A COMPUTABLE UNIVERSEUnderstanding and Exploring Nature as Computation8306.9789814374293-tp.indd 1 14/9/12 8:49 AMJune29,2012 9:45 WorldScienticReviewVolume-9inx6in-8306AComputableUniverse AComputableUniverseThis page intentionally left blank This page intentionally left blankNE WJ E RSE Y L ONDON SI NGAP ORE BE I J I NG SHANGHAI HONGKONG TAI P E I CHE NNAI World ScientifcUniversity of Shefeld, UK& Wolfram Research, USAA COMPUTABLE UNIVERSEUnderstanding and Exploring Nature as ComputationEditorHector Zenil Foreword by Sir Roger Penrose8306.9789814374293-tp.indd 2 14/9/12 8:49 AMBritishLibraryCataloguing-in-PublicationDataA catalogue record for this book is available from the British Library.Forphotocopyingofmaterialinthisvolume,pleasepayacopyingfeethroughtheCopyrightClearance Center, Inc., 222 Rosewood Drive, Danvers, MA 01923, USA. In this case permission tophotocopy is not required from the publisher.ISBN 978-981-4374-29-3All rights reserved. This book, or parts thereof, may not be reproduced in any form or by any means,electronic or mechanical, including photocopying, recording or any information storage and retrievalsystem now known or to be invented, without written permission from the Publisher.Copyright 2013 by World Scientific Publishing Co. Pte. Ltd.PublishedbyWorld Scientific Publishing Co. Pte. Ltd.5 Toh Tuck Link, Singapore 596224USA office:27 Warren Street, Suite 401-402, Hackensack, NJ 07601UK office:57 Shelton Street, Covent Garden, London WC2H 9HEPrinted in Singapore.ACOMPUTABLEUNIVERSEUnderstanding and Exploring Nature as ComputationChinAng - A Computable Universe.pmd 9/4/2012, 8:56 AM 1June29,2012 9:45 WorldScienticReviewVolume-9inx6in-8306AComputableUniverse AComputableUniverseToElenavJune29,2012 9:45 WorldScienticReviewVolume-9inx6in-8306AComputableUniverse AComputableUniverseThis page intentionally left blank This page intentionally left blankAugust30,2012 13:35 WorldScienticReviewVolume-9inx6in-8306AComputableUniverse AComputableUniverseContentsForeword xiiiR.PenrosePreface xxxviiAcknowledgements xliii1. IntroducingtheComputableUniverse 1H.ZenilHistorical,Philosophical&FoundationalAspectsofComputation 212. OriginsofDigitalComputing: AlanTuring,CharlesBabbage,&AdaLovelace 23D.Swade3. Generating,SolvingandtheMathematicsofHomoSapiens. E.PostsViewsonComputation 45L.DeMol4. Machines 63R.TurnerviiAugust30,2012 13:35 WorldScienticReviewVolume-9inx6in-8306AComputableUniverse AComputableUniverseviii Contents5. Eectiveness 77N.Dershowitz&E.Falkovich6. Axioms for Computability: DoTheyAllowaProof ofChurchsThesis? 99W.Sieg7. TheMathematiciansBiasandtheReturntoEmbodiedComputation 125S.B.Cooper8. IntuitionisticMathematicsandRealizabilityinthePhysicalWorld 143A.Bauer9. WhatisComputation?ActorModelversusTuringsModel 159C.HewittComputationinNature&theRealWorld 18710. ReactionSystems: ANatural ComputingApproachtotheFunctioningofLivingCells 189A.Ehrenfeucht,J.Kleijn,M.Koutny&G.Rozenberg11. Bacteria,TuringMachinesandHyperbolicCellularAutomata 209M.Margenstern12. ComputationandCommunicationinUnorganizedSystems 231C.Teuscher13. TheManyFormsofAmorphousComputationalSystems 243J.WiedermannAugust30,2012 13:35 WorldScienticReviewVolume-9inx6in-8306AComputableUniverse AComputableUniverseContents ix14. ComputingonRings 257G.J.Martnez,A.Adamatzky&H.V.McIntosh15. LifeasEvolvingSoftware 277G.J.Chaitin16. ComputabilityandAlgorithmicComplexityinEconomics 303K.V.Velupillai&S.Zambelli17. BlueprintforaHypercomputer 333F.A.DoriaComputation&Physics&thePhysicsofComputation 34518. Information-Theoretic Teleodynamics in Natural andArticialSystems 347A.F.Beavers&C.D.Harrison19. DiscreteTheoreticalProcesses(DTP) 365E.Fredkin20. TheFastestWayofComputingAllUniverses 381J.Schmidhuber21. TheSubjectiveComputableUniverse 399M.Hutter22. WhatIsUltimatelyPossibleinPhysics? 417S.Wolfram23. Universality,TuringIncompletenessandObservers 435K.SutnerAugust30,2012 13:35 WorldScienticReviewVolume-9inx6in-8306AComputableUniverse AComputableUniversex Contents24. AlgorithmicCausalSetsforaComputationalSpacetime 451T.Bolognesi25. TheComputableUniverseHypothesis 479M.P.Szudzik26. TheUniverseisLawlessor Pantonchrematonmetronanthroponeinai 525C.S.Calude,F.W.Meyerstein&A.Salomaa27. IsFeasibilityinPhysicsLimitedbyFantasyAlone? 539C.S.Calude&K.SvozilTheQuantum,Computation&Information 54928. WhatisComputation?(How)DoesNatureCompute? 551D.Deutsch29. TheUniverseasQuantumComputer 567S.Lloyd30. QuantumSpeedupandTemporalInequalitiesforSequentialActions 583M.Zukowski31. TheContextualComputer 595A.Cabello32. AGodel-TuringPerspectiveonQuantumStatesIndistinguishablefromInside 605T.BreuerAugust30,2012 13:35 WorldScienticReviewVolume-9inx6in-8306AComputableUniverse AComputableUniverseContents xi33. WhenHumansDoComputeQuantum 617P.ZizziOpenDiscussionSection 62934. OpenDiscussiononAComputableUniverse 631A.Bauer,T.Bolognesi,A.Cabello,C.S.Calude,L.DeMol,F.Doria,E.Fredkin,C.Hewitt,M.Hutter,M.Margenstern,K.Svozil,M.Szudzik,C.Teuscher,S.Wolfram&H.ZenilLivePanelDiscussion(transcription) 67135. WhatisComputation?(How)DoesNatureCompute? 673C. S. Calude, G. J. Chaitin, E. Fredkin, A. J. Leggett,R.deRuyter,T.Tooli&S.WolframZusesCalculatingSpace 72736. CalculatingSpace(RechnenderRaum) 729K.ZuseAfterwordtoKonradZusesCalculatingSpace 787A.German&H.ZenilIndex 795August30,2012 13:35 WorldScienticReviewVolume-9inx6in-8306AComputableUniverse AComputableUniverseThis page intentionally left blank This page intentionally left blankAugust30,2012 13:35 WorldScienticReviewVolume-9inx6in-8306AComputableUniverse AComputableUniverseForewordRogerPenroseMathematical InstituteUniversityofOxford,UKI am most honoured to have the privilege to present the Foreword to thisfascinatingandwonderfullyvariedcollectionofcontributionsa,concerningthe nature of computation and of its deep connection with the operation ofthosebasiclaws, knownoryetunknown, governingtheuniverseinwhichwe live. Fundamentally deep questions are indeed being grappled with here,andthefactthatwendsomanydierentviewpointsissomethingtobeexpected, since, in truth, we know little about the foundational nature andorigins of thesebasiclaws, despitetheimmenseprecisionthat wesoof-tenndrevealedinthem. Accordingly, it is not surprisingthat withintheviewpointsexpressedhereissomeunabashedspeculation,occasionallyborderingonjustpartiallyjustiedguesswork, whileelsewherewendagooddealofprecisereasoning,someintheformofrigorousmathematicaltheorems. Bothoftheseareasshouldbe,forwithoutsomeinspiredguess-work we cannot have new ideas as to where look in order to make genuinelynewprogress,andwithoutprecisemathematicalreasoning,nolessthaninpreciseobservation,wecannotknowwhenwearerightor,moreusually,whenwearewrong.Theyearofthepublicationofthisbook,2012,isparticularlyapposite,inbeingthecentenaryyearof AlanTuring, whosetheoretical analysisofthenotionof computingmachine,togetherwithhiswartimeworkinde-ciphering Nazi codes, has had a huge impact on the enormous developmentof electronic computers, and on the consequent inuence that these devicesaFootnotestonamesinthenextpagesarepointerstothechaptersinthisvolume(AComputableUniversebyH.Zenil),Ed.xiiiAugust30,2012 13:35 WorldScienticReviewVolume-9inx6in-8306AComputableUniverse AComputableUniversexiv R.Penrosehavehadonourlivesandonthewaythatwethinkaboutourselves. Thisimpactisparticularlyevidentwiththeapplicationofcomputertechnologytotheimplicationsof knownphysical laws, whethertheybeatthebasicfoundational level, or at a larger level such as with uid mechanics or ther-modynamicswhereaveragesoverhugenumbersofelementaryconstituentparticles again lead to comparatively simple dynamical equations. I shouldhere remark that from time to time it has even been suggested that, in somesense, the laws thatweappeartondinthewaythattheworldworksareall ofthisstatistical character, andthat, atroot, thereare no basicunderlyingphysical laws(e.g. Wheelers lawwithoutlaw,43Sakharovsideasof inducedgravity,31etc.,andwendthisgeneraltypeofviewex-pressedalsointhisvolumealsob). However,Indithardtoseethatsucha viewpoint can have much chance of yielding anything like the enormouslyprecisenon-statistical dynamics32andgreat mathematical sophisticationthatwendinsomuchof20thcenturyphysics. Thispointaside,wendthat in reasonably favourable circumstances, computer simulations can leadto hugely impressive imitations of reality, and the resulting visual represen-tations maybealmost indistinguishablefromthereal thing, afact thatisfrequentlymadeuseofinrealisticspecialeectsinlms,asmuchasinserious scientic presentations. When we need precision in particular impli-cationsofsuchequations,wemayrunintothedicultissuespresentedbychaoticbehaviour, wherebythedependenceoninitial conditionsbecomesexponentiallysensitive. Insuchcasesthereisaneectiverandomnessintheevolvedbehaviour. Nevertheless, thecomputational simulations willstillleadtooutcomesthatwouldbephysicallyallowable,andinthissenseprovideresultsconsistentwiththebehaviourofreality.Computational simulations canhavegreat importanceinmanyareasother than physics, such as with the spread of epidemics, or with economics(where the mathematical ideas of game theory can play an important role),cbutIshall herebeconcernedwithphysical systems, specically. Theim-pressiveness of computational simulations is oftenmost evident whenitissimply17thcenturyNewtonianmechanicsthatisinvolved, initsenor-mouslyvarieddierentmanifestations. TheimplicationsofNewtoniandy-namical laws canbe extensivelycomputedinthe modellingof physicalsystems, evenwheretheremaybehugenumbersof constituentparticles,suchasatomsinasimpliedgas,orparticle-likeingredients,suchasstarsinglobular clusters or eveninentiregalaxies. It mayberemarkedthatbseeCalude.cVelupillai.August30,2012 13:35 WorldScienticReviewVolume-9inx6in-8306AComputableUniverse AComputableUniverseForeword xvcomputationalsimulations are normallydone ina time sense where the fu-ture behaviour is deduced from an input which is taken to be in the past. Inprinciple, onecouldalsoperformcalculationsinthereverse teleologicaldirection, becauseof thetime-reversibilityof thebasicNewtonianlaws.dHowever, becauseof thesecondlawof thermodynamics, wherebytheen-tropy(orrandomness) of aphysical systemincreases withtimeinthenaturalworld,suchreverse-timecalculationstendtobeuntrustworthy.When Newtonian laws are supplemented by the Maxwell-Lorentz equa-tions, governingthebehaviourofelectromagneticeldsandtheirinterac-tionswithchargedmaterialparticles,thenthescopeofphysicalprocessesthatcanbeaccuratelysimulatedbycomputational proceduresisgreatlyincreased,such as with phenomena involving the behaviour of visible light,or with devices concerned with microwaves or radio propagation, or in mod-elling the vast galactic plasma clouds involving the mixed ows of electronsand protons in space,which can indeed be computationally simulated withconsiderablecondence.Thislatterkindofsimulationrequiresthatthosephysicalequationsbeused, that correctly come from the requirements of special relativity, whereEinsteins viewpoint concerning the relativity of motion and of the passageoftimeareincorporated. Einsteinsspecialrelativityencompassed,encap-sulated, andsupersededtheearlierideasofFitzGerald, Lorentz, Poincareandothers, butevenEinsteinsownviewpointneededtobereformulatedand made more satisfactory by the radical change of perspective introducedby Minkowski, who showed how the ideas of special relativity come togetherinthenatural geometrical frameworkof 4-dimensional space-time. WhenitcomestoEinsteinsgeneral relativity, inwhichMinkowskis4-geometryis fundamentallymodiedtobecomecurved, inorder that gravitationalphenomenacanbeincorporated, wendthatsimulationsofgravitationalsystems can be made to even greater precision than was possible with New-toniantheory. Theprecisionof planetarymotionsinourSolarSystemisnow at such a level that Newtons 17th century theory is no longer sucient,and Einsteins 20th century theory is needed. This is true even for the oper-ation of the global positioning systems that are now in common use,whichwouldbeuselessbutforthecorrectionstoNewtoniantheorythatgeneralrelativityprovides. Indeed, perhapsthemostaccuratelyconrmedtheo-reticalsimulationseverperformed,namelythetrackingofdoubleneutron-starmotions, wherenotonlythestandardgeneral-relativisticcorrectionsdBeavers.August30,2012 13:35 WorldScienticReviewVolume-9inx6in-8306AComputableUniverse AComputableUniversexvi R.Penrose(perihelionadvance, rotational frame-draggingeects, etc.) toNewtonianorbital motion need to be taken into account, but also the energy-removingeectsof gravitational waves(ripplesof space-timecurvature)emanatingfromthesystemcanbetheoreticallycalculated, andarefoundtoagreewiththeobservedmotionstoanunprecedentedprecision.Theothermajorrevolutioninbasicphysicaltheorythatthe20thcen-turybroughtwas, ofcourse, quantummechanicswhichneedstobecon-sideredinconjunctionwithitsgeneralizationtoquantumeldtheory,thisbeingrequiredwhentheeectsof special relativityhavetobetakenintoaccount together withquantumprinciples. It is clear frommanyof thearticlesinthisvolume, thatquantumtheoryis(rightly)consideredtobeof fundamental importance, when it comes to the investigation of the basicunderlyingoperationsof thephysical universeandtheirrelationtocom-putation. There are manyreasons for this, anobvious one being thatquantumprocessesareundoubtedlyfundamental tothebehaviourof thetiniest-scaleingredientsof ouruniverse, andalsotomanyfeaturesof thecollectivebehaviourof many-particlesystems, thesehavingacharacteris-ticallyquantum-mechanicalnaturesuchasquantumentanglement,super-conductivity, Bose-Einsteincondensation, etc. However, thereisanotherbasicfeatureof quantummechanicsthatmaybecountedasareasonforregardingthis scheme of things as beingmore friendlytothe notionofcomputationthanwas classical mechanics, namelythat there is abasicdiscreteness thatquantummechanicsintroducesintophysical theory. Itseems that in the early days of the theory,much was made of this discrete-ness,withitsimpliedhopeofa granular natureunderlyingtheoperationof thephysical world. Ahopehadbeenexpressed30,32thatsomehowthedominationof physical theorybytheideasof continuity anddierentia-bilitywhichgohand-in-handwiththepervasiveuseof thereal-numbersystemmighthaveatlastbeenbroken, viatheintroductionofquantummechanics. Accordingly, itwashopedthattheideasof discretenessandcombinatoricsmight soonbeseentobecomethedominant drivingforceunderlyingtheoperationof ouruniverse, ratherthanthecontinuityanddierentiabilitythatclassicalphysicshaddependeduponforsomanycen-turies. Adiscreteuniverseisindeedmuchmoreinharmonywithcurrentideasofcomputationthanisacontinuousone,andmanyofthearticlesinthis volumeeargue powerfully from this perspective, and particularly in thecontextofcellularautomataf.eBolognesi,Chaitin,Wolfram,Fredkin,andZenil.fMartnez,Margenstern,Sutner,WiedermannandZuse.August30,2012 13:35 WorldScienticReviewVolume-9inx6in-8306AComputableUniverse AComputableUniverseForeword xviiTheverynotionof computability thatarosefromtheearly20thcen-turyworkof various logicians Godel, Church, Kleene andmanyothers,harkingbackeventothe19thcenturyideasofCharlesBabbageandAdaLovelace,gand which was greatly claried by Turings notion of a computingmachine, and by Posts closely related ideas, indeed depend on a fundamen-taldiscretenessofthebasicingredients. Thevariousverydierent-lookingproposalsforanotionofeectivecomputabilitythattheseearly20thcen-turylogiciansintroducedallturnedouttobeequivalent tooneanother,afactthatiscentral toourcurrentviewpointconcerningcomputation, andwhichprovides us withtheChurchTuringthesis, namelythat this pre-cisetheoreticalnotionof computability doesindeedencapsulatetheideaof whatweintuitivelymeanbyanidealized mechanical procedure. Wendthisissuediscussedatsomedepthbynumerousauthorsinthisvol-umeh. Formyownpart, IamhappytoaccepttheChurchTuringthesis,in this original sense of this phrase, namely that the mathematical notion ofcomputabilityas dened by what can be achieved by Churchs -calculus,or equivalently by a Turing machineis indeed the appropriate ideal math-ematical notionthat we require for our considerations of computability.Whetherornottheuniverseinwhichweliveoperatesinaccordancewithsuch a notion of computation is then an issue that we may speculate about,orreasonaboutinonewayoranother(see,forexample,Refs.20,45).Nevertheless, Icanappreciatethatthereareotherviewpointsonthis,andthat somewouldprefer todenecomputationinterms of what aphysical object can(inprinciple?) achievei. Tome, however, this begsthequestion,andthissamequestioncertainlyremains,whichevermaybeourpreferenceconcerningtheuseoftheterm computation. Ifweprefertouse thisphysicaldenition, thenall physical systemscomputebydenition, andinthatcasewewouldsimplyneedadierentwordforthe(original Church-Turing) mathematical concept of computation, sothattheprofoundquestionraised, concerningtheperhapscomputablenatureof thelaws governingtheoperationof theuniversecanbestudied, andindeedquestioned. Accordingly,I shallhere use the termcomputation inthismathematical sense, andIaddressthisquestionofthecomputationalnatureofphysicallawsinaseriouswaylater.Returning,now,totheissueofthediscretenessthatcamethroughtheintroduction of standard quantum mechanics, we nd that the theory, as wegDeMol,Sieg,Sutner,SwadeandZuse.hDeMol,Sieg,Dershowitz,Sutner,BauerandCooper.iDeutsch,Teuscher,BauerandCooper.August30,2012 13:35 WorldScienticReviewVolume-9inx6in-8306AComputableUniverse AComputableUniversexviii R.Penroseunderstandittoday, hasnot developedinthisfundamentallydiscretedi-rection that would have tted in so well with our ideas of computation. Thediscreteness that Max Planck revealed, in 1900, in his analysis of black-bodyradiation (although not initially stated in this way) was in eect a discrete-ness of phase spacethat high-dimensional mathematical space where eachspatialdegreeoffreedom,inamany-particlesystem,isaccompaniedbyacorresponding momentum degree of freedom. This is not a discreteness thatcouldapplydirectlytoourseeminglycontinuousperceptionsofspaceandtime. Nonetheless, variouscontributorstothisvolumejhaveventuredinthat more radical direction, arguing that some kind of discreteness might berevealedwhenwetrytoexaminespatialseparationsofaroundthePlancklengthlP(approximately1035m)andtemporalseparationsofaroundthePlancktime(approximately1043s). Theseseparationsareabsurdlytiny,smallerbysome20ordersof magnitudefromscalesof distanceandtimethatarerelevanttotheprocessesofstandardparticlephysics. SincethesePlanckscalesareenormouslyfarbelowanythingthatmodernparticleac-celeratorshavebeenabletoexplore, it canbereasonablyarguedthatagranularityintheverystructureof space-timeoccurringattheabsurdlytinyPlanckscaleswouldnothavebeennoticedincurrentexperiments. Inadditiontothis, ithaslongbeenarguedbysometheoreticians, mostno-tably by the distinguished and highly insightful American physicist John A.Wheeler,42thatourunderstandingofhowaquantum-gravitytheoryoughttooperate(accordingtowhichtheprinciples of quantummechanics areimposeduponEinsteinsgeneraltheoryofrelativity)tellsusthatwemustindeedexpectthatatthePlanckscalesofspace-time,somethingradicallynewoughttoappear,wherethesmoothspace-timepicturethatweadoptin classical physics would have to be abandoned and something quite dier-entshouldemergeatthislevel. Wheelersargumentbasedonprinciplescomingfromconventional ideasof howHeisenbergsuncertaintyprinciplewhenappliedtoquantumeldsinvolves us inhavingtoenvisage wildquantumuctuations thatwouldoccuratthePlanckscale,providinguswithapictureof aseethingmessof topological uctuations. Whilethispicture is not at all similar to that of a discrete granular space-time, it is atleast supportive of the idea that something very dierent from a classicallysmooth manifold ought to be relevant to Planck-scale physics, and it mightturnoutthatadiscretepictureisreallythecorrectone. ThisisamatterthatIshallneedtoreturntolaterinthisForeword.jBolognesiandLloyd.August30,2012 13:35 WorldScienticReviewVolume-9inx6in-8306AComputableUniverse AComputableUniverseForeword xixWhen it comes to the simulation of conventional quantum systems (notinvolvinganythingofthenatureofPlanck-scalephysics)then, aswasthecasewithclassical systems, wendthatweneedtoconsiderthesmoothsolutionsof a(partial)dierential equationinthiscasetheSchrodingerequation. Thus, justaswithclassical dynamics, wecannotdirectlyapplytheChurchTuringnotionofcomputabilitytotheevolutionofaquantumsystem, anditseemsthatwearedriventolookforsimulationsthataremere approximations tothe exact continuous evolutionof Schrodingerswavefunction. Turinghimself wascareful toaddressthiskindof issue,39whetheritbeintheclassicalorquantumcontext,andheargued,ineect,that discrete approximations when they are not good enough for some par-ticular purpose can always be improved upon while still remaining discrete.It isindeedoneof thekeyadvantagesof digital asopposedtoanaloguerepresentations, thatanexponential increaseintheaccuracyof adigitalsimulationcanbeachievedsimplybyincorporatingadditional digits. Ofcourse,thesimulationcouldtakemuchlongertorunwhenmoredigitsareincludedintheapproximation,buttheissuehereiswhatcaninprinciplebe achieved by a digital simulation rather than what is practical. In theory,so the argument goes,the discrete approximations can always be increasedinaccuracy, sothatthecomputational simulationsof physical dynamicalprocesscanbeaspreciseaswouldbedesired.Personally,Iamnotfullyconvincedbythistypeofargument,particu-larly when chaotic systems are being simulated. If we are merely asking forour simulations to represent plausibleoutcomes, consistent with all the rel-evantphysicalequations,forthebehaviourofsomephysicalsystemunderconsideration,thenchaoticbehaviourmaywellnotbeaproblem,sincewewould merely be interested in our simulation being realistic, not that it pro-duces the actual outcome that will in fact come about. On the other hand,ifasinweatherpredictionitisindeedrequiredthatoursimulationem-phis to provide the actual outcome of the behaviour of some specic systemoccurringintheworldthatweactuallyinhabit, thenthisisanothermat-teraltogether, andapproximationsmaynotbesucient, sothatchaoticbehaviourbecomesagenuinelyproblematicissue.kItmaybenoted,however,thattheSchrodingerequation,beinglinear,doesnot,strictlyspeaking,havechaoticsolutions. Nevertheless,thereisanotionknownas quantumchaos, whichnormallyreferstoquantumsys-tems that are the quantizations of chaotic classical systems. Here the issuekMattersrelevanttothisissuearetobefoundin.12,44,46August30,2012 13:35 WorldScienticReviewVolume-9inx6in-8306AComputableUniverse AComputableUniversexx R.Penroseof quantumchaos is asubtleone, andisall tiedupwiththequestionof whatwenormallywishtousetheSchrodingerequationfor, whichhastodowiththefraughtissueof quantummeasurement. Whatwendinpractice, in a general wayand I shall need to return to this issue lateristhat theevolutionof theSchrodinger equationdoes not provideus withtheuniqueoutcomethatwendtohaveoccurredintheactualworld,butwithasuperpositionof possiblealternativeoutcomes, withaprobabilityvalue assignedtoeach. The situationis, ineect, nobetter thanwithchaoticsystems, andagainourcomputational simulationscannotbeusedtopredicttheactual dynamical outcomeof aparticularphysical system.As with chaotic systems, all that our simulations give us will be alternativeoutcomesthatareplausibleoneswithprobabilityvaluesattachedandwill not normally give us a clear prediction of the future behaviour of a par-ticular physical system. In fact, the quantum situation is in a sense worsethanwithclassical chaoticsystems, sinceherethelackof predictivenessdoesnotresultfromlimitationsontheaccuracyofthecomputationalsim-ulations that can be carried out, but we nd that even a completely precisesimulationoftherequiredsolutionoftheSchrodingerequationwouldnotenableustopredictwithcondencewhattheactual outcomewouldbe.The unique history that emerges,in the universe we actually experience,isbutonememberofthesuperpositionthattheevolutionoftheSchrodingerequationprovidesuswith.lEventhis precisesimulation isproblematictosomeconsiderablede-gree. We again have the issue of discrete approximation to a fundamentallycontinuous mathematical model of reality. But with quantum systems thereis alsoanadditional problemconfrontingprecisesimulation, namelythevastsizeof theparameterspacethatisneededfortheSchrodingerequa-tionofamany-particlequantumsystem. Thiscomesaboutbecauseofthequantumentanglements referredtoearlier. Everypossibleentanglementbetweenindividual particles of the systemrequires aseparate complex-numberparameter, sowerequireaparameterspacethatisexponentiallylarge, intermsof thenumber of particles, andthisrapidlybecomesun-manageableifwearetokeeptrackofeverythingthatisgoingon. ItmaylThe questionmaybe raisedthat the seemingrandomness that arises inchaoticclassical dynamics might betheresult of adeeper quantum-level actual randomness.However, this cannot be the full story, since quantumrandomness also occurs withquantizedclassical systemsthatarenotchaotic. Nevertheless, onemaywell speculatethatinthenon-linearmodicationsofquantummechanicsthatIshallbelaterarguingfor, such a connection between chaotic behaviour and the probabilistic aspects of present-dayquantumtheorycouldwellbeofrelevance.August30,2012 13:35 WorldScienticReviewVolume-9inx6in-8306AComputableUniverse AComputableUniverseForeword xxiwell bethatthefuturedevelopmentofquantumcomputers wouldnditsmainapplicationinthesimulationof quantumsystems. Wendinthiscollection,somediscussionofthepotentialofquantumcomputers,thoughthere no consensus is provided as to the likely future of this interesting areaofdevelopingtechnology.mWe see that despite the discreteness that has been introduced intophysicsviaquantummechanics,ourpresenttheoriesstillrequireustoop-erate with real-number (or complex-number) functions rather than discreteones. Thereare, however, proposals(e.g.4) inwhichthenotionof com-putationis takeninasenseinwhichit applies directly toreal-numberoperations,therealnumbersthatareemployedinthephysicaltheorybe-ingtreatedasreal numbers, ratherthan, say, rational approximationstoreal numbers (suchas nitelyterminatedbinaryor decimal approxima-tions). Inthis way, simulations of physical processes canbecarriedoutwithoutresortingtoapproximations. This, however, canrequirethattheinitial data for a simulation be given as explicitly known functions, and thatmay not be realistic. Moreover, there are various dierentconcepts of com-putabilitywithreal numbers,4,5,28,33,41which, unlikeinthesituationthatarosefordiscrete(integer-valued)variables,wheretheChurch-Turingcon-ceptappearstohaveprovidedasinglegenerallyaccepteduniversalnotionof computation,therearemanydierentproposalsforreal-numbercom-putabilityandnosuchgenerallyacceptedsingleversionappearstobeinevidence. Moreover,we unfortunately nd that,according to a reasonable-lookingnotionof real-number computability, the actionof the ordinarysecond-orderwaveoperatorturnsouttobenon-computableincertaincir-cumstances(seee.g. Refs. 28,29). Whatevertheultimateverdictonreal-number computabilitymight be, it appears not tohavesettleddowntosomethingunambiguousasyet.Thereisalsothequestionof whetheranexacttheoryof real-numbercomputabilitywouldhavegenuinerelevancetohowwemodelthephysicalworld. Sinceour measurements of realityalways containsomeroomforerrorwhetherthisbeinalimittotheprecisionof ameasurementorinaprobability that adiscreteparameter might takeoneor another value(assometimesisthecasewithquantummechanics)itisuncleartomehowsuchanexacttheoryofreal-numbercomputabilitymightholdadvan-tagesoverourpresent-day(ChurchTuring)discrete-computational ideas.Although the present volume does not enter into a discussion of these mat-mSchmidhuber,Lloyd,Zukowski.August30,2012 13:35 WorldScienticReviewVolume-9inx6in-8306AComputableUniverse AComputableUniversexxii R.Penroseters,Idoindeedbelievethattherearesignicantquestionsofimportanceherethatshouldnotbeleftaside(forexample,see5,20,29,41).Several articles inthis volumeaddress theissueof whether, insomesense, theuniverseactuallyis acomputer.nTome, this seems tobeasomewhatstrangeidea. AlthoughI canmore-or-lessunderstandwhatitmightmeanforittobepossibletohave(theoretically)acomputationalsimulationofalltheactionsofthephysicaluniverse,owhichinvolvessomesort of constructivist assumptionpfor the operation of the physical world,Inditmuchlessclearwhatitmightmeanfortheuniversetobeacom-puter. Various images come to mind, maybe suggested by how one choosestopictureamodernelectroniccomputerinoperation. Ourpicturemightperhaps consist of anumber of spatiallyseparatednodesconnectedtooneanotherbyasystemof wires,wheresignalsofsomesorttravelalongthewires,andsomeclear-cutrulesoperateatthenodes,concerningwhatoutput is toarisefor eachpossibleinput. Therealsoneeds tobesomekindof direct access toaneectivelyunlimitedstoragearea(this beingan essential part of the Turing-machine aspirations of such a computer-likemodel). However, suchadiscretepictureandaxedcomputergeometrydoes not very much resemble the standard present-day models that we haveof thesmall-scaleactivityof theuniverseweinhabit. Thediscretenessofthispictureisperhapsalittleclosertosomeofthetentativeproposalsforadiscretephysical universe, suchasthe causal setsqthatIshall brieyreturntolater, whichrepresentsomeattemptsatradical ideasforwhatspace-timemightbe like atthePlanckscale.Yet,therearesomepartialresemblancesbetweensuchacomputer-likepictureandour(verywellsupported)present-dayphysicaltheories. Thesetheories involve individual constituents,referred to as quantum particles,where each would have a classical-level description as being spatiallypoint-likethough persisting in time, providing a classical space-time picture ofa 1-dimensional world-line. If these world-lines are to be thought of as thewires intheabovecomputer-inspiredpicture, thenthe nodes couldbethoughtofastheinteractionplaces(orintersectionpoints)betweendier-entparticleworld-lines. Thiswouldbenotaltogetherunlikethecomputerimage described above, though in standard theory, the topological geometryoftheconnectionsofnodesandwireswouldbepartofthedynamics,andnLloyd,Deutsch,TurnerandZuse.oBolognesiandSzudzik.pBauer.qBolognesi.August30,2012 13:35 WorldScienticReviewVolume-9inx6in-8306AComputableUniverse AComputableUniverseForeword xxiiinotxedbeforehand. Perhapsthelackofaxedgeometryoftheconnec-tionswouldprovideapicturemoreliketheamorphous typeof computerstructurealsoconsideredinthisvolume,rthanaconventional computer.However,itisstillnotclearhowthe directaccesstoaneectivelypoten-tiallyunlimitedstoragearea istoberepresented. Moreseriously, thisismerely the classical picture that is conjured up by our descriptions of small-scale particle activity,where the quantumpicture would consist (more orless)of asuperpositionof all theseclassical pictures, eachweightedbyacomplexnumber. Sucha picture perhapsgetsalittleclosertothewaythataquantumcomputersmightberepresented, butagaintherearethecrucial issuesraisedbythetopologyof theconnectionsbeingpartof thedynamicsandtheabsenceof an unlimitedstoragearea, inthephysicalpicture, which seem to me to represent fundamental dierences between ouruniversepictureandaquantumcomputer. Inadditiontoallthis,thereisagainthematterofhowonetreatsthecontinuuminacomputationalway,whichinquantum(eld)theoryismoreproperlythecomplexratherthantherealcontinuum. Over-ridingallthisisthematterofhowoneactuallygets information out of a quantum system. This requires an analysis of themeasurementproblemthatIshallneedtocometoshortly.I think that, all this notwithstanding, when people refer to the universebeing acomputer, theimagethattheyhaveisnotnearlysospecicasanything like that suggested above. More likely, for ourcomputer universetheymightsimplyhaveinmindthatnotonlycantheuniversesactionsbe precisely simulated in all its aspects, but that it has nootherfunctionalquality to it, distinct from this computational behaviour. More specically,for our computer universe therewouldbelikelytobesomeparametert (presumablyadiscreteone, whichcouldberegardedas takingintegervalues) whichis todescribe the passage of time (not averyrelativisticnotion!),andthestateoftheuniverseatanyonetime(i.e. t-value)wouldhavesomecomputationaldescription,andsocouldbecompletelyencodedbyasinglenatural numberSt. ItwouldbetheuniversesjobtocomputeSt fromStwhenevert

>t, andtheuniversewouldbeconsideredtobeacomputerprovidedthatnotonlyisitablealwaystoachievethis,butmoreimportantlythatthisisthesolefunctionoftheuniverse. Itseemstomeif,ontheotherhand,theuniversehasanyadditionalfunction,suchastoassignarealitytoanyaspectof thisdescription, thenitwouldnotsimply be a computer, but it would be something more than this, succeedingrHewitt,Teuscher,MargensternandWiedermann.sSchmidhuber,Margenstern,Zukowski.August30,2012 13:35 WorldScienticReviewVolume-9inx6in-8306AComputableUniverse AComputableUniversexxiv R.Penroseinprovidingus withsome kindof ontology that goes beyondthe merecomputationaldescription.ToconcludethisForeword, I wishtopresentsomethingthatismuchmoreinlinewithmyownviewsastotherelationbetweencomputationandthenatureof physical reality. Tobeginwith, Ishouldperhapspointout that my views have evolved considerably over the decades, but withoutmuchinthewayof abruptchanges. EarlyonIhadbeenof afairlyrmpersuasion that there should be a discrete or combinatorial basis to physics,perhapssomewhatalongthelinesexpressedinsomeofthearticlestinthisvolume. In1967ErwinKronheimerandI publishedapaper14,18onthekindof causal sets referredtoearlier inthis Foreword, where the basicrelationshipsbetweentheelementsarethoseof causalityu, mirroringthecausal relations betweenevents incontinuous space-time, but where nocontinuityorsmoothnessisassumed, andwhereonecouldevenenvisagesituationsofthiskindwherethetotalnumberoftheseelementsisnite.AlthoughI alsohaddierent reasons tobeinterestedinspaces withastructuredenedsolelybycausal relationspartlyinviewof theirroleinrelationtosingularitytheorems11,17(for thestudyof blackholes andcosmology)the causality relations not necessarily being tied to the notionof asmoothspace-timemanifold, Ididnothavemuchof anexpectationthatthetruesmall-scalestructureof ouractual universeshouldbehelp-fully described in these terms. I had thought it much more probable that adierentcombinatorialidea,thatIhadbeenplayingwithagooddealear-lier, namelythatofspin-networks (seeRef.19)mighthavetruerelevancetothebasisofphysics(andindeed, muchlater, aversionofspin-networktheorywastoformpartof theloop-variableapproachtoquantumgrav-ity,1althoughtherolethatspin-networksacquireinloop-variablegravityissomewhatdierentfromwhatIhadoriginallyenvisaged).Spin-network theory was based on one of the most striking parts of stan-dard quantum mechanics, where a fundamental notion that is continuous inclassical mechanics,is discrete in quantum mechanics,namely angularmo-mentum (or spin). In fact, many of the most basic and counter-intuitive fea-turesofquantummechanics,suchasdiscretenessand(Bell)non-locality,varemostpowerfullyexpressedintermsofquantum-mechanical spin. Thepuzzlingrelationbetweenthecontinuousarrayof possibilitiesforthedi-rectionof aspinaxisinourclassical space-timepicturesandthediscretetBolognesi,Schmidhuber,Lloyd,Wolfram,Zuse,FredkinandZenil.uBolognesi.vBreuer,Cabello,SchmidhuberandZenil.August30,2012 13:35 WorldScienticReviewVolume-9inx6in-8306AComputableUniverse AComputableUniverseForeword xxv(or granular)natureofthequantumideaof spin-axisdirection hadal-ways maintainedafundamental fascinationfor me. This andthe basicnon-localityof informationinquantummechanicscometogetherinspin-networktheory, wheretheclassical ideaof aspatial direction doesnotarise ina well-denedwayuntil verylarge spin-networkstructures arepresent inorder provideagoodapproximationtothecontinuous sphereofpossiblespatialdirections. Specicmathematicaldevicesforcalculatingthe often extremely complicated expressions were developed, but everythingremainscompletelydiscrete, andcomputational intheconventional senseof theword, continuityarisingonlyinthelimit of largenumbers. Theneedtogeneralizetheideaofspin-networksinorderthatthegeometryof4-dimensional space-time might be described, rather than just the sphere ofspatial directions, nally found some satisfaction in the ideas of twistor the-ory(seeRefs.18,23,Chapter33). Thisprovidedadierentwayoflookingatspace-timegeometryfromwhatisusualbutnowtheideaofdiscrete-nessunderlyingthebasisofphysicsbegantofade,andbecamesupersededbythemagicofcomplexgeometryandanalysis.One normally thinks of the space-time 4-manifold as being composed ofevents(i.e. space-time points), which are the basic elements of the geome-try. Instead, twistor theory takes its basic elements to be modelled on entirehistoriesof masslessspinningparticlesinfreeight. Byacareful combi-nationof ideasfromspace-time4-geometryandthequantum-mechanicalstructureofrelativisticangularmomentumformasslessparticles,thecon-ceptof twistoralgebra wasdeveloped.18,19Inspecialrelativity,thebasicconcept of a twistor, which describes the kinematical structure of a spinningmasslessparticle, ndsitsmathematical descriptionasanelementof thecomplex 4-vector space T referred to as twistor space. The geometry of Trelates to the real geometry of Minkowski 4-space M by means of an explicitgeometrical correspondence, relating M directly to the complex 3-geometryof the projectivetwistor space PT. It turns out that the complex numbers ofquantum mechanics dovetail with those of the complex geometry of twistortheory in surprising ways, and that there is an intriguing interplay betweenthe non-locality that naturally arises in the twistor description of quantumwavefunctionsandthenon-localitythatweactuallyndinquantumphe-nomena.24Inrecentyears, twistortheoryhasfoundconsiderablevalueinthe calculation of high-energy scattering processes, where the rest-masses oftheparticlesinvolvedcanbeignored,(See,forexample,Ref.2)butmanyofthedeeperissuesconfrontingtwistortheoryremainunresolved.It has always been an aim of twistor theory (still only partially fullled)August30,2012 13:35 WorldScienticReviewVolume-9inx6in-8306AComputableUniverse AComputableUniversexxvi R.Penrosethatitshouldformavehicleforthenatural unicationof quantumme-chanicswithgeneral relativity. Bythis, Idonotmean quantumgravityin the conventional sense in which this term is used. What is usually meantbyquantumgravityissomeschemeinwhichtheideasofEinsteinstheoryofgravitynamelygeneralrelativity(orelseperhapssomemodicationofEinsteins theory)is brought under the umbrella of quantum eld theory.Thisviewpointistotakethelawsofquantumeldtheoryasbeinginvio-late, and that the ideas of general relativity must yield to those of quantumtheoryviasomeappropriateformof quantization. Myownviewhasal-ways been dierent from this, as I believe that quantum theory itself, quiteapartfromitsneedtobeuniedwithgeneralrelativitytheory,isbasicallyself-inconsistentand that some help is needed from outsidethe normal rulesofquantum(eld)theory. Theviewhereisthattheunderlyingprinciplesofgeneralrelativityshouldhelptosupplythisoutsideassistance.This inconsistency is a very fundamental one, and is in a clear sense com-pletelyobvious(the elephantintheroom!) asweshallsee. Asremarkeduponearlier,wetaketheevolutionofaquantumsysteminisolationtobegovernedbytheSchrodingerequationor,inmoregeneralterms,unitaryevolutionand for which I use the symbol U. But, as was remarked uponearlier, the reality of the world that we actually observe taking place aboutus tends notto be described directly by the solution of this equation thatwegetbythisU-evolution, butwhenanobservationor measurement isdeemedtohavetakenplace,isconsideredto jump tojustonememberrofafamilyofsuperposedalternativesolutions = 11 +22 +. . . +nn(1)where the respective squaredmoduli of the complex-number weightings1, 2, . . . , n,supplytherespectiveprobabilitiesofeachrbeingthere-sult(thequantitiesrbeingassumedtobeall normalizedandmutuallyorthogonal). The evolutionprocess wherebyisreplacedbythepartic-ularrthathappenstocomeaboutisthereductionofthestate(collapseofthewavefunction)andIdenotethisprocessbytheletter R.wOfcourse, therewill bemanysuchdecompositions, foragiven, de-pendingonthechoiceof basis thatissupposedtobedeterminedbythechoiceof measuringdevice. Indeed, wemustallowthatthismeasuringwInVonNeumannsclassicbookMathematical Foundationsof QuantumMechan-ics,16heintroduced R and U undertherespectivenames processI and processII.August30,2012 13:35 WorldScienticReviewVolume-9inx6in-8306AComputableUniverse AComputableUniverseForeword xxviideviceisalsopartoftheentiresystemunderconsideration,andsoshouldhaveaquantumstatethatbecomesentangledwiththequantumsystemunderexamination. Neverthelessthereisstilltakentobea jump inthesystemasawholeassoonasthemeasurementisconsideredtohavebeenmade,wherethedierent pointerstates ofthedeviceareentangledwiththedierentpossiblersthatcanresult. Itisobviousthatthis jumpingfromthestateofthesystem(consistingofboththemeasuringdeviceandsystemunderexamination, togetherwiththeentirerelevantsurroundingenvironment),frombeforemeasurementtoaftermeasurement,isnormallynotevencontinuous,let alone a solutionofthe Schrodinger equation: so RblatantlyviolatesU(inalmostallcircumstances).Whydophysicistsnotnormallyconsiderthistobeacontradictioninquantummechanics? Therearemanyresponses, usuallyinvolvingsomesubtleissueofinterpretation, accordingtowhichphysicists trytocir-cumventthis(seeming?) contradiction. Hereiswherethe many-worldsviewpointofHughEverettIIIisofteninvoked,6,8wherebyitisconsideredthatall alternativeoutcomessimply(!) co-exist inquantumsuperposition,andthatitisperhapssomehowafeatureofourconsciousperceptionpro-cessesthatwealwaysperceiveonlyoneofthesealternatives. Despitethisideaspopularityamongmanyphilosophicallymindedphysicists(orphysi-cally well-educated philosophers),I nd this viewpoint very unsatisfactory.I would agree that it is indeed where we are led, if we regard the U-processasinviolate,buttomethisistobetakenasareductioadabsurdumandaclearindicationthatweneedtoseekanimprovementincurrentquantummechanics. Toputthisanotherway,evenifthemany-worldsviewpointisinsomesense correct,itisstillinadequateasadescriptionofthephysi-calworld,forthesimplereasonthatitdoesnot,asitstands,describetheworldthatweactuallyobserve,inwhichwe ndthatsomethingextremelywell approximatedbytheR-process actually takes placewhenquantumsuperpositionsofstatesthataresucientlydierentfromoneanotherareinvolved.WhatdoImeanby sucientlydierent?Itisclearthatmerephysi-cal distanceapart,forthedierentmaterialdisplacementsinvolvedinthesuperposed states, is not the correct criterion, because there have been well-conrmed experiments in which photon states tens of kilometres apart stillmaintain their quantum entanglements with one another, so that their var-iouspossibledierentpolarizationstatesremaininquantumsuperpositionwitheachotherevenoversuchdistances.35However,therearereasonstoexpect,fromvariousfoundationalprinciplesofEinsteinsgeneraltheoryofAugust30,2012 13:35 WorldScienticReviewVolume-9inx6in-8306AComputableUniverse AComputableUniversexxviii R.Penroserelativity, (see7,21,25) that when mass displacements between two quantum-superposedstatesgetlarge,thensuchsuperpositionsbecomeunstableandoughttodecay, inaroughlycalculabletime, intooneortheother, sothatclassicalbehaviourbeginstotakeoverfromquantumbehaviour. Theestimateofisgivenbytheformula

EG(2)where EG is the gravitational self-energy of the dierencebetween the massdistributionsineachoftwoquantumstatesunderconsideration, eachbe-ingassumedtoconstituteastationarystateif onitsown. Suchadecaywouldrepresent adeviationfromthestandardlinearity of Uandmightperhaps evenbetheresult of somekindof chaoticbehaviour arisinginsomenon-lineargeneralizationof present-dayquantummechanics. Thereare experiments currently under development that are aimed at testing thisproposal, and we may perhaps anticipate results over the next several years(seeRef.15).For various reasons, partlyconcernedwiththe quantumnon-localityreferredtoearlier(whichonlybeginstopresentsubstantial problemsforquantum realism when R is involved, treated as a realphenomenon), I wouldexpect this change incurrent quantummechanics torepresent amajorrevolutionandwouldnotbeatall easytoarriveatsimplyby tinkeringwiththeSchrodingerequation. Indeed, myexpectationsarethatsuchatheorywouldhavetobenon-computable insomeverysubtleway. WhyamImakingsuchanassertion? Themainreasonsareratherconvoluted,and I quite understand why some people regard my proposals as somewhatfanciful. Nevertheless, Iamoftheviewthatthereisagoodfoundationalrationale for a belief that something along these lines may actually be true!ThebasicreasoncomesfromGodelsfamousincompletenesstheorems,whichIregardasprovidingastrongcaseforhumanunderstandingbeingsomethingessentiallynon-computable. Thecentralargumentisafamiliarone, andI still ndit dicult tocomprehendwhysomanypeople areunwilling to take on board what would seem to be its fairly clear implicationin this regard. In simple terms, the argument can be applied to our abilitiestodemonstratethetruthofcertainmathematicalpropositionswhichwecantaketobeoftheformof1-sentences. A1-sentenceisanassertionthatsomeproposedTuringcomputationneverterminates(examplesbeingWiless Fermats last theorem and Lagranges theorem that every naturalnumberisthesumoffoursquares). Wemighttrytoencapsulate, withinAugust30,2012 13:35 WorldScienticReviewVolume-9inx6in-8306AComputableUniverse AComputableUniverseForeword xxixsomealgorithmicprocedure A, allpossibletypesofargumentthatcan, inprinciple, beusedtoestablish1-sentences, accordingtohumaninsightandunderstanding. This argument might beaproof withinsomegivenformal system F, where Awouldbeanalgorithmforcheckingwhetheraproposedproofusingtherulesof Fhadbeencorrectlycarriedout, givingtheanswerYESafteranitenumberof stepsif thisisindeedthecase.What the Godel(Turing) theorem shows, in this context, is that if we havetrust in A (and therefore in the soundness of such an F, with regard to 1-sentences)thata proved 1-sentenceisindeedtruewhenever AassertsYES, thenonecanexplicitlyexhibita1-sentence GwhereourtrustinAextendsalsotoatrustinthetruthofG, eventhough Aitselfisshowntobeincapableof directlyestablishingG. Inthecaseof Godelssecondincompleteness theorem, the 1-sentenceG(= G(F)) would be an assertionoftheconsistencyof F, andGwouldbethe1-assertionthatamongthetheoremsof F, therewouldbenonewhosenegationisalsoatheoremofF. Althoughourtrusttellsusthat AwouldbeunabletoestablishG(i.e.theconsistencyofF), ourtrust that Gisactuallytruefollowsfromourtrust in A(whichdepends on Fsconsistencyotherwise Fwouldbeabletoestablish2=3, aconclusionwhichwecertainlywouldnottrust). Ourtrustintheuseof Fasameansofestablishingthetruthof1-sentencesthereforecarriesusbeyondthedirectcapabilitiesof F, andenablesustoassertthatG(F)istrue, onthebasisof thatsametrust, despitethefactthat FdoesnotcontainG(F)amongitstheorems.Thisisbasicallythethrustof Godelsattackonformalism. Althoughtheformalizationof variousareasof mathematicscertainlyhasitsvalue,allowingustothetransferdierentaspectsof humanunderstandingandinsightintocomputational procedures, Godel showsusthattheseexplicitprocedures,onceknownandtrustedcannotcovereverythinginmathe-maticsthatisaccessibletounderstandingandinsight.xAnd, indeed, thisappliesalreadyfortherelativelylimitedareaof1-sentences. Yet, acasecancertainlybearguedthat this does not yet provideademonstrationthathumaninsightis,atroot,anon-algorithmicprocedure,andIlistherewhatappeartobethemainargumentsinsupportofthatcase,i.e. ofcrit-icisms of the above claim that the Godel-type arguments show that humanunderstandingisnon-computational;(1) Errors argumenthuman mathematicians make errors, so rigorousGodel-typeargumentsdonotapply.xZizzi.August30,2012 13:35 WorldScienticReviewVolume-9inx6in-8306AComputableUniverse AComputableUniversexxx R.Penrose(2) Extremecomplicationargumentthe algorithms governing humanmathematical understandingaresovastlycomplicatedthattheirGodelstatementsarecompletelybeyondreach.(3) Ignoranceof thealgorithmargumentwedonotknowthealgo-rithmic process underlying our mathematical understanding, so wecannotconstructitsGodelstatement.I have tried to argue elsewhere20that (1), (2), and (3) do not invalidatetheconclusionthatourconsciousunderstandingsareveryunlikelytobeentirelytheproductofcomputationalactions,anditisnotmypurposetorepeatsuchdetailedargumentshere. NeverthelessIbrieysummarisemycounter-arguments,inwhatfollows.The main point, with regard to (1) is that human errors are correctable.We are not so much concerned with the often erroneous gropings that math-ematiciansemployintheirsearchfortruth,butmoretheidealsthattheygropefor and, moreimportantly, measuretheirachievementsagainst. Itistheirabilitytoperceivetheseidealsthatweareconcernedwith, ifonlyinprinciple,anditisthisabilitytoperceiveidealmathematicaltruththatweareconcernedwithhere, nottheerrorsthatweall makefromtimetotime. (It maybeevident fromthesecomments that I do regardmath-ematical truthespeciallywithregardtomatters sostraight-forwardas1-sentencesassomethingabsolute,andexternaltoourselves. ButIap-preciatethatothersyaresometimeslesssympathetictothiskindofview-point. I do not believe, however, that ones philosophical standpoint in thisrespect signicantly aects the arguments that I am putting forward here.)With regard to (2) the point is somewhat similar. If the algorithms were inprincipleto be known, then their size or complication is of no real concern.Thisappliestoagreatmanymathematical arguments. InEuclidsproofoftheinnityofprimes,forexample,weneedtoconsiderprimesthataresolargethattherewouldbenowaytowritethemdownexplicitlyintheentireobservableuniverse, andtocalculatetheproductofthemall uptosomesuchsizeinpractice,isevenmoreoutofthequestion. Butallthisisirrelevantfortheproof. Similarpointsapplyto(2).Theargument(3)is, however, muchmorerelevanttothediscussion,and was basically Godels own reservation (referred to in the commentariesherez)withregardtomakingthestrongconclusionthatIamarguingforhere. Ratherthanpushingthelogicalargumentfurther,whichiscertainlyyDeMol.zSieg.August30,2012 13:35 WorldScienticReviewVolume-9inx6in-8306AComputableUniverse AComputableUniverseForeword xxxipossibletodo(seeRef.20). Ishallheremerelyindicatetheextraordinaryimprobabilityoftheneededalgorithmicactionarisinginourheads,bytheprocess of natural selection. Such an algorithm would have to have extraor-dinary sophistication, so as to be able to encapsulate, in its eective formalsystem manystepsof Godelization. Asanexample, Ihavepointedoutelsewhere22that whereas Goodsteins theorem,9whosemeaningaais eas-ilyaccessibleeventothosewithlittlemathematicalknowledgeotherthanbasicnumerical notation, hasbeenshownbyKirbyandParis13tobein-accessiblebyrst-orderPeanoarithmetic(withouta Godelization step,thatis), yetthistheoremcanbereadilyseentobetruethroughmathe-matical understanding. Ifourmathematical understandingisachievedbysome(unknowable, butsound)algorithmicprocedure, itwouldbeatotalmysteryhowitcouldhavearisenthroughnatural selection, whentheex-periencesofourremoteancestorscouldhavegainednobenetwhatsoeverfrom having such a sophisticated yet totally irrelevant algorithm planted intheirbrains!If,then,it is accepted that our understanding of mathematics is not analgorithmicprocess,wemustaskthequestionwhatkindofprocesscanitbe?Akeyissue,itseemstome,isthatgenuineunderstanding(atleastinournormalsenseofthisword)issomethingthatrequiresawarenessasitwould seem to me to be a misuse of the word understanding if it could begenuinelyappliedtoanentitythathadnoactualawarenessofthematterunder its consideration. Awareness is the passive form of consciousness, so itseems to me that it was the evolutionary development of consciousness thatis the key, and that such a quality could certainly have come about throughnatural selection, beingable toconfer anenormous selective advantageonthosecreaturespossessingit. Insayingthis, Iamexpressingtheviewthat consciousness is indeed functional and is not anepiphenomenon thatsimplyhappenstoaccompanycertainkindsof cognitiveprocesses. Thisviewiscertainlyanimplicationofthequalityof understanding requiringconsciousawareness,sinceunderstandingiscertainlyfunctional.IshouldmakeclearthatIammakingnoclaimtoknowortobeableto denewhat consciousness actually is, but its role in underlyingunder-standing (whatever that is) seems to me to be of great evolutionary value,and could readily arise as a product of natural selection. I should also makeclearthatIamregardingtheconsciousnessissueasascienticone, andthatIdonot taketheviewthatthesearemattersthatareinaccessibletoaaVelupillai.August30,2012 13:35 WorldScienticReviewVolume-9inx6in-8306AComputableUniverse AComputableUniversexxxii R.Penrosescientic investigation. I also take it that healthy wakeful human brains (aswellaswhateverotherkindofanimalbrainsmayturnoutalsotobesimi-larly capable) are able, somehow, to evoke consciousness by the applicationofthoseverysamephysicallawsthatarepresentthroughouttheuniverse,eventhoughconsciousnessitselfcomesaboutonlyintheveryspecial cir-cumstancesoforganizationthatareneededtopromoteitsappearance.What kind of circumstance could that be, if we are asking for some sortofnon-computableactiontocomeaboutwhenwebearinmindthatthedeterministicdierentialequationsofclassicalorquantumphysicsseemtobeofanessentiallycomputablenature?Myresponsetothisqueryisthatthe non-computability must lie in hitherto undiscovered laws that could beof relevancehere. (Iamignoringtheissue, referredtoearlier, of thedis-cretecomputationalsimulationofacontinuousevolution. Yet,Idoacceptthattheremightbesomequestionsof genuinerelevanceherethatoughttobefollowedupmorefully.) AsfarasIcansee, theonlybigunknown,inphysicallaws,thatcouldhavegenuine relevancehere,is the U/Rpuzzleofquantummechanics,referredtoabove. Inalmostallprocessesthattakeplace,wehavenoneedofthepresumedNewTheorythatistogobeyondcurrent quantum mechanics, mainly because its eects would go un-noticed,being swamped by the multifarious random inuences of environmental de-coherence. But, inthebrain, theremight berelevant structuresabletopreserve quantum coherence up to a length of time at which the previouslymentioned /EG criterion actually becomes relevant. Then, the normalpurelyprobabilisticactionthatstandardquantumtheorysR-processpro-videsuswithistobereplacedbysomesubtlenon-computationaldecisionastowhichchoicethestatereductionleadsto. Withasophisticatedbrainorganization, wherethesynapticresponsesaresensitivetothesechoices,wecanimaginethattheoutputofthebraincouldindeedbeusefullynon-computational. This, indeed, is the basis of theorchestratedobjectivereduction (Orch-OR)schemethatStuartHameroandI haveproposedsomeyearsago, wheretheabove relevantstructures wouldbeneuronalmicrotubulesoftheappropriatetype(seeRefs.10,21,27).Itishardlysurprisingthatsuchaproposalhasmetwithsomeconsid-erablescepticism, mainlyfortheveryunderstandablereasonthattohavebody-temperature quantum coherence at anything like the level required isenormouslyfarbeyondtheexpectationsof standardphysical calculationsappliedtosimpliedmodels of cells.34Nevertheless, biological cells are,August30,2012 13:35 WorldScienticReviewVolume-9inx6in-8306AComputableUniverse AComputableUniverseForeword xxxiiiinfact, highlysophisticatedstructures,bbandonemayreasonablyexpectthat, whenthestructuresofcertaincell partsarededicatedintheappro-priate directions, their behaviour might exhibit quite unusual quantum-mechanical properties.ccInfact, recentexperimentscarriedoutinJapanbyAnirbanBandyopadhyay3andhis co-workers appear tohavedemon-stratedthat highlyintriguingquantum-coherent eects do actuallytakeplaceinbody-temperatureneuronal microtubules. Theseresults are, asof now, preliminary, buttheydoappeartoprovidesomeencouragementfortheOrch-ORscheme, anditwillbeveryinterestingtoseehowthingsdevelop.Evenifall ofthisisaccepted, wemaystill askwhatwouldbetheuseof alittlebitof non-computableaction, fromtimetotime, fortheoper-ationof thebrain? Indeed, therewouldnotbemuchvalueinthisunlessthequantumcoherenceisofaveryglobal character, involvinglargeareasofthebrain, andtheprocesswouldhavetoactinsomegloballycoherentway. ThisisindeedtheOrch-ORpicture,andwetakeitthatmomentsofconsciousnessoccurwhenstatereductionoccursatmanysites(inmicro-tubules)atonceinanorchestratedway,sothatthesynapsestrengthsareinuencedinmanyplacesandaconcertedinuenceresults, aswouldbeexpectedforconsciousactions. Theresultsofparticularactsofconsciousunderstandingwouldbeunlikelytobeusuallyanythingsimple,andwoulddependupontheexperienceofmemoriesaswellasonlogic. Butthenon-computableingredient is takentobeessential, for thereasons describedabove. According to this view, our conscious actions are calling upon partsof physicsencompassed in a New Theory that is presently unknown in de-tail. The impact of this theory on processes notorganized in this way wouldnotbeevident. Butitwouldmakeitsmarkonsystemssuchaswakefulhealthyhumanbrainswhereitemergesasconsciousactionsandpercep-tions. The non-computable eects of this New Theory would emerge in thiswayandresultinactionsthataredescribedas hypercomputational.How far outside the normal scheme of computational physics would thesehypercomputational actions be? Since the Godelianinsight that allowsus totranscendagiventrustedformal systemFprovides this insight intheformof a1-sentence, namelyG(F), wemightexpectthatwecouldmodel suchhypercomputational actions inthe formof aTuringoracle-machine,ddwheretheoracleis abletoassert thetruthor falsityof 1-bbMargenstern,Ehrenfeuchtetal.,RozenbergandZenil.ccZizzi.ddChaitin,Dershowitz.August30,2012 13:35 WorldScienticReviewVolume-9inx6in-8306AComputableUniverse AComputableUniversexxxiv R.Penrosesentence. Howeverthiswouldnotbesucient(nordoesitappeartobenecessary), as we can apply a Godel-type diagonalization insight again on1-sentence-oraclemachinestotranscendthesedevicesalso. Inarecentarticle,26Iconsideratypeof oraclethatIrefertoasa cautiousoracle,whichisintendedtomodelalittlemorecloselythekindofthingthatonemight consider idealized human mathematicians might be capable of, wherethe cautious oracle canexamine a n-sentence(for any naturalnumber n)andeitherrespond true or false (necessarilytruthfullyineachcase),orelseconfesstobeingunabletosupplyanansweror, failinganyof these,simply continue pondering indenitely without ever providing an answer atall. AgainaGodel-typediagonalizationallowsustheinsighttotranscendanysuchadevices capabilities! Whatever kindof hypercomputationalcapabilitiessucha NewTheory mightconfer,itappearstobesomethingvery subtle. It is some sort of never-ending capability of being able tostandback andcontemplatewhateverstructurehadbeenconsideredpreviously.Thisseemstobeaqualitythatconsciousnessisabletoachieve, buthowone incorporates this kind of thing into a physical theory is hard to imagine,asourpresent-daytheoriesstand.References1. Ashtekar, A. and Lewandowski, J. Background independent quantum gravity:astatusreport.Class.QuantumGrav.21,R53.[gr-qc/0404018],2004.2. Arkani-Hamed, N., Cachazo, F., Cheung, C., andKaplan, J. TheS-MatrixinTwistorSpace.arXiv:0903.2110v2[hep-th],20093. Sahu,S.,Ghosh,S.,Hirata,K.,Fujita,D.andBandyopadhyay,A.Ultrafastmicrotubulegrowththroughradio-frequency-inducedresonantexcitationoftubulinandsmall-moleculedrugs,toappearinNatureMaterials.4. Blum, L. AlanTuringandtheOtherTheoryof Computation(onTuringsRounding-oErrors inMatrixProcesses) inSBarryCooper andJanvanLeeuwen(eds),AlanTuringHisWorkandImpact.Elsevier,2012.5. Bridges, D.S. Can constructive mathematics be applied in physics?J. Philos.Logic28,43953,1999.6. DeWitt, B.S. andGraham, R.D., eds. TheMany-Worlds InterpretationofQuantumMechanics.PrincetonUniv.Press,Princeton,1973.7. Di osi, L. Models for universal reduction of macroscopic quantum uctuations.Phys.Rev.A40,116574,1989.8. Everett, H. Relative Stateformulation of quantummechanics. In J.A.WheelerandW.H.Zurek(eds),QuantumTheoryandMeasurement.Prince-ton Univ. Press,Princeton,1983),originally in Revs.ofModernPhysics,29,45462,1957.August30,2012 13:35 WorldScienticReviewVolume-9inx6in-8306AComputableUniverse AComputableUniverseForeword xxxv9. Goodstein,R.L.Ontherestrictedordinaltheorem.J.SymbolicLogic9,3341,1944.10. Hamero,S.R.andPenrose,R.Consciouseventsasorchestratedspace-timeselections.J.ConsciousnessStudies3,3663,1996.11. Hawking, S.W. andPenrose, R. Thesingularities of gravitational collapseandcosmology,Proc.Roy.Soc.,London,A314,529548,1970.12. Israeli, N. andGoldenfeld, N. Computational irreducibility and the pre-dictabilityofcomplexphysicalsystems.Phys.Rev.Lett.92,074105,2004.13. Kirby, L.A.S. and Paris, J.B. Accessible independence results for Peano arith-metic,Bull.L.ond.Math.Soc.14,28593,1982.14. KronheimerE.H. andPenrose, R. Onthestructureof causal spaces. Proc.Camb.Phil.Soc.63,481501,1967.15. Marshall,W.,Simon,C.,Penrose,R.andBouwmeester,D.TowardsQuan-tumSuperpositionsofaMirror,Phys.Rev.Lett.,Vol.91,Issue13,2003.16. von Neumann, J. Mathematical Foundations of Quantum Mechanics. (Prince-tonUniv.Press,Princeton),1955.17. Penrose, R. Gravitational collapseandspace-timesingularities, Phys. Rev.Lett.14,5759,1965.18. Penrose,R.Twistoralgebra,J.Math.Phys.8,34566,1967.19. Penrose, R. Angularmomentum: anapproachtocombinatorial space-time.In T. Bastin (ed), Quantum theory and Beyond, Cambridge University Press,Cambridge,1971.20. Penrose, R. Shadows of theMind; AnApproachtotheMissingScienceofConsciousness,OxfordUniv.Press,Oxford,1994.21. Penrose, R. Ongravitysroleinquantumstatereduction. Gen. Rel. Grav.28,581600,1996.22. Penrose, R. Canacomputerunderstand? InRose, S. (ed), FromBrainstoConsciousness? EssaysontheNewSciencesof theMind, AllenLane, ThePenguinPress,London)154179,1998.23. Penrose, R. The Road to Reality: AComplete Guide to the Laws of theUniverse,JonathanCape,London,2004.24. Penrose, R. Thetwistorapproachtospace-timestructures. InA. Ashtekar(ed), 100Years of Relativity; Space-timeStructure: EinsteinandBeyond,WorldScientic,Singapore,2005.25. Penrose, R. Black holes, quantumtheory and cosmology (Fourth Inter-national WorkshopDICE2008) J. Physics, Conf. Ser. 174, 012001. doi:10.1088/1742-6596/174/1/012001,2009.26. Penrose, R. On attempting to model the mathematical mind, in Cooper, B.S.andHodges,A.(eds),TheAlanTuringYear-TheOnceandFutureTuring,CambridgeUniversityPress,2012.27. Penrose, R. andHamero, S. ConsciousnessintheUniverse: Neuroscience,Quantum Space-Time Geometry and Orch OR Theory. Journal of Cosmology,Vol.14,2011.28. Pour-El, M.B. andRichards, I. Thewaveequationwithcomputableinitialdatasuchthat its unique solutionis not computable, Adv. inMath. 39,215239,1981.August30,2012 13:35 WorldScienticReviewVolume-9inx6in-8306AComputableUniverse AComputableUniversexxxvi R.Penrose29. Pour-El,M.B.andRichards,I.ComputabilityinAnalysisandPhysics.Per-spect.Math.Logic,(Springer-Verlag,Berlin,Heidelberg),206pp.,2003.30. Russell,B.TheAnalysisofMatter(AllenandUnwin;reprinted1954,DoverPubl.Inc.,NewYork),192731. Sakharov, A.D. VacuumQuantumFluctuations inCurvedSpaceandTheTheoryof Gravitation, Sov. Phys. Dokl., 12, 1040[Dokl. Akad. NaukSer.Fiz.177,70].Reprinted: (2000)Gen.Rel.Grav.,32365367,1968.32. Schrodinger, E. ScienceandHumanism: PhysicsinOurTime.(CambridgeUniv.Press,Cambridge),1952.33. Stannett,M. Computation and Hypercomputation. MindsandMachines,13,11553,2003.34. Tegmark, M. Importanceof quantumcoherenceinbrainprocesses, Phys.Rev.E,61,pp.41944206,2000.35. Tittel,W.,Brendel,J.,Gisin,B.,Herzog,T.,Zbinden,H.,andGisinN.Ex-perimental demonstrationof quantum-correlationsovermorethan10kilo-metersarXiv:quant-ph/9707042v3,2008.36. Turing,A.M.Oncomputablenumbers,withanapplicationtotheEntschei-dungsproblem,Proc.Lond.Math.Soc.(ser.2)42,230265;acorrection43,544546,1937.37. Turing, A.M. Computabilityand1-denability. J. Symb. Log., 2, 153163,1937.38. Turing, A.M. Systemsof logicbasedonordinals. P. Lond. Math. Soc., 45(2),161228,1939.39. Turing, A.M. Intelligentmachinery, withAMScorrectionsandadditions.Pages numbered137, with2un-numberedpages of references andnotes.Page1hasMSnotebyR.O.Gandy,Turingstypeddraft.n.d.,1948.40. Turing, A.M. Computing machinery and intelligence, Mind 59 no. 236;reprintedinD.R. HofstadterandD.C. Dennett(eds), TheMindsI, BasicBooks,Inc.;PenguinBooks,Ltd;Harmondsworth,Middx.1981,1950.41. Weihrauch, K. ComputableAnalysis: AnIntroduction.TextsinTheoreticalComputerScience,Springer,2000.42. Wheeler, J.A. Geometrodynamics(Societ` a Italiana Fisica: Questioni di sicamoderna,V.1,andAcademicPress,Inc.,NewYork),1982.43. Wheeler, J.A. Lawwithoutlaw, inQuantumTheoryandMeasurement. InJ.A. Wheeler, J.A. and Zurek, W.H. (eds), Princeton Univ. Press, Princeton,pp.182213,1983.44. Wolfram,S.ANewKindofScience.WolframMediaInc,2002.45. Zenil, H. and Delahaye, J.-P. On the Algorithmic Nature of the World. In G.Dodig-CrnkovicandM.Burgin(eds),InformationandComputation,WorldScientic,2010.46. Zenil, H, Soler-Toscano, F., andJoosten, J. J. Empirical EncountersWithComputational Irreducibility and Unpredictability, Minds and Machines, vol.21,2011.August30,2012 13:35 WorldScienticReviewVolume-9inx6in-8306AComputableUniverse AComputableUniversePrefaceSimpliedRoadmapof(Un)computableWorld-ViewsTodayinformationandcomputationplayamajorroleinmodernphysics,bothasasourceof newunifyingtheories, andalsoof soundapproachesto aspects of current mainstream theories such as statistical mechanics andthermodynamics, where it has proven to be of great use. Indeed it is centraltomanyphysicalconceptsnowadays.Compelled as I was by all these questions, it was a privilege tohave the opportunity of co-organising and being involved in severalevents aroundthis topic. In2008, together withAdrianGermanandGerardo Ortiz, we organised the second Midwest NKS Conference atthe University of Indiana, Bloomington, featuring an impressive set ofparticipants andspeakers (see http://www.cs.indiana.edu/~dgerman/2008midwestNKSconference/) includingBennett, Calude, Chaitin, Csic-sery, Deutsch, Fredkin, Grover, Leggett, Lloyd, Rowland, de Ruyter,Szudzik, Tooli and Wolfram. The momentum generated by the conferencecontributedtotherealisationofthisproject,includingthetranscriptionofDeutschscontributiontothisbookandthetranscriptionofthepaneldis-cussiononthesubject, whichis alsoincluded. In2010, withTommasoBolognesi (and mostly thanks to him) the JOUAL (Just One Universal Al-gorithm) Workshop(seehttp://fmt.isti.cnr.it/JOUAL2009/) was or-ganisedin2009, toconsiderquestionsaroundtheconceptsof emergence,space-timeandnatureincomputationalsystems. ItfeaturedRenateLoll,StephenWolframandJuergenSchmidhuber,amongotherspeakers.Animportantquestiontowhichthisvolumemaysuggestanansweriswhethertheseviewsarematureenoughtobeengagedwithanddiscussedatlengthandindepth. Ihavehadtheprivilegeofbeingabletoleadtheeort to undertake such a challenge, and the result, I think, is a comprehen-sive volume in which most, if not all current trends are represented in somefashion. InpreparingthisvolumeIhavemadesuretoincludedissentingxxxviiAugust30,2012 13:35 WorldScienticReviewVolume-9inx6in-8306AComputableUniverse AComputableUniversexxxviii Prefacevoices, theviewpointsof thosenotinagreementwiththemainthesisofthisvolume(anontological view, orapragmaticapproachtoa(Turing)computable universe), notably dissenting voices from the important eld ofquantum mechanics (beginning with Penrose, who has written the Forewordtothisvolume, andincludingaswell Zizzi, Lloyd, DeutschandCabello).Also included are those embracing some notion of hypercomputation in onewayoranotherandundersometermoranother(Doria, Cooper, Penroseand Zizzi again), and thinkers representing a novel trend, proponents of analgorithmicallyrandomworld(Calude, Meyerstein, SalomaaandSvozil),whichhappenstobethediametricaloppositeofmyownalgorithmicview.In an eort to provide a useful roadmap to these viewpoints, I have groupedthemintoafewcategories,fullyrealisingthatIruntheriskofoversimpli-cation. Someof thesecategoriesopposeeachotherorareabifurcationof alargercategory: e.g. digital vs. quantum, deterministicvs. random.This is onlymypersonal simpliedaccount, andbynomeans necessar-ilyrepresentstheviewstheseauthorswouldentertain, eitheroftheirownworkoroftheworkofothers. Thisismerelyintendedtohelpthereadercompare viewpoints, and perhaps orient him- or herself vis-`a-vis the severalhypotheses: The(Turing)ComputableUniverseHypothesis.(orsomeformofcomputationalism) (e.g. Schmidhuber, Hutter). Also DigitalPhysics Hypothesis. Supportedbythe various versions (exceptperhaps the original one) of the Church-Turingthesis as estab-lishedbyKleene. Oftenepistemologicalinnature,contrarytothecommon belief, it advances the idea that the universes upper com-putational power is that of Turing universality, which doesnt meanby any means that these authors advance the idea that the world isa(universal)Turingmachine. Itcangofromthepureontologicalposition(e.g. Wolframor Bolognesi aimingat providingabasisforphysicsasanemergentpropertyof reality, orFredkin(inher-itedfromthe CellularAutomatonHypothesis subcategory))totheepistemological formalism(e.g. Szudziks). Inthis categorypositionssuchasWheelers(itfrombit)andperhapsFeynmanswould be placed. One thing is certain under this category, that na-tureiscapableofTuringcomputationasattestedbytheexistenceof digital computersandnatureseemstobehavelikeif computa-tionalismweretrueaswehavemanagedtocapturemostnaturalphenomenainincreasinglyencompassingtheoriesdescribinglargeAugust30,2012 13:35 WorldScienticReviewVolume-9inx6in-8306AComputableUniverse AComputableUniversePreface xxxixpartsoftheworldbehaviour. The Cellular AutomatonHypothesis. Asub-categoryrstsuggestedbyZuseandthenadoptedandfurther developedbyFredkinunder his programof Digital Philosophy. Withtraditionally little support but proven to provide foundationalconcepts for the subeld of physicsofcomputation(e.g. ques-tionsrelatedtologicalandphysicalreversibility). TheMathematical StructureHypothesis. Asub-categoryoftheComputational Hypothesis. SuggestedbyMaxTegmarkandunder theComputableUniverseHypothesis giventhatTegmarkhasmentionedthatbyamathematicalstructurehemeansacomputableone(theuncomputableversioncanbegrouped under the Non-Turing Computable Universe Hypoth-esis). TheInformational UniverseHypothesis. (e.g. Wheeler)Most, ifnot all, authors of models of quantum gravity may fall into this cat-egory, even if the authors may not place or ask themselves whetherthey are doing so, as they place information as the ultimate reality(Zeilingerbeingtheextremecase). OtherauthorssuchasScottAaronsonmayalsofall intothis category, takingquantumme-chanicsasatheoryof unknowns, of probabilitymagnitudes, andultimately(unknown)information. TheComputational PragmaticHypothesis. Modelsbelongingtothis categoryaremostlyagnosticwithregardtoanyontologicalcommitment concerningthe ultimate structure of the world. Itis a weakformof computationalism, heldbyvirtuallymodre-searchers in the practice of science. They are pragmatic in their ap-proach to nature-like phenomena and seek real applications (e.g. inthisvolumeEhrenfeuchtetal. Rozenberg, Martinez, Adamatzky,Teuscher, Velupillai andZambelli). Mostpracticeof scienticre-searchfallsintothiscategoryasphysical lawscanbesolvedwithextraordinary precision up to unknown but increasingly more accu-rate levels, up to the point to believe that we can arrive to a ToE, asingle (in a large sense computable) formula (not necessarily mean-ingcompletepredictabilitypower). Thispragmaticapproachhasturnedtobeunreasonablyusefulinitsexplanatoryandpredictivepower and certainly has propelled both The Informational UniverseandThe(Turing)ComputableUniversehypotheses.August30,2012 13:35 WorldScienticReviewVolume-9inx6in-8306AComputableUniverse AComputableUniversexl Preface TheNon-TuringComputableUniverseHypothesis(aviewrepre-sentedinthisvolumebyDoria, andindierentways, andunderdierentlabelsorapproaches,byCaludeandCooper),whichisinoppositiontotheChurch-Turingthesisinitsvariousversions. Itsproponentseithersupporttheviewthatnatureiscapableof en-coding(andsolving)thehaltingproblemandthereforeembracingtype-ncomputation(orhyper-computation)forn > 1(inKleenenotation),orbelieveintheexistenceofuncomputableregularities,thatis, theyholdthatbysomemeansanirregularity(i.e. anon-computable pattern) can be construed as a regularity, suggestingthatthere exists innature something (e.g. the brain)thatis capa-bleofndingpatternswherecomputerscannot, eveninprinciple(inunboundedtime). Anexample wouldbeseeingaChaitinOmegawithoutthemachinethatgeneratesit, thoughwearenotevencapableofrecognisingpatternsincomputablenumberssuchas). OtherseriousproponentsmayincludescholarssuchasJ.FelixCosta,MarkBurginandSelmerBringsjord. TheRandomUniverseHypothesis. (Calude,Meyerstein,Sa-lomaa, McAllister). Couldbeviewedasasub-branchof theNon-TuringComputableUniverseHypothesis, butwithdif-ferent arguments anda novel approachthat uses the the-oryof algorithmicrandomness, alsopotentiallyembracingastrong interpretation, albeit mainstream, of quantum mechan-ics (Copenhagen), but needs no recourse to quantum mechan-icstomakeitscase. TheNon-ComputableUniverseHypothesis. (e.g. Penrose)Arguments belonging or in combination of the The Non-TuringComputable Universe Hypothesis andthe QuantumHypothesis. Otherpossibleproponents(withvariations)areLucas, Zizzi, andSearle. Opposedtomostotherviewsbutperhaps tangential to either The Non-Turing ComputableUniverseHypothesisandaversionoftheQuantumHypothe-sis,orboth. TheAlgorithmicInformationHypothesis. (Zenil)opposedtotheRandomhypothesis. Itadvancestheexplanationthatmost,ifnotall, thestructureintheworldisdescribedbyalgorithmicproba-bilitywhichinturnisbasedinTuringuniversalcomputationanddescribesthedistributionof patternsinnature. ItcansupersedeAugust30,2012 13:35 WorldScienticReviewVolume-9inx6in-8306AComputableUniverse AComputableUniversePreface xlithe upper level but it is not meant to describe the ultimate buildingblocks of theuniverse(or its computational power) unlikeotherhypotheses, but rather toexplainstructureas theoutput of al-gorithmic(computational) processes. It isalsointendedtobeavalidationtheory, potentiallyprovidingstatistical evidencefor astronger version up to supporting a Computable Universe Hypoth-esis,inparticularaTuringComputableUniverseHypothesis. The Standard Quantum Universe Hypothesis (Lloyd, Deutsch, Ca-bello): Physics andcomplexityfrom/basedinquantummechan-ics, particularly those taking as a departure point the Copen-hagen interpretation of Quantum Mechanics. In the cases of Lloyd,Deutsch, Cabello also acknowledging the importance of informationandcomputationbutontopofquantummechanics.Thevolumeincludeshistorical andphilosophical accounts(Swade, DeMol, Turner), alternative models and approaches to that of Turing (Hewitt,De Mol), profound investigations of the nature of mathematics and compu-tation(Turner, Sieg, DershowitzandFalkowich, Cooper, BauerandHar-rison, Sutner, Beavers, Ehrenfeuchtetal.), pragmaticapproachestonewforms of computation and real-world applications (Wiedermann,Martnez,Margenstern, Teuscher, Velupillai, Zambelli)andof computationinrela-tion to quantum reality (Cabello, Zizzi, Zukowski, Lloyd, Deutsch, Breuer).There is also an illuminating panel discussion on the question to which thebookisdevoted(WhatisComputation? (How)DoesNatureCompute?),featuringasubsetof thecontributorsandafewotherauthors, includingtheNobelprizewinner(Physics)TonyLeggett.HectorZenilSheeld,S.Yorkshire,UK,2012http://www.algorithmicnature.org/zenilAugust30,2012 13:35 WorldScienticReviewVolume-9inx6in-8306AComputableUniverse AComputableUniverseThis page intentionally left blank This page intentionally left blankAugust30,2012 13:35 WorldScienticReviewVolume-9inx6in-8306AComputableUniverse AComputableUniverseAcknowledgementsThevolumeopenswithamasterfulForewordbyRogerPenrose,towhomI amverygrateful, anddelightedtohavebeenabletoworkwith. I amof courseverygrateful totheauthors, all pioneersandbrilliantthinkers.Tohavetheirworkcollectedwithinasinglevolumewasitselfachallenge.Anovelfeatureofthebookisthediscussionsection(arstversionoftheideaforwhichcamefromCrisCalude), whereauthorsengagedinalivelyexchangeof ideasintheformof questionsandanswers, I wanttothankthemforhavingparticipated. Thiswasanenrichingprocess, Iservedattimesasalegitimateinquirerandatothertimesaskingquestionswhoseanswers bothI myself andsome of the authors couldhave anticipated,butwhichpromptedthosequestionedtosummarisetheirviewsinawayIhopedthatreaderswouldndhelpful, sometimesdisclosingcommitmentsthatmaynototherwisehavebeenevident.ThevolumealsocontainsarevisededitionofZusesCalculatingSpace,completelyrewritteninLATEX, aproject realisedthanks tothe drive ofAdrianGerman,whomIwantespeciallytoacknowledgeforhismanycon-tributionstothiseort,includingthetranscriptionsoftheroundtabledis-cussionandDeutschscontributiontothisvolume.Muchofmyownjourneyuptothispointhasbeentheresultofinter-actionswithsomeof thedirectandindirectcontributorstothisvolume.I attendedEdFredkinslecturesat CarnegieMellonwhileI wasavisit-ingscholar duringtheSpringsemester of 2008. I alsoattendedCharlesBennettslecturesattheInstituteHenri PoincareinParis, andhavehadtheopportunitytointeract withGregChaitinonseveral occasions. In-deedGregultimatelybecameamemberofmyPhDexaminingcommittee.StephenWolframhasbeenanimportantinuence,havingworkedforhimat his personal oce as a consultant (and senior research associate) at Wol-framResearchforalmost6years.I must also mention other mentors, colleagues and friends such asJean-Paul Delahaye, Cris Calude, Matthew Szudzik, Fred Meinberg,xliiiAugust30,2012 13:35 WorldScienticReviewVolume-9inx6in-8306AComputableUniverse AComputableUniversexliv AcknowledgementsToddRowland, SelmerBringsjord, WilfriedSieg, JeanMosconi, JacquesDubucs, BernardFrancois, GenaroJ. Marti nez, JoostJoosten, Francisco-Hernandez-Quiroz, James Marshall andothers whoseoeuvres, ideas andinteractionswithmehavehelpedbuildandenrichmyownwork. I alsowanttospeciallythankElenaVillarreal, whohelpedmeatvariousstagesintheeditingofthisvolume,andformanyotherreasons.June29,2012 9:45 WorldScienticReviewVolume-9inx6in-8306AComputableUniverse AComputableUniverseChapter1IntroducingtheComputableUniverseHectorZenilDepartmentofComputerScience,TheUniversityofSheeld,UKIHPST(Paris1Pantheon-Sorbonne/ENSUlm/CNRS),France&WolframScienceGroup,WolframResearch,USA1. Understanding Computation & Exploring Nature as ComputationSincethedays of NewtonandLeibniz, scientists havebeenconstructingelaborate views of the world. Inthe past century, quantummechanicsrevolutionisedourunderstandingofphysicalrealityatatomicscales,whilegeneral relativitydidthesamefor our understandingof realityat largescales.Somecontemporaryworldviewsapproachobjectsandphysicallawsinterms of informationandcomputation,25towhichtheyassignultimateresponsibilityforthecomplexityinourworld, includingresponsibilityforcomplex mechanisms and phenomena such as life. In this view, the universeandthethingsinitareseenascomputingthemselves. Ourcomputersdonomorethanre-programapartoftheuniversetomakeitcomputewhatwewantittocompute.Someauthorshaveextendedthedenitionofcomputationtophysicalobjects and physical processes at dierent levels of physical reality, rangingfromthedigitaltothequantum. Mostoftheleadingthinkersinvolvedinthiseortarecontributorstothisvolume,includingsomewhoopposethe(digital)approach,preferringtoadvancetheirown.The computational/informational view (sometimes identied as compu-tationalism)isrootedinpioneeringthinkingbyauthorssuchasJohnA.WheelerandRichardFeynman. AquotationfromFeynmansMessengerLectures, deliveredat Cornell Universityin1964, distills his sense thatnaturemost likelyoperatesat averysimplelevel, despiteseemingcom-1June29,2012 9:45 WorldScienticReviewVolume-9inx6in-8306AComputableUniverse AComputableUniverse2 H.Zenilplextous, andmarksashifttounderstandingphysicsintermsofdigitalinformation.It always bothers me that, accordingtothe laws asweunderstandthemtoday,ittakesacomputingmachineaninnitenumberoflogicaloperationstogureoutwhatgoesoninnomatterhowtinyaregionof space, andnomatterhowtinyaregionoftime... SoIhaveoftenmadethehypothesisthatultimatelyphysicswill notrequireamathematical statement, that inthe endthe machinerywill berevealed, andthelawswill turnouttobesimple,likethechequerboardwithallitsapparentcomplexities.His view was probably inuenced by his thesis advisor John A. Wheeler,thecoinerof thephrase itfrombit, suggestingthatinformationconsti-tutes the most fundamental level of physical reality.18Another pioneerwasKonradZuse, aneweditionof whoseCalculatingSpace(RechnenderRaum) wearepleasedtobeabletopublishinthis volume. WeveputgreateortintotranslatingintomodernLATEX thescannedversionoftheoriginal translationcommissionedbyEdFredkinandpublishedbyMIT.Fredkinishimselfanotherpioneer, havingfoundedtheeldthatistodayknownas digital physics, andis acontributor tothis volume. StephenWolfram, astudent of Feynman, has alsobeenbuildinguponthis view,spearheadingaparadigmshiftfacilitatedbytodaysavailabilityofincreas-inglygreater andcheaper computational resources. Evenmorerecently,somemoderntheoriesofphysicsunderdevelopmenthavebeenattemptingto unify quantum mechanics and general relativity on the basis of informa-tion(tHooft, Susskind, Smolin), adevelopmentrepresentedintheworkofsomecontributorstothisvolume.Thesecontemporaryviewsofacomputationaluniversearealsodeeplyrelated, viatheconceptof information, toacontemporaryeldof math-ematical researchcalledalgorithmic informationtheory (AIT). AITre-searchers think that the true nature of nature can only be unveiled by study-ingthenotionof randomness(GregChaitin, LeonidLevin, CrisCalude).Afewpapers aredevotedtothis topic, includingtwofromChaitinandCaludea.aWhere author names are providedwithout areference, the authors inquestionarecontributorstothisvolume.June29,2012 9:45 WorldScienticReviewVolume-9inx6in-8306AComputableUniverse AComputableUniverseIntroducingtheComputableUniverse 31.1. Whatiscomputation?Howdoesnaturecompute?Zuse suggested early on that the world was possibly the result of determin-isticdigital computation, inparticularacellularautomaton. EdFredkinwouldlaterdeveloptheideafurther(seehiscontributiontothisvolume).Inhisminimalisticapproach,Wolframarguesthattheworldmayturnoutto be the result of very simple rules,perhaps even a single one,from whichtheapparent complexityweseearoundus emerges. If theuniverseis acomputableone, wecouldjustrunauniversal Turingcomputeroneverypossibleprogramtogeneratenotonlyourownuniversebuteverypossibleone, asWolframandSchmidhuberhavesuggested(botharecontributorstothisvolume). Thequestionwouldthenbehowtodistinguishouruni-versefromanyother. Wolframhaspointedout that therewill besomeuniverses that are obviously dierent from ours, and many others that maylookverysimilar, inwhichcasewecanaskwhetherourswill turnouttobespecialoruncommoninanysense,whether,forexample,itwouldrankamongthe rstintermsofdescriptionsize,i.e.be amongthosehavingtheshortest description. If it is simple enough, AIT would then suggest that itis also frequent, the result of many programs generating the same universe.Thisisaconclusionbasedonalgorithmicprobability, whichdescribesthedistribution of patterns, and establishes a strong connection to algorithmiccomplexity.AccordingtoSchmidhubersapproach, itwouldseemthatacomputergenerating every possible universe would necessarily have to be several timeslarger than the universe itself. But if the programs are short enough, insteadof running every program a step at a time, beginning with the smallest andproceedinginincreasingorder of size, it maybe possible tostart withsomeplausibleuniversesandrunthemforlongertimes, asWolframhassuggested, checkingeachprogramandallowingthosethatareapparentlycomplex to unfold (and eventually leading to the physical properties of ouruniverse).Therearealsothosewhobelievethatnaturenotonlyperformsdigitalcomputation(asistheviewofZuse, FredkinorWolfram, all contributorstothis volume), but is itself theresult of quantumcomputation(Lloyd,DeutschandCabello, alsoauthorsofchaptersinthisvolume). Accordingtothem,theworldwouldinthelastinstanceberootedinphysics,partic-ularlyquantummechanics, andwouldreectthepropertiesofelementaryparticlesandfundamentalforces. Lloyd,forexample,askshowmanybitsthereareintheuniverse, oeringaninterestingcalculationaccordingtoJune29,2012 9:45 WorldScienticReviewVolume-9inx6in-8306AComputableUniverse AComputableUniverse4 H.Zenilwhich, giventhepropertiesof quantumparticles, theuniversecannotberenderedinadescriptionshorterthanitselfsimplybecauseeveryelemen-taryparticlewouldneedtobesimulatedbyanotherelementaryparticle.Acomputertosimulatetheuniversewouldthereforeneedtobethesizeoftheuniverse,andwouldrequiretheenergyoftheactualuniverse,hencemaking it undistinguishable from a (quantum) computer, the computer andthecomputedbeinginperfectcorrespondence.If thegoal isnottodescribewiththegreatestaccuracyouruniverseinthestateitisin,onemayaskwhetherauniverseofsimilarcomplexitywouldrequireaverycomplicateddescription. Thediscussionseemsthere-foretorevolvearoundthepossibledescriptionoftheuniverse, whetheritcanbe writteninbits or inqubits, whether it canbe shorter thantheuniverseornot. Ifdigital informationunderliesthequantum, however, itmayturnoutthattheshortestdescriptionoftheuniversewouldbemuchshorter than the universe itself, contrary to the views of, for example, Lloyd,Deutsch, Cabello or Calude et al., and could then be compressed into a sim-pleshortcomputerprogram, asZuse, Wolfram, Fredkin, Bolognesi andImyselfbelieve.Infact,onecanthinkofthegoalofdigitalphysicsasaminimalmodeldescribingtheuniverse,equivalenttothegoalofphysicsinitsquestforauniedtheory, adescriptiveformulagoverningall forcesandparticlesyettobediscovered, if anysuchremain. Whether suchaprogramexists isanopenquestion, justasitisanopenquestionwhetherthereisatheoryof everything(ToE). But it is nolonger anopenquestionwhether suchunicationcanbeachievedforverylargeportionsof physics. Itisafaitaccompli ineldssuchasgravitationandmovement, electricityandmag-netism, electromagnetismandmostelectronuclearforces, areasof physicsthattodaymodelwithfrighteningaccuracylargeportionsofnatureusingsimplelawsthancanbeprogrammedinacomputerandcaninprinciple(andoftendefacto) provideperfect prescriptions andpredictions aboutthe world. In fact we have physical laws and computer programs for prettymuch everything; what we lack is a single theory that encompasses all othertheories.2. TheAlgorithmicApproachIftheworldisinfactnotadigitalcomputer,itcouldneverthelessbehavelike one. Thus whether or not it is adigital computer, one couldtestwhether theoutput of processes intheworldresembles theoutput thatJune29,2012 9:45 WorldScienticReviewVolume-9inx6in-8306AComputableUniverse AComputableUniverseIntroducingtheComputableUniverse 5one wouldexpect fromrunningaprogram. There are manylaws thatcomputers mayfollow. Or if youprefer, theyfollowaspecicsubset ofphysicallaws relatedtoinformationprocessing,notablythedistributionofpatternsdescribedbyalgorithmicprobability.Frommypointofview, informationcanonlyexistinourworldifitiscarried by a process; every bit has to have a corresponding physical carrier.Eventhoughthiscarrierisnotmatter, ittakestheformofaninteractionbetweencomponentsofmatteranatominteractingwithanotheratom,oraparticleinteractingwithanotherparticle. Atthelowestlevel, however,themost elementaryparticles, just likesinglebits, carrynoinformation(theShannonentropyofasinglebitis0becauseonecannotimplementacommunication channel of 1 bit only, 1 and 0 having the same possible infor-mation content if taken in isolation). Isolated particles may have no causalhistory, beingmemorylesswhenisolatedfromexternal interaction. Whenparticlesinteractwithotherparticlestheyappeartobelinkingthemselvesto a causal network and seem to be forced to dene a value as a result of thisinteraction(e.g. ameasurement). Whatsurprisesusaboutthequantumworldispreciselyitslackof apparentcausality, whichweseeeverywhereelse and are so used to. But it is the interaction and its causal history thatcarriesall thememoryof thesystem, withthenewbitappearingasif ithadbeendenedatrandombecauseourtheoriesof quantummechanicsonlyprovideprobabilityamplitudes. Linkingabittothecausal networkmay seem tantamount to producing a correlation of measurements betweenseeminglydisconnectedpartsofspace,whileinfacttheymayhavealwaysalreadybeenconnected, if theworldweretakenasdeterministic(aviewimplicitinso-calledhiddenvariablesmodels).Levins universal distribution12basedonalgorithmic probabilityde-scribesexpectedoutputfrequenciesinrelationtotheircomplexity. Apro-cess that produces a string s with a program p when executed on a universalTuring machine Thas probability PrT(s) = 2|p|where [p[ is the length oftheprogramp.The coding theorem7,8connects the frequency Pr(s) with which a strings is producedto its algorithmic complexity C(s). The so-calledsemi-measure mhasalsotheremarkablepropertyof dominatingPrTforanyuniversal TuringmachineT. Roughlyspeaking, m(s) establishes that ifthere are many long descriptions of a certain string, then there is also a shortdescriptionwithlowalgorithmiccomplexityC(s), thatism(s) 2C(s).As neither C(s) nor m(s) is computable, no program can exist which takes astring s as input and produces m(s) as output. However, we have proven10June29,2012 9:45 WorldScienticReviewVolume-9inx6in-8306AComputableUniverse AComputableUniverse6 H.Zenilthatnumericalapproximationsarepossibleandthatreasonablenumericalevaluationsproducereasonableresults. Justasstringscanbeproducedbyprograms, wemayaskaftertheprobabilityof acertainoutcomefromacertainnatural phenomenon, if thephenomenon, justlikeacomputingmachine,isaprocessratherthanarandomevent. Ifnootherinformationabout the phenomenon is assumed, we can see whether m(s) says anythingabout a distribution of possible outcomes in the real world.22In a world ofcomputableprocesses, m(s)wouldindicatetheprobabilitythatanaturalphenomenon produces a particular outcome and tell us how often a certainpatternwouldoccur. Co