TheHaltingProblem ModuleHomePage TitlePage JJ II J I Page 2 of 12 Back FullScreen Close Quit 351 Introduction Someproblemsareunsolvable Weretalkingabout importantandinterestingproblems pr ID: 99924
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Introduction TheHaltingProblem ModuleHomePage TitlePage JJ II J I Page 2 of 12 Back FullScreen Close Quit 35.1. Introduction Someproblemsareunsolvable. We'retalkingabout { importantandinterestingproblems, { precisely-denedproblems(notvagueproblemssuchas`composeapoemontopicT')forwhich { nomatterhowcleverweare, { nomatterhowmuchexecutiontime,memory,etc.weallowthereisprovablynoalgorithmtosolvetheproblem. Ourterminology:computableandnon-computable. Herearesomeexamplesofnon-computableproblemsthatwebrushedupagainstearlierinthemodule: { TilingProblemsfortheintegergrid(lecture1); { TakeinaprogramP;returnYESifPwilleverperformadivisionbyzero,elsereturnNO(lecture10); { TakeinaprogramP;returnanadequatesetoftestdataforprogramP(lecture10); { TakeinaproblemspecicationPSandaprogramP;returnaproofthatPsolvesPS(ifitdoessolveit),elsereturnfail(lectures10&22). Butwe'regoingtolookrstatthemostfamousofallnon-computableproblems,theHaltingProblem. Introduction TheHaltingProblem ModuleHomePage TitlePage JJ II J I Page 3 of 12 Back FullScreen Close Quit 35.2. TheHaltingProblem 35.2.1. WhatistheHaltingProblem? Aswehavediscussedpreviously,compilersdetecterrorsinprograms(priortotrans-latingthesourceprogramintoatargetprogram).Inparticular,theydetecterrorsinthesyntaxandthestaticsemantics.Theycannot,ingeneral,detecterrorsinthedynamicsemantics.Thereisagoodreasonforthis:it'simpossible.TheHalt-ingProblemisoneoftheproblemswemightlikecompilerstosolve,butwhichnocompilercaneversolve. Itistraditionaltoexplainthisproblem(andcomputabilityingeneral)usingsome-thingcalledaTuringmachine.However,wewillexplainthistopicusingMOCCAprograms,whichisjustasgood.(Infact,wecoulduseJavaprograms,orCpro-grams,orPascalprograms,or...,andthesewouldbejustasgoodalso.)We'llcomebacktoTuringmachinesinafewlectures'time,andthat'swhenwecanshowthatusingMOCCA(orJava,etc.)isjustasgood. Thefollowingproblemisnon-computable: Problem35.1. TheHaltingProblemParameters:AMOCCAprogramP,andapotentialin-putxtoP.Returns:YESifPwouldhaveterminatedhadwerunitoninputx;NOotherwise.(WeareassumingthatPexpectsjustoneinputx.Againnothinghangsonthis.Indeed,ifPrequiresmorethanoneinput,wecanthinkofitastakinginasingleinput:theconcatenationoftheindividualinputs.) Introduction TheHaltingProblem ModuleHomePage TitlePage JJ II J I Page 4 of 12 Back FullScreen Close Quit E.g.AsolutiontotheHaltingProblemshouldreturnYESwhengiventhefollowingprogramandinputx=11: whilex6=1fx:=x�2;g E.g.AsolutiontotheHaltingProblemshouldreturnNOwhengiventhepreviousprogramandinputx=12. Question:Whycan'twesimplyrunPonxandseewhathappens? 35.2.2. Preliminaries WewillprovethattheHaltingProblemisnon-computable. Wewanttoprove: ThereisnoMOCCAprogramwhich,uponacceptinganypairhP;xicon-sistingofthetextofalegalMOCCAprogramPandastringofsymbolsx,terminatesaftersomeniteamountoftime,andoutputsYESifPhaltswhenrunoninputxandNOifPdoesnothaltwhenrunoninputx. Akeyobservationisthatsuchaprogram,ifitexists,mustworkforeverypairhP;xi.And,ofcourse,suchaprogram,ifitexists,isitselfalegalMOCCAprogram,soithastoworkonitself. Weshallprovethenonexistenceofsuchaprogrambycontradiction. Introduction TheHaltingProblem ModuleHomePage TitlePage JJ II J I Page 7 of 12 Back FullScreen Close Quit Here'sImpossagaininamoretextualform: Algorithm:Imposs(Q)ans:=HP(Q;Q);ifans=YESfwhiletruefggelsefreturnwhatever;g Andhere'sanexplanationofprogramImpossinwords. { ImpossisaMOCCAprogramwhichtakesinasingleinput,Q,whichitselfshouldbethetextofaMOCCAprogram. { ImpossbeginsbymakingacopyofQ. { ImpossthencallsHP.RecallthatHPexpectstwoinputs,sowhenImpossactivatesHP,itsuppliesthetwocopiesofQ. { HPmusteventuallyreturneitherYES(i.e.programQdoesterminateforinputQ)orNO(i.e.programQdoesnotterminateforinputQ). { Impossreactsasfollows: IfHPreturnedYES,Impossentersaninniteloop. IfHPreturnedNO,Impossterminates(theoutputbeingunimportant). Introduction TheHaltingProblem ModuleHomePage TitlePage JJ II J I Page 8 of 12 Back FullScreen Close Quit WewillnowshowthatImpossisalogicalimpossibility. Specically,wewillshowthatthereisacertaininputonwhichImposscannotter-minate,butitalsocannotnotterminate! TheinputthatcausestheproblemisImpossitself! We'llspellouttwocases { We'llsupposeImpossappliedtoitselfdoesterminate. { Thenwe'llsupposeImpossappliedtoitselfdoesnotterminate. SupposeImposs,whengivenitselfasinput,doesterminate. Whathappens?TwocopiesofImpossaremade.ThesearefedtoHP,whichreturnsananswer.Ittellsus,inthiscase,whetherImpossterminatesonImposs.Weareassumingthatitdoes.SoHPwouldreturnYES.However,atthispoint,weentertheinniteloop,soImpossneverterminates. Butthismeansthat,whenweassumethatImpossdoesterminateonImposs,thenImpossdoesnotterminateonImposs! Introduction TheHaltingProblem ModuleHomePage TitlePage JJ II J I Page 10 of 12 Back FullScreen Close Quit Fromthesetwocases,wehavefoundthat { Imposscannotterminatewhenrunonitself,and { Imposscannotnotterminatewhenrunonitself!SomethingisverywrongwithImposs. ButImposswasconstructedquitelegally.So,theonlypartofImpossthatcanbeheldresponsibleisHP. WeconcludethataprogramHP,solvingtheHaltingProblem,simplycannotexist. (HowmuchdidthisdependonMOCCA?Notatall.) Introduction TheHaltingProblem ModuleHomePage TitlePage JJ II J I Page 11 of 12 Back FullScreen Close Quit Acknowledgements Thistreatmentismostlybasedon[ Har92 ].Iacknowledgealsothein uenceofAchimJung'sModelsofComputationcoursenotes[ Jun ].ClipArt(ofheadwithbomb)licensedfromtheClipArtGalleryonDiscoverySchool.com. Introduction TheHaltingProblem ModuleHomePage TitlePage JJ II J I Page 12 of 12 Back FullScreen Close Quit References [Har92] D.Harel.Algorithmics:TheSpiritofComputing.Addison-Wesley,2ndedition,1992. [Jun] A.Jung.ModelsofComputation(CourseNotes).