Gosper'sGliderGuncreating"gliders"inthecellularautomatonConway'sGameofLife[1]

CellularautomatonFromWikipedia,thefreeencyclopedia

Acellularautomaton(pl.cellularautomata,abbrev.CA)isadiscretemodelstudiedincomputabilitytheory,mathematics,physics,complexityscience,theoreticalbiologyandmicrostructuremodeling.Cellularautomataarealsocalledcellularspaces,tessellationautomata,homogeneousstructures,cellularstructures,tessellationstructures,anditerativearrays.[2]

Acellularautomatonconsistsofaregulargridofcells,eachinoneofafinitenumberofstates,suchasonandoff(incontrasttoacoupledmaplattice).Thegridcanbeinanyfinitenumberofdimensions.Foreachcell,asetofcellscalleditsneighborhoodisdefinedrelativetothespecifiedcell.Aninitialstate(timet=0)isselectedbyassigningastateforeachcell.Anewgenerationiscreated(advancingtby1),accordingtosomefixedrule(generally,amathematicalfunction)thatdeterminesthenewstateofeachcellintermsofthecurrentstateofthecellandthestatesofthecellsinitsneighborhood.Typically,theruleforupdatingthestateofcellsisthesameforeachcellanddoesnotchangeovertime,andisappliedtothewholegridsimultaneously,thoughexceptionsareknown,suchasthestochasticcellularautomatonandasynchronouscellularautomaton.

Theconceptwasoriginallydiscoveredinthe1940sbyStanislawUlamandJohnvonNeumannwhiletheywerecontemporariesatLosAlamosNationalLaboratory.Whilestudiedbysomethroughoutthe1950sand1960s,itwasnotuntilthe1970sandConway'sGameofLife,atwodimensionalcellularautomaton,thatinterestinthesubjectexpandedbeyondacademia.Inthe1980s,StephenWolframengagedinasystematicstudyofonedimensionalcellularautomata,orwhathecallselementarycellularautomatahisresearchassistantMatthewCookshowedthatoneoftheserulesisTuringcomplete.WolframpublishedANewKindofSciencein2002,claimingthatcellularautomatahaveapplicationsinmanyfieldsofscience.Theseincludecomputerprocessorsandcryptography.

Theprimaryclassificationsofcellularautomata,asoutlinedbyWolfram,arenumberedonetofour.Theyare,inorder,automatainwhichpatternsgenerallystabilizeintohomogeneity,automatainwhichpatternsevolveintomostlystableoroscillatingstructures,automatainwhichpatternsevolveinaseeminglychaoticfashion,andautomatainwhichpatternsbecomeextremelycomplexandmaylastforalongtime,withstablelocalstructures.Thislastclassarethoughttobecomputationallyuniversal,orcapableofsimulatingaTuringmachine.Specialtypesofcellularautomataarereversible,whereonlyasingleconfigurationleadsdirectlytoasubsequentone,andtotalistic,inwhichthefuturevalueofindividualcellsdependonthetotalvalueofagroupofneighboringcells.Cellularautomatacansimulateavarietyofrealworldsystems,includingbiologicalandchemicalones.

Contents

1Overview2History3Classification

3.1Reversible3.2Totalistic

TheredcellsaretheMooreneighborhoodforthebluecell.

3.2Totalistic3.3Relatedautomata

4Elementarycellularautomata5Rulespace6Biology7Chemicaltypes8Applications

8.1Computerprocessors8.2Cryptography8.3Errorcorrectioncoding

9Modelingphysicalreality10Seealso

10.1Specificrules10.2Problemssolved10.3Seealso

11Referencenotes12References13Externallinks

Overview

Onewaytosimulateatwodimensionalcellularautomatoniswithaninfinitesheetofgraphpaperalongwithasetofrulesforthecellstofollow.Eachsquareiscalleda"cell"andeachcellhastwopossiblestates,blackandwhite.Theneighborhoodofacellisthenearby,usuallyadjacent,cells.ThetwomostcommontypesofneighborhoodsarethevonNeumannneighborhoodandtheMooreneighborhood.[3]Theformer,namedafterthefoundingcellularautomatontheorist,consistsofthefourorthogonallyadjacentcells.[3]ThelatterincludesthevonNeumannneighborhoodaswellasthefourremainingcellssurroundingthecellwhosestateistobecalculated.[3]ForsuchacellanditsMooreneighborhood,thereare512(=29)possiblepatterns.Foreachofthe512possiblepatterns,theruletablewouldstatewhetherthecentercellwillbeblackorwhiteonthenexttimeinterval.Conway'sGameofLifeisapopularversionofthismodel.AnothercommonneighborhoodtypeistheextendedvonNeumannneighborhood,whichincludesthetwoclosestcellsineachorthogonaldirection,foratotalofeight.[3]Thegeneral

equationforsuchasystemofrulesiskks,wherekisthenumberofpossiblestatesforacell,andsisthe

numberofneighboringcells(includingthecelltobecalculateditself)usedtodeterminethecell'snextstate.[4]Thus,inthetwodimensionalsystemwithaMooreneighborhood,thetotalnumberofautomata

possiblewouldbe229,or1.34 10154.

Itisusuallyassumedthateverycellintheuniversestartsinthesamestate,exceptforafinitenumberofcellsinotherstatestheassignmentofstatevaluesiscalledaconfiguration.[5]Moregenerally,itissometimesassumedthattheuniversestartsoutcoveredwithaperiodicpattern,andonlyafinitenumberofcellsviolatethatpattern.Thelatterassumptioniscommoninonedimensionalcellularautomata.

TheredcellsarethevonNeumannneighborhoodforthebluecell,whiletheextendedneighborhoodincludesthepinkcellsaswell.

Atorus,atoroidalshape

JohnvonNeumann,LosAlamosIDbadge

Cellularautomataareoftensimulatedonafinitegridratherthananinfiniteone.Intwodimensions,theuniversewouldbearectangleinsteadofaninfiniteplane.Theobviousproblemwithfinitegridsishowtohandlethecellsontheedges.Howtheyarehandledwillaffectthevaluesofallthecellsinthegrid.Onepossiblemethodistoallowthevaluesinthosecellstoremainconstant.Anothermethodistodefineneighborhoodsdifferentlyforthesecells.Onecouldsaythattheyhavefewerneighbors,butthenonewouldalsohavetodefinenewrulesforthecellslocatedontheedges.Thesecellsareusuallyhandledwithatoroidalarrangement:whenonegoesoffthetop,onecomesinatthecorrespondingpositiononthebottom,andwhenonegoesofftheleft,onecomesinontheright.(Thisessentiallysimulatesaninfiniteperiodictiling,andinthefieldofpartialdifferentialequationsissometimesreferredtoasperiodicboundaryconditions.)Thiscanbevisualizedastapingtheleftandrightedgesoftherectangletoformatube,thentapingthetopandbottomedgesofthetubetoformatorus(doughnutshape).Universesofotherdimensionsarehandledsimilarly.Thissolvesboundaryproblemswithneighborhoods,butanotheradvantageisthatitiseasilyprogrammableusingmodulararithmeticfunctions.Forexample,ina1dimensionalcellularautomatonliketheexamplesbelow,theneighborhoodofacellxitis{xi1t1,

xit1,xi+1t1},wheretisthetimestep(vertical),andiistheindex(horizontal)inonegeneration.

History

StanislawUlam,whileworkingattheLosAlamosNationalLaboratoryinthe1940s,studiedthegrowthofcrystals,usingasimplelatticenetworkashismodel.[6]Atthesametime,JohnvonNeumann,Ulam'scolleagueatLosAlamos,wasworkingontheproblemofselfreplicatingsystems.[7]VonNeumann'sinitialdesignwasfoundeduponthenotionofonerobotbuildinganotherrobot.Thisdesignisknownasthekinematicmodel.[8][9]Ashedevelopedthisdesign,vonNeumanncametorealizethegreatdifficultyofbuildingaselfreplicatingrobot,andofthegreatcostinprovidingtherobotwitha"seaofparts"fromwhichtobuilditsreplicant.Neumannreadapaperentitled"Thegeneralandlogicaltheoryofautomata"attheHixonSymposiumin1948.[7]Ulamwastheonewhosuggestedusingadiscretesystemforcreatingareductionistmodelofselfreplication.[10][11]NilsAallBarricelliperformedmanyoftheearliestexplorationsofthesemodelsofartificiallife.

UlamandvonNeumanncreatedamethodforcalculatingliquidmotioninthelate1950s.Thedrivingconceptofthemethodwastoconsideraliquidasagroupofdiscreteunitsandcalculatethemotionofeachbasedonitsneighbors'behaviors.[12]Thuswasbornthefirstsystemofcellularautomata.LikeUlam'slatticenetwork,vonNeumann'scellularautomataaretwodimensional,withhisselfreplicatorimplementedalgorithmically.Theresultwasauniversalcopierandconstructorworkingwithinacellularautomatonwithasmallneighborhood(onlythosecellsthattouchareneighborsforvonNeumann's

cellularautomata,onlyorthogonalcells),andwith29statespercell.[13]VonNeumanngaveanexistenceproofthataparticularpatternwouldmakeendlesscopiesofitselfwithinthegivencellularuniversebydesigninga200,000cellconfigurationthatcoulddoso.[13]Thisdesignisknownasthetessellationmodel,andiscalledavonNeumannuniversalconstructor.[14]

Alsointhe1940s,NorbertWienerandArturoRosenbluethdevelopedamodelofexcitablemediawithsomeofthecharacteristicsofacellularautomaton.[15]Theirspecificmotivationwasthemathematicaldescriptionofimpulseconductionincardiacsystems.Howevertheirmodelisnotacellularautomatonbecausethemediuminwhichsignalspropagateiscontinuous,andwavefrontsarecurves.[15][16]AtruecellularautomatonmodelofexcitablemediawasdevelopedandstudiedbyJ.M.GreenbergandS.P.Hastingsin1978seeGreenbergHastingscellularautomaton.TheoriginalworkofWienerandRosenbluethcontainsmanyinsightsandcontinuestobecitedinmodernresearchpublicationsoncardiacarrhythmiaandexcitablesystems.[17]

Inthe1960s,cellularautomatawerestudiedasaparticulartypeofdynamicalsystemandtheconnectionwiththemathematicalfieldofsymbolicdynamicswasestablishedforthefirsttime.In1969,GustavA.Hedlundcompiledmanyresultsfollowingthispointofview[18]inwhatisstillconsideredasaseminalpaperforthemathematicalstudyofcellularautomata.ThemostfundamentalresultisthecharacterizationintheCurtisHedlundLyndontheoremofthesetofglobalrulesofcellularautomataasthesetofcontinuousendomorphismsofshiftspaces.

In1969,GermancomputerpioneerKonradZusepublishedhisbookCalculatingSpace,proposingthatthephysicallawsoftheuniversearediscretebynature,andthattheentireuniverseistheoutputofadeterministiccomputationonasinglecellularautomaton"Zuse'sTheory"becamethefoundationofthefieldofstudycalleddigitalphysics.[19]

Inthe1970satwostate,twodimensionalcellularautomatonnamedGameofLifebecamewidelyknown,particularlyamongtheearlycomputingcommunity.InventedbyJohnConwayandpopularizedbyMartinGardnerinaScientificAmericanarticle,[20]itsrulesareasfollows:Ifacellhastwoblackneighbors,itstaysthesame.Ifithasthreeblackneighbors,itbecomesblack.Inallothersituationsitbecomeswhite.Despiteitssimplicity,thesystemachievesanimpressivediversityofbehavior,fluctuatingbetweenapparentrandomnessandorder.OneofthemostapparentfeaturesoftheGameofLifeisthefrequentoccurrenceofgliders,arrangementsofcellsthatessentiallymovethemselvesacrossthegrid.Itispossibletoarrangetheautomatonsothattheglidersinteracttoperformcomputations,andaftermucheffortithasbeenshownthattheGameofLifecanemulateauniversalTuringmachine.[21]Itwasviewedasalargelyrecreationaltopic,andlittlefollowupworkwasdoneoutsideofinvestigatingtheparticularitiesoftheGameofLifeandafewrelatedrulesintheearly1970s.[22]

StephenWolframindependentlybeganworkingoncellularautomatainmid1981afterconsideringhowcomplexpatternsseemedformedinnatureinviolationoftheSecondLawofThermodynamics.[23]Hisinvestigationswereinitiallyspurredbyaninterestinmodellingsystemssuchasneuralnetworks.[23]HepublishedhisfirstpaperinReviewsofModernPhysicsinvestigatingelementarycellularautomata(Rule30inparticular)inJune1983.[2][23]TheunexpectedcomplexityofthebehaviorofthesesimplerulesledWolframtosuspectthatcomplexityinnaturemaybeduetosimilarmechanisms.[23]Hisinvestigations,however,ledhimtorealizethatcellularautomatawerepooratmodellingneuralnetworks.[23]Additionally,duringthisperiodWolframformulatedtheconceptsofintrinsicrandomnessandcomputationalirreducibility,[24]andsuggestedthatrule110maybeuniversalafactprovedlaterbyWolfram'sresearchassistantMatthewCookinthe1990s.[25]

In2002Wolframpublisheda1280pagetextANewKindofScience,whichextensivelyarguesthatthediscoveriesaboutcellularautomataarenotisolatedfactsbutarerobustandhavesignificanceforalldisciplinesofscience.[26]Despiteconfusioninthepress,[27][28]thebookdidnotargueforafundamentaltheoryofphysicsbasedoncellularautomata,[29]andalthoughitdiddescribeafewspecificphysicalmodelsbasedoncellularautomata,[30]italsoprovidedmodelsbasedonqualitativelydifferentabstractsystems.[31]

Classification

Wolfram,inANewKindofScienceandseveralpapersdatingfromthemid1980s,definedfourclassesintowhichcellularautomataandseveralothersimplecomputationalmodelscanbedivideddependingontheirbehavior.Whileearlierstudiesincellularautomatatendedtotrytoidentifytypeofpatternsforspecificrules,Wolfram'sclassificationwasthefirstattempttoclassifytherulesthemselves.Inorderofcomplexitytheclassesare:

Class1:Nearlyallinitialpatternsevolvequicklyintoastable,homogeneousstate.Anyrandomnessintheinitialpatterndisappears.[32]Class2:Nearlyallinitialpatternsevolvequicklyintostableoroscillatingstructures.Someoftherandomnessintheinitialpatternmayfilterout,butsomeremains.Localchangestotheinitialpatterntendtoremainlocal.[32]Class3:Nearlyallinitialpatternsevolveinapseudorandomorchaoticmanner.Anystablestructuresthatappeararequicklydestroyedbythesurroundingnoise.Localchangestotheinitialpatterntendtospreadindefinitely.[32]Class4:Nearlyallinitialpatternsevolveintostructuresthatinteractincomplexandinterestingways,withtheformationoflocalstructuresthatareabletosurviveforlongperiodsoftime.[33]Class2typestableoroscillatingstructuresmaybetheeventualoutcome,butthenumberofstepsrequiredtoreachthisstatemaybeverylarge,evenwhentheinitialpatternisrelativelysimple.Localchangestotheinitialpatternmayspreadindefinitely.Wolframhasconjecturedthatmany,ifnotallclass4cellularautomataarecapableofuniversalcomputation.ThishasbeenprovenforRule110andConway'sgameofLife.

Thesedefinitionsarequalitativeinnatureandthereissomeroomforinterpretation.AccordingtoWolfram,"...withalmostanygeneralclassificationschemethereareinevitablycaseswhichgetassignedtooneclassbyonedefinitionandanotherclassbyanotherdefinition.Andsoitiswithcellularautomata:thereareoccasionallyrules...thatshowsomefeaturesofoneclassandsomeofanother."[34]Wolfram'sclassificationhasbeenempiricallymatchedtoaclusteringofthecompressedlengthsoftheoutputsofcellularautomata.[35]

Therehavebeenseveralattemptstoclassifycellularautomatainformallyrigorousclasses,inspiredbytheWolfram'sclassification.Forinstance,CulikandYuproposedthreewelldefinedclasses(andafourthonefortheautomatanotmatchinganyofthese),whicharesometimescalledCulikYuclassesmembershipintheseprovedundecidable.[36][37][38]Wolfram'sclass2canbepartitionedintotwosubgroupsofstable(fixedpoint)andoscillating(periodic)rules.[39]

Reversible

Acellularautomatonisreversibleif,foreverycurrentconfigurationofthecellularautomaton,thereisexactlyonepastconfiguration(preimage).[40]Ifonethinksofacellularautomatonasafunctionmappingconfigurationstoconfigurations,reversibilityimpliesthatthisfunctionisbijective.[40]Ifa

cellularautomatonisreversible,itstimereversedbehaviorcanalsobedescribedasacellularautomatonthisfactisaconsequenceoftheCurtisHedlundLyndontheorem,atopologicalcharacterizationofcellularautomata.[41][42]Forcellularautomatainwhichnoteveryconfigurationhasapreimage,theconfigurationswithoutpreimagesarecalledGardenofEdenpatterns.[43]

Foronedimensionalcellularautomatathereareknownalgorithmsfordecidingwhetheraruleisreversibleorirreversible.[44][45]However,forcellularautomataoftwoormoredimensionsreversibilityisundecidablethatis,thereisnoalgorithmthattakesasinputanautomatonruleandisguaranteedtodeterminecorrectlywhethertheautomatonisreversible.TheproofbyJarkkoKariisrelatedtothetilingproblembyWangtiles.[46]

Reversiblecellularautomataareoftenusedtosimulatesuchphysicalphenomenaasgasandfluiddynamics,sincetheyobeythelawsofthermodynamics.Suchcellularautomatahaverulesspeciallyconstructedtobereversible.SuchsystemshavebeenstudiedbyTommasoToffoli,NormanMargolusandothers.Severaltechniquescanbeusedtoexplicitlyconstructreversiblecellularautomatawithknowninverses.Twocommononesarethesecondordercellularautomatonandtheblockcellularautomaton,bothofwhichinvolvemodifyingthedefinitionofacellularautomatoninsomeway.Althoughsuchautomatadonotstrictlysatisfythedefinitiongivenabove,itcanbeshownthattheycanbeemulatedbyconventionalcellularautomatawithsufficientlylargeneighborhoodsandnumbersofstates,andcanthereforebeconsideredasubsetofconventionalcellularautomata.Conversely,ithasbeenshownthateveryreversiblecellularautomatoncanbeemulatedbyablockcellularautomaton.[47][48]

Totalistic

Aspecialclassofcellularautomataaretotalisticcellularautomata.Thestateofeachcellinatotalisticcellularautomatonisrepresentedbyanumber(usuallyanintegervaluedrawnfromafiniteset),andthevalueofacellattimetdependsonlyonthesumofthevaluesofthecellsinitsneighborhood(possiblyincludingthecellitself)attimet1.[49][50]Ifthestateofthecellattimetdoesnotdependonitsownstateattimet1thenthecellularautomatonisproperlycalledoutertotalistic.[50]Conway'sGameofLifeisanexampleofanoutertotalisticcellularautomatonwithcellvalues0and1outertotalisticcellularautomatawiththesameMooreneighborhoodstructureasLifearesometimescalledlifelikecellularautomata.[51][52]

Relatedautomata

Therearemanypossiblegeneralizationsofthecellularautomatonconcept.

Onewayisbyusingsomethingotherthanarectangular(cubic,etc.)grid.Forexample,ifaplaneistiledwithregularhexagons,thosehexagonscouldbeusedascells.Inmanycasestheresultingcellularautomataareequivalenttothosewithrectangulargridswithspeciallydesignedneighborhoodsandrules.Anothervariationwouldbetomakethegriditselfirregular,suchaswithPenrosetiles.[53]

Also,rulescanbeprobabilisticratherthandeterministic.Suchcellularautomataarecalledprobabilisticcellularautomata.Aprobabilisticrulegives,foreachpatternattimet,theprobabilitiesthatthecentralcellwilltransitiontoeachpossiblestateattimet+1.Sometimesasimplerruleisusedforexample:"TheruleistheGameofLife,butoneachtimestepthereisa0.001%probabilitythateachcellwilltransitiontotheoppositecolor."

Acellularautomatonbasedonhexagonalcellsinsteadofsquares(rule34/2)

Theneighborhoodorrulescouldchangeovertimeorspace.Forexample,initiallythenewstateofacellcouldbedeterminedbythehorizontallyadjacentcells,butforthenextgenerationtheverticalcellswouldbeused.

Incellularautomata,thenewstateofacellisnotaffectedbythenewstateofothercells.Thiscouldbechangedsothat,forinstance,a2by2blockofcellscanbedeterminedbyitselfandthecellsadjacenttoitself.

Therearecontinuousautomata.Theseareliketotalisticcellularautomata,butinsteadoftheruleandstatesbeingdiscrete(e.g.atable,usingstates{0,1,2}),continuousfunctionsareused,andthestatesbecomecontinuous(usuallyvaluesin[0,1]).Thestateofalocationisafinitenumberofrealnumbers.Certaincellularautomatacanyielddiffusioninliquidpatternsinthisway.

Continuousspatialautomatahaveacontinuumoflocations.Thestateofalocationisafinitenumberofrealnumbers.Timeisalsocontinuous,andthestateevolvesaccordingtodifferentialequations.Oneimportantexampleisreactiondiffusiontextures,differentialequationsproposedbyAlanTuringtoexplainhowchemicalreactionscouldcreatethestripesonzebrasandspotsonleopards.[54]Whentheseareapproximatedbycellularautomata,theyoftenyieldsimilarpatterns.MacLennan[1](http://www.cs.utk.edu/~mclennan/contincomp.html)considerscontinuousspatialautomataasamodelofcomputation.

Thereareknownexamplesofcontinuousspatialautomata,whichexhibitpropagatingphenomenaanalogoustoglidersintheGameofLife.[55]

Elementarycellularautomata

Thesimplestnontrivialcellularautomatonwouldbeonedimensional,withtwopossiblestatespercell,andacell'sneighborsdefinedastheadjacentcellsoneithersideofit.Acellanditstwoneighborsformaneighborhoodof3cells,sothereare23=8possiblepatternsforaneighborhood.Aruleconsistsofdeciding,foreachpattern,whetherthecellwillbea1ora0inthenextgeneration.Therearethen28=256possiblerules.[4]These256cellularautomataaregenerallyreferredtobytheirWolframcode,astandardnamingconventioninventedbyWolframthatgiveseachruleanumberfrom0to255.Anumberofpapershaveanalyzedandcomparedthese256cellularautomata.Therule30andrule110cellularautomataareparticularlyinteresting.Theimagesbelowshowthehistoryofeachwhenthestartingconfigurationconsistsofa1(atthetopofeachimage)surroundedby0s.Eachrowofpixelsrepresentsagenerationinthehistoryoftheautomaton,witht=0beingthetoprow.Eachpixeliscoloredwhitefor0andblackfor1.

Rule30

Rule30cellularautomaton

currentpattern 111 110 101 100 011 010 001 000newstateforcentercell 0 0 0 1 1 1 1 0

Rule30exhibitsclass3behavior,meaningevensimpleinputpatternssuchasthatshownleadtochaotic,seeminglyrandomhistories.

Rule110,liketheGameofLife,exhibitswhatWolframcallsclass4behavior,whichisneithercompletelyrandomnorcompletelyrepetitive.Localizedstructuresappearandinteractinvariouscomplicatedlookingways.InthecourseofthedevelopmentofANewKindofScience,asaresearchassistanttoWolframin1994,MatthewCookprovedthatsomeofthesestructureswererichenoughtosupportuniversality.Thisresultisinterestingbecauserule110isanextremelysimpleonedimensionalsystem,anddifficulttoengineertoperformspecificbehavior.ThisresultthereforeprovidessignificantsupportforWolfram'sviewthatclass4systemsareinherentlylikelytobeuniversal.CookpresentedhisproofataSantaFeInstituteconferenceonCellularAutomatain1998,butWolframblockedtheprooffrombeingincludedintheconferenceproceedings,asWolframdidnotwanttheproofannouncedbeforethepublicationofANewKindofScience.[56]In2004,Cook'sproofwasfinallypublishedinWolfram'sjournalComplexSystems(Vol.15,No.1),overtenyearsafterCookcameupwithit.Rule110hasbeenthebasisforsomeofthesmallestuniversalTuringmachines.[57]

Rulespace

Anelementarycellularautomatonruleisspecifiedby8bits,andallelementarycellularautomatonrulescanbeconsideredtositontheverticesofthe8dimensionalunithypercube.Thisunithypercubeisthecellularautomatonrulespace.Fornextnearestneighborcellularautomata,aruleisspecifiedby25=32bits,andthecellularautomatonrulespaceisa32dimensionalunithypercube.Adistancebetweentworulescanbedefinedbythenumberofstepsrequiredtomovefromonevertex,whichrepresentsthefirstrule,andanothervertex,representinganotherrule,alongtheedgeofthehypercube.ThisruletoruledistanceisalsocalledtheHammingdistance.

Cellularautomatonrulespaceallowsustoaskthequestionconcerningwhetherruleswithsimilardynamicalbehaviorare"close"toeach.Graphicallydrawingahighdimensionalhypercubeonthe2dimensionalplaneremainsadifficulttask,andonecrudelocatorofaruleinthehypercubeisthenumberofbit1inthe8bitstringforelementaryrules(or32bitstringforthenextnearestneighborrules).DrawingtherulesindifferentWolframclassesintheseslicesoftherulespaceshowthatclass1rulestendtohavelowernumberofbit1's,thuslocatedinoneregionofthespace,whereasclass3rulestendtohavehigherproportion(50%)ofbit1's.[39]

Rule110

Rule110cellularautomaton

currentpattern 111 110 101 100 011 010 001 000newstateforcentercell 0 1 1 0 1 1 1 0

Conustextileexhibitsacellularautomatonpatternonitsshell.[59]

Forlargercellularautomatonrulespace,itisshownthatclass4rulesarelocatedbetweentheclass1andclass3rules.[58]Thisobservationisthefoundationforthephraseedgeofchaos,andisreminiscentofthephasetransitioninthermodynamics.

Biology

Somebiologicalprocessesoccurorcanbesimulatedbycellularautomata.

Patternsofsomeseashells,liketheonesinConusandCymbiolagenus,aregeneratedbynaturalcellularautomata.Thepigmentcellsresideinanarrowbandalongtheshell'slip.Eachcellsecretespigmentsaccordingtotheactivatingandinhibitingactivityofitsneighborpigmentcells,obeyinganaturalversionofamathematicalrule.[59]Thecellbandleavesthecoloredpatternontheshellasitgrowsslowly.Forexample,thewidespreadspeciesConustextilebearsapatternresemblingWolfram'srule30cellularautomaton.[59]

Plantsregulatetheirintakeandlossofgasesviaacellularautomatonmechanism.Eachstomaontheleafactsasacell.[60]

Movingwavepatternsontheskinofcephalopodscanbesimulatedwithatwostate,twodimensionalcellularautomata,eachstatecorrespondingtoeitheranexpandedorretractedchromatophore.[61]

Thresholdautomatahavebeeninventedtosimulateneurons,andcomplexbehaviorssuchasrecognitionandlearningcanbesimulated.[62]

Fibroblastsbearsimilaritiestocellularautomata,aseachfibroblastonlyinteractswithitsneighbors.[63]

Chemicaltypes

TheBelousovZhabotinskyreactionisaspatiotemporalchemicaloscillatorthatcanbesimulatedbymeansofacellularautomaton.Inthe1950sA.M.Zhabotinsky(extendingtheworkofB.P.Belousov)discoveredthatwhenathin,homogenouslayerofamixtureofmalonicacid,acidifiedbromate,andacericsaltweremixedtogetherandleftundisturbed,fascinatinggeometricpatternssuchasconcentriccirclesandspiralspropagateacrossthemedium.Inthe"ComputerRecreations"sectionoftheAugust1988issueofScientificAmerican,[64]A.K.Dewdneydiscussedacellularautomaton[65]developedbyMartinGerhardtandHeikeSchusteroftheUniversityofBielefeld(WestGermany).ThisautomatonproduceswavepatternsthatresemblethoseintheBelousovZhabotinskyreaction.

Applications

Computerprocessors

CellularautomatonprocessorsarephysicalimplementationsofCAconcepts,whichcanprocessinformationcomputationally.Processingelementsarearrangedinaregulargridofidenticalcells.Thegridisusuallyasquaretiling,ortessellation,oftwoorthreedimensionsothertilingsarepossible,but

notyetused.Cellstatesaredeterminedonlybyinteractionswithadjacentneighborcells.Nomeansexiststocommunicatedirectlywithcellsfartheraway.[66]Onesuchcellularautomatonprocessorarrayconfigurationisthesystolicarray.Cellinteractioncanbeviaelectriccharge,magnetism,vibration(phononsatquantumscales),oranyotherphysicallyusefulmeans.Thiscanbedoneinseveralwayssonowiresareneededbetweenanyelements.Thisisveryunlikeprocessorsusedinmostcomputerstoday,vonNeumanndesigns,whicharedividedintosectionswithelementsthatcancommunicatewithdistantelementsoverwires.

Cryptography

Rule30wasoriginallysuggestedasapossibleBlockcipherforuseincryptography.Twodimensionalcellularautomataareusedforrandomnumbergeneration.[67]

Cellularautomatahavebeenproposedforpublickeycryptography.TheonewayfunctionistheevolutionofafiniteCAwhoseinverseisbelievedtobehardtofind.Giventherule,anyonecaneasilycalculatefuturestates,butitappearstobeverydifficulttocalculatepreviousstates.

Errorcorrectioncoding

CAhavebeenappliedtodesignerrorcorrectioncodesinthepaper"DesignofCAECCCellularAutomataBasedErrorCorrectingCode",byD.RoyChowdhury,S.Basu,I.SenGupta,P.PalChaudhuri.ThepaperdefinesanewschemeofbuildingSECDEDcodesusingCA,andalsoreportsafasthardwaredecoderforthecode.

Modelingphysicalreality

AsAndrewIlachinskipointsoutinhisCellularAutomata,manyscholarshaveraisedthequestionofwhethertheuniverseisacellularautomaton.[68]Ilachinskiarguesthattheimportanceofthisquestionmaybebetterappreciatedwithasimpleobservation,whichcanbestatedasfollows.Considertheevolutionofrule110:ifitweresomekindof"alienphysics",whatwouldbeareasonabledescriptionoftheobservedpatterns?[69]Ifanobserverdidnotknowhowtheimagesweregenerated,thatobservermightendupconjecturingaboutthemovementofsomeparticlelikeobjects.Indeed,physicistJamesCrutchfieldhasconstructedarigorousmathematicaltheoryoutofthisidea,provingthestatisticalemergenceof"particles"fromcellularautomata.[70]Then,astheargumentgoes,onemightwonderifourworld,whichiscurrentlywelldescribedbyphysicswithparticlelikeobjects,couldbeaCAatitsmostfundamentallevel.

Whileacompletetheoryalongthislinehasnotbeendeveloped,entertaininganddevelopingthishypothesisledscholarstointerestingspeculationandfruitfulintuitionsonhowcanwemakesenseofourworldwithinadiscreteframework.MarvinMinsky,theAIpioneer,investigatedhowtounderstandparticleinteractionwithafourdimensionalCAlattice[71]KonradZusetheinventorofthefirstworkingcomputer,theZ3developedanirregularlyorganizedlatticetoaddressthequestionoftheinformationcontentofparticles.[72]Morerecently,EdwardFredkinexposedwhathetermsthe"finitenaturehypothesis",i.e.,theideathat"ultimatelyeveryquantityofphysics,includingspaceandtime,willturnouttobediscreteandfinite."[73]FredkinandWolframarestrongproponentsofaCAbasedphysics.

Inrecentyears,othersuggestionsalongtheselineshaveemergedfromliteratureinnonstandardcomputation.Wolfram'sANewKindofScienceconsidersCAthekeytounderstandingavarietyofsubjects,physicsincluded.TheMathematicsoftheModelsofReferencecreatedbyiLabs[74]founderGabrieleRossianddevelopedwithFrancescoBertoandJacopoTagliabuefeaturesanoriginal2D/3Duniversebasedonanew"rhombicdodecahedronbased"latticeandauniquerule.Thismodelsatisfiesuniversality(itisequivalenttoaTuringMachine)andperfectreversibility(adesideratumifonewantstoconservevariousquantitieseasilyandneverloseinformation),anditcomesembeddedinafirstordertheory,allowingcomputable,qualitativestatementsontheuniverseevolution.[75]

Seealso

Specificrules

Brian'sBrainLangton'santWireworldRule90Rule184vonNeumanncellularautomataNobilicellularautomataCodd'scellularautomatonLangton'sloopsCoDi

Problemssolved

FiringsquadsynchronizationproblemMajorityproblem

Seealso

AutomatatheoryBidirectionaltrafficCellularautomatainpopularcultureCycliccellularautomatonExcitablemediumMirek'sCellebrationMovablecellularautomatonQuantumcellularautomataSpatialdecisionsupportsystemTurmites

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