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Turing’sLegacy:DevelopmentsfromTuring’sIdeasinLogic AlanTuringwasaninspirationalfigurewhoisnowrecognizedasageniusofmodern mathematics. In addition to being a leading participant in the Allied forces’ code- breaking effort at Bletchley Park in World War II, he proposed the theoretical foun- dationsofmoderncomputing,andanticipateddevelopmentsinareasfrominformation theory to computer chess. His ideas have been extraordinarily influential in modern mathematics and this book traces such developments by bringing together essays by leadingexpertsinlogic,artificialintelligence,computabilitytheory,andrelatedareas. Together,theygiveinsightintothisfascinatingman,thedevelopmentofmodernlogic, and the history of ideas. The articles within cover a diverse selection of topics, such asthedevelopmentofformalproof,differingviewsontheChurch–Turingthesis,the developmentofcombinatorialgrouptheory,andTuring’sworkonrandomnesswhich foresawtheideasofalgorithmicrandomnessthatwouldemergemanyyearslater. rod downey is Professor of Mathematics at Victoria University of Wellington, NewZealand.Hismainresearchinterestslieinalgebra,logicandcomplexitytheory. Downeyhasreceivedmanyprofessionalaccoladesthroughouthiscareer,includingthe SchoenfieldPrizeoftheAssociationforSymbolicLogicandtheHectorMedalofthe RoyalSocietyofNewZealand,alongwithnumerousfellowshipstolearnedsocieties andinstitutessuchastheIsaacNewtonInstitute(Cambridge)andtheAmericanMath- ematicalSociety. LECTURENOTESINLOGIC APublicationfor TheAssociationforSymbolicLogic This series serves researchers, teachers, and students in the field of symbolic logic,broadlyinterpreted.Theaimoftheseriesistobringpublicationstothe logiccommunitywiththeleastpossibledelayandtoproviderapiddissemina- tionofthelatestresearch.Scientificqualityistheoverridingcriterionbywhich submissionsareevaluated. EditorialBoard H.DugaldMacpherson,ManagingEditor SchoolofMathematics,UniversityofLeeds JeremyAvigad, DepartmentofPhilosophy,CarnegieMellonUniversity VolkerHalbach, NewCollege,UniversityofOxford VladimirKanovei, InstituteforInformationTransmissionProblems,Moscow ManuelLerman, DepartmentofMathematics,UniversityofConnecticut HeinrichWansing, DepartmentofPhilosophy,Ruhr-UniversitätBochum ThomasWilke, InstitutfürInformatik,Christian-Albrechts-UniversitätzuKiel Moreinformation,includingalistofthebooksintheseries,canbefoundat http://www.aslonline.org/books-lnl.html LectureNotesinLogic42 Turing’s Legacy: Developments from Turing’s Ideas in Logic Editedby ROD DOWNEY VictoriaUniversityofWellington ASSOCIATIONFORSYMBOLICLOGIC UniversityPrintingHouse,CambridgeCB28BS,UnitedKingdom CambridgeUniversityPressispartoftheUniversityofCambridge. ItfurtherstheUniversity’smissionbydisseminatingknowledgeinthepursuitof education,learningandresearchatthehighestinternationallevelsofexcellence. www.cambridge.org Informationonthistitle:www.cambridge.org/9781107043480 ©AssociationforSymbolicLogic2014 Thispublicationisincopyright.Subjecttostatutoryexception andtotheprovisionsofrelevantcollectivelicensingagreements, noreproductionofanypartmaytakeplacewithoutthewritten permissionofCambridgeUniversityPress. Firstpublished2014 PrintedintheUnitedKingdombyCPIGroupLtd,CroydonCRO4yy AcataloguerecordforthispublicationisavailablefromtheBritishLibrary LibraryofCongressCataloging-in-PublicationData Turing’slegacy:developmentsfromTuring’sideasinlogic/editedbyRodDowney, VictoriaUniversityofWellington. pagescm.–(Lecturenotesinlogic;42) Includesbibliographicalreferencesandindex. ISBN978-1-107-04348-0(hardback) 1. Computationalcomplexity. 2. Machinetheory. 3. Turing,AlanMathison, 1912-1954.I. Downey,R.G.(RodG.),editorofcompilation. QA267.7.T872014 510.92–dc23 2014000240 ISBN978-1-107-04348-0Hardback CambridgeUniversityPresshasnoresponsibilityforthepersistenceoraccuracy ofURLsforexternalorthird-partyinternetwebsitesreferredtointhispublication, anddoesnotguaranteethatanycontentonsuchwebsitesis,orwillremain, accurateorappropriate. CONTENTS RodDowney,editor Turing’slegacy: developmentsfromTuring’sideasinlogic.......... vii JeremyAvigadandVascoBrattka Computabilityandanalysis: thelegacyofAlanTuring............. 1 LenoreBlum AlanTuringandtheothertheoryofcomputation(expanded)....... 48 HarryBuhrman TuringinQuantumland........................................... 70 RodDowney Computabilitytheory,algorithmicrandomnessandTuring’s anticipation...................................................... 90 EkaterinaB.Fokina,ValentinaHarizanov,andAlexanderMelnikov Computablemodeltheory......................................... 124 CameronE.Freer,DanielM.Roy,andJoshuaB.Tenenbaum Towardscommon-sensereasoningviaconditionalsimulation: legaciesofTuringinArtificialIntelligence.......................... 195 ThomasC.Hales MathematicsintheageoftheTuringmachine...................... 253 StevenHomerandAlanL.Selman Turingandthedevelopmentofcomputationalcomplexity.......... 299 CharlesF.MillerIII Turingmachinestowordproblems................................ 329 AnilNerode MusingsonTuring’sThesis........................................ 386 DagNormann HighergeneralizationsoftheTuringModel........................ 397 WilfriedSieg Stepbyrecursivestep: Church’sanalysisofeffectivecalculability.... 434 v vi contents RobertIrvingSoare Turingandthediscoveryofcomputability.......................... 467 P.D.Welch Transfinitemachinemodels........................................ 493 TURING’SLEGACY: DEVELOPMENTSFROMTURING’SIDEASINLOGIC §1. Introduction. The year 2012 was the centenary of the birth of one of themostbrilliantmathematiciansofthe20thcentury. Thereweremanycele- brationsofthisfact,andmanyconferencesbasedaroundTuring’sworkand lifeduring2012. Inparticular,therewasahalfyearprogram(SyntaxandSe- mantics)attheNewtonInstituteinCambridge,andmany“Turing100/Cen- tenary”conferencesthroughouttheyear. Theseeventsincludedtrulymajor meetings featuring many of the world’s best mathematicians and computer scientists(andevenGaryKasparov)aroundhisactualbirthdayofJune23, includingTheIncomputable,ACMA.M.TuringCentenaryCelebration,How theWorldComputes(CiE2012),andTheTuringCentenaryConference. There arealsoanumberofpublicationsdevotedtoTuring’slife,workandlegacy. Tothegeneralpublic,TuringisprobablybestknownforhispartinBletchley Parkandthewar-winningeffortsofthecode-breakersatHut8. Tobiologists, Turingisbestknownforhisworkonmorphogenesis,thepaper“AChemical BasisforMorphogenesis”beinghismosthighlycitedwork. To logicians, and computer scientists, Alan Turing is best known for his work in computation, arguably leading to the development of the digital computer. Thisdevelopmenthascausedalmostcertainlythemostprofound change in human history in the last century. Turing’s work in computation grew from philosophical questions in logic. Thus it seems fitting that the AssociationforSymbolicLogicsponsoredthisvolume. Theideaforthisvolumeisto(mainly)lookatthevariouswaysTuring’sideas inlogichavedevelopedintomajorprogramsinthelandscapeofmathematics andphilosophyinthearly21stCentury. Thatis,wheredidtheseideasgo? A numberofleadingexpertswereinvitedtoparticipateinthisenterprise. Allof thepaperswerereviewedbothforreadabilitybynon-expertsandforcontent byotherexperts. §2. Turing’swork. ThereisanexcellentarchiveofTuring’sworkin http://www.turing.org.uk/sources/biblio.html. JackCopeland(sometimeswithDianeProudfoot)havehistoricalarticlesand books such as [1, 2, 3]. Below we give a few brief comments and refer the ResearchsupportedbytheMarsdenFundofNewZealand. vii viii RODDOWNEY readertotheseandthebooksDavis[4]andtoHerken[5]formorehistorical comments,aswellasthearticlesinthisvolumebyNerode,SiegandSoare. Turing [7] worked famouslyon theEntscheidungsproblem, thequestionof thedecisionproblemforvalidityoffirstorderpredicatecalculus. Hisworkand thatofChurch, Kleene, Postandotherssolvedtheproblem. Turing’spaper of 1936 laid the foundations for the advent of stored program computers. His 1936 paper had the key idea of stored program computers via universal machines. Turingknewofthepossibilitiesoflargescaleelectroniccomputers followingthegroundbreakingideasofFredFlowerswithhisworkonColossus inthesecondworldwar. Turing’sanalysisofcomputationandhisintroduction ofuniversalmachinesarediscussedinbothSieg’sandSoare’sarticles. Inalectureof1947,TuringsaidofhisdesignofACE(automatedcomputing engine) “Thespecialmachinemaybecalledtheuniversalmachine; itworks in the following quite simple manner. When we have decided what machinewewishtoimitatewepunchadescriptionofitonthetape of the universal machine ... . The universal machine has only to keeplookingatthisdescriptioninordertofindoutwhatitshoulddo ateachstage. Thusthecomplexityofthemachinetobeimitatedis concentratedinthetapeanddoesnotappearintheuniversalmachine properinanyway... . [D]igitalcomputingmachinessuchastheACE ... areinfactpracticalversionsoftheuniversalmachine.” In1943,McCullochandPittusedTuringideastoshowthecontrolmech- anismforaTMcouldbesimulatedbyafinitecollectionofgateswithdelays. TheseideaswerelaterdevelopedbyvonNeumannandotherswhichleadto ENIAC in 1943. A friend of von Newmann who worked with him on the Atomic bomb project was Stanley Frankel (see [3]) who is quoted as saying thefollowing: “vonNeumannwaswellawareofthefundamentalimportanceofTur- ing’spaperof1936‘Oncomputablenumbers... ’,whichdescribesin principlethe‘UniversalComputer’... Manypeoplehaveacclaimed vonNeumannasthe‘fatherofthecomputer’(inamodernsenseof theterm)butIamsurethathewouldneverhavemadethatmistake himself. Hemightwellbecalledthemidwife,perhaps,buthefirmly emphasized to me, and to others I am sure, that the fundamental conceptionisowingtoTuring.” TuringdesignedtheACE.Whilstneverbuilt,Turing’sdesignwasthebasisof thearchitectureofseveralcomputers. Forexample,Huxley’sG15computer, thefirstPC(aboutthesizeofafridge)wasbasedonit,withabout400sold worldwide,andremaininginuseuntil1970(!). Theworld’sfirstprogrammablecomputerwasbuiltinManchesterbyTur- ing’s lifelong friend Max Newman (who Turing met in 1935). Turing was