove– Ifthestateq iseitherha orhrwesaythatthismovecausesT tohalt• Onceithalts,itcannotmove

– WereturntodrawingtransitiondiagramssimilartobutmorecomplicatedthanthediagramsforFAs

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AGeneralModelofComputation(cont’d.)

• Thetransition(p,X)=(q,y,D)willberepresentedbythefollowingdiagram– IftheTMattemptstomovethetapeheadtotheleftwhenitisonsquare0,wewillsaythattheTMhaltsinstatehr ,leavingthetapeheadinsquare0andleavingthetapeunchanged

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AGeneralModelofComputation(cont’d.)

• NormallyaTMbeginswithaninputstringstartinginsquare1andallothersquares(square0andalltheonesfollowingtheinputstring)blank

• Inanycase,thesetofnonblanksquaresonthetapemustalwaysbefinite

• Wedescribethecurrentconfiguration ofaTMbyasinglestringxqywhereq isthecurrentstate,x isthestringofsymbolstotheleftofthecurrentsquare,y iseithernullorstartsinthecurrentsquare,andeverythingafterxy onthetapeisblank

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AGeneralModelofComputation(cont’d.)

• Wetraceasequenceofmovesbyspecifyingtheconfigurationateachstep

• Ifq isanon‐haltingstateandr isanystate,wewritexqy T zrw orxqy T* zrw tomeanthatTmovesfromthefirstconfigurationtothesecondinonemove,orinzeroormoremoves,respectively

• Theinitialconfigurationcorrespondingtoinputx isgivenbyq0x

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TuringMachinesasLanguageAcceptors

• Definition7.2:IfT=(Q,,,q0,)isaTMandx *,x isacceptedbyT ifq0 x⊢ T*whay forsomew,y ( {})*

• AlanguageL *isacceptedbyT ifL =L(T)={x *|x isacceptedbyT}

• ThefollowingtransitiondiagramsshowanFAandaTMthatacceptthesamelanguage

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TuringMachinesasLanguageAcceptors(cont’d.)

• Ifthelanguagewerenotregular,theTMcouldnotmoveitstapeheadtotherightoneverymove.

• The(b)diagramonthepreviousslidedoesnotshowanyofthemovestotherejectstate– Theyallhavethesameform,andthereisonefromeachofthestatesp,q,ands(thenonhaltingstatesotherthanq0 thatcorrespondtononacceptingstatesintheFA),asshownbelow

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TuringMachinesthatComputePartialFunctions

• ATuringmachinethatproducesanoutputstringforeverylegalinputstringissaidtocomputeapartialfunctionon*

• We’llalsoconsiderTMsthatcomputepartialfunctionson(*)k,i.e.,functionsofk variables

• Themostimportantissueiswhatoutputstringsareproducedforinputstringsinthedomainoff

• However,wewanttheTMtoacceptonlyinputsinthedomainoff,inordertobeabletosaythatitcomputesf andnotsomeotherfunctionwithlargerdomain

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TuringMachinesthatComputePartialFunctions(cont’d.)

• Definition7.9:– LetT=(Q,,,q0,)beaTuringmachine,k anaturalnumber,andf apartialfunctionfrom(*)k to*

– WesaythatT computesf ifforevery(x1,x2,…,xk)inthedomainoff,q0 x1 x2… xk ⊢T*haf(x1,x2,…xk)

andnootherinputthatisak‐tupleofstringsisacceptedbyT

• Apartialfunctionf isTuring‐computableifthereisaTMthatcomputesf

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TuringMachinesthatComputePartialFunctions(cont’d.)

• Forourpurposes,itwillbesufficienttoconsiderpartialfunctionson kwithvaluesin

• Wewilluseunarynotationfornumbers• TheofficialdefinitionissimilartoDefinition7.9,exceptthattheinputalphabetis{1},andtheinitialconfigurationlookslikeq01n11n2…1nk

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CombiningTuringMachines

• Justasalargealgorithmcanbedescribedasanumberofsubalgorithmsworkingincombination,wecancombineseveralTuringmachinesintoalargercompositeTM

• Inthesimplestcase,ifT1 andT2 areTMs,wecanconsiderthecompositionT1T2:“firstexecuteT1,thenexecuteT2 ontheresult”– ThesetofstatesofT1T2 istheunionofthesetsofstatesofT1 andT2 (relabeledifnecessary)

– TheinitialstateistheinitialstateofT1

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CombiningTuringMachines(cont’d.)

• ThetransitionsofT1T2 includeallofthoseofT2 andallofthoseofT1 thatdon’tgotoha

• AtransitioninT1 thatgoestoha isreplacedbyasimilartransitionthatgoestothestartstateofT2

• ItisimportantthattheoutputofT1 beavalidinputconfigurationforT2

• WemayusetransitiondiagramscontainingnotationssuchasT1 T2, inordertoavoidshowingallthestatesexplicitly

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CombiningTuringMachines(cont’d.)

• Wemightuseanyoftheabovenotationstomean“instatep,ifthecurrentsymbolisa,thenexecutetheTMT”

• Similarlywemightuseanyofthefollowingtomean“executeT1,andifT1 haltsinhawithcurrentsymbola,thenexecuteT2”

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MultitapeTuringMachines

• SomealgorithmscanbeunwieldytoimplementonaTM,becauseofthebookkeepingnecessary

• Awayofsimplifyingthemistohavemultipletapeswithindependentheads– Thisisadifferentmodelofcomputation,amultitapeTuringmachine

– Itwillturnoutthatthat,justasnondeterminismand‐transitionsdonotincreasethepowerofFAs,allowingaTuringmachinemultipletapesdoesnotincreaseitspower

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MultitapeTuringMachines(cont’d.)

• WewillshowthatforeverymultitapeTMT thereisasingle‐tapeTMthatacceptsexactlythesamestringsasT,rejectsthesamestrings,andproducesexactlythesameoutputforeveryinputstringitaccepts

• Tosimplifythediscussionwewillonlyconsidertwo‐tapemachines,butitiseasytoseethattheprinciplesarethesameiftherearemorethantwo

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MultitapeTuringMachines(cont’d.)

• A2‐tapeTMcanalsobedescribedbya5‐tupleT =(Q,,,q0,),wherethistime :Q ( ∪ {})2 (Q ∪ {ha,hr}) ( ∪ {})2 {R,L,S}2

• Asinglemovecanchangethestate,thesymbolsinthecurrentsquaresonbothtapes,andthepositionsofthetwotapeheads

• Wewillrepresentaconfigurationbya3‐tuple(q,x1a1y1,x2a2y2)whereq isthecurrentstate,xiaiyiisthecontentsoftapei,andai isinthecurrentsquareoftapei

• Theinputconfigurationforinputxwillbe(q0, x,)

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MultitapeTuringMachines(cont’d.)

• Theorem7.26:Forevery2‐tapeTMT=(Q,,,q0,)thereisanordinary1‐tapeTMT1=(Q1,,1,q1,1),with 1,suchthat:– L(T)=L(T1)– Foreveryx *,if(q0, x,)⊢T*(ha,yaz,ubv),thenq1x ⊢T1*yhaaz

• Proof:byconstruction– Thetapeisdividedintotwo“tracks”,assuggestedbythisdiagram:

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MultitapeTuringMachines(cont’d.)

• Insquare0,thereisamarkertomakeiteasytolocatethebeginningofthetape– Itwillalsobehelpfultohaveamarkerattheotherendofthenonblankportionofthetape

• Startingwiththeinitialconfigurationq1xwithinputx=a1a2…an,T1 placesthe$markerinsquare0,insertsblanksbetweenconsecutivesymbolsofx andplacesthe#markerafterthelastnonblanksymbol,toproducethetape

$a1a2…an#

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MultitapeTuringMachines(cont’d.)

• Fromthispointon,the#ismovedifnecessarytomarkthefarthestrightthatT hasmovedoneitherofitstapes

• TheonlysignificantcomplicationinT1 simulatingT isthatitmustbeabletokeeptrackofbothofthetapeheads

• WehandlethisbyincludinginT1’stapealphabetanextracopy’ofeverysymbol (including)thatcanappearonT’stape

• Theprimedsymbolgivesthelocationofthehead

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MultitapeTuringMachines(cont’d.)

• ByusingextrastatesT1 can“remember”thecurrentstateofT andalsowhattheprimedsymbolonthefirsttrackiswhileitislookingfortheprimedsymbolonthesecondtrack(orvice‐versa)

• ThestepsT1 makestosimulateamoveofT are:– Movethetapeheadlefttothe$,thenrighttoaprimedsymbol’;remember andmovebackto$

– Moverighttoaprimedsymbol’onthesecondtrack;ifthemovethatisnowdeterminedis(p,,)=(q,1,1,D1,D2)thenrejectifq=hr,elsechange’to1 andmoveindirectionD2 totheappropriatesquare

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MultitapeTuringMachines(cont’d.)

• ThestepsT1 makes(cont’d,)– Ifinmovingthetapeheadthisway,the$isencountered,thenreject,otherwise(movingthe#ifnecessary)changethesymbolinthenewsquareontrack2tothecorrespondingprimedsymbolandmovebacktothe$

– Locate’onthefirsttrackagain,changeitto1,andmovethetapeheadindirectionD1 totheappropriatesquareonthefirsttrack

– Afterallowingforeithermarker,asabove,changethenewsymbolonthefirsttracktothecorrespondingprimedsymbol

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MultitapeTuringMachines(cont’d.)

• AslongasT hasnothalted,iteratingthesestepsallowsT1 tosimulatethemovesofT correctly

• IfT finallyaccepts,thenT1 mustcarryoutthesestepsinordertoendupintherightconfiguration:– Deleteeverysquareinthesecondtracktotheleftof#– Deletebothend‐of‐tapemarkers– Movethetapeheadtotheprimedsymbol,changeittothecorrespondingunprimedsymbol,andhaltinhawiththetapeheadonthatsquare

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MultitapeTuringMachines(cont’d.)

• Corollary7.27:– Everylanguagethatisacceptedbya2‐tapeTMcanbeacceptedbyanordinary1‐tapeTM,andeveryfunctionthatiscomputedbya2‐tapeTMcanbecomputedbyanordinaryTM

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TheChurch‐TuringThesis

• TosaythattheTMisageneralmodelofcomputationimpliesthatanyalgorithmicprocedurethatcanbecarriedoutatall,byahumancomputerorateamofhumansoranelectroniccomputer,canbecarriedoutbyaTM– ThestatementwasformulatedbyAlonzoChurchinthe30’s

– ItisreferredtoasChurch’sthesisortheChurch‐Turingthesis

– Itisn’tastatementthatcanbeproved,butthereisalotofevidenceforit

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TheChurch‐TuringThesis(cont’d.)

• ThenatureofthemodelmakesitseemthataTMcanexecuteanyalgorithmahumancan

• EnhancementstotheTMhavebeenshownnottoincreaseitspower

• OthertheoreticalmodelsofcomputationproposedhavebeenshowntobeequivalenttoaTM

• NoonehaseversuggestedanykindofcomputationthatcannotbeimplementedonaTM

• Fromnowon,wewillconsiderthatbydefinition,an“algorithmicprocedure”iswhataTMcando

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NondeterministicTuringMachines

• WecanaddnondeterminismtoTuringmachines:asusual,(q,a)becomesasubset,notanelement

• Theorem7.31:ForeverynondeterministicTMTthereisanordinary(deterministic)TMT1 withL(T1)=L(T)

• Proof:– Theideaistouseanalgorithmthatcantest,ifnecessary,everypossiblesequenceofmovesofT onaninputstringx

– Seebookfordetails

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UniversalTuringMachines

• TheTMswehavestudiedsofarhavebeenspecial‐purposecomputerscapableofexecutingasinglealgorithm

• Wecanconsidera“universal”Turingmachine,whichiscapableofexecutingaprogramstoredinitsmemory– Itreceivesaninputstringthatspecifiesboththealgorithmitistoexecuteandtheinputthatistobeprovidedtothealgorithm

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UniversalTuringMachines(cont’d.)

• Definition7.32:Auniversal TuringmachineisaTuringmachineTu thatworksasfollows– Itisassumedtoreceiveaninputstringoftheforme(T)e(z),whereT isanarbitraryTM,z isastringovertheinputalphabetofT,ande isanencodingfunctionwhosevaluesarestringsin{0,1}*

– ThecomputationperformedbyTu onthisinputstringsatisfiesthesetwoproperties• Tu acceptse(T)e(z)ifandonlyifT acceptsz• IfT acceptsz andproducesoutputy, thenTu producesoutpute(y)

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UniversalTuringMachines(cont’d.)

• Wediscussasimpleencodingfunctione,andthensketchoneapproachtoconstructingauniversalTM

• Theencodingfunctionhasseveralcrucialfeatures:– Itispossibletodecidealgorithmically,foranarbitrarystringw {0,1}*,whetherw isalegitimatevalueofe

– Astringw shouldrepresentatmostoneTM,oratmostonestring

– Thereshouldbeanalgorithmfordecodingstringsoftheforme(T)ore(z)andreconstructingtheTMorstringitrepresents

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UniversalTuringMachines(cont’d.)

• Statelabelswillbereplacedbynumbers,andwewillbasetheencodingonthesenumbers

• Inordertousenumbertoencodesymbolsaswell,weadoptthisconvention:thereisaninfinitesetS={a1,a2,…}ofsymbols,includinga1 = ,suchthatthetapealphabetofeveryTMT isasubsetofS

• TheideaoftheencodingistorepresentaTMasasetofmovesandeachmoveisassociatedwitha5‐tupleofnumbers– Eachnumberisinunaryfollowedbya0

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UniversalTuringMachines(cont’d.)

• Definition7.33:IfT=(Q,,,q0,)isaTMandz isastring,definethestringse(T)ande(z)asfollows– Firstassignnumberstoeachstate,tapesymbol,andtapeheaddirectionofT;n(ha)=1,n(hr)=2,andn(q0)=3• Theotherelementsofq getdistinctnumbers 4,andn(R)=1,n(L)=2,n(S)=3

– Foreachmovem oftheform(p,)=(q,,D),e(m)=1n(p)01n()01n(q)01n()01n(D)0

– ListthemovesofT asm1,…,mk(theorderisarbitrary),andlete(T)=e(m1)0e(m2)0…0e(mk)0

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UniversalTuringMachines(cont’d.)

• Definition7.33:(cont’d.)– Ifz=z1z2…zj isastring,thene(z)=01n(z1)01n(z2)0…01n(zj)0

• Theorem7.36:LetE={e(T)|T isaTM}– Thenforeveryx {0,1}*,x E ifandonlyif:

• xmatchestheregularexpression(11*0)50((11*0)50)*,sothatitisasequenceof5‐tuples

• Notwosubstringsofx representing5‐tupleshavethesamefirsttwoparts

• Noneofthe5‐tupleshavefirstpart1or11• Thelastpartofeach5‐tuplemustbe1,11,or111

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UniversalTuringMachines(cont’d.)

• Thoseconditionsdon’tguaranteethatthestringrepresentsaTMthatcarriesoutameaningfulcomputation– ButtheydoensurethatwecandrawatransitiondiagramcorrespondingtotheencodedTM

• Testingastringtodeterminewhetheritsatisfiestheseconditionsisstraightforward,sowehaveverifiedthate satisfiestheminimalrequirementsforsuchafunction

• Detailsofthesimulationareinthebook

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